[    {        "title": "0804.4059v2.Long_range_antiferromagnetic_interactions_in_ZnFe2O4_and_CdFe2O4.pdf",        "content": "arXiv:0804.4059v2  [cond-mat.mtrl-sci]  15 Jul 2008Long-range antiferromagnetic interactions in ZnFe 2O4and CdFe 2O4\nChing Cheng\nDepartment of Physics, and National Center for Theoretical Sciences,\nNational Cheng Kung University, Tainan 70101, Taiwan\n(Dated: October 24, 2018)\nFor the ﬁrst time, the Fe-Fe interactions in the geometrical ly frustrated antiferromagnetic systems\nof zinc and cadmium ferrites are determined quantitatively by the ﬁrst-principles methods of density\nfunctional theory. Both the generalized gradient approxim ation (GGA) as well as GGA plus the one-\nsite Coulomb interaction (GGA+U) are considered for the exc hange-correlation energy functional.\nThe interactions up to third neighbours are found to be all an tiferromagnetic for both materials,\nregardless of which approximation scheme ( GGA or GGA+U) is u sed. Surprisingly, the third-\nneighbour interactions are estimated to be much stronger th an the second-neighbour interactions\nand on the same order in magnitude as the ﬁrst-neighbour inte ractions.\nPACS numbers: 61.50.-f,71.20.-b,75.50.Ee\nThe geometrically frustrated antiferromagnets, dis-\ntinct from other magnetic systems, exhibit a number of\nexceptional features whose formation, to a great extent,\ndepends on the range as well as the nature of the interac-\ntions between the magnetic ions in the systems[1, 2, 3].\nBoth zinc and cadmium ferrites (ZnFe 2O4and CdFe 2O4\nshorted as ZFO and CFO hereafter) are well known as\none type of the geometrically frustrated systems[4, 5].\nMagnetic neutron scattering measurements have been\ncarried out on both ZFO and CFO to determine the in-\nteractions between the Fe ions. Surprisingly, it was con-\ncluded that the frustration in CFO[6] is driven mainly\nby the strong nearest-neighbour antiferromagnetic inter-\naction in contrast to the antiferromagnetically coupled\nthird-neighbour interaction in ZFO[7]. In this Letter, we\nstudy the magnetic interactions between the Fe ions in\nthese two compounds by considering structures with dif-\nferent collinear magnetic distributions as well as cation\ndistributions using ab initio methods.\nThe structure of spinel ferrites are constructed by ﬁll-\ning one-eighth of the tetrahedral sites and half of the\noctahedral sites (denoted as A and B site respectively\n113\n32\n24\n11\n22 3 \n34\n44\nSC1 44\n13\n32\n12\n3 4\n422\n311\nSC2 \n113\n32\n24\n11\n22 3 \n34\n44\nSC3 \n113\n32\n24\n11\n22 3 \n34\n44\nSC4 \nFIG. 1: The four B-site cation distributions considered in\nthis study. The numbers denote the nth layer when one\nviews along the (001) direction of the cubic cell. The B-site\ncations are divided into two groups (B1 and B2 in Tab. I),\none denoted in green (light grey) and the other in blue (dark\ngrey). The two types of B-site cations correspond to diﬀeren t\ncations, e.g. Fe and Zn, in the inverse spinel or oppositely\npolarized Fe ions in the normal spinel.hereafter) in the FCC sublattice of oxygen. Spinel fer-\nrites are usually categorized into two types, i.e. normal\nand inverse spinel, according to the distribution of the\ndivalent and trivalent cations in the A and B sites. A\nnormal spinel corresponds to the structure with all the A\nsites being occupied by the divalent cations, e.g. Zn (Cd)\nof ZFO (CFO), and the B sites by the trivalent cations,\ni.e. the Fe ions. In an inverse spinel structure, the di-\nvalent cations occupy half of the B sites while the other\nhalf as well as all the A sites are occupied by the Fe ions.\nBothZFOandCFOareconsideredformingnormalspinel\nthough coexistence of both phases has been reported[8].\nTo investigate the magnetic interactions between the Fe\nions, we consider three diﬀerent B-site cation distribu-\ntions (named SC1, SC2, and SC3 as shown in Fig. 1)\non the basis of a cubic cell consisting of eight formulas\n(56 atoms) of the materials. Inclusion of the possible\ncation distributions on the A and B sites as well as the\ndiﬀerentmagneticordersofthe Fe ionsleadstothe struc-\ntures summarized in Tab. I, i.e. four conﬁgurations for\nnormal spinel and 6 conﬁgurations for inverse spinel. A\nfourth B-site cation distribution, i.e. SC4, is also in-\ncluded to estimate the magnitude of the more distant in-\nteractions between Fe cations in the normal spinel as dis-\ncussedlater. Therehavebeenﬁrst-principlescalculations\nfor other spinel compounds, e.g. studies of MnFe 2O4,\nFe3O4, CoFe 2O4, and NiFe 2O4by Szotek et al[9] and\nACr2X4(A=Zn, Cd, Hg; X=O, S, Se) by Yaresko[10].\nNote that the only conﬁguration considered in the pre-\nvious studies [4, 11] corresponds to that of nb1 in Tab.I,\ni.e. the normal spinel phase with an equal number of\noppositely polarized B-site Fe ions distributed as those\nin SC1 of Fig. 1. We shall show that the nb3 is a more\nstable structure compared to that of nb1 and similarly\nib3 to ib1 for both ZFO and CFO.\nTo study the extent of the interactions between the\ncations in these systems, the distances and numbers of\nneighbouring cations for the A- and B-site cations with\nneighbouring distances up to around 6.7 ˚Aare listed in\nTab.II. If we assume that the interactions between the Fe\nions in the normal spinel can be approximately modelled2\nTABLE I: The ten conﬁgurations considered in the present\nstudy and their corresponding cation distributions on the A\nand B sites as well as the magnetic orders of the Fe ions.\nsite A B1 B2conﬁgurationsnormal cation Zn(Cd) Fe Fe\nspinel magnetic 0 + + na1=na2=na3\nconﬁgurations 0 + −nb1,nb2,nb3\ninverse cation Fe Zn(Cd) Fe\nspinel magnetic + 0 + ia1,ia2,ia3\nconﬁgurations + 0 −ib1,ib2,ib3\nTABLE II: The distances (d) and numbers (N) of neighbour-\ning cations in the spinel systems are listed in the order of\nincreasing distances. For example, the number of the neares t-\nneighbour B-site cations for an A-site cation is twelves as i n-\ndicated in the second line. The distances are written in term s\nof the squared values (in unit of a0, i.e. the cubic lattice con-\nstant)andthelength(inunitof ˚A)assuming a0= 8.5˚A. They\nare further grouped according to the order of neighbouring.\nThe nth-neighbour interactions between the B-site Fe ions i n\nnormal spinel, i.e. J nof the Ising model, are also denoted.\n(d/a0)2d(˚A) whena0=8.5˚A N\nB(B) J 1 8/64 3.0 6\nA(B) 11/64 3.5 12\nB(A) 11/64 3.5 6\nA(A) 12/64 3.7 4\nB(B) J 2 24/64 5.2 12\nA(B) 27/64 5.5 16\nB(A) 27/64 5.5 8\nA(A) 32/64 6.0 12\nB(B) J 3 32/64 6.0 12\nB(B) J 4 40/64 6.7 12\nby the Ising model, i.e. H=−(1/2)/summationtext\nn/summationtext\niJnSiSi+n,\nthe values of E0,J1,J2, andJ3of these materials can\nbe obtained from the calculated total energies of the four\nconﬁgurations of na, nb1, nb2, and nb3. The approxi-\nmations made by using this simple model to describe the\ninteractions in these compounds and to obtain the corre-\nsponding interaction energies are supported by the fact\nthat the moments on the Fe ions in ZFO and CFO are\nsizable and considerably localized. Similar approaches\nhavebeen usedpreviouslytostudythemagnetismin, e.g.\nspinel MnO 2[12] and YCuO 2.5[13]. Studies using ﬁrst-\nprinciples calculations for these systems with collinear\nmangetic distributions should provideinformation on the\nsize and the extent of the interactions between the mag-\nnetic cations in these systems. These information are, on\nthe other hand, important references to further under-\nstand the possible phases of the materials through the\nstudies of the corresponding classical three-dimensional\nHeisenberg model.\nAll electronic calculations in this study are based on\nthe spin-polarizeddensity functional theory[14, 15]. The\ngeneralized gradient approximation (GGA) proposed by\nPerdew, Burke, and Ernzerhof [16] for the non-local cor-rection to a purely local treatment of the exchange-\ncorrelation energy functional are used. The on-site\nCoulomb interaction U [17] for Fe ions is also included\nin the second stage of the investigation to study its ef-\nfect on the physical properties of these systems. The\nvalues of U and J are taken from the previous studies as\n4.5eV[18] and 0.89eV[17] respectively. The interactions\nbetween the ions and valence electrons are described by\nthe projector augmented-wave(PAW) method [19] in the\nimplementation of Kresse and Joubert [20] and the num-\nbersofthe treatedvalenceelectronsare12,8and6forZn\n(Cd), Fe and O atoms respectively. The single-particle\nKohn-Sham equations [21] are solved using the plane-\nwave-basedVienna ab-initio simulation program (VASP)\ndeveloped at the Institut f¨ ur Material Physik of the Uni-\nversit¨ atWien [22]. The energy cutoﬀs used for the plane-\nwavebasisis500eVandasetof(666)k-pointssampling\naccording to Monkhorst-Pack[23] is used for the integra-\ntion over the ﬁrst Brillouin zone, unless speciﬁed other-\nwise. Relaxationprocessesinoptimizingstaticstructures\nare accomplished by moving oxygen atoms to the posi-\ntions at which the atomic forces are smaller than 0.02\neV/˚A. The cell volumes are also relaxed under the con-\nstraint that the systems remain cubic.\nThe calculated energies (relative to the corresponding\nlowest-energy one), lattice constants and magnetic mo-\nments of Fe ions of the studied conﬁgurations (Tab.I)\nfor both ZFO and CFO using either GGA or GGA+U\napproach are listed in Tab.III.\nThe experimental lattice constants for ZFO and CFO\nare around 8.52 ˚A[7] and 8.72 ˚A[6] respectively. The cal-\nculated lattice constants areall within 1% deviation from\nthe experimental values except for the ia1, ia2, and ia3\nof ZFO[24]. One also notices that the energies of these\nthree ferromagnetic inverse-spinel conﬁgurations are sig-\nniﬁcantly larger than the rest of the considered conﬁgu-\nrations. Therefore the eﬀect of U is not applied to the\nia1, ia2, and ia3 conﬁgurations. From now on we shall\nfocus the discussions on the rest seven conﬁgurations.\nThelocalmagneticmomentsforFeionsintheseconﬁg-\nurations were found to be within the range of 3.7 ∼3.9µ0\nand 4.1∼4.2µ0for GGA and GGA+U calculations re-\nspectively. The GGA+U results are closer to the exper-\nimental estimates of the magnetic moments, i.e. 4.22[25]\nand 4.44[6] µ0for ZFO and CFO respectively.\nThe orders in energy of these conﬁgurations are diﬀer-\nent for ZFO and CFO which demonstrates the underly-\ning diﬀerences in the properties of these two compounds.\nHoweverthelowestenergyconﬁgurationsareidenticalfor\nthese two compounds, i.e. they are ib3 and nb3 in the\nGGA and GGA+U calculations respectively. Alhough\nthe lowest energy conﬁguration is an inverse spinel in the\nGGA calculations, it is a normal spinel in the GGA+U\ncalculations. The on-site Coulomb interaction U is usu-\nally considered as essential for properly describing the\nelectronicand magneticpropertiesofthe Feionsin spinel\nferrites. Note that the lowest energy conﬁguration of the\nGGA+U results, i.e. nb3, is diﬀerent from the conﬁgu-3\nTABLE III: The calculated energies (in unit of eV per 8 formul as of the materials) relative to the corresponding lowest-e nergy\nconﬁguration, the lattice constants (in ˚A) and the local magnetic moments of Fe ions (in unit of Bohr mag neton) for the ten\nconﬁgurations of Tab.I using either GGA or GGA+U (data in par enthese) approach.\nZFO na nb1 nb2 nb3 ia1 ia2 ia3 ib1 ib2 ib3\n∆E 4.28(0.54) 2.21(0.18) 2.71(0.14) 1.98(0.00) 5.18 4.89 4 .12 0.38(2.18) 0.90(2.68) 0.00(1.68)\na08.52(8.53) 8.50(8.52) 8.51(8.52) 8.50(8.52) 8.35 8.36 8.3 5 8.45(8.49) 8.45(8.50) 8.45(8.49)\nµ 3.9(4.2) 3.8(4.2) 3.9(4.2) 3.8(4.2) 3.7/1.5 3.7/1.4 3.7/1 .5 3.7(4.1) 3.7(4.1) 3.7(4.1)\nCFO na nb1 nb2 nb3 ia1 ia2 ia3 ib1 ib2 ib3\n∆E 3.50(0.76) 0.99(0.15) 1.74(0.27) 0.79(0.00) 8.32 7.71 6 .89 1.91(4.94) 1.89(4.41) 0.00(2.81)\na08.82(8.82) 8.78(8.81) 8.80(8.81) 8.79(8.81) 8.77 8.15 8.7 7 8.73(8.76) 8.75(8.78) 8.72(8.75)\nµ 4.0(4.2) 3.8(4.2) 3.9(4.2) 3.8(4.2) 3.8 3.9 3.8 3.7(4.1) 3. 7(4.1) 3.7(4.1)\nTABLE IV: The nth-neighbour interaction energy (in unit of\nmeV) between the B-site Fe ions obtained from the calculated\nenergies of the normal spinel conﬁgurations of na, nb1, nb2\nand nb3. The J1andJ2obtained from considering merely the\nenergies of nb1, nb2 and nb3 are also listed. Unless speciﬁed\naccordingly, the calculations use an energy cutoﬀ of 500eV\nand the Monkhorst-Pack mesh of (6 6 6) for the k-points\nsampling.\nZFO 4 J14J24J34J1a4J2b\nPBE -220.4 -19.0 -23.6 -126.2 +28.1\nPBE+Uc-97.7 -7.0 -16.5 -31.6 +26.0\nPBE+Ud-43.7 -3.2 -12.2 +5.0 +21.1\nPBE+U -39.6 -2.7 -12.3 +9.7 +22.0\nPBE+Ue-39.7 -2.6 -12.4 +9.8 +22.2\nPBE+Uf-38.1 -3.5 -12.7\nPBE+Ug-40.2 -2.4 -12.5\nCFO 4 J14J24J34J1a4J2b\nPBE -275.9 -18.7 -21.8 -188.8 +24.8\nPBE+U -71.1 -2.2 -10.6 -28.9 +18.9\naJ1= (nb1−nb2)/2bJ2= (nb1−nb3)/4cU=2.5eV\ndEcutoff=400eV kpoint set=(4 4 4)\neEcutoff=600eV kpoint set=(8 8 8)\nfobtained from na, nb1, nb2, nb4\ngobtained from na, nb1, nb3, nb4\nration considered in all the previous studies, i.e. nb1.\nThe interaction energies between the B-site Fe ions up\nto third neighbours can be determined by applying the\nIsing model to the calculated energies of the four normal\nspinel conﬁgurations, i.e. nb1, nb2, nb3 and na, and the\nresults are presented in Tab. IV. They all turn out to be\nantiferromagnetic, no matter whether the eﬀect of U is\nincluded or not. The eﬀect of U reduces the magnitudes\nofJ1andJ2substantially, but only halves the magni-\ntudes of J3. The universal reduction in magnitude of Jn\nafter including U is implied in the density of states of\nthese systems. The most prominent eﬀect of U in ZFO\nis localizing the ﬁve d electrons of Fe ions which are pre-\nviously well hybridized with the other valence electrons,\nand therefore weakening the interactions between them\nwhile at the same time widening the band gap of the sys-\ntem. Similar results take place in CFO except that thestrongly localized ten d electrons of the Cd ions locate\nat the lower energy range than the localized ﬁve d elec-\ntrons of Fe ions even after the eﬀect of U is included.\nThat how the exchange interactions depend on the val-\nues of U have been studied previously, e.g. in FeSbO 4[26]\nand in NiGa 2S4[27]. The variations of J due to U were\nshown being monotonically. Our results of using U=0\nand U=4.5 suggest that for both ZFO and CFO with\nthe on-site Coulomb interaction within the ragne of 0 to\n4.5 all lead to having antiferromagnetic interactions up\ntothird neighboursandstrongerthird-neighbourinterac-\ntions than the second-neighbour ones. These conclusions\nareconsistentwith the testedresultsofcalculationsusing\nU=2.5eV as shown in Tab.IV.\nAs the magnitude of J2in the GGA+U results is con-\nsiderably smaller than those of J1andJ3, the numerical\nerrors due to using ﬁnite basis sets and discrete k points\nare examined. The lower (400eV) and higher (600eV)\nenergy cutoﬀs for the plane-wave basis are considered as\nwell as the sparser (4 4 4) and denser (8 8 8) sets of\nMonkhorst-Pack sampling. The results in Tab. IV in-\ndicate that the signiﬁcant ﬁgure for the values of these\ninteractions, i.e. 8 Jn(8 is the number fomulas in the cu-\nbic cell used in the calculations), can be considered as in\nmeV. The magnitude of J3is at least three (ﬁve) times\nlarger than that of J2and about one third (sixth) of J1\nfor ZFO (CFO) in the GGA+U calculations. This, i.e.\ntheJ3interaction, correspondsto an interaction between\nB-site Fe ions ofat least 6 ˚Aapart which is a considerably\nlong-rangeinteraction. Iftheinteractionsbeyondthe2nd\nneighbours are negligible, the values for J1andJ2can be\nreadily obtained from merely considering the energies of\nnb1, nb2 and nb3. These results are also listed in Tab.\nIV. Under this assumption, both J1andJ2of ZFO in\nthe GGA+U calculations are switched to ferromagnetic\ninteractions. This is not a plausible result when con-\nsidering the experimentally identiﬁed antiferromagnetic\nphase for ZFO. To estimate the magnitude of the next\ndistant interaction, i.e. J4, which is not included in the\nprevious discussion, the total energy of the SC4 conﬁgu-\nration in the normal spinel, i.e. nb4, was also calculated.\nThe magnitude of J4can not be obtained from consider-\ning the ﬁve conﬁgurations of na, nb1, nb2, nb3 and nb4\nas their energy fomulas are linearly dependent. How-4\never, the magnitudes of the ﬁrst three neighbour interac-\ntions can be evaluated from considering either na, nb1,\nnb2 and nb4 or na, nb1, nb3 and nb4 whose results are\nalso listed in Tab.IV. It is obvious that the eﬀect of the\nlonger-distance interactions does not change the ﬁndings\nwe shall conclude next, i.e. existing antiferromagnetic\ninteractions up to the third neighbours, much weaker J2\ncompared to those of J1andJ3and that J1andJ3are\nat the same order of magnitude. Similar characters take\nplace in CFO except that the nearest-neighbour interac-\ntion, when compared to J3, is much more dominant than\nthat in ZFO which is consistent with the experimental\nsuggestions[6, 7].\nThere have been numerous theoretical studies for the\ngeometrically frustrated antiferromagnetic systems using\nthe classical three-dimensional Heisenberg model[1, 2, 3].\nStudies including up to the 2nd neighbour interactions\nsuggested that with the present estimated values of J1\nandJ2, i.e.J1<0 andJ2/J1<0.5, the system re-\nmains paramagnetic down to 0K[2]. A mean-ﬁeld ap-\nproach to magnetic ordering in the highly frustrated py-\nrochlores also concluded that no long-range order can be\nestablished for a system with antiferromagnetic J1andJ3interactions but negligible J2andJ4interactions[3].\nExperimentally there has been suggestion that no long-\nrange order can be established down to 0.1K for CFO\neven with an applied ﬁeld of up to 9T[6] and similarly\nZFO was found remaining disordered even at the lowest\nobservable temperature of 1.5K[7].\nIn summary, we have generated diﬀerent conﬁgura-\ntions, in both cation distribution and magnetic order, to\nstudy quantitatively the Fe-Fe interactions in ZFO and\nCFO using GGA as well as GGA+U approach. The in-\nteractions bewteen the B-site Fe ions are found to extend\nup to 3rd neighbours and all in antiferromagnetic nature.\nThe 2nd-neighbour interaction is estimated to be much\nsmaller in magnitude than those of the ﬁrst-neighbour\nand the third-neighbour interactions.\nAcknowledgments The author gratefully acknowledge\nChing-Shen Liu for generating SC2 and SC4, GY Guo\nand HT Jeng for helpful discussions, and the support of\nNCTS. This work was sponsored by the National Science\nCouncil of Taiwan. The computer resources were mainly\nprovided by the National Center for High-Performance\nComputing in HsinChu of Taiwan.\n[1] N. Reimers, Phys. Rev. B 45, 7287 (1992); R. Mossener\nand J. T. Chalker, Phys. Rev. Lett. 80, 2929 (1998).\n[2] A. J. Garcia-Adeva and D. L. Huber, Phys. Rev. B 65,\n184418 (2002).\n[3] J. N. Reimers, A. J. Berlinsky, and A.-C. Shi, Phys. Rev.\nB43, 865 (1991).\n[4] P. W. Anderson Phys. Rev. 102, 1008 (1956)\n[5] J. Ostorero, A. Mauger, M. Guillot, A. 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B 76140406 (2007)."    },    {        "title": "2306.03804v1.Stress_evolution_in_plastically_deformed_austenitic_and_ferritic_steels_determined_using_angle__and_energy_dispersive_diffraction.pdf",        "content": "Stress evolution in plastically deformed austenitic and ferritic ste els determined using \nangle - and energy -dispersive diffraction  \n \nM. Marciszko -Wiąckowska1,*, A. Baczmański2, Ch. Braham3, M. Wątroba4, S. Wroński2, \nR. Wawszczak2, G. Gonzal ez5, P. Kot6,M. Klaus7, Ch. Genzel7 \n1AGH University of Krakow , ACMIN, al. Mickiewicza 30, 30 -059 Kraków, Poland  \n2AGH University of Krakow , WFiIS, al. Mickiewicza 30, 30 -059 Kraków, Poland  \n3Arts et Métiers -ParisTech, PIMM, CNRS UMR 8006, 151 Bd de l’Hôpital, 75013 Paris, France  \n4EMPA, Swiss Federal Laboratories for Materials Science and Technology, Labratory for Mechanics of  \nMaterials and Nanostructures, Feuerwerkerstrasse 39, 3602 Thun, Switzerland  \n5Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma  de México, Circuito Exterior \nS/N, Cd. Universitaria, A.P. 70 -360, Coyoacán, C.P. 04510, Mexico  \n6NOMATEN Centre of Excellence, National Centre of Nuclear Research, A. Sołtana 7, 05 -400 Otwock -\nŚwierk, Poland  \n7Abteilung fürMikrostruktur - und Eigenspannungsanalyse, Helmholtz -Zentrum Berlin fürMaterialien und \nEnergie, Albert -Einstein -Str. 15, Berlin 12489, Germany  \n \n \n \n \n*Corresponding author: Marianna Marciszko -Wiąckowska ( marciszk@agh.edu.pl ), \n+48126175309  \n \nAbstract  \nIn the presented research, the intergranular elastic interaction and the second -order plastic \nincompatibility stress in textured ferritic and austenitic steels were investigated by means of \ndiffraction. The lattice strains were measured inside the samples by the multiple reflection \nmethod using high energy X -rays diffraction during uniaxial in situ tensile tests. Comparing \nexperiment with various models of intergranular interaction, it was found that the Eshe lby-Kröner model correctly approximates the X -ray stress factors (XSFs) for different reflections \nhkl and scattering vector orientations.  \nThe verified XSFs were used to investigate the evolution of the first and second -order \nstresses in both austenitic and  ferritic steels. It was shown that considering only the elastic \nanisotropy, the non -linearity of sin2ψ plots cannot be explained by crystallographic texture. \nTherefore, a more advanced method based on elastic -plastic self -consistent modeling (EPSC) \nis req uired for the analysis. Using such methodology the non -linearities of cos2φ plots were \nexplained, and the evolutions of the first and second -order stresses were determined. It was \nfound that plastic deformation of about 1 - 2% can completely exchange the st ate of second -\norder plastic incompatibility stresses.  \n \nKeywords  \nX-ray diffraction, stress analysis, elastic anisotropy, intergranular interaction, austenitic \nstainless steel, ferritic steel  \n \n1. Introduction  \nKnowledge of the residual stress state is essential to understanding the mechanical behavior \nof polycrystalline materials.  The element may be damaged or strengthened when external \nstresses are added to this stress, which develops during mechanical or thermal processing. \nTherefore, t he macroscopic residual stress (first-order stress ) in the element's subsurface layer \nis important; for instance, compressive stress slows crack initiation and propagation , while \ntensile stress typically speeds it up . Also , the so -called second -order residual stresses  [1], \ncharacterizing t he heterogeneity of the stresses on the scale of polycrystalline grains, may affect \nthe plastic deformation process of the material [2,3] . These stresses, defined as  the deviations of the stresses for individual polycrystalline grains from the mean macroscopic value (first -\norder stress), are caused by anisotropy or heterogeneity  of the processes occurring  for different \ngrains and , as a result , lead to a mismatch in their shape or volume . For example, during plastic \ndeformation, second -order  stresses result from the difference in activation of the s lip system or \ntwin phenomena occurring in grains belonging to different phases or having different  lattice \norientations  [2–9]. \nIn recent years, much e ffort has been put into studying the phenomena occurring in individual \ngrains using synchrotron diffraction measurements  (e.g. [10–17]). These measurements make \nit possible to determine the state of stress for individual grains during the deformation of the \nsample. For example, in the work  [10], the stress state in the grains of a titanium sample was \ndetermined using high -energy X -ray diffraction microscopy. However, the tests were carried \nout for a limited number of single grains , and some dispersion of results was found. The first \ndirect determination of the stress state for different grains in the AZ31 magnesium alloy was \ncarried out by neutron diffraction [18]. In situ , measurements were made for several preferred \ntexture orientations. Such experiments are promising due to the high  statistics of the grains for \nwhich the measurement is performed. However, both methods have some limitations ; direct \ngrain stress scanning with a microscopic synchrotron beam can be performed for large grains \nwith dimensions of several tens of µm, and due  to the long measurement time and complicated \ndata processing, it can be performed for a limited number of specific grains. On the other hand, \nneutron measurements of grain stresses for groups of grains can only be performed for a highly \ntextured sample.  \nDiffraction methods based on the measurement of lattice deformations are commonly used to \ndetermine first -order stresses in a polycrystalline material, both textured and quasi -isotropic  \n(i.e. having a random orientation distribution ). The stress tensor can  be calculated from the \nmeasured e lastic strains only when the so -called XECs or XSF s (X-ray Elastic Constants  or X-ray Stress Factors [19,20]) relating lattice elastic deformation with the first-order stresses (mean \nstresses for considered volume), are known. The values of XSFs (or XECs) are determined \ndirectly from the experiment or calculated from well -known grain interaction models  [9,19,21] , \ne.g., Reuss  [22], Voigt [23], Eshelby -Kröner [24–26], and f ree-surface [9]. The concepts of the \nfree-surface model are provided in [9], whereas the standard models, i.e. , Reuss, Voigt, and \nEshelby - Kröner  approaches, are extensively discussed in the liter ature (e.g. [19–21,27]).  \nHowever, a given model 's applicability is usually  not well -argued, considering  such effects as \nrelaxation of the forces perpendicular to the s urface, grains size , and shape. In the previous \nworks , usually , the Eshelby -Kröner was regarded a s the closes to the real material ; however , it \nis not a general rule as was shown , for example , in the case of near-surface volume [28,29] , in \nthe case of textured samples  [27,30]  or for the columnar microstructure of grains in the co ating s \n[21]. Therefore, t he first objective of this study is to verify the accuracy of XSF 's model in stress \nanalysis  using representative information gained from the experiment, i.e. , using different hkl \nreflections  and many orientations of the scatterin g vector . So, the Voigt , Reuss , and Eshelby -\nKröner model s are  compared with experimental XSFs determined for austenitic and ferritic \nsteel. A further requirement for the model's applicability is the accuracy of the fit between the \ntheoretical and measured lattice strains.  \nThe important  aim of the present study  is to apply the verified XSFs to study the  development \nof the first- and the second -order stress  in textured austenitic  and ferritic  steel subjected  to a \nuniaxial tensile test . The stress was determined using the multireflection method  [3,31 –33]. \nSecond -order plastic incompatibility stresses are the fluctuation around mean stress resulti ng \nfrom differences in plastic deformation of crystals having different orientations. T heir effect \nmay be observed as non -linearities or change s in the slope of the  <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. 𝑠𝑖𝑛2𝜓 for \ndifferent orientations of the scattering vector ( where 𝜓 angle is between scattering vector and \nnormal to the surface and 𝜑 is a rotation angle about the normal, c.f.  [30] measured for polycrystalline samples subjected to elastoplastic d eformation  [34–36]. The elastic anisotropy \nof the crystallites within textured samples would be the first cause of the non -linearities ; \nhowever , in many cases , the effect of the second -order  stresses  can be even more significant  \n[8,9,31] . The elastic anisotropy  can be incorp orated in stress analysis if the XSF values are \nknown from the experiment or computed using the suitable  grain interaction model, as has been \ndone in this work. The calculations of XSFs, based on the crystallographic texture and single \ncrystal elastic constants, allow for the prediction of the character of sin2ψ  plots for a sample \nunder applied or residual m acrostresses. Nonethele ss, the interpretation of non -linearities based \nsolely on elastic anisotropy is typically insufficient.  Therefore, the first - and second -order \nplastic incompatibility stresses analysis requires a more advanced method  based on modeling \nthe plastic deformation process [9,25,37 –41]. It is worth noting that in many works , the \npresence of the second -order plastic incompatibility stresses was observed as the changes in the \ntendency of the lattice strains  dependence vs. macroscopic stress, measured in the direction of \napplied load and transverse direction  (the experimental lattice strains were compared with \nelastic -plastic self -consistent model)  [40,42] . However , the distribution of th ese stresses in \nEuler space or their mean amplitude was not determined.  Previously ,  we determined the mean \nvon Mises second -order plastic incompatibility stresses using multiple reflection methods  \nduring tensile for duplex steel [3] and pearlitic steel [32]. A similar analysis is done in this \nstudy, but additionall y the distribution of second -order stresses in Euler space for single -phase \naustenitic and ferritic steels  is determined and presented . The methodology presented below  \nwas applied to in situ  measurements of lattice strains during a tensile test . The diffraction \nexperiment was performed in transmission mode using high -energy synchrotron  radiation  for \naustenitic and ferritic steel . The XSFs were calculated using the Eshelby -Kröner model, \naccounting for texture  [21], and the model values were verified in this experiment during the unloading of the sample. Hence , the complete stress state analysis  has been done using correctly \ndetermined XSFs and including  the second -order stresses.  \nThe basic equation  relating  the first -order stresses 𝜎𝑖𝑗𝐼 with corresponding elastic lattice \nstrain <𝜀(ψ,φ)>ℎ𝑘𝑙𝑒 can be written in th e following form:  \n<𝑑(ψ,φ)>ℎ𝑘𝑙𝜎−𝑑ℎ𝑘𝑙0\n𝑑ℎ𝑘𝑙0 =<𝜀(ψ,φ)>ℎ𝑘𝑙𝑒=𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,𝜑,𝑓)𝜎𝑖𝑗𝐼    (1) \nwhere:  <𝑑(ψ,φ)>ℎ𝑘𝑙𝜎  is the value s of interplanar spacings  measured using hkl reflection in a \ndirection  characterized by angles ψ and φ  [19,30]  for a material subjected to  the first-order  \nstress 𝜎𝑖𝑗𝐼,  𝑑ℎ𝑘𝑙0 is the stress free interplanar spacing , 𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,φ,𝑓) are the XSFs  and 𝑓 \ndenotes ODF (Orientation Distribution Function) characterizing crystallographic texture . \nThis equation represent s an elastic response of the lattice to the residual or applied stress \n𝜎𝑖𝑗𝐼 and it can be used to determine the values of  XSFs experimentally . For example, if a known \nincrement  of uniaxial stress ΔΣ11=Σ11(2)−Σ11(1) applied to the sample  during the tensile test, the \nfactor 𝐹11(ℎ𝑘𝑙,ψ,𝜑,𝑓)  can be calculated from the corresponding lattice strain, measured as \nthe change in the interplanar spacings : \n𝐹11(ℎ𝑘𝑙,ψ,𝜑,𝑓)=<𝜀(ψ,φ)>ℎ𝑘𝑙𝑒\nΔΣ11=<𝑑(ψ,φ)>ℎ𝑘𝑙Σ2−<𝑑(ψ,φ)>ℎ𝑘𝑙Σ1\n<𝑑(ψ,φ)>ℎ𝑘𝑙Σ1 ΔΣ11                            (2) \nwhere:  <𝑑(ψ,φ)>ℎ𝑘𝑙Σ2 and <𝑑(ψ,φ)>ℎ𝑘𝑙Σ1  correspond respectively to the interplanar  \nspacings measured under uniaxial loads Σ11(2) and Σ11(1) applied to the sample . \nThe above formula enables  the determination of the XFSs from measured in situ  changes in \ninterplanar spacings using ap propriate experimental techniques  for the increments of the \nstresses ΔΣ11 in the elastic range of sample loading or unloading . Then, the model calculated \nXSFs 𝐹11𝑚𝑜𝑑(ℎ𝑘𝑙,ψ,𝜑,𝑓) can be compared with the experimental ones  𝐹11𝑒𝑥𝑝(ℎ𝑘𝑙,ψ,𝜑,𝑓). It \nshould be emphasized that in this incremental method , the influence of the residual stresses on \nthe experimentally determined X SFs is avoided because the lattice strains corresponding to these stresses are canceled when  the differences  <𝑑(ψ,φ)>ℎ𝑘𝑙Σ2−<𝑑(ψ,φ)>ℎ𝑘𝑙Σ1 are \ncalculated [28,43] . \nVerification of the XSFs , presented in our previous work [28], can be done by analyzing the \nresults of residual stress measurements. In such a case , the accuracy of fitting theoretical lattice \nparameter values to measured ones is evaluated for several XSFs  models.  Assuming that the \neffect of the 𝜎𝑖𝑗𝐼 on the measured <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎 is dominating (i.e. , the plastic incompatibility \nis not significant , and the influence of stacking faults is negligible),  the values of the first-order \nstresses 𝜎𝑖𝑗𝐼 and value of strain -free parameter 𝑎0 can be determined using  least square method \nbased on  the well-known  equation:   \n<𝑎(ψ,φ)>ℎ𝑘𝑙𝜎=𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,𝜑,𝑓)𝜎𝑖𝑗𝐼𝑎0+𝑎0                                            (3) \nwhere the values <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎=√ℎ2+𝑘2+𝑙2 <𝑑(ψ,φ)>ℎ𝑘𝑙𝜎 are determined \nexperimentally  for cubic  lattice .  \nThe verified XSFs can be used  to determine residual or imposed  sample stresses . In this \nwork , the Eshelby -Kröner XSF model is applied to study the variation of the stress state within  \nthe austenitic  and ferritic  sample during elastic -plastic deformation. The diffraction \nmeasurements of the lattice strains were done inside the sample (transmission method) during \na tensile test.   \nIn a plastically deformed material, the lattice strains  <𝜀(ψ,φ)>ℎ𝑘𝑙 can be expressed as a \nsuperposition of strains induced by macrostresses <𝜀(ψ,φ)>ℎ𝑘𝑙𝑒 and the lattice strains \ninduced by  second -order  incompatibility stress es <𝜀(ψ,φ)>ℎ𝑘𝑙𝑝𝑖: \n<𝜀(ψ,φ)>ℎ𝑘𝑙=<𝜀(ψ,φ)>ℎ𝑘𝑙𝑒+<𝜀(ψ,φ)>ℎ𝑘𝑙𝑝𝑖=<𝑎(ψ,φ)>ℎ𝑘𝑙−𝑎0\n𝑎0  (4) \nwhere: <𝜀(ψ,φ)>ℎ𝑘𝑙𝑒=𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,𝜑)𝜎𝑖𝑗𝐼. \nThe elastic -plastic self -consistent (EPSC) model has already been presented and proven to \ndetermine  both types of stresses  [41,44,45] . In this method , it is assumed that <𝜀(ψ,φ)>ℎ𝑘𝑙𝑝𝑖=𝑞<𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖, where <𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖 can be calculated by the self -consistent model and q \nis a fitting parameter scaling  the magnitude of plastic strains . \nTherefore , the experimental lattice parameters <𝑎(ψ,φ)>ℎ𝑘𝑙 obtained from the diffraction \nmethod can be expressed as  (cf. Eq. 4 ) [3,5,8,9,46,47] : \n<𝑎(ψ,φ)>ℎ𝑘𝑙=[𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,𝜑,𝑓)𝜎𝑖𝑗𝐼+𝑞<𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖]𝑎0+𝑎0         (5) \nIn the present interpretation , both terms of Eq. 4 are considered . Next, using information from \nthe EPSC model , the magnitude of the residual stress 𝜎𝑖𝑗𝐼𝐼,𝑝𝑖 and its dependence on the crystal \norientation may be determined : \n𝜎𝑖𝑗𝐼𝐼,𝑝𝑖=𝑞 𝜎̃𝑖𝑗𝐼𝐼,𝑝𝑖                                               (6) \nwhere 𝜎̃𝑖𝑗𝐼𝐼,𝑝𝑖 are the plastic incompatibility stresses calculated from the EPSC model.  \nIn the case when diffraction elastic constants are known, strains are theoretically predicted , and \nlattice spacings  are measured, all other unknown quantities from Eq. 5 can be found using the \nleast square  procedure, based on minimi zing the merit function called  𝜒2, which is defined as:  \n𝜒2=1\n𝑁−𝑀∑ (<𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑒𝑥𝑝−<𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑐𝑎𝑙\n𝛿𝑛)2\n𝑁\n𝑛=1                                             (7) \nwhere <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑒𝑥𝑝 and <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑐𝑎𝑙 are the experimental and calculated lattice \nparameters,  respectively,  𝛿𝑛=(<𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑒𝑥𝑝) is the measurement uncertainty (standard \ndeviation) of <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎,𝑒𝑥𝑝 for the n-th measurement, N and M are the number s of measured \nlattice parameters and fitting parameters, respectively.  \nThe described methodology is applied in the presented study  to simultaneously calculate  first-\norder  stresses  and plastic incompatibility  second -order stresses  in the single -phase austenitic  \nand ferritic  steel using energy - and angle -dispersive diffraction . To predict the theoretical \nsecond -order  stresses caused by plastic incompatibilities , the Elasto -Plastic S elf-Consistent \n(EPSC) model based on the work of  Berveiller and Lipińsk i [44] was used.  In model prediction , the sample is represented by a number of grains , having  a distribution of \norientations reproducing the initial experimental textures. First, t he model sample is subjected \nto elastoplastic deformation ; next, the external stresses are unloaded.  So performed modeling \nwas carried out  to det ermine the value s of 𝜀̃(ψ,φ), which are  used in stress analysis using Eq. \n5.  \n \n2. Experimental  \n2.1. Material  \nThe presented study investigated two materials, austenitic and ferritic stainless steel  \n(composition given in Table 1 ), with high elastic anisotropy (Zener  ratio A, given in Table 2). \nThe single -crystal elastic constants ( Cij) used in this work for the investigated sample s are \ngathered in Table  2.  \n \nTable 1. Composition of investigated stainless steals samples  (wt.%) (ASS:  austenitic \nstainless steal 316L (Z2CND17 -12) and FSS:  ferritic stainless steal AIPI 5L X65  \n Fe Cr Ni Mo Mn Cu Si P S C N Co \nASS bal. 16.63  11.14  2.03 1.31 0.35 0.52 0.022  0.025  0.02 0.03 0.18 \nFSS bal. 0.034  0.399  0.023  1.38 0.193  0.285  0.008  0.001  0.031  -- -- \n \nTable 2. Single crystal elastic constants (used for XSF calculation) [48] together with the Zener \nratio. \nMaterial  C11 (GPa)  C12 (GPa)  C44 (GPa)  A \nFe-austenite  197 122 124 3.3 \nFe-ferrite  231 134.4  116.4  2.4 \n \n A dog -bone -shaped tensile specimen with the following gauge dimensions: 5 mm in width, 3 \nmm in thickness, and 33 mm in length was made of a  hot-rolled austenitic steel sheet . In the \ncase of a cold -rolled ferritic steel sheet, the dimensions of the sample having a similar shape \nwere 1.5 mm in width, 1.5 mm in thickness, and 12 mm in length . The initial microstructure \nwas analyzed using a Tescan Mira scanning electron microscope (SEM) equipped with  an \nEDAX DigiView electron back-scattered diffraction (EBSD ) camera.  An EBSD analysis was \nperformed on two cross -section s of each specimen, perpendicular to the rolling direction  and \nperpendicular to the normal direction  (Fig.1) . The samples were prepared for microscopic \nobservations using standard metallographic preparation  steps . EBSD maps  were collected at 25 \nkV, from 300 μm x 300 μm area with a step size of 0.25 µm. The average grain size, \ncrystallographic orientation  maps  were analyzed using TSL OIM™ Analysis software.  A single  \ngrain was defined as a set of at least 5 measurement points surrounded by a continuous grain \nboundary segment with a misorientation of at least 15˚. In Fig. 1 the Inverse Pole Figure (IPF) \nmaps for austenitic and ferritic samples were presented. A uniform microstructure with \napproximately  equiaxed recrystallized grains in the austenitic sample  with some \nrecrystallization twins  is seen . In the case of the ferritic sample , the grains show defected \nmicrostructure and more complex shapes but without significant elongation in one direction . \nTherefore in model calculations , the grains in both phases  were approximated by spherical \nEshelby inclusions, exhibiting s tatistically isotropic interaction with the matrix .  The average \ngrain size in austenitic and ferritic sample is 14.2 ± 7.5  µm and 9.3 ± 5.9 µm, respectively . \nThe crystallographic texture was characterized by the X -ray diffraction method using Co \nradiation  (Empyrean XRD Diffractometer, Malvern Panalytical) . To do this , the pole figures \n110, 200, 211 for ferrite and 111 , 200, 220 for austenite  were me asured. The ODFs calculated \nfrom pole figures using the WIMV method  [49] are shown in Fig. 2., where the orientation of the sample coordinate system respectively to the main directions of the rolling process is \ndefined.  \n \nFig. 1. The EBSD -IPF orienta tion map s show the microstructure of austenitic (a , c) and ferritic \n(b, d) steel  samples .  The ma ps were measured at two different  cross sections : (a, b) plane \ndetermined by  rolling direction (RD) and normal  direction ( ND); (c, d) plane determined by \nrolling direction (RD) and transverse direction (TD) . \n \nFig. 2. Normalized orientation distribution function (ODF) determined using Co radiation  for \naustenitic (a) and ferritic (b) steel . For fe rritic steel , only the initial tex ture was presented as the \ndeformation for this sample was small.  The sections through Euler space with the step of 5° are \npresented along the ϕ2. The Euler angles are defined with respect  to sample axes  RD, TD and ND \n(as in standard presentation for rolled sheet  [50]). The levels express multiples -of-random -\ndistribution . \n \n2.2. Measur ements  \nThe X -ray diffraction method was used to measure in situ  lattice strains during the tensile test.  \nTwo separate measuring methods : energy -dispersive (ED) diffraction for austenitic steel and \nangle -dispersive (AD) diffraction for ferritic steel , were used to determine the lattice strains in \nsitu during a tensile test. As presented in Fig. 3, an experimental setup contained  a dog -bone -\nshaped specimen stretched along the transverse direction (TD). Then, d uring the tensile test , \nthe actual deformation in the elastic range and for small plastic  deformation  was measured by \na gauge placed on the sample.  \nIt should be emphasized that during deformation of austenitic 316L stainless steel, a dynamic \nphase transitions  and twinning process can occur. However, in the diffraction patterns collected \nduring the entire experiment, only diffraction peaks corresponding to the pure austenitic phase \nwere found, which means that even if transformations occurred, the volume fraction s of the \nnew phases are insignificant compared to the austenitic phase. In addition, twinning is more \nlikely at higher strains than those used in our experiment, i.e. above 5 - 10% [51–53]. Theref ore, \nit was assumed that crystallographic slip is the dominant deformation process. The same \nassumption was made for ferritic steel, for which the presence of a pure ferritic phase was found \nin the diffractograms collected during the tensile test.  \n \n \nFig. 3. An experimental setup used for lattice strain measurement in the case of austenitic steel \nby ED diffraction (a) and ferrit ic steel by AD diffraction (b). In the case of ED method the \nmeasurements were done for positive angles  φ, wh ile the positive φ+ and negative φ- are \navailable from AD diffraction rings recorded by 2D detector.  The stress  tensors and orientation \nof the scattering vector are defined with respect to X coordinates for which x1 || TD, x 2 || RD and \nx3 || ND . \n \n \n \n \n2.2.1 Energy -dispersive diffraction  measu rements  \nFor austenitic steel , the stress measurements performed in situ during the tensile test were made \nusing  synchrotron ED diffraction at BESSY (EDDI@BESSYII beamline, HZB, Berlin) using \na white beam (wavelength in the range λ: 0. 18 - 0.3 Å) [54,55] . The primary be am cross -section \nwas equal to 1 x 1  mm2, and a double slit system restricted the angular divergence in the \ndiffracted bea m with apertures of 0.1  x 5 mm2 to Δ𝜃 ≤ 0.005°  (Fig. 3a).  \nGathered  diffraction line profiles and calculations of the lattice strains for various scattering \nvector orientations were  used to determine the stresses . Diffractograms were collected with the \nsteps of  0.1 vs.  cos2φ (Fig. 3),  within the range of φ = (0°, 90°),  in symmetrical transmission \nmode for a constant 2θ = 10° scattering angle. Note that in th e presented  experiments , the \nevolution of interplanar spacings or dependence of 𝐹𝑖𝑗(ℎ𝑘𝑙,ψ,φ,𝑓) are presented versus cos2φ \ninstead  of sin2ψ, as usual . This is because of the specific geometry of the measurements in \nwhich the  tilt of the scatterin g vector from normal to the sample direction is given by constant \nangle ψ. However, it can be easily s hown that in the case of the quasi -isotropic  sample with \nnegligible second -order stresses , the cos2φ plots s hould be linear , and the deviations from \nlinearity are caused by crystallographic texture or/and second -order stresses.  In the present \nwork, t he measurement w as carried out for the non -loaded sample  (initial) , next for specific  \ntensile loads  (characterized by  Σ11) applied to the sample  and finally during elastic unloading \nof the load. To do this, a load rig (from Walter+Bai AG) with a maximum load of 20kN \nmounted on a Newport quarter circle cradle segment was employed for the in situ mechanical \ntest. During plastic deformation , diffraction data were collected for fixed sample strain  (E11) \nafter stabilization of the load applied to the sample.   \nDiffraction peaks  were fitted with the pseudo -Voigt function  to determine their positions 𝐸ℎ𝑘𝑙 \nversus energy scale.  The interplanar spacings <𝑑>ℎ𝑘𝑙 were evaluated using the following \nequation : <𝑑>ℎ𝑘𝑙 = ℎ𝑐\n2𝑠𝑖𝑛𝜃1\n𝐸ℎ𝑘𝑙     (8) \nwhere: c - speed of light and h -Planck constant.   \nAs a result, the interplanar spacings for many diffraction hkl lines were  simultaneously \nmeasured for given  values of cos2φ. Such measurement enable s to determine 𝜎11𝐼 component of \nthe stress tensor in the direction of the applied stress Σ11. \n \n2.2.2 Angle -dispersive diff raction     \nFor ferritic steel , the in situ stress  measurements were performed during the tensile test using \nAD diffraction at the ID15 synchrotron beamline (ESRF, Grenoble, France). The applied high -\nenergy synchrotron radiation with wavelength λ =  0.14256  Å and a beam size of 100 μm x 100 \nμm enabled transmission measurements in the interior of the samples having a square cross -\nsection with a side length of 1.5 mm (Fig. 3b). A square CCD detector (Thales PIXIUM 4700) \nwas used to capture two -dimensional diffraction patterns during 10 -second exposures separated \nby 5-second intervals. It was possible to conduct diffraction measurements in situ throughout a \ncontinuous  tensile test due to the short data collection period.  \nThe Fit2D software [56] was used to handle the collected data by the integration of 2D sectors \nwith an angular size equal to  Δφ = 2° and converting them into the 1D ones composed of \nintensity dependence vs. 2θ scattering  angle . The theoretical functions were then fitted to the \n1D diffractograms using Multifit software  [57]. The positions of the diffraction peaks were \nfound by adjusting the pseudo -Voigt function  vs. 2θ, and the interplanar spacings <𝑑>ℎ𝑘𝑙  for \ndifferent  {hkl}  planes wer e determined from the Bragg law:  \n<𝑑>ℎ𝑘𝑙 = 𝜆\n2𝑠𝑖𝑛𝜃ℎ𝑘𝑙             (9) \nSimilarly , as in the previous experiment ( ED diffraction for austenitic sample) the <𝑑>ℎ𝑘𝑙 \nspacin gs can be determined simultaneously for many reflections  hkl, for given values of cos2φ (with a small ste p of Δφ = 2°) . Therefore, the 𝜎11𝐼 component of stress tensor (in the direction \nof the applied load Fig. 3b) can be determined with small  increments  ∆Σ11 corresponding to \n10-second exposures separated by 5 -second intervals  during the continuous tensile test .  \nIt should be emphasized that the performed experiments were done using both ED and AD  \ntechniques . This allows to verify whether the proposed second -order stress testing methodology \ncan be applied to these techniques and to ensure that the observed phenomena are actually \ncaused by the processes occurring in the material, and not the artefacts leading to changes in \nthe 2θ angle in the AD method or energy shifts  in the ED method. An important question \nconcerning the performed experiment is whether enough information about second order \nstresses can be obtained and whether the experimental data are represent ative for the studied \nsamples . To check this, the analysis of  grains ' contribution to the di ffracted beam  and the \nobtained  results was done. At first , the integration path s in the Euler space w ere found for each \nexperimental point , and the contribution of the grains to diffracted beam intensity was \ncalculated considering  the ODF function. Th is procedure of integration path determination is \nthe same as when the pole figure s from ODF are calculated [50]. Then the measure of \norientations' contribution  to the diffraction peak was determined , taking into account  the ODF \nvalue s along the determined path. The traces in Euler space were found  for all experimental \npoint s, and the contribution measure of grains  was summed over all measuring points. The so -\nobtained function of orientation contribution was normalized in the same way as ODF [50] and \npresented in Fig . 4 for both studied materials . As seen in Fig. 4 a, the experimental information \nis obtained mostly for preferred orientations (which contribution is dominating  - cf. Fig . 2a), \nand on the contribution function , irregular  spots corresponding to more informative regions  are \nseen. This is the effect of the distribution of integration paths in the Euler space. In the case of \nferritic steel, the step of φ angle was very small  (step of Δφ = 2°) , and as a result , 46 point s were \nobtained for each  <𝑎(𝜓,𝜑)>ℎ𝑘𝑙 vs. 𝑐𝑜𝑠2𝜑 plot, while in the case of austenitic sample only 11 points were measured for each plot. Therefore , the orientation contribution function is very \nsmooth and  it is very similar to the ODF presented  in Fig. 2b ; however , it is seen that the regions \nwith a small value of Φ - Euler angle  are more representative . It can be concluded that for both \ntested samples, the preferred orientations contribute the most to the experiment ; however, \ninformation from w eak orientations is also included in the results. It is also seen tha t the \ninformative region is  distributed over a large  part of the Euler space ; therefore , the obtained \nresults are  representative for the majority of grains in the studied samples .   \n \n \nFig. 4. The normalized orientation contribution function calculated for the integration path s \ncorresponding to experimentally measured diffraction peaks with the  weights given by ODF s \nfor (a) austenitic and (b) ferritic  samples.  The levels express multiples -of-equal -contributions . \n \n3. Results  \n3.1 Validation  of the XSF  \nIn order to perform a correct stress analysis, it is necessary to appropriately determine the elastic \ndiffraction constants, especially for elastically anisotropic crystallites.  Thus, in the first step , \nthe theoretically predicted constants using the grain interaction models (Reuss, Voigt, Kr öner) \nand the crystallographic texture were compared  with the experimental results . Elastic constants \nof austenite  and ferrite  and anisotropy of elastoplastic deformation  were verified by analy zing \nthe lattice strains measured for different hkl reflections . To do this, the interplanar spacings \nwere measured before and after a considerable change in the applied load (corresponding to \nincrement ΔΣ11). The F11 constants were calculated from Eq. 2. I n the case of austenite , the \nsample was subjected to elastoplastic deformation up to the applied stress Σ11(1)= 360 MPa  and \nthen completely unloaded to Σ11(2)= 0 MPa  (i.e. ΔΣ 11 = -360 MPa ). As shown in Fig. 2a, t he \ntexture did not change significantly during the tensile test ; therefore,  it was insignificant in the \ncalculation o f the XSF. The ferritic sample broke after the last diffraction measurement at 5% \nstrain because , in the experiment , a stress control mode was used with a constant step of the \napplied load (due to the low  hardening of the ferritic sample, the strain increased sharply in the \nlast step of increasing load).  Thus, t he F11 factors  were calculated based on  the experimental \ndata for sample loading  from  a small load applied to fix the sample  Σ11(1)= 5 MPa up to Σ11(2)= \n352MPa  (i.e., ΔΣ11 = 347MPa) . Model calculations were performed considering  the initial \ntexture shown in Fig. 2b. The calculated and experimental  results of  F11 vs. cos2φ are presented \nin Fig s. 5 and 6.  \nFig. 5. F11 vs. cos2φ curves for experimentally obtained  data for austenitic steel , using ED \ndiffraction . Six different reflecting planes  were used  together with the following grain ela stic \ninteraction models: Reuss, Voigt,  and Eshe lby-Kröner. The experimental points (dots) \ncorrespond  to the stress increment  ΔΣ11 = -360 MPa , during sample  unloading . \n \nFig. 6. F11 vs. cos2φ curves for experimentally obtained data for ferritic steel, using AD \ndiffraction. Five different reflecting planes were used together with the following grain el astic \ninteraction models: Reuss , Voigt , and Eshelby -Kröner. The experimental points (dots) \ncorrespond to the stress increment  ΔΣ11 = 347.3 MPa  during elastic loading . \n \nIn Figs. 5 and 6, it can be c learly seen that the values of F11 (XSF) calculated by the Eshelby -\nKröner model fit best the experimental results  for both sampl es. Therefore, t he stress measured \nby X -ray diffraction was determined using the XSF derived by the Eshelby -Kröner model with \nthe ODF function shown in Fig. 2 . It should be noted that despite crystallographic texture the \nnon-linearities of the F11 vs. vs. 𝑐𝑜𝑠2𝜑 curves are very small , at least for the chosen direction \nof lattice strains  measurement.  This is confirmed both by the experimental and theoretical \nresults.  \nIt should be emphasized that the in situ test of factors  F11 performed with multiple hkl reflections \nis very rigorous because many groups of crystallites with different orientations are involved in \nthe diffraction experiment  (cf. orientation contribution functions  show n in Fig . 4). It proves that \nthe used single crystal elastic  constants and grain interactions are correct ly approximated .  \nIt is worth noting that , in our work , the careful verification of the applicability of the model for \ncalculation XS Fs is necessary because to evaluate the effect of second -order stresses, at first \nthe effect of the stress applied to the sample must be known. It is well know n that for textured \nsample s, the nonlinearities of the <𝑎(𝜓,𝜑)>ℎ𝑘𝑙 vs. 𝑐𝑜𝑠2𝜑 (or <𝑎(𝜓,𝜑)>ℎ𝑘𝑙 vs. 𝑠𝑖𝑛2𝜓 in \nstandard stress analysis ) can be caused by  nonlinearities of  F11 vs. 𝑐𝑜𝑠2𝜑 dependence  (for the \nuniaxial test). Thus to minimize such an effect , the orientation of the scattering vector w as \nchanged between RD and TD because , in this case , the nonlinearities on F11 vs. vs. 𝑐𝑜𝑠2𝜑 \ncurves were small, as shown in Figs. 5 and 6. Therefore we avoided the problem of overlapping \nthe nonlinearities coming from the anisotropy of X SF (due to texture) and th ose caused by the \nsecond -order stresses.  \n \n \n 3.2 Stress evolution during elastoplastic deformation  \nNext,  the results of stress measurements during the controlled tensile test were  analyzed  for \nboth samples using the verified XSF . The interplanar spacings (Eqs. 8 and 9)  were measured in \nsitu for each certain number of loads  in the elastic and plastic deformation range  during loading \nand unloading . As already mentioned,  the ferritic sample fractured  during the test before the \nunloading step . The force was applied along TD for both samples . Applying Eq. 3 and the fitting \nprocedure for the experimental data, the set of quantities: first-order stress  component  𝜎11𝐼, 𝑎0 \nand the q-factor were determined  from Eq. 5 using  the least square procedure.  In fitting all \nlattice parameters <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎 measured  for different hkl reflections were used \nsimultaneously for a given load or unloaded sample.  The 𝜎22𝐼, 𝜎33𝐼 and shear first -order stress \ncomponents were assumed equal to zero because  uniaxial  stress was imposed during the tensile \ntest. The measurement of the <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎 for orientations of scattering vector changed  \nbetween RD  (x2) and TD (x1) is enough to determine 𝜎11𝐼 and 𝑎0, similarly as in standard X -ray \nmeasurement with the assumption of z ero normal stress  (𝜎33𝐼=0). It is worth noting that, , the \nadjust ement  of the q-factor is based on model -calculat ed lattice strains <𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖 fitted \nto experimental n onlinearities of  the <𝑎(ψ,φ)>ℎ𝑘𝑙𝜎 vs. 𝑐𝑜𝑠2𝜑 plots . The lattice strains \n<𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖 are caused by all stress  components of second -order stresses  𝜎̃𝑖𝑗𝐼𝐼,𝑝𝑖, and \ninformation about the varia tions  of these components and the form of the stress tensor for \nindividual grains is determined by the model . The factor q multiplies the magnitude of lattice \nstrains <𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖 and can be used  as the scaling factor for all  components of the model \nstress tensor   𝜎̃𝑖𝑗𝐼𝐼,𝑝𝑖, to find their magnitude s in the real sample  𝜎𝑖𝑗𝐼𝐼,𝑝𝑖 (for all grains \nsimultaneously) . Therefore , knowing the values of the q-factor , the second -order  stresses 𝜎𝑖𝑗𝐼𝐼,𝑝𝑖 \n(full tensor) were  calculated from Eq. 6  for each orientation of the grain lattice.  The self -consistent model of elastoplastic deformation  elaborated by Lipinski and Berveiller \n[44] was used to calculate the values <𝜀̃(ψ,φ)>ℎ𝑘𝑙𝑝𝑖 needed to interpret the experimental \nresults  and to determine the second -order  stresses  from Eq. 6. The input file generated for EPSC \nmodel , contain ed 10000 spherical inclusions ( representing grains ) with equal volume fraction \nand distribution of orientation s determined based on  initial experiment al ODF  (Fig. 2) . Single \ncrystal elastic constants defined with respect to the crystal lattice  and the residual stresses 𝜎11𝐼,𝑟𝑒𝑠 \nmeasured for the initially non -loaded sample  were assigned to each inclusion . The input files \ncreated in this way were used to simulate  the elastic -plastic tensile deformation of ferritic and \naustenitic samples using the EPSC model.  \nIn calculations , the isotropic hardening matrix and the linear hardening law were assumed . All \nslip systems in all grains (for the given sample) had the same initial Critical Resolved Shear \nStress (CRSS) . The parameters characterizing slip system activation ( CRSS) and hardening ( H) \n[44] were optimized to adjust the theoretical macroscopic stress -strain curves to the \nexperimental ones. The values of optimal parameters for both sample s are given in Table 3, \nwhile the fitted (model) and experimental macroscopic mechanical (obtained during in situ \ntensile  test) curves will be presented in Figs. 11 and 12, together with the plots  obtained from \ndiffraction . \nPreviously obtained values of CRSS  and H parameter s optimized by fitting  the EPSC model to \nan experimental mechanical stress -strain plot  for the austenitic sample are given by Neil at al. \n[40], and they are not far from th ose given in Table 2, i .e., τc0  = 93 MPa and H = 375 MPa  \n(represented by θ0 in that work) . Certainly , these values  are not the same because they depend \non the material microstructure , chemical composition and performed treatment . Concerning \nstudied ferritic steel, we have not found analogous results, but it is well known that τ c0  is related \nto the yield stress which could be very different for different types of microstructure and material processing. However , a very small work hardening ( H→0, at the beginning of plastic \ndeformation , c.f. Table 3) is characteristic for ferrite, as was found , e.g., in pearlitic steel [32].     \nTable 3. Input parameters characterizing the initial microstructure of the investigated \nmaterials  \nStructure  Slip systems  Initial CRSS τc0 \n(MPa) Hardening \nparameter  \nH (MPa)  \nbcc \n(ferrite)  <111>{1 -10} \n<111>{11 -2} \n<111>{12 -3} 190 0 \n \nfcc \n(austenite)  <110>{111}  84 270 \n \n \nIn the stress analysis using  the least square  method , seven reflections: 200, 220, 222, 311, 331, \n400, 420, and five reflections: 110, 200, 211, 310, 220 were considered for austenitic and ferritic \nsteel, respectively . The analysis  was carried out for the initial non -loaded samples and for the \nsample s subjected to various uniaxial loads applied  during in situ  measurements . Finally , the \nstresses were determined in the unloaded samples. In the case of the fractured  ferritic specimen, \nthe measurements were performed  in a different location than during the tensile test to avoi d \nthe effect of stress heterogeneity close to the fracture surface.  \nThe first example of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ curves  is shown for the initial (non -loaded) \nsamples in Fig. 7. The theoretical plots are presented for two assumptions, i.e. when the plastic \nincompatibility stresses are not present (dashed lines, q=0) and when their influence is taken \ninto account . The q parameter is determined from Eq. 5 (continuous lines).   \n  \nFig. 7. Comparison of the experimental data (points)  with the calculated <𝑎(ψ,φ)>ℎ𝑘𝑙vs. \ncos2φ plots obtained for adjusted q-parameter  (solid line) and the q = 0 (dashed line) . The results \nare shown  for the initial  non-loaded (a) austenitic sample  (Σ11 = 0 MPa ; both lines , solid and \ndashed , overlap each other ) and (b) ferritic sample (Σ 11 = 5 MPa, the small load was applied to \nfix the sample).  \n \nIn the case of initial austenitic sample , the q-fitting method had no impact on the results . Thus \nthe small non-linearities in the experimental <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ data do not coincide \nwith those simulated for the uniaxial tensile elastoplastic deformation.  It means  that the residual \nsecond -order  stresses generated during sample preparation , cannot be determined for the initial \nsample , and only the value 𝜎11𝐼 of tensile residual first-order stress in the measured gauge \nvolume was found.  On the contrary , when second -order stress is taken into account in stress \nestimat ion for the initial ferritic  sample  (with adjusted q parameter) , the result s of the least \nsquare fitting  are much improved , as shown in Fig. 7 b. This means that the residual second -\norder  stresses in the initial state of the  ferritic sample  well coincide with those predicted by the \nEPSC model for plastic deformation occurring during tensile test.  Moreover, the  first-order \ncompressive  stress 𝜎11𝐼 was obtained for the measured gauge volume . This sample contains \nresidual stresses due to  the cold rolling procedure followed by sample preparation.  \nThe second example of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ curves is shown for the loads applied in \nsitu during  purely elastic  tensile  deformation  (Fig. 8). In this case , the slopes of the plots \nsignificantly changed , showing the tensile character of the first -order stresses for both studied \nsamples . However, the analysis of the second -order  stresses shows the same results a s for initial \nsamples, i.e. , no correlation between model and experiment non-linearities in the case of the \naustenitic sample (adjustment of q parameter does not improve the quality of fitting) and very \ngood correlation  of non -linearities in the case of the ferritic sample , as shown  in Fig. 8 a and 8 \nb, respectively . That means th at the state of second -order  stresses did not change significantly \nduring the elastic loading of both samples.  \nThe next example of the  <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ curves was chosen for the significant  plastic \ndeformation of both sample s under applied load  (Fig. 9). In this case , the slopes of the plots increased , indicat ing the increase of applied tensile load . Also, interesting evolution of non -\nlinearities was observed in the case of the austenitic sample  (Fig. 9 a). Indeed, the solid  lines \nrepresenting results with adjustment of the q parameter started to match the experimental points. \nIt implies  that the non -linearities of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ plots predicted by the model  \nmore  closely match those o btained from measur ements. Therefore  the plastic incompatibility \nsecond -order  stresses  (i.e., the second -order stresses induced by plastic deformation) can be \ndetermined  using Eq. 6. It can also be  concluded that the stress state for the grains was \ntransformed entirely  due to  the plastic deformation , i.e. , second -order  stresses occurring  \ninitially in the prepared sample  have been completely  replaced with stresses  generated due to \nplastic deformation during  the tensile test . In the case of a ferritic sample , similarly as in the \ninitial sample , the non-linearities in the sample subjected to plastic deformation are still well \nreproduced by model data  (the fitting of the solid line is unquestionably better than that of the \ndashed line , as shown in Fig.  9 b. It means that no significant modification  of the second -order  \nstresses occurred due to plastic deformation during the tensile test. \nFinally, the last example  of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ curves  referring to the results for the \nunloaded austenitic sample and the fractured  ferritic sample is presented in Fig. 10. When \ncomparing the initial (Fig. 7 a) and unloaded state  (Fig.  10 a) for the austenitic sample , it can \nbe seen that the dep ende nce <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ changed significantly . This proves that \nthe state of residual second -order plastic incompatibility stress was c hanged entirely  during the \nplastic deformation of the sample . The essential  improvement of fitting quality  for the unloaded \nsample,  when the q parameter is adjusted , confirms  that the analysis is  carried out correctly \nwhen the model data from the EPSC model for the tensile test are used. The results obtained \nfor the fractured  ferritic sample (Fig. 10 b) compared with the initial state (Fig. 7 b) show that \nno significant change occurred in the state of second -order stresses.  \n  \nFig. 8. Comparison of the experimental data (points)  with the calculated <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. \ncos2φ plots obtained for adjusted q-parameter (solid line) and the q = 0 (dashed  line). The results \nare shown for the elastic range of deformation for (a) austenitic sample (Σ 11 = 140 MPa) and (b) \nferritic sample (Σ 11 = 352 MPa).  The solid and dashed lines overlap each  other in the case of \naustenite.  \n \nFig. 9. Comparison of the experimental data (points)  with the calculated <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. \ncos2φ plots obtained for adjusted q-parameter (solid line) and the q = 0 (dashed line) . The results \nare shown for the load causing plastic deformation for (a) austenitic sample (Σ 11 = 360 MPa ) \nand (b) ferritic sample (Σ 11 = 585 MPa ). \n \nFig. 10. Comparison of the experimental data (points)  with the calculated <𝑎(ψ,φ)>ℎ𝑘𝑙  vs. \ncos2φ plots obtained for adjusted q-parameter (solid line) and the q = 0 (dashed line) . The results \nare shown for the (a) unloaded austenitic sample (Σ11 = 0 MPa)  and fractured ferritic sample \n(Σ11 = 0 MPa) . \n \nBy applying Eq. 5 and by fitting the results from the model to the experimental data, the values \nof first-order stress 𝜎11𝐼 and the q factor could be  found. Then the values  of second -order stress  \n𝜎𝑖𝑗𝐼𝐼,𝑝𝑖 for each polycrystalline grain was determined  using Eq. 6 in which the model stresses ase \nmultiplied by q factor . Finally, the mean value of von Mises  stress  𝜎𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑖̅̅̅̅̅̅̅̅ over all gr ains was \ncalculated  for the initial , loaded , and unloaded/ fractured  samples  (for details , see [8,9] ). The so \ndefined 𝜎𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑖̅̅̅̅̅̅̅̅ is a good  measure  of mean magnitude of second -order plastic incompatibility \nstresses  because the hydrostatic stress computed from σij𝐼𝐼,𝑝𝑙 is equal to zero for purely plastic \ndeformation, as it was also verified using EPSC model . In Figs. 1 1 a and 1 2a, the so-obtained \nvalues of stresses  𝜎11𝐼 and 𝜎𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑖̅̅̅̅̅̅̅̅  are presented vs. sample strain E11. To determine the evolution \nof the stress state during the loading process , the first- and second -order stress are presented as \nthe function  of the superposition of the  imposed  stress ( Σ11) and residual stress in the initial \nnon-loaded sample  (𝜎11𝐼,𝑟𝑒𝑠), i.e.: Σ11 + 𝜎11𝐼,𝑟𝑒𝑠  (Figs. 1 1 b and 1 2 b). These  graphs illustrate the \ncomparison of the first-order stress evolution in the irradiated  volume during sample loading \ndetermine d in two ways: as the s um of the residual  stress  (𝜎11𝐼,𝑟𝑒𝑠) superposed with the imposed \nstress  (Σ11) and as the stress determined directly from the diffraction experiment ( 𝜎11𝐼) for the \ncorresponding load. Certainly , the two values should be equal  if the sample and stress state is \nhomogenous . The difference between them indicates heterogeneity of the stress distribution \nacross the sample along the direction x2 (Fig. 3) caused by the material processing and \npreparation of the dog -bone -shaped samples . It should be emphasized that due to the much \nsmaller size of the synchrotron beam spot compared to the sample  width (in direction x2), the \nlocal stress state in the center of the samples was determined in both measured initial samples . \nAs mentioned before i n the case of the experiment performed for ferritic sample (the high \nenergy synchrotron radiation with wavelength λ = 0,14256 Å  at ESRF ) the beam size of 100 μm x 100 μm enabled transmission measurements in the interior of the sample having 1.5 mm \nin width. The primary beam cross -section for the EDDI experiment using a white X -ray beam \nwas equal to 1 x 1 mm2, and again the spot size was smaller than the sample's width equal to 5 \nmm. \nThe evolution of the second -order stresses 𝜎𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑖̅̅̅̅̅̅̅̅  vs. (𝜎11𝐼,𝑟𝑒𝑠+ Σ11) are also shown in Figs. 1 1 \nb and 1 2b. In addition , in Figs. 11 and 1 2, the results of the EPSC prediction corresponding to \nthe experimental data are presented ( solid lines). The prediction starts from the initial small \nresidual stress  𝜎11𝐼,𝑟𝑒𝑠 (tensile for austenitic sample seen in Fig. 11; and compressive for ferritic \nsample  seen in Fig. 1 2) and ends at the residual stress es value  remained after the tensile test , so \nafter sample unloading.  It should be emphasized that the parameters of the model (Table  3) \nwere adjusted to reproduce the dependence of the first-order stress  (in the information volume \nseen by diffraction)  determin ed as 𝜎11𝐼,𝑟𝑒𝑠+ Σ11 on the sample strain E11 (squares shown in Figs. \n11 a and 1 2 a). \nIn order to confirm  the correct ness of the  chosen model (Eshelby -Kröner) for the calculation of \nthe stress factor 𝐹11 the experimental values 𝜎11𝐼,𝑟𝑒𝑠+ Σ11 are compar ed with the first -order \nstress 𝜎11𝐼 determined by diffraction using Eshelby -Kröner, Voigt , and Reuss model  (Figs. 11 a \nand1 2 a). An excellent  agreement was obtained for the Eshelby -Kröner model , while the use of \nboth other models leads to a significant discrepancy between the stress calculated from the \napplied load (𝜎11𝐼,𝑟𝑒𝑠+ Σ11) and that measured by diffraction (𝜎11𝐼) for both investigated  \nmaterials . It supports the choice of the Eshelby -Kröner  model  for this study , which has already \nbeen  verified in section 3.1. It should be emphasized that the perfect agreement between 𝜎11𝐼 \nand 𝜎11𝐼,𝑟𝑒𝑠+ Σ11 stress es were obtained for the elastic range of deformation  (see Figs. 1 1 b and \n12 b), where the solid re d line (obtained from the EPSC model ) illustrates the equality 𝜎11𝐼 = \n𝜎11𝐼,𝑟𝑒𝑠+ Σ11. Slight deviation s of the diffraction results from the solid re d line are observed for the plastic deformation range, especially for the austenitic sample . This effect  can be explained \nby a modification  of the first-order residual stress  distribution (heterogeneity) along the x2 axis, \nleading to different residual stress 𝜎11𝐼 in the measured volume after the tensile test compar ed \nto the initial value  before loading  (cf. Fig. 1 1 b, loading and unloading).  \n \nFig. 11. Evolution of the experimental first-order stresses expressed in two ways: 𝜎11𝐼 and \n𝜎11𝐼,𝑟𝑒𝑠+ Σ11 compared with 𝜎11𝐼 predicted by the EPSC model , using parameters given in Table  \n3, for an austenitic sample subjected to tensile deformation. The evolution of determined  \nsecond -order stresses mean von Mises stress (σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙̅̅̅̅̅̅̅̅) is also presented . The plots vs. sample \nstrain  E11 (a) and first -order stress  Σ11+𝜎11𝐼,𝑟𝑒𝑠 (b) are shown . \n \n \n Fig. 12. The analogous evolutions as presented for the ferritic sample  in Fig. 1 1.  \n \nImportant results were obtained when the second -order stresses (characterized by σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙̅̅̅̅̅̅̅̅) were \nanalyzed . The evolution of σij𝐼𝐼,𝑝𝑙 stresses should be discussed together with the changes in the \nfigure of merit 𝜒2 (Eq. 8) describing the least square fitting  quality of the  calculated  lattice \nparameters <𝑎(ψ,φ)>ℎ𝑘𝑙 to the experimental results .  \nIn Fig. 1 3, the variation of  𝜒2 vs. sample total strain E11 is shown for two ways of data treatmen t. \nFirst assum es q = 0 (the influence of  σij𝐼𝐼,𝑝𝑙 is neglected) , while the second takes  into account  \nthe adjust ed q-parameter . Both for austenitic (Fig. 1 3 a) and ferritic (Fig. 1 3 b) samples , the \nfitting is much better , i.e., 𝜒2 is much  lower when the q is adjusted  in Eq. 5 (excluding  the two \nfirst points for the austenit ic sample ).  \n \nFig. 1 3. Evolution of parameter 𝜒2 in the function of sample strain E11 for austenitic (a) and \nferritic (b) samples subjected to a tensile test. A dashed vertical line separates the elastic range \nfrom the plastic deformation rang e.  \n \nIn Fig. 14 the 𝑎0 stress -free parameter obtained in stress analysis was shown for different \ndeformation of the sample , using the same range for 𝑎0 as for <𝑎(ψ,φ)>ℎ𝑘𝑙 in Figs. 7 and \n10. Comparing the variation of  <𝑎(ψ,φ)>ℎ𝑘𝑙 in Figs. 7 and 10 with the changes of 𝑎0  in \nFig. 14, it can be concluded that the value 𝑎0 obtained from the analysis is almost constant \nduring the tensile test , and their insignificant changes do not influence results concerning \ndetermined first- and second -order stresses .  \nE110.00 0.02 0.04 0.06 0.08 a0  (nm)\n0.35920.35930.35940.35950.35960.35970.35980.35990.36000.36010.3602\nAustenite\na)\nE110.00 0.01 0.02 0.03 0.04 0.05 0.06 a0  (A)\n0.28660.28670.28680.28690.28700.28710.2872\nFerrite\nb)\n \nFig. 14. The 𝑎0 stress -free parameter determined for loaded and unloaded samples vs. sample \nstrain . The same vertical scale as for the initial  (Fig. 7) and unloaded /fractured  samples was \napplied for better comparison  (Fig.  10). \n \nFinally , it is interesting to present the dependence of the plastic incompatibility second -order \nstresses on grain orientation  (i.e. σij𝐼𝐼,𝑝𝑙 obtained from Eq. 6  using our method ). Because the full \ntensor σij𝐼𝐼,𝑝𝑙 is buil t from 6 independent components , its presentation in Euler space is difficult . \nTherefore, the von Mises stress σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 calculated from the tensor σij𝐼𝐼,𝑝𝑙 for each orientation can \nbe shown as the measure of the magnitude of the second -order plastic incompatibility stresses . \nThe so calculated σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 is shown in Euler space for austenitic  (Fig. 15 a) and ferritic (Fig. 15 \nd) samples subjected to  tensile deformation followed by  unloading /fracture . It was found  that \nthere the minima of von Mises second -order stresses σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 corresponds to maxima of ODF  \nobtained by EPSC for austenitic steel  (compare Fig. 15 a with Fig.15 b ); however , no such \ncorrelations  were found in the case of the ferritic sample . What is more, in some regions  of \nEuler space , minima of  σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 correspond to minima of  ODF  – compare Fig. 15 d with Fig.1 5 \ne. Comparison of Fig s. 15 b and e with Fig. 3 shows that the texture  change during the \nperformed deformation  is very small  for both  studied  samples .  \nFinally , the distribution of second -order stresses σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 can be compared with the maximum \nvalue of the resolved shear stress RSS for all potentially active slip systems at given lattice \norientation s. It can be noticed that low value s of the maximum RSS correspond to small value s \nof von Mises second order plastic incompatibility stresses  in the case of austenitic sample, c.f. \nFig. 15 a and 15 c . In the case of ferritic samples , the correlation is similar but not so strong, \ni.e., for the low  values of  maximum RSS , the values of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 are also low, but not always small \nvalue of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 correspond to a low value of maximum RSS , c.f. Fig. 15 d and 15 f .  \nFig. 15. Second -order plastic incompatibility stress (von Mises value  σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙) obtained using the \ninitial experimental texture and initial residual stresses (a, d),  final ODFs obtained from model  \n(b, e) and a maximum value of resolved shear stress (RSS) from all potentially active slip \nsystems assuming uniaxial macrostress state (c, f). The resul ts are presented in Euler space for \naustenitic (a,  b, c) and ferritic (d,  e, f) steels  subjected to tensile deformati on and unloaded . \n \n4. Discussion  \nBased on the tests performed and the findings presented in section 3.1, it can be said that the \nEshelby -Kröner model for the sample's interior, in which the lattice strains were measured for \nthe elastic range of sample deformation, accurately predicts the XSF values (Fig.  6). This \nimplies that the ellipsoidal Eshelby inclusion in the effective matrix representing the \nmacroscopic sample can be used to predict the intergranular interactions, which are the elastic \nresponse of the grains to the applied load. For the tested textured samples, slight non -linearities \nof the determined F11 vs. cos2φ plots were found when the load was applied along TD , and the \norientation of the scattering vector changed between TD and RD . It was confirmed by both the \nexperimental and calculated F11 vs. cos2φ plots . An important takeaway  of this result is that the \nnon-linearities <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ plots measured experimentally (Figs. 7-10) cannot be \nexplained by the elastic anisotropy of the crystallites and textured sample.  \nAfter XSF verification, the methodology based on the results of the EPSC model was used to \ndetermine the first and second -order stresses in the initial and in situ deformed sample s. The \nproposed approach of data interpretation made it possible to identify the reason of non -\nlinearities in the  <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ plots , which can be  explained as the effect of second -\norder plastic incompatibility  stress es. Such stresses arise as a result of plastic anisotropy of \nindividual grains, leading to mismatch with the neighboring  ones and they can be simulated by \nthe EPSC model. The very good agreement of the calculated and experimental non -linearities \nconfirms the correctness of the model results.  \nUsing the methodology de scribed in the Introduction (Eqs. 5 and 6), a zero value of the plastic \nincompatibility was found for the initial austenitic  sample and when the load was applied within \nthe elastic range of deformation (two first points in Fig s. 11 and 13 a). In this case,  despite \nadjusting q, high values of 𝜒2 were obtained , approximately equal to those obtained assuming \nq = 0 (cf. Fig. 1 3 a). However, t his does not mean that the plastic incompatibility  stresses σij𝐼𝐼,𝑝𝑙 \nin the non-loaded  sample are negligible, but their variation with orientation does not coincide \nwith that predicted  by the EPSC model.  Therefor e the stresses σij𝐼𝐼,𝑝𝑙 cannot be determined for  \nthe initial sample and the sample subjected to the load within the elastic deformation range . Then , when plastic deformation beg an, the value of 𝜒2 decrease d at about E11 = 1.5 % (for the \ncase of adjusted  q) and stabiliz ed at about E11 = 1.5 % of sample strain. Starting from this point , \nthe analysis of σij𝐼𝐼,𝑝𝑙 can be performed because the fitted  data matche d the experimental point s \nthat was already observed for the  example plots <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ shown in Fig s. 8 a \nand 10 a. Therefore, t his means that the evolution of the value σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙̅̅̅̅̅̅̅̅ was qualitatively \ndetermine d, showing its progressive increase with plastic deformation (above approximately \nE11 = 1 %) until the beginning of unload ing, as presented in Fig. 1 1 b. Then the value  of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙̅̅̅̅̅̅̅̅ \nstresses  remain s unchanged  during sample unloading . It should be concluded that the sample \nstrain of about E11 = 1% - 2 % is enough to  generate the  incompatibility stresses σij𝐼𝐼,𝑝𝑙 \ncorresponding to tensile plastic deformation , which replac ed the previous σij𝐼𝐼,𝑝𝑙 stresses  present \nin the initial austenitic sample. This result agrees well  with the behavior of the second -order \nplastic incompatibility stresses determined recently during the tensile test in such materials as \nmagnesium alloy [33], ferrite in pearlitic steel  [32], and both phases in duplex st eel [3]. \nContrary to austenite, non -linearit ies of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ plots measured in the \ninitial non-deformed ferritic sample coincide well with model prediction causing significant \nimprovement of the fitting , so the decrease of 𝜒2, when the  parameter q is adjusted  (Fig. 1 3 b). \nThis significant ly smaller  value of  𝜒2 for the analysis taking into account second -order stresses \n(q adjusted) compared to the assumption of  q =0 is then observed during the elastic a nd plastic \ndeformation of the sample . It was found that the character of the second -order stresses σij𝐼𝐼,𝑝𝑙 in \nthe non -loaded sample ( subjected primarily to a cold roll ing process ) is similar to th at which \ncorrespond s to tensile plastic deformation (as predicted by the model). It is also seen that the \ntensile test do es not significantly change the character of the second -order stresses in the ferritic \nsample during the tensile test but the value of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙̅̅̅̅̅̅̅̅ increased durin g deformation (cf. Fig. 12 \nb). Interesting results were obtained from the comparison of  σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 distribution with the ODF \nfunction  (cf. Fig. 15) . Despite  the correlation between minima of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙  with maxima of ODF  \nin the case of austenite, such correlation was not confirmed by the ferritic sample. It leads to \nthe conclusion drawn  by Daymond et al.  [42], based on  lattice strain evolution in an austenitic \nsample during in situ tensile test , that the second -order stresses do not correlate strongly  with \nthe crystallographic texture. This fact can be explained by the Eshelby -type interaction  in which \nthe ellipsoidal inclusion interact s with the mean matrix  representing polycrystalline material . \nCertainly , the deformation of the matrix depend s on the texture , but this is not a very significant \neffect compar ed to differences in the plastic behavior of grain s (inclusions) , which are caused  \nby the activation of slip systems  depending on the lattice orientation with respect to the applied  \nload. In turn, the resulting plastic incompatibilities between the grain s and the matrix mostly \ndepend on the RSS values on different slip systems . This was confirmed in the present work, \nwhere the  correlation between the low values of the minimum  Schmid factor (corresponding to \nuniaxial load)  with the minima of σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 stresses were found for both studied samples  (cf. Fig. \n15). Similar conclusions concerning the correlation between the Schmid factor for basal system \nand σ𝑀𝑖𝑠𝑒𝑠𝐼𝐼,𝑝𝑙 stresses was also observed  for Mg -alloy studied in [18].     \n \n5. Summary and conclusions  \nThis paper deals with the reasons for the non-linea rity of the <𝑎(ψ,φ)>ℎ𝑘𝑙  vs. cos2φ plots \nobtained  by diffraction methods used for stress  measurement s. High energy synchrotron beam \ndiffraction in transmission mode and prediction from the EPSC model allowed for studying  the \nimpact of crystallographic texture and second -order plastic incompatibility stresses on the \nresults of stress measurements during in situ tensile tests carried out for ferritic and austenitic \nsteels.  The main findings of the work are the following:  - The F11 XSF constants determined in situ from diffraction measurements carried out for \nthe applied stress increments  (elastic loading or unloading) are almost linear vs. cos2φ \nwhen they are measured between TD and RD  for both textured materials . This linear \nbehavior was confirmed by models for XSF determination in which crystallographic \ntexture is taken into account.  \n- The Eshelby -Kröner model used for XSF calculations agrees best with the experimental \nresults.  \n- It is possible  to determine the plastic incompatibility second -order stress only when the \nmode of deformation process is known . In this case , the model prediction of anisotropy \nof lattice strains due to second -order plastic incompatibility stresses is possible and the \nfirst-order as well as  second -order stress es can be simultaneously determined.  \n- For the unloaded austenite  sample , the non -linearities of the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ \nplots  are small , while a significant undulation of the plots  was observed for ferrite.  For \nferrite , it was possible to determine the second -order incompatibility stresses because \nthe non -linearities correspond with the model prediction . For  austenite , it was not \npossible  because there are no such correlations.  \n- During the tensile test , the second -order incompatibility  stresses  remained unchanged  \nin the purely elastic range, but they immediately transformed  for the sample strain above \n1 - 2%. Although these stresses changed significantly in the austenitic  sample, such \ntransformation did not occur for the ferritic sample. This is  because the residual  plastic \nincompatibility in the initial ferritic  sample  had a character similar to those generated \nduring the tensile test.  \n- It was found  that the second -order stresses are generated or modified only during plastic \ndeformation, while they remain unchanged  for a purely elastic sample deformation . The \nsecond -order incompatible stresses  remaining in the samples after unloading as residual stresses were the reason for the non -linearities in the <𝑎(ψ,φ)>ℎ𝑘𝑙 vs. cos2φ plots for \nboth investigated samples.  \n- The orientation distribution of second -order plastic incompatibility stresses is not \ndirectly correlated with crystallographic  texture  but correlates with the maxim um value \nof Schmid factor calculated for all potentially active slip systems.  \n \nAcknowledgments  \nWe would like to thank Helmholtz -Zentrum Berlin für Materialien und Energie for the \nbeamtime provided at 7T -MPW -EDDI beamline (BESSY II).  \n \nFunding  \nThis work was financed by grants from the National Science Centre, Poland (NCN): UMO -\n2021/41/N/ST5/00394 . Research project was partly supported by the program “Excellence \ninitiative – research university\" for the AGH University of Science and Technology . \n \nConflict of interest  \nThe authors declare no conflict of interest.  \n \n \n \n Authorship contribution statement  \nM.M -W. Conceptualization, Methodology, Software, Validation, Data curation , Formal \nanalysis, Investigation, Writing - Original draft preparation , Writing - Review & Editing, \nProject administration  \nA.B.  Conceptualization, Methodology, Writing - Review & Editing, Funding acquisition  \nC.B. Sample preparation, Formal analysis , Review & Editing  \nM.W.  Formal analysis, Investigation , Review & Editing  \nS.W.  Investigation  \nR.W.  Sample preparation  \nG.G.  Investigation, Review & Editing  \nP.K. Software, Review & Editing   \nM.K.  Conceptualization, Investigation, Methodology , Review & Editing  \nCH. G.  Conceptualization, Investigation, Methodology, Review & Editing  \n \n \nData availability  \nThe raw data required to reproduce these findings are available to download from \nhttps://data.mendeley.com/datasets/trgrhm83hs/draft?a=a2b7f7eb -5533 -4864 -a5e8 -\n22fc17babb0c. The processed data required to reproduce these findings are available to \ndownload from https://data.mendeley.com/datasets/b6v7w9rpdg/draft?a=f247ff02 -764f-43e4 -\nb59d -57ae4297fb49.  Bibilography  \n[1] E. Macherauch, H. Wohlfahrt, U. Wolfstieg, Zur zweckmäßigen Definition von \nEigenspannungen, HTM Journal of Heat Treatment and Materials. 28 (1973) 2 01–211. \nhttps://doi.org/10.1515/htm -1973 -280305.  \n[2] P. Kot, A. Baczmański, E. Gadalińska, S. Wroński, M. Wroński, M. Wróbel, G. \nBokuchava, C. Scheffzük, K. 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Hilairet, Multifit/Polydefix:  A framework for the analysis of polycrystal \ndeformation using X -rays, Journal of Applied Crystallography. 48 (2015) 1307 –1313. \nhttps://doi.org/10.1107/S1600576715010390.  \n "    },    {        "title": "1907.03703v1.Effect_of_elasto_plastic_compatibility_of_grains_on_the_void_initiation_criteria_in_low_carbon_steel.pdf",        "content": "Effect of elasto -plastic compatibility  of grains  on the void initiation criteria \nin low carbon steel  \nAnish Karmakar1*, Kaustav Barat2 \n1Indian Institute of Technology, Roorkee , Uttrakhand -247667, India  \n2CSIR -National Aerospace Laboratories, Bangalore -560017 , Karnataka, India  \nCorresponding author email id: anishfmt@iitr.ac.in  \n \n \nAbstract:  \nThe present study evidences the role of ferrite grain size distributions on the occurrence of \nvoid initiation in a low carbon steel. Various thermomechanical treatments were  done to \ncreate ultrafine, bimodal and coarse range of ferrite grain distributions. A two parameter \ncharacterization of probable void initiation sites is proposed; elastic modulus difference and \ndifference in Schmid factor of the grains surrounding the voi d. All microstructures were \ncategorized based on the ability to ease or resist void nucleation. For coarse grains, elastic \nmodulus difference as well as the Schmid factor difference is  highest, intermediate for  \nultrafine and lowest for bimodal microstructu re.  \n \n \nTremendous  drives are there from automotive, naval  and oil/gas sectors for the development \nof steels with optimum combination of strength, ductility, toughness and weldability [1]. In \nrecent years, demand for electric vehi cles has taken the mandate  for lightweight steel to a \nnew pedestal. Mechanical  properties can be improvised  depending upon the tuning of the \nmicrostructural constituents [2,3] . The requirement of strength is undoubtedly very much \nessential from the viewpoint of fatigue and crash resistance of the componen t [4]. Alongside  \nthe requirement of good ductility , formability /weldability  can’t be neglected [5,6] . In order to \ndesign a better weldable and formable component, fully ferritic microstructure is an essential \ncriterion  [2,3,7]. Steels with u niform distribution of fine microalloyed carbides / cementite in \nthe ultrafine grained ferrite matrix is develop ed for optimum balance of strength and ductility \n[8,9] . Role of carbide particles in the failure behaviour of ferrite -carbide structures  has been  \ndiscussed elaborately in literature [10–12]. At larger carbide contents , the dispersion rate \ndecreases with a coarse distribution of carbides. These carbides crack in front of the stress \nconcentration front as well as facilitating cracks to propagate via grain boundaries [12]. \nNanoscale carbide dispers ed NanoHiten and BHT steels have been especially designed for \ncar-chassis [13]. The degeneration of carbides has been studied for a long time considering \nthe pile -up of the dislocations against carbide particles, generating stress concentrations in \nfront of the particles  [11,14] . The results in [10] evidenced tha t the toughness measured in \nnotched specimens is mainly determined by the grain sizes, which define the local fracture \nstress (\n. Size of carbide particle plays a minor role  there . However, on the contrary, in pre -\ncracked specimens the toughnes s is sensitive to the carbide sizes, which affect the critical \nplastic strain  (\n  for initiating a crack nucleus . The role of ferrite matrix on the phenomena \nof fracture has  been subdued  under the presence of carbides. Overall the finer carbides  with a moderate volume fraction  can control the properties better . The effect of ferrite grain size \nwith a bimodal distribution on the tensile properties of low carbon steels have already been \nstudied by several researchers [15–17]. Damage initiation can be expressed from the elasto -\nplastic incompatibility created during the  straining. The e lastic incompatibility prior to \nstraining can be assessed by the elastic modulus difference of microstructural domains  getting \ndeformed. Similarly, the plastic incompatibility can be given by the Schmid factor difference \nof the grains involved in deformation.  Present study additionally shows that the steel having \ncarbides less than 20 vol. %, the elastic modulus and Schmi d factor of different grains around \na void can be used to predict potent damage nucleation sites and based on that a criterion for \nvoid initiation  can be defined . Previous  work depicted the concept of incompatibility \nmodulus to explain the elasto -plastic s train evolution in continua containing dislocation  [18]. \nA finite strain can be decomposed to a compatible and incompatible part and if it is a elasto -\nplastic strain, this can be decoupled to a plastic and elastic part. Experimental determination \nof this incompatibili ty modulus is yet to be attempted , therefore;  in the present study  this \nproblem has been taken up with the help of two readily derivable q uantities from EBSD data, \ni.e. e lastic modulus and Schmid factor difference. As described by Baczmanski’ et al [19], the \nvariation of elastic grain stresses is caused by the misfit of the e lastic strains in grains in \nrelation to the surrounding polycrystalline aggregate. Now, coming to plastic deformation, in \norder to predict the same, the contribution of slip on crystallographic planes must be \nconsidered. The essential criterion used in the  model is the Schmid criterion [Eqn . 1] \naccording to which the set of active slips is selected.  \nslip\nhkl uvwslip\nhkl uvw  ] [ ] [ \n      (1) \nA 6 mm thick  steel strip has been used  in this present study  with a nominal composition of \nFe- 0.10 C -0.33 Si-1.42 Mn -0.01 P -0.003 S -0.035 Al -0.05 Nb -0.05 V -0.007 N ( wt. %). The \nalloy is  soaked at 1100 °C for 30 minutes followed by different thermomechanical processing \nschedules as depicted in Fig.  1a. Heavy deformation in the metastable austenite region \n(~830  °C - between Ae 3 and Ar3) followed by air-cooling  produces ultrafine ferrite -carbide \nstructures. Heavy warm deformation (~ 600 °C) followed by rapid intercritical annealing \ngenerates bimodal ferrite -carbide structure w hereas the  air cooled sample  after 80 % cold \nrolling and a nnealing (600 °C for  8 h) is responsible for the coarse one. The mechanisms \nbehind the formation of the microstructures are given elsewhere [15,16,20 –22]. The \nmicrostructures of the  heat treated samples are prepared following standard metallographic \ntechniques.  The micro graphs  are characterized by scanning electron microscope (Zeiss -\nAuriga) with an attachment of electron backscattered diffraction (EBSD). For EBSD analysis, \nthe sample s are electropolished with a solution of methanol and perchloric acid (80:20 ratios) \nkept at -40 °C.  Grain size of each samples are measured by equal circle diameter  (ECD) \nmethod described in [23,24]  considering at least five secondary electron images of each \nsample . All these structures are having carbide <25 vol. % . Elastic  modulus and Schmid \nfactor  of different grains arou nd a void have been calculated from TSL-OIM software after \nEBSD scanning . The standard tensile samples have been machined  following ASTM E -8 \nstandard  [25] and tested uniaxially in Instron -8862 Universal Testing Machine (10 ton \ncapacity ). The band contrast images generated after EBSD scanning of the different ferrite -carbide structures are shown in Fig. 1(b -d). Inset in Fig. 1 (c) is showing the bimodal \ndistribution of ferrite grain sizes. Average grain sizes for ultrafine and coar se grain regions \nare 2.2±0.5 and 10±1 µm , respectively . The grain size s measured from the unimodal samples \nare in the same range with the ultrafine and coarse grain regions in the bimodal one. True \ncurves  of the samples considering the domain between yield  stress (YS) and ultimate tensile \nstress (UTS) have be en plotted in Fig. 2 (a). The strength dominance is quite clear for \nultrafine grain materials while the coarse grains are showing the curve of highest ductility , \nFig. 2(a). Also the same figure evidences  the rate and ability of strain hardening behaviour as \nplotted by dashed lines. The ultrafine grained sample showed the maximum values of strain \nhardening rate compared to other s, Fig. 2 (a).  \nIn order to assess the participation of each phase  in deformatio n qualitatively , the yield level \napproach of Polak  has been followed  [26]. This method is based on the probability density \nfunction of the internal critical stresses f(σic) which was deformed to the strain ε for a \npolycrystal loaded axially with a stress value σ.  This critical internal stress followed a \ndistribution in the deformation volumes loaded parallel and can be expressed with the help of \na Probability distribu tion function f(σic) mentioned in Appendix 1. \nFor further analysis on the nature of the deformation which is supposed to be dominant by \nferritic matrix, second derivative of the tensile curves ( d2σ/dɛ2) have been done. The ratio of \nthe peak width values ha ve been taken from the second derivative curve for all the three \ninvestigated samples , depicted in  Fig. 2 (b). In Fig . 2(b), sharp yield levels can be seen in the \nultrafine grained samples. So, there is a low effective stress as the centroid of this almost \nbinormal distribution is situated at lowest true strain value. This does also mean that due to \navailability of tiny micro sized stressed domains the internal stress distribution starts from the \nlowest value. Peaks are also symmetric  in nature  showing that equal stress levels co existing \nin the material. Now, coming to the bimodal, the peaks are not so distinct like ultrafine \ngrained steel  (Fig. 2 (b). The peaks are rather diffused but still one assymetric distribution can \nbe figured out. There are two much c oherent type of domains exist that can take strain. The \ndomain having greater peak width is situated at lower true strain value and that of lower peak \nwidth and higher intensity situated at higher true strain side. These cosharing of similar yield \nlevels i n which the init ial lie pretty within the ultra fine levels. It is evident with increasing \nstrain the second yield level emerged and thus taking most of the strain when the true strain \ncrosses 0.06. The course grain nature is also like ultrafine  but the str ain distribution is  much \ndiffused here.  The peaks are not sharp  but two yield level  distributions are clearly  evident in \ncase of bimodal microstructure.  \nA comparative analysis of this peak width has been done using the f ull width half maxima  \n(FWHM)  of this  probabil ity distribution function. The r esults are depicted in Fig . 2(b). It \nshows that in bimodal microstructure there exist almost equal strain sharing  yield  levels \ndepicting more or less iso str ained type dual phase behaviour  and in coarse grained ther e is a \nhuge difference between the two levels, ultrafine lies in between.  \nThe Schmid factor and the elastic modulus map of the multiple ferrite grains surrounding a \nvoid has been shown in Fig. 3. Inset of Fig. 3 (a) is showing the enlarged band contrast map  of \nthe void portions surrounded by ferrite grains. The colour legends  in both the map  are \nindicating the maximum and minimum values of the Schmid factor (0.3 – 0.6) and elastic modulus (90 – 235). For this particular void, the  variations  in the elastic mo dulus values of \nthe grains surrounding the voids are  more compared to the Schmid factor values, as \nconfirmed by the colour codes in the figures, Fig. 3. Similar observations have been carried \nout with statistically significant  void nucleation sites  (10-15 for each sample ). All the \nobservations depicted that voids either nucleated on the grain boundaries of two adjacent \ngrains or from the grain boundary triple point s. The Schmid factor differences and elastic \nmodulus differences of the void bounded grains ha ve been plotted in Fig. 4.  \nFrom the study of local damage initiation phenomena, it is evident that elastic modulus \nmismatch gives rise to l ocal stress concentration. Quite recently Shakerifard et al [27] \nquantified the damage initiation in advanced high strength steel and obtained a conclusion \nthat orientation dependence in void initiat ion may be blurred by presence of different phases \nin a multiphase microstructure. But here, only a single phase with some prominent granular \narchitecture has been chosen to show the mutual dependence of elastic and plastic \nincompatibilities. This work has  been aimed at formulating a local crystallographic criterion  \nfor steels having  wide range and distribution of ferritic grains. It is very much understandable \nthat the grain boundaries between incompatible grains give rise to void nucleation. A \npotential c rystallographic description for incompatible grains has been attempted to be \nformulated and th ree microstructures (i.e. ultra fine, bimodal and coarse g rained) are ranked \nbased on this incompatibility criterion, formulated as the combined contribution of el astic \nincompatibility and plastic incompatibility. The elastic and plastic incompatibilities  have \nbeen characterized by local elastic m odulus difference and local difference in Schmid factors , \nrespectively . Now, from Fig. 4, it is evident that the sensitiv ity towards void nucleation in \nterms of elasto -plastic incompatibility is highest in bimodal microstructure. Slight difference \nin elastic modulus and Schmid factor gives rise to void initiation  in this . From the perspective \nof void initiation, bimodal stru cture is highly unstable. In ultrafine grain microstructure, it \nrequires  grains with higher  and moderate difference in  elastic modulus to initiate voids  \ncompared to the bimodal one . This microstructure shows a lower degree of instability. Coarse \ngrained mi crostructure shows lowest degree of instability as it require grains with higher \nelastic modulus and Schmid factor mismatch to create voids. A probable reasoning can be \ngiven for this behaviour based on the processing schedules for the formation of these \nstructures, Fig.  1. Earlier studies showed that t he bimodal microstructure is having coarse \ngrains with a dominance of alpha fibre  texture (<110 ||RD>)  whereas the fine grains \nembedded in it shows  texture randomization  [16]. The ultrafine grain sample  possessed \nrandom texture  [20] while cold rolling and annealing texture is expected in coarse grain \nstructure.  Considering all the investigated samples, texture randomization occurred because \nof the retransformation of austenite grains to ul trafine ferrite. The cold / warm rolled ferrite \ngrains got recrystallized followed by growth during the processing schedule (Fig. 1a), leading \nto the formation of coarse grains. From Fig 4, it is evident that the bimodal grain structures \nare most sensitive  for void formation followed by ultrafine and coarse grain materials. The \nfollowing factors can be considered in order to address the phenomena ; (i) grain  boundary \nmisorientation  adjacent to void  (ii) local strain and texture within the grains and (iii) si ze \ndifference of the grains surrounding void .In the present investigation, all the samples are \nrecrystallized with more than 80 percent high angle grain boundaries. So, the randomization \nof texture along with grain size should have played a major role for  the initiation of voids. The probability of texture randomization of grains surrounding the void is highest in case of \nultrafine structure. In case of bimodal structure, there is also a decent probability  of random \ntexture as the coarse grains are heterog eneously distributed along with the ferrite grains in the \nbimodal matrix. The probability is lowest in case of coarse grain samples as it possesses \npredominantly alpha and ga mma fibre texture. The more the texture randomization within the \nneighbourhood grains, the less will be the compatibility among those grains. In this proposed \nconcept , the ultrafine structure should have show ed the high sensitivity of the void formation, \nbut the size difference of the grains surrounding the voids played a major role here. \nCombining both,  i.e. grain size and neighbouring grain orientation , the bimodal grain \nstructure showed the highest sensitivity towards void nucleation (Fig. 4 ). Under unia xial \ntension, randomly textured grain aggregates with different  sizes will have different \ndislocation distribution and pileup stresses along with different alignments from the loading \naxis. Th e combined effect of two mechanisms probably boosts the bimodal structure to \nnucleate voids very quickly compared to others. The ultrafine grains  are having lowest \nelongation to failure and the highest UTS. Therefore, the available strain accumulation time \nto initiate void is lower here and random ly oriented  grains enh ances the modulus difference \nor Schmid factor difference  requirement . Coarse grain shows maximum stability  in this \naspect  as it is having  very little amount of random texture and largest grain sizes among all. \nThis will accommodate more dislocations within  it during straining, delaying the void \nnucleation . Similar type of textures in the neighbouring grains enhances the compatibility of \nthe coarse grain structure . \nIn the present case, the grains qualifying for void initiation should have an elastic modulus \ndifference 60 -80 MPa or if it is lower than 60 MPa it should have a Schmid factor difference \nof more than 0.1. Therefore , either higher elastic or higher plastic incompatibility is favoured \nfor void nucleation.  The aforementioned grain incompatibility has a lower bound also below \nwhich no void initiation has been identified . This combinatorial approach of treating elastic \nand plastic modulus represents a complex dependence of local crystallographic parameters \npresent in the grains. The current  representatio n also clarifies the fact that crystallographic \nconstraints constitute some role behind damage initiation; it is not purely random as classical  \ncontinuum mechanics describes.  \n \n \nAcknowledgment : The authors would like to acknowledge Research & Development, T ATA \nSteel, Jamshedpur for supplying the materials and support from Central Research Facility -IIT \nKhargapur to carry out experiments.  \n \n \n \n \n \n \n \n \n Bibliography:  \n[1] R. Song, D. Ponge, D. Raabe, J.G. Speer, D.K. Mat lock, Materials Science and \nEngineering: A 441 (2006) 1 –17. \n[2] T. Gladman, The Physical Metallurgy of Microalloyed Steels, illustrate, Institute of \nMaterials, 1997.  \n[3] R.E. Reed -Hill, Physical Metallurgy Principles, Van Nostrand, 1973.  \n[4] M.S. Rashid, A nnual Review of Materials Science 11 (1981) 245 –266. \n[5] M.Y. Demeri, Advanced High -Strength Steels: Science, Technology, and \nApplications, ASM International, 2013.  \n[6] R. Kuziak, R. Kawalla, S. Waengler, Archives of Civil and Mechanical Engineering 8 \n(2008) 103 –117. \n[7] R.W. Cahn, P. Haasen, Physical Metallurgy, Elsevier Science, 1996.  \n[8] A. Karmakar, S. Sivaprasad, S. Kundu, D. Chakrabarti, Metallurgical and Materials \nTransactions A: Physical Metallurgy and Materials Science 45 (2014).  \n[9] Rathikanta Pra dhan, Effect of Thermo -Mechanical Processing Parameters on the \nFormation of Nano -Precipitate in Advanced High -Strength Steel, Indian Institute of \ntechnology, Kharagpur, 2016.  \n[10] J.H. Chen, G.Z. Wang, Q. Wang, Y.G. Liu, International Journal of Fracture 1 26 \n(2004) 223 –241. \n[11] S. Rezaee, C. Berdin, Stress Distribution in Carbides of Spheroidised Steel Using a \nPolycrystalline Modeling, 2009.  \n[12] J. Pacyna, L. Witek, Steel Research 59 (1988) 68 –74. \n[13] K. Seto, Y. Funakawa, S. Kaneko, (n.d.).  \n[14] A.R. Ro senfield, G.T. Hahn, J.D. Embury, Metallurgical and Materials Transactions B \n3 (1972) 2797 –2804.  \n[15] D. Chakrabarti, Development of Bimodal Grain Structures and Their Effect on \nToughness in HSLA Steel, University of Birmingham, 2007.  \n[16] A. Karmakar, A. Karani, S. Patra, D. Chakrabarti, Metallurgical and Materials \nTransactions A 44 (2013) 2041 –2052.  \n[17] S. Patra, S.M. Hasan, N. Narasaiah, D. Chakrabarti, Materials Science and \nEngineering: A 538 (2012) 145 –155. \n[18] S. Amstutz, N. Van Goethem, Proceedings  of the Royal Society A: Mathematical, \nPhysical and Engineering Science 473 (2017) 20160734.  \n[19] A. Baczmański, N. Hfaiedh, M. François, K. Wierzbanowski, Materials Science and \nEngineering: A 501 (2009) 153 –165. \n[20] S. Patra, S. Roy, V. Kumar, A. Haldar, D. Chakrabarti, Metallurgical and Materials \nTransactions A 42 (2011) 2575 –2590.  \n[21] A. Karm akar, Study on the Ferrite Grain Structure and Precipitation in \nThermomechanically Tailored HSLA Steel, Indian Institute of Technology, \nKharagpur, 2017.  \n[22] S. Roy, S. Patra, S. Neogy, A. Laik, S.K. Choudhary, D. Chakrabarti, Metallurgical \nand Materials T ransactions A 43 (2012) 1845 –1860.  \n[23] D. Chakrabarti, C. Davis, M. Strangwood, Materials Characterization 58 (2007) 423 –\n438. \n[24] G.F. Vander Voort, J.J. Friel, Materials Characterization 29 (1992) 293 –312. \n[25] A. Standard, Standard Test Methods for Ten sion Testing of Metallic Materials (2001).  \n[26] J. Polak, Materials Science Monographs 63 (1991) 15 –308. \n[27] B. Shakerifard, J. Galan Lopez, F. Hisker, L.A.I. Kestens, IOP Conference Series: \nMaterials Science and Engineering 375 (2018) 12022.  \n  \nFig: 1: (a) Schematic diagrams of the heat treatment schedules for the development of \ndifferent ferrite grain sizes.  Band contrast images of ferrite -carbide structures  with (b) \nultrafine, (c ) bimodal and (d) coarse ferrite grain sizes. The coarse grain regions are \nhighlighted separately  in the bimodal structure  in (c) . Number and area frequency evidenced \nthe b imodal grain size distributions in the inset of ( c). [AC-air cooling, FC -furnace cooling, ɛ-\namount of strain, έ - strain rate].  \n \n \n \n \n \n \n \n(c) \n(b) \n(a) \n(d)  \nFig. 2: (a) True stress (solid lines) and work hardening rate (dashed lines) of different grain \nstructures have been plotted against true strain, (b) ratio of peak widths ( 1st peak : 2nd peak \ntaken from d2σ/dɛ2 vs. ɛ plot) for different ferrite structures . The plot of second derivative of \nthe samples are shown n the inset of 2b.  \n \n \nFig. 3: (a) Schmid factor and (b) elastic modulus map of the respective grains  around a void \ndepicted by the black spot in the figures. Inset in (a) showing the enlarged view of the void \nzone. Attached colour legends are showing the minimum and maximum value of schmid \nfactor and elastic modulus.  \n \n \n(a) \n (b) \n(a) \n(b)  \nFig. 4: Plot between elastic modu lus difference and Schmid factor difference showing the \nvoid initiation regions for different grain structures along with the no initiation zone.  \n \n \nTable 1. Chemical composition (wt. %) of the investigated sample.  \nC Mn Si P S Nb Ti V Al N \n0.1 1.42 0.33 0.01 0.003  0.05 0.01 0.05 0.035  0.007  \n \n \n \n \n \n \n \n \n \n \n \n Appendix 1: \nAn elegant way to understand this continuous distribution of yield is through use of statistical \ntechniques such as probability density function (PDF). A number of previous investigators \nhave used PDF method to explain the local alterations in  the yield levels. The method \nbasically describes the distribution of yield levels ( σic) of the elements (assuming the material \nis composed of several parallel elements, each having a different yield level σic. Further \ndetails of continuous yield probabili ty can be described below  by a probability density \nfunction (PDF), f(σic) which upon normalization would yield  \n1 )(\n0\nic icd f\n \nThus \n\n0)(ic icd f represents the area fraction of elements with yield stresses in the interval \nbetween \nic and \nic ic d  \nTotal stress experienced by an axially loaded polycrystalline  aggregate is  \nic eff itotal \n \nAccording to the statistical approach, the macroscopic stress σ can be expressed as  \n \n \nEic ic ic icE\nic d f E d f\n\n )( )(\n0\n \nWhere f(σ ic) is the probability distribution of the internal stresses in the polycrystalline \naggregate showing the Elastic stiffness E.  \nNow, solving that Integral equation the probability distribution function can be derived, it \ntakes a form like  \n22\n21) (\n\n\nEEfeff\n \n So, the second derivative of any stress strain response should depict the \nProbability distribution of possible constituents level those are deformed due to straining.   "    },    {        "title": "0906.2928v1.Microwave_Technologies___Determination_of_Magnetic_and_Dielectric_Materials_Microwave_Properties.pdf",        "content": " \n 1\n \n \n        \n Disclaimer: The techniques in this summa ry; each of them works very well for the \nsuitable subjects and they are well practiced by  real time measurements of microwave \nmaterials such as magnetic and dielectrics . However, there will be no responsibility\napplicable for inappropriat e applications and relevant losses. June 16, 2009.  \n 2\nAbstract \n \nIn this research summary, four different micr owave measurement techniques are presented \nfor materials characterization. They are the fo llowing. 1) Rectangular waveguide measurement \ntechnique for normal microwave materials micro wave properties such as permeability and \npermittivity. This technique re moved guess parameter and dispers ive effect issues of the old \nwaveguide measurement techniques forty years hi story. As such it projects a new route for \ndetermination of any microwave materials magnet ic and dielectric properties without using any \nguesses. 2) Coaxial probe measurement technique for the liquid and biological tissues dielectric \npermittivity. This coaxial probe technique has an ad vantage which is to atta in the highest reflected \nsignal from the coaxial probe tip, so  that it is a fast and very sensit ive technique to differentiate lossy \nmaterials dielectric permittivity. This technique coul d be useful non destructive detections for tumors \nin hospital and non destructive detections for chemical liquids as well. 3) A microstripline \nmeasurement technique for oxides microwave meas urement at low frequency spectra where the \nwaveguide technique becomes robot and cumbersome.   4) A new methodology is presented for the \nrectangular waveguide technique to determine microwave metamaterials refractive index, \npermeability and permittivity using the rectan gular waveguide. In summary, the presented \ntechniques are capable enough to de termine magnetic and non magnetic solid state materials, liquids, \npowders and biological tissues microwave propertie s from the broad microw ave frequencies spectra \nbetween 1 GHz to 50 GHz. Appropriate advices and us eful micro codes of measurement processes will \nbe provided to interested agent, base d on their appropriate requests.   \nLastly, a special ferrite’s microwave properties are also presented in this research summary \nsince the negative refractive index from the insulato r materials such as ferrite is an interesting \nsubject.  \n \n \n \n \n \n \n \n \n \n These researches was done when Dr. Mahmut  Obol is with Electrical and Computer \nEngineering Department at Tufts University, MA, USA  \n 3\n \nCONTENTS \n \nI. Microwave Materials Permeability and Permittivity \nMeasurements by T/R Technique in Rectangular Waveguide-  4 \nII. Coaxial Probe Technique for Microwave Characterizations of  \nLiquid and Biological Tissues--------------------------------          9 \nIII. Permeability and Permittivity Measurement Technique by \nPropagation and Impedance of Microstriplines   -------------     18 \nIV. Full Band Microwave Isolator of Rectangular Waveguide \nHaving Periodic Metal Strips and Ferrites-----------------------  26  \nV. An Unusual Internal Anisotropy Field of Spiral Magnetized \nCrystal Compound of Sr 1.5Ba0.5Zn2Fe12O22   -------------------   31 \nVI. Biography and CV------------------------------------------------     36 \n \n          \n  \n 4\nMicrowave Materials Permeability and Permittivity Measurements by \nT/R Technique in Rectangular Waveguide \n \n \nAbstract  — There is a huge demand to accurately determine the \nmagneto-electrical properties of so lids, and particles in the nano \nsized regime due to the modern IC technology revolution and \nbiomedical application science.  In this paper, one presents a \nmicrowave waveguide measurement technique for complex permeability and permittivity of expensive nano sized magnetic \npowder materials.  In the meas urement process, Agilent’s 8510C \nvector network analyzer was used to have a standard TRL calibration for empty space inside the waveguides.  In order to \nmaintain the recommended insertion phase range, a very thin \nprepared sample was loaded inside the calibrated waveguide.  The loaded material’s magnetic and dielectric effects were also \nconsidered into the cutoff w avelength calculation of the \npropagation constant of the TE\n10 wave from the geometrical \ndimensions of the waveguides.  These considerations make the \nmeasured permeability and perm ittivity more reliable than \ncommonly used techniques.  However,  at this time this technique \nis capable enough to determine microwave properties for the \nthick samples up to 0.5cm.  \nIndex Terms —  ferrites, dielectrics, waveguide technique, \npermeability, permittivity.  \nI. INTRODUCTION  \n Nicolson and Ross [1] developed a broadband \nsimultaneous measurement technique of transmission and reflecti on (T/R) by using forward \nand backward energy scattering in coaxial \ntransmission line.  Weir\n [2] extended it into \nwaveguide transmission line by using a Hewlett-Packard vector network an alyzer.  J. Baker-Jarvis\n \n[3] has solved the phase ambiguity problem in [2] \nby using a reasonable guessing parameter to detect reasonable permittivity in relatively thick samples, in which the samples can be both lossless and lossy \nmaterials.  The other attempts were also studied in [4] to remove inaccurate reflection peaks in complex permittivity measurements.  Since then the transmission and reflection technique was also used in the free space measurement technique [5, 6].  The S-matrix analysis was also deployed for the coaxial\n \n[7] and rectangular [8, 9] waveguide techniques to \ndetermine the complex permeability and complex permittivity of specific materials.  These measurements of the complex permeability and permittivity are very reliable.  However, the minimum diameter requirement of Gaussian beams in free space measurement always requires a wavelength for operating at a central frequency. In order to avoid diffracti on errors from target \nmaterials, the surface diamet er of the target material \nwas at least three times larg er than the diameter of \nthe Gaussian beam. This implies that in order to determine a material’s electro-magnetic properties at lower frequencies, a large amount of material is needed in the measurement process.  Due to the relatively expensive and nom inal nature of nano \nmaterials, a cost effective measurement technique requires a minimal sample size. For example, coaxial transmission line technique [10] was deployed to measure the complex permeability and permittivity of nano materials.  The waveguide technique also fits the requirements for this and \ndelivers very accurate results.  It is obvious that each waveguide has a limited frequency band; however, they do not suffer radiation losses like free space measurements except for the attenuation losses of specific modes in the waveguide.  In order to remove any unwanted losses from the waveguide, we applied a standard TRL calibration\n \n[8] technique to achieve a zero reference plane for \nthe measurements inside the waveguide.  After a great number of trials using TRL calibration, we \nnoticed that loaded material’s permittivity and permeability effects into the cutoff wavelengths from the free space waveguide need to be accounted for.  Nevertheless, we have not seen this phenomenon explicitly expressed in any relevant works.  In addition, the loaded lossy material’s \nthickness should be as thinner as possible for lower frequency measurements in order to maintain the recommended insertion phase regime by Agilent.  The transparent ticker sample is hold to transparent signal, providing very accu rate S-parameters and no \nphase ambiguity.  It was also reported\n [3] that \nthinner sample loading t echnique was a source of \nerrors due to uncertainty in reference plane \npositions. However, we noted that using electrical \ndelay function of VNA, one  could eliminate those \nerrors. To account for the thin and thick loaded \nsample, one presents a modified reflection and \ntransmission formulation for in-waveguide  \n 5\nmeasurements.  Also presented is a modified \npermeability and permittivity formulation from the modified propagation constant of the loaded waveguide.  The studies show that these modifications are necessa ry and known data is \npresented to confirm the accuracy of the measurement technique.  The derived permeability and permittivity data is very reliable and not effected by the scattering voltage ratios of the vector network analyzer.   \nII. THEORY  AND  MEASUREMENT  \n  A propagating electroma gnetic wave inside the \nwaveguide is being reflected, S 11, and transmitted, \nS21, by the loaded material.  A diagram of this setup \ncan be seen in Fig.1.  The known electric and magnetic polarization of  propagating waves in \nwaveguides is very useful in analyzing the physical properties of certain materials.   It is also very useful \nin determining the dielectric and magnetic complex permeability and permittivity of these materials.  \n \nFig.1 Schematic diagram of powder sample in \nwaveguide \n \nIn this waveguide measurement technique, the standard TRL calibration applies to the zero reference plane findings.  The zero reference planes were realized by the typical quarter wavelength \ndifference (\n)l between thru and line in air. The \nrecommended insertion phase ranges from 200 to \n1600 are retained by prope r TRL calibration. In \norder for the insertion phase  contributions from air \nto be removed from the actual transmission line for the loaded material measurements, the reasonable thin and thick prepared targ et materials were loaded \ninside the waveguide and mounted onto the zero \nreference planeside.  The modified S parameters are as follows: 2 2\n02 2\n0\n) (\n21 21) 0(\n11 11\n~~\ncc\nkk dljkk j\neS SeS S\n− ×−− ×\n==               (1) \nReturn losses of less than -50 dB from the air inside \nthe waveguide are easily achieved using the calibration techniques described.  This enables us to neglect any unwanted reflections from the inner walls of the waveguide when analyzing the S parameters.  The reflecti on and transmission by the \nscattering parameters inside the waveguide, in which the transmission and reflection may be \nresembled by free space formulations, can now be presented as follows: \nΓ + −Γ− +=+ −=− ±=Γ\n)~ ~(1~ ~~21~ ~1\n21 1121 11112\n212\n112\nS SS STSS SKK K\n                 (2) \nThe transmission coefficient through the material \nmay also be written as follows: dj de eT) ( β α γ +− −= = .  \nThe propagation constant through the material \ninside the waveguides can be derived to be: \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ −+ =dnjdTT\nTEϕπγ2)1ln(\n10    (3) \nNormally, a sample thickness of less than one \nquarter wavelength is desirable in this calibration, because it will make n = 0.  In order to achieve our goal and derive the complex permeability and permittivity for the loaded material inside the waveguide, we must de termine the propagation \nconstant through the materials in the waveguide.  To achieve this one must solve Maxwell’s equations with respect to E\ny for the TE 10 mode as seen in \nFig.2.  \n \nFig.2 Propagating TE 10 wave inside waveguide and \nthe loaded material   \n 6\n0 ) (2\n2 2= +\n′∂∂+\n′∂∂\nyE\ny xβ      (4) \nWhere\nμε1xx=′  and\nμε1yy=′ . \n \nSolving Maxwell’s equation leads to the following: \n) cos() sin( y x C Ey x y′ ′ = β β     (5) \nC is a constant to be determined from boundary \nconditions.  The boundary condition tells us that the \npropagation constant com ponents may be presented \nas follows: \n002\nλπγ= , μεπβan\nx=  and μεπβbm\ny= . \nThis results in the following relationship for the \ntotal propagation constant  through the material \ninside the waveguide: 2 2 2\n02\ny xβ β γ γ − − = . The \npropagation constant of the TE 10 can subsequently \nbe written as follows: \n2 2\n0002 2\n0\n21 1221 12\n1010 10\n⎟\n⎠⎞⎜\n⎝⎛−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛== ⋅⎟\n⎠⎞⎜\n⎝⎛−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛=\najaj\nTETE TE\nλπ γημγ μελπ γ\n     (6) \n \nThe complex permeability and permittivity associated with the propagation constant are:  \n⎟⎟⎟⎟⎟⎟⎟\n⎠⎞\n⎜⎜⎜⎜⎜⎜⎜\n⎝⎛\n⎟\n⎠⎞⎜\n⎝⎛−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− +\n⎟\n⎠⎞⎜\n⎝⎛⎟\n⎠⎞⎜\n⎝⎛\nΓ−Γ+−= =2 2\n00\n21 1) 2()1ln(\n21\n11\n1010\nanj\nT\ndj\njT\nTETE\nλϕπ\nπ γηγ\nμ    (7)  \nIn our waveguide measurement technique the \npropagating wave inside the waveguide was assumed to be the TE\n10 mode. This implies that the \npropagating wave detects the permeability directly. However, the permittivity is detected indirectly and can be derived as follows: \n()⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n− =\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−= =− 2 2\n02\n02 1\n22\n02241 1\n41a\naZZ\nn\nTETE λλ\nημ\nλμ με                                                                                              \n                                                                  (8) \nWhere η=Γ−Γ+= =11\nair\nTEload\nTE n\nTEZZZ  2 2 2\n0111 1 1ln( ) (2 )12 2Tcjj nfd T aεπ ϕπλ⎛⎞⎛⎞ ⎛⎞ ⎛⎞ −Γ⎛⎞ ⎛⎞ ⎛ ⎞ ⎜⎟=− + − − ⎜⎟ ⎜⎟ ⎜⎟⎜⎟ ⎜⎟ ⎜ ⎟ ⎜⎟ ⎜⎟ +Γ⎝⎠ ⎝⎠ ⎝ ⎠⎝⎠ ⎝⎠ ⎝⎠⎝⎠                 \n                                                          \n                                                                 (9) \nThe above permeability and permittivity equations can now be re-written by following configurations \ntoo. \neq eqnjZ−=μ                                              (10) \neqeq\nZn\nj−=ε                                                 (11) \nWhere equivalent impeda nce inside rectangular \nwaveguide, and equivalent refractive index inside \nrectangular waveguide  are the follows. \n21\n2\n0\n21−\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− =aZeqλη                              (12) \n21\n2\n0\n0 0 21~ 2\n1010\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟\n⎠⎞⎜\n⎝⎛− =⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛= =an nTETE\neqλγλπ\nγγ   (13) \nΓ−Γ+=11η  and jn n +=κ~ \n \nIt is also possible to define the medium impedance inside waveguide by follows where one deployed simple algebra between the transmission line matrixes A, Z, and S.  \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−=\n21\n212\n11) 1() sinh(2\n10\nSSSdTEγ\nη                             (14) \nThe permeability and permittivity inside waveguide \ncan also be figure out by following formats too. \nr eq eq nj njZ μ η μ = −= −=~                     (15) \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− −= −=2\n0\n21~\nanjZn\nj\neqeq λ\nηε                (16) \nThe equation (16) firmly shows that the effective \npermittivity inside waveguide is ) 1(2\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− =ffc\nrεε . \nThe \nrcacf\nε2=  is the cutoff frequency of medium \nloaded waveguide. So that relative permittivity of  \n 7\nthe medium inside rectangul ar waveguide is to the \nfollows: 2\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+=ffair\nc\nrε ε                       (17) \nHowever, loaded medium’s relative and effective \npermeability inside waveguide are equal and same, and they don’t have disper sive effect issues from \nthe rectangular waveguide (see equation (15). The \nequations (15) and (17) are used to calculate the complex relative permeability and relative permittivity of samples inside the waveguide. Also, \nthe \nacfair\nc2= is the cutoff frequency of empty \nwaveguide.  In order to validate this technique, the \nsimultaneous complex permeability and permittivity measurement was presented by using this technique for the known material YIG and Phenyloxide.  \nyittirium iron garnet\n0246810121416\n8 9 10 11 12\nFrequency (GHz)Relative valuesreal permeability\nimaginary permittivity\nreal permittivity\nimaginary permittivity\n \nFig.3 Complex permeability and permittivity of \nYIG, sample thickness 0.5 mm \nPhenyloxide\n00.511.522.5\n8 9 10 11 12\nFrequency (GHz)Relative valuesreal permittivity\nimaginary permittivity\n \nFig.4 Complex permittivity of Phenyloxide, sample \nthickness 0.5 cm The available samples of YIG and Phenyloxide \nwere tested at sample X-band, and the sample thickness is 0.5 mm of YIG and 0.5 cm of phenyloxide and the shim thickness of TRL is 9.54mm. The YIG samples permittivity was \nrecorded as \n414.4 10r j ε−≅− by pacific microwave \nceramics brochure.  \nAerogel from Silica with Alumina\n Density, 0.0875g/cm^3\n00.20.40.60.811.21.4\n18 20 22 24 26\nFrequency in GHzRelative valuesreal permeability\nimag permeability\nreal permittivity\nimag permittivity\n \nFig.5 nano porous aerogel materials permittivity \nmeasurements, sample thickness 0.452 cm \nAerogel from Silica with Alumina and with Iron Oxide \nDensity 0.0158g/cm^3\n00.20.40.60.811.21.4\n18 20 22 24 26\nFrequency in GHzRelative valuesreal permeability\nimag permeability\nreal permittivity\nimag permittivity\n \nFig. 6 iron substituted nano porous aerogel \nmaterials permittivity measurements, 0.452 cm \nLooking at Figs.3, 4, 5, and 6 one is able to see that \nthis method will not have the divergence problem like Weir; also one does not need initial guess for the permittivity determination like Baker-Jarvis. The need to obtain an appropriate reflection \ncoefficient is crucial in determining accurate permittivity measurements. The excited ferrites permittivity and permeability determination, this method could be useful, but it is needed to be clarified some classical concepts. This part of  \n 8\ndetermination will be presented in resonate \nmediums permittivity and permeability determination section. Now one also presents the measured complex permeability and permittivity of nano ferrite materials of magnetite using this waveguide technique.  \n \nFig.7 Complex permeability and permittivity measurements of nano spinel ferrite of magnetite\n \n \nOne can recall previous studies using waveguide \nmeasurements where cutoff wavelengths were not accounted for in the permeability and permittivity measurements. The techni que proposed in this \npaper demonstrates this techniques ability to \nsuccessfully account for the effect of the cutoff frequencies and obtain the potential permeability and permittivity as shown above. \n  \n  \n  \n \nFig.8 the cutoff wavelength s over contributions to \nthe propagation and attenuation  III. Conclusion \n Compared to previously published data, the data \nobtained using the new propos ed technique shall be \nsuperior by any means. This is because this technique successfully avoi ded the phase ambiguity \nand reference plane uncertainties in the measurement process. Thr ough the analysis one can \nhave noticed that the recommended phase can be maintained for extreme high permeability and permittivity materials by loading thinner samples. Other than that this technique even cable to measure sample thicknesses up to half centimeter. It implies that this technique is now good for thin and thick samples. It also implies that the measurement could be very successful for films such as carbon nanotubes, due to the fact that the partial insertion \nphase region is able to prove the reflection and transmission theories. Lastly, this technique claims that it has no guess and disp ersive issues for any \nmicrowave measurements by waveguide. \nACKNOWLEDGEMENT  \nThe authors wish to ackno wledge the support of US \nArmy contract, National Ground Intelligence Center. \nREFERENCES  \n[1]  A. M. Nicolson and G. F. Ross, IEEE Trans. \non Instrumentation and Measurement, Vol. IM-19, No.4, pp. 377-382, Nov. 1970. \n[2]   W. B. Weir, Proceedings of th e IEEE, vol. 62, No.1, pp. 33-36, January \n1974. \n[3]   J. Baker-Jarvis, E. J. Venzura and W. A. Kissick, IEEE Trans. \nMicrowave Theory Tech., vol. 38, No.8, pp. 1096-1103, August 1990.  \n[4]    Abdel-Hakim Boughriet , Christian Legrand, and Alain     \n       Chapoton, IEEE Trans. Microwave Theory Tech., Vol.45,  \n       No.1, pp. 52-57,January 1997.   \n[5] D. K. GHODGAONKAR, V. R. VARADAN, and V. K.  VARADAN, \nIEEE transactions on instrumentation and    measurement, Vol. 39, NO. 2, April 1990. \n[6]  Rene Grignon, Mohammed N. Afsar, Yong Wang and Saquib  Butt, \nIMTC 2003-Instrumentation and Measurement Technology Conference, Vail CO, USA, 20-22  May 2003. \n[7]  Madhan Sundaram, Yoon Kang, S. M. Shajedul Hasan, and Mostofa K. \nHowlader, SNS-CONF-ENGR-133 \n[8]  Y. Wang and M. N. Afsar, Progress In Electromagnetics Research, PIER \n42, 131-142, and 2003. \n[9]  Achmad MUNIR, Noriaki HAMANAGA, Hiroshi KUBO, and Ikuo \nAWAI, IEICE Trans. Electron., Vol. E88-C, NO.1     JANUARY 2005. \n[10]  Tang X, Zhao BY, Tian Q, et al., Journal of Physics and          Chemistry of Solids 67 (12): 2442-2447DEC 2006. \n[11] Patrick Queffelec, Marcel Le  Floc’h, and Philippe Gelin,    \n        IEEE Trans. Microwave Theory Tech., Vol.47, No.4,     \n        April 1999.  \n 9\n \nAbstract — A key advantage of using a coaxial probe for \nmicrowave characterization of  biological media is the non-\ninvasive nature of the technique.  Coaxial probes are being used \nextensively for the complex permittivity measurements of \nmaterials in the microwave region.  Usually, these types of \nmeasurements require electromagnetic full-wave analysis or a \ncalibrated reflection coefficient, S11, of the material being \ntested. In this paper, a new coaxial probe technique is \npresented, which features the microwave characterization of \nbiological tissues based on the calibrated reflection coefficient \nS11 of a known dielectric material. Using this technique, which \nsimply requires distilled water as a reference material, the \ncomplex permittivity of normal tissues from animals and both \nnormal and cancerous tissues from human bodies were \nmeasured over a broadband micro wave region. The biological \ntissue measurements show that the complex permittivity \nmeasured by this method is in concurrence with other \ninvestigations. It is a cost effective, non- invasive and practical \ntechnique, which may make it useful for diagnostic and \ntherapeutic biomedical applications in which microwave \npermittivity is considered.  \n \nIndex Terms —complex permittivity, wa ter, biological tissue, \nand coaxial probe \nI.  INTRODUCTION \niological substances have been studied \nextensively using the co axial probe technique, \nand the use of known diel ectric materials as a \nreference in the calibr ation of coaxial probe \nmeasurements is common. Various problems related to coaxial probe measurements have been \nstudied and related improvements were proposed specific to each case \n[1, 2, 3, 4, 5, 6, and 7]. The specific \ncontributing papers are as follows. First, an error \ncorrection method was proposed for correcting \nerrors from liquid reference cases [1]. Second, the \ntemperature sensitivity of coaxial probes was \nstudied for in-vivo  applications [2]. Third, a \nthrough reflection and evanescent mode analysis \nwere made for cases with air gaps between coaxial probe and test materials \n[3]. Fourth, an admittance \nmodel for the coaxial probe was carried out by \nnumerical calculations to overcome the unknown coaxial probe radiations \n[4]. Fifth, an improved \ncalibration technique for co axial probes, as well as \nits applications for finite and infinite half space slabs, was studied \n[5]. Sixth, ever since this \ntechnique has been app lied, the methods that \ninvolve the TEM mode propagating through a coaxial probe, as well as electromagnetic full-\nwave analysis and numerous rigorous \nmathematical modeling elsewhere, have been used to detect the permittivity of test materials by coaxial probe methods \n[6]. Lastly, besides the \npreviously mentioned procedures, some other \ncalibration techniques for co axial probes, such as \nbilinear calibration, were also reported elsewhere \n[7]. As such, the VNA based, non-destructive \ncoaxial probe applications were extensively employed for detecting permittivity worldwide. These methods consist of measured calibrated reflection coefficients a nd specific software for \nnumerical calculations in  order to detect the \ndielectric properties of specific materials. It is obvious that some of the above techniques have application ranges that are narrow and time consuming due to large scale mathematical modeling.  In most cases, the coaxial probe technique would be signifi cantly more reliable if a \nhigher level of reflection was retained from the calibrated plane of the probe tip where the tested material is placed. This is because the lesser the \nreflection from the probe tip, the greater the error source of phase from the less reflected signal of the probe tip.  Usually, the coaxial probe measurement technique is considered to be less accurate and less repeatable in real measurements that involve radiation from  the coaxial probe tip to \nair. Bearing these factors in mind, a simple circuit model for the coaxial probe is presented, in which the largest signal content shall be reflected back \nfrom the load to the coaxial probe tip.  Based on this simple circuit model, formulas were developed that are capabl e of producing repeatable \ncomplex microwave permittivity data for biological tissue samples. This is a fast detecting technique for real time measurements and it differentiates between the dielectric properties of \nvarious biological and liquid materials while showing reasonable data.  Previously, the higher order mode excitations from the coaxial probe geometry such as TM\n01, TM 02, and TE 01 were \nconsidered in coaxial probe measurements. For example, Baker-Jarvis measurements were made considering the highest order modes possible \n[3]. Coaxial Probe Technique for Microwave Characterizations of \nLiquid and Biological Tissues  \nB  \n 10\nHowever, in this study, a coaxial probe with an \nouter diameter as small as 1.45 mm was implemented. The deployed operating frequency spectrum, from 2 to 18 GHz, implies that the potential higher order mode s in this coaxial probe \nmay be less critical in the computation of the dielectrics using such a smaller diameter probe. This factor greatly simplifies the approach presented here. The attempt to use the larger diameter coaxial probes for detecting the permittivity of test materials by using this technique may reduce the reliability of this technique, because it may demand one to consider the potential evanescent modes excitements in the coaxial probe measurement process. For some other cases, such as if one is interested in measuring small samples by using a coaxial probe, it will require the c onsideration of wave \ndistribution on the surface of  the sample, as well \nas penetration distances through the sample\n [8]. In \nthis paper, a technique is presented for measuring complex permittivity of the materials such as biological tissues. It is not necessary to know the wave's actual penetration distances for the material that is examined. \nII. CIRCUIT AND THEORY \nIn using a coaxial probe technique, the examined material needs to be placed on the probe tip during the measurement process. The greatest challenge of the coaxial probe measurement may be retaining a sufficient reflected signal from the coaxial probe tip. Since the measured dielectric properties of the examined materials are based on their interaction with electromagnetic waves, an increase in electromagne tic wave radiating from \nthe coaxial probe makes it  more difficult to obtain \na reliable phase from the reflected signal. This will affect the process of obt aining accurate dielectric \nproperties of the target material. The VNA (see Fig.1) used in this study is incorporated with the usual coaxial probe for th e dielectric measurement \nof biological tissues. A 50 ohm match load for the conventional SOL (short, open, load) was \narranged for the calibration procedure, in which \nthe reflections from the coaxial cable onto the reference plane of the pr obe tip could be fully \nremoved. \n \nFig.1  A sketch of the Agilent ’s 8510C VNA and a 2.4mm coaxial \nprobe can be seen on the right.  \nNow suppose an ideal load mεis placed between \nthe 50 ohm transmission lines and air (see Fig.2). \nThe circuit model below was implemented to measure the dielectric cons tants of distilled water \nand biological substances in the real experiment and measurements. \n \nFig.2 Circuit of an open-ended co axial probe with an ideal load \nmaterial, 0 56 jm − ≅ε .  \nIn the real measurement por tion of this paper, it \nshould be understood that the materials to be \nmeasured were placed on the ideal load (see Fig.2). The formulations for the complex \npermittivity derivations based on the circuit model \nas seen in Fig. 2 will now be presented. The voltage and current s hould be continuous at \n0=xand lx=according to the matched boundary \nconditions of the transm ission lines. Also, the \nwave propagation constant through the load may \nbe defined as β αγ j+= . Normally, it is relatively \neasy to record the S11 parameter at the terminal, x \n= 0, but it is not as simple to determine the \nreflection coefficient )(xΓ through the actual load \nbetween x = 0  and x = l ranges.  Thus, various \nelectromagnetic modeling and calibration methods  \n 11\nare required to determine the realistic microwave \ncomplex permittivity of the testing materials. Let the definition for reflection coefficients be \nexpressed as \n)(lm mΓ=Γ , which represents the \nreflection coefficient at the receiving end of the \nnetwork at lx=. According to Fig. 2, )(xmΓ may \nbe defined as follows: \nl x\nmxl\nm m eelx elx xγ γ γ 2 2 )(2) ( ) ( )(− − −= Γ= = Γ= Γ      (1)  \nFor the ideal load that was imagined in Fig. 2, the \nreflection coefficient at 0=x may also be defined \nas l\nm m elx xγ2) ( )0 (−= Γ= = Γ . In general, according to \nelectromagnetic properties, the coaxial probe \nproblems such as those in Fig.2 are used to make \nthe TEM mode analysis for propagation along the transmission line. Therefore, the transmission lines can be considered as a one port microwave network, in which the microwave power can enter or leave through only one transmission line.  We assume that this one port network can be fully understood with a scattering parameter, S\n11, by \nusing the coaxial probe (s ee Fig.2). An ordinary \ntext book presents a useful formula for these types \nof circuits, which is the following [9]: \n  l\nml\nm\neeSγγ\n2 22\n111)1 (\n−−\nΓ−− Γ=                                           (2) \nThe equation (1) may be derived from the \nfollowing formats too: \nl\nmm\nl\nml\nm\nmell\nelell xSγ γγ\n2 22\n2 22 2\n11)( 1)( 11\n)( 1)1 )(()()0 (− −−\nΓ−Γ−+−=\nΓ−− Γ= Γ=  (3)                  \nl\nmm\nmelll xSγ2 22\n11)( 1)( 11)()0 (−Γ−Γ−=+ Γ=                      (4)      \nAccording to equation (3), one would need to \nobtain a reference plane with an ideal reflection, \n1 )( ±≈ Γlm . This may be possible for the coaxial \nprobe experiments. As for an ideal load case \n( 0 56 jm − ≅ε  in Fig.2), it plays the role of attaining \na matched network. There are to be 50 ohms from \nx=0 to x=l, and the matched network would create a perfect transition, with no reflection between x=0 and x=l in the circuit. By using the reflection definition of equation (1) and the circuit structure in fig.2, one obtains the following: \niv\nm mmlxZ xZZ\nΓ−Γ+\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= == ~1~1\n) ()0 (2\n0ε                  (5) \nwhere  l l\nil l\nv\nel el lel el l\nγ γγ γ\n2 2 22 2 2\n)( )()(~)( )()(~\n− −− −\nΓ− Γ+Γ=ΓΓ+ Γ+Γ=Γ                    (6) \nTo be an ideal load, the load (mε) has to be in \nquarter wave lengths, and \nOhm ZlxZ xZ\nmm m m 50120) ( )0 ( = = == ==\nεπ. \nIt is clear that the maximum reflection is now \npossible for the ideal load, and it can be expressed \nas 1 )( ±≈ Γlm . This is because π1200=Z  ohms and \n50)(=lZm  ohms, so one obtains the \nfollows: 1 )]( )][( [)(1\n0 0 ≈ + − = Γ−lZ ZlZ Z lm m m . As \nsuch, equation (4) simplifies to the following: \n    01)()0 (11 ≈+ Γ= l xSm                              (7)                             \nBy using the impedances relationships of the \nnetwork (see Fig. 2) one may also write the following configurations as well: \n)0 ( 1)0 ( 1\n)( 1)( 1)(\n1111\n50 = += −−=Γ−Γ+=xSxS\nll\nZlZ\nmm m      (8) \n) tanh(1 )(\n50 l ZlZm\nγ=                                 (9) \nBy combining (8) and (9), one can obtain the \nfollowing relationships.  \n) tanh(1\n)0 ( 1)0 ( 1 1\n11112\n500\nl xSxS\nZZ\nm γ ε⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= += −−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛   (10) \n) tanh()0 ( 1)0 ( 1\n11112\n500lxSxS\nZZ\nm γ ε⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= −= +\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−=      (11) \nBy now, it should also be convincing that the \nreflection coefficient is caused by the two way travel of propagating wa ves, so the propagation \nconstants may be defined as follows: \n11 10log2068.81\n21Sl ⎟\n⎠⎞⎜\n⎝⎛= =lα α               (12) \n()1121Sl ϕ β β = =l                                  (13) \nThis approach of the ideal  load should be useful \nfor the  dielectric measurement of a real material media using a coaxial probe, since the measured S\n11 can always be reasonable at x=0, which may \ninclude the circuit mismat ches at their boundaries. \nAs for the biological medium, the permittivity equation, equation (11), mu st be rescaled with \nregard to the permittivity of the known materials such as de-ionized or distilled water for this study. This is because the biological medium is similar to water. It is practical to use the distilled or de-ionized water as a reference material for the  \n 12\nmeasurements of biological tissues. Research also \nshows that the traceable “known” complex permittivity is crucial for the reference liquids’ purpose of measuring biological tissues \n[10, 11].  \nThat also supports the rescaling of the permittivity equation (11) for water. The rescaling process is \nvery simple:  \n)( )(\n)()(\n0 0\n00ω ωε\nω εω εAm refref\n= and )()()()(\n00ωωεω εω ε A\nmref\nref = , \nwhere A represents the right hand side of equation \n(11) for the frequency spectra. As such, one obtains the following: \n) tanh()0 ( 1)0 ( 1 )()(\n11112\n500 0lxSxS\nZZ\nmref\nw\nw γεω εωε⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= −= +\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−≅   (14) \nAs for the reference water measurements, the \nempirical rescaling factor was needed and was implemented as follows:  \n[] fj\nm mref\nw) 1(01.0180 )(0+ −⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛=ε εω ε. The empirical \nrescaling factor of frequency dependence was \nrequired in order to have better agreements with \nwater data of earlier measurements published \nelsewhere. Perhaps, it implies that this method is still needed to consider  neglected higher order \nmodes when frequencies increase in the measurements. However, this kind of rescaling is sufficient for the measurements in this paper. As such, this rescaling factor is in accordance with the real permittivity of water for up to 18 GHz.  As for the imaginary permittivity of water, it is in accordance up to 10 GHz, but becomes a little less reliable in the 10 GHz to 18 GHz range, compared \nto previous researchers. It may still be that the \nhigher order modes are aff ected by the level of the \nsignal at the higher frequency end.  Once measurements for the referenced materials such as de-ionized and distilled water are established, it \nwill be fairly simple to determine the dielectric properties of the biologi cal tissues by repeating \nthe method of equation (14).  This can be seen below.   \n) tanh()0( 1)0( 1\n)()( )()(\n11112\n500 0\n1111\nlxSxS\nZZ\nS\nwS\nt\nmref\nw\nt γ\nωεωε\nεωεωε⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= −= +\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−≅                                                                          \n                                                             (15) \nIt is obvious that the permittivity of the test \nmaterial, )(11ω εS\nt , is unknown, as well as the \npermittivity of reference water, )(11ω εS\nw , without measuring the permittivity of those biological \nsubstances. However, it is possible to determine \ntheir ratios ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n)()(\n1111\nω εω ε\nS\nwS\nt by using their phase \nmeasurements from calibrated reflections of S11 at \nlower frequencies. This is done by considering the ratio of penetration distances of traveling waves in the different biological media. It is also possible to find the permittivity ratios from the propagation constants. In doing so, one may determine the reasonable ratios between penetration thicknesses that are supported by the Cole-Cole method. That process may be written as follows: \n  \n) ( ) (00i rj j j j j ε εεμω α β β αγ − = − = +=       (16) \nBy squaring both sides of  equation (16), one can \narrive at the following:   \n 2\n222\n11\n2\n22\n22\n22\n12\n12\n1\n) () (\nll\nll\nrr\nεε\nα βα β=\n−−                                         (17) \n 2\n2222\n112\n2\n2222\n111\n2 21 1\n22\n22\nll\nll\nii\nl lll\nεωεω\nβαβα\nβαβα= =                            (18) \nHere, 1land 2l are penetration di stance of wave \npropagation through the different medium.  \nThe1α,2α, 1β, and 2β are the attenuation and \npropagation constants of  propagating waves in \nrespect to the different media. Also, the following notes were used in the calculation: \n111llα α= , 222llα α= , 111llβ β= , and 222llβ β= . As \nsuch, one obtains the ratios of penetration \nthicknesses as: \n  21\n21\n12\n21\n2 21 1\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n=\nllll\nii\nll\nβαβα\nεε                                    (19) \nThe equation (17) is used to determine the initial \nand final permittivity of test materials with regards to the reference permittivity. It is based on the reasonable assumption that the penetration thickness would be infin ite at both zero and \ninfinite frequencies. That makes us eligible to detect the permittivity differences between the test and reference materials at the lossless region according to the Cole-Cole plot. The ratio of penetration thickness for the entire frequency range is calculated using equation (19), and is needed for the imaginary permittivity component of the test material frequency dependence from the Cole-Cole method as follows:  \n 13\n⎟⎟⎟⎟⎟\n⎠⎞\n⎜⎜⎜⎜⎜\n⎝⎛\n+−+ =∞\n∞\n000\n1ffjimagi ε εε ε                             (20) \nHere, 00ε is the permittivity at zero frequency, ∞ε \nis the permittivity at infinite frequency, and 0f is \nthe relaxation freque ncy of medium.  \nIII. MEASUREMENTS OF WATER AND BIOLOGICAL \nSUBSTANCES \nUsually, the coaxial probe measurement technique \nrequires a simple calibration procedure, such as SOL.  The standard open , short and match load \n(SOL) were used in this paper to remove potential \nreflections from the coaxial cable to the probe tip (see Fig.3). This is because of the potential reflection from coaxial cab le and because its phase \ncontributions have to be removed before starting the measurements. The phase removing procedure fully complies with Agilent’s 8510C VNA built in SOL (short, open, and load) procedure.   \n \nFig. 3 The short, open, load sta ndards and the coaxial probe were \nused in this measurement methodology \nIt implies that the 50 oh m match load that was \nimplemented in the SOL calibration finds a plane that produces reflections that are typically lower \nthan -50dB (\ndB S 50 log2011 10 −≤ ). Afterwards, the \nmatch load is simply replaced by the 50 ohm \ncoaxial probe (see Fig.3), and the coaxial probe replacement finds another plane with maximum \nreflections, i.e. \ndB S 0 log2011 10 ≈ . At this point a \nplane with maximum reflection in regards to \nobtained magnitude of S 11 has been found. In the \nreal time measurement process, the S11 coefficient \nfrom the coaxial probe itself was recorded first as a reference.  It was then used to normalize the S\n11 \nmagnitude and to find the S11 phases from the \nmaterial under test.  To determine the normalized S\n11 magnitude and its phases from the material under test, the simple algebra of the least squares \nmethod of polynomials was used for smoothing the data of the S\n11 parameters as part of the \ncalibration procedure. This simple reflection removal procedure provides an arrangement which is capable of returning the highest signal content from the coaxial probe tip attached to the test materials.  In order to demonstrate the validity of this theory and the calibration procedure, the measurements of Methanol (HPLC grade, 99.9%) were presented, with the distilled water as the reference (see Fig. 4 and Fig. 5). The measured permittivity of methanol in Fig. 5 matches well with Agilent’s measurements up to 10 GHz, but beyond 10 GHz the imaginary permittivity measurements achieved by this method are lower than the Agilent results \n[12]. Next are the \npresentations for the measurements of water and biological tissues. From these measurements in Fig.4, Fig.5, and Fig.6, there is a visible difference between the complex permittivity of deionized water and that of tap water, and these measurement differences are consistent for multiple trials. The measurements show that this technique is capable of differentiating the degree \nof the purity of the water (see Fig.6), and therefore \ncan be considered a useful technique for detecting \nboth normal and malignant substances in biological tissues.  As for demonstrating the differentiation capability of this technique, the \nmeasured real and imaginary parts of complex dielectric permittivity were presented for commercial (available in grocery stores) beef and chicken materials using distilled water as a \nreference material. This new data is shown in Fig. 7 and Fig. 8. \n \nFig. 4 S11 phase measurements of distilled water and Methanol over \nthe frequency range 0. 045- 18 GHz.  \n 14\n \nFig. 5 Microwave permittivity spectra for the real and imaginary \nparts of complex dielectric permittivity of methanol and distilled \nwater. The frequency is shown on a logarithmic scale.  \n \nFig. 6 A comparison of real and imaginary parts of complex permittivity spectra for deionized water and tap water. This technique reveals the difference between pure and tap water \ncontents according to this measurement. \n \nFig. 7 the measured phase spec tra for the scattering parameter 11S \nfor distilled water, commercial beef, and chicken (available in food \nstores).  The typical retained water content for the beef and chicken \nis about one percent.  \n \nThe tested beef and chicken meats were purchased from a usual food store and they were labeled as 1% retained water content, but not identified for the content of salt. Thus, unfortunately, the salt contents could not be presented in this paper. Fig. \n7 compares the spectra for the phase of the scattering parameter S\n11 for distilled water and \ncommercially available beef and chicken. The experimental measurements of the above samples were carried out at room temperature in a microwave measurements lab.  The reproducibility of the data is reasonab le (see Fig.8). Following \nthese tests, the measured complex permittivity of normal and malignant breast tissues were from the human body was presented in Fig.9 and Fig.10.  The specimens from human body were collected from the Tufts New England Medical Center (Department of Pathology), and were formalin treated (10%). Specimens were then brought in to \nthe microwave measurement laboratory for dielectric measurement purposes. The specimens have not been identified for race or age at this time. The measured results  agree very well with \nthe results that were reported by various other researchers (former experiments as mentioned\n in \nreferences) [16]. The results in this study were \npresented against the results and predictions of previous researchers, and in doing so; a useful table was created for co mparison purposes (see \nTable 1).  The table di splays the dielectric \nproperties of some of th e materials measured in \nthis experiment (see Fig. 8), in addition to references to previous experiments.  \n \nFig. 8 Comparison of real and imaginary parts of complex \ndielectric permittivity spectra for distilled water, beef and chicken \n(available in grocery stores). Both beef and chicken contain about \none percent water.  \n \nFindings include malignant tissue having more calcium content, maki ng the real part of \npermittivity lower than that of water. Thus, one expects to see a higher imaginary part of permittivity for malignant tissues.\n  \n 15\n \nFig. 9 the spectra showing  S11 phase measurements of 10% \nformalin treated normal and maligna nt breast tissues from humans. \nNo information about the source is known yet. \n \nFig. 10 Comparison of real and imaginary parts of complex dielectric permittivity spectra for 10 percent formalin treated \nnormal and malignant breast tissue from human body.  \n \nUsually, the following mathematical curve fitting is used to apply for smoothing the data and errors analysis are employed for experimental measurements: \n()\n()22\n1\nmeanfit\nyyyy\nR\n−−\n−=        (13) \nHere, y stands for measured parameters, y fit being \nthe fitted parameters from the least squares of \npolynomials, realy being the real part of the \nmeasured parameters, and y mean being the average \nvalue of the measured parameters.  The value of R is always brought as close to one as possible, so that the best fit is achieved. A relative error analysis may be viewed in Fig.11. As for the errors analysis for other measurements, such as biological tissues, this study presents average absolute deviation (AAD) from the actual \nmeasurements, that is \n∑\n=− =N\nimean iyyNAAD\n11. The iy stands for the measured data of each frequency \nspectrum. Fig.12 displays the AAD. \n \n \nFig.11 Relative error analysis of  the complex permittivity of \ndistilled water. The measured errors of relative real and relative \nimaginary parts were on a reas onable scale for supporting the \nreliability of this technique. \n \nFig.12 Average absolute deviations  of the complex permittivity of \nsix different substances which we re measured by this technique \n \nTable 1 Tabulated Permittivity and Conductivity Results \nTissues ′\nrε ″\nrε sσ GHz Ref \nBeef 43 \n47.7 42.7 13.5\n \n13.4 \n16.5 - \n- \n- 2.45 \n2.8 2.8 [12] \n[14] \nours \nChicken 52.3 \n37 \n37.6 17.7 \n5.0 \n23.8 - \n- - 2.4 \n12.0 12.0 [12] \n[12] \nours \nNormal breast 10~3\n5 \n24 - \n9.5 0.15~0.46 \n- 6.0 \n6.0 [15] \nours \nMalignant \nbreast 54 \n50.1 - \n24 0.7 \n- 6.0 \n6.0 [15] \nours \nFat%(0-30) \nFat%(85-100) 48.4 \n4.7 - \n- 0.7 \n0.036 average \naverage [16] \n[16] \n \nIt is important to differentiate errors by human, \ninstrumentation, and system in a study with \nscientific measures. However, in this study, the \nabove errors analysis is su fficient at this stage of \nthe study. This is becaus e the reported relative \nerrors were already on a reasonable scale (within \n5% fluctuations) for this report. See Fig.11 for \ndistilled water and Fig. 12 for average absolute  \n 16\ndeviations for each real and imaginary permittivity \nmeasurements by using this technique. Also, \nFig.12 show the absolute average deviations, as \nwell as displays that the liquid substances of water \nand methanol have larger average absolute \ndeviation in comparison to the tissue substances \nfrom both the human body and cows and chickens. \nLogistically, the liquids have less interference \nproblems compared to semi-solid substances such \nas tissues. As such, this source of error may not be \nmechanical from interference between coaxial \nprobe tip and substance. Perhaps, system errors \nfrom calibration procedure may be reconsidered to \neliminate the uncertainty region for its calibration \nstandards.  However, the porous media, such \ntissues, are usually composed of the materials of a \nsolid matrix, a gaseous phase, and liquid water. \nFurthermore, the liquid water phase is sometimes \nsubdivided into free water and bound water, which \nmay be restricted in its mobility by specific \nstructures, such as tissues. If so, the errors may be \nfundamental rather than caused by \ninstrumentation. In eith er case, those kinds of \nerrors analysis works should be reported in \nseparate experiments. Pe rhaps that kind of study \nwould help to differentiate the errors from different cell structures in tissues.  \nIV. DISCUSSION AND CONCLUSION \nAs for the measured data for water, one should \nnote that this technique is  capable of showing the \ndifference between de-ionized water and distilled water. The measurements show that a significant difference exists between tap water and de-ionized water in terms of the imaginary components of their permittivity. The complex permittivity measurements show th at there are larger \ndifferences between beef and chicken from a food store. It is interesting th at the white chicken meat \nhas less fat content compare to the red beef meat that is consistent to the analysis of permittivity\n [17]. \nThe measurements are in concurrence with previous studies\n of the permittivity of liquids, beef \nand chicken [13, 14, and 15]. These results certainly \nconfirm the validity of this measurement technique. The normal and malignant breast tissue \nmeasurements also agree with previous studies\n [16, \n17]. However, the normal breast tissue measurements showed some irregular behaviors at \nthe lower frequencies of the spectrum. This phenomenon is attributed to the potential porosity of the normal breast tissue. The breast tissues from the human body were preserved in formalin liquid. The malignant breast tissu e is a hard biological \nsubstance that would not absorb the formalin, but the normal breast tissue may have absorbed some amounts of formalin, creating certain porosity in the biological substance. This can be seen by the dispersion of the real permittivity (see Fig. 10) at lower frequencies in the spectrum. Also, it appears that the inference proble m of this technique may \nneed to be addressed in the near future.  Finally, it is apparent that this technique will not have any conflicts with Agilent’s st andard probe technique. \nAlternatively, this method provides a new perspective for the microw ave characterization of \nbiological tissues and liquid substances. \nACKNOWLEDGMENT  \nThis research is supported by a contract from the US Army National Ground Intelligence Center. \nREFERENCES  \n[1] Adiseshu Nyshadham, Christopher L. Sibbald, and Stainislaw S. \nStuchly, “Permittivity Measuremen ts Using Open - Ended Sensors \nand Reference Liquid Calibration – An Uncertainty Analysis”, IEEE Transactions on Microwave Theory and Techniques, Vol. 40, NO. 2, \nFebruary 1992. \n[2] Bruce G. Colpitts , “Temperature Sensitivity of Coaxial Probe Complex \nPermittivity Measurements: Experimental Approach”, IEEE \nTRANSACTIONS ON MICROWAVE THEORY AND \nTECHNIQUES, VOL. 41, NO. 2. FEBRUARY 1993. \n[3]  J. Baker-Jarvis, M. D. Janezic, P. D. Domich, and R. G. Geyer,  \n“Analysis of an Open-Ended Coax ial Probe with Lift-Off for Non-\nDestructive Testing” IEEE Tran sactions on Instrumentation and \nMeasurement, Vol. 43, pp. 711–718, Oct. 1994. \n[4] C Gabriel, T Y A Chan and E H Grant, “Admittance models for open \nended coaxial probes and their place in dielectric spectroscopy”, Phys. Med Biol. 39 (1994) 2183-2200, Printed in the UK. \n[5] Stoyan I. Ganchev, Nasser Qaddoumi, Sasan Bahtiyiari, and Reza \nZoughi, “Calibration and Measurement of Dielectric Properties of Finite Thichness Composite Sheets with Open –Ended Coaxial \nSensors” , IEEE Transactions on Instrumentation and Measurement, \nVol. 44, No. 6, December 1995. \n[6] Ching-Lieh Li and Kun-Mu Chen , “Determination of electromagnetic \nproperties of materials using fla nged open-ended coaxial probe full \nwave analysis”, IEEE Transactions on Instrumentation and Measurement, Vol. 44, No.1, February 1995. \n[7]   Kjetil Folgero and Tore Tjomsl and, “Permittivity Measurements of \nThin Liquid Layers Using Open – Ended Coaxial Probes”, Meas. Sci. Tecnol. 7 (1996) pp.1164-1173, Printed in the UK.  \n[8] S. Hoshina, Y. Kanai, and M. Miyakawa, “A numerical study on the \nmeasurement region of an open-ended coaxial probe used for complex permittivity measurement” Magnetics, IEEE Transactions on Volume 37, Issue 5, Part 1, Sept. 2001 pp. 3311 – 3314. \n[9] Carmine Vittoria, “Elements of mi crowave networks”, World Scientific \nSingapore, 1998. \n[10] A. P. Gregory, M. G. Cox, a nd R. N. Clarke, “Improved Monte Carlo \nuncertainty modeling with cross-frequency correlation for microwave \ndielectric reference liquid data”, 2008 CPEM confer ence proceedings,  \n 17\npp.526-528. \n[11] M.Obol, et.al, “Direct broadba nd microwave characterization of \nbiological tissues using the co axial probe technique” 2008 CPEM \nconference proceedings, pp 530-531. \n[12]   http://cp.literature.ag ilent.com/litweb/pdf/5989-2589EN.pdf  \n[13] F. Tanaka, P. Mallikarjunan, Y.  C. Hung, “Dielectric properties of \nshrimp related to microwave freq uencies: from frozen to cooked \nstages”, Journal of food process engineering, Vol. 22, Issue 6. Page \n(455-468), December, 1999. \n[14] Ken-ichiro, Murata Akio Hanawa, and Ryuske Nozaki, “Broadband \ncomplex permittivity measurement tec hnique of materials with thin \nconfiguration at microwave frequenc ies”, Journal of Applied Physics \n98, 084107 (2005). \n[15] J. G. Lyng, L. Zhang, N. P. Brunton, “A survey of the dielectric \nproperties of meats and ingred ients used in meat product \nmanufacture”, Meat Science 69 (2005) 589-602. \n[16] Mark Converse, Essex J. Bond,  Susan Hagness, and Barry D. Van \nVeen, “Ultrawide – Band microwave space-time beamforming for \nhyperthermia treatment of breast cancer: a computational feasibility \nstudy”, IEEE Transactions on Microwave and Techniques. Vol. 52, \nNo. 8, August 2004. \n[17] Carey Rappaport, “A dispersive microwave model for human breast \ntissue suitable for FDTD computation”, IEEE Antennas and Wireless \npropagation letters, Vol. 6, 2007. \n \n \n                                \n                         \n \n                        \n 18\n  \nAbstract —Numerous wireless communication devices such as \nintegrated circuits and micro biochips are emerging in the 1 to 4 GHz frequency range.  Thus it has become crucial to \naccurately determine effective permeability and permittivity of \nsuch oxide materials and devices in this spectral range. \nTraditional techniques such as waveguide and free space are \ngood measurement techniques at higher frequencies. However, \nat lower frequencies it is extremely challenging to apply these \nmethods.  In this paper, we present a propagation and \nimpedance technique for the microstripline. Three different \nsubstrates (Alumina, Silicon di oxide and Beryllium Oxide) are \nemployed to design TRL sets for measurement purposes. The \ncustom-designed TRL sets return losses up to -50 dB which is \nthe same as a standard waveguide TRL calibration. Thin low-\nloss oxide samples were placed on the top side of a zero \nreference plane of an L microstripline. For the upper frequency \nband, we reported the permeability and permittivity \nmeasurements of the materials by waveguide without using any \nguess parameter. Here, we report effective permittivity and \npermeability measurements for oxides without using any guess \nparameter by microstripline. In the paper we present results of \nYIG, Nickel ferrite, and Glass from 1 to 4 GHz. Also, a \nnonlinear excitation of YIG is presented for demonstration \npurpose. \n \nIndex Terms —Permeability, permittivity, and microstripline \nTRL calibration.  \nI. INTRODUCTION \noday’s rapid developments in integrated \ncircuits for various el ectronic, microwave and \nbiomedical applications are based on thin film materials. The thin films are grown on various dielectric and magnetic s ubstrates. They may be \nsingle layer or multilayer structured. For the bulk of available materials, th e waveguide technique is \nsuitable to determine their relative permeability and relative permittivity in air. The machined thin substrates have dies of  various microstripline \ncircuits on their surfaces. Since the relative permittivities of substrates are known, the impedance and effective relative permittivity of microstripline circuits may be configured by \nWheeler equations \n[1] for circuit design purposes. \nMicrostripline was widely used and recognized for microwave technology applications\n [2, 3] for \nmagnetic materials, such as YIG, in the past few decades. The application of the microstripline technique and the TRL (thru-reflect-line) \n                                                \n \n  calibration method for micr ostriplines are not new.  \nHowever, there is a new perspective proposed here \nfor applying a direct measurement technique by \nusing microstriplines to find the permittivity and \npermeability of thin films of single and \nmultilayered materials. This technique allows for the detection of effective permittivity and permeability of oxides, including magnetic oxides at frequencies between 1 GHz to 4 GHz. The use \nfor such an application was observed previously, when a microstripline technique for the simultaneous measurement of permittivity and permeability of materials was revealed \n[4]. \nHowever, this technique featured a full wave analysis for the microstripline, where the full wave propagating through the micr ostripline may not be \nin a TEM mode. Consequently, a numerical optimization process was used to find global \nminima in the squared error analysis, which may need more testing in order to confirm the validity of the algorithm. However, some papers\n \nconsidered the accuracy of the quasi TEM mode configuration for microstr iplines in a propagation \nand impedance analysis \n[5, 6]. Based on quasi TEM \nmode assumptions, measurement techniques by \nmicrostripline were also presented elsewhere to measure permeability of ferrites, but these techniques require a micr ostripline fabrication on \nthe material in question \n[7, 8, 9]. In order to make the \nmicrostripline technique more capable and flexible for measuring permeability and permittivity simultaneously, a new microstripline technique is presented in this paper. It is capable of simultaneously measuring the effective permeability and permittivity of thin materials without requiring a micr ostripline fabrication on \neither the thin oxide slab of the target or on films. In order to effectively determine the permeability and permittivity of the sample being tested, this paper articulates the valuable microstripline concepts\n and transmission line theory that have \nbeen published by others working in this area [10, \n11]. Also in this paper is an attempt to see the \nnonlinear excitations from the YIG samples. Nonlinear excitations were  studied and reviewed Permeability and Permittivity Measurement Technique by \nPropagation and Impedance of Microstriplines  \nT  \n 19\nfor YIG materials to reveal the fundamentals of \nnonlinear excitations of the YIG samples [12, 13, 14, \nand 15]. Evidently, this technique will also provide a \nreasonable platform fo r studies of nonlinear \nexcitations in magnetic thin films as well. \nII. THEORY AND MEASUREMENT TECHNIQUE \nAlthough numerous techniques, such as in-waveguide, cavity perturbation and free space are available for the permeability and permittivity measurements at higher microwave frequencies, the applicability of those techniques at lower frequencies is challeng ing due to the physical \nconstraints of measurement equipment and instruments. The waveguide dimension becomes cumbersomely robust, t hus requiring several \npounds of materials for any reasonable measurement. However, at present there is a huge commercial interest\n for operating frequencies \nlying between 1 GHz and 4 GHz or even broader frequencies up to 10 GHz for various wireless communication applications \n[16, 17]. Thus, a \nreliable, cost- effective and fast measurement technique may be needed for the effective permeability and permittivity measurements of materials at lower frequency spectra. For this purpose, an obvious choice is a microstripline technique that does not n eed complex fabrication \ntechnology for the TRL standards. The custom designed standard TRL sets (see Fig.1 and Fig.2) of different dielectric subs trates were fabricated at \nMicrofab Incorporation (NH, USA). In this technique, the characteristic impedance of line standards is known prior to TRL calibration. This helps determine the charac teristic impedance and \npropagation constant of th e loaded material that \nhas to be found by employing S-parameters of materials or devices being tested in a constructed network. The physical dime nsions of the TRL sets \nare shown in Table.1. Since the requirement of loaded materials on the line standard are far less in volume compared to the substrates of the microstripline, a reliable reflection coefficient, \nΓ, \nis not expected from the loaded material on the \nmicrostriplines.  \n \n \nFig. 4 A set of TRL BeO calibration substrates. The loaded \nmagnetic material was placed on the microstripline for testing. \nIn general, for the cases of permeability and \npermittivity measurements using two port networks by S\n11 and S 21, two different methods are \navailable [10, 19]. The first is based on the concept \nof the existence of multiple reflections inside lossless media. Baker-Jarvis has presented the following, which is closely consistent with the well-known Weir equations\n in nature [18, 19]. That \nis equation set (1). This set of equations (1) has been in practice for decades in respect to the workable subjects.  \n \nFig. 5 A sketch showing geometrical  dimensions of microstripline \nand material under test by  a constructed network. \n \nTable.1 Physical dimens ions and electrical \nparameters for various micr ostripline substrates (1 \nmil =0.0254mm) \nSubstrate effε Z0(Ω) w(mil) h(mil) (L-T) \n(mm) \nSiO 2 3.07 50 46.6 25 5.95 \nBeO 4.58 50 29.9 25 3.96 \nAlumina 6.56 50 20.1 25 2.75 \n \n  \n 20\n001121222\n21222\n11\n11) 1(1) 1(\nZZZZSSe TTTSTTS\niil\niiii iiii i\n+−=ΓΓ−= =Γ−Γ−=Γ−−Γ=\n−γ               (1) \nOn the other hand, based on the transmission line \ntheory, Vittoria has presented a new set of equations, which is as follows \n[10]. \nZ ZZ ZSSe TS e STTS\neel\neele ee e\n+−=ΓΓ+= =Γ + =Γ−− Γ=\n−−\n00112111 212 22\n11\n1) 1(1)1 (\nγγ\n             (2) \nThe question now is how far the equations from \nset (1) to set (2) are.  Based on the definitions of \ntheir first reflections, it seems that e i Γ−=Γ . From \nthis equality, the equation  set (1) and (2) should be \nthe same in nature. However, the problem is not so simple, since one must also consider where the reflections are coming  from. Upon doing so, \naccording to the definitions\n of first reflections (iΓ \nandeΓ), one would correlate them and end with \nthe relationship el\nie Γ =Γ−γ2. This implies that if the \nnetwork is lossless and th e used transmission line \ndistance between thru and line equals a quarter wavelength, then the James and Vittoria equations hold and are equivalent in nature. For the low \nfrequency measurements using the microstripline \nin this study, the equation set (2) was only partially used, the obvious reason bring that the loaded materials on the transmission lines of TRL calibration standards used in this study were not capable of generating relia ble multiple reflections \nfor the quarter wavelength  differences between the \nthru and line of the microstripline for this frequency spectra. The Ba ker-Jarvis equation for \nthe waveguide measurement technique was used in that study\n [20], because the transmission line \ndifferences between the thru and line of TRL standards of waveguides were on the order of quarter wavelengths, and the loaded materials \ninside the waveguide are ab le to generate reliable \nreflections \n[20]. In summary, there is the option to \nchoose between equation sets (1) and (2) based on which problems are less pertinent. As for the perfectly calibrated matched transmission line, the relationship between the transmission coefficients \nwould be \n21ST=.  However, the target material \nloaded transmission line is capable of generating a \nnegligible amount of refl ection from the scattering \ncoefficient11S, so the transmission coefficient \nthrough the material needs to be normalized to its \nloss of 2\n112\n21 1S S −= . Bearing this in mind, it is \nnoted that the reasoning behind the normalization \nhas to be supported by reasonable methods.  According to the set of e quations (2) we have the \noption to deploy its transmission as follows: \n( )1\n11 211− −Γ + = =LdS S eTγ. Due to the nature of the \nproblem being addressed in this paper, reliable \nreturn reflection from the samples that were loaded on the top side of the transmission lines \nwas unattainable. Also ignor ed were the arbitrary \nphases of small reflection errors of very small magnitude of the loaded sample in the calculations. Allow the nor malization factor be \ndefined as follows\n: \n( )()1\n111\n11*\n21 21* 21 1− −−Γ + Γ + = =ψ ψ j\nLj\nL e S e S SS TT T  \n                                                                         (3) \nManipulating the above equation with simple \nalgebra produces the following: \n( )1\n112\n112\n212) cos( 2 1−\nΓ + Γ + = ψL L S S S T              (4) \nFor equation (4) shown above, the higher order \nmagnitude of reflections can be dropped due to negligible magnitudes from reflection. Also, one \nwould assume\nπ ψ m2→ , (m= 0, 1, 2, etc) which \nallows for the removal of arbitrary phases of 11S \nandLΓ, in order to reach the following: \n2\n212\n111 2\n112\n212) 1( ) 1( S S S S T −≅ + ≅−         (5) \nNow it is easy to obtain the magnitude of the \ntransmission coefficient: \n212\n11) 1( S S T ⋅ − =                                     (6) \nNow the transmission coefficient of this \nmicrostripline may be determined as follows:  \n212\n11) (1 S S e eTdj d×⎟\n⎠⎞⎜\n⎝⎛− = = =+− − β α γ            (7)  \n 21\nOne can then extract the propagation constant in \nloaded material through the transmission coefficient as follows: \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n− +⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n= ) 2(1ln1\nT njT dϕ π γ                            (8) \nAccording to the equivalent circuit (see Fig. 3), \nthe A 11 element of A-matrix may be represented \nby the elements of the S-matrix.  \n \nFig. 6  Equivalent circuit di agram of microstripline. The A 11, A12, \nA21 and A 22 are elements of the A- matrix.  Z 0 is the characteristic \nimpedance, V 1, V2, I1 and I 2 represents voltages and currents \nNow the relation between the elements of the A \nmatrix and the S-matrix are presented, in accordance with the constructed network in Fig.3. The circuit may be repres ented as having a single \nvoltage. \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+−=⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+ −\n= =+++\n+\n+−\n− +\n=21\n212\n11\n212\n2\n11\n1 1\n0 21\n111\n21\n21\n2SSS\nVVVV\nVVV V\nVVA\nI\n   \n                                                                                        \n                                                            (9)  \n0 21\n11 0 21\n11\n0 21\n21\n2 2 22\n= = == = =\nI I IVV\nZ VV\nVI\nVIA      (9a) \nThe currents in the networks were considered to \nbe equal and in opposite directions, making the \ntotal current to be zero in the network, it is because of load oxides are no grounded in the circuit. According to the relationship between Z and the S matrix of the reciprocal network, one \nwould have the relationship of: \n21 12 22 1121 12 22 11\n0 111\n) 1)( 1() 1)( 1() )( (\nSS S SSS S SZ ZSISI Z\n− − −+ − +=− +=−\n                 (9b) \nThe concentration of this paper was on the \nmeasurements of isotropic media that can stay in a reciprocal state in th e network, meaning S\n11 = S 22 \nand S 12 = S 21. By the combination of (9), (9a), and \n(9b), A 21 may be derived as follows: \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−=21\n212\n11\n021) 1(1SSS\nZA                 (9d) It is obvious that it is convenient for us to use an A \nmatrix rather than an S-matrix alone to determine the permeability and permittivity from the microstripline technique. We employed a reasonable method for corre lating the two matrices \nbased on reciprocal network concepts\n that were \npresented by former investigators [6, 7, 8, 9, 10, and 11]. \nThe relationship of the A-matrix and the S-matrix \nyields the result22 11 A A= , for example. As for the \nA12, we need to solve the determinant of the A \nmatrix, 21 12 22 11 1) det( AA AA A − ==  which produces a \nrelationship of0\n21112\n0\n21 124ZSS ZA A = − . It is now \npossible to obtain the full elements of the A matrix \nwith the following: \n⎟⎟⎟⎟⎟\n⎠⎞\n⎜⎜⎜⎜⎜\n⎝⎛\n+−\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−++−\n=⎟⎟⎟\n⎠⎞\n⎜⎜⎜\n⎝⎛\n=\n21\n212\n11\n21\n212\n11\n021\n212\n11 0\n21\n212\n11\n1 ) 1(2) 1(\n21\n21) cosh() sinh(1) sinh( ) cosh(\nSSSSSS\nZSSS ZSSSd dZd Z d\nA\nLL\nγ γγ γ\n       \n(10) \nThe normalized characteristic impedance of loaded materials may then be written as follows:  \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−==\n21\n212\n11) 1() sinh(\nSSSdZn\nLγη                      (11) \nHere,LZ is the load impedance on the \nmicrostripline, n\nLZ is the normalized impedance on \nthe microstripline of target materials, and η is the \nmedium impedance, in ohms. Based on a quasi \nTEM mode assumption of the microstripline on the substrate, a derivation of the relative permittivity and permeability of loaded materials to the L standards of the substrate may be done: \nn\nLr\neffn\nLr\neff\nZfcjZ fcj\nπγμπγε\n21\n2\n−=−=\n                                      (12) \nHere,cfj jπ\nλπγ2 2\n00 = = , and c is the speed of light. \nThe impedance of microstriplines fabricated on \nsubstrates is known, so it is possible to derive the \nrelative permeability and relative permittivity of oxides by using (12).      \n 22\nIII. EXPERIMENTAL MEASUREMENTS \nIn this measurement process, the widths of the \nloaded materials are shorter than the line difference between L and T of TRL sets from \ndifferent substrates. The samples are placed on the center of calibrated L lines. Locating the sample on the exact reference plane side of S\n11 is always \ncrucial for obtaining reliable phases of reflections, which help to derive accurate data for permittivity and permeability. For the measurements on SiO\n2 \nand BeO substrates, the loaded material dimensions are width = 3.5mm, height = 0.5mm \nand baseline = 7mm. For the measurements on Alumina substrates, the loaded material dimensions are width = 1.26mm, height = 0.5mm \nand baseline = 2.52mm. In the final data extrapolation, the propaga tion contributions from \nthe L lines (\n) (TL dl −Δ−=δ ) of the substrates were \nsubtracted from the total propagation constant. \nThey were recorded us ing a vector network \nanalyzer. Next, the simultaneous permeability and permittivity measurements of thin ferrite disks and glass on the BeO substrate were done by this microstripline technique (see Fig.4, 5, and 6). The simple table below (Table 2) compares the data for real permittivity by microstriplines and by the in-waveguides of YIG, Nickel  ferrite, and Glass on \nBeO substrate at 4 GHz for comparison purposes.  \n \nTable.2 A comparison of permittivity data at 4GHz for YIG, nickel \nferrite and glass measured by two methods \nMaterials line microstrip\neffε  waveguide\nrelativeε  \nYIG 6.8 13.0 \nNickel ferrite 6.8 12.0 \nCorning® 1737 \nglass 5.5 6.0 \n \nIt was already noted that the measurement figures from figs 4 to 6 show very smooth data for permeability and permittivity. Consequently, the measurement data show some unphysical parts for \nboth permittivity and permeability. Also, the figures show some freque ncy dependent increases \nfor the real part of the permittivity. Let it be mentioned that the equation (7) was based on a bold approximation of the magnitude of S\n11 being \nvery small. However, the reflection from any sample is increased when the frequency increases. As such, the approximation appears bold, but when the frequency increases, perhaps this will \ncause some frequency dependent phenomena for the permittivity in the figures above. Conversely, a method of least squares of polynomial for presenting smooth data on those figures has been \napplied. In general, the following mathematical curve fitting for smoothing the data is deployed for the experimental measurements.   \n( )\n()22\n1\nmeanfit\nyyyy\nR\n−−\n−=  and \nrealmean\nreal yyy\nyyerr−=Δ=      (13) \nHere, y stands for measured parameters, y fit stands \nfor fitted parameters from the least squares of \npolynomials, realy is the real part of the measured \nparameters, and y mean stands for the average value \nof the measured parameters.  In order to achieve the best fit, the value of R is made as close to one as possible. However, fitting that is too good creates some data for the imaginary components of the permeability and permittivity. Therefore, errors within 5% would be acceptable for our measurements. We attribute these 5% errors to system errors, instrumentation errors, and human errors, but the quantified differentiations of error sources will not be analyzed at this time. The 5% errors stand for compromising between best curve fitting and the errors that are allowed due to indications of unphysical parts measurements of the permeability and permittivity. This kind of curve fitting may also app licable in finding direct \nerrors from the scattering parameters of S\n11 and \nS21 if it is necessary to do so. When using this \ncurve fitting, the permeability and permittivity \nmeasurement figures show little unphysical \nranges, and they are within the acceptable spectra that were presented in figs.7 and 8. Although this technique works very well for the simultaneous permeability and permittivity measurements of several materials on BeO substrate, the measurements of permittivity on the Alumina and SiO\n2 substrates were not successful with this \ntechnique. In contrast, the complex permeability of the materials is detectable by using this microstripline on Alumina, BeO and SiO2 substrates. For a demonstration, nonlinear excitations obtained fro m YIG by using this \nmicrostripline technique on BeO substrate are presented in fig.9.  \n 23\n \nFig. 4   Complex magnetic permeability and dielectric permittivity \nmeasurements of YIG on BeO substrate \n \nFig. 5 Complex magnetic permeability and dielectric permittivity \nmeasurements of nickel ferrite on BeO substrate \n \nFig. 6 Complex magnetic permeability and dielectric permittivity \nmeasurements of glass (Corning® 1737) on BeO substrate \nFig.7 Relative errors of real parts of measured permeability and \npermittivity, where mu represents r eal permeability, ep represents \nreal permittivity and err represents errors. YIG (yttrium iron \ngarnet), nickel is nickel ferrite , and glass is corning glass 1735.  \nFig. 8 Relative errors of imaginar y parts of measured permeability \nand permittivity, where mui represents imaginary permeability, epi \nrepresents imaginary permittivity and err represents errors. The \nYIG (yttrium iron garnet), nickel  is nickel ferrite, and glass is \ncorning glass 1737.  \n \nAs for the nonlinear excitation analysis of YIG, numerous excellent stud ies have already been \nreported in various journals  for specific cases of \nYIG microwave measurements in the past several decades. For example, some of the nonlinear analysis and studies for YIG materials were reported\n to understand the nonlinear dynamical \nbehaviors of YIG [12, 13, 14, and 15]. In this study, a \nYIG sample was simply placed on the microstripline and it was subjected to non-uniform magnetic field. To see the nonlinear excitations from YIG, the excited nonlinearity was correspondingly attributed  to excited MSM cases \n(magneto static modes), according to the findings of former investigators. It is clear that the nonlinear phenomenon of YIG has been known  \n 24\nfor five decades, so a detailed discussion on the \nnonlinear phenomenon will not be presented at this time. It is beyond the scope of this paper, but \nit would appear that the nonlinear excitations of YIG might also be obs erved by using this \ntechnique without further complicated experimental setup. Perh aps it will provide a new \nscope to study the nonlin ear excitations of YIG \nsince the YIG has had a vital role in various technological applications for decades.   \n \nFig. 9 A comparison of spectra showing the real and \nimaginary parts of magnetic permeability of YIG specimen. A YIG rectangular disk was placed in non-uniform fringe \nexternal fields of 500 Oe \n \nIV. DISCUSSION AND CONCLUSION \n \nFor the higher permittivity substrate such as \nAlumina, the propagation wave through the microstripline may not be a quasi TEM mode. As for the loaded magnetic specimen, perhaps, one may have to take into account its demagnetization \neffect on wave propagation. Since the measurements showed that a microstripline on Alumina substrate severely weakens permittivity measurements of the oxides at lower frequencies spectra, the quasi TEM mode approximation for \nmicrostripline on the Alumina substrate may not \nbe true in this case.  The microstripline technique for SiO\n2 showed that higher imaginary \ncomponents for complex permittivity of the materials, which we attribute to the conductive carriers in SiO\n2. However, in this design, the \nfabricated microstripline on BeO fitted the quasi TEM mode approximation very well, which can be seen in the presented data on the measured effective complex permeability and permittivity of \nYIG, nickel ferrite, and glass material. The permittivity and permeability measurements are slightly different between the waveguide technique\n and the microstripline technique on \ndielectric substrates [20]. In an inside waveguide \nmeasurement technique, the materials were positioned in air, so that the measured relative permittivity and relative permeability were in relation to air.  However, with the microstripline measurement technique, the measured permittivity and permeability were relative to the substrate and air. Thus, the presented complex permeability and permittivity can be understood as effective permeability and effective permittivity due to the averaging effect between the load, dielectric substrate and air. \n \nACKNOWLEDGMENTS \n \nThis research is supported by a contract from the US Army National Ground and Intelligence Center. \nREFERENCES  \n[1] H.A. Wheeler, “Transmission-line properties of a strip on a dielectric \nsheet on a plane” IEEE Trans. Microwave Theory Tech. MTT-25, \npp.631-647, (1977). \n[2] Waguih S. Ishak, “Magnetosta tic wave technology: A Review”, \nProceedings of the IE EE, Vol.76, No. 2, pp. 171-187, February 1988. \n[3] J. D. Adam, “Analog signal pr ocessing with microwave magnetics”, \nProceedings of the IE EE, Vol.76, No. 2, pp. 159-170, February 1988. \n[4] Patrick Queffelec, Marcel Le Floc’h, and Philippe Gelin, “Broad band \ncharacterization of magnetic and deielectric thin films using a \nmicrostrip line”, IEEE Trans. In strumentation and Measurement, \nVol.47, No.4, pp.956-963, August 1998. \n[5] Dylan F. Williams and Roger B. Marks, “Transmission line capacitance \nmeasurement”, IEEE Microwave and guided wave letters, Vol.1, \nNo.9, pp.243-245, September 1991. \n[6] Roger B. Marks and Dylan F. Williams, “Characteristic impedance \ndetermination using propagation constant measurement”, IEEE \nmicrowave and guided wave letters, Vol.1, No.6, pp.141-143, June 1991. \n[7]  M. Obol, C. Vittoria, “Measurement of permeability of oriented Y-type \nhexaferrite”, JMMM, pp.290-295, 265 (2003). \n[8] Masakatsu Senda and Osamu Ishii,  “Magnetic properties of RF diode \nsputtered CoxFe100-x alloy thin films”, IEEE Transactions on \nMagnetics, Vol.31, No.2, pp.960-965, March 1995. \n[9] Y. Q. Wu, Z.X. Tang, B. Zha ng, and Y. H. Xu, “Permeability \nmeasurement of ferromagnetic materials in microwave frequency \nrange using support vector machine regression”, Progress In Electromagnetics Research, PIER 70, pp.247-256, and 2007. \n[10] C.Vittoria, Elements of Micr owave Networks, World Scientific, \nSingapore, 1998. \n[11] L.F.Chen, C.K. Ong, C.P. Neo, V.V. Varadan, V.K. Varadan, \nMicrowave Electronics, Measurement and Materials Characterization, \nJohn Wiley & Sons Ltd (2004). \n[12] D. J. Mar, L. M. Pecora, F. J. Rachford, and T. L. Caroll, “Dynamics of \ntransients in yittrium-iron-garnet”,  an Interdiscip linary Journal of \nNonlinear Science -- December 1997 -- Volume 7, Issue 4, pp. 803-\n809.  \n 25\n[13] V.E.Demidov et. al., “Spin-wave  eigenmodes of a saturated magnetic \nsquare at different precession angl es”, Phys. Rev. Lett., vol. 98, \np.157203 (2007). \n[14] D. D. Stancil, Theory of Magne tostatic Waves, Springer-Verlag, New \nYork, (1993). \n[15] A.B.Ustinov, G. Srinivasan, B. A. Kalinikos, “Ferrite-ferrolectric \nhybrid wave phase shifter”, Appl. Phys. Lett. 90,  p. 031913 (2007). \n[16] M.Obol et.al, “Permeability a nd permittivity measurements using \npropagation and impedance of TRL calibrated microstripline”, 2008 \nCPEM conference pro ceedings, pp.528-529. \n[17] M. Hirose and S. Kurokawa, ‘Com pact extended port capable of full 2-\nport calibration using optical techniques”, 2008 Conference on \nPrecision Electromagnetic Measurements, CPEM 2008, pp. 508-509. \n[18] William B. Weir, “Automatic measurement of complex dielectric \nconstant and permeability at microw ave frequencies”, Proceedings of \nthe IEEE, Vol. 62, No. 1, pp.33-36, January 1974. \n[19] J. Baker-Jarvis, E. J. Venzura, and W. A. Kissick, “Improved technique \nfor determining complex permittiv ity with transmission/reflection \nmethod”, IEEE Trans. Microwave Theory Tech., vol. 38, No.8, pp. \n1096-1103, August 1990. \n[20] Nawaf N. Al-Moayed, Mohammed N. Afsar, Usman A. Khan, Sean \nMcCooey, Mahmut Obol,  “Nano ferrites microwave complex \npermeability and permittivity measur ements by T/R technique in \nwaveguide”, IEEE, Transactions on Magnetics, Vol.44, No.7, July \n2008,pp.1768-1772.  \n \n                         \n \n       \n                         \n \n                        \n 26\nFull Band Microwave Isolator of Rectangular Waveguide Having \nPeriodic Metal Strips and Ferrites  \n \nAbstract — This study presents a fu ll band microwave isolator \nin the X-band spectrum that features ferrite samples coated in \nuniformly spaced strips of metal wire. The objective is to use \nthe wire strips to obtain effective negative permittivity levels at \nthe desired frequencies within the spectrum, as well as to create \nnegative permeability with the ferrite-wire configuration. In \npractice, we were capable of controlling the wire-covered \nferrite samples by using an ext ernal field as small as 100 mT. \nThe controllable permeability of ferrites in this experiment \nallowed us to create unidirectional wave propagation over the \nentire X-band frequency spectrum. This nonreciprocal circuit \nis a challenge to the traditional methods of defining refractive \nindex, permittivity and permeab ility from the S-parameters of \na vector network analyzer. We therefore propose a novel \nmethod to define the refractive index for nonreciprocal circuits. \nIndex Terms — metal wires, fe rrites, negative permeability, \nnegative permittivity, refractive index and metamaterials \nI. INTRODUCTION  \nThe microwave isolator has been in practice for \nseveral decades.  The use of ferrites in waveguides \nfor isolator application is  now a well-established \nconcept in the field of  microwave technology. \nAlthough this type of study has a long history with \nregard to its known pract ical applications and \ntheoretical completeness, we plan to construct a full band microwave isolator in the X-band spectrum using the wire-coated ferrite samples.  The resonance loss properties of ferrites are used to design one-way transmission lines. These lines have a large percentage of energy propagation \nabsorption by the ferrite in one direction, while permitting nearly lossless transmission in the opposite direction.  The r eal consequence of such \nresults depends on the reac tion to large external \nmagnetic fields while operating in the high frequency region of the spectrum. Although some hexaferrites are effective in reducing the external field influence at hi gher frequencies, the \nproduction of high-quality hexaferrites for general \nmicrowave spectra often involves extra costs and a comprehensive knowledge of hexaferrite crystal structures. By using high- quality yet less costly \nferrites, including YIG a nd Nickel, as well as \napplying a weaker external magnetic field to the microwave isolators in the waveguide, we present \na method for deriving negative permittivity from \nlined metal wire arrays on the microwave isolator. The basic notions of negative permittivity were gathered from Pendry’s notable findings [1]. The \nferrites used were YIG and Nickel and the metal wire was a thin aluminum tape. The metal wires, which were 2mm wide, were taped along the width of the rectangular ferrite pieces with 0.5 mm \nbetween each strip. The wr apped ferrite disks were \nthen loaded into an X-band rectangular waveguide. The S-parameter measurements showed that the wrapped ferrite disks were fully capable of annihilating the S\n21 transmission for the \nfull X-band spectrum; that is to say that the achieved annihilation was up to 4 GHz bandwidth. We attribute this phenomenon to effective negative permittivity from the periodic metal wire array. During the application of a 100 mT external magnetic field that was perpendicular to the RF magnetic field of the TE\n10 mode in the rectangular \nwaveguide, the annihilated S 21 transmission was \ndramatically recreated for the full X-band spectrum. This phenomenon is due to the effective negative permeability of the ferrites. Also observed in this experime nt was a unidirectional \npropagating wave mode. Th e opposite direction of \npropagation became evanescent with the application of a 100 mT external field. These results are most likely due to the left-handed properties of metamaterials\n [2, 3]. Thus, we were \nable to generate broa dband effective negative \npermittivity using metal wires, and effective negative permeability by using a 100mT external magnetic field perpendicular to the RF magnetic field of the TE\n10 mode in the rectangular \nwaveguide. The successful microwave isolator was used in an X-band waveguide and its isolation frequency band was 4 GHz, ranging from 8 GHz to 12 GHz. The isolation st retched as wide as 40 \ndB. In this paper, the negative permittivity and negative permeability generated by the microwave isolator in the X-band are presented with the scattering parameters measured by a vector network analyzer (the Agilent 8510C vector \nnetwork analyzer). The microwave isolator presented in this paper is used to apply the concepts of negative permittivity and permeability for practical microwave device applications. As  \n 27\nfor the negative refractive index creation with \nmetal wire and with ferrite materials, it may be useful for device miniat urization [3]. Although \nNicolson-Ross and Weir [4] and Baker-Jarvis [5] presented ways of defining the refractive index, permeability, and permittivity in a waveguide, their methods require initial estimated parameters and integer values for phases of propagating waves in normal microwave materials. It is often \nhard to be accurate with those estimations when using a metamaterial. As for determining metamaterial permittivity and permeability, a method has been presented by Smith, Schultz and Markos [2], who proposed that the constructed network or circuits should be in reciprocal states \nin order to do so. The circuit featured in this paper used neither normal materials nor remained in a reciprocal state. In the past, we presented a T/R method [6] for normal materi als that does not need \ninitial parameter estimations, and based on that method [6], we present a novel method for the special case of nonrecipr ocal circuits with \nmetamaterials. Consequently, the method presented in this paper is successful in determining the permeability and permittivity of both normal materials and of  reciprocal circuits \nfrom metamaterials. \nII. MEASURED S-PARMETERS OF YIG AND NICKEL \nFERRITE IN A WAVEGUIDE \nIn the experiment, the wires were made with \nrectangular aluminum foil strips. The dimensions of the wires are such that for w (width), L (length), and t (thickness), they satisfy the following condition: L > w >> t. The media is the lined wire array taped onto the ferrite disk and placed inside the waveguide. There we re nine 2mm wide \naluminum foil strips attached to the ferrite disks; with 0.5mm gaps separating them from one another in order to cr eate effective negative \npermittivity (see Fig.1). Since the desired effective negative permittivity came from good conductors such as aluminum foil wires, the corresponding effective negative permeab ility was achieved by \napplying different external magnetic fields to the \nferrite disks. In this experiment, the external field H\n0 was applied along the RF electric field of the \nTE10 mode in the rectangular waveguide. Thus, \nthe H 0 field is parallel to the wires on the ferrite \ndisk.   \n \nFig. 7 Ferrite disk taped with aluminum wire strips. Aluminum \nwire width is 2 mm, the gap be tween wires is 0.5 mm, and the \nlength of the disk is 2.3 cm. \nThe scattering parameters from the wire-coated \nferrite samples in the waveguide may be viewed in Figures 2 and 3. \n \nFig.2 Measured scattering parameters of S 21 and S 12, in dB, from 8 \nto 12 GHz for aluminum wires ta ped on YIG with and without an \nexternally applied magnetic field (90mT). Aluminum wire width is \n2 mm, the gap between the wires is  0.5 mm, and length of the YIG \ndisk is 2.3 cm. \nFor the case of external field, H 0 = 0 Oe, the S 21 \ntransmission data shows that the transmission drops below -50dB (see Fig.2 and Fig.3). This indicates that the metal wires achieved negative permittivity and annihilated the transmission agent of S\n21. While applying the exte rnal fields of 90 mT \nand 100mT to the wire-coated ferrite samples, the transmission agent S\n21 was created back in the \nwaveguide. This is simply because the negative permeability plays a role in changing the attenuated wave to the propagated wave again in \nthe waveguide. Moreover, the S\n12 gets a small \nadvantage from the applied external fields. It should be noted that th e right and left-handed \npermeability depends on the magnetic polarization by the external field, H\n0, and its spin is caused by \nRF driving forces to the ferrite disks.  Although \nwe have 10 dB insertion losses, we have reached a very broad band of isolation between S\n21 and S 12. \nThe insertion loss can either be amplified by  \n 28\nmicrowave amplifiers or improved upon using \nsimulation designs. As wa s noted earlier, none of \nthe available techniques are capable of determining the refractive index, permeability and permittivity of this kind of nonreciprocal circuit. It \nis now necessary to pres ent a technique that is \ncapable of deriving negative permittivity and permeability for such nonreciprocal circuits. \n \nFig.3 Measured scattering parameters of S 21 and S 12, in dB, from 8 \nto 12 GHz for aluminum wires ta ped on nickel ferrite with and \nwithout an externally applied ma gnetic field (100mT). Aluminum \nwire width is 2 mm, the gap be tween the wires is 0.5 mm, and \nlength of the nickel ferrite disk is 2.3 cm. \nIII. CAUCHY -RIEMANN EQUATIONS FOR THE PROPOGATION \nCONSTANT AND ITS APPLICATION TO THE REFRACTIVE \nINDEX DETERMINATION  \nThe phase shift and attenuation from the \ntransmission line of the waveguide [6] can be \nwritten as follows: \nP Lj njk ljknj ϕ ϕ ϕ ϕ γ + = + = − =Ψ0 0 0 ) (   (1) \nApplied to equation (1), the Cauchy-Riemann \nequation may be used as follows. \nn kn k\nL PP L\n∂∂−=∂∂∂∂=∂∂\nϕ ϕϕ ϕ\n         (2) \nBased on equations (1) and (2), we obtain the \nfollowing: \n⎥⎦⎤\n⎢⎣⎡\n∂∂\n∂∂−∂∂⎟\n⎠⎞⎜\n⎝⎛\n∂∂=−\nωϕωϕ\nωϕ k\nknnP\n01\n0        (3) \n⎥⎦⎤\n⎢⎣⎡\n∂∂−∂∂=∂∂\nωϕ\nωϕ\nϕ ω0\n01kkL                  (4)  \n 1−\n⎟\n⎠⎞⎜\n⎝⎛\n∂∂\n∂∂=∂∂\nωϕ\nωϕL P\nkn                        (5) By using equations (3), (4 ), and (5), one can be \neasily arrive at the following: \n1−\n⎟\n⎠⎞⎜\n⎝⎛\n∂∂\n∂∂=ωϕ\nωϕL Pkn             (6) \nIn order to validate this measurement \nmethodology, the normal material phenyloxide was tested first (see Fig.4). \n \nFig. 4  Measurement of comple x refractive index of normal \nmaterial phenyloxide in wa veguide using Cauchy-Riemann \nimplementation. \nAs for the refractive index from the wire-coated \nnickel ferrite (see Fig.5) , the measurement showed \na negative refractive index up to 8.7 GHz.  \n \n \nFig.5 Measurements of complex refractive index for wire-coated \nnickel ferrite in waveguide. \nMoreover, the positive refractive index beyond the \n8.7 GHz spectra shows that the refractive index continues to increase with increases in frequency. This may be reasonable based on the theoretical \ncalculation of the negative  permeability of nickel  \n 29\nferrites in a 100mT field, for example. At this \npoint, we present a brief theoretical analysis for the permeability of ferrite and the permittivity of periodic metal strips.  In this waveguide \nmeasurement of this experiment, the external field H\n0 applies along RF elect ric field of the TE 10 \nmode, that is to say the external field H 0 field is \nparallel to the wires on the ferrite disk. So, one easily obtains the effectiv e permeability of this \nferrite disk inside waveguide by follows.  \nμκ μμ2 2−=eff             (7) \nwhere2 2\n00\n) (1\nGm\nfjf fff\nΔ+ −+=μ ,2 2\n0 ) () (\nGG m\nfjf ffjff\nΔ+ −Δ+=κ ,  \n \n) 4 (0 0 0 sM HH f π γ + = ,s m M f πγ4= , kOe GHz / 8.2=γ ,  \n \nGfΔ is a microwave loss from the ferrite disk and \nit is in frequency unit which is usually described \nby the Gilbert damping. Perhaps it represents the intrinsic spin misalignments as well as crystal \nanisotropy field of ferrite disk. The \nsMπ4  is the \nsaturation magnetization of the ferrite disk and f \nis an operational frequency from the RF source, \nwhich is from the vector network analyser.  A modeled permeability for a ferrites disk inside the waveguide may be seen in Fig.6. \n \nFig.6 The theoretical calculation of complex permeability of ferrite \ndisk inside rectangular waveguide , in which the parameters were \nassumed as follows: the external field, mT H 1000=  , saturation \nmagnetization, kOe Ms 8.1 4 = π , GHz fG 1.0= Δ  \nIn order to understand the permittivity \ncontribution from the periodic metal wires, a sample of periodic metal wires arrays were prepared on Scotch tape. The prepared sample was \ntested with the frequency spectrum of the X-band (see Fig.7).  It is now n ecessary to point out that \nthe all samples were in the same geometrical dimensions. The designed sample dimensions were the same as those of the cross section surface geometry of the rectangular waveguide. A standard rectangular waveguide of X-band was also used in this experi ment. The following figure \nis the result of the S\n21 measurement from the \nvector network analyser.  \n \n \nFig. 7 Measured scattering parameters of S 11 and S 21 in dB from 8 \nto 12 GHz.  Periodic metal wires taped on Scotch tape. Each \naluminum foil tape wire width is  2 mm, the gap between the wires \nis 0.5 mm and the total length of the cell is 2.3 cm. \nThe measured S 11 and S 21 parameters (see Fig.7) \nfor periodic metal wires applied to Scotch tape revealed that there was deeper absorption near the higher frequency end of the waveguide, while the lower frequency range allowed more transmission to pass. As such, this kind of structure of periodic metal strips provide enormously enhanced effective mass that helps to obtain the Plasmon frequency within the GHz  frequency spectrum, \nand it appears around 12 GHz for this case (see Fig.7). It implies that the effective permittivity from periodic metal wires on Scotch tape may not be negative; but it’ll be negative when they were \narrayed on the ferrite disk. This part of the research may be further explained by the obtained \npermeability and permittivity of this wire-coated ferrite samples. Those works will be reported in the extended versions of this report.  \n 30\nIV. DISCUSSION  \nAccording to the S-parameter measurements, a \nfull band microwave isolator was achieved using ferrite samples coated with metal strips. The refractive index of the nonreciprocal circuit was \nderived using equation (6), which was implemented with the Cauchy-Riemann equations. Measurements of phenyloxide were taken using this novel method, and we re then matched against \nknown phenyloxide measurements. Until now, we have no problems to determine excited mediums refractive index. However, this research has been seen the positive imaginary parts of permeability and permittivity excited mediums for some time now. This kind of phenomena was also reported elsewhere, and various arguments are existed by other investigators propos als. This kind of \nanomaly is serious problem for classical electrodynamics analysis while others are even \nclaiming the proof of energy conservation of \nelectrodynamics in medium may be enough for such microwave metamaterials cases. However, it is a plan now to report this problem by deploying \nquantum treatment such as WKB approximation since the resonant states  responsibly created the \nmicrowave metamaterials. By deploying quantum treatment, it allows the positive imaginary parts \ncan be at large while cl assical dynamics does not \nsupport that kind of idea. E ither case, the relevant \nresearches will be reported in near future. Perhaps, it’ll be appeared in regular transactions of Microwave Theory and Techniques then.  \nACKNOWLEDGEMENT  \nThe research is supported by a contract from United States \nArmy National Ground Intelligence Center.  \nREFERENCES  \n [1] J. B. Pendry, A. J. Holden, D.  J. Robbins, and W. J. Stewart, \nMember, “Magnetism from conductors and enhanced \nnonlinear phenomena”, IEEE, Transactions on microwave \ntheory and techniques, Vol. 47, No. 11, November 1999. \n [2] D.R. Smith, S. Schultz, P. Mar kos, C.M. Soukoulis, \n“Determination of effective permittivity and permeability of \nmetamaterials from reflection and transmission coefficients”, \nPhys. Rev. B 65 (2002) 195104. \n[3]  Y. He, P. He, S. D. Yoon, P.V. Parimi, F.J. Rachford, V.G. \nHarris, C. Vittoria, “Tunable ne gative index materials using \nyttrium iron garnet”, Journa l of Magnetism and Magnetic \nMaterials 313 (2007) 187–191.  \n[4] W. B. Weir, “Automatic measurement of complex dielectric \nconstant and permeability at  microwave frequencies,” Proceedings of the IEEE, Vol. 62, No.1, pp. 33-36, January \n1974. \n[5]  J. Baker-Jarvis, E. J. Venz ura, and W. A. Kissick, “Improved \ntechnique for determining complex permittivity with the \ntransmission/reflection method, ” IEEE Trans. Microwave \nTheory Tech., vol. 38, No.8, pp. 1096-1103, August 1990. \n[6]  N. N. Al-Moayed, M. N. Afsar, U. A. Khan, S. McCooey, M. \nObol, “Nano Ferrites Microwave Complex Permeability and \nPermittivity Measurements by T/R Technique in Waveguide”, \nIEEE, Transactions on Magneti cs, Vol.44, No.7, July 2008.  \n 31\nAbstract \n \nThe spiral spin structured magnetic materials display important magneto-electric properties that have lead them to become known  \nas frustrated magnets. Recently, an insulator single crystal compound of Sr 1.5Ba0.5Zn2Fe12O22 was also found to be a material that has \nmagneto-electric (ME) effects. This is a rare magnetoelectric material, with its N eel temperature, at 326 K, above room tempera ture. \nAs such, we measured ferromagnetic resonance (FMR) of this material at 9.5 GHz in 1.5 kOe, and the corresponding ferromagnetic \nresonance response showed  that this compound needs - 0.5 kO e unusual internal fields to hold  ferromagnetic resonance condition for \nthis crystal compared to the ferromagnetic resonance of normal planar hexaferrites.  To justify this strange internal field nec essity, \nsubsequent measurements from a SQUID (superconducting quantum interference device ) magnetometer and a vibrating sample \nmagnetometer (VSM) were also performed. Those measurements were capable of showing the saturation magnetization as 4 πMs = \n1800 Gauss, the internal plan e anisotropy field as H a = 9.8 kOe, and the line width of ferromagnetic resonance as around 50 Oe. It is \ninteresting that this unusual internal anisotropy field helps to estimate existence of domain walls for this crystal which is s imilar to \nBloch’s wall, in which the magnetization rotates through the c- axis of this crystal; the esti mated domain wall thickness is \napproximately 0.27 micron meter of this crys tal.    This kind of domain wall existenc e in insulator ferrite may indicate that i t is a \nsource of distinct electric polarizations, which means this crystal belongs to the magneto-electric property material.   \nIndex Terms — Ferrites, ferromagnetic resonance, internal anisotropy fields, and permittivity \nPACS: 75.50.Gg \nI. Introduction \nhe ferromagnetic crystal structure of the \nhexaferrite compound Ba 2-xSrxZn2Fe12O22 is \nknown to be  helimagnetic. It  belongs to a group of \nfrustrated ferrimagnets, as named by former \ninvestigators, and recently, it has been reported to \nhave ferroelectric properties [1, 2, and 3]. This implies \nthat the material could potentially have a magneto-\nelectric property that is very useful in modern \ntechnology, such as aiding in the electrical tuning \nfor magnetic systems. Theoretical microscopic \ncalculations show that these types of materials may \nhave frustrated exchange in teractions, and that they \nmay induce the electric polarizations inside of \ncrystals [3, 4, and 5]. Traditionally, this type of helical \nhexaferrite is also identified as a class of \nantiferromagnetic materials [6]. Both the past and \nrecent reports of these materials encourage the \ninvestigation of magnetic and microwave properties \nthrough different methods. Thus, the classical \nmeasurements of FMR, VSM, and the SQUID were \ncarried out on a single cr ystal hexagon disk of Ba 2-\nxSrxZn2Fe12O22. In order to interpret the \nexperimental measurements, the traditional free \nenergy model was used to define the internal fields \nof this material [6, 7]. The results of this study show \nthat an internal field of about – 0.5 kOe was needed \nto hold the ferromagnetic re sonance condition of the \nnormal linear microwave mode analysis. This \nirregularity in the behavi or of the compound may be due to its unusual spin st ructure coming from layer \ninteractions between the ferromagnetic and \nantiferromagnetic phases.  According to definition, \nthese types of helical spin  systems feature exchange \ninteractions between neares t-neighbor planes that \nare ferromagnetic, while those between second \nnearest-neighbor planes expe rience interactions that \nare antiferromagnetic [6]. Therefore, the classical \nmicrowave mode analysis, or  Kittel mode, of this \nsystem calls for a previously undiscovered internal \nfield to explain experimental measurements. As \nsuch, an anisotropy field, namely an internal \nanisotropic field or demagne tizing damping field, is \nconsidered to fill this role.  \nII. Ferromagnetic resonance condition \nfor the crystal composition of spin spirality \nin easy plane magnetization ⊥ to the c-axis \nThe composite of (Sr,Ba)Zn 2Fe12O22 was studied in \nthe early 1960’s for its helic al spin configuration, \nand it was because of the identified internal anisotropy fields in this composite that further analysis was also carried out to understand the internal anisotropy fields for other such composites. \nUntil now, there has been no convincing argument that explains the force that  induces the helical spin \nalignments that disturb planar magnetizations in an anisotropy field with high magnitude (on the order of one Tesla). Although there were already several practical arguments in existence for the reason behind the helimagnetic status of this composition, An Unusual Internal Anisotropy Field of Spiral Magnetized \nCrystal Compound of Sr 1.5Ba0.5Zn2Fe12O22    \nT  \n 32\nhere the free energy terms are rewritten and \npresented for the helical hexaferrite crystal disk in this paper. Supposing one can follow the classical approach for the planar helical hexagonal crystal disk’s free energy \n[7], it is expanded using a power \nseries of cosine as follows:  \nϕ ϕ ϑ ϑϕ ϑ ϕ ϑϕ ϑ\nϕ ϑ 6cos cos cos ) cossin sin cos sin (21cos sin\n2 22 2 2 2 2 2\nK K K MNMN MNHM F\na s zs y s xs h\n− − − ++ +−=\n  (1)                                                                                                            \nIn equation (1), ]7[ϑ polar angle, ]7[ϕazimuth angle, \nand N x, N y, and N z are the demagnetizing factors, \naKis introduced to this crystal in this paper which \nrepresents a force that causes the spiral \nmagnetization in this syst em. It may be understood \nthat it is an exchange interaction between ferromagnetic and anitferromagnetic layers in this system. One proposes that the hexagonal crystal disk of this composite is found to have the following layer sequence: (AFM) (FM) (FM) (AFM) (FM) (FM) (AFM) al ong the c-axis. If so the \npotential domain walls conf iguration inside crystal \nbehave like Bloch walls. It implies that the magnetization rotates through the c-axis of this compound. This kind of macroscopic ordering generates a fairly strong repulsion force between the \nferromagnetic layers, a force that is strong enough \nto cause the level of spiral magnetization that is seen in the composite. In such a scenario, the internal anisotropy f iled may be estimated \n[8]. The \neffective anisotropy fields may be defined as follows \n[9]: \nssh\nMKHHM KF\nϑ\nϑϑ ϑ\nϑϑ ϑ ϑϑ\n20sin sin cos 2\n=≈= =∂∂\n              (2)                                                                                 \n        \nsa\nas sa seff\nsah\nMKHMK\nMK\nHH HM HMK KF\n≈= ≈≈+ = =+ =∂∂\nϕ ϕ\nϕϕϕ\nϕϕϕϕ ϕϕ ϕϕ\n36\nsin6sin 60sin) ( sin6sin 6 sin\n            (3) \nNow the ferromagnetic resonance condition is \npresented. For this special case, we assume the c-axis of the crystal is positioned along the z-axis, the \nDC magnetic field strength, H is along the y-axis, and the RF magnetic field (TEM) propagates along the y-axis.  The effective static and microwave fields inside crystal disk may be represented as follows: \nz\nseff\nz\nszz zz\nix\nseff\nxx\nii\nmMH\nmMHmN h hmMH\nh hH H\n− − − =− ==\nϑ                     (4) \nUsing the above equations (2 ), (3), and (4) with the \nmotion equation for magnetization [6, 9], one obtains \nthe ferromagnetic resonance condition: \n) 2 )( (s a ahM H H HH H HH πγω\nϑ ϕ ϕ + + + + + + ≈  (5)                  \nIn the derivation proce ss for equation (5), \napproximations for the special hexagon disk were used. That is, the aspect ratios\n that are responsible \nfor the demagnetization of this single crystalline \nhexaferrite disk were meas ured to be the following: \n0 4 ≈− ≈ −bh\nahN Nx y π and π2≈ −x zN N . \nClearly, if the demagnetization of magnetic \nmaterials induced spiral structure, then all magnetic materials would be capable of forming such a spiral configuration. Demagnetiza tion could therefore not \nbe the means by which the spiral structure is formed. \nIII. The measurements of A crystal of planar \nhelical hexaferrite \nThe Bell Laboratories, Kimura provided a platelet \ncrystal of Sr 1.5Ba0.5Zn2Fe12O22 [1]. The properties of \nthe crystal were mass m = 4.9mg and geometrical dimensions a ≈ 1.5mm, b ≈ 1.3mm and h = \n0.73mm. The Neel temperat ure for this composition \nwas reported as T\nN = 326K and magnetically \ninduced ferroelectricity was up to 130 K, an observable phenomenon perpen dicular to the c-axis \nof the crystal \n[1]. In the study, the easy \nmagnetization plane (external field perpendicular to the c-axis) and hard magne tization plane (external \nfield parallel to the c-axis) temperature-dependent magnetization phase diagrams were measured by SQUID at various temper atures, ranging from 5 \nKelvin to 300 Kelvin (see Fig.1 and Fig.2).   \n 33\n \nFig.8 Temperature dependence of  saturation magnetization in the \nplane by SQUID measurements at various temperatures, where the \nexternal fields were applied perpendicular to the c-axis of the crystal \ndisk. \n \nFig. 9 Temperature dependence of  saturation magnetization in the \nplane by SQUID measurements at various temperatures, where the \nexternal fields were applied parallel to the c-axis of the crystal disk \nIn Fig. 1, the basal plane SQUID measurement \nshows that the temperature-dependent magnetizations were quite non-linear in  behavior \naround the temperature of 100 Kelvin. However, in Fig. 2 there are no specific differences in the linearity of magnetizations at different temperatures. According to Figures 1 and 2, it appears that the force causing spiral magnetization may be laid in the plane of magnetization, which may be perpendicular to the c-axis of the hexaferrite \ndisk. If so, it further suppor ts the existence of the \nunusual internal an isotropy, field \naH .In equation \n(1) and (3) it was written in terms of azimuth angle, \nand ϕ was expressed in the physical sense. In \naddition, the SQUID measurement helps determine \nthe saturation magnetization in the plane of this crystalline disk at room temperature. It was measured to be as 4 πM\ns = 1800 Gauss (see Fig.1). \nIn order to further understand internal anisotropy fields and saturation magnetizations at room temperature, we have also made the measurements of VSM and FMR for this crystalline disk. The VSM measurement confirms the SQUID measurement in terms of saturation magnetization, and it also shows that the in-plane anisotropy field was around 9.8kOe (see Fig.3). The FMR measurement further confirms the in-plane internal anisotropy field measurement by VSM and it helps to determine the internal anisotropy field of this material as well (see Fig.4).   \n \nFig. 3 VSM measurements of this single crystalline for the hard and \neasy axes of the magnetizations at room temperature \n \nFig. 4 FMR measurements at 9.5GHz, A -0.5 kOe anisotropy field was needed to hold the normal ferromagnetic resonance condition \npoint, which occurred at an external field of 1.5 kOe for the \ncrystalline disk \n \nIn general cases, the basal plane anisotropy field of planar hexaferrites, H\nφ, are very small and within \nthe order of Oe. This allows the ferromagnetic resonance condition of this crystalline sample may \nto be simplified to the following:  \n 34\n) 2 )( (s a ahM H HH HH πγω\nϑ+ + + + ≈      (6) \nIn order to interpret the above ferromagnetic \nresonance of this sample at 9.5 GHz in an external field of H = 1.5 kOe, the estimated unusual anisotropic field magnitude, H\na, may be found as \nfollows, kOe M K Hs a a 5.0 / −≈ = . It is our \nunderstanding that this fo rce may be understood by \nMalozemoff findings [8].  As such this study finds it \nas follows, \nhex\natKAKϑ 2= . Usually, the exchange \nstiffness constant by follows, \ncm erg Aex / 105.16−× ≈ and this study finds the \ncrystallography anisotropy energy toward c-axis by \nfollows, 3 6/ 1075.0 cm erg K × ≈ϑ . In order to generate \nthe unusual 500 Oe anisotr opy field, the estimated \ndomain wall thickness along the c- axis has to be \nm th μ27.0=  if Malozemoff finding is acceptable to \nreaders. This is an amazingly good data to be \naccepted for this crystal and this study suggests taking credit from the research\n [8]. According to \nhistorical investigations [10, 11, 12], one could assume \none of several possibilities for the force that causes the spiral structure of the magnetization of this crystal. The first is the possibility of oscillating states from lattice parameters due to the discrete magnetizations seen in the easy magnetizations plane \n[10]. The next suggests that there may be \nmagnetostriction along the c-axis due to the easy \nmagnetization hardness th at was observed along the \nc-axis [11]. The last possibility is that the larger \nnegativity of this field coul d be the frustration factor \nsource from the exchange interactions between ferromagnetic and antiferromagnetic layers. Considering macroscopic reasons, the force may also be the result of interaction between moments in different micromagnetic domains due to the quasi-continuous spiral magnetiz ation structure of the \ncrystalline disk. The negativity of this field may \nalso be due to the fact th at the interaction between \nmicromagnetic domains (postulated multi-domains for negative anisotropy) may be such that they align themselves into internal energy minimizing directions, like a demagn etizer, along the c-axis \n[13]. \nAccording to the findings of this research and the model of the free energy for this crystalline disk from this paper, this strange internal field may be due to domain walls betw een antiferromagetic and ferromagnetic layer sequences was imagined for \nthis disk, that is: (AFM) (FM) (FM) (AFM) (FM) (FM) (AFM) is the earlier  stated sequence of the \ndisk. Most likely the domain walls formation similar Bloch’s wall along c-axis and the magnetization rotate through the c-axis of this crystal. After all, this extra field is needed to hold this crystalline material’s FMR measurement at 9.5GHz for the linear microwave mode (Kittel’s mode) analysis of this crystalline sample. Based on those suggestions and free energy terms for the crystal from this paper, as well as the level of the strange internal field that was observed, the \nferromagnetic resonance in Fig.4 shows that the repelling force between th e ferromagnetic layers of \nthe crystal would be able  to cause such spiral \nmagnetization in the easy magnetized plane. In contrast to that, the othe r possibilities of causing \nsuch a result are ineffectual, based on their magnitudes, in causing such spiral magnetization for this hexaferrite disk.    \nIV. Domain wall’s appli cation of this crystal \n \nIf this crystal could be induced to the electric dipoles when applying the magnetic field perpendicular to the c-axis \n[1], it implies that the \ncrystal domain walls lose its stable state, which is perpendicular to the external magnetic field. In such circumstances the oscillation forces are as follows: \n       \nx K Fm⋅ −=                          (7) \nThemK is the restoring force of displacement of \ndipoles from domain wall os cillation (it may be the \nforce which was seen in the ferromagnetic resonance in Fig. 4) and x is the displacement vector. The field’s nega tivity is then inducing \ndemagnetizing behavior from domain walls against external magnetic field exertion.  An applied appropriate periodic ex ternal pulse field, H\nd, may \nmaintain the oscillatory states in the crystal. If so one obtains the following: \ndex\ndmdmd m\njxjH jxmK\nvvHmexmKvjevH xKvm\nω ωω\nγωω\n−=+−\n=−= +−= +\n2&\n       (8) \nWhere v is the velocity of oscillatory waves caused \nby disturbing external fiel ds toward domain wall, γ  \n 35\nis the gryromagnetic ratio, d d Hγ ω=  is resonator \nfrequency from the pulse field, m is the effective \nmass of vibrating domain  walls of the crystal \nandmKm\nex=ω  is intrinsic frequenc y.   It is obvious \nnow that there may be an effective conductive \ncurrent through the domain  walls along the c-axis, \nwhich is related to the term of these vibrating states. This may be written as follows: \ndex\ndex\nvjPjjNexj Nev Jω ωω\nω ωω\n−−=−−= −=2 2\n     (9) \nFor checking to see this crystal system’s effective \npermittivity, one may rewrite the Maxwell equation, which may be as follows: \n            \n)) (1(22\nEP\njEjjPjEjJtDH\ndexdexv\nrrsrrrrr\nεω ωωωωεω ωωωε\n−− =−− =+∂∂= ×∇\n                (10) \nFrom the above formulas one can conclude that the \neffective permittivity of this crystal system is as follows. \n)) (1(2\nEP\njdex\neffεω ωωωε ε−− =                   (11) \nEquation (11) reveals an important potential \napplication of the effective permittivity in future technology, that being that the permittivity of this crystal system can be tune d by external electric as \nwell as magnetic fields if the crystal has the ferroelectric property P \n[1]. It urges one to believe \nthat a possible negative refractive index may also be possible in insulator ferrites such as Sr\n1.5Ba0.5Zn2Fe12O22.  \nV. Conclusion \nIn this article, an unusual internal anisotropic field, \nHa, was presented, which has a negative value of -\n0.5 kOe. The basal plane anisotropy field (H φ) was \nregarded as negligible in this study due to the \npresence of large anisotropy fields, for example, \nϕH Ha>> andϕH He>> , so it would not be used in \nfulfilling the ferromagnetic resonance condition for \nthe planar hexaferrite compound of this single \ncrystalline disk. Beyond that, a reiterated \nmethodology is for deriving the ferromagnetic resonance condition for hexaferrites using Kittel’s torque technique is presented. This methodology \neasily applies to the case of  single helical crystalline \nhexaferrite disk as well . Both static and dynamic \nconsiderations on the sy stem are accounted for, \nmaking it more useful than Smit and Beljers’ ferromagnetic resonance condition derivation, as they defined the resonance condition only on pure static magnetization systems. The origin of the negative anisotropic field was briefly discussed in \nthis paper for further study purposes. Moreover, an important possible applicati on for this material was \nproposed through reasonable formulas.  \nACKNOWLEDGEMENT  \n This study is partially s upported by a contract from \nthe US Army National Ground Intelligence Center.  \nREFERENCES  \n1 T. Kimura, G. Lawes and A.P.Ramires, Physical Review Letters, PRL94,    \n   137201 (2005), April 8, 2005. \n2 G.T.Rado and V.J. Folen, Phys. Rev. Lett. 6 , 609 (1961). \n3 K.Knizek, P.Novak, and M.Kupfer ling, Physical Review B73,  \n   153103(2006). \n4 I.A.Sergienko and E. Dagotto, , Phys. Rev. B73, 094434 (2006). \n5 H.Katsura, N.Nagaosa, and A.V.Balatsky, , Phys.Rev.Lett.95, 057205  \n   (2005). \n6 Soshin Chikazumi , Physics of Ferromagnetism, Second Edition, Oxford  \n   Science Publication, 1997. \n7 Mahmut Obol, Carmine Vittoria, J ournal. Magn. Ma gn. Mater, 272- \n   276(2004) E1799- E1800. \n8 A. P. Malozemoff, Phys. Rev. B35, 3679-3682(1987). \n9 C.Kittel, Phys. Rev, Vol. 73, NO.2, 155-161(1948).    \n10 U. Enz, Journal of Applied Physics, Supplement to Vol. 32, No. 3, March,  \n   1961. \n11  E. W. Lee, Proc. Phys. Soc., 1964, Vol. 84. \n12 M. R. Fitzsimmons, et.al, Physical Review B, Vol. 65, 134436, March,  \n     2002.   \n13  J. O. Artman, Physical Review, Vol. 105, No.1, January 1. 1957. \n \n \n \n  \n \n  \n \n \n  \n 36\n36Mahmut Obol, Ph.D. \n \n \nEMAIL: mahmut.obol@gmail.com \n \nCAREER GOALS \n \nTo work in ceramics and biophysics R&D sector devel oping magnetic functional mate rials for next generation \nceramics technologies as well as their applications into biophysics technologies.  \n \nEDUCATION \n \nPh.D.  Electrical and Computer Engineering, No rtheastern University (Boston, MA)  2004 \nM.S.    Physics, Northeastern University (Boston, MA)                   2000 \nM.S.     Physics, Institute of Atomic Energy of China (Beijing, China)                 1991  \nB.S.     Physics, Xin Jiang University (Urumqi, China)                   1988 \n \n WORK EXPERIENCE AND HANDS ON TECNIQUES  \n \n1) Magnetic and dielectric materials microwave characteriz ations in waveguide. This technique is in state of \ndetermine materials refractive index,  impedance, permeability and permi ttivity etc without using any guess \nparameters and this technique  is free any divergence from the waveguides at all. This technique has also several \nprofessional tactics to determine normal material, resonant material, and metamaterials microwave characterizations in terms of refractive index, im pedance, permeability and permittivity. \n2)  Single and poly crystalline ferrites growth by flux melt and sintering methods and their crystal axis orientation \ntechnique. This technique is in state of grow and pr oduce numerous single and poly hexaferrites which includes \nmagneto-electric strontium rich planar hexaferrite. The str ontium rich planar hexaferrite  is going to have potential \nmicrowave applications such as sensors. 3) Coaxial probe technique for sensing complex permittiv ity and conductivity of liquid and semi liquid materials \nsuch as biological tissues. This is fast detecting t echnique for non magnetic materials microwave properties. \n4) Microstripline measurement technique in terms of detecting permeability and pe rmittivity of oxides. This \ntechnique is in state to detect materials microwave pr operties from 1 GHz to 4GHz spectra where the waveguide \ntechnique is challenged by its physical  dimension. This technique’s capability is great to compensate covering such \nfrequencies spectra.  \nINSTRUMENTATION AND COMPUTER SKILLS \n \n• Deploying microwave measurements on waveguide, coax ial probe, cavity and microstripline with Agilent’s \nVector Network Analyzer at broad GHz frequencies spectra. \n• Characterized magnetic and structural properties of materials using XRD, SQUID, SEM, VSM and FMR. \n• Professional computational tool is for me MATLAB. \n• Designed passive devices using electromagne tic simulation tools including Sonnet\nTM\n. \n• Familiar publishing tools are Micros oft Office and Photoshop etc. \n \nPERSONAL AND SPIRIT \n \nTo devote in fast faced work spirited environments, admi re people but can’t stand talks too much, and practicing \npositive progressive attitude in research and develo pment. Respect individual and team efforts. \n  \n 37\n37LANGUAGES  \n \nEnglish, Chinese, and Uyghur (native) \n \nPROOFS OF HANDS ON TECHNIQUES \n \nThe hands on techniques will be presented for the inte rested research sectors and companies by practice. \n  \n EMPLOYMENT \n \n2006-2010 Research Staff, Tufts University (Medford, MA) \n2005-2006 Associate Professor, Xinjiang No rmal University (Urumqi, China) \n2000-2004 Research Assistant, North eastern University (Boston, MA) \n1998- 2000 Teaching Assistant, Physics Department, Northeastern University (Boston, MA)  \n1991-1998 Lecturer, Physics Department, Xin Jiang University (Urumqi, China) \n \nTEACHING CAPABILITY \n \nQuantum mechanics, Electrodynamics, Classical Mechanics, and Circuits fo r Undergrade and Graduate Levels "    },    {        "title": "1812.09031v2.Structural_and_Magnetic_Characterization_of_Spin_Canted_Mixed_Ferrite_Cobaltites__LnFe0_5Co0_5O3__Ln___Eu_and_Dy_.pdf",        "content": "1 \n Structural and Magnetic Characterization of Spin Ca nted Mixed \nFerrite-Cobaltites: LnFe 0.5 Co 0.5 O3 (Ln = Eu and Dy) \nAshish Shukla 1, Oleg I. Lebedev2, Md. Motin Seikh 3 and Asish K. Kundu 1* \n1Discipline of Physics, Indian Institute of Informat ion Technology, Design & Manufacturing Jabalpur, Du mna \nAirport Road, Madhya Pradesh–482005, India \n2Laboratoire CRISMAT, ENSICAEN UMR6508, 6 Bd Marécha l Juin, Cedex 4, Caen-14050, France \n3Department of Chemistry, Visva-Bharati University, Santiniketan, West Bengal –731235, India \n \nAbstract \n The mixed ferrite-cobaltites LnFe 0.5 Co 0.5 O3, with Ln = Eu & Dy have been prepared \nby a sol-gel method and the samples have been chara cterized using X-ray diffraction and \nelectron microscopy. The magnetic investigations re veal that both samples ordered in canted \nantiferromagnetic structures near room temperature.  The Dzyaloshinskii-Moriya or \nantisymmetric exchange interaction induces weak fer romagnetism due to canting of the \nantiferromagnetically ordered spins. In the case of  Ln-Fe-Co orthoferrites, two magnetic \nsublattices (Ln 3+ -4f and Fe 3+ /Co 3+ -3d) generally align in opposite directions and int eresting \ntemperature dependent phenomena: e.g. uncompensated  antiferromagnetic sublattices and \nspin-reorientations, are observed in the system. Th e existence of hysteresis at low \ntemperature region has been explained in terms of t he strength of magnetic interactions \nbetween Fe 3+  and Co 3+  ions with different A-site rare earth cations. \n \n \nKeywords:  Antiferromagnet; Hysteresis loop; Sol-Gel method; Weak ferromagnetism \n*Corresponding author E-mail: asish.k@gmail.com/asi sh.kundu@iiitdmj.ac.in 2 \n 1. Introduction \n \nOrthoferrite perovskites LnFeO 3 have attracted researchers for decades due to thei r \ninteresting magnetic, electrical and gas sensing pr operties [1-5]. Among orthoferrites, \nparticular interest is focused on doped phases beca use this structural framework can \naccommodate a wide variety of cations on the 12-coo rdinated A-site and 6-coordinated B-\nsite. Such structural flexibility provides large fr eedom to prepare new phases and/or \nmodification of existing materials with new propert ies. These captivating physical properties \nhave encouraged researchers to design various elect rical and chemical sensors [4,5], catalysts \n[6,7], solid oxide fuel cells [8,9] and numerous ot her devices based on utility. Moreover, the \nmagnetic interactions between lanthanide (Ln 3+ -4f) and transition metal (Fe 3+ -3d) in these \nperovskites are still an open research topic since they greatly influence the multiferroic \nbehavior [2,10-13]. However,  the 3d–4f coupling is difficult to investigate expe rimentally \nsince it is generally much weaker compared to 3d–3d  interaction in most perovskite \nstructures [14]. In order to realize the 3d-4f coup ling at the cost of the 3d-3d coupling in \northoferrites, the iron cation has been partially r eplaced by other magnetic cation on the B \nsites. Therefore, the B-site doping effect in LnFe 0.5 Co 0.5 O3 orthoferrites have been recently \ninvestigated for their interesting physical propert ies, and these materials have also been found \nto be promising materials for multiferroics [12,13, 15,16]. The first member of the series i.e. \nLaFe 0.5 Co 0.5 O3 prepared by soft chemical synthesis, crystallizes in orthorhombic structure \n(Pbnm space group) and it is a canted antiferromagn et with T N~ 370 K [15], which is higher \nthan the solid state synthesized sample [16]. There fore, we adopted the similar route of \nsynthesis and reported a unique nano-scale ordered layered phase for SmFe 0.5 Co 0.5 O3 [13], \nwhich also exhibits antiferromagnetism around room temperature (T N~ 310 K). Nevertheless, \nlatter perovskite depicts intrinsic magneto-dielect ric coupling at low temperature. Recently, \nLohr et al [12] have reported the synthesis and phy sical properties for LnFe 0.5 Co 0.5 O3 3 \n perovskites with Ln = Tb, Dy, Ho, Er and Tm. These compounds exhibit antiferromagnetic \ntransition (T N) around the temperature range of 254-249K and ther e is an appearance of \nelectric polarization in the lower size lanthanide (TmFe 0.5 Co 0.5 O3).  It is worth mentioning \nthat with decreasing the lanthanide cation T N value decreases, which also supported by the \ndata from our group [13,15]. Nevertheless, the tran sition temperature T N is almost 50 K lower \nfor LnFe 0.5 Co 0.5 O3 series and also the temperature variation with siz e is significantly weaker \ncompared to La and Sm-phases [13,15]. Therefore, we  considered it important to investigate \nthe missing member i.e. EuFe 0.5 Co 0.5 O3 in the present series along with the reported \nDyFe 0.5 Co 0.5 O3 compound, which shows higher T N (~300 K) in the present investigation. \nDespite the interest in doped orthoferrites, their synthesis in pure phase remains a challenge, \nprincipally because high synthesis temperatures gen erally required for preparing perovskite \northoferrites. Also, entropy plays a dominant role in determining the pure phase and stable \nstructure of reaction products. Previous investigat ions on the doped orthoferrites have \nmotivated us to synthesize doped perovskite with sm aller size of lanthanides in similar \nconditions [13,15]. Therefore, we have utilized sol -gel route synthesis to avoid higher \ntemperature as it has explicit inherent advantages over the solid-state reaction synthesis \nmethods. This controls the reaction pathways on a m olecular level during the transformation \nof the precursor materials to the final product. Bo th the EuFe 0.5 Co 0.5 O3 and DyFe 0.5 Co 0.5 O3 \nperovskites were prepared below 1000 °C and detaile d variable temperature magnetic \nmeasurements were performed. The results of these i nvestigations include the unusual and \ncomplex magnetic behavior of DyFe 0.5 Co 0.5 O3 with a T N of 300 K, which is much higher than \nthe reported value i.e. 252 K [12]. Likewise, ortho ferrite EuFe 0.5 Co 0.5 O3 also exhibits \nantiferromagnetism around room temperature. In the case of orthoferrites the two magnetic \n(Ln and Fe-Co) sublattices generally ordered magnet ically at different temperatures and show \ninteresting phenomena, which have been investigated  in details. 4 \n 2. Experimental procedure \nSoft chemical route method sol–gel was used for syn thesizing LnFe 0.5 Co 0.5 O3 (Ln = \nEu and Dy) orthoferrites. The metal nitrates used i n the synthesis were dissolved in distilled \nwater along with citric acid (molar ration 1:2). Th e resulting solutions were heated for 3 \nhours at 60 ºC and evaporated at 100 ºC for gel for mation. The gel mixture was dried at 150 \nºC for 12 h. Thereafter the resulting powders were allowed to decompose in air at 250 ºC for \n24 h. The resulting powders sintered at several tem peratures ranging from 400 °C to 800 °C \nwith intermediate grinding. The powder samples were  again ground thoroughly and pressed \ninto pellets, and finally sintered for 48 h in air at 927 ºC (for EuFe 0.5 Co 0.5 O3) and 950 ºC (for \nDyFe 0.5 Co 0.5 O3) respectively. The sintered pellets were used for various measurements and \nground again to form fine powder for recording XRD patterns in the 2 θ range of 10°-100° \nwith a step size of 0.02°.  The powder X-ray diffra ction (PXRD) data were recorded with a \nPhilips X-Pert diffractometer employing Cu-K α radiation (λ = 1.5418Å). The phases were \nindexed by performing Rietveld  method using the FullProf program [17]. The transm ission \nelectron microscopy (TEM) including electron diffra ction (ED) and high resolution TEM \n(HRTEM) experiments were performed using aberration  double-corrected JEM ARM 200F \nmicroscope operated at 200 kV equipped with CENTURI O EDX detector. TEM samples \nwere prepared by dropping the colloidal solutions o f powders in methanol onto a holey Cu \ncarbon grid followed by the evaporation of the solv ent. The field and temperature dependence \nof magnetization under various conditions was inves tigated using a physical property \nmeasurement system (PPMS, Quantum Design) and the s ystem was cleaned after each \nmeasurement for remanent field. 5 \n 3. Results and discussion \nThe powder X-ray diffraction (PXRD) patterns of EuF e 0.5 Co 0.5 O3 and DyFe 0.5 Co 0.5 O3 \ncould be indexed with the orthorhombic symmetry in the Pnma  space group, as concluded \nfrom Rietveld refinement. The refined structural pa rameters of samples based on XRD \npatterns are shown in Fig. 1 and the corresponding lattice parameters are presented in Table-\n1. The difference in the values of lattice paramete rs of two samples could be attributed to the \nsize differences between the rare earth cations. \nFrom PXRD study, we obtained information about the crystal structure however the \ndistribution of different phases, crystallinity of grains etc., cannot be determined from it. In \norder to determine those factors, Transmission Elec tron Microscopy (TEM) investigations \nhave been carried out. The electron diffraction (ED ) patterns (Fig. 2 & Fig. 3a) confirm the \ncrystalline nature and orthorhombic Pnma  structures of both the samples. Both mixed ferrite -\ncobaltites show similar size of grains, which are c learly visible in the grain size distribution at \ndifferent scale (Fig. 2 & Fig. 3b).  High angle ann ular dark field scanning TEM (HAADF-\nSTEM) was acquired in parallel with the energy-disp ersive X-ray spectroscopy (EDX) \nmapping (Fig. 3c) shows that the distribution of th e Dy, Fe and Co elements is uniform with \nan atomic ratio close to nominal composition.  High  resolution TEM (HRTEM) imaging (Fig \n3d) confirms good crystallinity of the grains, whic h are free from any defects. \nFig. 4 shows the temperature dependent magnetizatio n, M(T), for EuFe 0.5 Co 0.5 O3 in the \nzero field cooled (ZFC), field cooled cooling (FCC)  and field cooled warming (FCW) \nconditions in the fields of 50 Oe and 1000 Oe. The ZFC-FCW (FCC) data show large \nthermomagnetic irreversibility below 300 K with a c lear signature of magnetic transition \n(TN~300 K) and finally they merge above room temperatu re. The ZFC data exhibits a peak \naround the transition point (~300 K). The FC-magnet ization value increases gradually with 6 \n decreasing temperature and depicts a sharp transiti on at T N. The maximum value of magnetic \nmoment is merely 0.019 µ B/f.u. (Fig. 4b), signifying an antiferromagnetic (A FM) type \ninteraction between the Fe 3+ and Co 3+  cations similar to the reported LnFe 0.5 Co 0.5 O3 mixed \nferrite-cobaltites [12]. \nIn order to correlate the effect of Dy 3+  cation (considering the contribution of 4f electro ns) \non magnetism, we have also carried out magnetizatio n, M(T), investigation for \nDyFe 0.5 Co 0.5 O3 mixed ferrite-cobaltites. Fig. 5 shows the temperat ure dependent ZFC, FCC \nand FCW magnetization for DyFe 0.5 Co 0.5 O3 in the applied fields of H = 50 Oe and 1000 Oe, \nthe curves exhibit a broad magnetic transition near  the room temperature akin to \nEuFe 0.5 Co 0.5 O3 perovskite. The M(T) data for DyFe 0.5 Co 0.5 O3 perovskite exhibits various \nmagnetic transitions down to lowest temperature in the low applied field (Fig.5a) and \ncorroborate the behavior of higher field as reporte d earlier [12]. The 50 Oe FCC/FCW \nmagnetization value increases rapidly below T N (~ 300 K) and a sharp drop in magnetization \nappears around 90 K. Further decreasing temperature  (< 70 K) the moment increases and \nshows broad valley around 60 K. We have also notice d the thermal hysteresis below T N \nbetween the FCC and FCW data (Fig. 5a) with some un usual behavior as far as the thermal \nhysteresis is concerned. They exhibit higher value of magnetization for FCC compared to \nFCW data in the thermal hysteresis region (60-300 K ), which could give rise to metastability \nin the system. The higher field ZFC and FCC/FCW dat a (Fig. 5b) show weak thermal \nhysteresis and follow similar behavior in the 10-40 0 K temperature range.  \nThe complicated magnetic behavior of both mixed fer rite-cobaltites in measured \ntemperature range could arise due to the two inequi valent magnetic ordering at low \ntemperature between the cations in different magnet ic sublattices, i.e. 4f electron based Ln \nsublattice and 3d electron based Fe-Co sublattice, which are arranged anti-parallelly [11,12]. \nMoreover, the AFM ordering of Fe-Co sublattice conf orms with the AFM behaviour of Fe 3+ -7 \n O-Co 3+ , Fe 3+ -O-Fe 3+  and Co 3+ -O-Co 3+  interactions [12,13,15]. The magnetic transition n ear \n300 K for both perovskites may be due to canting of  the antiferromagnetically aligned \nmagnetic spins (which is well known for orthoferrit es). This leads to weak ferromagnetic \nstate related to non-collinear antiferromagnetism i n the Fe sublattice similar to \nDzyaloshinskii-Moriya interaction [2,18]. However, Ln-sublattice order \nantiferromagnetically at relatively low temperature s [11,12]. Here, it is important to mention \nthat the T N for present DyFe 0.5 Co 0.5 O3 compound is much higher than the reported value of \n252 K [12]. This is because the synthesis methods p lay a vital role in determining the crystal \nstructure and physical properties of the same compo und, which we have reported earlier for \nmixed ferrite-cobaltites LaFe 0.5 Co 0.5 O3 [15].  \nFor understanding the exact behaviour of magnetic i nteractions below room \ntemperature, the field dependent isothermal magneti zation needs to be investigated at \ndifferent temperatures. Fig. 6 exhibits the field d ependent magnetization, M(H), at 10 K and \n100 K for EuFe 0.5 Co 0.5 O3. The sample depicts hysteresis loop with low reman ent \nmagnetization (0.01 µ B/f.u.) and large coercivity (4.2 kOe) at 10 K. Impo rtantly, the \nsaturation in the magnetization value is not attain ed in the applied field and the graph \nindicates almost linear behaviour with increasing f ield. The highest obtained magnetization \nvalue is only 0.13 µ B/f.u. for H = 50 kOe. Nevertheless, the M(H) behavi or at 100 K exhibits \nhigher values of remnant (0.013 µ B/f.u.) and  coercivity (6.5 kOe) compared to 10 K. This \nmay be the consequence of large domain wall pinning  of magnetic spins at 100 K. Thus, the \nmagnetic hysteresis loops indicate that EuFe 0.5 Co 0.5 O3 is a canted antiferromagnet in which \nthe canting of magnetic spins results in uncompensa ted magnetic moment, leading to a weak \nferromagnetic behavior similar to reported LnFe 0.5 Co 0.5 O3 mixed ferrite-cobaltites [12,13,15]. \nWe have also compared the isothermal magnetization,  M(H), for DyFe 0.5 Co 0.5 O3 perovskite \nunder three different temperatures (Fig. 7) to corr elate its AFM behaviour with 8 \n EuFe 0.5 Co 0.5 O3 perovskite. In contrast to latter perovskite, the M(H) curves exemplify weak \nhysteresis at 100 K (M r~ 0.011 µ B/f.u. and H c ~ 520 Oe) and 80 K (H c ~ 320 Oe). However, \nthere is no hysteresis at 10 K (inset Fig. 7a). The  M(H) data show linear field dependency at \nhigher field, without exhibiting magnetic saturatio n (the maximum magnetization value being \n2 µB/f.u. at 10 K). The weak hysteresis below T N suggests that the DyFe 0.5 Co 0.5O3 perovskite \nalso manifests canted AFM type interactions among t he cations similar to EuFe 0.5 Co 0.5 O3 \nperovskite, which is clearly in accordance with the  previously reported results [11-13,15,16]. \nImportantly, the H C values are higher for Eu-phase compared to Dy-phas e, which could be \ndue the AFM contribution of Dy 3+  cations at low temperature [18] (ground state of E u 3+  to be \nnon-magnetic[14]). Both the perovskites exhibit a w eak ferromagnetic behavior below the \nroom temperature, arising from a canting of the AFM  spins. The DM interaction could be the \nbasis of weak ferromagnetism in these mixed ferrite -cobaltites, which has been well \nestablished for perovskite orthoferrites [1-3,11-13 ].  \n4. Conclusions \nIn this work on Ln-Fe-Co-O systems (Ln = Eu & Dy), the sol-gel synthesis method \nenables the orthorhombic structure to be obtained a s a pure phase. The magnetic properties \ninvestigation of these oxides show that both the co mpositions exhibit near room temperature \nmagnetic transitions. The canted antiferromagnetic state is realized at low temperature for \nboth the perovskites, which is most likely due to t he ordering of 3d and 4f sublattices as \nreported in the literature for Ln-Fe-Co-O systems [ 12]. We hope our finding will motivate the \nresearcher to obtain room temperature magneto-elect ric material, which is of current interest \nfor future technological applications. \nAcknowledgements \nAKK and MMS thank the Science and Engineering Resea rch Board (SERB), India for \nfinancial support through the project grant # EMR/2 016/000083  and Prof. B. Raveau for his 9 \n valuable comments and suggestions. We would also li ke to thank the reviewers for their \nconstructive guidance during evaluation process.  \nReferences: \n  \n1.  M. Eibschutz, S. Shtrikman, D. 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Pomiro, R. D. Sánchez, V. Pomjak ushin, G. Aurelio, A. Maignan, C. \nMartin, R. E. Carbonio, Spin reorientation and meta magnetic transitions in RFe 0.5 Cr 0.5 O3 \nperovskites (R = Tb, Dy, Ho, Er), Phys. Rev. B 98 ( 2018) 134417. \n12.  J. Lohr, F. Pomiro, V. Pomjakushin, J. A. Alonso, R . E. Carbonio, R. D. Sánchez, \nMultiferroic properties of RFe 0.5 Co 0.5 O3 with R = Tm, Er, Ho, Dy, and Tb, Phys. Rev. B \n98 (2018) 134405. \n13.  A. Shukla, A. Singh, M. M. Seikh, A. K. Kundu, Low temperature Magneto-dielectric \ncoupling in nanoscale layered SmFe 0.5 Co 0.5 O3 perovskite, J. Phys. Chem. Solids 127 \n(2019) 164.  \n14.  A. K. Kundu, V. Hardy, V. Caignaert, B. Raveau, Int erplay between 3d–3d and 3d–4f \ninteractions at the origin of the magnetic ordering  in the Ba 2LnFeO 5 oxides, J. Phys.: \nCondens. Matter 27 (2015) 486001. \n15.  V. Solanki, S. Das, S. Kumar, M. M. Seikh, B. Ravea u, A. K. Kundu , Crucial role of \nsol–gel synthesis in the structural and magnetic pr operties of LaFe 0.5 (Co/Ni) 0.5 O3 \nperovskites, J. Sol-Gel Sci. Technol. 82 (2017) 536 .  \n16.  D. V. Karpinsky, I. O. Troyanchuk, H. Szymczak, M. Tovar, others, Crystal structure and \nmagnetic ordering of the LaCo 1-xFe xO3 system, J. Phys. Condens. Matter. 17 (2005) 7219.   \n17.  J. Rodriguez-Carvajal, An introduction to the progr am FullProf 2000, Lab. Leon \nBrillouin, CEA-CNRS Saclay, Fr. (2001).  \n18.  A. Stroppa, M. Marsman, G. Kresse, S. Picozzi, The multiferroic phase of DyFeO3: an ab \ninitio study, New J. Phys. 12  (2010) 093026. \n \n \n \n \n \n \n \n \n \n \n \n 11 \n Table. 1. Lattice parameters of LnFe 0.5 Co 0.5 O3 (Ln = Eu, Dy) perovskites. Here a, b, c are the lattice parameters, \nRb and R f are the Bragg factor and fit factor, respectively.  \n \nPerovskite EuFe 0.5 Co 0.5 O3 DyFe 0.5 Co 0.5 O3 \nSpace group Pnma  Pnma  \na (Å) 5.488(4) 5.505(6) \nb (Å) 7.577(5) 7.504(6) \nc (Å) 5.315(3) 5.239(4) \nV (Å 3) 221.074 216.421 \nRb(%) 13.6 11.3 \nRf(%) 32.7     19.9 \nChi-factor 3.95 2.87 \n \n \n \n \n \n \n \n \n \n 12 \n -10000 010000 20000 30000 40000 50000 60000 70000 \n20 40 60 80 100 010000 20000 30000 40000 50000 60000 \n2θ (a)EuFe 0.5 Co 0.5 O3 Intensity (arb. unit) \n (b)DyFe 0.5 Co 0.5 O3 \n \n \nFig.1.  Rietveld analysed XRD pattern for (a) EuFe 0.5 Co 0.5 O3 and (b) DyFe 0.5 Co 0.5 O3. Open symbols are \nexperimental data and the solid and vertical lines represent the difference curve and Bragg positions \nrespectively.  \n 13 \n \n \n \nFig.2. ED patterns along two main zone axis of EuFe 0.5 Co 0.5 O3 sample indexed based on Pnma  structure \ndetermine by PXRD (Table 1). Two representative low  magnification bright field TEM images of grains sh ape \nand size distribution are presented in lower panel.  \n 14 \n \n \n \nFig.3.  (a) ED patterns along main zone axis [010], [001] and [101]. All ED patterns indexed based on Pnma  \nstructure and parameters determined by PXRD (Table 1); (b) Low magnification bright field TEM image of  \nDyFe 0.5 Co 0.5 O3 sample; (c) HAADF-STEM image and parallel acquired  EDX mapping of Co K (purple), Fe K \n(yellow) and Dy L (light blue) elements. (d) HRTEM image of DyFe 0.5 Co 0.5 O3 grain viewed along [101] \ndirection.  \n 15 \n 0 50 100 150 200 250 300 350 400 0.000 0.005 0.010 0.015 0.020 0.000 0.004 0.008 0.012 0.016  \n (b) H = 1000 Oe \nT(K)  (a) H = 50 Oe \n ZFC \n FCC \n FCW M(µB/f.u. ) \n  \n EuFe 0.5 Co 0.5 O3\n \nFig.4. Temperature dependent ZFC (open symbol), FCC(half solid) and FCW (solid symbol) magnetization, M, \nplot for EuFe 0.5 Co 0.5 O3 in the applied fields of (a) H=50 Oe and (b) H=100 0 Oe.  \n \n0.000 0.004 0.008 0.012 0.016 \n0 50 100150200250300350400 0.00 0.04 0.08 0.12 0.16 \nT(K) FCC \nFCW  \n  \n DyFe 0.5 Co 0.5 O3(a) H = 50 Oe \nZFC \n M(µB/f.u.)\n(b) H = 1000 Oe \n \n   ZFC \n FCC \n FCW \n \nFig.5. Temperature dependent ZFC (open symbol), FCC(half s olid) and FCW (solid symbol) magnetization, M, \nplot for DyFe 0.5 Co 0.5 O3 in the applied fields of (a) H=50 Oe and (b) H=100 0 Oe. \n 16 \n -50 -40 -30 -20 -10 0 10 20 30 40 50 -0.10 -0.05 0.00 0.05 0.10 \nT=100K  \n  \n EuFe 0.5 Co 0.5 O3\nH(kOe) M(µB/f.u.)T=10K \n \nFig.6. Field dependent isothermal magnetic hysteresis, M( H), loops at two different temperatures for \nEuFe 0.5 Co 0.5 O3. \n \n-20 -10 0 10 20 -2 -1 012\n-3000 -1500 0 1500 3000 \n-0.2 -0.1 0.0 0.1 0.2  \n  \n H(kOe) M(µB/f.u.)DyFe 0.5 Co 0.5 O3\n 10 K \n 80 K \n 100K \nM(µB/f.u. )\nH(Oe) \n \n \nFig.7. Field dependent isothermal magnetic hysteresis, M( H), loops at three different temperatures for \nDyFe 0.5 Co 0.5 O3. \n "    },    {        "title": "1505.01200v1.Asymmetric_Band_Diagrams_in_Photonic_Crystals_with_a_Spontaneous_Nonreciprocal_Response.pdf",        "content": "Asymmetric Ba nd Diagrams in Photonic Crystals with a \nSpontaneous Nonr eciprocal Response  \nFilipa R. Prudêncio (1),  , Sérgio A. Matos(2) and Carlos R. Paiva(1) \n \n1Department of Electrical and Computer Engineering – Instituto de Telecomunicações,  \nInstituto Su perior Técnico – University of Lisbon  \nAvenida Rovisco Pais, 1, 1049 -001 Lisboa, Portugal  \n2Department of Information Science and Technology – Instituto de Telecomunicações  \nInstituto Universitário de Lisboa – ISCTE  \nAvenida das Forças Armadas, 1649 -026 Lisboa , Portugal  \n \nAbstract  \nWe study  the propagation of electromagnetic waves in  layered photonic crystals \nformed by materials  with a spontaneous nonreciprocal response , such as  Tellegen \n(axion)  media  or topological insulators . Surprisingly , it is prove n that stratified \nTellegen photonic crystals that break simultaneously the space inversion  and time \nreversal symmetries have  always  symmetric dispersion  diagrams . Interestingly, we \nshow  that by combining chiral and nonreciprocal materials the photonic band \ndiagrams  can exhibit  a spectral asymmetry such that \n kk . Furthermore, it \nis demonstrated that in some conditions two juxtaposed  Tellegen medium layers have \nan electromagnetic response analogous to that of a biased ferrite  slab. \n \n \n                                                 \n Corresponding author: filipa.prudencio@lx.it.pt   I. Introductio n \nPhotonic crystals are inspired by the geometry of crystalline natural materials, \nsuch as  semiconductor crystals [1], [2]. Periodic layered dielectric structures are  of \nparticular importance in the realization of filters, resonant cavities, and light sources \n[3], [4]. The propagation of light in crystals with nonreciprocal media has also \nreceived considerable attention in the literature [5]-[7]. In particular, recent works \nhave demonstrated how by exploring interference phenomena in different  magneto -\noptical photonic crystalline structures it is possible to boost the nonreciprocal \nresponse and achieve optical isolation, optical switching, one -way extraordinary \noptical transmission, and a giant Faraday rotation [5]-[17]. These efforts are partly \nmotivated by the contemporary interest in on-chip miniaturi zation of all -optical \ncircuits, which requires  on-chip optical signal isolation . \nNotably, having a nonreciprocal response and asymmetric power flows i s far \nfrom trivial in photonics. Indeed, the Lorentz reciprocity theorem establishes that a \ntwo-port network formed by conventional dielectrics and metals is intrinsically bi -\ndirectional, such that the roles of the source and load can be interchanged witho ut \naffecting the amount of power transmitted through the system. This contrasts with \nelectronics wherein the rectification functionality is provided by diodes and \ntransistors, which effectively behave as unidirectional couplers for electrons. A \nnonreciproc al response can be obtained with an external  bias magnetic field (e.g.  \nusing ferrimagnetic materials such as  ferrites  at microwaves [5] or with bismuth iron \ngarnet at optics [ 8]) with the temporal refractive -index modulation [ 9], or by using \noptomechanical effects (e.g. with moving media [ 10]). Even though the first solution \nis quite established, the requirement of an external biasing is inconvenient because it \nlimits the range of applications and because the ass ociated biasing circuit may be bulky.  In general, devices made of nonreciprocal media can allow  for highly \nasymmetric transmissions [5]-[7]. A related effect can also  occur in more restricted \nconditions in chiral structures [18]-[19]. Moreover, photonic crystals made of chiral \nmedia may be useful for polarization state conversion [18], [19].  \nTellegen media are a subclass of bi -isotropic media with magnetoelectric \ncoupling [20], [21]. The key characteristic of Tellegen media is the combination of an \nisotropic response with a spontaneous  nonreciprocal magnetoelectric effect which  \n[20], does not  require any external biasing . This idea was put fo rward by Tellegen in \n1948, in connection with his proposal of a new nonreciprocal circuit element [22]. He \nsuggested that a mixture of randomly distributed particles with glued permanent \nelectric and magnetic di pole moments may have a  magnetoelectric isotropic response. \nSome time ago an artificial Tellegen medium was implemented using essentially this \nrecipe [23]. Notably , the physics of the Tellegen medium is a partic ular form of axion \nelectrodynamics  [24]. Axions were originally introduced by F. Wilczek as an attempt \nto explain the missing dark matter of the universe [25]. As not ed by several  authors, \nthe magnetoelectric coupling due to a axion field with a time independent axion -\ncoupling term  is equivalent to the Tellegen constitutive relations  [24], [26]-[28]. For \ncompleteness , we provide an explicit proof of this result in the Appendix A . \nThere are  natural ly available  media with a spontaneous  nonreciproc al \nmagnetoelectric response , being the most prominent example  chromium oxide Cr2O3 \n[29]. It was experimentally verified that the spontaneous magnetoelectric effect in \nCr2O3 creates a polarization rotation in the  reflected wave, which is the fingerprint of \na nonreciprocal magnetoelectric coupling [30]. In the last decade there has been a \nresurge nce in the interest in materials with a strong magnetoelectric response [31]-\n[34]. In particular, it has been suggested that the new class of crystalline solids known as electronic  topological insulators [32], [33], may be characterized by axion -type \nelectrodynamics, which as previously discussed is equivalent to the Tellegen medium \nresponse [28]. In most topol ogical insulators an observation of the magnetoelectric \neffect may require adding a time -reversal breaking perturbation, e.g. a permanent \nmagnet [32], i.e. a biasing element. Interestingly, it has been suggested  that \nantiferromagnetic topological insulators can enable a spontaneous magnetoelectric \neffect, similar to Cr2O3 [34] . Moreover, in [31] it was theoretically demonstr ated that \nthere are several crystal structures and corresponding arrangements of magnetic \nmoments compatible with a purely isotropic linear spontaneous nonreciprocal \nmagnetoelectric coupling. Therefore, it is relevant to explore possible applications of \nmaterials with an isotropic spontaneous nonreciprocal response  in the context of  \nelectromagnetic  wave propagation.  It is important  to make clear that the topological \nmagnetoelectric effect discussed above is totally unrelated to “photonic topological \ninsulat ors”, which have received significant attention in the recent literature [35]-[36]. \nThe topological magnetoelectric effect is a consequence of the nontrivial topologi cal \nnature of the electronic  band structure of a bulk solid -state material [33]. Moreover, in \nthis article we do not attempt to characterize the topology of the band structure of \nTellegen photonic crystals.  \nMoti vated by the above discussed developments  in material science , here we \ninvestigate the consequences  of a spontaneous nonreciprocal response in stratified \nperiodic structures. The isotrop y of a Tellegen medium  implies that  the dispersion of \nthe photonic sta tes is doubly degenerate and symmetric , so that  \n kk . Thus, \nin a certain sense the nonreciprocal response is hidden. Only in composite structures \nformed by different materials (e.g. a photonic crystal) it is possible to unveil the \nnonreciprocal character of the Tellegen media. In this work , we want to explore under which conditions it is possible to have asymmetric band diagrams in  photonic crystal s \nwith Tellegen media and which physical symmetries of the photonic crystal need to \nbe broken to achieve this.  \nSurprisingly, it is found  that the band structure of layered Tellegen photonic \ncrystals is always symmetric for propagation along the stratification  direction , \nnotwithstanding  the space inversion and time reversal symmetries are bro ken. We \nexplain this result by identifying another symmetry transformation that protects the \nspectral symmetry of the band diagrams. Importantly , it is shown  that by putting \ntogether chiral media and Tellegen media, it is possible to obtain asymmetric band  \ndiagrams,  \n kk , and highly asymmetric group velocities. Furthermore, we \nprove that in some conditions the role of two Tellegen layers in the photonic crystal is \nessentially equivalent to  a biased ferrite layer. We would like to  highlight that other \nauthors  have shown that a  proper arrangement of ferrites and anisotropic dielectrics \nmay yield asymmetric band diagrams [5]-[6]. However, here we demonstrate  for the \nfirst time that the same result can be obtained without a biasing static field  based on \nmedia with a spontaneous nonreciprocal response.  \nThis article is organized as follows. In Sect . II we star t by characterizing  the \nnonreciprocal effects in  the scattering by Tellegen stratified structures using  the \nscattering matrix formalism [37], [38]. In Sect . III we present a similar study for the \ncase of stratif ied ferrite structures. In Sect . IV we apply the developed formalism to \ndetermine the band structures of Tellegen and fer rite photonic crystals. Finally, in \nSect. V the conclusions  are drawn .  II. Stratified Structures  with Bi -isotropic Media  \nAs a starting point, we characterize t he scattering of waves by stratified \nstructures formed by bi -isotropic media. The geometry of the structure is represented \nin the Fig. 1a wherein each slab is either a  Tellegen material , or a chiral material  or a \nconventional  dielectric. The stratification direc tion is assumed to be the z direction \nand the slabs have thicknesses \n12, ,...,N d d d . The longitudinal propagation constants \n(along the z-direction) are denoted by \n12, ,...,N    . The input and output regions a re \nassumed to be  the vacuum or air.  \n \nFig. 1 (Color online) The system represented in panel (a) is equivalent to the four-port network  \nshown  in panel  (b). Even though there are only two physical channels, because of the polarization \ndegrees of fre edom the equivalent network is formed by four ports.  \nThe electromagnetic response of b i-isotropic media is determined by the well -\nknown constitutive relations  [20]: \n 11\n00 ,         c i c i            D E H B H E\n,  (1) \nwhere \n  is the relative permittivity, \n  is the relative permeability, \n0  and \n0  are the \nvacuum permittivity and permeability , and \n00 1c  is the speed of light  in \nvacuum . The plane waves  in an unbounded  bi-isotropic medium can be decomposed  \ninto two independent wave field components  corresponding to two circularly \npolarized electromagnetic waves . Specifically, the electric field and the magnetic fields \nE and \nH  can be written in the forms \nE E E  and \nH H H , \nrespectively , where the “+”  (“-”) signal is associated with a  right  (left)  circularly \npolarized wave (RCP /LCP ). The corresponding wave  number s are  \n, 0 ,k k n    , \nwhere \n2\n,n      are the refractive indices . The chirality and Tellegen \nparameters, \n  and \n , are resp onsible for the ma gnetoelectric coupling of bi -isotropic \nmedia.  \nA. The s cattering problem  \nWe consider first  that a plane wave propagates in  the air region ( e.g. \n0z ) \nand illuminates the  stratified  structure . The transverse electric an d magnetic fields that \npropagate in a generic Tellegen  or dielectric  medium slab are given by (the variation \nof the fields along the x and y directions , if any,  is omitted)  \n T,         i z i z\nxy z e E E z e           E H Y E\n,  (2) \nwhere “T” represent s the transpose of the vector , and the superscripts \n  indicate if \nthe wave propagates along the \n  or \n  z-direction . The characteristic impedance  \nmatrices  are defined such that \n zz   E Z H  and \n   E Z H  and are given \nby \n22\n0 0\n22 2 2 2\n0 0x y y\nx x yk k k k\nk k k k nk    \n        Z\n.   (3) \nThe admittance matrices are the inverse of the impedance matrices \n1YZ  and \n1YZ\n. The longitudinal propagation constant in the pertinent slab  is \n2 2 2 2\n0 xy k n k k  \n where  \n2n   is the refractive index , \n0 / kc , and ,xykk are the transverse wave  numbers det ermined by the incidence angle. For a \nconventional dielectric one should use \n0  in Eq.  (3). \nOn the other hand, for a chiral slab the propagation constants depend on the  \nwave  polarization  state and thus the formulas for the transverse fields are more \nintricate. To avoid complicating excessively the formalism, we only show the \nformulas for the particular case of normal incidence  (\n0xykk ), which is the focus \nof our study . The f ields in a chiral slab for \n0xykk  can be written as:  \n\n i z i z\nPM\ni z i z\nPMz e E e E\niz e E e E\n\n\n  \n\n  \n\n E e e\nH e e\n,  \n\n i z i z\nPM\ni z i z\nPMz e E e E\niz e E e E\n\n\n   \n\n   \n\nE e e\nH e e  (4) \nwhere  \n00      is the  chiral medium wave  impedance. For a chiral slab th e \nlongitudinal wave  numbers satisfy \n0kn  with the refractive indices \nn \n. A chiral medium is thus a birefringent medium with two distinct \nrefractive indices . For propagation along the +z-direction the normaliz ed wave fields  \nmay be taken equal to \nˆˆie x y  and \nˆˆie x y , and  propagate with the exponential \nfactors \nize  and \nize , respectively. Note that for propagation along the –z direction \nthe plane waves associated with the same eigenstates \ne  and \ne have the propagation \nfactors \nize  and \nize , respectively.  \nThe complex valued field ampli tudes in each slab \n,PMEE  are linked at the \ninterfaces (\ni zz ) through the usual continuity equations:  \n\n11\n11i i i i i i i i\ni i i i i i i iz z z z\nz z z z   \n\n   \n  \n  E E E E\nH H H H\n,  \n0,...,iN   (5) where \n0 1 1 2 1 20, ,z z d z d d    , etc. Thus, if the amplitudes of the incident waves \nare known (\n0E  in case of incidence from the left, and  \n1N\nE  in case of incidence from \nthe right), the remaining field amplitudes can be easil y determined by solving a \nstandard linear system. For incidence from the left (with \n10N\n E ), we define the \nreflection and transmission matrices, \nR  and \nT , through  the formulas \n0 0 0 0z z z z    E R E\n and \n1 0 0NNz z z z\n    E T E . The \nR  and \nT  \nmatrices may be written explicitly as  \n11 12 11 12\n21 22 21 22,R R T T\nR R T T      \n   RT\n,    (6) \nwhere the coefficients \nijR  and \nijT  are complex valued numbers.  \nB. Scattering matrix  \nThe wave scattering in stratified media can be conveniently studied using t he \ntheory of microwave networks  [38]. In this context, eac h channel of propagation is \nassociated with a port. For normal incidence there are two physical channels, one \nassociated with the propagation in the left -hand side air region and another with the \npropagation in the right -hand side air region ( Fig. 1a). However, because there are \ntwo allowed polarization states the system is equivalent to a four -port network, as \nshown in Fig. 1b. \nThe scattering matrix provides a  complete description of the network response \nand relates the incident and scattered waves, \nincE  and \nscattE , respectively, as \nscatt scatt inc E S E\n. In our case we have \n  0, 0, 1, 1,T\ninc x y N x N y E E E E   \n E  and \n  0, 0, 1, 1,T\nscatt x y N x N y E E E E   \n  E\n, where the transverse fields 0 0 1 ,NN z z z z\n EE are defined as in the previous subsection. Thus, the \nscattering (four -by-four) matrix for normal incidence is such that:  \nLR\nscatt LR\nRTS\nTR\n,     (7) \nwhere \n,LLRT  are the reflection and the transmission matrices for an incident wave \npropagating from the left to the right (L -R). Similarly, \n,RRRT  are the reflection and \nthe transmission matrices for an incident wave propagating from the right to the left \n(R-L). The superscripts “L” and “R” indicate from which side (left or right) the \nincident wave comes from. The matrices \n,LLRT  are calculated as detailed in the \nprevious subsection  and \n,RRRT  can be found in a similar manner.  \nA microwave network is reciprocal if and only if the correspond ing scattering \nmatrix satisfies  [38]: \nT\nscatt scatt SS\n,    (8) \nwhere \nT\nscattS  is the transpose of \nscattS . For lossless media, the concept of \nnonreciprocity is strictly linked with the breaking of the time -reversal symmetry in \nelectrodynamics.  Thus, the network associated wi th the scattering matrix  defined in \nEq. (7) is reciprocal if  \nT T T, ,   L L R R L R  R R R R T T\n.   (9) \nIt is useful to note that because all the slabs are isotropic and invariant to \nrotations about the z-axis the response of the structure must remain qualitatively the \nsame, independent of the direction of the incoming electric field. This implies that for \nnormal incidence all the reflection and transmission dyadics \n  , , ,L R L RT T R R  are \nnecessarily o f the type  11 12\n12 11aa\naaA.    (10) \nThus, the diagonal elements must be equal and the anti -diagonal elements must be \nadditive inverses . For future reference, it is noted that that the sum of two matrices \n,abAA\n of the form  defined in Eq. (10) is still a matrix of the same type. T he inverse \nof \nA  is also a matrix of the same type. Moreover, two matrices \n,abAA  of the form  \npresented  in Eq. (10) commute, \na b b a  A A A A , and t he product is still a matrix of \nthe same type.  \nC. Eigenstates of the scattering problem  \nThe eigenstates of the scattering problem illustrated in Fig. 1 are circularly \npolarized waves. To show this  we note that the eigenvectors of a matrix \nA  of the type  \npresented in Eq. (10) are \nT1ie , \nT1ie , such that: \na  A e e\n,  \na  A e e ,   (11) \nwith \n11 12 a a i a  and \n11 12 a a i a . Because the reflection and transmission \nmatrices are of the form  defined in Eq. (10), it follows that when the incident field is \ninc inc EEe\n then the corresponding reflected and transmitted waves are \nref incEREe\n and \ntx incETEe , respectively, where \n11 12 T T iT  and \n11 12 R R iR\n. The polarization states \ne  are evidently associated with circularly \npolarized waves. Hence, when \ninc inc EEe , it is seen that the  transmitted wave has \nalways the same polarization s tate as the incident wave, whereas the reflected wave is \nalso circularly polarized but has handedness opposite to that of the incoming wave.  \nWhen the incoming wave propagates from the left to the right (L -R) the \npertinent transmission and reflection matric es are \n,LLTR  and the corresponding eigenvalues are  \n11 12L L LT T iT  and \n11 12L L LR R iR . Similarly, for propagation from \nthe right to the left (R -L) the relevant transmission and reflection matrices are \n,RRTR\n and the corresponding eigenvalues are  \n11 12R R RT T iT  and \n11 12R R RR R iR . \nNote that \ne  and \ne  correspond to right and left circularly polarized waves (RCP and \nLCP) for a wave propagating along  the +z direction. On the other hand, for a wave \npropagating along  the -z direction, \ne  and \ne  are associated with LCP and RCP \nwaves, respectively. In summary, an incident RCP /LCP wave illuminating the \nstratified structure originates a reflected LCP/RCP wave and a transmitted RCP/LCP \nwave.  \nThe reciprocity condition s in Eq.  (9) can be expressed  in terms of \n, RLTT  and \n, LRRR\n. It is straightforward to show that  the conditions  in Eq. (9) are equivalent to:  \nLLRR\n,  \nRRRR , \nLRTT\n .   (12) \nIn other words, for reciprocal systems the reflection coefficients for LCP and RCP \nwaves must be the same for incidences from the same side of the structure. On the \nother hand, the transmission coefficient is required to be same for waves with the  \nsame handedness, and incidence from opposite sides of the structure. Thus, as \nillustrated in Fig. 2, incident RCP/LCP waves coming from the  L-R and R -L \ndirections  must have the same transmission coefficien ts in case of  stratified structures \nformed by conventional dielectrics  and/or chiral media .   \nFig. 2 (Color online)  Symmetric transmission of circularly polarized waves incident on the front \nand back sides of plana r structure s made of reciprocal materials, such as, conventional dielectrics  \nand/or chiral media.  \nD. Stratified Tellegen media \nNext , we consider the particular case wherein all the slabs are either conventional \ndielectrics or Tellegen materials. In Appendix B, it is proven that , independent of the \nnumber of layers  of a stratified Tellegen structure,  the transmission matrix for \nincidence L -R is exactly the same as the transmission matrix for incidence R -L: \nLRTT\n.      (13) \nIn particular, the transmission coefficients for circularly polarized waves satisfy \nRLT T T  \n and \nRLT T T   . \nTo understand the role of the nonreciprocal response of Tellegen m edia in  the \nwave  scattering,  next we present a few  illustrative numerical examples. First, we \nconsider the case wherein a single Tellegen slab stands alone in free -space (this \ncorresponds to \n2... 0N dd    in Fig. 1a). In Fig. 3 we depict  the reflection and \ntransmission coefficients for an incoming circularly polarized wave  for the \nconstitutive parameters \n2 , \n1 , \n0.1  and thickness \nd . The absolute values \nof the transmission and reflection coefficients for a wave propagating  in the directions  \nL-R and R -L, \n,LLRR , \n,RRRR , \nRLT T T    and \nRLT T T    , are depicted in Fig. 3a. The phases of the same scattering parameters  are shown in Fig. 3b. Consistent \nwith the fact that the time -reversal symmetry is broken in Tellegen media, it is found \nthat \nLLRR  and \nRRRR  [see Eq. (12)]. Thus, the nonreciprocal response of the \nstructure may be detected by inspection of the reflection coefficients. As seen  in Fig. \n3, we have \nTT  and thus, despite the nonreciprocal response of the Tellegen \nmaterial, the reciprocity criterion \nLRTT\n  [see Eq. (12)] is still satisfied . In this \nexample, the property \nLRTT\n  is a simple consequence of the “left -right” symmetry \nof the structure. Indeed, we numerically verified that the condition \nLRTT\n  is always \nobserved for spatially symmetric structures of the type Air-T1-T2-T3-(…)-TN-(…)-T3-\nT2-T1-Air, where “T i” corresponds to some Tellegen me dium.  \n \nFig. 3 (Color online)  Reflection and transmission  coefficients for an incoming wave propagating in \nthe L -R and R -L directions  for a Tellegen slab , with thickness \nd , standing in vacu um. (a) Amplitude \nof the reflection and transmission coefficients \n,LLRR  ,\n,RRRR  and \n,TT . (b) Phase of the reflection \nand transmission coefficients \n,LLRR  ,  \n,RRRR  and \n,TT .  \nWe also found out that generally one can have \nT T T  even for \nasymmetric stratified structures (formed by several slabs), provided each and e very \nsubset of three materials  (including the air regions)  have linear dependent constitutive \nparameters, such that:  1 1 1\n2 2 2\n3 3 3det 0  \n  \n  \n.    (14) \nIn such conditions, we say that all the materials are in the same  equivale nce class  \n[39]. It was shown  by us  in Ref. [39] that in this case  the structure is reducible under a \nsuitable duality transformation to a stratified structure formed by conventional \ndielectrics  with \n0 , i.e. the Tellegen parameter can be formally eliminated . In \nAppendix C, we present a brief overview of duality transformations. Thus, because \nthe transmission matrix \nT  in the transformed problem  reduces to a scalar, it follows \nthat it also reduces to the same scalar in the original structure, and hence \n120 T  and \nthus \nT T T . Note that this argument only works f or the transformed transmission \nmatrix and it does not apply to the transformed reflection matrix. The reason is that \nthe transmission coefficient for the electric field (defined as \nT incTEE  ) is the same \nas the transmission coefficient for the magnetic field (defined as \nT incTHH  ), and \nthus the transmitted electric and magnetic fields are transformed in the same manner \nin a scattering problem involving only conventional dielectrics. On the other hand, the \nreflection co efficients for the transverse electric and magnetic fields are the \nsymmetric of one an other. This difference  is of crucial importance  because a duality \ntransformation mixes the electric and magn etic fields  (see Appendix C). \nIn the second example [see Fig. 4], we characterize  the wave scattering by  two \njuxtaposed slabs with thicknesses \n1d  and \n2d  embedded in vacuum. The parameters of \nthe Tellegen media and the parameters of the vacuum are chosen such that the \nstructure is not reducible to conventional dielectrics, i.e., the material parameters do \nnot satisfy Eq. (14). Because of this, it is possible to  simultaneously  break all the \ncriteria in Eq. (12) and detect the nonreciprocal response of the involved materials both in transmission and reflection . The simplest structur e that can reveal  these effects  \ncorresponds to a Tellegen slab juxtaposed to a dielectric slab. This is illustrated in  Fig. \n4a for the case  \n12 0.5 d d d  and for materials with  the constitutive parameters  \n12\n, \n11 , \n10.1  and \n22 , \n21 .  Figure 4 b shows that  by juxtaposing two  \nTellegen slabs with constitutive param eters \n12 , \n11 , \n10.1  and \n22.5 , \n21.5\n, \n20.5 , and with  \n12 0.5 d d d  it is possible to enhance the nonrec iprocal \neffects . Finally, Fig. 4c shows how the strength of the Tellegen parameter \n1  \ndetermines the asymmetric response.  \n \nFig. 4 (Color online)  Amplitude of reflection and transmission coefficients for an incoming wave \npropagating  in the L -R and R -L directions  for stratified structures with Tellegen media.  The total \nthickness is \n12 d d d  with \n12dd . (a) Amplitude of the reflection and transmission coefficients  \n,LLRR\n ,\n,RRRR  and \n,TT  for a Tellegen  slab ju xtaposed to a dielectric  slab. (b) Amplitude of the \nreflection and transmission  coefficients  \n,LLRR  ,\n,RRRR  and \n,TT  for two juxtaposed Tellegen slabs. (c) Transmission coefficients  \n,TT  for the example (a ), but for different values of \n1 . \nIII. Stratified Structures with Ferrites  \nIt is interesting to compare the scattering provided by  media  with a \nspontaneous nonreciprocal response , with the more conventional ferrites, which as \npreviously discussed require an external biasing  and thus have an anisotropic \nresponse . A ferrite  biased with a static magnetic field oriented along the z-direction is  \ncharacterized by the following constitutive relations  [38] \n00,   ff     DΕ B H\n,    (15) \nwhere \nf  is the relative permittivity and \nf  is the relative permeability matrix given \nby \n0\n0\n0 0 1ff\nf f fi\ni\n  \n\n.    (16) \nIn case of negligible material los s, the parameters \n,ff  depend  on frequency as  \n2 2 2 21 ,   R m m\nff\nRR       \n,    (17) \nwhere \nR  is the precession or Larmor  frequency and \nm  is the is the ele ctron Larmor \nfrequency at the saturation magnetization of the ferrite [38].  \nHere, w e are interested in structures with a geometry analogous to that of Fig. \n1a, but  we allow some  of the slabs to be ferrites biased along the z-direction. For \npropagation along the stratification direction  the transverse electric and magnetic \nfields in a generic ferrite  slab can be written as a superposition of circularly polarized \nwaves propagati ng along the positive and negative z-directions :  \n ,          \n,       i z i z i z i z\nP M P M\ni z i z i z i z\nP M P Miiz e E e E z e E e E\niiz e E e E z e E e E   \n   \n   \n        \n   \n\n         \n   \n   \n    \nE e e H e e\nE e e H e e(18) \nwhere the longitudinal wavenumbers satisfy \n0kn  with the refractive indices \nwritten as \nf n  with \nff    and \n00 f      . Similar to the \nprevious section,  the reflection and transmission matrices can be found from the \nsolution of a linear system, and the eigenstates of the scattering problem are  circularly \npolarized waves .  \nIn Fig. 5 we show the absolute values of the reflection and transmission \ncoefficients  for a single ferrite slab embedded in  a vacuum. As seen,  the \nnonreciprocity is detected  in both the reflection  and trans mission  coefficients because  \nLLRR\n, \nRRRR , \nLRTT  and \nRLTT . In our simulations  it was  \nassumed t hat \n4Rdc  and \n2mdc , and, f or simplicity, the relative permittivity \nwas taken equal to  \n1f . \nFig. 5 (Color online)  Amplitude of the transmission and reflection coefficients for a single ferrite slab  \nwith thickness \nd . (a) Amplitude of the transmission coefficients , \n,LLTT , \n,RRTT , for a wave \npropagating in the L -R and R -L directions , respectively . (b) Amplitude of the reflection coefficients , \n,LLRR\n, \n,RRRR , for a wave propagating in the L -R and R -L directions , respectively .   Hence  the nonreciprocity of a single ferrite slab is mani fested  in both  the \nreflected and transmitted waves , different from what  happen s in stratified Tellegen \nstructures with media in the same  equivalence  class [see Sect.  II.D], and in particular \ndifferent from a single Tellegen slab . Thus, we conclude that to mimic the behaviour \nof a single ferr ite slab, we need  at least two juxtaposed Tellegen slabs  (or alternatively \na Tellegen slab an d a standard dielectric)  in such a way that the two Tellegen media \nand the vacuum are not in the same  equivalence  class.  \nIV. Photonic Crystals  with Nonreciprocal  Media \nWith the aim of further exploring the opportunities created by  a spontaneous  \nnonreciprocal response, next we investigate the photonic band diagrams of periodic \nlayered structures formed by Tellegen  media or ferrites combined with  chiral  media  \nand/or conve ntional dielectrics. It is prove n that in some conditions the photonic band \ndiagrams can exhibit  an asymmetry such that \nzzkk . We restrict our \nattention to Bloch modes with \n0xykk , which corresponds to propagation  along \nthe direction of stratification.  \nA. Dispersion equations for the Bloch waves  \nPeriodic structures may be seen as a cascade of identical multi -port networks. In \nparticular, in case of normal incidence a stratified (layered) structure [ see Fig. 6] is \nequivalent to a 4 -port network, as discussed previously. Thus, each unit cell, \nindependent of its complexity or of the involved materials, is fully characterized by \nthe reflection and transmission matrices for L -R and R -L incidences, i.e., \n,LLRT  \nand \n,RRRT  respectively. In particular, the electric fields at the front and back \ninterfaces of the n-th interface are related by:  1 1 1,        n L n R n n L n R n\nscatt inc inc scatt inc inc          E R E T E E T E R E.  (19) \n \nFig. 6 (Color online)  A periodic layered structure formed by different  media.  \nFor Bloch waves  it is necessary that \n1zik a nn\nscatt inc e EE  and \n1zik a nn\ninc scatt e EE , where \nzk  \nis the propagation constant of the periodic structure and a is the spatial period. \nSubstituting this result into Eq. (19) one obtains  after simple manipulations  \n0\n0z\nzik a n LR\ninc t\nik a n LR\nscatt te\ne       E T I R\nE R T I\n,   (20) \nwhere \ntI  is the identity 2 2 dyadic . To have a nontrivial solution, the determinant of \nthe above matrix must vanish  [2]: \ndet 0z\nzik a LR\nt\nik a LR\nte\ne   T I R\nR T I\n.   (21) \nTo make further progress, we note that because the polarization eigenstates of our \nstratified structures are circularly polarized, the Bloch eigenstates must be such that \nnn\ninc inc\nnn\nscatt scattE\nE\n      \n   Ee\nEe\n with \nT1ie  and \nT1ie . Thus, f rom Eq. (20) we can \nfactorize the dispersion equation into two independent equations:  \n  1 1 0,     1 1 0z z z zik a ik a ik a ik a L R L R L R L RT e T e R R T e T e R R\n              \n.  (22) \nNote that the reflection and transmission coefficients are functions of \n . For periodic structures formed by conventional dielectrics one has \nLLRR , \nRRRR\n, \nR L L RT T T T    , and thus the  two dispersion equations are coincident. In \nthat case, the Bloch eigenmodes are degenerate. However, both for chiral and \nTellegen media in general the two equations yield modes with different dispersions. \nThe modes associated with the “+” (“ -”) sign correspond to a superposition of RCP \n(LCP) waves propagating in the + z-direction and LCP (RCP) waves propagating in \nthe -z-direction. It is easy to check using Eq. (12) that for reciprocal materials ( e.g. \nchiral media) the dispersion of the modes associated with “+” sign, \nzk , and the \ndispersion of the modes associated with “” sign, \nzk , are such that \nzzkk\n. Thus, as could be expected, for reciprocal materials the photonic \nband diagrams are always symmetric.   \nB. Band diagrams for Tellegen periodic structures  \nLet us now consider the particular case wherein all the materials are either \nTellegen media or conventional dielectrics. It was proven in Se ct. II.D that in this \nsituation \nLRTT , and thus we can put  \nRLT T T    and \nRLT T T   . Thus, the \ncharacteristic equations defined in Eq. (22) reduce to  \n11cos 1 ,    cos 12 2 2 2L R L R\nzzTTk a R R k a R RTT\n   \n     \n.       (23) \nIn particular, it is evident that \nzzkk  and hence, despite the nonreciprocal \nresponse of Tellegen media, the band diagrams exhibit  always  spectral symmetry.  \nIn general, the Bloch modes are not deg enerate and \n . However, when all \nthe materials in the unit cell are in the same  equivalence  class, the photonic crystal is \nreducible under some duality transformation to a photonic crystal containing only \nconventional dielectric m edia. One important property of  duality transformations  is that they  only act over the electromagnetic fields, leaving the spatial coordinates \n,,x y zr\n and the time coordinate t invariant  (see Appendix C). As a consequence, \nthe band stru cture of a photonic crystal is invariant under a duality mapping, i.e the \ndispersion diagrams \n vs.  k  are precisely the same for the original crystal and for a \nduality -transformed crystal  [39]. This shows th at the Bloch modes of a Tellegen \nphotonic crystal with media in the same equivalence class must be degenerate \n\n for propagation along the stratification direction.  \nTo illustrate the discussion, we consider a photonic crystal such that the unit cell \nis formed by an air slab with thickness \n00.5 da  and a Tellegen slab with thickness \n10.5da\n. The constitutive parameters of the Tellegen medium are given by \n12 , \n11\n and \n10.8 . The computed band structure is depicted in Fig. 7a. As seen, \nbecause any two  Tellegen  materials define a n equivalence  class the dispersions of the \nmodes “+” an d “” are exactly coincident.  \nIn the second example, we consider a crystal such that the unit cell is formed \nby three  material slabs: an air slab with thickness \n00.2 da , a Tellegen slab and a \ndielectric slab with thicknesses \n10.4da  and \n20.4 da . The constitutive parameters \nof the two media are \n12 , \n11  and \n10.8  and \n22 , \n21 , respec tively . \n  \nFig. 7 (Color online) B and diagrams of periodic structures formed by bi-isotropic media.  The lattice \nconstant is  \na. (a) Unit cell with Tellegen media in the same class. (b) Unit cell with Tellegen media in \ndifferent classes.  (c) Unit cell with a dielectric medium and a chiral medium . (d) Unit cell with a \ndielectric medium , a chiral medium and a Tellegen medium.  \nIt is clear from Fig. 7b that fo r this photonic crystal the band diagrams \ncorresponding to the two polarization eigenstates \ne  and \ne  are different. This \nhappens because the considered media are not simultaneously reducible to \nconventiona l dielectrics.  \nFor completeness, we mention that for symmetric unit cells of the type Air-T1-\nT2-T3-(…)-TN-(…)-T3-T2-T1-Air one also obtains degenerate modes (\n ) even \nwhen the different media do not belong to the same  equivalence  class. Indeed, in such \na case we have \nTT  [see Sect . II.D] and because of the symmetry of the problem it \nis evident that \nLRRR\n .  C. Sufficient conditions for spectral symmetry  \nThe band diagrams \nzk  depicted in the Fig. 7a and Fig. 7b have the  spectral \nsymmetry, \nzzkk . It is well known that such  a property  is characteristic of \nstructures that either satisfy the Lorentz reciprocity theorem or that are invariant under \na spatial -inversion transformation \nrr . However, in general a stratified photonic \ncrystal formed by Tellegen media is not protected by such transformations. Hence, it \nis quite remarkable that our numerical simulations give  \nzzkk . In what \nfollows, we prove that such result is not accidental and that the spectral symm etry of \nthe considered photonic crystals is protected (for propagation along the z-direction) by \nanother symmetry of the system.  \n \nFig. 8 (Color online) T ransformation theory for stratified periodic crystals  with bi -isotropic and ferrite \nmaterials . (a) Original photonic crystal. (b) -(e) Structure of the photonic crystal after the \nelectromagnetic fields are subjected to the indicated operation. (b) Time reversal operation, \nT . (c) \nSpatial inversion trans formation  \n,,x y zI  \n:rr . (d) Composition of time reversal and y-spatial \ninversion , \ny\n TI . (e) Rotation of 180º about the y-axis, \nyR\n : , , x y z  r .  \nFigur e 8 illustrates  how different  symmetry  transformations of the \nelectromagnetic fields affect the structure of a stratified  photonic crystal.  All the transformations map \nzk  into \nzk . Thus, if the photonic crystal stays invariant under \none of these transformations then the spectral symmetry is protected by the \ntransformation.   \nNext , we focus on the action of the transformation sketched in Fig. 8d and prove \nthat the spectral symmetry of the  Tellegen  photonic crystals  is a consequence of this \nsymmetry transformation. T hus the ferrite  and the chiral  slabs in  Fig. 8d should be \nignored  in the fo llowing discussion . Let \n, , ,D E B H  be the electromagnetic fields \nassociated with a Bloch mode characterized by the wave  number (for propagation \nalong the z-direction) \nzk  with the oscillation frequency \n . Let us define the auxiliary \nfields:  \n\n**\n** ,\n,y y y y\ny y y y     \n     D r D r E r E r\nB r B r H r H rI I I I\nI I I I\n,  (24) \nwhere \nyI\n : , , , ,x y z x y z  represents an inversion of the y-coordinate. Th e \nabove  transformation is equiva lent to a composition of the time -reversal operator and \nthe inversion of the y-coordinate operator. It can be checked that the primed fields \nsatisfy the Maxwell equations (with the same oscillation  frequency  \n) in a \ntransformed ph otonic crystal characterized by the material matrix:  \n *00\n00yy\ny\nyy            M r M rIIIII\n.   (25) \nThe material matrix \n Mr  characterizes the original Tellegen photonic crystal and is \ngiven by  \n\nMr\n.     (26) \nClearly, because of the conjugation operation, the primed fields define a Bloch wave \nassociated with the wave  number \nzk . Because our (stratified along z) photonic crystal has parameters  independent of the y coordinate, and because the parameters of \nTellegen media are real -valued, it can be checked that \n* M r M r M r . In \nother words, a stratified photonic crystal formed by  lossless  Tellegen media stays \ninvariant under the ap plication of the composition of time -reversal followed by an \ninversion of the y-coordinate ( Fig. 8d). Note that the photonic crystal has neither the \ntime reversal symmetry (because Tellegen media are nonreciproc al) nor the y-\ninversion symmetry (because a change in the coordinates \n , , , ,x y z x y z  \ntransforms the Tellegen parameter as \n ), but  it has the symmetry \ncorresponding to the composition of the two operations. This means that the primed \nfields are Bloch solutions of the Maxwell equations in the original photonic crystal \nassociated with the parameters \n,zk . This evidently implies that the band \nstructure of the photonic crystal has a  spectral symmetry for wave  propagation along \nthe z-direction, as we wanted to prove.  \nD. Band diagrams for periodic structures with bi-isotropic media \nThe spectral symmetry of photonic crystals with chiral media is not protected by \nthe transformation defined in  Eq. (24) because for such structures \n* M r M r M r\n. Indeed, chiral media stay invariant under the time -reversal \noperation but the chirality parameter is transformed as \n  under the y-inversion \noperation. This  suggests that photonic crystals formed by both Tellegen and chiral \nmedia may have asymmetric band diagrams. This possibility motivated the study of \nband diagrams of general photonic crystals formed by bi -isotropic media, which is the \ntopic of this subsection. \nTo begin with, we consider a photonic crystal  with a unit cell  formed  by a \nlossless chiral slab and an air layer . In Fig. 7c we depict a representative band diagram and, as expected, it exhibits spectral  symmetry because this symmetry  is \nprotected by the invariance under time reversal. In our calculations we assumed that \nthe dispersion of the chiral material is described  by the Condon model [40]. Within \nthis framework, the con stitutive parameters of a lossless chiral medium \n , \n  \nand \n  depend on the frequency \n  as: \n  \n 2 2 2 2 2 2\n22,   ,   \n                           .b R R b R\nRR\n           \n          \n \n (27) \nIn the plot of  Fig. 7c the parameters of the Condon model are chosen as: \n1b , \n1.5b\n, \n2Rac , \n4 , \n0.5  and \n1.6 . The thicknesses of the air and \nchiral regions are \n00.5 da  and \n10.5da , respectively.  \nNext, we consider periodic structures formed by both Tellegen and chiral media. \nTo have a spectral asymmetry we need at least three different material slabs in the \nunit cell. Indeed any layered photonic crystal formed by uniquely two  isotropic  slabs \nin the unit cell stays invariant under a rotation of 180º about the y-axis. Such  a \ntransformati on does not affect the material parameters of either Tellegen or chiral \nmedia (unlike an inversion) ( Fig. 8e), and thus it follows that the invariance under an \n180º rotation about the y-axis also protects the sp ectral symmetry.  \nTherefore, we consider a periodic structure wherein the unit cell is formed by an \nair slab, a chiral medium slab and a Tellegen medium slab with thicknesses \n0d , \n1d \nand \n2d , respectively. The chiral medium has the same constitutive parameters as in \nthe example of  Fig. 7c. The Tellegen medium is  characterized  by the constitutive \nparameters \n22 , \n21  and \n20.8 . The slab thicknesses are \n00.25 da , \n10.5da\n and \n20.25 da . The computed band diagram is shown in Fig. 7d. Although periodic structures formed by only Tellegen media or by only chiral \nmedia exhibit symmetric band diagrams ( Fig. 7a, b and c), it is seen  that the symmetry \nis broken in a crystal formed by both chiral and Tellegen media. Indeed, this type of \nphotonic crystals is neither protected by time reversal invariance nor by the \ntransformation defined in Eq. (24). \nAn immediate consequence of the spectral asymmetry is that the g roup velocities \n(\ngzvk  ) of the Bloch waves that propagate in  the forward and backward \ndirections are not the symmetric of one another. This is illustrated in Fig. 9a for the \nphotonic crystal considered in the Fig. 7d, which represents the group velocities for \nthe two eigenmodes, \ne  and \ne  as a function of frequency. In Fig. 9b we show the \ndifference of the absolute values of the group velocities of waves propagating in \nopposite directions, \ng g gv v v  \n , for each eigenvector, \ne  and \ne . The group \nvelocities associated with the arrow “\n ” (“\n”) correspond to wave propagation \nalong the  \nz (\nz) –direction.  \n \nFig. 9 (Color online) G roup velocities of the Bloch waves. (a) The blue (dark gr ay) line represents \nthe group velocity of the wave  \ne and the green (light gray ) line corresponds to  the group velocity of \nthe wave  \ne. (b) Difference of the absolute values of the group velocities for counter -propagating \nwaves.  It is interesting to investigate how strong is the spectral asymmetry in a \nrealistic photonic crystal wherein the Tellegen parameter \n  has magnitude \ncomparable to the magnetoelectric parameter of \n23Cr O  [41]. Hence, next we consider \nthat the constitutive parameters of the Tellegen medium are  \n2 , \n1  and \n310\n. The remaining structural and material parameters are the same as in the \nprevious example. The band diagram of this more realistic structure is plotted in the \nFig. 10a and the frequency range near the band gap is zoomed in Fig. 10b and Fig. \n10c. As se en, the asymmetry is tiny but detectable near the band gaps.  \n \nFig. 10 (Color online)  (a) Band diagram of BI periodic structures formed by a Tellegen material \nwith \n310 . (b) and (c) Zooms of the band dia gram near the band -gap. E. Band d iagrams for periodic structures with ferrites and chiral \nmedia \nFinally, w e analyze periodic structures formed by nonreciprocal ferrite materials, \nconventional dielectrics and chiral media. The dispersion of the Bloch waves in ferrite \nphotonic crystals is also found by solving  Eq. (22).  \n In the first example,  we suppose  that the unit cell is formed by a  dielectric  slab \nand a ferrite slab with thicknesses \n12 0.5 d d a . The calculated  band diagram is \nshown  in Fig. 11a. The constitutive parameters of the dielectric  are \n12  and \n11  \nand the dispersion of the lossless ferrite material i s modelled by Eq. (17). In this \nsimulation it was assumed that  \n4Rac  and \n2mac , \n1f . It is clear from \nFig. 11a that for this periodic structure the band diagrams corresponding to the two \npolarization eigenstates \ne  and \ne  are different. Interestingly, consistent with the \ndiscussion at the end of S ect. III, it is seen that a unit cell formed by  a single ferrite \nslab and a dielectric is equivalent to a  three material  Tellegen unit cell wherein the \nmedia are in different equivalence classes  [see Sect. IV B]. In particular, it is seen that \nthe band diagrams are symmetric. This property  can be readily explained with the \nhelp of  the transformation theory outlined in Fig. 8. From Fig. 8c it is readily checked \nthat this photonic crystal is invariant under a spatial inversion (\n,,x y zI ) because the \ndielectric has vanishing  chiral  and Tellegen  parameter s. This implies that \nzzkk\n.   \nFig. 11 (Color online) B and diagram of a stratified periodic structure . (a) Unit cell formed by a \ndielectric  slab and a ferrite slab.  (b) Unit cell formed by a chiral medium slab  and a ferrite slab . \nInterestingly, a photonic crystal whose unit cell is formed by a ferrite slab and a chiral \nslab is not protected by any of the symmetry transformations considered in Fig. 8. \nHence, this suggests that such  a confi guration can allow for asymmetric band \ndiagrams. This is indeed the case, as shown in Fig. 11b. In this second example, the \nunit cell contains a chiral slab and a ferrite slab with thicknesses \n12 0.5 d d a . The \nconstitutive parameters o f the ferrite material are as in the previous example, while \nthe chiral med ium parameters are as in Sect.  IV.D. This example and the results of  \nSect. IV.D further confirm that the role of a single ferrite slab can be mimicked by \ntwo Tellegen slabs (one of the  Tellegen  slabs may be a regular dielectric).  \nV. Conclusions  \nWe investigated the scat tering by stratified structures formed by Tellegen  (e.g. \ntopological insulators or Cr 2O3) or ferrite materials . It was  found that if the Tellegen \nstructures  cannot be simultaneously reduced under a duality transformation to \nconventional dielectrics, then i t is possible to have asymmetric transmissions of waves \nincident on opposite  sides of the stratified structures. Otherwise, nonreciprocal effects \nare only manifested  in the reflected waves.  The propagation of electromagnetic waves in periodic stratified s tructures \nformed by chiral media and Tellegen or ferrite materials  was studied . In case of \nTellegen photonic  crystals, it was  proven  that in general the eigen modes  are not \ndegenerate  for propagation along the stratification direction , except if the structu re \nexhibits  a two-fold rotation symmetry  (about the y-axis)  or, alternatively, if all the \ninvolved media are in the same  equivalence  class. Surprisingly, we discovered  that \nnotwithstanding the nonreciprocal response of Tellegen media, the photonic  band \ndiagrams  are always  spectral ly symmetr ic. We explained this result by showing that \nthe spectral symmetry is protected by a symmetry transformation that corresponds to \nthe time-reversal operation followed by the y-inversion operation.  \nWith the motivation of fi nding a bi -isotropic crystal wherein the spectral \nsymmetry is broken , we considered  periodic stratified structures with  both Tellegen \nand chiral media. The band structure of such crystals can be highly asymmetric and in \nparticular the group velocities of c ounter -propagating waves are different. We \ninvestigated how pronounced this effect can be in photonic crystals formed by a \nTellegen material with parameters consistent with those of Cr2O3 and found that \nalthough very tiny the asymmetry is revealed.  Further more, it was proven  that \nperiodic structures formed by chiral media and ferrites also have asymmetric band \ndiagrams . In general , the role of a ferrite slab biased with a static field oriented along \nthe z-direction can be mimicked by two juxtaposed Tellegen  media slabs with a \nspontaneous nonreciprocal response.  \n Because the electromagnetic response of  electronic  topological insulators may \nbe equivalent to the response of Tellegen media, our findings suggest exciting \napplications of these materials  in novel p hotonic couplers  with asymmetric \ntransmissions. This topic will be further explored in future work.  Appendix A:  Axion electrodynamics and the Tellegen medium  \nThe Maxwell  equations in presence of a time independent axion -coupling term  (\n) \nare [25]: \n, t BE\n     (A1) \n(),e\nt       BEjE\n    (A2) \nwhere \n  is a coupling constant.  Substituting  \n        E E E  into Eq. \n(A.2) and using  Eq. (A.1) one obtains:  \n()e\ntt           B E BjE\n.    (A3) \nAfter some manipulations  Eq. (A.3) can be rewritten as:  \n()e\nt        BE E B j\n.    (A4) \nHence, introducing the electric displacement field vector, \nD , and the magnetic field \nvector, \nH , defined as  \n,     BH E D E B\n,     (A5) \nwe get  \n()e\nt  DHj\n.       (A6) \nEquation (A.5) is coincident with Eq. (1) of [28]. Clearly, the constitutive relations in \nEq. (A.5) can be rewritten as : \n11\n00 , cc           B H E D E H\n,    (A7) \nwhere \n2 1\n00,, c                and the refractive index  is  \n2\n00 n         \n. Hence, the axion electrodynamics lead s to the constitutive relations of a Tellegen medium. From this result it follows that if the \naxion parameter \n  is piecewise constant in space, then at an interface ( i.e. at the \nsurface wherein \n  is discontinuous) the tangential electric field ( E) and the tangential \nmagnetic field ( H) are required to be continuous [24]. On the other hand, the \ntangential component of \nB  is in general discontinuous.  \nAppendix B: Transmission matrices for stratified Tellegen media  \nIn this Appendix we prove that , independent of the number of lay ers, the \ntransmission matrices associated with  Tellegen  stratified  structures satisfy \nLRTT . \nTo begin with, we consider the simple case wherein the stratified structure is formed \nby a single Tellegen slab (air -Tellegen -air) with thick ness \n1d . The transmitted electric \nfield vector can be formally calculated based on  a (vector) reflection diagram that \naccounts for the multiple wave reflections/transmissions at the two interfaces. The \nreflection diagrams for incid ence from the left -hand side (propagation L -R) is shown \nin Fig. 12a. A similar  reflection diagram can be constructed for  incidence from the \nright -hand side (propagation R -L) (not shown) . \nFig. 12 (Color online)  (a) Reflection diagram for the scattering of a plane wave by a Tellegen slab for \nwave propagation L -R. (b) A set of N-Tellegen slabs is juxtaposed to a single material Tellegen slab. \nEach structure is characterized by the respective glo bal reflection and transmission matrices.  In both cases, the transmitted electric field vectors \nL\nTE  and \nR\nTE  are a superposition of \nmultiple transmitted waves, as shown in the reflection diagram of Fig. 12a. Hence, the \ntotal transmitted field for an incoming wave propagating in the L -R direction, \nL\nTE , is \ngiven by the following geometric series  \n\n 1 1 1 1\n11\n11\n113\n10 01 1 10 10 10 01 1\n5\n10 10 10 10 10 01 1\n10 01 1 2\n10 10    ...\n1    1ik d ik d L L L L R L L\nT\nik d L R L R L L\nik d LL\nik d RLee\ne\nee\n\n       \n       \n   E T T E T R R T E\nT R R R R T E\nT T ERR\n   (B1) \nwhere \n 01 10,LLTT  are the transmission matrices  associated with the air -Tellegen and \nTellegen -air interfaces, respectively, for a wave propagating in the L -R direction. On \nthe other hand, \n 10 10,LRRR  are the reflecti on matrices  associated with the Tellegen -air \ninterface for a wave propagating in the L -R and R -L directions, respectively. In a \nsimilar manner, it is possible to prove that the total transmitted field for an incoming \nwave propagating in the R -L direction i s \n\n 1 1 1 1\n11\n11\n113\n10 01 1 10 10 10 01 1\n5\n10 10 10 10 10 01 1\n10 01 1 2\n10 10    ...\n1    1ik d ik d R R R R L R R\nT\nik d R L R L R R\nik d RR\nik d LRee\ne\nee\n\n       \n       \n   E T T E T R R T E\nT R R R R T E\nT T ERR\n   (B2) \nwhere \n 01 10,RRTT  are the transmission matrices  for the air -Tellegen and Tellegen -air \ninterfaces, respectively, for a wave propagating in the  R-L direction. Thus, from Eqs.  \n(B1) and (B2) the global transmission matrices \n,LRTT  for the Tellegen slab satisfy  \n1 1 1 1\n1 1 1 110 01 10 01 22\n10 10 10 1011,        11ik d ik d L L L R R R\nik d ik d R L L Reeee        T T T T T TR R R R\n.   (B3) \nUsing arguments analogous to those of  Sect. II.B it readily follows that \n01 10 01 10 10 10, , , , ,L L R R L RT T T T R R\n are necessarily matrices of the form  presented in Eq. (10). Thus, all the relevant matrices commute and hence to prove that \nLRTT  it suffices to \nshow that \n10 01 10 01L L R R  T T T T . The transmission matrices \n  01 10 01 10, , ,L L R RT T T T  may be  \nexplicitly  written  as a function of the characteristic admittance dyadics \n  0 0 1 1, , ,   Y Y Y Y\n (defined as  explained  below Eq. (3)) as \n \n 11\n01 0 1 0 0 10 1 0 1 1\n11\n01 0 1 0 0 10 1 0 1 1,        \n,        LL\nRR       \n                    \n             T Y Y Y Y T Y Y Y Y\nT Y Y Y Y T Y Y Y Y\n.     (B4) \nThe characteristic admittances are also of the  generic  form  of Eq. (10) and hence \nbecause the sums, inverses, and  products of such matrices commute, it follows that \n10 01 10 01L L R R  T T T T\n. This concludes the proof for the case of a single material slab.  \nNext, we generalize the result \nLRTT  for the case of N- juxtaposed Tellegen \nslabs. The p roof is done by induction in the number of slabs. Consider a generic \nstructure formed by N+1 slabs, as depicted in Fig. 12b. This structure can be regarded \nas a collection of N slabs with total thickness \n1d  and another slab with thickness \n2d . \nIt is convenient to imagine that the two s tructures stand in a ir and  are separated by a \ndistance \n0d . In the end, we will consider the limit of a vanishing \n0d . The global \ntransmission and reflection matrices for the first structure ( i.e., the collection of N \nslabs) are denoted by \n  , , ,L R L R\na a a aT T R R  whereas for the second slab they are \n  , , ,L R L R\nb b b bT T R R\n. Because of the i nduction hypothesis, we can assume that \nLR\naaTT  \nand \nLR\nbbTT . Next, we compute the global transmission matrices for the set of N+1-\nslabs.  \nTo do this, we use again reflection diagrams analogous to those of  Fig. 12. In \nthis manner, we find that for an incoming wave propagating in the L -R direction the \ntransmitted electric field vector \nL\nTE  is given by  00\n0021\n1ik d L L L inc\nT b a a ik d RL\nabee   E T T ERR   (B5) \nwhereas for an incoming wave propagating in the R -L direction one obtains  \n00\n0021\n1ik d R R R inc\nT a b b ik d LR\nbaee   E T T ERR\n.   (B6) \nThus, letting the thickness between the structures to approach zero, \n00 d , using the \ninduction  hypothesis, and the fact that because of the isotropy of the involved \nmaterials all the matrices are necessarily of the form  presented in Eq. (10) and thus \ncommute, it follows that the global transmission coefficients for the set of t he N+1 \njuxtaposed slabs \n,LR\nccTT  satisfy:  \n11\n11L L L R R R\nc b a a b c R L L R\na b b a         T T T T T TR R R R\n.  (B7) \nBy definition, the global  transmission  matrices are such that \nL L inc\nT c aE T E  and \nR R inc\nT c bE T E\n. Thus, thi s result proves that the induction hypothesis also holds for \nthe set of N+1-slabs, and thus it is valid for arbitrary N, as we wanted to show.  \nAppendix C: Duality transformations  \nIn this Appendix we briefly review duality transformations applied to Tellege n \nmedia.  Duality transformations are linear mappings of the electromagnetic fields of \nthe form  [42]: \n00d\nd         EESHH\n, with \n11 12\n21 22ss\nssS ,  (C1) \nwhere \nS  is a 22 real -valued matrix with constant elements (independent of the \nspatial coordinates) and \n11 12 21,,s s s  and \n22s  are real -valued  parameters . It is well -\nknown that the duality transformed fields \ndE  and \ndH  are solutions of the Maxwell’s equations in a transformed structure described by the transformed material matrix  \n[39]: \nT11detd  M S S M S\n,    (C2) \nwhere \n\nM  is the material matrix of a Tellegen medium defined so that  \n0\n0/\nc\n   E DMH B\n.    (C3) \nIn a previous work, we demonstrated that electromagnetic structures formed by \nTellegen med ia can be transformed, in some conditions, into simpler structures \nformed by simple isotropic media using a duality mapping [39]. This can be rather \nuseful because if a solution for the duality transformed problem can be found,  then the \nsolution of the original problem can be readily obtained by applying an inverse \nduality mapping [42]-[44]. For example, if a Tellegen photonic crystal is reducible to \na cry stal formed by only simple isotropic media using a duality transformation, then \nthe dispersion diagrams of the original structure are exactly the same as the diagrams \nof the transformed photonic crystal [39]. It was shown in [39] that typically three \ndifferent Tellegen media cannot be transformed into simple isotropic media  using the \nsame duality transformation, or in other words, they do not belong to the same \nTellegen equivalence class  [39]. 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Obradorsa \n \naInstitut de Ciència de Materials de Barcelona (ICMAB -CSIC), Campus UAB, Bellaterra, 08193, Catalonia, Spain  \nbInstitut des Nanotechnologies de Lyon (INL) CNRS -Ecole Centrale de Lyon, 36 avenue Guy de Collongue, 69134 Ecully, \nFrance   \ncInstitut d’Electronique et des Systemes (IES), CNRS, Universite Montpellier 2 860 Rue de Saint Priest  34095 Montpellier  \n \n \nABSTRACT  \n \nWhile piezoelectrics and ferroelectrics are playing a key role in many everyday applications, there are \nstill a number of open questions related to the physics of those materials. In order to foster the \nunderstanding of piezo electrics and ferroelectric and pave the way to future applications, the nanoscale \ncharacterization of these materials is essential. In this light, we have developed a novel AFM based \nmode that obtains a direct quantitative analysis of the piezoelectric co efficient d 33. This nanoscale tool  \nis capable of detecting and reveal  piezo -charge generat ion through the direct piezoelectric effect at the \nsurface of the piezoelectric and ferroelectric materials.  We report the first nanoscale images of the \ncharge gene rated in a thick single crystal of Periodically Poled Lithium Niobate (PPLN) and a Bismuth \nFerrite (B iFO 3) thin film by applying a force and recording the current produced by the materials. The \nquantification of both d 33 coefficients for PPLN and BFO  are 1 3 ± 2 pC/N and 46 ± 7 pC/N \nrespectively, in agreement with the values reported in the literature. This new mode can operate \nsimultaneously with PFM mode providing a powerful tool for the electromechanical and piezo -charge \ngeneration characterization of fer roelectric and piezoelectric materials . \n \nINTRODUCTION  \n \n \nThe piezoelectric effect , which  consists in the dielectric polarization of non -centrosymmetric crystals \nunder a mechanical stress , was discovered by the Curie brothers in 18801. The following year, from \nthermodynamic considerations, G . Lippmann predicted the converse effect, i.e. that a piezoelectric \nmaterial would be mechanically strained by an applied electric field2 and the Curies readily measured \nit3. These findings spawned more research  which eventually led to the discovery of ferroelectricity  in \npolar piezoelectrics4. Since those  early discoveries, the unique ability of piezoelectrics  and \nferroelectrics for interconvert ing mechanical and electrostatic energies5 has endle ssly inspired  \ntechnological developments and these materials , which  represent  nowadays a billion euro \nindustry , are found in many everyday applications6–12: ultrasound generators for echography \nscanners, shock detectors within airbags, accelerometers, diesel injection valves, tire pressure \nsensors, vibration dampers , oscillators, improved capacitors,  or new dynamic access ran dom \nmemories , to just cite a few. Moreover, the p rospects for  future applications  in new markets are  bright , \nincluding energy harvesting , CMOS replacement switches , or photovoltaics and photocataly sis13–16. \nYet, in spite of such industrial relevance and the amount of past and present research , the basic \nunderstanding of  piezoelec tricity and ferroelectricity is  challenged  and reshaped  by findings that come \nalong with new developments  in the characterization of materials. This is well illustrated by  the \nadvances in atomic force microscopy , which  brought a new perspective  of ferroelectric domain walls17–\n20. The development of new modes and an  improve d spatial resolution  have  revealed the domain wall complexity  and its intrinsic properties  21–23 and have  also opened the door to get more insight in long -\ndate issues such as the extrinsic contributions to dielectric permittivity and piezoelectricity due to \ndomain wall pinning  at dislocations and grain24. In this direction, P iezoresponse Force Microscopy \n(PFM) is the most widely used technique for the nanoscale and mesoscale cha racterization of \nferroelectric and piezoelectric materials25–28. PFM method is based on the converse piezoelectric effect \nand consists in measur ing the material deformation  under an AC  electric field  applied through the \ncontacting AFM tip . In this technique the sample vibration is determined by an optical beam deflection \nsystem, which is an indirect measurement29, making the accurate determination of the piezoelectric \ncoefficient  challenging. Moreover, the quantitative piezoelectric measurements by P FM30, are \nfurther complicated by  the difficulty of disentangling , from the electromechanical response , the \ncontributions of the piezoelectric response and other  physical phenomena  such as, ionic motion  \nand charging, electrostatic or thermal effects18,31 –33. Indeed, the increasing awareness about these \nissues a mong the scientists of the field34 prompts the need for new developments in scanning \nprobe microscopies , which remain  a unique tool for the characterization of piezoelectric and \nferroelectric materials at the  nanoscale.  \n \nTo address this need , here we introduce a new SPM tool that  exploits the direct piezoelectric \neffect to  obtain a quantitative measurement of the piezoelectric constant in piezoelectric s. This \ntechnique , that we call Direct Piezoelectric Force Microscopy (DPFM)  uses a specific amplifier \nand a conductive tip which simultaneously strains a piezoelectric material and collects the charge \nbuil t up by the direct piezoelectric effect. The amplifier is an ultralow input bias current (<0.1 fA) \ntran simpedance  capable of converting electric charge into a voltage signal, readable by any \ncommercial microscope (see figure 1a). As a consequence, the developed setup has a very low \nleakage current, and thus all the charges generated by the piezoelectric mat erial can be read by \nthe amplifier.  Just by maintaining the tip on the surface of the films and sequentially applying \ndifferent force values with  the AFM tip, the charges generated by the material are measured  and \nthe direct piezoelectric coefficient can be readily calculated from the applied stress and the \ncollected compensation charge . Interestingly, by combining this tool with PFM measurements, a \ncomplete electromechanical and piezo -charge generation characterization can be achieved .  \nMeasuring the direc t piezoelectric effect with an AFM is a challenge that has n ot been addressed \nso far due to the impossibility of performing reliable measurements of tiny amounts of  generated  \ncharge . An AFM probe can apply a user predefined force with picoNewton precision , up to \nmaximum values of hundreds of microNewtons35–37. Applied to a piezoelectric material, such \nforce will generate a charge, which  can be collected to obtain currents of different intensity \ndepending on the sampling  time. For instance, we can estimate that the 1fC  charge generated by \napplying a 100 μN force into a 10 pC/N piezoelectric material38 , will produce a current of 1 fA if \ngenerated in 1 s, 2 fA if generated in 0.5 s and so on.  With such requirements, an amplifier \ncapable of measuring 1fA with a BandWidth (BW) of 1 Hz is needed. More importantly, the \ncharge that the amplifier l eaks has to be well below that desired threshold  of 1fA , otherwise a \nsubstantial part of the current will be lost during measurements . Since t hese requirements were \nnot met by any AFM manufacturing companies, a special amplifier was employed.  \n \nEXPERIMENTAL SETUP  \n \nThe complete  setup to perform measurements according to  the proposed method is depicted in Figure \n1a. The amplifier consists of three different commercially available Operational Amplifiers (OA), \nwhich were supplied by Analog Devices Inc. The amplification process is divided into two stages, a \ntransimpedance stage and a voltage amplifier stage. The transimpedance stage was configured with a feedback resistor of 1TeraOhm which yields a Current -to-Voltage gain of -1x1012 V/A39. The voltage \namplifier stag e adds an additional gain of 72 ,25. Following  standard amplifier theory, the final gain of \nboth concatenated stages is the multiplicatio n of each stage gain, which results in a  gain of -72,25 x \n1012 V/A40. Even t hough theoretical gain calculation is precise , we experimentally calibrated the \namplifier twice wit h a test resistor of 40 ± 0,4 GOhm giving an experimental gain of -16,9 ± 1,0 x 1012 \nV/A ( see Figure S1 of Supplementary Information ). The leakage current through the amplifier induces  \nan error, which will be responsible of charge losses  while measuring. Such current was provided by \nAnalog Devices as being as low as 0 ,1 fA, which can be  considered small compared to the generated \npiezocharge41 to be measured, whi ch is in the order of several fA . An intrinsic property of the setup is \nthat both tip and back -surface of  the sample are connected to ground, which enables  the study of high \nleakage ferroelectric films.  \n  \nWith such setup the charge generated by a piezoelec tric material can be recorded with an AFM tip. The \nphysics underlying the generated current is depicted in Figure 1b, c and d . Two different cases are \nconsidered, when the tip scans from left to right (Trace) and from right to left (Retrace). While in trace \nscanning, Fig. 1b , the moving tip creates a strained area on the right side of the tip apex, while the area \non its left sid e is unstrained. When  an up domain polarization is scanned, a positive charge (+Q) is \ngenerated in the strained region implying a positive flowing current. In contrast,  a charge of opposite \nsign is created ( -Q) in the unstrained area on the left side of ti p apex. The charges generated at the \nstrained and unstrained regions cancel out, yielding a zero net charge, because the strained and \nunstrained charge generation processes are compensated. Nevertheless, the situation is completely \ndifferent  at domain wall s. Once the tip apex  is located on the domain wall, the strained region, which \nnow has a downward s polarization, will generate a negative charge  (-Q). The unstrained up polarization  \nregion  will remain  unchanged, generating a negative charge  (-Q). Thus , a negative charge is generated, \nwhich  can be quantified by  measur ing the negative current flowing through the tip. In this case, the \nmeasured current corresponds to the tip loading the down polarization and unloading the up \npolarization. Similarly,  when the tip scans from right to left  the unstrained region corresponds  now to \nthe down polarization state, and hence, a positive charge (+Q) is generated  (see Fig 1c ). At the same \ntime, the strained region, which is in the up polarization state, will gen erate a positive charge (+Q). As \na result , a positive charge is generated at the domain wall and a positive current can be measured  by the \nAFM tip . Again,  no net charge results from scanning a single domain , as the strained and unstrained \nregions will gene rate charge s of opposite  signs. Spectroscopy experiments can also be performed , see \nFig 1d , as the tip exerting a force generates a positive charge (+Q), if an up domain is loaded, or a \nnegative charge ( -Q) if a down domain is loaded. By the contrary, the unloading process  generate s a \nnegative charge ( -Q) for an up domain and a positive charge (+Q) for a  down domain . As current is \nbeing recorded,  the rate at which the force is applied rate  is crucial, as the current increases with force \nrate. Throughout the manuscript it is considered that a positive force -straining force - will generate a \npositive current if applied into a positive (up) poled domain.  \n \n \nIn the experiments  we used a commercial probe with reference RMN -25PT200H. The tip is made out \nof a solid platinum wire consisting in an ultra stiff cantilever, with spring constant of 250 N/m. Such \nfully metal lic tip ensures that its conductivity nature is preserved  while applying a high load and only a \ndecrease in resolution can eventually occur . We tested t he new m ode on a typical  reference material  for \nPFM experiments which is a commercially available Periodically Poled Lithium Niobate (PPLN)42 in \nthe form of a thick crystal . This material ha s been widely studied and its d 33 piezoelectric constant is in \nthe range of 6 -16 pC/N43. Before starting the measurements, the sampl e was scanned with the \nconductive tip in order to discharge its surface from screening charges and minimize  their effects44,45. \n RESULTS  \n \nThrough the aforementioned setup and the proposed physical explanation, we have been able to \nperform the first mapping of piezoelectricity at the nanoscale. The output signal of the amplifier was \nboth recorded at the Trace ( Figure 2a ) and Retrace ( Figure 2 b) scans. The images consist of a \n256x128 pixels frames , 15 μm x 30 μm obtained at a speed of 0.01 l ines/s (ln/s)  (0,66 μm/s) and were \nrecorded with a loading force of 234 μN. We use d a particularly low speed to avoid scrapping  surface \nscreening charge which could  interfere with the collected charge46. With th ese imaging par ameters \nbandwidth needed to record current is 5 Hz, which is in accordance with what our amplifier can \nperform.  The obtained images  (see Figure 2 a,b), show  that the current is only recorded at the domain \nwalls in accordance with the proposed physical model. A peak current of 15 fA is generated at the \ndomain walls while its sign depend s on the direction of the tip scan . We labeled  the “Trace” image, \nfrom left to right, as “DPFM -Si”, for Direct P FM Signal input, and the “ Retrace ” image , from right to \nleft, as “DPFM -So”, for Direct PFM Signal output. We have also tried to perform both PFM and \nDPFM methods, simultaneously . In order to do so, the back of the PPLN crystal was connected to the \nAC gener ator of the AFM, so an AC voltage signal was applied to the bottom surface of the sample  \nmaintaining a DC coupled ground . The PFM phase image is shown in Figure 2c  and PFM amplitude \nimage is shown in Figure 2d . The simultaneous acquisition of the four imag es of Figure 2 shows how \nthe DPFM mode can complement the standard PFM measurements providing, as we will discuss \nbelow, the data to quantify the piezoelectric coefficient of the material . Moreover, standard t opography \nimage and friction image, obtained fr om contact mode operation are recorded ( see Figure S2 of SI ). \nFrom DPFM -Si and DPFM -So images it is observed that there is a little gradient in the single domains \nareas, this will imply the collected current is not exactly zero. This could be due to different processes \noccurring simultaneously with  piezoelectric c harge generation as, for instance,  surface screening \nrecharging47. However its contribution  is negli gible compared to the peaks recorded at domains walls \n(see Figure S3 of SI ). \n \nIn order to obtain strong evidence  of the piezoelectric origin of the current signal from the amplifier we \nprepare d a full set of experiments related to the dynamics of piezoelectric charge generation. The \ncharge generated from piezoelectric effect is known to be linear with the applied force5. This is a key \naspect  to distinguish piezoe lectric charge from other possible charge generation  phenomena46,48. The \nrelationship between current and applied load was tested by scanning the PPLN sample under  different  \napplied loads, starting from a low loading force of 9 μN which was stepwise  increased until reaching a \nmaximum force of 234 μN. The recorded DPFM -Si and DPFM -So images are plotted in Figure 3a and \nFigure 3b, respectively. The tip speed was maintained constant along the whole image at a rate of 0,55 \nμm/s. We can observe that at the lowest load, no charge was collected by the amplifier , which was not \ncapable of read such a small current, i.e between 0,1 -0,3 fA  for an applied force  of 9 μN. The area \nrecorded with the minimum force loading  is also interesting to assess the  influence of surface charge \nscreening in the recorded current s. Before DPFM experiments , the sample was scanned with the same \ntip, at a tip sp eed 100 times faster , in order to fully discharge the sample surface from surface screening \ncharge. The area scanned with 9 μN confirms that surface screening charges do not play an important \nrole in the collected charge. If removal of surface screening ch arge through a scrapping process was \nimportant we should see a current in the 9 μN region, as the applied force is two-fold that needed  to \nstart the scrapping process44. Once the force is increased, the current recorded by the amplifier \nincreases as well, as it should be expected from a piezoelectric generated charge. More importantly, the \nwidth of the curre nt line generated at domain walls does not substantially  increase  with applied load. \nThe size of this line is not related to the domain wall thickness, but to a convolution  effect caused by \nthe tip49,50 (see S4  in SI ). \n In order to elucidate if the generated charge is proportional to the force we have analyzed the peak \ncurrent values for DPFM -Si and DPFM -So frames , for each applied load. The maximum cu rrent values  \nof a scan l ine were multiplied  by the specific time constant of one pixel, which is 0,39 s, so the most \npart of the piezoelectric charge is fully integrated. Finally, a relation between the collected Charge vs \nApplied load is found, which is plotted  in Figure 3c . A l inear fit was used for both positive and \nnegative charge generated confirming the linear relatio nship between the generated charge and the \napplied force with Pearson's R of 0,99 and -0,93 for each linear fitting. From the slope of this linea r fit, \nan appro ximation of the d 33 piezoelectric constan t of the material can be found with a value of 8,2 \npC/N. The value obtained  is an underrated  approximation , as there is a part of the current generated \nthat it is not being  considered , as only the peak current is integrated . The current profile shape for each \napplied load was also analyzed, which are plotted in Figure 3d . The profiles provide information on \nthe dynamics of the charge generation at the nanoscale as the tip passes throughout the domain wall. It \nis found that the piezoelectric current has a Gaussian -like shape, where the area be low the  Gaussian  \ncurve is the piezo -generated charge. The profiles, evidence  that the increased generated charge for \nhigher loads is related to the maximum current peak, rather than to  the width of the Gaussian -like curve \nshape. This is in accordance with the fact that the tip does not significantly increase its radius with the \napplied load. Once the origin of the generated charge has been proved to be the direct piezoelectric \neffect, we can now perform a mapping of the piezopower generation at the nanoscale  with images of \nFigure 2 ( see S5 in SI ).  \n \nObtaining quantitative  values of piezoelectric and ferroelectric materials through an easy and reliable \nmethod is a high pursued targ et in the scientific community51,52. In order to test if the method can be \nquantitative, we performed a zoomed -in image of a domain wall, recording both DPFM -Si and DPFM -\nSo signals, see Figure  4a and Figure 4b. The images were performed with a tip speed of 0,22 μm/s \nand an applied load of 234 μN. The zoomed in images were sufficiently  precise to fully integrate the \ngenerated current. In order to reduce thermal noise53, the mean average profile for the total number of \nlines composing  the image was obtained for both cases, see Figure 4c  and Figure 4d. The resulting \nprofile corresponds to the piezoelectric generated charge vs distance ( μm) which divided by  the tip \nvelocity value  can be converted into charge vs time. With such experimental profiles, see Figure 4c \nand 4d , we can perform a gaussamp fit of the obtained curves to estimate the area beneath the curve. \nWe have found that the piezoelectric charge generated is 5,7 ± 0,4 fC for DPFM -Si and 6,5 ± 0,5 fC for \nthe DPFM -So profiles . In order to see if the collected charge is a function of the tip  speed we studied \nthe evolution of the recorded charge versus tip speed ( see S 6 in SI ). The measured charge corresponds \nto a loading and unloading mechanism, and hence to find the piezoelectric charge we must divide this \ncharge by a factor of two. The exac t force exerted was calculated using a Force -vs-Distance curve, (see  \nS7 in SI ) and with such deflection sensitivity and the cantilever spring constant, the applied force was \nobtained. To diminish the error associated to the applied force , we have calculated the exact force \nconstant of the probe used in the experiment, throu gh a formula provided by the tip manufacturer and \nthe real dimensions of the cantilever. Upon c alculations, we found that the applied  load is 234 μN, \nwhich yield s a piezoelectric  constant of 12,1 pC/N and 13,8 pC/N,  for DPFM -Si and DPFM -So, \nrespectively. This is in accordance with the value found in the bibliography, w here the d 33 constant of \nPPLN is in the order of 6 -16 pC/N. In fact, we have evaluated the error that corresponds to the \nproposed method. The force error was found to be ± 9 μN, mainly caused by the determination of the \nspring constant of the cantilever. The charge measurement error was calculated as the sum of the \nstatistical error, the error created from the amplifi er leakage current and the error obtained from the \nelectrical calibration. The total error is ± 0,7 fC for DPFM -Si and ± 0,9 fC  for DPFM -So profiles .  \nSumming all the error s, we found that the d 33 piezoelectric constant of our sample is  12 ± 3 pC/N and \n14 ± 4 pC/N for DPFM -Si and DPFM -So respectively. As we are crossing the very same domain, we \ncan use both q uantities to acquire the final d 33 constant of the material as being 12,9 ± 2,4 pC/N standard error.  \n \nSpectroscopy experiments were performed  to elucidate if the method could also be employed not only \nfor imaging, but also as a tool of characterizing the piezoelectric response  outside the ferroelectric \ndomain walls or  in non-ferroelectric  piezoelectric s. For such purpose,  the tip was placed in the middle \nof a ferroelectric domain and the current recorded while a  Force -vs-Distance curve was obtained . The \ncurve starts with a loading force of 5 μN and it  is increased to a maximum value of 258 μN to go back \nto the initial 5 μN load. The current reco rded from the amplifier was measured for different applied \nforce rates, see Figure 4e . As the force/time rate is increased, the re corded current increases as well \nconfirming its direct relationship . Different spectroscopy  events were obtained, see Figure 4 f; top \nwhich corresponds to a spectroscopy for up domain area and bottom for down domain area. It is found \nthat for the up domain case, a loading curve will generate a positive current;  however the current sign is \nthe opposite in the case of a down polariz ation domain. The spectroscopy curves started with the tip in \ncontact with the surface under  an applied load of 5 μN, to avoid collecting charge  generated by \nelectrostatic effects wh ile the tip is moved  from air to the sample surface. For both curves a forc e/time \nratio of 53,2 μN/s was employed.  \n \nThe feasibility of the method has been  successfully  demonstrated  for a thick ferroelectric crystal with a \nlow-intermediate piezo electric d 33 constant. In order to check  if the performance of the DPFM method \non other materials  it was also tested on a 400nm -thick BFO ferroelectric layer over platinum , \ncommercially available  from MTI Corp . The sample was previously scanned using PFM in order to \nrecord a pattern in its surface -the pattern is sho wn in PFM phase image  of Figure 5a , where DC \nvoltages of +45 VDC and -45VDC were applied to the bot tom contact  of the sample  in order to pol l the \ndomains. The same area was scanned using normal PFM mode in order to see if the domains can be \nread. Once recorded, DPFM -Si and DPF M-So images were performed, which are shown in Figure 5b  \nand Figure 5c. It is found that the cur rent generated is only present at  domain walls, however with a \nsimilar scanning parameters, it is found that the peak current is near 25fA. BFO is a well-charac terized \nferroelectric that has a young modulus of 170 G Pa and a surface screen charge of 80 μC/cm²54. These \nvalues are comparable to those of  the previously tested PPLN43. However, the piezoelectric constant  of \nBFO  is significantly larger,  between 16 -60 pC/N54. These differences in the measured d 33 constants can \nbe used to explain the larger current that is recorded for B FO compared to  PPLN. In order to discard \nimaging artifacts, t he same pattern w as reread in DPFM mode but rotating the scan direction , which \nrotates the image motives as well (see S8 in SI) . It was found that the generated charge had its \nmaximum value where the tip passes from a full polarized area to the opposite polarization direct ion. \nThe capability  of the mode  to be quantitative  was again test ed by determining  the d 33 value for the BFO \nsample. The same procedure as explained for Figure 4c  and Figure 4d  were employed for Figure 5b  \nand Figure 5c . The squared area in 5b and 5c were used to obtain an average of the lines composing \nsquares resulting in the average profile of  Figure 5d . The top part corresponds to the A square and the \nbottom part corresponds to the B square . The  profiles were  fitted with a gauss -amp curve s, red and  blue \nlines respectively . The values obtained for the fitting curves are 24,1 ± 1,7 fC and -26,6 ± 3,4 fC, which \ndivided b y the  applied force, yield a  d33 values  of 44,1 ± 6,9 pC/N and 48,7 ± 1 2,7 pC/N for DPFM -Si \nand DPFM -So profiles, wh ich, averaged, result in a  d33 value of 46,4 ± 7,2 pC/N.  Such value is \naccordance with what is found in the literature, which ranges between 16 and 60 pC/N54, confirming \nthe feasibility of the mode as a tool to quantify the piezoelectric coefficient.  \n \nCONCLUSIONS  \n \nThe measurement of charges generated by the direct piezoelectric effect with nanoscale  resolution  has \nbeen demonstrated  through the use of a novel AFM based method. The new mode, which we call Direct Piezoelectric Force Microscopy (DPFM), is based in the direct piezoelectric measurement \nprinciple where the piezogenerated charg e is collected by applying  forces in the μN range  to a \npiezoelectric sample  with a  conductive AFM tip . We studied  the feasibility of this new mode by \nexploring the piezogenerated charge dynamics of Periodically Poled Lithium Niobate and Bismuth \nFerrite fer roelectric s. The simultaneous acquisition of DPFM and standard  Piezoresponse Force \nMicroscopy images, can provide a new tool for a better understanding of the electromechanical and \npiezocharge generation dynamics at the nanosca le. The method was also appli ed in spectroscopy \nexperiments which allow the determination of the  piezoelectric  response  outside the domain walls and \nin non -ferroelectric s. We have demonstrated that the new mode is quantitative by measuring  the d33 \nconstants for  PPLN and BTO , which were in accordance with the values previously reported . The \nspecific nature of AFM, with its high force precision, plus the use of a n ultra-low-leakage high \nprecision amplifier makes this mode a promising tool as an accurate, fast and reliable ferroelectric \nmaterial characterization technique.  \n \nACKNOWLEDG MENT S \n \nICMAB acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, \nthrough the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV - 2015 -0496) ,  and the \nprojects Consolider NANOSELECT (CSD 2007 -00041).  and  MAT2014 -51778 -C2-1-R project, co -\nfinanced with FEDER , as well as the Generalitat de  Catalunya (project 2014SGR213) .The authors \nthank ICMAB Scientific and Technical Services.  We thank Oliver Anderson from Rocky Mountain \nNanotechnology LLC for discussion on how to calibrate cantilever spring constant.  \n \nREFERENCES  \n \n1. Curie, J. & Curie, P. Development, via compression, of electric polarization in hemihedral \ncrystals with inclined faces. 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Choi, Y., Tong, S., Ducharme, S., Roelofs, A. & Hong, S. Charge collection kinetics on \nferroelectric polymer surface using charge gradient microscopy. Sci. Rep.  6, 25087 (2016).  \n48. Hiruma, Y., Imai, Y., Watanabe, Y., Nagata, H. & Takenaka, T. Large electrostrain near the \nphase transition temperature of (Bi[sub 0.5]Na[sub 0.5])TiO[sub 3] –SrTiO[sub 3] ferroelectric \nceramics. Appl. Phys. Lett.  92, 262904 (2008).  \n49. Grütter, P., Zimmerm ann-Edling, W. & Brodbeck, D. Tip artifacts of microfabricated force \nsensors for atomic force microscopy. Appl. Phys. Lett.  60, 2741 –2743 (1992).  \n50. Seidel, J. et al.  Conduction at domain walls in oxide multiferroics. Nat. Mater.  8, 229–234 \n(2009).  \n51. Jungk, T., Hoffmann, Á. & Soergel, E. Quantitative analysis of ferroelectric domain imaging \nwith piezoresponse force microscopy. Appl. Phys. Lett.  89, 1–4 (2006).  \n52. Harnagea, C., Pignolet, A., Alexe, M. & Hesse, D. Piezoresponse Scanning Force Microscopy: \nWhat Quantitative Information Can We Really Get Out of Piezoresponse Measurements on \nFerroelectric Thin Films. Integrated Ferroelectrics  44, 113–124 (2002).  \n53. Vaseghi, S. V. Advanced Digital Signal Processing and Noise Reduction . (Wiley, 1988).  \n54. Catalan, G. & Scott, J. F. Physics and applications of bismuth ferrite. Advanced Materials  21, \n2463 –2485 (2009).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1a,  Setup used to record the piezoelectric charge generated by the material through the use of a \nspecial current -to-voltage transimpedance amplifier. The amplifier maintains a reasonable bandwidth \nof 4-5 Hz with a n ultra low input -bias current consumption of less than 0.1 fA. Fig. 1b,  Qualitative \nmodel explaining the charge  generated in a ferroelectric material  during the scan of an AFM tip in \ncontact mode . As the tip moves from left to right, material is strained at the right side of the tip apex \nand  is in an unstrained  state at the left part of tip apex. While the tip scans a single domain, the strained \nand unstrained regions  generate  charges  of opposite sign s and hence the net current is zero. When the \ntip scans the domain wall region, the generated charge present the same sign and  hence a \npiezogenerated charge is created. Fig. 1c, Physical model of the charge generated sign once the tip \nscans from right to left. The strained region is located at the left part of the tip apex, while the \nunstrained region is at the right side. Charge generation occurs again at domain walls, but with an sign \nopposite to that of a   right -left scan. Fig 1d , Spectroscopy sweep model obtained when the tip \nperforms a Force -vs-distance sweep. While the tip exerts a force on the sample a   strain is created. \nOnce the force is released, “unstrain” occurs. The strain and unstrain processes create positive  or \nnegative charge s generated depending  on the polarization  of the domain.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2a   Piezo -generated current map obtained when the tip scans from left to right -trace (DPFM -\nSi). Fig 2b  Piezo -generated current map obtained when the tip scans from right to left -retrace (DPFM -\nSo).  Current is generated at domain walls, orange and blue vertical lines, where the tip strains and \nunstrains opposite domains. Inside the domains, a near zero current si gnal is observed, however a little \ncontrast is present that can be due to surface screening recharging process. Fig 2c  PFM phase image \nand Fig 2d  PFM amplitude image of the same sample, obtained simultaneously with DPFM signals. In \norder to obtain DPFM sig nals a n AC bias was connected to the back electrode of the specimen.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3a  DPFM -Si and Figure 3b  DPFM -So of the proposed PPLN test sample, obtained at different \napplied forces. In order to demonstrate the origin of the recorded current, different forces where applied \nduring the scan -see red line dot.  The current recorded increases  with the applied f orce,  as expected \nfrom a piezoelectric charge generation. Figure 3c  Charge vs Force spectroscopy sweep obtained from \nthe profiles of Figure 3a. The current profiles where integrated with a time constant of 390 ms in order \nto obtaine the charge generated at  a specific pixel. The linear relation displayed between force and \ncharge collected confirms the piezoelectric nature of the generated charge. Figure 3d  current profiles \nextracted from Fig. 3a for different applied forces. Applying 9 μN is not enough to re ad the current \ngenerated as it lies  below the current threshold of the amplifier. As the force is increase d, the amplifier \nresponds to the generated charge, within a symmetric gaussamp like curve.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 4a  DPFM -Si and 4b DPFM -So images of a zoomed region of the PPLN sample recorded in \norder to fully integrate the charge generated. The mean profile average from the images was obtained \nin order to reduce noise. The resulting profiles are plotted in Figure 4c  and Figure 4d . A symmetric \ngauss -amp fitting curve was performed in order to estimate the charge production of the PPLN material \nas the integral of the fitting curve (crossed filled area). Figure 4e  Current -vs-Force spectroscopy sweep \nperformed in the Up d omain configuration, where different Force sweep rates where applied. The \ncurrent generated increases with the increasing force rate. Its sign is the opposite for approach -when \nforce increases - and retract -when force decreases. Figure 4f  Current vs Force s pectroscopy sweep for \nan Up domain (top) and a Down domain (bottom). For an Up domain increasing the force will generate \na positive current while the opposite occurs for a Down domain.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 5a  PFM phase image obtained in a prerecorded 400 nm thick BiFeO 3 (BFO) ferroelectric \nsample. Figure 5b  DPFM -Si image and Figure 5c  DPFM -So image of the prerecorded area. The inner, \nsmallest square recorded has the highest current output, as we are crossing t wo distinct fully polarized \ndomains. However, largest recorded square produces less current output, as we are crossing from a \nvirgin state to a fully polarized domain. Figure 5d  Mean -profile of the dot -line square d area A - in \nFigure 5b - top and Mean profil e of the dot -line square B of Figure 5c. Both profiles where fitted with a \nGaussAmp curve in order to obtain the charge generated by the material and hence its piezoelectric \nconstant.  \n \n \n \n \n"    },    {        "title": "1504.01500v1.Impedance_Measurements_of_the_Extraction_Kicker_System_for_the_Rapid_Cycling_Synchrotron_of_China_Spallation_Neutron_Source.pdf",        "content": "Impedance Measurement s of the Extraction Kicker System for the \nRapid Cycling Synchrotron  of China Spallation Neutron  Source * \nLiangsheng Huang1,2, Sheng Wang1,2, Yudong Liu1,2,#, Yong Li1,2, Renhong Liu3, \nOuzheng Xiao3 \n1 Dongguan Campus , Institute of High Energy Physics (IHEP), Chinese Academy of Sciences \n(CAS)，Dongguan , 523803, China  \n2 Dongguan Institute of Neutron Science (DINS), Dongguan , 523808, China  \n3 Institute of High Energy Physics, Chinese Academy of Science, Beijing , 1000 49, Chin a \nABSTRACT  \nThe Rapid Cycling Synchrotron  (RCS)  of China Spall ation Neutron  Source  (CSNS) \nincludes eight  modules of the ferrite kicker magnets with window -frame geometry. The \nfast extraction kicker system is one of the most important accelerator components, \nwhose inner structure will be the main source of the impedance in the RCS . It is \nnecessary  to understand  the kicker  impedance before it s installation  into the tunnel . The \nconventional  and improved  wire method s are employed for the bench marking \nimpedance measurement  results . The experimental result on the kicker’ s impedance  is \nexplained by comparison with the CST PARTICLE STUDIO  simulation . The \nsimulation  and measurements confirm  that the window -frame  ferrite  geometry  and the \nend plate  are the important structure s causing the coupling impedance. It is prove d in \nthe measurements that the mismatching  among  the long cable, the termination  and the \npower form network  lead a serious oscillation sideband in  the longitudinal  and vertical \nimpedance, which can be restricted by the ferrite  absorb ring. The total impedance of \nthe eight modules  systems is determined  by the scaling law from the measurement  and \nthe impedance measurement of the kicker system is summarized . \n* Supported  by National Natural Science Foundation of China (11175193 , 11275221 )  \n# liuyd @ihep.ac.cn  Key words:  CSNS, RCS, impedance , measurement, extraction , kicker , mismatch  \nPACS: 29.27.Bd.  \n1. Introduction  \nThe driving terms of beam instabilities in accelerator depend on the interaction \nbetween the charged particles and the surroundings which are usually described by the \ncoupling  impedance. The l ongitudinal coupling impedance may lead to beam energy \nloss, thus heating of components  and energy spread [1]. The transverse impedance  \nattribute s to the instability  when the off -axis beam passing through  vacuum components.  \nFor every  new particle accelerator, careful establishment of an “impedance budget” is \na prerequisite for achieving expected performance. Therefore, theoretical analyses, \nsimulation and measurement on the bench of the coupling impedance are crucial tasks \nin the accelerator design, development, and research. The analytical formulas of some \nvacuum components are given in references [2][3][4], and some developed code can be \nused to simulate the coupling impedance, such as ABCI, HFSS  [5] and CST  \nPARTICLE STUDIO [6], but the coupling impedance of very complicate components \nis difficult to study through the analytical formula and the simulation code, so the \nimpedance measurement on the bench is useful at that time.  \nThe conventional wire -method has been widely employed in the coupling impedance  \nmeasurement on the bench, which are insert on -axis wire  (coaxial -wire method)  into \nthe Device Under Test (DUT) to measure the longitudinal impedance and insert two \nparallel wire s with out of phase wire current  (twin -wire method)  into the DUT  to \nmeasure  the transverse impedance.  The longitudinal  coupling impedance measurement \nwith the coaxial wire method was firstly achieved by A. Faltens in 1971  [7]. M. Sands \nand J. Rees exhibited  the coupling impedance measurement in  the time domain  [8], but \nthe impedance measurement in the frequency domain is widely used as the development  \nof technique . G. Nassibian  measured the transverse impedance  in 1978 by the loop \nmethod  [9], the origin  of the twin-wire method.  Mr Walling measured  the transverse \nimpedance  by the twin-wire method in 1987  [10] and he gave th e famous formula of the distributed  component  impedance  - the log arithmic  formula.  F. Caspers and H. \nHahn gave some useful  and comprehensive document for the impedance bench \nmeasurement  [11][12]. Moreover, the validity of the measured longitudinal impedance \nby the coaxial -wire has been interpreted  by H. Hahn  [13]. There has been a lon g history \nin the development of  the wire method and  the improvement of its accuracy  in both \ntheory and technique , the wire method, therefore, is called standard method for the \nimpedance  measurement . In addition, the twin-wire method is developed recently since  \nthe longitudinal coupling impedance measurement by the twin-wire metho d with in \nphase wire current (common -mode signal) is reported  [14], so it is convenient  for the \ntwin-wire method to measure the longitudinal and transverse impedance together . The \nmeasured method and example of  the impedance  is shown in many papers  [15]-[20]. \nThe basic principle of the impedance measurement with the wire method is to use \nthe wire current to simulate the beam current. In this equivalent, there are two \napproximations  [8]: first, there must be similarity on electromagnetic field produced by  \nthe wire current and the beam current. Second, the distortion  on the wire current can \nrepresent the energy variety of the  beam current. Additionally, the algebraic expression \nof the impedance should be decided for the bench measurements . The error of the bench \nmeasurement, therefore , comes from the wire current approximation and the \napproximate  expression of the impedance  measurement.  \nThe China Spallation Neutron Source (CSNS) accelerator s consist of an H- linac , a \nproton Rapid Cycling Synchrotron (RCS), and two beam transport  lines  [21]. The RCS \nis designed to accelerate the proton beam  from 80  MeV to 1.6  GeV with the repetition \nrate of 25  Hz. The beam power of the CSNS accelerator is 100  kW, and it will be \nupgrad ed to 500  kW. Due to the high beam intensity and high repetition rate, the ratio \nof beam loss must be controlled to a very low level.  In the case of  the RCS, the \nextraction kicker represents the most critical impedance item  [22], so it  should be  \nmeasured on the bench. The bunch length in the RCS is shorten ed from 420  ns to 80  ns. \nThe frequency range of interest to the CSNS covers fre quencies, from ~100  MHz down to be low 1  MHz, with emphasis on the 1  - 12 MHz range. The c oaxial -wire method  is \nused to measure longitudinal coupling impedance . The transverse coupling impedance \nis firstly measured by  the twin-wire method. Applying the twin -wire method to the low \nfrequency part entails difficulties  [23], especially below 10  MHz, the signals become \nvery weak and the noise and drift of the instrument lead to results which may no longer \nbe valid , thus, o ne turn loop is  been adopted  to measure the transverse  impedance on \nlow frequency , and the input impedance  is measured .  \nThe extract kicker sys tem is very complicate, the P ulse Form Network (PFN) , the \n12.5 Ω termination  and the connection long cable contribute  to the longitudinal and \nvertical  impedance , so the kicker system i s firstly introduced in the section 2 . The \nlongitudinal and transverse coupling impedances  measurements are  exhibited  in section \n3 and section 4, respectively. Section 5 is the summary  of the kicker impedance  study.  \n2. The extraction kicker system of CSNS/RCS  \nThere are eight kickers in CSNS/RCS  [24][25], a tot al kick  strength of 20 mrad  for \nthe kickers will create about 130  mm central orbit displ acement at the entrance of the \nlambertson magnet, which is required by the separation of the extracted  beam, the \ncycling beam and the septum thickness. The parameters o f the kickers are shown in  \nTable 1. The schematic view of a  kicker system is shown in  Fig 1, and the main part of \nthe kicker system is the window -frame geometry  in Fig 2 and its material is CMD5005 \nferrite. The side strap  at the side of the magnet in Fig 1 connects the upper and lower \nbusbar plates, the busbar and window -frame  ferrite is locate d in a vacuum vessel, 0.58  \nm length . The busbar is fully isolated from the vessel by the ceramic block  of the \nfeedthru . The eddy current strip  (ECS)  locate d in vertical center of the ferrite block to \nreduce the longitudinal impedance  is invisible in Fig 2. The total inductance  of a kicker \nis about 0.9  μH. The capacitance is about 30 pF mainly from the end plate in Fig 1.  \nThe power of the magnet comes from  the PFN [26] via the end plate, the feedthru  \nand the 130 m length cable. The characteristic impedance of the cable is 12.5  Ω, and \nthe impedance of the PFN is 6.25 Ω. The PFN matches t he cable  which  parallel s the 12.5 Ω termination during the time of the beam extraction . The saturated reactor is \ndesigned to cut the PFN when the beam is accelerated, at that time, the cable connects \non the termination, and it  matches.  Due to the error of the cable and the distributed  \neffect of the termination  [17][27], the cable, the terminati on and the PFN mismatch  \nactually , which affect s the longitudinal and vertical impedance.  Therefore, the \nmeasurement  divides  two parts: the kicker without the long cable, the PFN and the \ntermination  (naked kicker) and the kicker  connects  the cable, the PFN and the \ntermination  on (kicker system) . \nThe No. 2 kicker (K2) is firstly manufactured . Based on the kicker , the longitudinal \nand transverse impedances are measured.  \nTable 1: The parameters of the eight kickers  \nNumber  kicker 1  Kicker  2 Kicker  3 - 4 Kicker  5 - 6 Kicker  7 - 8 \nStrength  (T) 0.0558  0.0559 0.0526 0.0571 0.0609 \nAngle (mrad ) 2.6 2.2 2.99 2.9 2.76 \nLength  (m) 0.4 0.32 0.45 0.41 0.36 \nTop time  (ns) >550  >550  >550  >550  >550  \nWidth (mm) 155 212 155 157 168 \nGap (mm) 136 144 153 141 132 \nRise time  (ns) ≤250 ≤250 ≤250 ≤250 ≤250  \nFig 1: The schematic view of the CSNS/RCS kicker system  \n \nFig 2: The schematic view of the window -frame  geometry  \n3. The l ongitudinal impedance measurement  \nThe schematic setup of the coaxial -wire method  is shown in  Fig 3. The kicker  with \na thin copper wire on its beam axis can be regards as a two -port microwave circuit, of \nwhich the forward scatter coefficient can be measured with the Vector Network \nAnalyzer (VNA).  The measurement constantly requires two independent, consecutive \nmeas urements of the forward scatter coefficient S 21 of the DUT and a smooth reference \nbeam pipe (REF)  of equal length . The longitudinal coupling impedance can be found \n[10] from the measured transmission coefficients  S21 as \n \n21,\n21,2 ln( ),DUT\nc\nREFSZRS  (1) \nhere, S21, DUT and S21, REF are the forward scatter coefficient of the DUT  and the REF, \nrespectively. Rc is characteristic impedance formed by the reference pipe with the \ncopper  wire as a coax ial transmission line structure.  The radius of beam pipe and the \nwire are 125 mm and 0.25 mm, respectively . Matching the system impedance  of the \nVNA  (50 Ω) to Rc of the reference line  (372 Ω) is achieved by adding a series metal \nresistor  (Rs , 320 Ω) at the each end of the line . The matching resistor is shielded by the \n35 mm SUCOBOX  [28] on the right. The type of the network analyser  - Agilent \nE5071C is used in the measurement.  In order to decrease the temperature dr ift, the \ntemperature is stable at 25 degrees in the experiment.  \n  \nFig 3: The schematic setup of the longitudinal coupling impedance measurement  \nThe forward transmission coefficient s for the REF, the naked  kicker and the kicker \nsystem  are measured . Replac ing the coefficient into Eq. (1), the lon gitudinal coupl ing \nimpedance of the naked  kicker  and the kicker system  are obtained and shown in  Fig 4 \nand Fig 5. Two peaks on about 18  MHz and 30  MHz in Fig 4 come  from the end plate  \nand the window -frame  ferrite  geometry , respectively. The contribution of the end plate  \nis expor ted by the 130 meters length cable, so the peak on 18 MHz disappears  in Fig 5 \nwhen the cable , the PFN and the termination connect on the naked kicker , but it is clear \nto see that a n oscillation appears at that time . The space of the oscillation is 0.72 MHz . \nThe space  of the reflection from the long cable  can be expressed theoretically as \n \n,2cfL  (2) \nhere, L is the length of the cable . The speed factor of the cable medium , α, is 1.6 for  the \npolythene . The space  is 0.72 MHz, which is absolutely consistent with the result of the \nmeasurement. Therefore, it is roughly certain that the oscillation comes from the \nmismatch  among  the long cable, the termination  and the power form network . It \nmatch es perfect ly on low frequency and the oscillation sideband is invisible , which is \nthe character istic of distribution effect of the termination.  \nTo improve  the validity of the measured longitudinal impedance, the kicker \nimpedance is simulated by CST PARTICLE STUDIO [6]. Due to excessive memories \nand CPU time, it is mostly impossible to sim ulation the real kicker system with the 130 \nm coaxial cable . Therefore, the impedance  of the naked kicker is simulated. A highly \nsimple CST simulation model is constructed, which only includes the window ferrite, \nthe busbar, the end plate, the feedthru and  the vacuum tank. Fig 4 gives the comparison \non the measurement and the simulation  impedances of the naked  kicker . The measured \nlongitudinal impedance agrees well with the simulation results . \n  \nFig 4: The l ongitudinal coupling impedance of the naked  kicker  \n \nFig 5: The l ongitudinal coupling impedance of the kicker system  \nOne of the missions in the measurement is to decrease the impedance, so 8C12 ferrite \nring [29][30] is added in the feedthru to restrict the oscillation. The oscillation  sideband  \nof the longitudinal impedance  mostly disappears  in Fig 6 when the ferrite ring is applied. \nTherefore, the ferrite ring is useful to absorb the reflection.  \n \nFig 6: The  longitudinal impedance  of the kicker system  with and without the 8C12 \nferrite ring  \nThe entire CSNS/RCS fast extraction kicker system consists of eight individual \nkickers. The kicker construct ion of these magnets is similar  although different \ngeometrical dimensions in Table 1. Base on CST PARTICLE STUDIO simulation by \nchanging the scale size of the window -frame geometry, the approximate expression of \nthe longitudinal coupling impedance of others kicker is found as  \n \n2\n2\n2, 1,3,...8.i\ni\niLSZ Z iSL  (3) \nHere, Si is the area of inner surface of the window -frame geometry  of ith kicker, Li is \nthe length of the window and Zi is longitudinal impedance of the ith kicker. S2 and L2 \nare the area of inner surface and the length of the window of the K2 kicker, and Z2 is \nthe measured longitudinal impedance  of the K2 in Fig 5. \nThe longitudinal  average impedance of the eight extraction kicker systems is shown \nin Fig 7, the peak impedance  of the extraction kicker system is about 53+j45 Ω. \n \nFig 7: The longitudinal average impedance of the eight kicker systems  \n4. The transverse  impedance measurement  \nFor the transverse coupling impedance measurement with the wire method, one \nstandard way insert two parallel wires with out of phase signal  (differential -mode)  into \nthe DUT in order to produce a dipole  current moment , which is called  the twin-wire \nmethod . The forward scatter coefficient , S21, of the twin-wire can be measured by the \nnetwork analyzer. The schematic view of the transverse coupling impedance \nmeasurement is shown in Fig 8. The spacing  (2d) of the 0.5 mm diameter copper wire \nis 40 mm, and  the characteristic impedance  of the twin-wire (Rc) is 603 Ω [31] , thus the \n250 Ω metal resistors  are used to match . The differential -mode signal is produced by \nthe MINI hybrid - ZFSCJ -2-1 [32]. The isolation of the hybrid is almost  bigger than 30  \ndB, so the common -mode  error  is extremely very weak  and it is ignored . Four 6 dB \nattenuators between the hybrid and the DUT absorb the reflection from the port.  The \ntemperature is also stable at 25 degrees  \n \nFig 8: The schematic view of the twin-wire transverse impedance measurement  \nThe transverse coupling impedance  of the kicker can be calculated from the \nmeasured scatter coefficients S21, DUT and S21, REF as  \n \n21,\n2\n21,2 ln( ),(2 )DUT\nc\nREFS cZRdS  (4) \nwith the angle frequency ω. \nThe measured  vertical  coupling impedance  is given  in Fig 9. Due to big error below \n10 MHz, the vertical impedance  is shown  from 10 MHz to 100 MHz . It is clear to say \nthat there is an impedance  peak  in the left figure  of the naked kicker  on about 18 MHz \nas the result of  the window -frame  ferrite  geometry  and the end plate  based on the CST \nsimulation,  but it disappears  in the measurement of the kicker system  in right figure . \nMoreover, t he oscillation of the vertical impedance appears for the kicker system in the \nright picture . The oscillation is similar to the one of the longitudinal impedance , and its \nspacing is also 0.72 MHz, which also  comes from the weak mismatch  among  the kicker , \nthe PFN and the termination .  \n     \nFig 9: The measured transverse  coupling impedance of the naked kicker  (left)  and the \nkicker system  (right) by the twin -wire \nThe transverse impedance below 10 MHz by the twin-wire is not g ood since the error \nis serious. To extend the measured frequency, one turn loop  is adapted  to measure \ntransverse impedance on low frequency . The schematic view of the  loop method is \nshown in Fig 10, and t he input impedance s of the DUT and the REF are  also measured, \nso the transverse impedance  can be expressed as  [33] \n \n2,\n2DUT REF\nTc Z ZZ\nd  (5) \nwith the input impedance  of DUT \nDUTZ  and REF \nREFZ . \n \nFig 10: The schematic view of  the transverse impedance  measurement  by the loop \nThe horizontal and vertical impedance s of the naked kicker  are measured by the loop. \nThe horizontal coupling impedances of the naked kicker  and the kicker system are \nshown in  Fig 11, the real part of the horizontal impedance  of the kicker system  is almost \nsimilar to the one of the naked kicker, and the imagine parts are thin ly difference . \nTherefore, the horizontal coupling impedance is mostly not affected  by the cable, the \nPFN and  the termination.  \nThe vertical impedance of the naked kicker  and the kicker system  is shown in Fig 12. \nThe peak on about 18 MHz  in the left figure  also appears  for the naked kicker , which  \nagrees well with the result of the twin-wire in Fig 9, and it also disappears for the kicker  \nsystem  in the right figure . The oscillation  in the right figure , which  comes from the \nreflection  of the mismatch amon g the cable,  the PFN and the termination , is excited \nwhen the kicker connects on the long cable, the PFN and the termination, and the \nspacing of the oscillation sideband is also 0.72 MHz, which is consistent with the result \nof Eq. (2). \n \nFig 11: The measured h orizontal impedance of the kicker  by loop  \n    \nFig 12: The measured vertical impedance of the  naked  kicker  (left)  and the kicker \nsystem  (right)  by loop  \nTo find the start point  of the oscillation, a n improve method of the loop in Fig 13 is \napplied. In order to increase the signal, only the kicker with the long cable (cable open) \nis measured, and the vertical impeda nce is shown in Fig 14. It is easy to show that the \nstart point of th e oscillation is about 0.35 MHz (Owing  to common -mode error, the \nvalue of measured impedance may be incorrect, it is not serious, the point of the \nimprove method  only confirms  the start frequency of the oscillation) . The oscillation of \nthe mismatch from the PFN system  theoretically  is expressed as  \n \n.4startcfL  (6) \nThe start point in Eq. (6) is 0.36 MHz, which agrees well  with the measured result. \nTherefore, it is certain that the oscillation of the vertical impedance of the kicker system \ncomes from the mismatch among the cable , the PFN and the termination . \n \nFig 13: The i mprove loop method of transverse impedance  measurement  \n \nFig 14: The measured vertical impedance of the naked kicker with the cable open  by \nthe improved loop  \nThe transverse impedance of the naked kicker is also simulated by CST PARTICLE \nSTUDIO, and the  vertical and horizontal  results of the simulation and the measurement  \nare shown in Fig 15. It is easy to show that t he difference between measure ment  and \nsimulation is also small . \n   \nFig 15: The simulation and measurement transverse  imped ance of naked kicker  \nThe type of  8C12 ferrite ring  is also adopted  to restrict the oscillation of the vertical \nimpedance . The vertical impedance s of the kicker system with and without  the ferrite \nring are measured by the twin -wire. The oscillation is also extremely decr eased  in Fig \n16. Therefore, the ferrite ring is also useful to absorb the reflection  of the vertical \nimpedance .  \n  \nFig 16: The vertical impedance of the kicker system  with and  without  the ferrite ring  \nThe transverse impedance  of the others kicker  can be expressed [17] as \n \n2\n2\n2 2(), 1,3,...8.()i\nihZ Z ih  (7) \nHere , \niZ  is the transverse impedance of ith kicker, and hi is its height. \n2Z and h2 \nare the measured impedance and the height  of the K2, respectively.  The transverse  \nimpedance of eight kicker systems  are shown in Fig 17, the total vertical impedance is \nabout 5+j10 kΩ/m  and the h orizontal impedance totally is about 3+j22 kΩ/m . \n  \nFig 17: The total transverse impedance of the eight kickers  \n5. Summary  \nThe CSNS/RCS includes eight modules of the ferrite kicker magnets with the \nwindow -frame geometry. The l ongitudinal and transverse coupling impedances of the  \nK2 fast extraction kicker are measured by the coaxial -wire, the twin-wire and the loop \nmethod. The measured  impedances are explained by comparison with  the simulation of  \nCST PARTICLE STUDIO , and they agree well . The simulation  and the measurement  \nindicate  that the window -frame  ferrite  geometry  and the end plate  are the  important \nstructure s causing the coupling impedance.  The mismatching among the cable, the \ntermination  and the PFN affects the longitudinal and vertical coupling impedance , and \n8C12 ferrite ring is useful to absorb the reflection of the mismatching.  Based on the \nimpe dance measurement  and simulation  of the K2 kicker, the total coupling impedance \nof the eight  kicker system s is confirm ed by the scaling law. The longitudinal  average \nimpedance  of the eight kicker systems  is about  53+j45 Ω, the total vertical and \nhorizontal  impedance  are 5+j10 kΩ/m  and 3+j22 kΩ/m , respectively.  \nAcknowledgements  \nWe would like t o acknowledge  the support of many colleagues, especially  helpful \nassistance from prof. Li Shen, prof. Hong Sun, Jun Zhai, Lei Wang, Jiyuan  Zhai, Lin \nLiu and Ahong  Li. The author would also like to thank Prof. Yoshiro Irie and Prof. Fritz \nCaspers  for many discussions and comments in the measurement.  \nReference:  \n[1] A. Chao, Physics of Collective Beam Instabilities in High Energy Accelerator, \nWiley, New York (1993) . p. 81, 164,  174. \n[2] Bruno W Zotter, Semyon A Kheifets, Impedances and Wakes in High -Energy \nParticle Accelerators, World Scienti fic, Singapore (1998) . 73 – 368. \n[3] S. Y. Lee, Accelerator Physics, (World Scientifi c, Singapor e, 2004) . p. 216, 369.  \n[4] A. Chao and M. Tigner, Handbook of Accelerator Physics and Engineering, World \nScientific, Singapore (1998).  p. 194.  \n[5] HFSS, http://www.ansys.com , 2014.11.  \n[6] CST PARTICLE STUDIO, http://www.cst.com,  2014.11. \n[7] A. Faltens, et al. , An Analog  Method  for Measuring the  Longitudinal  Coupling  \nImpedance  of a Relativistic  Particle Beam with Its Environment , Proc . of the 8th \nInternational Conference on High -Energy Accelerators, Geneva,  1971.  p. 338 . \n[8] M. Sands and J. Rees, A Bench Measurement of the Energy Loss of a Stored Beam \nto a Cavity, PEP -95, August 1974.  \n[9] G. Nassibian et al. , Methods for Measurement Transverse Coupling Impedance in \nCircular Accelerator, Nucl. Instr. and Methods. A, 159 (1979) . 21 - 27. \n[10] L. S. Walling, et al. , Nucl. Instr. and Methods . A, 281 (1989). p. 433.  \n[11] F. Caspers, BENCH ME THODS FOR BEAM -COUPLING IMPEDANCE \nMEASUREMENT, CERN PS/88 -59，Geneva, 1988.  \n[12] H. Hahn, F. Pedersen , On Coaxial Wire Measurement of the Longitudinal Coupling \nImpedance, BNL50870, 1978.  \n[13] H. Hahn, Validity of coupling impedance bench measurements ，PRST  - AB, 3, \n122001  (2000).  [14] Takeshi Toyama, et al. , Coupling  Impedance  of the  J-PARC K icker  Magnets , Proc . \nof HB2006, Tsukuba, Japan,  2006.  p. 140 \n[15] Huang Gang, et al. Longitudinal  Broadband  Impedance  Measurement  System  by \nCoaxial  Line Methods , Proc . of the  PAC’01, Chicago,  2001.  p. 2060.  \n[16] H. Hahn, D. Davino, M easured  Transverse  Coupling Impedance  of RHIC I njection  \nand Abort Kickers , Proc . of the  PAC’01, Chicago, IEEE , 2001.  p. 1829.  \n[17] H. Hahn, Impedance measurements of the Spallation Neutron Source extraction \nkicker sy stem, PRST  - AB, 7, 103501 (2004).  \n[18] B. Podobedov and S. Krinsky, Transverse impedance of axially symmetric tapered \nstructures , PRST - AB, 9, 054401 (2006).  \n[19] Y . Shobuda, et al. , Horizontal  Impedance  of the  Kicker  Magnet  of RCS at J-PARC, \nProc. of IPAC’10, Kyoto, Japan,  2010. p. 2024.  \n[20] Benoit Salvant, Impedance Model of the CERN SPS and Aspects of LHC Single -\nBunc h Stability, Ph. D Thesis, CERN , 2010.  \n[21] S. Wang, et al., An Overview of Design for CSNS/RCS Accelerators and Beam \nTranspor t, Scientific China Physics Mechanics & Astronomy, V ol. 54, (2011) . p. \n239. \n[22] Yudong Liu, Impedance and Beam Instability in RCS/CSNS, High Power Laser \nand Particle Beams, 25  (2), 2013 . p. 465.  \n[23] H. Hahn, Direct Transverse Coupling Impedance Measurements of Kicker \nMagnets, BNL/SNS TECHNIC AL NOTE, NO. 120, BNL, 2003.  \n[24] J. Y . Tang, et al., Extraction  System  Design  for the  CSNS/RCS, Proc . of EPAC \n2006, Edinburgh, Scotland,  2006.  p. 1777.  \n[25] W. Kang, et al. , Design and Prototype Test of CSNS/RCS Injection and Extraction \nMagnets, IEEE T rans.  on App. Sup., Vol. 20, NO . 3, 2010 . p. 356. \n[26] CHI Yun -Long, WANG Wei, Design of Pulse Power Supply for CSNS Extraction \nKicker Magnet, Chinese Physic s C (HEP  & NP), V ol. 32 (Z1). 2008 . p. 25.  \n[27] Y . Shobuda, et al., Measurement scheme of kicker impedances via beam -induc ed voltages of coaxial cables, Nucl. Instru. & Meth. A , 713  (2013) . p.52. \n[28] http://de.farnell.com/huber -suhner/fbb -cb-50-0-1-e/sucobox -prototy -\nbox/dp/4162 950. 2013.11 . \n[29] F. Caspers, private contact, Oct., 2013.  \n[30] http://www.ferroxcube.com/FerroxcubeCorporateReception/datasheet/FXC_HB2\n013.pdf, 2014.2.16 . p. 174.  \n[31] A.W. Gent, Capacitance of Shielded Balanced -Pair Transmission Line, Electrical \nComm., 33 (1956) . p. 234. \n[32] Hybrid, http://www.minicircuits.com , 2013.11.  \n[33] A. Mostacci, et al. , Bench  Measurements  of Low Frequency  Transverse  Impedance , \nProc. of the  PAC’03, 2003. p. 1801.  "    },    {        "title": "0910.5789v1.Resonant_x_ray_scattering_in_3d_transition_metal_oxides__Anisotropy_and_charge_orderings.pdf",        "content": "Resonant x-ray scattering in 3d-transition-metal oxides: \nAnisotropy and charge orderings \nG. Subías1, J. García1, J. Blasco1, J. Herrero-Martín2 and M. C. Sánchez1  \n1 Instituto de Ciencia de Materiales de Aragón, Departamento de Física de la \nMateria Condensada, CSIC-Universidad  de Zaragoza, 50009-Zaragoza, Spain \n2 European Synchrotron Radiation Facility, 38043-Grenoble-Cedex, France \n \nE-mail: gloria@unizar.es \n \nAbstract . The structural, magnetic and electronic properties of transition metal oxides \nreflect in atomic charge, spin and orbital degrees of freedom. Resonant x-ray scattering \n(RXS) allows us to perform an accurate i nvestigation of all these electronic degrees. \nRXS combines high-Q resolution x-ray diffraction with the properties of the resonance \nproviding information similar to that obtained by atomic spectroscopy (element \nselectivity and a large enhancement of scattering amplitude for this particular element and sensitivity to the symmetry of the elect ronic levels through the multipole electric \ntransitions). Since electronic states are coupled to the local symmetry, RXS reveals the occurrence of symmetry breaking effects such as lattice dist ortions, onset of electronic \norbital ordering or ordering of electronic charge distributions. We shall discuss the \nstrength of RXS at the K absorption edge of 3d transition-metal oxides by describing \nvarious applications in the observation of local anisotropy and charge \ndisproportionation. Examples of these resonant effects are (I) charge ordering \ntransitions in manganites, Fe\n3O4 and ferrites and (II) forb idden reflections and \nanisotropy in Mn3+ perovskites, spinel ferrites and cobalt oxides. In all the studied cases, \nthe electronic (charge and/or anisotropy) orderings are determined by the structural distortions. \n1.  Introduction \nThe discovery of new electronic properties, including high-T C superconductivity, colossal \nmagnetoresistance and multiferroic modifications , in transition-metal oxides has fuelled a \nresurgence of interest in atomic charge, spin a nd orbital degrees of freedom in systems of highly \ncorrelated electrons. X-ray spectroscopy tec hniques are highly sensitive to these electron \ndegrees of freedom. However, they probe short- distance correlations only. In contrast, x-ray \ndiffraction reveal long-range ordered static correlations. Resonant x-ray scattering (RXS) \ncombines absorption and diffraction as they have in common the x-ray atomic scattering factor \n(ASF), f, which is usually written as: f = f0 + f ’+ if ” [1]. It contains an energy independent part, \nf0, corresponding to the classical Thomson scattering and two energy-dependent terms, f ‘ and  f \n“, also known as the anomalous ASF. RXS occurs when the x-ray energy is tuned near the \nabsorption edge of an atom in the crystal. In this case, the anomalous ASF strongly depends on \nthe photon energy, which manifests in marked variations of the scattered intensity. This \ndependence of the scattered intensity appears in any Bragg reflection when crossing the \nabsorption edge of a constituent atom.  \nThe study of the energy-dependent modulation of the diffraction intensity of intense Bragg \npeaks is the scope of the diffraction anomalous fine structure technique (DAFS) [2,3]. This technique allows the determination of the local structural information around the anomalous atom that is chemical and valence specific sim ilar to that of Extended X-ray Absorption Fine Structure (EXAFS). The advantage of DAFS is that it is spatially and site-selective. Our interest \nhere is focused on the study of either weak-allo wed or forbidden reflections that appear in a \nphase transition. In the first case, the anomalous  scattering contribution is comparable to the \nThomson one and the peculiar characteristics of RXS can be studied. The intensity of weak \nsuperlattice reflections can be due to either a st ructural modulation, contributing to the structure \nfactor as a Thomson term or because the anomalous ASF of atoms, now in different crystallographic sites, differ in some energy range. Normally, these differences are larger for \nphoton energies close to an absorption edge, s howing an enhancement (or strong decrease) of \nthe scattered intensity just around the absorption threshold. In the case of symmetry forbidden \nreflections, only the resonant term is present. Since RXS involves virtual transitions of core \nelectrons into empty states above the Fermi le vel, the excited electron is sensitive to any \nanisotropy around the anomalous atom so that the anomalous ASF has a tensorial character. \nThus, a reflection can be observed on resonance if any of the components of the structure factor \ntensor is different from zero. In the case of weak superlattice reflections, the resonant atoms \noccupy different crystallographic sites and have different local structures. Thus, the anomalous \nASF are different at energies close to the abso rption edge and the scattered intensity shows a \nresonance reflecting the differences of the ASF between these non-equivalent atoms. RXS \nintensity is then observed coming from the differe nce between the diagonal terms of the ASF. In \nthis case, there is a Thomson contribution and the analysis of the spectral shape must include it. \nOn the other hand, resonances observed in forbidde n reflections are related to atoms that occupy \nequivalent crystallographic sites. Symmetry elements with translation components (screw axes and glide planes) of the crystal space group transf orm an atom into another equivalent in the \nlattice, but with a differently oriented atomic  surrounding. This makes that some of the off-\ndiagonal terms change sign after these symmetry operations, giving rise to structure factor \ntensors that contain non-vanishing off-diagonal te rms. Templeton & Templeton [4] first noticed \nsuch reflections that show a local polarization anisotropy of the x-ray susceptibility and they are \nnow known as ATS reflections. We note that ATS reflections have strong polarization and \nazimuth dependences. \nThe vector potential of the electromagnetic field in the matter-radiation interaction term can \nbe developed in a multipolar expansion in such a way that the symmetry of the excited levels \ncan be chosen. Thus, we can speak on dipol ar-dipolar transitions, dipolar-quadrupolar \ntransitions, etc. Consequently, resonant met hods are also sensitive to the symmetry of the \nelectronic shells, which compose the intermedia te states. For electric dipole-dipole (E1E1) \ntransitions the wave vectors ki and kf of the incident and scattered x-rays do not enter the \nscattering amplitude. The strength of the x-ray resonances associated with electric quadrupole-\nquadrupole (E2E2) and electric dipole-quadrupole transitions (E1E2) are less intense than for E1E1 process but must content the k\ni and kf dependence. A complete polarization and azimuthal \nanalysis of the RXS experiment is needed to  assign the multipolar origin of the RXS signal. \nIn this contribution, after a recall of the RXS amplitude, we will discuss several examples \nthat correlate with the charge and orbital orde ring concepts. We restrict to the metal K edges \nand mainly to the dipolar-dipolar channel, which describes most of the phenomenology. Two types of reflections are observed. The first one originates from the different anomalous scattering factors of non-equivalent atoms in the crystal. The occurrence of this type of resonance is correlated to the energy shift of th e absorption edge (chemical shift) and it has been \nconsidered as a proof of charge ordering (CO). The second one arises from the anisotropy of the anomalous scattering factor of an atom at crystallographic equivalent sites. These ATS \nreflections have been assigned to orbital ordering (OO). However, the first type of reflections \ncan also exhibit anisotropic behaviour. In all th e studied cases, we conclude two main results: (i) \nthe charge segregation is much smaller than one  electron and consequently, it would be better \ndescribed as a charge modulation and (ii) the anisotropy is present in the p-empty density of states and it seems not to be correlated with a real d-orbital ordering.  \n2.  Resonant x-ray scattering tensor \nIn RXS, the global process of photon absorption,  virtual photoelectron excitation and photon re-\nemission, is coherent through the crystal, gi ving rise to the usual Bragg diffraction condition  \n) (\" '\n0 if f f ej Fj j jjR Q i+ + ∑=⋅rr\n    (1) \nwhere Rjr is the position of the j-th scattering atom in the unit cell, i fk k Qrrr\n− =  is the \nscattering vector (  and  are the wave vectors of the incident and scattered beams) and \n is the Thomson scattering part of the at omic scattering factor. The resonant part, , \nis given by the expression [5] ikrfkr\njf0\" 'if f+\n \n∑Γ− − −〉 〉〈 〈 −\n= +\nn n\ng ngi\nn nf\ng g ne\ni E EO O E Emfif\n2) (ˆ ˆ ) (1* 3\n2\" '\nωψ ψ ψ ψ\nωhhh (2) \nIn this expression, ωh is the photon energy,  is the electron mass, emgψ describes the \ninitial and final electronic state with energy  and and gEnEnΓ are the energy and inverse \nlifetime of the intermediate excited states nψ. The interaction of the electromagnetic radiation \nwith matter is expressed by the operators  and . By multipole expansion of these \noperators up to the electric quadrupole term, we have [6]: iOˆ *ˆfO\n \n)211 ( ˆ ) ( ) ( ) (r k i r Of i f i f i rrrr⋅ − ⋅ = ε     (3) \nHere rr is the electron position measured from the absorbing atom and  is the \npolarization of the incident(scattered) beam. Correspondingly, we get that there are three \ncontributions to the resonant scattering factor: dipole-dipole ( dd), dipole-quadrupole ( dq) and \nquadrupole-quadrupole ( qq). ) (f iεr\nUsing the Cartesian reference coordinate system defined in figure 1 for the photon \npolarization and wave vector, we can develop the scalar product of equation (3) and the resonant \nscattering amplitude of equation (2) can be written in the form: \n \n  \n⎥⎦⎤\n⎢⎣⎡∑∑ ∑ + − −∑Γ− − −−\n= +\nβ α αβγ αβγδαβγδ δ γ β α βαγ γ αβγ γ β α αβ β α ε ε ε ε ε εωω\n,* * * *3\n2\" '\n41) (22) () ( 1\nQ k k I k I kiDi E EE E mfif\ni f i f f i i f i fn n\ng ng n e\nhhh\n(4) \n \nwhere α, β, γ, δ are indexes that vary independently over the three Cartesian directions x, y, z, \nand the transition matrix elements ,  and  associated to dd, dq and qq \ncontributions are characterized by the following Cartesian tensors of second, third and fourth \nrank, respectively: αβDαβγIαβγδQ\n \n〉 〉〈 ∑〈 =〉 〉〈 ∑〈 =〉 〉〈 ∑〈 =\ng n\nnn gg n\nnn gg n\nnn g\nr r r r Qr r r Ir r D\nψ ψ ψ ψψ ψ ψ ψψ ψ ψ ψ\nδ γ β α αβγδγ β α αβγβ α αβ\n   (5) Figure 1. Scheme of a general \nRXS experiment (vertical \nscattering plane) \n \nIt is important to determine the symmetry properties of the D, I and Q tensors [7] that depend \non two factors: (i) the transformation properties of a space group itself and (ii) the local \nsymmetry of the resonant atoms position. Referring to non-magnetic samples, the structure \nfactor tensor is a second rank symmetric tensor (parity-even) with six independent components \nin the dd approximation. This number reduces upon taking into account the local site symmetry \nof the atom. The qq term is also symmetrical under the inversion symmetry, whereas the dq \ncontribution is only antisymmetrical. As a result, if an atom sits in an inversion centre, the I \ntensor must be zero and the dipole-quadrupole tr ansition is only allowed for atoms breaking the \ninversion symmetry. \nIn the following we shall describe some significant RXS experiments following the two \ntypes of resonant reflections, weak-allowed and forb idden. In the first case, we shall relate them \nto the charge modulation, intimately joined to the transition metal-ligand bond distances and in \nthe second case we shall relate them to the anisot ropy of the geometrical local structure around \nthe resonant atoms. Since the charge is a time-re versal invariant quantity, we shall deal either \nwith pure electric dd and qq transitions that are even under parity or with parity-odd electric dq \ntransitions. Time-reversal odd events that are related to the magnetic properties of the system will be not considered. \n3.  Resonant effects due to different cry stallographic sites (charge) ordering \nUnderstanding the charge state in the high and low temperature phases of mixed valence \ntransition-metal oxides is of fundamental interest  in the context of metal-insulator transitions \nthat are assumed to be driven by CO. The sens itivity of RXS to the CO relies on the fact the \nenergy values of the absorption edge for the two different valence states of the transition-metal \natom are slightly different (known as chemical shift). If long-range order of these valence states \nexists, superlattice reflections due to the contrast between the atomic scattering factors of the two valence states will exhibit a resonance enhancement. The point is that CO is intimately \ncorrelated with the associated crystal distortions  coming from the structural transitions that \naccompanied the metal-insulator ones. Thus, a qu estion arises whether the electronic CO is the \ncause or the effect of the lattice distortions. \nThe concept of CO in solids was first applied by E. J. W. Verwey to the metal to insulator \ntransition that occurs in magnetite (Fe 3O4) at T v ∼ 120 K, now known as the Verwey transition \n[8]. Above T v, Fe 3O4 has the inverse spinel cubic AB 2O4 structure, where A and B are the \ntetrahedral and octahedral Fe sites, respectivel y. Verwey originally proposed that the hopping of \nvalence electrons on the octahedral B-site subl attice is responsible for metallic conductivity. In \nthe insulating phase, spatial localization of the va lence electrons on these B-sites gives rise to an \nordered pattern of Fe3+ and Fe2+ ions in successive [001] planes (cubic notation). The B-site \nsublattice, shown in the inset of figure 2, can be regarded as a diamond lattice of tetrahedra of \nnearest-neighbour Fe atoms sharing alternate corners. This simple model was implying the \nobservation of (0, k,l) reflections with k+l=4n+2 in the low temperature phase. These reflections are forbidden above T v because of the diamond glide plane of the spinel structure. The first RXS \nexperiments in magnetite showed th at (002) and (006) reflections be longing to this type of cubic \nforbidden reflections originates from the local stru ctural anisotropy of the Fe atoms at the B-\nsites that have a trigonal point symmetry ( 3m) [9-11]. These works discarded the Verwey’s CO \nmodel but they did not guarantee the lack of CO with other periodicities of the cubic unit cell. \nIn order to investigate other possible CO pe riodicities, we need to start from the low \ntemperature crystallographic structure. The symmetry lowering Cc m Fd→3  generates 8 and \n16 non-equivalent Fe sites at tetrahedral and octahedral positions, respectively; each one can \nhave its own local atomic charge. However, th e exact structure is not yet perfectly known [12-\n14]. A good approach to the real structure consists of a  cell with lattice parameters c P/ 2\nc c c a a a2 2 / 2 /× × ≈ , ac being the cubic cell parameter [12,14]. Complexity is greatly \nreduced because there are only 6 non-equivalent Fe atoms, two in tetrahedral sites (A 1 and A 2) \nand four in octahedral sites (B 1, B 2, B 3 and B 4). It can be noticed that atomic displacements in \nthis low temperature structure result in two ma in types of superlattice reflections, which are \nindexed in the cubic notation: (I) ( h,k,l) reflections such that h+k=even resulting form the loss of \nthe  translation that give rise to charge modulations with wave vector fcc c q) 1 , 0 , 0 ( =r and (II) \nhalf-integer ( h,k,l+1/2) reflections arising from the doubling of the cell along the c axis that \ncorresponds to charge modulation with c q ) 2 / 1 , 0 , 0 ( =r. We examine now the limit for the \npossible charge segregation over the octahedral atoms along the c axis given by the sensitivity \nof the RXS technique in a highly stoichiometric single crystal of Fe 3O4 (TV=123.5 K) [15].  \nFigure 2 (left panel) shows the energy dependence of the intensities for some characteristic \nBragg and forbidden reflections at the Fe K- edge and at 60 K compared to the fluorescence \nspectrum. Experimental data have been corrected for absorption. \n \n7,10 7,11 7,12 7,13 7,14 7,15020406080100\n(0 0 7/2)σ-π(0 0 1)σ-σ(1 1 0)σ-σ(4 4 3/2)σ-σ\nx 103\n  Intensity (arb. units)\nEnergy (keV) \n00.20.40.60.811.2\n116 118 120 122 124 126 128(4,0,1/2)\n(0,0,7/2)\n(0,0,1)I (T) / I (115 K)\nT (K)\nFigure 2. The left panel shows some of the experime ntal RXS spectra around the Fe K-edge in \nFe3O4, corrected for absorption. The right pane l shows the temperature dependence of the \nintegrated intensities, on and off-resonance, normalized to the low-temperature value. \n \n(a) Energy scans of the permitted (0,0,1) and (1,1 ,0) reflections show three resonant peaks, a \nfirst one at the Fe K threshold (7118 eV) and the other two around 7124-7129 eV, which \ncorresponds to the white line in the fluorescence sp ectrum. The observed strong resonant effect \nis a consequence of electronic and structural differences among the Fe atoms at B1 and B2 \noctahedral sites. This can be parameterised  in terms of valence as a charge segregation δ =0.23 e. \nThis result confirms the lack of ionic CO in terms of Fe2+ and Fe3+ ions, in agreement with \nprevious RXS [16-18] and synchrot ron powder diffraction [14] studies. (b) Resonant intensity is only observed in the σ−π’ channel for the (0,0,7/2) reflection, \nindicating that this is a forbidden reflection in the low temperature phase. The energy scan \nshows a three-peak structured resonance nearly  at the same energies as the (0,0,1) and (1,1,0) \nreflections. In this case, the electronic anisot ropy comes from interference among equivalent \ncrystallographic sites.  Six different sites are present for the Fe atoms so up to six different terms \ncould contribute to the resonant signal [19]. Th e superlattice (4,4,3/2) corresponds to the same \nperiodicity along c as the forbidden (0,0, l/2) reflections. However, it displays hardly any \nresonant effect opposite to what is expected to occur at those very weak reflections. Therefore, \nwe can conclude that no charge segregation ex ists with (0,0,1/2) periodicity and the ATS \n(0,0,l/2) reflections have its origin in the loss of octahedral and tetrahedral symmetry originated \nby the structural phase transition. \nIn order to establish the correlation between the lattice distortion and the charge segregation \nand anisotropy orderings, we have measured the temperature dependence of the intensity of the \nfollowing superlattice reflections: (4,0,1/2) off-res onance (E=7.1 keV) and (0,0,1) and (0,0,7/2) \non resonance (E=7.125 keV), which is reported in  figure 2 (right panel). The resonant and non-\nresonant signals simulta neously disappear at T v (±0.5 K) and the intensity of all these reflections \nis zero at temperatures above 125 K. This result shows that RXS in Fe 3O4 comes from the \nordering of local distortions at the structural tran sition, which leads to an ordered formal charge \nsegregation and electronic anisotropy at the Fe atoms [15].   \nWe will comment now on the half-doped manganites such as Nd 0.5Sr0.5MnO 3 and \nBi0.5Sr0.5MnO 3. It was proposed the ordering of an alternating pattern of Mn3+ and Mn4+ ions \nwas predicted leading to the onset of superlattice reflections doubling the b axis of the \northorhombic Pbnm (Ibmm) cell [20]. The observed modulation wave vectors are (0, k,0) and \n(0,k/2,0) with k  odd for the ordering of charge and orbital degrees of freedom, respectively. The \noccurrence of orbital order (OO) is also predicted independently from the CO because Mn3+ are \nJahn-Teller distorted ions. We have measured the energy dependence of the intensity at the Mn \nK edge at Q=(0,3,0) and Q=(0,5/2,0) for both, Nd :Sr [21] and Bi:Sr [22] single crystals. Figure \n3 summarizes the results obtained.  \n \n6,54 6,55 6,56 6,570,00,40,81,21,62,0\n  Intensity (arb. units)\nE (keV) Bi0.5Sr0.5MnO3\n Nd0.5Sr0.5MnO3 \n6,54 6,55 6,56 6,570,00,20,40,60,81,01,2 Bi0.5Sr0.5MnO3\n Nd0.5Sr0.5MnO3\n  Intensity (arb. units)\nE (keV)\nFigure 3. The left panel shows the normalized intens ity for the non-rotated polarization channel \nat the (0,3,0) peak associated to CO whereas the right panel shows the rotated polarization \nchannel at the (0,5/2,0) p eak associated to OO in Nd 0.5Sr0.5MnO 3 and Bi 0.5Sr0.5MnO 3.  \n  \nNon-resonant intensity can be observed away from the K-edge of Mn and a broad resonance \nis found at the absorption edge at (0,3,0) reflections in the non-rotated ( σ-σ’) channel. On the \nother hand, a strong Gaussian-shaped resonance is  observed at energies close to the K-edge of \nMn at (0,5/2,0) reflections only in the rotated ( σ-π’) channel, which identifies these half-integer \nreflections as structurally forbidden ones. The variation with azimuth of these resonances shows \na characteristic oscillation with π periodicity. We note that in this case, the two kinds of behaviour mixed since the marked different anisot ropy of the two types of atoms. The resonant \nscattering at either (0, k,0) or (0, k/2,0) reflections does not depend on the  wave vector and it \narises from E1 (1s→  np) transition. The temperature dependence of the resonant intensities \nshows that the two types of reflections disappear at the metal-insulator phase transition. k\nThe complete analysis of the energy line-sh ape and the azimuth and polarization dependence \nof the resonant intensities is carried out using a semi-empirical structural model [23]. The \ncheckerboard arrangement of two types of crysta l Mn sites gives an excellent agreement with \nthe experimental data. These two sites differ in their local geometric structure: one site is \nanisotropic  (tetragonal distorted oxygen octa hedron) and the other site is isotropic  (nearly \nundistorted oxygen octahedron). Estimates of th e valence modulation between the two Mn \natoms are slightly variable depending on the ma nganite but all fall below the ideal charge \nsegregation of ±0.5e. As shown in [21] and [22], ±0.08 for Nd 0.5Sr0.5MnO 3 and ±0.07 for \nBi0.5Sr0.5MnO 3, respectively. \nCO for x different from 0.5 is in general not we ll defined. We have also explored further the \nBi1-xSrxMnO 3 system toward the Mn3+-rich side, that is for x=0.37 [24]. Superlattice reflections \nof (0,k ,0) and (0,k /2,0) types with k odd were observed at the Mn K-edge. This is to be \ncompared with the observation of the K-edge resonances in Bi 0.5Sr0.5MnO 3. Figure 4 shows an \nexample of the resonant enhancement as observed at (0,3,0) and (0,7/2,0) reflections in Bi\n0.63Sr0.37MnO 3. \n \n1.522.533.54\n00.511.5\n6.53 6.54 6.55 6.56 6.57 6.58Intensity σ-σ'Intensity σ-π'\nE (keV)x 200(0,3,0)\n00.20.40.60.811.2\n-60 -30 0 30 60 90 120 150 180\nϕ (º)E=6552.6 eV\nσ-σ'Intensity (arb. units) \n6540 6550 6560 65700,00,20,40,60,81,0\n-60 -40 -20 0 20 40 60 80 100 120 140 160 1800,00,20,40,60,81,0 E=6551.6 eVIntensity (arb. units)\nϕ (deg)(0, 7/2 ,0)\n  Intensity (arb. units)\nE (keV)\nFigure 4. RXS results in Bi 0.63Sr0.37MnO 3 at the Mn K-edge. (a) Polarization analysis of the \n(0,3,0) reflection as a function of photon energy. (b) energy dependence of the forbidden \n(0,7/2,0) reflection for the σ-π’ channel. The insets show th e respective azimuthal angle \ndependence on resonance. \n \nThe energy dependence of the intensity for both, weak-allowed (0, k,0) and forbidden \n(0,k/2,0) reflections is identical to that observe d in the half-doped bismuth manganite (figure 3). \nThe dependencies of the resonant intensities w ith azimuthal angle reveals identical twofold \nsymmetry too. We note that the σ-σ’ intensity of the (0,3,0) peak approaches the non-resonant \nintensity at the minimum opposite to the constant  evolution expected for a pure CO reflection. \nThese results indicate that the checkerboard orderi ng of two types of Mn atoms in terms of the \nlocal structure in the ab plane in a ratio 1:1 is strongly stable and extends to doping \nconcentrations x<0.5. Intermediate valen ce states lower than +3 .5 are deduced for \nBi0.63Sr0.37MnO 3, where the charge disproportion is found to be ±0.07. \nThe last example is provided by La 0.33Sr0.67FeO 3, which shows a charge modulation that is \ncommensurate with the carrier concentration (n=1-x) and corresponds to a wave vector \nq=(2 π/ap) (1/3,1/3,1/3), being a p the primitive cubic lattice parameter. Ordered layers of Fe3+ \nand Fe5+ ions in a sequence of …Fe3+Fe3+Fe5+… along the cubic [111] direction were originally proposed to explain the metal-paramagnetic to insulator-antiferromagnetic transition at 200 K \n[25]. We have investigated this three-fold CO using the RXS technique [26]. Superlattice \n(h/3,h/3,h/3) reflections in the pseudocubic cell notation were observed in the non-rotated σ-σ’ \npolarization channel for h=2, 4 and 5. These reflections are forbidden in both cubic ( Pm-3m ) \nand rombohedral ( R-3c ) symmetries.  However, they exhibit a resonant behavior at energies \nclose to the Fe K edge on top of a non-resonant signal, as shown in figure 5. \n \n7,10 7,11 7,12 7,13 7,14 7,15051015202530354045505560\n-90 -60 -30 0 30 60 900123(4/3,4/3,4/3)\nσ-σ'\n  Integr. Intens. (a.u.)\nAzimuthal angle ( °)\n(5/3,5/3,5/3)(2/3,2/3,2/3)(4/3,4/3,4/3)\n  Intensity (electron units)\nE (keV)Figure 5. Energy \nscans of the σ-σ’ \nintensity for the \n(h/3,h/3,h/3) \nreflections measured \nat 10 K and corrected for absorption. The inset shows no dependence of the \nresonant intensity \n(7129.5 eV) on the \nazimuth angle for \n(4/3,4/3,4/3) reflection. \n \nThe non-resonant intensities are explained due to  a periodic structural modulation of both Fe \nand Sr(La)O 3 atomic planes along the cubic [111] direction. To account for the strong variation \nin the Thomson scattering among the different satellite reflections, the Fe and Sr(La)O 3 atomic \nplanes must move in opposite senses, keeping the inversion symmetry of the supercell. These \nshifts differentiate two crystallographic sites for the Fe atoms and produce an ordered sequence \nof two compressed and one expanded FeO 6 octahedra. The compressed octahedron is \nasymmetric with three short and three long Fe-O bonds whereas the expanded one is regular. \nConcerning the local charges of the two types of  Fe cations, the chemical shift between the \nexpanded and compressed Fe atoms was found to be 0.7 ±0.1 eV. This is to be compared to the \nenergy shift of 1.26 eV between Fe3+ and Fe4+ [27]. Assuming a linear relationship between \nenergy shift of the K absorption edge and local charge, we determined that the amount of charge \nsegregation is 0.6 e. Since the resonant intensity is constant as a function of the azimuth angle \nover a range of 180 º, as shown in the inset of figur e 5, the Fe atoms have not a local anisotropy \nin this case. This result indicates the presence of  a pure charge density wave ordering in the \nmixed-valent perovskite La 0.33Sr0.67FeO 3. \n4.  Resonant effects due to anisotropy (orbital) ordering \nThe direct observation of OO is a difficult task because it is accompanied by other effects such \nas lattice distortions and charge segregations. The study of the RXS and the anisotropic \ncharacter of the x-ray scattering were developed in  crystallography more than 20 years ago [4]. \nIn particular, Dmitrienko [28,29] developed a general theoretical treatment of the “forbidden” \nATS reflections, deriving new extinction rul es valid near the absorption edge. In the \nmicroscopic point of view, these ATS reflections are due to the presence of the absorbing atom in an anisotropic chemical environment, whic h brings about the orientation of unoccupied \nelectronic levels. \nThe experimental application of RXS to the study of OO started in 1998, when Murakami \nand co-workers [30] investigated the CO and OO in La 0.5Sr1.5MnO 4 at the Mn K-edge. It was concluded the successful observation of OO of the  electrons ( - and -\ntype orbital) on Mnge ) 3 (2 2r z− ) (2 2y x−\n3+ sites by detecting the (2n +1/4,2 n+1/4,0) forbidden reflections. The \nangular dependence around the scattering vector of the intensity (azimuthal dependence) \nconfirms that they result from the anisotropic ch aracter of the anomalous part of the atomic \nscattering factor. However, it was later demonstr ated that the Jahn-Teller distortion of the \noxygen octahedra surrounding the Mn atoms is suffi cient to reproduce the experimental results \nat the K edge without invoking any contribution of  the 3d-OO [31].After this first observation, \nRXS is widely used to study the orbital degree of freedom in several manganites [32-36] and \nother transition-metal oxides [37-40]. In par ticular, the following experiments provide key \ninformation for the mechanism of RXS and OO. \n(1) The parent compound of colossal magnetoresistive manganites, LaMnO 3, was expected \nto show an alternating ordering of  and  orbitals in the ab plane below \n780 K due to the greatly distorted MnO) 3 (2 2r x− ) 3 (2 2r y−\n6 octahedron and the A-type  antiferromagnetic order \n[41]. The observation of resonance in the σ-π’ scattered intensity of the (3,0,0) forbidden \nreflection at the Mn K-edge was initially interpreted as a direct probe of this OO [32]. Moreover, the increase of the resonant signal with decreasing temperature as T\nN was \napproached from above was interpreted as a dir ect correlation of the magnetic order with OO. \nRecently, we have revisited the RXS at the Mn K-edge of LaMnO 3, both experimentally and \ntheoretically [36]. We have observed two inde pendent forbidden reflections, (0,3,0) [or, \nequivalently (3,0,0)] and (0,0,3), which ar e related to two different nonzero off-diagonal \nelements of the second-rank atomic scattering factor  tensor. The different energy dependence of  \nthe RXS spectra for the two types of forbidde n reflections has been explained within the \nmultiple scattering theory in terms of long-range ordered structural distortions around Mn atoms \nusing a cluster that includes up to 63 atoms beyond the first oxygen neighbors [42], as demonstrated in figure 6(a). Since no change in either the intensity or its azimuthal dependence was observed when crossing T\nN at 140 K (see figure 6(b)), these forbidden reflections are \nascribed as ATS reflections in agreement with a structural origin and opposite to the OO interpretation.   \n00.51.11.62.1\n6.54 6.55 6.56 6.57 6.58Intensity (arb. units)\nE (keV)(003) (030) \n0 5 10 15 20 25 0.0 0.4 0.8 1.2 \n    \n absorption, theory  \n (030), theory  \n (003), theory  \n    \n absorption, theory  \n (030), theory  \n (003), theory  \n \n0 5 10 15 20 25 0.0 0.4 0.8 1.2 \n    \n     (030) \n (003)     \nE-E0 (eV) \n00.20.40.60.811.21.4\n0 50 100 150 200 250 300(003)\n(030)Intensity (arb. units)\nTemperature (K)TN\n00.20.40.60.811.2\n-180-160-140-120-100 -80 -60 -40 -20 0Calculation\n(030)\n(003)\nAzimuth angle (degree)Intensity (arb. units)\n \nFigure 6. (a) Experimental (circles) RXS of the (0,3 ,0) and (0,0,3) forbidden reflections in \nLaMnO 3. The inset shows the MXAN calculations done using the 63-atoms cluster (upper panel) \nand the 6-atoms cluster (lower panel). (b) Temperature dependence of the integrated intensity of \nthe (0,3,0) and (0,0,3) reflections on resonance,  normalized for comparison. The inset shows the \nrespective azimuthal angle dependence on resonance. \n  \n(2) Another example of the excitement of forb idden reflections on a cubic crystal structure \nthat deserves discussion are the (0,0,l) (l=4n+2) reflections in Fe 3O4. Once the Verwey model of CO was discarded, these reflections offer the possibility to study RXS excited by a dq-transition \nwhich is only allowed on the tetrahedral site wher eas the octahedral site allows, on the contrary, \nonly for dd and qq transitions. Two distinct resonant lines are observed in the RXS spectra of \n(0,0,2) and (0,0,6) forbidden reflections in Fe 3O4 at the pre-edge and at the main edge energies \nof the Fe K-edge [10]. The comparison of the energy and azimuthal dependence of these \nreflections at the Fe and Co K edges in Fe 3O4, CoFe 2O4 and MnFe 2O4 spinels is reported in \nfigure 7 [38]. No pre-edge resonance is ob served either at the Fe K-edge in MnFe 2O4 or at the \nCo K-edge in CoFe 2O4, whereas the main-edge resonance is observed at the Fe K-edge in both, \nCoFe 2O4 and MnFe 2O4, and at the Co K-edge in CoFe 2O4. \n7,10 7,11 7,12 7,13 7,14 7,1501234\n00.20.40.60.811.2\n7.7 7.72 7.74Intensity  (arb. units)\nE (keV)CoFe2O4\nCo K-edge\nCoFe2O4\nMnFe2O4\nFe3O4Fe K-edgeIntensity (arb. units)\nEnergy (KeV)Figure 7. Energy scans of the (0,0,2) \n(closed symbols) and (0,0,6) (open \nsymbols) forbidden reflections at the Fe K edge in the spinel ferrites. The inset \ncompares the Co K-edge RXS of the \n(0,0,2) and (0,0,6) reflections in CoFe\n2O4. \n   \nThese results confirm that the pr e-peak resonance originates from dq transitions at the \ntetrahedral Fe atoms and the main-edge resonance is due to the anisotropy of the trigonal ( ) \npoint symmetry of octahedral B sites of the spinel structure. The azimuthal dependence corroborates the dd character of the main-edge resonance, which does not depend on either the \ntype or the formal valency of the transition-me tal atom that occupies the octahedral site. We \nnote here that we have observed anisotropic RXS, mainly of a structural origin, in a system where the local atom environment is nearly isotropic since there is not a significant distortion of the ligand configuration [43]. \n(3) Recently, the correlation of OO and magnetic ordering has been studied on layered cobalt \noxides, RBaCo 2O5.5 with R=rare-earth [39,40]. Resonan ces have been observed at the Co K \nedge for (0,k ,0) reflections with k  odd in TbBaCo 2O5.5 whereas ( h,0,0) reflections with h odd \nwere not detected either on or off resonance in any phase transition [39]. The cusp of the \nresonant scattering is either up –(0,3,0) and (0 ,7,0)- or down –(0,1,0) and (0,5,0)- depending on \nthe k value. This behavior arises from displacemen ts of the Tb and Ba ions from the ideal \ntetragonal positions while the occurrence of RXS and its azimuthal dependence comes form the \npresence of two different environments for Co atoms (octhedron and pyramid), ordered along the b axis. Therefore, resonant  (0, k,0) reflections with k odd corresponds to ATS reflections and \nno further contribution due to OO has been deduced from this experiment [40]. On the other hand, superlattice (1,0, l) with l  even reflections were reported occurring at the metal-insulator \ntransition in GdBaCo\n2O5.5-x [40]. RXS was observed at these re flections, which is sensitive to \nthe difference in anisotropy of adjacent Co py ramids and/or Co octahedral and it has been \nassociated to the presence of OO.     5.  Conclusions \nRXS has demonstrated its potential to solv e old and new problems on strong correlated \ntransition-metal oxides. The results on the low temperature insulating phase of magnetite has re-\nopened the discussion on the origin of the Verwey  transition. The classical CO model has been \novercome and many recent theoretical papers gi ves a more realistic interpretation without \ninvoking to ionic ordering. In general, all these RXS results discard the description of mixed-\nvalence transition-metal oxides as a bimodal distribution of integer valence states and the so-called CO phase implies the segregation in different crsytallographic sites whose electronic \ndifferences are mainly determined by the structural distortions. These structural distortions \ninduces a different charge density on different crystallographic sites and in some cases, an electronic anisotropy by lowering the local symmetry. Finally, RXS has been also fundamental to determine the type of ordering in many cases, such as Bi\n0.67Sr0.33MnO 3 and La 0.33Sr0.67FeO 3.  \n \nAcknowledgement  \nThe authors thank financial support from the Spanish MICINN (FIS08-03951 project) and DGA (Camrads). We also acknowledge ESRF for granting beam time and technical support.  \nReferences \n[1] Materlik G, Spark C J and Fisher K 1994 Resonant Anomalous X-ray Scatterin g, Theory \nand Applications  (Amsterdam: North-Holland/Elsevier Science B. V.)  \n[2] Stragier H, Cross J O, Rehr J J, Sorensen L B, Bouldin C E and Woicik J C 1992 Phys. \nRev. Lett . 21 3064 \n[3] Proietti M G, Renevier H, Hodeau J L, García J, Bérar J F and Wolfers P 1999 Phys. Rev. \nB. 59 5479 \n[4] Templeton D H and Templeton L K 1980 Acta Cryst. 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B  78 054123 \n[40] García-Fernández M, Scagnoli V, Staub U, Mulders A M, Janousch M, Bodenthin Y, \nMeister D, Patterson B D, Mirone A, Tanaka  Y, Nakamura T, Grenier S, Huang Y and \nConder K 2008 Phys. Rev. B  78 054424 \n[41] Goodenough J B 1963 Magnetism and Chemical Bond  (New York:Interscience) \n[42] Benfatto M, Della Longa S and Natoli C R 2003 J. Synchrotron Radiat . 10 51 \n[43] Subías G, García J and Blasco J 2005 Phys. Rev. B  71 155103 \n "    },    {        "title": "1012.3017v2.Scaling_behavior_of_the_spin_pumping_effect_in_ferromagnet_platinum_bilayers.pdf",        "content": "Scaling behavior of the spin pumping e\u000bect in ferromagnet/platinum bilayers\nF. D. Czeschka,1L. Dreher,2M. S. Brandt,2M. Weiler,1M. Althammer,1I.-M. Imort,3G. Reiss,3\nA. Thomas,3W. Schoch,4W. Limmer,4H. Huebl,1R. Gross,1, 5and S. T. B. Goennenwein1,\u0003\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Walter Schottky Institut, Technische Universit at M unchen, 85748 Garching, Germany\n3Fakult at f ur Physik, Universit at Bielefeld, 33602 Bielefeld, Germany\n4Institut f ur Quantenmaterie, Universit at Ulm, 89069 Ulm, Germany\n5Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\nWe systematically measured the DC voltage VISHinduced by spin pumping together with the in-\nverse spin Hall e\u000bect in ferromagnet/platinum bilayer \flms. In all our samples, comprising ferromag-\nnetic 3 dtransition metals, Heusler compounds, ferrite spinel oxides, and magnetic semiconductors,\nVISHinvariably has the same polarity, and scales with the magnetization precession cone angle.\nThese \fndings, together with the spin mixing conductance derived from the experimental data,\nquantitatively corroborate the present theoretical understanding of spin pumping in combination\nwith the inverse spin Hall e\u000bect.\nSpin current related phenomena are an important as-\npect of modern magnetism [1{4]. For example, pure spin\ncurrents { a directed \row of angular momentum without\nan accompanying net charge current { can propagate in\nmagnetic insulators [5]. A spin current Jscan be de-\ntected via the inverse spin Hall (ISH) e\u000bect, where Js\nwith polarization orientation ^ sis converted into a charge\ncurrent Jc/\u000bSH(^ s\u0002Js) perpendicular to both ^ sand\nJs[2, 6]. Recently, Mosendz et al. [7] showed that the spin\nHall angle\u000bSHcan be quantitatively determined from so-\ncalled spin pumping experiments in ferromagnet/normal\nmetal (F/N) bilayers. Here, the magnetization of the F\nlayer is driven into ferromagnetic resonance (FMR) and\ncan relax by emitting a spin current into the adjacent\nN layer [8, 9]. Spin pumping can thus be understood\nas the inverse of spin torque [1, 10], and gives access\nto the physics of spin currents, magnetization dynamics,\nand damping. The present theoretical models [8, 11] sug-\ngest that spin pumping in conductive ferromagnets is a\ngeneric phenomenon, where the magnitude of Jsis gov-\nerned by the magnetization precession cone angle and the\nspin mixing conductance g\"#at the F/N interface. How-\never, most spin pumping experiments to date have been\nperformed in transition metals [9, 12, 13], so that generic\nproperties could not be addressed. In this letter, we pro-\nvide experimental evidence that the present theories for\nspin pumping are not limited to transition metal-based\nbilayers, but also apply to the ferromagnetic Heusler\ncompounds Co 2FeAl and Co 2FeSi, the ferrimagnetic ox-\nide spinel Fe 3O4, and the dilute magnetic semiconductor\n(Ga,Mn)As (DMS). We demonstrate this by simultane-\nous DC voltage and FMR measurements that yield the\ncorrelation of the inverse spin Hall voltage VISH/Jc\nalong Jcand the magnetization precession cone angle \u0002\nin FMR. Our experimental \fndings clearly con\frm the\nscaling behavior of VISHsuggested by theory.\nWe fabricated F/Pt bilayers, using Ni, Co, Fe,\nCo2FeAl, Co 2FeSi, Fe 3O4, and (Ga,Mn)As, for the F\nlayer. The Ni, Co and Fe \flms were deposited on oxi-dized silicon substrates via electron beam evaporation at\na base pressure of 1 \u000210\u00008mbar. The Heusler compounds\nwere sputtered on (001)-oriented MgO single crystal sub-\nstrates at an Ar pressure of 1 :5\u000210\u00003mbar, followed\nby annealing at 500\u000eC [14]. Epitaxial (100)- and (111)-\noriented Fe 3O4\flms were grown via pulsed laser depo-\nsition in argon atmosphere on (100)-oriented MgO and\n(0001)-oriented Al 2O3substrates, respectively, at a sub-\nstrate temperature of 320\u000eC [15]. The Ga 1\u0000xMnxAs\n(x= 0:04) \flms were grown via low temperature molecu-\nlar beam epitaxy on (001)-oriented GaAs substrates [16].\nAll F layers have a thickness tF= 10 nm, except for\nthe (111)-oriented Fe 3O4\flm, which has tF= 35 nm,\nand the (Ga,Mn)As \flms with tF= 200 nm, 175 nm and\n65 nm. As a high-quality, transparent interface is cru-\ncial for spin pumping [17], all F layers were covered in\nsitu withtN= 7 nm of Pt, except for the (Ga,Mn)As\n\flms, which were covered after exposure to ambient at-\nmosphere. All samples were cut into rectangular bars\n(lengthL= 3 mm, width w= 1 mm or 2 mm) and con-\ntacted on the short sides for electrical measurements as\nshown in Fig. 1(a).\nThe FMR and spin pumping experiments were per-\nformed in a magnetic resonance spectrometer at a \fxed\nmicrowave frequency \u0017MW= 9:3 GHz as a function of an\nexternally applied static magnetic \feld H, in the temper-\nature range from 2 K to 290 K. We took care to position\nthe respective sample on the axis of the TE 102microwave\ncavity, in order to locate it in an antinode of the mi-\ncrowave magnetic \feld and in a node of the microwave\nelectric \feld. The FMR was recorded using magnetic \feld\nmodulation and lock-in detection, so that the resonance\n\feldHrescorresponds to the in\rection point in the FMR\nspectra. The DC voltage VDCbetween the contacts in-\ndicated in Fig. 1(a) was measured with a nanovoltmeter.\nFigure 1 shows a selection of FMR and VDCspectra,\nrecorded for two magnetic \feld orientations in the \flm\nplane:\u001e= 0\u000ecorresponds to Hparallel to ^ x(blackarXiv:1012.3017v2  [cond-mat.mes-hall]  29 Jul 20112\nLVISH\nPt\nFerromagnetH\nxy\nφJs\nJc\nwz\nFIG. 1. (a) Sketch of the the coordinate system and the F/Pt\nbilayer sample. (b),(d),(f),(h),(j),(l) show the FMR signal\nof F/Pt bilayers, with F as quoted in the individual panels,\nrecorded with Hparallel (black full squares) and antiparallel\n(red open circles) to ^ x. DMS stands for 200 nm (Ga,Mn)As.\n(c),(e),(g),(i),(k),(m) show the DC voltage measured simulta-\nneously with the respective FMR traces.\nfull squares), while for \u001e= 180\u000e,His antiparallel to\n^ x(red open circles). All measurements in Fig. 1 were\ntaken at 290 K, except for the (Ga,Mn)As data recorded\natT= 10 K (Fig. 1(l) and (m)). Since the FMR is in-\nvariant with respect to magnetic \feld inversion, the FMR\ntraces for\u001e= 0\u000eand\u001e= 180\u000eshould superimpose, as\nindeed observed in experiment. The FMR signal of all\nsamples in Fig. 1 consists of a single resonance line, with\nthe exception of (Ga,Mn)As, in which several standing\nspin wave modes contribute to the FMR spectrum [18].\nTheVDCtraces show one clear extremum in VDCatHres;\nonly in (Ga,Mn)As, several VDCextrema corresponding\nto the spin wave modes can be discerned. The magnitude\nofVDCranges from a few 100 nV in Fe 3O4to a few 10\n\u0016V in Co 2FeSi and Fe (Fig. 2). In contrast to the FMR,\ntheVDCextremum changes sign when the magnetic \feld\nis reversed. It also is important to note that VDCal-\nways has a maximum ( VDC>0) for\u001e= 0\u000e, whereas a\nminimum (VDC<0) is observed for \u001e= 180\u000e.\n020406080100020406080100010203040506070µ\n0H (mT)78K90K106K130K26K2\n0K1\n0K2K FMR (arb. u.)2 90K1\n87K( a) F\ne/PtV\nDC (µV)µ\n0H (mT)26K2\n0K1\n0K2\nK78K90K106K130K290K1\n87K(b) FIG. 2. Temperature-dependent evolution of (a) the FMR,\nand (b) the VDCspectra of a Fe/Pt bilayer.\nWe furthermore studied the evolution of the FMR and\nVDCsignals as a function of temperature in several bilayer\nsamples. As an example, the FMR and VDCspectra of\nan Fe/Pt sample, recorded for a series of temperatures\n2 K\u0014T\u0014290 K are shown in Fig. 2. With decreasing\nT, the FMR broadens and shifts to lower Hres, as does\nthe peak in VDC.\nWe now turn to the interpretation of the experimental\ndata of Figs. 1 and 2. We attribute the peaks in VDC\nto spin pumping in combination with the inverse spin\nHall e\u000bect in the F/Pt bilayers [7{9]. This naturally ex-\nplains that VDCchanges sign when the Horientation is\ninverted from \u001e=0\u000eto 180\u000e, asHdetermines the orien-\ntation of the spin polarization vector ^ sinJc/(^ s\u0002Js).\nHence, Jcand thus also VDCis reversed if the magnetic\n\feld is inverted. We note that the Hresare well above\nthe coercive and the anisotropy \felds of the respective\nferromagnets, such that the magnetization MkHin\ngood approximation. Furthermore, the experimental ob-\nservation that VDCinvariably has the same polarity for a\ngiven \feld orientation \u001e, irrespective of the ferromagnetic\nmaterial used in the F/Pt bilayer and of the measure-\nment temperature, is fully consistent with spin pump-\ning theory [7, 8, 11, 19{21]. We note that other mecha-\nnisms for the generation of a DC voltage in conjunction\nwith FMR have been suggested and were observed in ex-\nperiment [22{24]. Microwave recti\fcation e\u000bects linked\nto the anisotropic magnetoresistance or the anomalous\nHall e\u000bect often are superimposed onto the spin pump-\ning signal, in particular if the sample is not located in\na node of the microwave electric \feld [7, 25]. However,\nwe rule out such mechanisms as the origin of the VDCob-\nserved in our experiments (Figs. 1 and 2) for two reasons.\nFirst, we have positioned the sample in a node of the\nmicrowave electric \feld, which minimizes recti\fcation-\ntype processes. Second, and more importantly, both the\nspontaneous resistivity anisotropy \u0001 \u001adetermining the\nanisotropic magnetoresistance and the anomalous Hall\ncoe\u000ecientRHare substantially di\u000berent in magnitude\nand in sign for the di\u000berent ferromagnetic materials in\nour F/Pt bilayers [26]. Nevertheless, for a given \u001e, we\ninvariably observe the same VDCpolarity, which is di\u000e-3\ncult to rationalize for recti\fcation e\u000bects.\nIn order to quantitatively compare our experimental\ndata with spin pumping theory, we start from\nVISH=\u0000e\u000bSH\u0015SDtanhtN\n2\u0015SD\n\u001bFtF+\u001bNtNg\"#\u0017MWLPsin2\u0002 (1)\nderived by Mosendz et al. [7, 21] for the inverse spin Hall\nDC voltage VISHarising due to spin pumping in permal-\nloy/N bilayers, assuming that the Nlayer is an ideal spin\ncurrent sink. Here, eis the electron charge, \u0015SDis the\nspin di\u000busion length in N, g\"#is the e\u000bective spin mix-\ning conductance [21], \u0002 is the magnetization precession\ncone angle (cf. inset in Fig. 3(a)), and Pis a correction\nfactor taking into account the ellipticity of the magneti-\nzation precession [21, 27]. For our samples, we calculated\n0:5\u0014P\u00141:3. Note that Eq. (1) has been adapted to\nour experimental con\fguration, and accounts for both\nthe conductivity \u001bNand\u001bFof the N and F layer con-\ntributing to the bilayer conductivity.\nSince invariably tN= 7 nm and N=Pt for all our bi-\nlayer samples, C\u0011\u000bSH\u0015SDtanh(tN=2\u0015SD) is a constant\nat a given temperature. In addition, the denominator in\nEq. (1) can be expressed in terms of the sample geom-\netryw=L and resistance R, measured in four point ex-\nperiments: \u001bFtF+\u001bNtN= (Rw=L )\u00001. We thus rewrite\nEq. (1) as\nVISH\n\u0017MWPRw=\u0000eCg\"#sin2\u0002: (2)\nThe theoretical models for the spin mixing conduc-\ntance [11, 28] suggest that g\"#of conductive ferromag-\nnet/normal metal interfaces is determined mainly by the\nN layer, i.e., the Pt layer in our case. In other words, g\"#\nshould be of comparable magnitude in all our samples.\nEquation (2) then represents a scaling relation for all\nF/Pt bilayers made from a conductive ferromagnet and a\nPt layer of one and the same thickness tN, irrespective of\nthe particular ferromagnetic material, its magnetic prop-\nerties, or the details of the charge transport mechanism\nsuch as band conduction or charge carrier hopping.\nWe now test the scaling relation of Eq. (2) against\nour experimental data. At ferromagnetic resonance,\nthe magnetization precession cone angle is \u0002 res=\n2hMW=(p\n3\u0001Hpp) [30], with the microwave magnetic\n\feldhMW= 0:12 mT as determined in paramagnetic res-\nonance calibration experiments. We extract the FMR\npeak-to-peak line width \u0001 Hppfrom the experimental\ndata, and use the measured DC voltage VDC;resat\nHresto determine VISH=VDC;res. Figure 3(a) shows\nVISH=(\u0017MWPRw ) versus sin2\u0002resthus obtained. Full\nsymbols indicate data measured at 290 K, while measure-\nments at lower temperatures are shown as open sym-\nbols. Data for permalloy/Pt (Py/Pt) extracted from\nRefs. [7, 21, 29] are also included in the \fgure. For the\nsake of completeness, data for Y 3Fe5O12/Pt (YIG/Pt),\nM(t)Heffθ\n10-710-610-510-410-310-210-110-1710-1610-1510-1410-1310-1210-1110-10\n0 50 100150200250300101810191020sin2(Θres) (Ga,Mn)As 200nm\n (Ga,Mn)As 175nm\n (Ga,Mn)As 65nm\n Fe3O4 (100)\n Fe3O4 (111)  Co2FeSi\nYIG, Kajiwara et al.\n(a) Fe\n Co\n Ni\n Co2FeAl\n Py, Mosendz et al.\n Py, Nakayama et al. VISH / (νMW P R w) (Vs/Ωm)\n1  g   (1/m2)\nT (K)(b)FIG. 3. (a) In all F/Pt bilayers made from conductive ferro-\nmagnets, the DC voltage VISHinduced by a collective mode\nFMR scales with sin2\u0002resto within a factor of 10, as indicated\nby the grey bar. The inset depicts the magnetization preces-\nsion around the e\u000bective magnetic \feld. (b) From the scaling\nanalysis, the spin mixing conductance g\"#can be quanti\fed\nas a function of temperature (see text). In panels (a) and\n(b), full symbols represent data taken at 290 K, open symbols\ncorrespond to data measured at lower T. The Py and YIG\ndata are taken from the literature, Refs. [5, 7, 21, 29].\ntaken from Ref. [5], are also shown. Since YIG is an in-\nsulator, however, g\"#is dominated by its imaginary part,\nin contrast to the mostly real g\"#for conductive ferro-\nmagnets [11, 31, 32]. Moreover, spin wave modes govern\nthe YIG FMR signal, impeding a straightforward anal-\nysis [33]. Thus, we here focus only on conductive ferro-\nmagnet/Pt bilayers. In these samples, VISH=(\u0017MWPRw )\nindeed scales as suggested by Eq. (2) to within a factor\nof 10 (grey bar in Fig. 3(a)). The deviations from per-\nfect scaling are due to a slight material dependence of\ng\"#, as detailed in the next paragraph. The scaling be-\nhavior is observed over more than four orders of mag-\nnitude inVISH=(\u0017MWPRw ) and sin2\u0002res, for samples\nmade from conductive ferromagnetic \flms with qualita-\ntively di\u000berent exchange mechanisms, transport proper-\nties, crystalline quality, and crystalline structure. More-\nover, F/N bilayer samples fabricated and investigated by4\ndi\u000berent groups are consistently described.\nTo quantify g\"#of a given bilayer, we write Eq. (2) as\ng\"#=\u0000VISH=\u0002\n\u0017MWPRweC sin2\u0002res\u0003\n. Using the room\ntemperature values \u000bSH= 0:013 and\u0015SD= 10 nm for\nPt [21, 34], Pcalculated as detailed in Ref. [21] (and lit-\nerature values for the conductivities of Py and Pt [7, 35]\nfor the data points extracted from Refs. [7, 21, 29]), we\nobtaing\"#as shown in Fig. 3(b). Clearly, the conjec-\nture thatg\"#is independent of the F layer properties\nis well ful\flled for highly conductive (\\metallic\") ferro-\nmagnets, such as the 3 dtransition metals, permalloy,\nor the Heusler compounds, which all are in the range\ng\"#= (4\u00063)\u00021019m\u00002. In the low-conductivity ferro-\nmagnet Fe 3O4,g\"#is about a factor of 6 smaller, but the\nlinear scaling is still observed. In (Ga,Mn)As, g\"#appears\nto be in between these two regimes. However, several\nspin wave modes contribute to the FMR in (Ga,Mn)As\n(cf. Fig. 1(l)) and a \ft with at least three Lorentzian lines\nwas required to reproduce the FMR and VDCdata. So\nthe assumption of a single, position-independent magne-\ntization precession cone angle \u0002 resis not warranted [18].\nFor standing spin waves, the magnetization precession\namplitude \u0002 res(z) changes as a function of zacross the\n\flm thickness, which in turn can qualitatively alter the\nmagnitude of VDC[33]. Since \u0002 res(z) moreover depends\non the particular spin wave mode excited, a more thor-\nough study of spin pumping due to spin wave modes is\nmandatory to evaluate g\"#in (Ga,Mn)As/Pt. In addi-\ntion, in systems with large spin-orbit coupling such as\n(Ga,Mn)As, the magnetization precession can even in-\nduce a charge current via an inverse, spin-orbit driven\nspin torque e\u000bect [36]. Taken together, the experimen-\ntal results summarized in Fig. 3(b) represent an incentive\nto theory to calculate g\"#for ferromagnets with di\u000berent\nconductivity magnitude, transport mechanisms, and in-\nhomogeneous spin texture.\nAnother interesting experimental observation from\nFig. 3(b) is that temperature has little in\ruence on g\"#.\nAccording to the present theoretical understanding, g\"#in\ndi\u000busive bilayers is governed by the conductivity \u001bN(T)\nof the normal metal [11, 28]. The weak temperature\ndependence of g\"#(Fig. 3(b)) thus suggests that \u001bN(T)\nof our Pt \flms also should not substantially change\nwith temperature. This is corroborated by resistance\nmeasurements, which show that \u001bN(T) increases by less\nthan a factor of 2 from 290 K to 2 K. Since \u000bSH/\n\u001b0:6:::1[34, 37] is governed by \u001bN(T), and since \u0015SDin\nPt increases by less than 50% from 290 K to 2 K [34],\nC=\u000bSH\u0015SDtanh(tN=2\u0015SD) changes by at most a factor\nof 3 in the whole temperature range investigated experi-\nmentally. This warrants the use of Cg\"#as an essentially\ntemperature-independent scaling constant in Eq. (2).\nIn summary, we have measured the DC voltage caused\nby spin pumping and the inverse spin Hall e\u000bect in\nF/Pt samples, with F made from elemental 3 dferro-\nmagnets, the ferromagnetic Heusler compounds Co 2FeAland Co 2FeSi, the ferrimagnetic oxide spinel Fe 3O4, and\nthe magnetic semiconductor (Ga,Mn)As. Although the\nmagnetic exchange mechanism, the saturation magneti-\nzation, the spin polarization, the charge carrier transport\nmechanism and the charge carrier polarity are qualita-\ntively di\u000berent in the di\u000berent samples, the DC voltage\nhas identical polarity for all bilayers investigated, and\nits magnitude is well described by a scaling relation of\nthe form of Eq. (2) within the entire temperature range\n2 K\u0014T\u0014290 K studied. Our experimental \fndings\nthus quantitatively corroborate the present spin pump-\ning/inverse spin Hall theories [7, 8, 11, 19{21], and are an\nincentive for quantitative calculations of g\"#(T) in various\ntypes of F/N bilayers.\nWe thank G. E. W. Bauer for valuable discussions. The\nwork at WMI was \fnancially supported via the Excel-\nlence Cluster \\Nanosystems Initiative Munich (NIM)\".\nA.T. and I.-M. I. are supported by the NRW MIWF.\n\u0003goennenwein@wmi.badw.de\n[1] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater.\n320, 1190 (2008).\n[2] S. Takahashi and S. Maekawa, Sci. Technol. Adv. Mat.\n9, 014105 (2008).\n[3] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[4] S. Maekawa, Concepts in Spin Electronics (Oxford Uni-\nversity Press, New York, 2006).\n[5] Y. Kajiwara et al. , Nature 464, 262 (2010).\n[6] J. E. Hirsch, Phys. Rev. Lett. 83, 1834 (1999).\n[7] O. Mosendz et al. , Phys. Rev. Lett. 104, 046601 (2010).\n[8] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B 66, 224403 (2002).\n[9] E. Saitoh et al. , Appl. Phys. Lett. 88, 182509 (2006).\n[10] J. A. Katine et al. , Phys. Rev. Lett. 84, 3149 (2000).\n[11] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[12] R. Urban, G. Woltersdorf, and B. Heinrich, Phys. Rev.\nLett. 87, 217204 (2001).\n[13] M. Costache et al. , Phys. Rev. Lett. 97, 216603 (2006).\n[14] D. Ebke et al. , J. Magn. Magn. Mater. 322, 996 (2010).\n[15] D. Venkateshvaran et al. , Phys. Rev. B 79, 134405\n(2009).\n[16] W. Limmer et al. , Phys. Rev. B 74, 205205 (2006).\n[17] O. Mosendz et al. , Appl. Phys. Lett. 96, 022502 (2010).\n[18] C. Bihler et al. , Phys. Rev. B 79, 045205 (2009).\n[19] Y. Tserkovnyak et al. , Rev. Mod. Phys. 77, 1375 (2005).\n[20] X. Wang et al. , Phys. Rev. Lett. 97, 216602 (2006).\n[21] O. Mosendz et al. , Phys. Rev. B 82, 214403 (2010).\n[22] W. G. Egan and H. J. Juretschke, J. Appl. Phys. 34,\n1477 (1963).\n[23] Y. S. Gui, N. Mecking, and C.-M. Hu, Phys. Rev. Lett.\n98, 217603 (2007).\n[24] N. Mecking, Y. S. Gui, and C.-M. Hu, Phys. Rev. B 76,\n224430 (2007).\n[25] H. Y. Inoue et al. , J. Appl. Phys. 102, 083915 (2007).\n[26] R. C. O'Handley, Modern Magnetic Materials : Princi-\nples and Applications (John Wiley, New York, 2000).5\n[27] K. Ando, T. Yoshino, and E. Saitoh, Appl. Phys. Lett.\n94, 152509 (2009).\n[28] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys.\nRev. Lett. 84, 2481 (2000).\n[29] H. Nakayama et al. , IEEE Trans. Magn. 46, 2202 (2010).\n[30] Y. Guan et al. , J. Magn. Magn. Mater. 312, 374 (2007).\n[31] K. Xia et al. , Phys. Rev. B 65, 220401 (2002).\n[32] K. Carva and I. Turek, Phys. Rev. B 76, 104409 (2007).[33] C. W. Sandweg et al. , Appl. Phys. Lett. 97, 252504\n(2010).\n[34] L. Vila, T. Kimura, and Y. Otani, Phys. Rev. Lett. 99,\n226604 (2007).\n[35] K. Ando et al. , Phys. Rev. Lett. 101, 036601 (2008).\n[36] K. M. D. Hals, A. Brataas, and Y. Tserkovnyak, Euro-\nphys. Lett. 90, 47002 (2010).\n[37] H. Nakayama et al. , J. Phys: Conf. Ser. 200, 062014\n(2010)."    },    {        "title": "0805.0859v1.Design_Methodology_and_Manufacture_of_a_Microinductor.pdf",        "content": "9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5  cro\nsatlNIBµµ=\nBI lo\nrc µ\nµ=Design Methodology and Manufacture of a \nMicroinductor  \nD. Flynn and M. P.Y. Desmulliez \nSchool of Engineering and Physical Sciences, Electrical and Electronic Engineering department, Mountbatten Building, Heriot-\nWatt University, Edinburgh, UK, EH14 4AS \n \nAbstract-  Potential core materials to supersede ferrite in the \n0.5-10 MHz frequency range are investigated. The performance of electrodeposited nickel-iron, cobalt-iron-copper alloys and the commercial alloy Vitrovac 6025 have been assessed through their inclusion within a custom-made solenoid microinductor.  Although the present inductor, at 500 KHz, achieves 77% \npower efficiency for 24.7W/cm\n3 power density, an optimized \nprocess predicts a power efficiency of 97% for 30.83W/cm3 \npower density. The principle issues regarding microinductor design and performance are discussed. \n  \nI.  I NTRODUCTION \nWhile the physical size of digital and analogue electronic \ncircuits has been drastically reduced over the past 20 years, the size of their associated power supplies has been reduced \nat a much slower rate.  As a result, the power supply \nrepresents an increasing proportion of the size and cost of electronic equipment.  One of the main difficulties in the miniaturization of power conversion circuits such as DC-DC converters is the construction of inductors and transformers [1-3].   \nRecent efforts to miniaturise the overall size of DC-DC \npower converters, has resulted in an increase in switching frequency of the power circuit from the 100-500 kHz range to the 1-10MHz range.  This effort has led to a reduction of the size of the energy storage components that dominate the \nconverter volume.  Several problems arise when frequencies \nare pushed into the MHz region.  Core materials commonly used in the 20-500 kHz region such as MnZn ferrites, have rapidly increasing hysteresis and eddy current losses as the frequency is increased.  Furthermore, eddy current losses in windings can also become a severe problem, as the skin depth in copper becomes small in relation to the cross section of wire used.  Even if these problems are adequately dealt with, the resulting transformer/inductor is still one of the physically largest and most expensive components in the circuit. \nVarious types of microinductors and transformers using an \narray of different thin-film metal alloys and winding \nconfigurations have been fabricated and reported in the literature over the past 25 years [4-9]. The results of these devices have produced mixed results with varying degrees of success due to the prominent challenge in the design and fabrication of such devices. The key to successful microinductor design is to achieve the optimal balance of the various design parameters for a given application e.g. inductance requirement, low winding resistance, set efficiency at high frequency, high power density etc whilst \nconsidering fabrication limitations. \nThe focus of this paper is to design and fabricate a \nmicroinductor that balances the various design tradeoffs yet displays adequate performance for a variety of power related \napplications. This includes using low-temperature \nfabrication techniques that permit integration with supporting IC circuitry. The component footprint will be minimized with a high winding packing density limiting the area used while still targeting reasonable inductance values within the 0.5-1 MHz, so that it could be used in a highly miniaturized switched mode power supply for stepping up and down a wide range of voltages. The impact of core geometry on core material properties is highlighted as is the degree of enhancement of the microinductor efficiency and power density following a simple optimization process.  \n \nII. I\nNDUCTOR DESIGN  \nA.  Magnetic Core Geometry  \nA decision on which category of component is optimal \nrequires a comparison across a variety of factors such as \npower handling, efficiency, practicality of fabrication, inductance requirement etc. Consider a core material for a \none-turn micro-inductor.  The saturation flux density,\nsatB, \n \n(1) \n  is a predetermined constant for a given material.  In the \nsame way, the current applied to the component would be known for a given application. Therefore, the only free parameter left will be either the relative permeability or the flux path length, since \n \n(2) \n \n \n \nThe core, typically a rectangle, triangle or a circle, has a \ncertain perimeter. Considering a rectangular core within a \none turn pot-core component, shown in Fig. 1 with the main flux path indicated. As frequency increases the skin depths of the magnetic material and winding decrease.  Therefore, to minimise skin depth losses and maintain the inductance value i.e. overall magnetic core area, the device becomes 9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5  thinner and wider.  The resulting increase in flux path length \nrequires an increase in relative permeability for the inductance value to be maintained.  An increase in relative permeability for the same resistivity value will make the material susceptible to skin depth losses at reduced frequency.  \n Magnetic Core     Conductor   Flux Path   \n \n    \n \nFig. 1. Simple rectangular one-turn micro-inductor. \n \nIf, for the same area and core material, the rectangular \ngeometry was changed to a triangular geometry, then the \nperimeter is reduced by one third.  The required permeability \nis then reduced by 1/3 and greater skin depth is achieved for the same resistivity.  \nA one turn micro-inductor with a triangular trench \nstructure was fabricated by Sullivan et al [10].  The fabrication of this V-groove micro-inductor required sputter deposition. Sputtering is a slow and costly process, and due to delamination can only be used for small core areas.  Section will demonstrate how limiting the number of laminate layers can have an adverse effect on component performance. \nA circular component would suffer from the same \ntechnical disadvantages as the triangular component.  Moreover, a circular component fabricated by micromachining methods would require non-planar construction.  Minimal footprint and low profile are prerequisites for present and future magnetic components.  Therefore, a circular component was not investigated.   \nDue to the aforementioned complications of triangular and \ncircular geometries. the investigated components herein are of rectangular planar form.  Disadvantages of this geometry will be overcome via tailored core properties and optimal component design.  \nAnother consideration is the tailoring of magnetic film \nproperties with magnetic field annealing.  Both pot-core and solenoid geometries can be used with magnetic field annealing; therefore, this is not a design restriction. \n \nB.  Winding Geometry  \nThe winding geometries generally encountered for the \ninductor are of the spiral, meander or solenoid forms. A single layer solenoid component allows near ideal performance [11]. The number of turns per single layer of traditionally wound solenoid inductors is limited by the core’s inner diameter and how tightly you can pack the turns together. To increase inductance, a larger core or multiple layers of windings would normally be needed. This would \nnormally lead to an increase in component size and parasitic \ncapacitance. UV-photolithography discussed within the manufacturing of the solenoid allows a higher winding \ndensity avoiding such methods.  \nSometimes an important issue in choosing a winding \nconfiguration is the magnitude and distribution of external magnetic fields generated by the transformer or inductor.  The conventional pot-core is regarded as performing more favorably in minimizing external magnetic fields, as the core encloses and so shields the windings.  The result is low external fields regardless of the distribution of primary and secondary windings inside the core.  Solenoid components \nalso have low external fields due to the winding distribution.  \nIn transformer applications the primary and secondary windings are customarily distributed evenly around the core, lying on top of each other with little space between.  Thus the currents cancel each other, so little external field results.  In micro-fabrication the ability to finely pattern windings makes it relatively easy to arrange the windings so that primary and secondary currents locally cancel.  Hence, external fields are not a primary concern in component development within this work.   \nPerhaps the most important factor in comparing the pot-\ncore, spiral type winding, and solenoid designs is the ease of \nmanufacture.  This parameter is the most difficult to quantify \nand obviously dependent on fabrication capability.  In the solenoid case, a difficulty often encountered is the interconnection of the top and bottom winding layers to encircle the core.  This requires accurate alignment between the layers to insure proper connections, requires low-resistance contacts and connection over a vertical distance greater than the thickness of the core.  A novel flip-chip bonding procedure overcomes the aforementioned difficulties [12].  \nIn the pot core design, the inter-layer connection \nrequirements are less severe because the windings are planar \nand manufactured in a single deposition process.  A complication arises if the inductor winding is spiral as access to the output pad is required.  This typically requires a fourth deposition step, or the use of external connections such as bond wires.  The thickness of the conductor layers is also typically larger than the thickness of the core layers, therefore, the vertical distance over which connections must be made is larger in the pot-core design.  Furthermore, to form a closed magnetic core the upper and lower sections of the core have to be joined.  The points where the two layers join is within the main flux path and can generate an \nunfavorable leakage flux leading to an increase in winding \nresistance, electromagnetic interference (EMI) and a reduction in component inductance.  \n  \nC.  Analytical Design FlowChart  \nFig. 2 displays the design flowchart for the development \nof a microinductor. The solenoid microinductor developed \nwithin this work formed a closed magnetic path. The \naccuracy of the analytical equations will be reduced with components that include air-gaps within the main flux path due to fringing effects. The equations within Fig.2 are used for the theoretical data within section III. \n 9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2 The Design Flowchart of the Microinductor \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Properties of \nmagnetic core \nmaterial: µr, \nHc, Bsat and ?. \nNumber of \nwinding turns, \nN\nIs the magnetic core \nanisotropic or contains an \nair-gap?\nCalculation of Isat to maximise \nflux density & avoid core \nsaturationCalculation of component \nInductanceCalculation of magnetic core \nreluctance.\ncroc\ncoreAlR\nµµ=Yes\nNoCalculation of magnetic core \nreluctance.\nOrgap air core total R R R− + =\naxis hard axis easy total R R R− −+ =\nRNLDC2\n=\nNRABIc sat\nsat=\nRac and optimal inter-winding \nspacing determined by:Calculation of winding skin depth:\nπσµµδfro1=\ntt t\nr rWS WF F+='\n42\n451 51 ϕ−+=pFrDetermination of magnetic core ac inductance and resistance: \nWinding height limited to 2xskin-\ndepth. Calculation of winding width \nto carry Isat.\nJIAsat\nw=\n\n+\n\n\n\n+\n\n\n× =\nδ δδ δ\nδb bb b\nbL LDC AC2cos2cosh2sin2sinh\n21\n\n\n+\n\n\n\n−\n\n\n× =\nδ δδ δ\nδωb bb b\nbL RDC AC2cos2cosh2sin2sinh\n21Calculation of optimal output \npower:Calculation of core power loss:Calculation of winding power loss:\nCalculation of component efficiency for \ngiven frequency:\nCalculation of component power \ndensity for given frequency:out out out I V P × =()\nspk\neddyhYX Bfn Pρπ\n2423 2\n=\n()c pk hys HBfV P 443=RI Pcu2=\n%100×+ +=\ncore cu outout\nP P PPη\nhwlPPout\ndensity×××=η9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5  \nIII.  COMPONENT CHARACTERISATION \nThe novel microinductor fabricated via flip chip assembly \nis shown in Fig.3. The manufacture process of the inductor and properties of the core materials utilized are described in detail within [13]. The (*) symbol is used to denote magnetically anisotropic films, as displayed in Fig.3. \n \n \n                                           5 mm (Hard Axis) \n  \n \n       1.5 mm \n    (Easy Axis) \n  \n \n \n \n \nFig. 3. Fabricated micro-inductor. The O-shape core is assembled between the \nwindings layers prior to flip-chip bonding. The component is approximately \n5mmx2mmx0.25mm ( lxbxh), respectively. When Ni-Fe is anisotropic the \norientation of the Easy and Hard axes are indicated.  \n \nA critical point in the performance of an inductor is the \nmeasurement of the inductance value as a function of frequency and the additional core losses. A Hewlett Packard \n4192A LF-impedance analyzer was used to record the \ninductance, resistance and Q-factor over a frequency range of 1 kHz-10MHz.  The resistance value shown in Fig.4 is for an air-core component. The reason for omitting the magnetic core was to remove core AC resistance from the recorded value. Therefore, the increase in resistance at high frequency from the R\nDC value is due to skin and proximity affects with increasing \nfrequency. The winding is 90µm thick and at 1MHz has a skin \ndepth value of approximately 66µm. Due to the winding \nconsisting of a single layer and inter-winding spacing optimized, the proximity effect is minimized and the main \ncontributor to the ac resistance is assumed to be the skin effect.  \n \n \n \n \nFig.4. Winding resistance vs. frequency for 90µm thick copper windings \n One important source of loss in a dc-dc converter is the dc \nwinding resistance,dcR, of the output inductor. This parameter \ndefines the minimal loss condition  of the inductor. For \nexample, if dcR= 1mΩ, and dc output current of  I=100 Amps \nthe inductor dc power loss is; \n)(102W RIPdc= =  \nIf the converter were to provide 100 Amps at 0.75 Volts, \nthe loss in efficiency due to the inductor dcRwould be 11.76 \n%. This result demonstrates that the converter would be less \nthan 89% efficient due to dcR alone. Hence, it is important to \nhave a winding construction that minimizes dcRwhilst still \nmeeting applied current and inductance criteria. \nThe resistance of the components with the magnetic core \nincluded are shown in Fig.5. The minor variation between experimental and analytical values may be due to the \nlimitations of the 2D model representing the intrinsically 3D \nnature of the leakage and eddy current flux. \n0246810121416\n02468 1 0\nFrequency(MHz)Resistance(Ohms )\nCoFeCu:Theo\nNiFe(PR):Theo\nNiFe(DC):Theo\nCoFeCu:Meas\nNiFe(PR):Meas\nNiFe:Meas\n \nFig. 5. Micro-inductor resistances as a function of frequency. The resistance \nrepresents the winding and core contribution. \n \n  \n \n \n \nFig. 6. Inductance vs. Frequency of the electrodeposited alloys \n 9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5  \nFig.6 displays the theoretical values (theo) calculated and \nmeasured values (meas) of inductance with increasing frequency.  \nThe self resonant frequency (SRF) of an inductor is \ndefined as the frequency at which the reactance becomes zero, (null Q-factor). The measured reactance increases linearly with frequency; hence the SRF induced by parasitic capacitance effects does not occur in the frequency range of interest. The Q-factor of the electrodeposited alloys is displayed in Fig.7. The maximum Q-factors of the alloys \noccurred in the 500 kHz-1MHz frequency range indicating \nthat the thin film dimensions are acceptable for operation within this range. Due to the properties of the CoFeCu core the 10µm thick core layer remains below one skin depth at 1MHz. The thicknesses of the components windings are under 2xskin depth therefore AC winding losses are maintained at acceptable levels. Therefore the ratio of the stored power to dissipative power of the component, the Q-factor, was optimal in this frequency range. \nThe Q-factor response of the commercial alloy \ndemonstrates the susceptibility of this particular film to skin depth effects. The skin depth effects increase the dissipative \nlosses within the component, therefore reduce the Q-factor. \nThe high Q-factor in the low frequency range is greatly reduced at 1 MHz with respect to the CoFeCu film. \nUsing the theoretical DC saturation current formula within \nFig.2. the maximum applied current can be determined. The performance of the prototypes is summarized in Table 1. K is taken to be equal to 4.44, sinusoidal waveform, and \nsatBis used to maximize the power density of the \ncomponent.  \n \n \n \n \n \n \n           \n \nFig. 7. Q-factor vs. frequency for the electrodeposited alloys \n \n \n \n \n \n \n TABLE 1 \nMICROINDUCTOR PERFORMANCE FOR RESPECTIVE CORES \nAT 0.5MHZ \n \n The saturation current of the Vitrovac material is only \n1.33mA. Hence, high efficiency is achieved at the expense of power density. An appropriate reduction in the number of windings and increase in core area via lamination will result in the same overall inductance. This would reduce the major Joule loss mechanism, winding loss, and allow windings of larger dimensions to be fabricated in order to apply greater \ncurrent. The 90µm thick windings limit the maximum \ncurrent to 180mA which is well beneath the saturation current of the CoFeCu components. An optimal component design process is therefore required. \n \nIV.  OPTIMAL DESIGN PROCESS\n \nA simplified optimal design procedure for the solenoid \ncomponent is outlined within this section. The optimization \nis performed in two circumstances in reference to the CoFeCu prototype in section III; (1) the aim is to maintain the inductance value whilst increasing the power density significantly and improve efficiency, and (2) the primary aims are two maintain inductance and improve efficiency. With good relative permeability, small coercivity, reasonable resistivity and the largest saturation current (1.24A), the CoFeCu alloy is selected for optimization. Once a frequency is selected, in this case 0.5 MHz, skin depth values for the windings and core laminate are 93µm and 17.4µm, respectively. For comparison with the prototype, the \ninductance required will be 0.3µH (Fig.12). The core area \nand numbers of turns are therefore: \n              \nBNA LIc =    (3) \n \nTherefore the area of the component is:   \n      \nBNLIAc=\n     ( 4 )  500kHz \nParameter NiFe NiFe* Vit NiFe \n(PR) CoFeCu \nIsat(mA) 100 137 1.32 66 180 \nL(µH) 1.07 0.86 15.5 1.87 0.28 \nINV  0.29 0.29 0.36 0.28 0.51 \nPout \n(mW)   29 40 0.47 18.48 91.8 \nPeddy \n(mW) 7.23 7.23 3.47 2.86 24.6 \nPhys \n(mW) 0.198 0.03 0.02 0.08 0.11 \nPcu  \n(mW) 0.5 0.9 0.0000\n8 0.21 1.62 \nEfficiency \n(%) 78 82 X 85 77 \nPower \nDensity \n(W/cm³) 7.8 11.3 X 5.46 24.7 9-11 April 2008 \n©EDA Publishing/DTIP 2008  ISBN: 978-2-35500-006-5   \nDesign constraints are included at this stage. Each \nCoFeCu laminate will equal ½ skin depth and the number of laminations will be restricted to 10. The number of laminations will affect component manufacturability and cost such that N is only 11 turns in our case. The maximum applied current is 1.24A, PCB current density is 10A/mm² and winding height is restricted to 2x δ. The dimensions and \nperformance of the optimized component can therefore be calculated and are given in Table 2. \n \nTABLE 2  \nOPTIMISED PERFORMANCE & PARAMETERS  \n \nDimensions Optimized \ncomponent (1) Optimized \ncomponent (2) Prototype \nComponent \nNumber  of turns 11 33 33 \nTurn thickness 180 µm 90 µm 90 µm \nTurn width 550 µm 200 µm 200 µm \nTurn spacing  55 µm 20 µm 20 µm \nLaminations 2 10 1 \nLamination thickness  5 µm 1 µm 10 µm \nLamination \nwidth  500 µm 500 µm 500 µm \nLamination \ninsulation 5 µm 5 µm 5 µm \nPerformance Optimized \ncomponent Prototype \nComponent Prototype \nComponent \ninV 0.17 0.51 0.51 \noutP 0.2108 (W) 0.0918 (W) 0.0918 (W) \npkB\n 1.4(T) 1.4(T) 1.4(T) \neddyP\n 5.54(mW) 0.22(mW) 24.6(mW) \nhysP\n 0.57(mW) 0.11(mW) 0.11(mW) \nCuP 184(mW) 1.62(mW) 1.62(mW) \nEfficiency (%) 83 97 77 \nPower density \n(W/cm³) 174.8 30.83 24.7 \n \nOptimized component (1) produces a 6% increase in \nefficiency and 7 fold increase in power density. Conventional inductors and transformers will normally operate with efficiency in the range of 90-95%. Hence, the application would have to greatly benefit from the increase in power density to tolerate the level of efficiency. Optimized component (2) uses the data from the prototype to identify eddy current core loss as the main loss mechanism. The core is reduced to ten 1µm laminates and via this adjustment improves efficiency by 20%. Comparing the two \noptimized components, a trade off between efficiency and \npower density is evident.  \nV.  CONCLUSION\n \nMicromachined inductors with different magnetic cores \nhave been fabricated on glass using micromachined techniques borrowed from the LIGA process.  The analytical results using the two dimensional model agree well with the \nexperimental values of inductance.  The challenge of developing inductors and transformers \nfor MHz DC-DC converter operation requires the development of suitable core materials and manufacturability of laminated layers. This fact is demonstrated by the results within Tables 1 and 2. The micro-inductor presented in this article has proved that an efficient component can be manufactured via an inexpensive UV LIGA and electroplating process. \n  \nACKNOWLEDGMENT \nThanks are extended to Prof Cywinski for the VSM data \nperformed at the University of Leeds, and to Raytheon \nSystems Limited for performing the resistivity measurements. This work was made possible through the funding of the Scottish Consortium in Integrated Micro Photonic Systems (SCIMPS) funded by the Scottish Funding Council under the Strategic Research Development Grant scheme. \n  \nREFERENCES \n[1]  Huljak, R.J.; Thottuvelil, V.J.; Marsh, A.J.; Miller, B.A.; \nApplied Power Electronics Conference and Exposition, 2000, “Where are power supplies headed?”. APEC 2000. Fifteenth \nAnnual IEEE, Volume 1, 6-10 Feb. 2000, pp.10-17, vol.1Digital \nObject Identifier 10.1109/APEC.2000.826076  \n[2]  International Technology Roadmap for Semiconductors 2001 \nEdition System Drivers (ITRS) \n[3]  Lotfi, A.W.; Wilkowski, M.A. 2001 “Issues and advances in \nhigh-frequency magnetics for switching power supplies”; \nProceedings of the IEEE Volume 89,  Issue 6,  June 2001 pp. \n:833 - 845  Digital Object Identifier 10.1109/5.931473 \n[4]  S.Ohnuma, H.J.Lee, N. Kobayashi, H. Fujimori, and T. \nMasumoto, July 2001, “Co-Zr-O Nano-Granular Thin Films with \nImproved High Frequency Soft Magnetic Properties”, IEEE Trans. Magnetics, Vol. 37, No 4. \n[5]  O.Dezuari, S.E. Gilbert, E. Belloy, M.A.M. Gijs, 2000, “High \ninductance planar transformers”, Sensors and Actuators 81, pp. 355-358.  \n[6]  Park J Y, Lagorce L K and Allen M G 1997 Ferrite-based \nintegrated planar inductors and transformers fabricated at low temperature IEEE Trans. Magn. 33, pp. 3322–5 \n[7]    Chong H. Ahn, and Mark G. Allen, Dec 1998, “Micromachined \nPlanar Inductors on Silicon Wafers for MEMS Applications”, IEEE Transactions on Industrial Electronics, Vol. 45, No. 6, pp. \n886-876 \n[8]            J. Park et al., “A Sacrficical layer Approach to Highly Laminated \nMagnetic Cores”, in Proc IEEE Int. Conf. MEMS, Las Vegas, \nNV, Jan. 2002, pp. 380-383.    \n[9]  D. P. Arnold et al , “Vertically Laminated Magnetic Cores \nbyElectroplating Ni-Fe Into Micromachined Si”, IEEE Trans. On \nMagnetics, Vol. 40, No. 4, July 2004.  \n[10]  L. Daniel, C.R. Sullivan, “Design of microfabricated inductors”, \nIEEE, Trans on Power Electronics, Vol. 14, No. 4, July 1999. \n[11] M. Seitz et al, “Squeeze More Performance Out of Toroidal \nInductors”, Power Electronics Technology August 2005 . \n[12]  L. Hua ,  D.Flynn, C.Bailey and M.Desmulliez , Sept 2006, “An \nAnalysis of a Microfabricated Solenoid Inductor”, ESTC, \nDresden Germany. \n[13]  D. Flynn, R.S. Dhariwal, M.P.Y. Desmulliez, IoP Journal of \nMicroengineering and Micromechanics (2007) “Study of a \nsolenoid microinductor operating in the MHz frequency range”, pp. 1811-1818. "    },    {        "title": "1406.5488v1.Linear_magnetoelectricity_at_room_temperature_in_perovskite_superlattices_by_design.pdf",        "content": "Linear magnetoelectricity at room temperature in perovskite superlattices by design\nSaurabh Ghosh, Hena Das, and Craig J. Fennie\u0003\nSchool of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA\n(Dated: June 27, 2021)\nDiscovering materials that display a linear magnetoelectric e\u000bect at room temperature is chal-\nlenge. Such materials could facilitate novel devices based on the electric-\feld control of magnetism.\nHere we present simple, chemically intuitive design rules to identify a new class of bulk magneto-\nelectric materials based on the `bicolor' layering of Pnma ferrite perovskites, e.g., LaFeO 3/ LnFeO 3\nsuperlattices for which Ln = lanthanide cation. We use \frst-principles density-functional theory\ncalculations to con\frm these ideas. Additionally, we elucidate the origin of this e\u000bect and show it\nis a general consequence of the layering of any bicolor, Pnma perovskite superlattice in which the\nnumber of constituent layers are odd (leading to a form of hybrid improper ferroelectricity) and\nGoodenough- Kanamori rules. Here, the polar distortions induce both weak ferromagnetism and a\nlinear magnetoelectric e\u000bect. Our calculations suggest that the e\u000bect is 2-3 times greater in mag-\nnitude than that observed for the prototypical magnetoelectric material, Cr 2O3. We use a simple\nmean \feld model to show that the considered materials order magnetically above room temperature.\nI. INTRODUCTION\nMultiferroics are materials in which ferroelectricity and\nmagnetism coexist. [1{3] Despite recent intense e\u000borts to\ndiscover new multiferroics, there are surprisingly few ma-\nterials that display this property at room temperature.\nFurthermore, the primary challenge remains to identify\nmaterials that have a functional coupling between an\nelectrical polarization and a magnetization at room tem-\nperature [4, 5]. Such materials may, for example, fa-\ncilitate technologically important devices based on the\nelectric \feld control of magnetism [6{11].\nOne way to design such cross-couplings is to start\nwith a paraelectric material that is magnetically or-\ndered and induce a ferroelectric lattice distortion. [12]\nFor example, it was shown [13] how a polar distortion\n{ in an antiferromagnetic{paraelectric (AFM-PE) ma-\nterial displaying linear magnetoelectricity { would in-\nduce weak-ferromagnetism in the LiNbO 3structure, e.g.,\nFeTiO 3[14] or MnSnO 3[15], and subsequently allow\nfor the electric-\feld switching of the magnetization by\n180\u000e. Alternatively a ferroelectric distortion in an AFM-\nPE material that displays weak-ferromagnetism can in-\nduce linear magnetoelectricity [5, 12]. Here, Bousquet\nand Spaldin recently realized that the orthorhombic per-\novskites, space group Pnma , are prime realizations and\nproposed epitaxial strain as a route to induce ferroelec-\ntricity [16]. They showed from \frst-principles that under\nlarge strain, Pnma CaMnO 3indeeds becomes ferroelec-\ntric. The polar lattice distortions lowers the symmetry\ntoPmc 21and a linear magnetoelectric e\u000bect is subse-\nquently induced.\nIn the present study, we take an alternate route to\nachieve ferroelectrically induced linear magnetoelectric-\nity in Pbnm (space group number 62, in standard setting\nwhich isPnma ) perovskites by taking advantage of a re-\n\u0003fennie@cornell.educent direction [17, 18] whereby the combination of rota-\ntions/tilts of the BO 6octahedra and A-site cation order-\ning facilitate ferroelectric order [19{21], without the need\nfor strain. We consider the rare-earth (La/Ln)Fe 2O6or-\nthoferrite superlattices in which the La and Ln cations\n(Ln = Ce, Nd, Sm, Gd, Dy, Tm, Lu and Y) are ordered in\nlayers along the crystallographic c-axis, where respective\nsupercells have been constructed asp\n2ap\u0002p\n2ap\u00022ap\n(here,apthe pseudocubic lattice parameter of Pbnm\nLaFeO 3). Note that similar results are obtained for\n(LaFeO 3)n/(LnFeO 3)mheterostructures of Pnma mate-\nrials when both nandmare odd [22, 23]. The choice\nof the orthorferrites was dictated by the fact that bulk\nLnFeO 3materials order magnetically above room tem-\nperature, with T Nas high as\u0018740 K for LaFeO 3[24{\n27].\nWe show from \frst-principles that the magnitude of\nthe linear magnetoelectric (ME) tensor in these het-\nerostructures is 2-3 times that of the canonical linear\nME, Cr 2O3. [28{32] This work provides a practical route\nto create a new class of multiferroic materials that dis-\nplay a linear magnetoelectric e\u000bect at room temperature\nwhereby octahedral rotations mediate a nontrivial cou-\npling between magnetism and ferroelectricity [18, 20, 33{\n35].\nII. COMPUTATIONAL DETAILS\nFirst-principles calculations have been carried out\nusing density functional theory [36] with projector\naugmented wave (PAW) potentials [37] and within\nLSDA+U [38], as implemented in the Vienna ab ini-\ntio simulation package (VASP) [39]. We have consid-\nered PAW potentials for Ln3+ions where f-states are\ntreated in the core, eliminating magnetic orderings as-\nsociated with the f-state magnetism which occurs at\nmuch lower temperature. For Fe3+ions, we have in-\ncluded the on-site d\u0000dCoulomb interaction parameter\nU=6.0 eV, and exchange interaction parameter J=1.0arXiv:1406.5488v1  [cond-mat.mtrl-sci]  20 Jun 20142\neV. The exchange-correlation part is approximated by\nPBEsol functional [40], which improves the structural de-\nscriptions over standard LDA or GGA [41]. The conver-\ngence in total energy and Hellman-Feynman force were\nset as 0.1\u0016eV and 0.1meV =\u0017A, respectively. All calcula-\ntions have been performed with a 500 eV energy cuto\u000b\nand with a \u0000-centered 6 \u00026\u00024 k- point mesh. conver-\ngence has been tested with higher energy cuto\u000b, k-mesh\nand found to be in agreement with the present settings.\nNon-collinear magnetization calculations were performed\nwith L-S coupling [42], whereas total polarization was\ncalculated with the Berry phase method [43] as imple-\nmented in VASP.\nIII. POLARIZATION, MAGNETIZATION AND\nSWITCHING\n[110] !\"\n[001] [110] \n[001] [110] !\"\n[110] \n[001] [110] \n[110] !\"(a) (b) (c) \nPm3m AFM _\t\r \nP4/mbm AFM \nIcmm wFM \nPbnm wFM \nCmcm AFM \nQRot (a0a0c+) (M3+) QTilt (a-a-c0) (R4+) QAFE (X5+) (d) (e) \nQRot (a0a0c+) QTilt (a-a-c0) \t\r \t\r \t\r \t\r -QAFE +M +QAFE +M \n+QAFE -M -QAFE -M \nFIG. 1. The `orthorhombic' ABO 3perovskite, space group\nPbnm . The structure, Glazer pattern a\u0000a\u0000c+, is described\nby three symmetrize basis modes of cubic Pm\u00163m:(a) Q Rot,\nin-phase rotation of BO 6octahedra about [001] (irrep. M+\n3),\n(b) Q Tilt, tilt of BO 6octahedra about [110] with A-site dis-\nplacement (irrep. R+\n4), (c) Q AFE anti-ferroelectric A-site dis-\nplacement (irrep. X+\n5). (d) The group-subgroup relation from\nhigh-symmetric Pm\u00163mto low-symmetric Pbnm . (e) Two di-\nmensional energy surface contour of LaFeO 3with respect to\nthe primary Q Rot(a0a0c+) and Q Tilt(a\u0000a\u0000c0) distortions.\nThe space group symmetry of the orthorhombic Pbnm\nstructure adopted by most perovskites [44, 45] is es-\ntablished by two symmetry-lowering structural distor-\ntions of the cubic Pm\u00163mperovskite structure: an in-\nphase rotation of the BO 6octahedra about the cubic\n[001] axis (transforming like the irreducible representa-\ntion M+\n3) QRotand an out-of-phase tilt of the BO 6oc-\ntahedra about the cubic [110] axis (transforming like the\nirreducible representation R+\n4), Q Tilt, as shown in Fig-ure 1a and Figure 1b, respectively. Together these two\ndistortions produce the Glazer rotation pattern a\u0000a\u0000c+.\nOther kinds of structural distortions are also allowed by\nsymmetry in the Pbnm structure. In particular, recent\nwork has shown that anti-polar displacements of the A-\nsite cations\u0000as shown in Figure 1c, the displacements\nare equal in magnitude but in opposite directions in ad-\njacent AO planes { play a crucial role in stabilizing the\nstructures of Pbnm perovskites [46, 47]. These anti-polar\ndisplacements (transforming like the irrep X+\n5) are cou-\npled to the two rotation distortions, i.e., there is a tri-\nlinear term in the free energies of cubic Pm\u00163mperovskites\nthat couples all three distortions [48], F= Q AFE QTilt\nQRot, where Q AFE, QTilt, and Q Rotare the amplitudes\nof the anti-polar distortion, tilt and rotation distortions\nrespectively. Hence, reversing the sense of either Q Tilt\nor Q Rotwill therefore reverse the direction of the anti-\npolar displacements, Q AFE.Pbnm orthoferrites, the Fe\nspins typically order in a G-type antiferromagnetic order-\ning pattern with weak ferromagnetism (wFM) along the\nPbnm orthorhombic c-axis. This wFM is in fact induced\nby the Q Tiltdistortion in Pbnm and hence the sense of\nthis particular rotation and the direction of the canted\nmagnetic moment are naturally coupled in a non-trivial\nway. These ideas are summarized in Figures 1d and e.\nThePbnm perovskites are thus a system in which\noctahedral rotations mediate a non-trivial coupling be-\ntween antiferroelectricity and magnetism. The recently\ndeveloped theory of hybrid improper ferroelectricity has\nshown how antiferroelectricity in Pbnm perovskites can\ngive rise to ferri-electricity in perovskite hetereostruc-\ntures, [21] such as (A/A')B 2O6double perovskites, Fig-\nure 2a. A simple picture [20] that elucidates the mecha-\nnism is the following: the two rotation distortions, Q Rot\nand Q Tilt, break inversion symmetry at the A-site of the\ncubic Pm\u00163mstructure whereas the A/A' cation order-\ning breaks B-site inversion symmetry, such that the A-\nsite displacements depicted in Figure 1c are no longer\nequal and opposite but instead give rise to a macro-\nscopic polarization, as shown in Figures 2b (in other\nwords, this A-site displacement mode becomes a zone-\ncenter polar mode in the cation ordered unit cell, which\nhasP4=mmm symmetry in the absence of any rotations,\nFigure 2a). The key is that since the (now polar) A-site\ndisplacements are coupled to the rotations as described\nabove, switching the direction of the polarization, Q P,\nwill switch the sense of one of the rotations. If it is Q Tilt\nthat switches, then the direction of the canted moment\nwill also switch, resulting in electric \feld control of the\nmagnetization. These facts are summarized in Figure 2c\nand d (it is highly instructive to compare Figures 1d and\n2c.)\nWe have used \frst-principles total energy calculations\nto consider the complete manifold of possible lower sym-\nmetry structures for this class of compounds and have\nidenti\fed the structure shown in Figure 2b as the low-\nest in energy. This structure has polar Pb21mspace\ngroup symmetry and displays both ferroelectricity (with3\na polarization along the orthorhombic y axis, P y) and\nweak ferromagnetism (with a net magnetization along\nthe z-axis, M z). The resultant magnetic con\fguration\nhas magnetic point group m0m20and consists of G-type\nAFM ordering with the easy axis along x, A-type AFM\nordering along the y-axis and a FM canting of spins along\nthez-axis (G x, Fz) [25{27, 49{52].\nSince, the origin of the polarization in our\n(La/Ln)Fe 2O6materials is a non-cancelation of the LaO\nand LnO layer polarizations. A simple way to increase\nthis non-cancellation and hence the polarization is to\nchoose a Ln cation whose tendency to o\u000b-center from\nthe ideal perovskite A-site di\u000bers greatly from that of\nLa. This is accomplished by choosing a Ln cation that\nis much smaller than La. In Figure 2e, we show that\nthe magnitude of P ymonotonically increases as the Ln\ncation becomes smaller, from 2.2 \u0016C/cm2for Ln=Ce to\n11.6\u0016C/cm2for Ln=Lu. Said another way, as the av-\nerage tolerance factor \u001cavgdecreases, the polarization in-\ncreases (note, \u001cABO 3= (rA+rO)=p\n2(rB+rO), where\nrA,rBandrO, are ionic radii of A, B and O atoms re-\nspectively). As discussed by Mulder et al. [21], such a\nsimple behavior is only true when one of the two A-site\ncations is the same for all compounds. Additionally, M z\nis roughly 0.07 \u0016B/f.u:for all compounds (there is a small\nincrease in M zas\u001cavgdecreases, 0.065-0.070 \u0016B/f.u, but\nis too small to be signi\fcant).\nThe direction of the polarization can in principle be\nswitched 180\u000ebetween symmetry equivalent states with\nthe application of an electric-\feld. In this process the\nsense of either Q Rotor Q tiltwill switch. The question\nas to which distortion would actually switch is a chal-\nlenging, dynamical problem, one for which today we still\ndon't have a satisfactory answer (please see Ref. [35] for\na nice discussion) and beyond the scope of this paper.\nWe know, however, that switching does depend in some\nway on the energy barriers between the energy minima\ndisplayed in Figure 2d. Understanding how to control\nany of the energy barriers is useful information, even if\nthe precise path is not known. Within this limited sense\nlet us brie\ry discuss the naive switching paths.\nWe found that the barrier height along Q Rot, \u0001E Rot=\nEPb21m- EPbmm , is about three times smaller than the\nbarrier height along Q Tilt, \u0001E Tilt=EPb21m- EP4=mbm\nfor all the compounds we considered. From this we con-\nclude that it is more likely that Q Rotwould switch when\nthe polarization switches. Since M zswitches only if Q Tilt\nswitches, the (La/Ln)Fe 2O6systems do not appear to\nbe likely candidates to pursue the electric-\feld switch-\ning of the magnetization. Furthermore, examination of\nFigure 2e shows that the ideal energy barrier to switch\nthe polarization, \u0001 EP\u0011\u0001ERot, increases dramatically\nas\u001cavgdecreases, as expected from the design rules of\nRef. [21]. In fact, it is not likely that the polarization\nin the majority of these materials could ever be switched\nunder realistic electric \feld strengths, other than perhaps\n(La/Ce)Fe 2O6, which has the lowest switching barrier (63\nmeV/f.u:along thea0a0c+rotation path).\nb [110] c [001] \na [110] !\"\nb [110] c [001] \na [110] !\"\nP4/mmm Pb21m (a) (b) \nP4/mmm PE, AFM \nP4/mbm PE, AFM \nPbmm PE, wFM \nPb21m FE, wFM, ME \nCm2m FE, AFM \nQRot (a0a0c+) (M2+) QTilt (a-a-c0) (M5-) QP (!5+) (c) \nQRot (a0a0c+) (in Å) QTilt (a-a-c0) (in Å) \n!\"!\"!\"!\"\n-P +M -\" +P +M +\" \n+P –M -\" -P –M +\" (d) \nY La Fe O \n0.060.0650.070.0750.080.0850.090.09503691215\n0.00350.0040.00450.0050.00550.0060.00650.0070.00750.0080.00850.00901002003004005006007001-τavg \n(1-τavg)2 Py(110) (µC/cm2) ΔEP (meV/f.u.) (e) (f) La/Lu La/Ce La/Tm La/Y La/Dy La/Gd La/Sm La/Nd EnergyPEP∆ΔEP PEnergy  FIG. 2. Structural and ferroelectric properties of\n(La/Ln)Fe 2O6superlattices. (a) (La/Y)Fe 2O6superlattice\nin high-symmetry P4=mmm structure, (b) lowest energy\nPb21mstructure with rotation and tilt of FeO 6octahedra,\n(c) Group-subgroup relation from P4=mmm to other lower\nenergy structures, (d) Two dimensional energy surface con-\ntours for (La/Y)Fe 2O6superlattice with respect to the pri-\nmary Q Rot(a0a0c+, irrep. M+\n2) and Q Tilt(a\u0000a\u0000c0, irrep.\nM\u0000\n5) distortions. In contour, Black dots represents the the\npossible minimum structures, each corresponding to di\u000berent\nsense of FeO 6octahedra rotations, Green triangles (down)\nindicate to the P4=mbm structures where Q Tiltis zero and\nOrange triangles (up) indicate to the Pbmm structure where\nQRotis zero. Variation of (e) Polarization P and (f) Switch-\ning barriers along Rotation (\u0001 ERot) with respect to (1- \u001cavg)\nand (1-\u001cavg)2, respectively.4\nIV. LINEAR MAGNETOELECTRIC COUPLING\nThe structural distortions associated with the sponta-\nneous polarization, however, induce by design a linear\nmagnetoelectric e\u000bect (which does not require switching\nof either the polarization or the magnetization),\n\u0001Mi= \u0006\u000bijEj (1)\n\u0001Pi= \u0006\u000bijHj; (2)\nwhere \u0001M i(\u0001P i) is the induced magnetization (polar-\nization) along the ith direction due to an electric (mag-\nnetic) \feld applied along the jth direction. The magnetic\npoint group of all (La/Ln)Fe 2O6compounds is m0m20,\ntherefore, the only non-zero components of the linear ME\ntensor are,\n\u000b=2\n40 0 0\n0 0\u000byz\n0\u000bzy03\n5 (3)\nwhere\u000byz6=\u000bzy(the ME process associated with these\ncomponents are schematically shown in Figure 3(a)).\nThe design strategy guarantees the existence of \u000b, but\nwhat is its magnitude? Here, we used the method de-\nscribed in Ref. [28] to calculate the lattice contribution\nof\u000b(although the linear ME response can have both\nlattice and electronic contributions, [12, 28] the method\nof Ref. [28] should give a reasonable order of magnitude\nestimate).\nA brief description of this method is the following: con-\nsidering only the lattice contribution to the energy, the\nenergy of the Pbnm crystal (U) under an applied electric\n\feld ( E) can be given by,\nU(qn;E) =U0+1\n2X\nnCnq2\nn\u0000X\nnqnpn\u0001E (4)\nwhere,qn, Cn, and pnare the amplitude, force constant,\nand dielectric polarity of the nthinfrared (IR) active force\nconstant eigenvector, ^qn, respectively. The dielectric po-\nlarity pnof thenth-IR active mode can be calculated as,\npn=@Pn=@qn, where P is the polarization (note that\nthe force constants and dielectric polarity are routinely\ncalculated from \frst principles. [28, 53]) Therefore, for a\ngiven electric \feld it is straightforward to calculate the\ninduced atomic displacements associated with each force\nconstant eigenvector, qn=qn^qn, whereqn=1\nCnpn\u0001E.\nSubsequently, the linear ME tensor,\n\u000bij=@Mi=@Ej; (5)\ncan be calculated by freezing in the total induced atomic\ndisplacements, u=P\nnqn^qn, then recalculating the net\nmagnetization.\nAs an example let us discuss the calculation of the lin-\near ME response for (La/Y)Fe 2O6(Ps\ny=9.0\u0016C/cm2and\nPx=Pz=0; Ms\nz= 0.13\u0016Band M x=My=0). It is useful\nto keep in mind that in Pb21mthe IR modes transform\n!Pz !My \nAFeO3 A!FeO3 AFeO3 A!FeO3 AFeO3 \nb [110] (y) c [001] (z) a [110] (x) !\"\nE !Mz !Py \nE \"zy \"yz (a) \nPsy Msz \n(b) FIG. 3. Linear Magnetoelectric response of (La/Y)Fe 2O6su-\nperlattice. (a) Superlattice with La/Y cation ordering along\nz-direction which is the crystallographic c-axis. The spon-\ntaneous HIF polarization is along y-direction and net mag-\nnetization M in the system along z-direction(Ps\ny, Ms\nz). The\nME tensor has two non-zero components. First, \u000byz, change\nin polarization along z-direction (\u0001P z) changes the magne-\ntization along y-direction (\u0001M y) and second \u000bzy, change in\npolarization along y-direction (\u0001P y) changes the magnetiza-\ntion along z-direction (\u0001M z), (b) Change in magnetization\n(with respect to the saturation magnetization) subject to the\nvariation of electric \feld ( E). Linear magnetoelectric compo-\nnents\u000bzyand\u000byzare obtained form the slope of M zvs. E y\nand M yvs. E zcurve, respectively. The magnitude of \u000bzyand\n\u000byzare given in Gaussian units (g.u.).\nas irreducible representations \u0000 1, \u00003, or \u0000 4, each leading\nto a polarization along the y,z, andxdirections, respec-\ntively. By symmetry, the \u0000 4modes do not mediate a ME\ne\u000bect. This corresponds to the fact that \u000bijfor anyiorj\n=xis zero by symmetry. For the purpose of the calcula-\ntion, we imagine the experiment in which an electric-\feld\nis applied and the resulting change in magnetization is\nmeasured.\nWith the application of an electric \feld along the y\ndirection, symmetry dictates that only the \u0000 1modes re-\nspond, i.e., pn\u0001E=pn\u0001^yE6= 0 forn2\u00001. The induced\natomic displacements\nu\u00001=X\nn2\u000011\nCnpnEy^qn; (6)5\nwere frozen into the equilibrium structure and the change\nin magnetization, which by symmetry is along the zdirec-\ntion, was calculated from \frst-principles. This procedure\nwas repeated for various magnitudes of the applied elec-\ntric \feld. These results are shown in Figure 3(b)), the\nslope of which gives the linear ME coupling \u000bzy. We \fnd\nthat the magnitude of \u000bzyis 3.54x10\u00004g.u., which is 2-3\ntimes larger than the transverse linear ME response of\nthe prototype ME compound Cr 2O3[28] at 0 K.\nWith the application of an electric \feld along the z\ndirection, symmetry dictates that only the \u0000 3modes re-\nspond, i.e., pn\u0001E=pn\u0001^zE6= 0 forn2\u00003. The induced\natomic displacements\nu\u00003=X\nn2\u000031\nCnpnEz^qn: (7)\ncan be calculated. Here we \fnd that the corresponding\nlinear ME response (see Figure 3(b)), measured by the\ncomponent \u000byzis equal to 0.72x10\u00004g.u., much weaker\nthan\u000bzy.\nV. ELECTRONIC STRUCTURE\n-8-6-4 -2 0 2 4-4-2024DOS (states/atom/eV)-8-6-4 -2 0 2 4-4-2024DOS (states/atom/eV)-8-6-4 -2 0 2 4Energy (eV)-4-2024DOS (states/atom/eV)Fe-3dO-2pFe-3dO-2pFe-3dO-2pO-2pFe-3d2.08 eV2.13 eVFe-3dO-2pO-2pFe-3d2.10 eVabcLaFeO3 YFeO3 (La/Y)Fe2O6 (a) (b) (c) \nFIG. 4. Calculated Density of States (DOS) for (a) LaFeO 3,\n(b) YFeO 3inPbnm and (c) (La/Y)Fe 2O6inPb21msymme-\ntry. For Fe+3ions, we have used the on-site d\u0000dCoulomb\ninteraction parameter U=6.0 eV and exchange interaction pa-\nrameter J=1.0 eV.\nWe have been discussing an approach to create room\ntemperature linear magnetoelectrics by layering Pbnm\nmaterials. From previous work, the robustness and uni-\nversality of the polar ground state is clear. There are,\nhowever, two important questions concerning the mag-netic state that need to be addressed in order to support\nthe claim of room temperature magnetoelectricity.\nThe \frst concerns the type of antiferromagnetic or-\ndering. As we discussed, we have found by direct \frst-\nprinciples calculations that all of the superlattices we\nconsidered order in a G-type pattern, which is required\nto observe the ME physics. There is a simple reason\nwhy this should be the case as the G-type antiferro-\nmagnetic ground state of the orthoferrites is driven by\nthe electronic con\fguration of the Fe+3ion. As an ex-\nample, let us brie\ry discuss the basic electronic struc-\nture ofPbnm LaFeO 3and YFeO 3and the superlattice\nmade out of them i.e. (La/Y)Fe 2O6inPb21msymme-\ntry. As shown in Figure 4(a) and (b), both LaFeO 3and\nYFeO 3are charge transfer insulators. The valence band\nis formed by majority Fe-3 dstates and O-2 pstates, while\nthe minority Fe-3 dstates are completely empty and form\nthe conduction band. Due to the d5electronic con\fgu-\nration only antiferromagnetic superexchange interactions\nbetween Fe+3ions via single O-2 porbitals are allowed.\nThe ferromagnetic contribution involving two perpendic-\nular 2porbitals is negligibly small as the Fe-O-Fe bond\nangle is close to 180\u000e. Therefore, the G-type AFM con-\n\fguration is universal for LnFeO 3systems. We have\nfound that the layered arrangement of La/Y cations re-\nsults in negligible changes to the basic electronic struc-\nture of the energy level positioning and band width (see\nFigure 4(c)). We therefore expect that the major compo-\nnent of Fe spins in the ground state of the (La/Y)Fe 2O6\nsuperlattice will be G-type (it is important to note that\nthe G-type AFM con\fguration in Pb21m, with the mag-\nnetic anisotropy along any crystallographic axis, allows\nlinear ME coupling).\nVI. ORDERING TEMPERATURE\nThe second important question that needs to be ad-\ndressed is whether or not we expect the spins to or-\nder at room temperature. A relatively straightforward\nmean \feld approach to calculating the N\u0013 eel temperature\n(TN) involves mapping total energy calculations onto a\nHeisenberg model, from which the magnetic exchange\ninteractions, Jij, can be extracted. Unfortunately, it\nis well know that the results obtained through this ap-\nproach depend sensitively on the particular value of Hub-\nbard U. Here we can take advantage of the fact that\nthe experimental values of T Nare known for the per-\novskite constituents of our superlattices. Table1, shows\nthe calculated values of T Nfor LaFeO 3, YFeO 3, and the\n(La/Y)Fe 2O6superlattice for a \fxed value of U, consid-\nering up to the third nearest neighbor exchange interac-\ntions. There are a few things to note. We found that the\ndominant interaction is only between nearest neighbor\n(nn) spins (the interaction pathways are shown in Fig-\nure 5) and the average nnexchange interaction ( Javg\nnn) of\n(La/Y)Fe 2O6is almost equal in value to that of LaFeO 3.\nAdditionally, when compared to the experimental values,6\nabcFe Fe \nFe Fe Fe Fe Opl Oap Oap Opl Opl Opl Fe - Oap- Fe à Jc  Fe - Opl- Fe à Jab  Fe \nFIG. 5. Exchange Interactions pathways, Jab\nnnimplies near-\nest neighbor Fe-Fe interactions (Fe- Fe distance 3.85 \u0017A inab\nplane) mediated by planer Oxygen (O pl), whereas Jc\nnnim-\nplies nearest neighbor Fe-Fe interactions (Fe-Fe distance 3.76\n\u0017A alongc-axis) mediated by apical Oxygen (O ap).\nTABLE I. Computed superexchange constants ( Jij) and mean\n\feld estimated magnetic transition temperature. Jab\nnnand\nJc\nnnrepresent nearest neighbor Fe-Fe interaction mediated via\nplanner and apical oxygen, respectively. The average interac-\ntion is given by Javg\nnn= (4\u0002Jab\nnn+ 2\u0002Jc\nnn)/6. See Figure 5).\nSystem Jab\nnnJc\nnnJavg\nnn TN(K)\nComputed Experiment\nLaFeO 3 5.81 5.20 5.61 1139 740 [25]\nYFeO 3 5.20 4.51 4.97 1009 655 [25]\n(La/Y)Fe 2O65.60 5.30 5.50 1117 {our calculations generally overestimated T N, however,\nthe calculated ratio of TLaFeO 3\nN andTY FeO 3\nN is in good\nagreement with the ratio of the measured values. Given\nthis fact, the TNof (La/Y)Fe 2O6superlattice is expected\nto be close to the magnetic transition temperature of\nLaFeO 3(\u0018740K). For the other (La/Ln) superlattices,\nwe have also calculated the corresponding TNand found\nthat for all the cases it is around the T Nof LaFeO 3. This\nleads us to propose that these superlattices are expected\nto order magnetically above room temperature.\nVII. CONCLUSION\nWe have used \frst-principles calculations to identify a\nfamily of (La/Ln)Fe 2O6superlattices that may display\na strong linear magnetoelectric e\u000bect at room tempera-\nture. Although the magnetoelectricity is ferroelectrically\ninduced, polarization switching is not required to observe\nthe e\u000bects studied here. 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Agarwal1, 2, 5\n1Institute for Quantum Science and Engineering, Texas A &M University, College Station, TX 77843, USA\n2Department of Physics and Astronomy, Texas A &M University, College Station, TX 77843, USA\n3Quantum Optics Laboratory, Baylor Research and Innovation Collaborative, Waco, TX 76704, USA\n4Department of Mechanical and Aerospace Engineering, Princ eton University, Princeton, NJ 08544, USA\n5Department of Biological and Agricultural Engineering,\nTexas A &M University, College Station, TX 77843, USA\n(Dated: May 28, 2020)\nThe cavity mediated spin current between two ferrite sample s has been reported by Bai et. al. [Phys. Rev.\nLett. 118, 217201 (2017)]. This experiment was done in the linear regi me of the interaction in the presence of\nexternal drive. In the current paper we develop a theory for t he spin current in the nonlinear domain where the\nexternal drive is strong so that one needs to include the Kerr nonlinearity of the ferrite materials. In this manner\nthe nonlinear polaritons are created and one can reach both b istable and multistable behavior of the spin current.\nThe system is driven into a far from equilibrium steady state which is determined by the details of driving\nﬁeld and various interactions. We present a variety of stead y state results for the spin current. A spectroscopic\ndetection of the nonlinear spin current is developed, revea ling the key properties of the nonlinear polaritons. The\ntransmission of a weak probe is used to obtain quantitative i nformation on the multistable behavior of the spin\ncurrent. The results and methods that we present are quite ge neric and can be used in many other contexts where\ncavities are used to transfer information from one system to another, e.g., two di ﬀerent molecular systems.\nI. INTRODUCTION\nIt is known from the quantum electrodynamics that an ex-\nchange of a photon between two atoms results in the long-\nrange interaction such as dipole-dipole interaction. This inter-\naction is responsible for transferring the excitations fro m one\natom to another1. In free space, however, such interactions\nare prominent only if the atoms are within a wavelength. This\nchallenge can be overcome by utilizing cavities and in fact i t\nhas been shown how the dispersive cavities can produce sig-\nniﬁcant interactions in a system of noninteracting qubits2–4.\nWhile much of the work has been done in the context of\nqubits, there have been experiments demonstrating how the\nexcitations can be transferred among macroscopic systems5.\nIn particular in a paper using macroscopic ferrite samples, Bai\net al. demonstrated transfer of spin current from one ferrite\nsample to another. Apart from the coupling to the cavity, the re\nis no interaction between the two bulks. Thus the cavity medi -\nates the transfer of spin excitation from one system to anoth er.\nThe demonstrations of excitations for the macroscopic sys-\ntems are fascinating, but have ignored any possible intrins ic\nnonlinearities of the macroscopic systems. It is known in ca se\nof ferrites that the nonlinearities arise from the anisotro pic in-\nternal magnetic ﬁelds which lead to a contribution to the en-\nergy proportional to higher powers of magnetization. As a\nsignature of this nonlinearity one observes the bistable na ture\nin the ferromagnetic material if it is pumped hard6,7. In this\nwork we study the nonlinearities in the transfer of spin exci ta-\ntions and in particular the nonlinear spin current. The magn on\nmodes in one of ferromagnetic sample are pumped hard while\nthe other one is undriven. Each sample is interacting with th e\ncavity. The spin excitation migrating from one to the other\nis studied for diﬀerent degrees of the microwave drive ﬁeld.\nUnder various conditions for drive ﬁeld, the spin current ca n\nexhibit a variety of nonequilibrium transitions to bistabl e to\nmultistable values. We work in the strong coupling regime ofthe caivty QED8–11. The basis for detecting these nonlinear\nbehavior of spin current is developed through the examinati on\nof the nonequilibrium response of the nonlinear system to a\nweak probe. From a theoretical view-point, the nonequilib-\nriumness violating the detailed balance is essential for cr eat-\ning the stationary nonlinearity responsible for the multis tabil-\nity and large-scale quantum coherence nature of the collect ive\nexcitations12–15.\nIt is worth noting that the ferromagnetic materials espe-\ncially the yttrium iron garnet (YIG) samples are increasing ly\nbecoming popular in the study of the coupling to cavities,\nthanks to their high spin density and low dissipation rate16–21.\nThis results in the advantage of achieving strong and even\nultrastrong couplings to cavity photons10,11,22–26. The cavity\nmagnon polaritons, as demonstrated by recent advance, be-\ncome powerful for implementing the building block for quan-\ntum information and coherent control in the basis of strong\nentanglement between magnons10,27, photons28–31, acoustic\nphonons32and superconducting qubits24,33.\nNotably, the generic nature of our work presented in this\narticle shows the perspective of extending the approach to t he\nexcitons in polyatomic molecules and molecular aggregates ,\nby noting the similar form of nonlinear coupling Ub†bb†b\nwhere Uquantiﬁes the exciton-exciton scattering and bis the\nexcitonic annihilation operator34,35. The multistable nature is\nthen expected to be observed in molecular excitons as scalin g\nup the parameters.\nThis paper is organized as follows. In Sec.II, we discuss\nthe theoretical model for the nonlinear spin current and int ro-\nduce basic equations for the cavity-magnon system. We write\nthe semiclassical equations for spin current in the YIG sphe re\nand present numerical results using a broad range of parame-\nters in Sec.III. In Sec.IV , we develop a spectroscopic detec tion\nmethod for the spincurrents based on the polariton frequenc y\nshift by sending a weak probe ﬁeld into the cavity. We discuss\nthe theory of nonlinear magnon polariton in the case of a sin-2\ngle and two YIG system. Further, we numerically obtain the\ntransmission spectra and the polariton frequency shift usi ng\nexperimentally attainable parameters and show the transit ion\nfrom bistability to multistability. We conclude our result s in\nSec.V .\nII. THEORETICAL MODEL\nTo control the spin wave of the electrons in ferromagnetic\nmaterials, we essentially place two YIG spheres in a single-\nmode microwave cavity, due to the fact that the collective\nspin excitations may strongly interact with cavity photons (see\nFig.1). The dispersive spin waves haven been observed in YIG\nbulks, involving two distinct modes: Kittel mode and magne-\ntostatic mode (MS)36,37. The Kittel mode has the spatially uni-\nform proﬁle as obtained in the long wavelength limit, wherea s\nthe MS mode has ﬁnite wave number so that it has distinct fre-\nquency from the Kittel mode. The technical advance on laser\ncontrol and cavity fabrication recently made the mode selec -\ntion accessible. In our model, we take into account the Kitte l\nmode strongly coupled to cavity photons, along the line of re -\ncent experiments in which the MS mode is not the one of in-\nterest. The Kittel mode is a collective spin of many electron s,\nassociated with a giant magnetic moment, i.e., M=γS/V,\nwhereγ=e/mecis the gyromagnetic ratio for electron spin\nandSdenotes the collective spin operator with high angular\nmomentum. This results in the coupling to both the applied\nstatic magnetic ﬁeld and the magnetic ﬁeld inside the cavity ,\nshown in Fig.1. The Hamiltonian of the hybrid magnon-cavity\nsystem is\nH//planckover2pi1=−γ2/summationdisplay\nn=1Bn,0Sn,z+γ22/summationdisplay\nn=1/planckover2pi1K(n)\nan\nM2nVnS2\nn,z\n+ωca†a+γ2/summationdisplay\nn=1Sn,xBn,x(1)\nassuming that the magnetic ﬁeld in cavity is along the xaxis\nwhereas the applied static magnetic ﬁeld B0is along the zdi-\nrection. The 2nd term in Eq.(1) results from the magnetocrys -\ntalline anisotropy giving the anisotropic ﬁeld. We thereby as-\nsume the anisotropic ﬁeld has zcomponent only, in accor-\ndance to the experiments such that the crystallographic axi s\nis aligned along the ﬁeld B0.ωcrepresents the cavity fre-\nquency. By means of the Holstein-Primako ﬀtransform38, we\nintroduce the quasiparticle magnons described by the oper-\nators mandm†with [ m,m†]=1. Considering the typical\nhigh spin density in the ferromagnetic material, e.g., yttr ium\niron garnet having diameter d=1mm in which the density\nof the ferric iron Fe3+isρ=4.22×1027m−3that leads to\nS=5N\n2=5\n2ρV=5.524×1018, the collective spin Sis of\nmuch larger magnitude than the number of magnons, namely,\nS≫/angbracketleftm†m/angbracketright. The raising and lowering operators of the spin\nare then approximated to be S+\ni=√2Simi,S−\ni=√2Sim†\ni\n(i=1,2 labels the two YIGs). In the presence of the external\nmicrowave pumping, we can recast the Hamiltonian in Eq.(1)\nFIG. 1: Schematic of cavity magnons. Two YIG spheres\nare interacting with the basic mode of microcavity in\nwhich the right mirror is made of high-reﬂection mate-\nrial so that photons leak from the left side. The static\nmagnetic ﬁeld producing Kittel mode in YIG1 is along z-\naxis whereas the static magnetic ﬁeld for YIG2 is tilted\nwith respect to z-axis. The microwave ﬁeld is along y-\naxis and the magnetic ﬁeld inside cavity is along x-axis.\ninto\nHeﬀ//planckover2pi1=ωca†a+2/summationdisplay\ni=1/bracketleftig\nωim†\nimi+gi/parenleftig\nm†\nia+mia†/parenrightig\n+Uim†\nimim†\nimi/bracketrightig\n+iΩ/parenleftig\nm†\n1e−iωdt−m1eiωdt/parenrightig(2)\nwhere the frequency of Kittel mode is ωi=γBi,0−\n2/planckover2pi1K(i)\nanγ2Si/M2\niViwithγ/2π=28GHz/T.gi=√\n5\n2γ√\nNB vac\ngives the magnon-cavity coupling with Bvac=√2π/planckover2pi1ωc/V de-\nnoting the magnetic ﬁeld of vacuum and Ui=K(i)\nanγ2/M2\niVi\nquantiﬁes the Kerr nonlinearity. The Rabi frequency is rela ted\nto input power PdthroughΩ=γ/radicalig\n5πρdPd\n3c. From Eq.(2) we ob-\ntain the quantum Langevin equations (QLEs) for the magnon\npolaritons as\n˙m1=−(iδ1+γ1)m1−2iU1m†\n1m1m1−ig1a+Ω+/radicalbig\n2γ1min\n1(t)\n˙m2=−(iδ2+γ2)m2−2iU2m†\n2m2m2−ig2a+/radicalbig\n2γ2min\n2(t)\n˙a=−(iδc+γc)a−i(g1m1+g2m2)+/radicalbig\n2γcain(t) (3)\nin the rotating frame of drive ﬁeld, where δi=ωi+\nUi−ωdandδc=ωc−ωd.γiandγcrepresent the\nrates of magnon dissipation and cavity leakage, respective ly.\nmin\ni(t) and ain(t) are the input noise operators associated with\nmagnons and photons, having zero mean and broad spec-\ntrum:/angbracketleftmin,†\ni(t)min\nj(t′)/angbracketright=¯niδi jδ(t−t′),/angbracketleftmin\ni(t)min,†\nj(t′)/angbracketright=(¯ni+\n1)δi jδ(t−t′),/angbracketleftain,†(t)ain(t′)/angbracketright=0 and/angbracketleftain(t)ain,†(t′)/angbracketright=δ(t−t′)\nwhere ¯ ni=[exp(/planckover2pi1ωi/kBT)−1]−1is the Planck distribution.3\nFIG. 2: Spin current signal obtained from Eq.(5) illus-\ntrating bistability-multistabbility transition. (a) ωd/2π=\n9.9869GHz,ωc/2π=10.078GHz and (b) ωd/2π=\n9.9989GHz,ωc/2π=10.078GHz; (c)ωd/2π=10GHz,\nωc/2π=10.06GHz and (d)ωd/2π=10GHz,ωc/2π=\n10.075GHz. Other parameters are ω1/2π=10.018GHz,\nω2/2π=9.963GHz, g1/2π=42.2MHz, g2/2π=33.5MHz,\nU1/2π=7.8nHz, U2/2π=42.12nHz,γ1/2π=5.8MHz,\nγ2/2π=1.7MHz andγc/2π=4.3MHz. In Fig.2(b), for\ndrive power=30mW, we observe three stable states given\nbyx=1.58×1014,x=5.6×1014andx=8.83×1014.\nIII. SPIN CURRENT IN NONLINEAR MAGNON\nPOLARITONS\nSince the YIG1 is driven by a microwave ﬁeld, one would\nexpect a spin transfer towards YIG2. This results in the spin\ncurrent which can be detected electronically through the ma g-\nnetization of the systems. Thus the spin current is deter-\nmined by the quantity /angbracketleftm†\n2m2/angbracketright, up to a constant in front. The\nspin migration eﬀect has been observed in Ref.5. However\nas indicated in the introduction, the nonlinearity of the sa m-\nple starts becoming important if the driving ﬁeld increases .\nThus we would like to understand the behavior of the spin\ncurrent when the dependence on Kerr nonlinearity in Eq.(3)\nbecomes important. As a ﬁrst step we will study the re-\nsulting behavior at mean-ﬁeld level, i.e., the quantum nois e\nterms in Eq.(3) are essentially dropped and the decorrela-\ntion approximation is invoked when calculating the mean val -\nues of the operators. In the steady state, these mean values\nO(0)=/angbracketleftO/angbracketright(O(0)=M1,M2,A;O=m1,m2,a) obey the non-\nlinear algebraic equations\n−(iδ1+γ1)M(0)\n1−2iU1|M(0)\n1|2M(0)\n1−ig1A(0)=−Ω\n−(iδ2+γ2)M(0)\n2−2iU2|M(0)\n2|2M(0)\n2−ig2A(0)=0\n−(iδc+γc)A(0)−i(g1M(0)\n1+g2M(0)\n2)=0.(4)\nFIG. 3: Spin current signal against drive power at di ﬀer-\nent values of cavity leakage. (a) γc<g1,2indicates strong\nmagnon-cavity coupling; (b,c) γc≃g1,2indicates the interme-\ndiate magnon-cavity coupling; (d) γc>g1,2gives rise to weak\nmagnon-cavity coupling. ωc/2π=10.078GHz,ωd/2π=\n9.998GHz and other parameters are the same as Fig.2.\nA manipulation of Eq.(4) yields to the following nonlinear\nequation for the spin transfer, i.e., magnetization from YI G1\nto YIG2 with x≡|M(0)\n2|2\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜δ1+2U1(δ2\nc+γ2\nc)\ng2\n1g2\n2/vextendsingle/vextendsingle/vextendsingle˜δ2+2U2x/vextendsingle/vextendsingle/vextendsingle2x/parenleftig˜δ2+2U2x/parenrightig\n−g2\n1g2\n2\n(δc−iγc)2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nx=5πg2\n1g2\n2γ2ρdPd\n3c(δ2c+γ2c)(5)\nwhere ˜δ1,2=δ1,2−iγ1,2−g2\n1,2\nδc−iγc. We ﬁrst note that in the\nabsence of Kerr nonlinearity, the spin current reads\nx=5πg2\n1g2\n2γ2(δ2\nc+γ2\nc)ρd\n3c|˜δ1˜δ2−g2\n1g2\n2|2Pd (6)\nwhich corresponds to the linear spin current measured in\nRef.5. This gives rise to the linear regime with lower drive\npower in Fig.2 and Fig.3.\nFig.2 depicts the spin current ﬂowing to YIG2 against var-\nious degrees of the drive power. One can observe a smooth\nincrease of the spin current obeying the linear law with the\ndrive power, under the weak pumping. When the drive be-\ncomes stronger, a sudden jump of the spin current shows up,\nmanifesting more e ﬃcient spin transfer between the two YIG\nspheres. When reducing the drive power, we can observe\nan alternative turning point, where a downhill jump of spin4\nFIG. 4: Schematic of detecting spin polarization migration\nbetween YIGs. Small panel shows the frequency shift, re-\nsulting from Kerr nonlinearity ampliﬁed by strong drive.\ntransfer is elaborated. By tweaking the magnon-light inter -\naction, a bistability-multistability transition is furth er mani-\nfested, wherein the latter is resolved by the two cascading\njumps. For instance, Fig.2(a,b) elaborate such transition by\nincreasing the frequency of the drive ﬁeld. The similar tran si-\ntion can be observed as well through increasing the cavity fr e-\nquency, shown in Fig.2(c,d). It is worth noting from Fig.2 th at\nthe multistability of magnon polaritons is accessible with in\nthe regime U1≪U2, whereas the multistable feature becomes\nless prominent with reducing the Kerr nonlinearity of YIG2,\nnamely, U1∼U2.\nSo far, the results has manifested the essential role of the\nnonlinearity in producing the multistable nature of the spi n\ntransfer between magnon modes. Next we plot in Fig.3 the ro-\nbustness of multistability for di ﬀerent degrees of cavity leak-\nage. The spin current manifests the multistable nature of\nmagnon polaritons within a broad range of cavity leakage\nrates. Given the low-quality cavity where g1,2≃γc≫γ1,2,\none can still see the multistability.\nNotice that the above results indicated |M(0)\ni|2≪2S≃\n1.1×1019which fulﬁlled the condition for the validity of the\neﬀective Hamiltonian in Eq.(2).\nIV . SPECTROSCOPIC DETECTION OF NONLINEAR\nMAGNON POLARITONS\nIn order to study the physical characteristics of a system,\nit is fairly common to use a probe ﬁeld. The response to the\nprobe gives the system characteristics such as the energy le v-\nels, line shape and so on. We adopt a similar strategy here\nthough we are dealing with a nonlinear & nonequilibrium sys-\ntem. We apply a weak probe ﬁeld to the cavity ans study how\nthe transimission spectra changes with increasing drive po wer,\nsee Fig.4. When turning o ﬀthe drive, the probe transmission\ndisplays two polariton branches in the limit of strong cavit y-\nmagnon coupling. As the drive ﬁeld is turned on, the nonlin-\nearity of the YIG spheres starts entering, which results in a\nsigniﬁcant change in the transmission of the weak probe. The\ntransmission peaks are shifted, besides the transmission b e-\ncomes asymmetric. To elaborate this, we will start o ﬀfrom a\nFIG. 5: (a) Transmission spectrum for a single YIG in a\nsingle-mode microwave cavity, as a function of scanning\nprobe frequency, according to Eq.(12). Blue line is for\nthe case when turning o ﬀthe drive ﬁeld. Figure 5(b) de-\npicts the spin polarization against the drive power. We ob-\nserve that, for drive power =90mW, there are two sta-\nble states at|M(0)|2=0.66×1015and|M(0)|2=2.55×\n1015. The green and red lines in ﬁgure 5(a) are for the\nsame bistates with input power Pd=90mW. Figure 5(c)\ndepicts the frequency shift of the lower polariton peak\nas a function of drive power. Parameters are ωc/2π=\n10.025GHz,ωm/2π=10.025GHz,ωd/2π=9.998GHz,\ng/2π=41MHz, U/2π=8nHz,γm/2π=17.5MHz\nandγc/2π=3.8MHz, taken from recent experiments7.\nsimple case including a single YIG sphere.\nA. Nonlinearity of a single YIG as seen in probe transmission\nFor a single YIG sphere in a microwave cavity as consid-\nered in Ref.7, the dynamics obeys the following equations\n˙M=−(iδm+γm)M−2iU|M|2M−igA+Ω\n˙A=−(iδc+γc)A−igM+Epe−iδt(7)\nperturbed by a weak probe ﬁeld at frequency ωandΩp(t)=\nEpe−iδt+c.c., whereEpis the Rabi frequency of the probe ﬁeld\nandδ=ω−ωd. The existence of nonlinear terms in Eq.(7)\nallows for the Fourier expansion of the solution such that\nM=∞/summationdisplay\nn=−∞M(n)e−inδt,A=∞/summationdisplay\nn=−∞A(n)e−inδt(8)\nwhere M(n)andA(n)are the amplitudes associated with the\nn-th harmonic of the probe ﬁeld frequency40. LetM0≡M(0)\nandA0≡A(0)denote the zero-frequency component, giving5\nthe steady-state solution when turning o ﬀthe probe ﬁeld. In-\nserting these into Eq.(7) one can ﬁnd the linearized equatio ns\nfor the components M±≡M(∓1)andA±≡A(∓1)\n(∆−δ)M++2UM2\n0M∗\n−+gA+=0\n2UM2\n0M∗\n++(∆+δ)M−+gA−=0\ngM++(∆c−δ)A+=−iEp\ngM−+(∆c+δ)A−=0,\n∆=δm+4U|M0|2−iγm,∆c=δc−iγc(9)\nwhich yields to\nA+=Ep\ni(∆c−δ)/bracketleftigg\n1+g2\n(∆c−δ)v/bracketrightigg\n(10)\nwhere\nv=∆−δ−g2\n∆c−δ−4U2(∆∗\nc+δ)|M0|2\n(∆∗c+δ)(∆∗+δ)−g2. (11)\nEq.(10) deﬁnes the 1st-order response function and hence th e\ncomplex transmission amplitude is given by\nT(δ)=−i\n∆c−δ/bracketleftigg\n1+g2\n(∆c−δ)v/bracketrightigg\n(12)\nwhich leads to the polariton frequency\nδ2=1\n2/bracketleftbigg\n(δm+4U|M0|2)2+δ2\nc+2g2−4U2|M0|2\n±/radicalig\nF+16U2δ2c|M0|2/bracketrightbigg(13)\nwith\nF=/parenleftig\n(δm+4U|M0|2−δc)2+4g2−4U2|M0|2/parenrightig\n×/parenleftig\n(δm+4U|M0|2+δc)2−4U2|M0|2/parenrightig\n.(14)\nFor a given drive power, we calculate |M0|2from Eq.(7) and\ninsert this value into Eq.(12) to obtain the transmission am pli-\ntude. The peak positions are given by Eq.(13). We plot the\ntransmission spectrum in Fig.5(a), employing the experime n-\ntally feasible parameters7. It shows the Rabi splitting between\nthe two polariton branches at zero input power. As the input\npower is switched on, the peak shift can be considerably ob-\nserved, resulting from the Kerr nonlinearity, as predicted from\nEq.(13). For a given drive power, the lower and higher po-\nlaritons correspond to the lowest and highest energy peaks o f\nthe transmission spectra at frequencies ωLPandωHPrespec-\ntively. This is further illustrated in Fig.5(b), where the t wo\nstable states are observed at Pd=90mW. Fig.5(c) depicts the\nfrequency shift of the peak of lower polariton as a function\nof input power, and the bistability of the magnon polaritons\nis therefore evident. Here the frequency shift of lower pola ri-\nton is deﬁned by∆LP≡ωLP−ω0\nLPwithω0\nLPgiving the lower\npolariton frequency in the absence of Kerr nonlinearity.\nFIG. 6: (a) Transmission spectrum for two YIG in a mi-\ncrowave cavity, as scanning probe frequency, according to\nEq.(21). Blue line is for the case without driving, while\ngreen, black and red lines are for triple states with input po wer\nPd=30mW. They represents the same three stable states de-\nscribed in ﬁgure 2(b). (b) Frequency shift associated with\nupper polariton peak, where δHP=ωHP−ωd. Other pa-\nrameters areωc/2π=10.078GHz,ω1/2π=10.018GHz,\nω2/2π=9.963GHz,ωd/2π=9.998GHz, g1/2π=42.2MHz,\ng2/2π=33.5MHz, U1/2π=7.8nHz, U2/2π=42.12nHz,\nγ1/2π=5.8MHz,γ2/2π=1.7MHz andγc/2π=4.3MHz.\nB. Detection of multistability in spin current via probe\ntransmission\nFor two YIG spheres interacting with a single-mode cavity,\nwe obtain the following equations for the system perturbed b y\na probe ﬁeld\n˙M1=−(iδ1+γ1)M1−2iU1|M1|2M1−ig1A+Ω\n˙M2=−(iδ2+γ2)M2−2iU2|M2|2M2−ig2A\n˙A=−(iδc+γc)A−i(g1M1+g2M2)+Epe−iδt.(15)\nApplying the Fourier expansion technique given in Eq.(8), w e\nﬁnd the linearized equations for the components associated\nwith the harmonic e±iδt\n(∆1−δ)M1,++2U1M2\n1,0M∗\n1,−+g1A+=0\n2U1M2\n1,0M∗\n1,++(∆1+δ)M1,−+g1A−=0\n(∆2−δ)M2,++2U2M2\n2,0M∗\n2,−+g2A+=0\n2U2M2\n2,0M∗\n2,++(∆2+δ)M2,−+g2A−=0\ng1M1,++g2M2,++(∆c−δ)A+=−iEp\ng1M1,−+g2M2,−+(∆c+δ)A−=0(16)\nwhich can be easily solved by matrix techniques. Eq.(16) can\nreduce to two linear equations with two unknowns\nv11v12\nv21v22M1,+\nM2,+=iEpα1\nα2 (17)6\n(c) (d)(b) (a)\nFIG. 7: Transition between bistability and multistability .\n(a)ωc/2π=10.078GHz,ωd/2π=9.9909GHz and (b)\nωc/2π=10.078GHz,ωd/2π=9.9989GHz; (c)ωc/2π=\n10.07GHz,ωd/2π=10GHz and (d)ωc/2π=10.085GHz,\nωd/2π=10GHz. Other parameters are the same as Fig.2.\nwith the coeﬃcients\nv11=∆ 1−δ−g2\n1\n∆c−δ+U1M2\n1,0\nU2M2\n2,0\n×g2\n1g2\n2−4U1U2(∆∗\nc+δ)(∆c−δ)M∗,2\n1,0M2\n2,0\n(∆c−δ)[(∆∗c+δ)(∆c+δ)−g2\n1]\nv12=g1g2\n∆c−δU1M2\n1,0\nU2M2\n2,0g2\n2−(∆c−δ)(∆2−δ)\n(∆∗c+δ)(∆∗\n1+δ)−g2\n1−1\nv21=g1g2\n∆c−δU2M2\n2,0\nU1M2\n1,0g2\n1−(∆c−δ)(∆1−δ)\n(∆∗c+δ)(∆∗\n2+δ)−g2\n2−1\nv22=∆ 2−δ−g2\n2\n∆c−δ+U2M2\n2,0\nU1M2\n1,0\n×g2\n1g2\n2−4U1U2(∆∗\nc+δ)(∆c−δ)M2\n1,0M∗,2\n2,0\n(∆c−δ)[(∆∗c+δ)(∆c+δ)−g2\n2](18)\nand\nα1=g1\n∆c−δ1−U1M2\n1,0\nU2M2\n2,0g2\n2\n(∆∗c+δ)(∆∗\n1+δ)−g2\n1\nα2=g2\n∆c−δ1−U2M2\n2,0\nU1M2\n1,0g2\n1\n(∆∗c+δ)(∆∗\n2+δ)−g2\n2(19)\nwhere∆j=δj+4Uj|Mj,0|2−iγj;j=1,2. Note that M1,0\nandM2,0are to be obtained from Eq.(4). Solving for A+we\nFIG. 8: Frequency shift of upper polariton against in-\nput power at diﬀerent values of cavity leakage. (a)\nγc<g1,2indicates strong magnon-cavity coupling; (b,c)\nγc≃g1,2indicates the intermediate magnon-cavity cou-\npling; (d)γc> g1,2gives rise to weak magnon-\ncavity coupling. All the parameters are same as Fig.3.\nﬁnd, with relatively little e ﬀort\nA+=Ep\ni(∆c−δ)/bracketleftigg\n1+(g1v22−g2v21)α1−(g1v12−g2v11)α2\nv11v22−v12v21/bracketrightigg\n(20)\nwhich leads to the transmission amplitude\nT(δ)=−i\n∆c−δ/bracketleftigg\n1+(g1v22−g2v21)α1−(g1v12−g2v11)α2\nv11v22−v12v21/bracketrightigg\n.\n(21)\nAll the information on nonlinear magnon polaritons are con-\ntained in Eq.(21).\nFig.6(a) illustrates the transmission spectra of the hybri d\nmagnon-cavity systems under various input powers. Here we\nhave taken into account the experimentally feasible parame -\ntersωc/2π=10.078GHz,ω1/2π=10.018GHz,ω2/2π=\n9.963GHz,ωd/2π=9.998GHz, g1/2π=42.2MHz, g2/2π=\n33.5MHz, U1/2π=7.8nHz, U2/2π=42.12nHz,γ1/2π=\n5.8MHz,γ2/2π=1.7MHz,γc/2π=4.3MHz39. First of\nall we observe at very weak input power three distinct peaks\npositioned at the same frequencies as the polariton branche s,\ntermed as lower (LP), intermediate (MP) and higher polari-\ntons (HP) in an ascending order of energy. With increas-\ning input power, the peak shift of magnon polaritons can\nbe observed from the transmission spectra, where the fre-\nquency shifts associated with the polariton states are deﬁn ed\nby∆σ=ωσ−ω0\nσ;σ=LP, MP and HP, respectively, where\nω0\nσdenotes the polariton frequency with no nonlinearity. This7\nshift is attributed to the Kerr nonlinearity given by the ter m\nU1|M1|4+U2|M2|4which is greatly enhanced as the strong\ndrive creates large magnon number. Since the weak Kerr non-\nlinearity in real ferromagnetic materials would lead to tin y\nfrequency shift only, we essentially plot the polariton fre -\nquency shift as a function of input power. The net hysteresis\nloop is thereby monitored through the frequency shift of the\nhigher polariton, ranging from 0 to 30MHz, shown in Fig.6(b) .\nThe same trends can be also demonstrated for the frequency\nshift of lower polariton, which will be presented elsewhere .\nThe multistability can then be clearly manifested by means\nof the two cascading jumps of frequency shift with increas-\ning input power. More interestingly as shown in Fig.7, the\nbistability-multistability transition in magnon polarit ons is re-\nvealed through tweaking either the frequency of microwave\ndrive (upper row of Fig.7) or the cavity-magnon detuning\n(lower row of Fig.7). Within the parameter regimes feasible\nfor experiments, the two magnon system shows in Fig.7(a,c)\nthe bistability that has been claimed in a single magnon in\nrecent experiments7. By either increasing drive or cavity fre-\nquency, the multistable feature is further observed as depi cted\nin Fig.7(b,d).\nFig.8 shows the robustness of multistability in magnon po-\nlaritons, against the cavity leakage. Clearly, the multist ability\nbecomes weaker when using the worse cavity. Indeed, the re-\nvisit of the hysteresis curves indicates that the multistab ility\nmay be achieved, even with lower-quality cavity giving rise to\nintermediate magnon-cavity coupling, where g1,2≃γc≫γ1,2\nyields to Fig.8(b,c). This regime is crucial for detecting t he\nmultistability and spin dynamics of magnons used in Ref.5,7,\nin that a spectrometer is desired to read out the photons im-\nprinting the magnon states information. The photons leakin g\nfrom the cavity will then undergo a Fourier transform throug h\nthe grating attached to the detector. This scheme requires\nthe much larger cavity leakage than the magnon dissipation,\nnamely,γc≫γ1,2, so that the magnon states remain almost\nunchanged when reading o ﬀthe photons from the cavity.\nV . CONCLUSION AND REMARKS\nIn conclusion, we have studied the nonlinear spin migra-\ntion between the massive ferromagnetic materials. Due to\nthe Kerr nonlinearity coming from the magnetocrystalline\nanisotropy, the multistability in the spin current between thetwo YIG spheres was demonstrated. This goes beyond the\nlinear regime of spin transfer studied before. We further de -\nveloped a transmission spectrum for resolving the spin pola r-\nization migration, through the response of nonlinear magno n\npolaritons to the external probe ﬁeld. Using a broad range of\nparameters, we showed that the spin current as a distinct sig nal\nof detection produced the results in perfect agreement with the\ntransmission spectrum. Our work elaborated the net hystere sis\nloop which manifested the bistability-multistability tra nsition\nin magnon polaritons. The multistability is surprisingly r obust\nagainst the cavity leakage: the multistable nature may pers ist\nwith a low-quality cavity giving intermediate magnon-cavi ty\ncoupling. This may be helpful to probing the multistable ef-\nfect in real experiments.\nIt is worth noting that our approach for multistability in\nmagnons may be potentially extended to condensed-phase\npolyatomic molecules and molecular clusters, along with th e\nfact of similar forms of nonlinear couplings Ub†bb†bandb†bq\nwhere bis the annihilation operator of excitons and qdenotes\nthe nuclear coordinate. With the scaled-up parameters, one\nwould anticipate to observe the multistablity in molecular po-\nlaritons. Notably, the two-exciton coupling in J-aggregat es\nand light-harvesting antennas is ∼0.3% of the magnitude of\nthe electronic excitation frequency41,42. This is much stronger\nnonlinearity than that in YIGs with Kerr coe ﬃcient being\n∼10−9of its Kittel frequency. Recent developments in ul-\ntrafast spectroscopy and synthesis have shown that the mole c-\nular polaritons may be beneﬁcial for the new design of molec-\nular devices43–45. Hence implementing the multistability in\nmolecules would be important for the study of molecular de-\nvices.\nACKNOWLEDGEMENTS\nWe gratefully acknowledge the support of AFOSR Award\nNo. FA-9550-18-1-0141, ONR Award No. N00014-16-\n13054, and the Robert A. Welch Foundation (Awards No. A-\n1261 & A-1943). J. M. P. N. was supported by the Herman\nF. Heep and Minnie Belle Heep Texas A&M University En-\ndowed Fund administrated by the Texas A&M Foundation.\nWe especially thank Dr. J. Q. You and his group for initial\ndata on experiments on nonlinear spin current.\nZ. D. Z. and J. M. P. N. contributed equally to this work.\n∗jayakrishnan00213@tamu.edu\n†zhedong.zhang@tamu.edu\n1C. Cohen-Tannoudji, J. Dupont-Roc and G. Grynberg, Photons\nand Atoms-Introduction to Quantum Electrodynamics , pp. 486\n(Wiley-VCH, 1997)\n2S. Haroche and J.-M. Raimond, Exploring the Quantum: Atoms,\nCavities, And Photons (Oxford University Press, Oxford, 2013)\n3G. S. Agarwal, Quantum Optics (Cambridge University Press,\nCambridge, 2012)\n4C. Song, et al. , Science 365, 574-577 (2019)5L. Bai, M. Harder, P. Hyde, Z. Zhang and C.-M. Hu, Phys. Rev.\nLett. 118, 217201-217205 (2017)\n6R. Bonifacio and L. A. Lugiato and M. Gronchi, Laser Spec-\ntroscopy IV , Springer Series in Optical Sciences, vol 21 (Springer,\nBerlin, Heidelberg, 1979)\n7Y .-P. Wang, G.-Q. 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Taylor, Introduction to Solid-State Theory\n(Springer-Verlag, Berlin, 1978)\n39The parameters were taken from recent ongoing experiments o f\ntwo YIG bulks implemented by You’s group at Zhejiang Univer-\nsity. Some parameters had been used in their bistability exp eri-\nments of single YIG which has been published7. The experimen-\ntal results of multistability in magnon polaritons will be r eleased\nsoon.\n40L. M. Narducci, R. Gilmore, D. H. Feng and G. S. Agarwal, Phys.\nRev. A 20, 545-549 (1979)\n41V . I. Novoderezhkin, J. M. Salverda, H. van Amerongen and R.\nvan Grondelle, J. Phys. Chem. B 107, 1893-1912 (2003)\n42V . I. Novoderezhkin, M. A. Palacios, H. van Amerongen and R.\nvan Grondelle, J. Phys. Chem. B 108, 10363-10375 (2004)\n43Z. D. Zhang, K. Wang, Z. Yi, M. S. Zubairy, M. O. Scully and S.\nMukamel, J. Phys. Chem. Lett. 10, 4448-4454 (2019)\n44A. D. Dunkelberger, B. T. Spann, K. P. Fears, B. S. Simpkins an d\nJ. C. Owrutsky, Nat. Commun. 7, 13504-13513 (2016)\n45C. Sch¨ afera, M. Ruggenthalera, H. Appela and A. Rubio, Prod .\nNatl. Acad. Sci. U.S.A. 116, 4883-4892 (2019)"    },    {        "title": "1406.1296v1.Magnetic_properties_of_epitaxial_Fe__3_O__4__films_with_various_crystal_orientations_and_TMR_effect_in_room_temperature.pdf",        "content": "arXiv:1406.1296v1  [cond-mat.mtrl-sci]  5 Jun 2014Magnetic properties of epitaxial Fe 3O4ﬁlms with various crystal orientations and\nTMR eﬀect in room temperature\nTaro Nagahama,1,a)Yuya Matsuda,1Kazuya Tate,1Shungo Hiratani,1Yusuke\nWatanabe,1Takashi Yanase,1and Toshihiro Shimada1\nGraduate School of Engineering, Hokkaido University, Kita 13 Nishi8, Kitak-ku,\nSapporo, 060-8628, Japan\n(Dated: 2 April 2018)\nFe3O4is a ferrimagnetic spinel ferrite that exhibits electric conductivity a t roomtem-\nperature (RT). Althoughthe material has been predicted to be ah alf metal according\nto ab-initio calculations, magnetic tunnel junctions (MTJs) with Fe 3O4electrodes\nhave demonstrated a small tunnel magnetoresistance eﬀect. No t even the sign of the\nTMR ratio has been experimentally established. Here, we report on t he magnetic\nproperties of epitaxial Fe 3O4ﬁlms with various crystal orientations. The ﬁlms ex-\nhibited apparent crystal orientation dependence on hysteresis c urves. In particular,\nFe3O4(110) ﬁlms exhibited in-plane uniaxial magnetic anisotropy. With resp ect to\nthe squareness of hysteresis, Fe 3O4(111) demonstrated the largest squareness. Fur-\nthermore, we fabricated MTJs with Fe 3O4(110) electrodes, and obtained an TMR\neﬀect of -12% at RT. The negative TMR ratio corresponded to the n egative spin\npolarization of Fe 3O4predicted from band calculations.\na)nagahama@eng.hokudai.ac.jp\n1Half metals that have 100% spin polarization (P) at the Fermi level ar e key materials to\nfabricate spintronic devices because their high spin polarization ena bles very large magne-\ntoresistance eﬀects. The most impressive case is in magnetic tunne l junctions (MTJs) with\nepitaxial MgO tunnel barriers1,2. As transport in MgO-MTJs is dominated by coherent\ntunneling of ∆ 1electrons with 100% spin polarization, the TMR ratio has reached 600 % at\nRT3. Such a large TMR ratio has allowed us to fabricate highly functional s pintronic devices\nlike magnetoresistive random access memories (MRAMs). However, MTJs with MgO have\nstringent limitations where the crystal orientation should be bcc (0 01) due to band structure\nmatching between MgO and the electrodes. Half metal is the solution to large TMR ratios\nwithout restricting the crystal structure or orientation. Thus f ar, many oxide materials have\nbeen proposed as candidates for half metals, e.g., CrO 24, La0.7Sr0.3MnO35, and Fe 3O46. Of\nthese materials, Fe 3O4has been considered to be the most promising as a half metal becaus e\nof its high Curie temperature of 858 K, which is an advantage in applica tions to spintronic\ndevices that require high Tc. The crystal structure is an inverse s pinel with Fe3+cations\noccupying tetrahedral sites (A sites) and Fe3+and Fe2+cations occupying octahedral sites\n(B sites). The magnetic couplings between A and B sites are antiferr omagnetic and the\ncouplings at A-A or B-B are ferromagnetic; consequently, it is a fer rimagnetic material. As\nFe3O4exhibits good electric conductivity at RT due to the hopping of electr ons between\nFe2+and Fe3+on the B sites7, the conduction electrons are 100% spin polarized. As hopping\nis frozen on cooling, conductivity greatly decreases at low tempera ture, which is known as\nVerwey transition. The transition temperature, T v, is 120K8. The saturation magnetiza-\ntion of bulk Fe 3O4is 510 emu/cc9. According to Julliere’s formula10, MTJs with Fe 3O4\nelectrodes are expected to exhibit very high TMR ratios due to large spin polarization.\nTo date, researchers have fabricated MTJs with Fe 3O4and measured magnetoresistance;\nhowever, the TMR ratios have been small. Although the reason for t his is not completely\nunderstood, such small TMR ratios can be attributed to imperfect antiparallel magnetic\nstates in MTJs11. The magnetization process of Fe 3O4ﬁlms should be improved to achieve\nclear parallel and antiparallel magnetic conﬁgurations. We prepare d epitaxial Fe 3O4ﬁlms\nwith various crystal orientations, and investigated their crystallin e qualities and magnetic\nproperties. We also fabricated MTJs with Fe 3O4electrodes and observed a negative TMR\neﬀect of -12%.\nThe Fe 3O4thin ﬁlms were prepared with three crystal orientations of (001), (110), and\n2(111) by using a molecular beam epitaxy (MBE) system. The sample st ructures were:\n1) an MgO(001) substrate/MgO (20 nm)/Fe 3O4(60 nm),\n2) an MgO(110) substrate/MgO (20 nm)/Fe 3O4(60 nm), and\n3) an Al 2O3(0001) substrate/Pt (20 nm)/Fe 3O4(60 nm).\nFollowing the deposition of MgO or Pt buﬀer layers, Fe 3O4thin ﬁlm was formed by reactive\ndeposition at a temperature (T sub) of 300◦C in an O 2atmosphere of 4 ×10−4Pa. Then,\nthe ﬁlms were annealed at 600◦C for 30 min in an O 2atmosphere. The partial pressure of\nO2gas was 1 ×10−4Pa during annealing. All the samples were fabricated under the same\ngrowth conditions to enable the quality of Fe 3O4ﬁlms to be compared. Epitaxial growth\nwas observed with reﬂection high energy electron diﬀraction (RHEE D) and the surface\nmorphology was observed with atomic force microscopy (AFM). We a lso investigated the\nmagnetization process at RT with a vibrating sample magnetometer ( VSM).\nFigs. 1(a)and(b)showtheRHEEDpatternsofFe 3O4(100)beforeandafterO 2annealing\nat 600◦C for 30 min. The electron beam was incident along [100]. Fig. 1 (c) is an a tomic\nforce micrograph (AFM) of Fe 3O4(100) after annealing. A streak RHEED pattern can be\nobserved in Fig. 1 (a) meaning the Fe 3O4ﬁlm grew epitaxially. In addition, p(1x1) surface\nreconstruction was observed1213. The streak pattern sharpened after annealing at 600◦C\nin the O 2atmosphere, as can be seen from Fig. 1 (b). A step-terrace stru cture can be\nconﬁrmed from the AFM in Fig. 1 (c). The roughness average, R a, was 0.12 nm, and the\nterrace width was 200 nm.\nFigs. 1 (d)-(f) show the RHEED patterns and AFMs of Fe 3O4(110) grown on MgO(110).\nThe incident electron beam direction was [-110]. A spotty pattern wa s obtained before an-\nnealing due to the island growth of Fe 3O4(110). However, the surface ﬂatness was improved\ndramaticallybyO 2annealingat600◦C,ascanbeseenfromtheRHEEDpatterninFig. 1(e).\nThe surface in the AFM of Fe 3O4(110) after annealing in Fig. 1 (f) had anisotropic shapes\nalong [100], which seemed to originate from the anisotropy of the MgO (110) substrate. R a\nwas estimated to be 0.39 nm.\nFinally, Figs. 1 (g)-(i) show RHEED patterns and AFMs of Fe 3O4(111). The direction\nof the incident electron beam was [11-20]. Fig. 1(g) shows RHEED pat terns of as-deposited\nFe3O4(111). It shows streak patterns that indicate a ﬂat surface and surface reconstruction.\nTerraceandstepstructurescanbeobservedintheAFMofFe 3O4(111)afterannealinginFig.\n1 (i); however, islands with a diameter of 200 nm and height of severa l tens of nanometers\n3FIG. 1. RHEED patterns and AFMs of epitaxial Fe 3O4(60 nm) ﬁlms. RHEED patterns were\ntaken after deposition at 300◦C and annealing at 600◦C. AFM observations were carried out after\nannealing. (a), (b), and (c) are for MgO(100)/Fe 3O4(100) (60 nm). (d), (e), and (f) are for\nMgO(110)/Fe 3O4(110) (60 nm). (g), (h), and (i) are for Al 2O3(0001)/Pt(111) (20 nm)/Fe 3O4(111)\n(60 nm).\nwere observed on the surface (not shown) in the AFM of a large are a. The R awas estimated\nat 2.40 nm, which was one order of magnitude larger than the other c rystal orientations.\nThe large roughness could be attributed to the lattice mismatch bet ween Fe 3O4and the Pt\nbuﬀer layer14. As the lattice constant of Fe 3O4was 0.8397 nm and that of MgO was 0.421\nnm, the lattice mismatch was about 0.3%. However, as the lattice con stant of Pt was 0.392\nnm, Fe 3O4lattice mismatch to the Pt buﬀer layer was 6.6%. Such large lattice mism atch\ncould give rise to a larger surface roughness for Fe 3O4(111) than that for Fe 3O4(100).\nThe magnetization curves at RT for the Fe 3O4ﬁlms are plotted in Fig. 2. The magnetic\nﬁeld was applied in plane. The diamagnetic components of the substra tes were subtracted\nunder the assumption that the magnetizations of the Fe 3O4were saturated at 5 kOe, which\nis the maximum ﬁeld of VSM. The magnetization curve of Fe 3O4(100) is in Fig. 2 (a). The\nsaturation magnetization (M s) was 330 emu/cc, the remanent magnetization (M r) was 100\nemu/cc, and the coercive ﬁeld (H c) was 80 Oe. The remanent magnetization ratio (M r/Ms)\nwas 0.30. Fig. 2(b) plots the magnetization curves of Fe 3O4(110) where the directions of\nthe magnetic ﬁeld were [001] and [-110]. The saturation magnetizatio n was 185 emu/cc for\nboth magnetic ﬁeld directions. M r, Hc, and M r/Msin the magnetic ﬁeld along [001] were\n4!\"## !$## #$## \"## \n!%### # %### \n!\"#$%&'\"$()*++%&!,-../&\"0%&\n!\"## !$## #$## \"## \n!%### # %### \n!\"#$%&'\"$()*++%&!\n!,..-/&\n,1--./&\"2%&\n!\"## !$## #$## \"## \n!%### # %### \n!\"#$%&'\"$()*++%&!\n!,-1-../&\n,--13./&\"+%&\nFIG. 2. Hysteresis curves obtained from VSM measurements at RT for epitaxial Fe 3O4ﬁlms with\nvarious crystal orientations. (a) is Fe 3O4(100), (b) is Fe 3O4(110), and (c) is Fe 3O4(111). Directions\nof magnetic ﬁelds are given in plots.\n30 emu/cc, 210 Oe, and 0.16, and those for [-110] were 100 emu/cc , 780 Oe, and 0.54. The\nmagnetization process strongly depended on the directions of the magnetic ﬁeld, viz., the\nsquareness and M r/Mswere larger for the [-110] magnetic ﬁeld than those for [100]. Never -\ntheless, the ﬁlms had an anisotropic shape along the [100] direction, as shown in Fig. 1(f),\nand the ﬁlms had a larger remanent ratio in the [-110] direction. Ther efore, the anisotropy\nin Fig. 2 (b) was attributed to the magneto-crystalline anisotropy in Fe3O4. Saturation\nmagnetization was 390 emu/cc, remanent magnetization was 290 em u/cc, and coercivity\nwas 300 Oe in the magnetization curve of Fe 3O4(111). The remanent magnetization ratio\nwas approximately 0.74, which was the largest value in the three crys tal directions. The\nmagnetic process was almost independent of the ﬁeld directions. Th ese values are summa-\nrized in Table 1. All the ﬁlms exhibited smaller saturated magnetization s than the value for\nbulk Fe 3O4of 510 emu/cc. The reason for this is that the external ﬁeld was no t suﬃcient\nto saturate the magnetic moments in the Fe 3O4ﬁlms. According to previous studies, Fe 3O4\nthin ﬁlms contain considerable numbers of antiphase boundaries (AP Bs)15that make the\nFe3O4hard to saturate magnetically due to antiferromagnetic coupling at the APBs.\nWe fabricated the MTJs with Fe 3O4(110) electrodes, and measured the tunnel magne-\ntoresistance eﬀect. The ﬁlm structure was MgO(110)/NiO(110) ( 5 nm)/Fe 3O4(110) (60\nnm)/Al 2O3(2.4 nm)/Fe (5 nm)/Co (10 nm)/Au (30 nm). The NiO layer was inserte d to\nsuppress the diﬀusion of Mg from the substrates. Junctions of 10 ×10µm2were fabricated\n5TABLE I. Magnetic characteristics of Fe 3O4ﬁlms with various crystal orientations.\nMs Mr Hc Mr/Ms\n(emu/cc) (emu/cc) (Oe)\nFe3O4(100) 330 100 80 0.30\nFe3O4(110) H//[001] 185 30 210 0.16\nFe3O4(110) H//[-110] 185 100 780 0.54\nFe3O4(111) 390 290 300 0.74\n/uni002D.14/uni0031.18/uni0030.17 /uni002D.14/uni0035.22 /uni0030.17\n/uni004D.46/uni0052.51/uni0020.1/uni0072.83/uni0061.66/uni0074.85/uni0069.74/uni006F.80/uni0020.1/uni0028.9/uni0025.6/uni0029.10 \n/uni002D.14/uni0031.18/uni0030.17/uni0030.17/uni0030.17 /uni0030.17 /uni0031.18/uni0030.17/uni0030.17/uni0030.17 \n/uni004D.46/uni0061.66/uni0067.72/uni006E.79/uni0065.70/uni0074.85/uni0069.74/uni0063.68/uni0020.1/uniFB01.112/uni0065.70/uni006C.77/uni0064.69/uni0020.1/uni0028.9/uni004F.48/uni0065.70/uni0029.10 /uni0020.1/uni0031.18/uni0030.17/uni006D.78/uni0056.55 \n/uni0020.1/uni0035.22/uni0030.17/uni0030.17/uni006D.78/uni0056.55 \n/uni0020.1/uni0031.18/uni0030.17/uni0030.17/uni0030.17/uni006D.78/uni0056.55 \nFIG. 3. TMR curve for MTJ of MgO(110)/NiO(110) (5 nm)/Fe 3O4(110) (60 nm)/Al 2O3(2.4\nnm)/Fe (5 nm)/Co (10 nm)/Au (30 nm) at RT. Red, blue, and black lines are TMRs with bias\nvoltages of 10, 500, and 1000 mV.\nby photolithography, Ar ion milling, and sputtering. The junctions de monstrated a clear\nTMR eﬀect of -12% at RT, as shown in Fig. 3. The negative MR agreed w ith the ab-initio\ncalculations that predicted negative spin polarization in Fe 3O4. To the best of our knowl-\nedge, these are the ﬁrst experimental results of a negative MR ra tio with an AlO barrier\nand Fe 3O4electrodes1617181920. The polarization of Fe 3O4deduced from the MR ratio based\non Julliere’s formula was -16%, in which the polarization of Fe/Al 2O3was assumed to be\n40%21. Although the polarization was much smaller than the predicted value , -16% is of the\nsame order as the reported values using various barrier materials2223.\nFe3O4epitaxial ﬁlms with various crystal orientations were fabricated by reactive MBE\nand all the ﬁlms grew epitaxially. The Fe 3O4(110) ﬁlms exhibited clear uniaxial magnetic\nanisotropy that originated from crystal anisotropy. The square ness of the hysteresis curves\nstrongly depended on the crystal orientation. A negative MR ratio of -12% was observed in\n6the MTJs with Fe 3O4(110) electrodes. Although the absolute value was small, the negat ive\nMR agreed with the theoretical predictions.\nWe would like to express our gratitude to Prof. Yamamoto’s group fo r the cooperation\nin microfabrications. This work was supported by JSPS KAKENHI Gra nt-in-Aid for Young\nScientists (A) Grant Number 23686006 and the Collaborative Resea rch Program of Institute\nfor Chemical Research, Kyoto University (grant 2014-75).\nREFERENCES\n1S. 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Tedrow, “Spin-polarized electron tunneling,” Ph ysics Reports 238,\n173–243 (1994).\n822G. Hu and Y. Suzuki, “Negative spin polarization of fe 3o4in magnetite/manganite-based\njunctions,” Physical review letters 89, 276601 (2002).\n23L. Alldredge, R. Chopdekar, B. Nelson-Cheeseman, and Y. Suzuki, “Spin-polarized con-\nduction in oxide magnetic tunnel junctions with magnetic and nonmag netic insulating\nbarrier layers,” Applied physics letters 89, 182504–182504 (2006).\n9"    },    {        "title": "1610.02489v1.Enhancement_of_Impedance_by_Chromium_Substitution_and_Correlation_with_DC_Resistivity_in_Cobalt_Ferrite.pdf",        "content": " \n1 \n Enhancement of Impedance  by Chromium Substitution  \nand Correlation with DC Resistivity  in  Cobalt Ferrite  \n \nSweety Supriya,1,a Sunil Kumar,1,b and Manoranjan Kar1,c \n \n1Department of Physics, Indian Institute of Technology Patna, Patna -8011 03, India  \n \nAbstract : Chromium substituted cobalt ferrite with grain size less than the single domain (~ 70 \nnm) has  been prepared by the sol -gel method. XRD analysis reveals that the samples crystallize \nto cubic symmetry with    ̅  spacegroup. Two transition temperatures ( TD~450 K and \nTM~600K)  have been observed from the impedance verses temperature measurement . TD \nincreases with the increase in frequency due to dipole response to the frequency. TM is \ncomparable  with the par a-ferrimagnetic transition temperature of cobalt ferrite, which is \nindependent of frequency. This result is well supported by the temperature dependent DC \nconductivity measurement. The modified Debye relaxation could be explained the impedance \nspectra of C oFe 2-xCrxO4. The grain and grain boundary effect on impedance spectroscopy has \nbeen observed from Cole -Cole analysis.  The ac  conductivity follows Arrhenius behavior at \ndifferent frequencies. Al l the samples exhibit the negative temperature coefficient of r esistance \nbehavior which reveals the semiconduct ing behavior  of the material. The Mott VRH model could \nexplain the DC electrical conductivity. Both ac impedance and DC resistivity are well co -related \neach other to explain the electron transport properties in Cr substituted cobalt ferrite.  The \nelectrical transport properties could be explained by the electron hop ping between different metal \nions via oxygen in the material.  \n \nKeywords: Cobalt Ferrite, XRD, Impedance spectroscopy, Electrical conductivity  \n \nIntroduction : \nThe miniaturization of magnetic and electronic devices needs advance functional materials of \nnanosize to have greater efficien cy in new forms. Nanocrystalline  ferrites are having remarkable \nphysical properties , which make it suitable for technolo gical applications such as , switching \ndevices, EMI shielding, actuators, recording tapes, microwave devices, transducers, sensors, high  \n2 \n quality digital printing, biotechnological applications, and high densit y magnetic information \nstorage .1-12 Owing to var ious significant electrical and magnetic properties, ferrites have very \nwide applications in electronic industries. These spinel ferrites function as a dielectric material in \nthe preferred region of frequency which ranges from radio frequency to microwave frequencies \nbecause of having a high resistivity and low  eddy currents and loss factor .7-8 Out of various \nferrites, CoFe 2O4 were the topic of demanding research in recent years. Cobalt Ferrite is a hard \nferrimagnetic material with a Curie temperature at ar ound 793 K  has distinctive properties like \nhigh magnetic sensitivity (5400 Oe), high magnetocrystalline anisotropy and moderate satu ration \nmagnetization (80emu/g) .5, 9-11 \nNow days there have been an intensive demand for dielectric and magneto -dielectric \napplication based materials.12 Ferrite  materials are insulating magnetic oxides which permit \nelectromagnetic field penetration as compared to metals, where there is limitation of penetration \nof the high freq uency field due to skin effect .13 Partial replaceme nt of Fe by metal ions in \nCoFe 2O4 proposed to tailor the structural, magnetic, electr ical and dielectric properties.14-19 This \nreplacement causes structural distortion and induces lattice strain and modifies the magnetic and \nelect rical properties significa ntly.16-19 There are abundance researches on cobalt ferrite for its \napplication as a magnetic material. Researchers have modified its magnetic property by \nsubstituting either in Co site and/or in Fe site. However, there is limited study of its dielectric \nproperties to explore the technological applications. In this regard, there are a few reports \navailable on dielectri c properties of cobalt ferrite.16-22 \nCobalt ferrite exhibits an inverse spinel structure represented by the formula AB 2O4 as \nCoFe 2O4. In whic h metal cations Fe3+ and Co2+ are at A (tetrahedral) and B (octahedral site) \ninterstitial sites respectively formed by the face centered cubic (FCC ) arrangement of oxygen  \n3 \n anions .23 Cobalt ferrite crystallizes to normal/inverse/mixed phase spinel ferrite. H owever, \nmostly the inverse spinel structure is obtained by the conventional method of preparation. In \ninverse spinel FCC lattice, half of Fe3+ ions are occupied by tetrahedral (A) sites of spinel \nstructure and half of Fe3+ and Co2+ ions are occupied by the  octahedral (B) sites. The distance \nbetween the two metal ions one on B site and another at A site is larger than the distance \nbetween metal ions in the same site. The electrical conduction mechanism in ferrites is mainly \nexplained by considering the polar on hopping between Fe3+ to Fe2+ions. 24 Mostly it arises due to \nlocal displacement of charge and results in polarization. The probability of electron hopping \nbetween B –B hopping is very high compared to that of B -A hopping. There are only Fe3+ ions at \nA sites for inverse spinel structure, hence there is no possibility of hopping between A –A sites. \nChromium ions have a preference for octahedral site, thus they reduce the Fe3+ ions in the Fe -site \nand in turn reducing Fe3+ - Fe2+ ions polaron hopping .25 Also, this concept has been extended to \nunderstand the dipole oscillation in spinel cobalt ferrite. It  is observed that, the charge hopes \nfrom Fe2+ to Fe3+ or vice -versa via oxygen. Also, there is possibility of charge hopping in \nbetween Co2+/3+-O-Fe2+/3+ or Co2+-O-Co3+. Due to the above assumptions, a dipole form between \nCo2+/3+/Fe2+/3+ and it oscillates.  \nThe electrical resistivity in the ferrites significantly depends on the density, porosity, grain \nsize, chemical composition and c rystal structure of the sample .26-27 Although the high dielectric \nloss of cobalt ferrite at low frequency confines their electrical applications, the dielectric \nrelaxation behavior of nanocrystalline cobalt ferrite which is reported recently, indicates its \npotential  application in phas e shifters .28 Moreover, the successive observation of relaxation \nbehavior in magnetic materials may break through their limitation in magnetic applications and \nextends widely to electrical aspect. Also, the factors involved in tailoring the magnetic and  \n4 \n electrical properties of polycrystalline materials are method of preparation, temperature, sintering \ntime, cation distribution, type and quantity of dopants and grain size.26-27 Mostly the surface \ncharacteristic strain at lattice site and charge distribution  in octahedral and tetrahedral sites \nmodifies to tune the physical properties. Decrease in resistivity is observed with the increase in \ntemperature only because of the decrease in grain boundary resistance. The electrical properties \nreveal the behavior of localized electric charge carriers and the phenomenon correlated to the \npolarization. The macroscopic electrical properties of polycrystalline ceramics can be well \nexplained  by the microstructure grain and grain boundary distribution in the sample. These \nfactors stimulate a deep study of frequency and temperature dependent impedance spectroscop y \nthrough experimental and theoretical modeling. As discussed above, the electrical properties of \nCoFe 2O4 can be controlled by suitable substitution at Co or Fe site as it will modify the charge \ndistribution as well  as lattice strain. Cr is a transition element with similar charge states of Fe  \nwith different ionic size . Hence, there are a few reports on Cr substitution on CoFe 2O4 to tune the \nmagnetic properties. It is observed that the hard ferrimagnetic cobalt ferrite becomes a soft \nmagnet ic material by Cr substitution .29 Hence, it is expected there will be a modification of \nelectrical properties on CoFe 2-xCrxO3. However, a literature survey  shows that, there is no det ail \nreport available to understand the electrical  transport properties of  Cr substituted cobalt ferrite. \nMoreover, researchers have  reported dielectric properties of different substituted cobalt ferrite or \nDC tran sport properties independently.  Hence, the study of correlation between ac impedance \nspectroscopy and DC resistivity is necessary to understand the electrical transport properties  on \nsubstituted cobalt ferrite. Also, it is expected  that the  magnetic ordering in the cobalt ferrite \naffects the electr ical transport properties. To address the above issue s the Cr substituted cobalt \nferrite has been prepared. The detailed temperature variation impedance spe ctroscopy and DC  \n5 \n resistivity have  been studied. Interestingly, it  is observed that the magnetic orde ring temperature \nwell reflected in both DC resistivity and ac impedance measurement. The magnetic ordering \ntemperature is independent of  applied ac signal frequency. The present study can help to \nunderstand the other substituted cobalt ferrite and opens a  window to tune the electrical \nproperties of cobalt ferrite by substitution for technological applications.  \n \nExperiments : \nNanocrystalline samples of CoFe 2-xCrxO4 (CFCO) ferrite for x=0.0, 0.1, 0.2, 0.3 and 0.4) were \nprepared by the modified sol -gel techniq ue.30-33 The precursor materials are Cobalt nitrate \n(Co(NO 3)2.6H 2O) (Merck), Iron nitrate (Fe(NO 3)3.9H 2O)(Alfa Aesar), Chromium nitrate \n(Cr(NO 3)3.9H 2O)(Merck ) and Citric acid (C6H8O7.H2O)(Merck) with 99.9% purity. The powders \nof the pre -sintered synthesize d material were ground in a mortar pestle and heated at 1173 K for \n1 hour with the heating rate of 5 K per minute. The samples were made into pellets having \ndiameter 10 mm and the cross -sectional area is 7.85 cm2. Phase identification and crystal \nstructure  of the materials were investigated by using the X -ray diffraction (XRD) technique with \nthe help of Rigaku X -ray Diffractometer (Model TTRX III). Measurements were carried out at \nroom temperature using CuK α radiation ( λ=1.5406Å). The density of the material measured by  \nemploying  the Archimedes' principle and with the help of a  balance Sartorious (Model No. \nCPA225D). Hardness of the material s was measured by the Micro Hardness Tester (Model No.  \nMICROTEST MTR 3/50-50/NI) using Vicker  indentors under a test force  2 N for 5 second.  \nElectrical measurements were carried out on the specially prepared samples. Briefly, the surfaces \nof the samples were well polished, silver paint was coated on the surface and then hea ted in a \nfurnace programmed as at 150 oC for an hour and 250 oC for half an hour to remove the content \nof binder (PVA -Polyvinyl alcohol, hot water soluble form HIMEDIA laboratories Pvt. Ltd.) used  \n6 \n for making pellets and removal of epoxy used for making ele ctrodes. Pellets with silver electrode \nwere sandwiched between two metal holders and connected through wires to N4L Impedance \nanalysis interface (PSM1735 NumetriQ). The sample assembly was kept in a furnace equipped \nwith a temperature controller to carry o ut the impedance measurements as a function of \nfrequency and temperature. The sample assembly constitutes the parallel plate capacitor \ngeometry with the CoFe 2-xCrxO4 as the dielectric. An ac signal of 2V and frequency (f) in the \nrange of 1 Hz to ~10 MHz wa s applied to the circuit using N4L Impedance analysis interface \nthrough PSM1735 NumetriQ. In addition, standard calibration was carried out before the \nmeasurement to avoid any stray capacitance, contact and lead resistance and, reset the device to \nremove t he history of data. The capacitance, dielectric, phase, loss tangent and impedance are \nrecorded with varying temperatures from 323 K to 673 K as a function of frequency ranging \nfrom 1 Hz –10 MHz to quest the dielectric dispersion and the response of Cr subs tituted CFO in \nthe conduction mechanism. The Complex dielectric, complex electric modulus and the \nconductivity were calculated at different temperatures using measured capacitance and phase \ndata at the fixed frequency. The temperature variation of DC resis tivity measurement was carried \nout by Keithley multimeter (2001) using two probe method.  \nResult s and Discussion : \nX-ray diffraction (XRD) patterns of CoFe 2-xCrxO4 for x=0.0, 0.1, 0.2, 0.3 and 0.4 are shown in \nthe figure 1(a). The observed peaks in the diffr action patterns in the XRD pattern could be \nindexed to \nmFd3  space group with cubic crystal symmetry. The extra peaks and intermediate \nphase formation were not observed within the X -ray diffraction pattern technique limit. The \nobserved bro adening of the diffraction peaks in the XRD pattern affirms that the particles are of \nultrafine nature and small in size. The crystallite size of compound CoFe 2-xCrxO4 for x=0. 0, 0.1,  \n7 \n 0.2, 0.3 and 0.4 was calculated using Debye -Scherrer’s formula .34 Avera ge crystallite sizes are \nfound to be in between 60-70 nm which are enlisted in table I. Hence particles are below the \nsingle domain size as the single magnetic domain size of CoFe 2O4 is ~ 70 nm .35 The (311) peak, \nwhich is highest intensity peak in CoFe 2O4 XRD pattern is shown in figure 1 (b) for all the \nsamples. It is observed that peak shifts towards higher angle. It reveals the decrease of lattice \nparameter enlisted in table 1 with the increase in Cr substitution. It is due to the smaller ionic \nradius of Cr to that of Fe and it also reveals the incorporation of Cr in the CoFe 2O4 lattice. The \nlattice parameters are comparable with those reported in the  literature for cobalt ferrite.36 As the \ndensity is very important for impedance study it was determined by  the Archimedes ' method and \nthose values are enlisted in table I. These values are comparable to those reported for cobalt \nferrite  i.e., 5.3g/cm-3 (approx) of theoretical value .36 Also values are almost constant for all the \nsamples ( ~4.6 g/cm3), and above 86% of the theoretical value (5.3 g/cm3). 37 Hence the variation \nof electrical properties is only due to Cr substitution as samples are well dense, which is \nexplained in the next section. Also the hardness of the samples was measured and those values \nare en listed in the table I. It is also observed that hardness for all the samples are close to each \nother (154 -160 HV) and just ified with the earlier reports .38 \n \nImpedance Analysis : \nImpedance spectroscopy is an important technique to characterize the electrical  properties of the \nmaterial  and their electrode interface .39 Impedance spectroscopy techniques employed to study \nthe electric properties of CoFe 2-xCrxO4 for x=0.0, 0.1, 0.2, 0.3 and 0.4. The contribution from the \nintergrain and intragrain effect due to het erogeneous distribution of charge and their influence on \nelectrical conductivity as a function of frequency and temperature has been studied extensively.  \n8 \n This technique also divides the resistive and reactive part of electrical component that gives the \nexact picture of electrical properties. The temperature and frequency dependent real ( Z') and \nimaginary parts ( Z'') of the Complex Impedance Z* are 40-41 described as,  \n'' ' *jZ Z Z\n                                    (1) \n 2'\n) (1ppp\nCRRZ\n                           (2) \n 22\n''\n) (1pppp\nCRCRZ\n\n                           (3) \n \nWhere Rp is the parallel resistance, Cp is the parallel capacitance, ω is the angular frequency.  \n \nMostly these behaviors explained by the Koop’s theory, which describes that the inhomogeneous \ndielectric structure consists of fine conducting grains and poor ly conducting grain boundaries .42 \nHence the conductivity and impedanc e of grains are high from that of the grain boundary. Figure \n2(a) - 2(e) shows the  frequency  dispersion of real parts of the impedance ( Z') of CoFe 2-xCrxO4 for \nx= 0.0, 0.1, 0.2, 0.3, and 0.4 samples  at some selected temperatures. The logarithmic scale of \nfrequency is taken for clarity. The frequency variation of Z' from temperature 473 K to 673 K are \nshown as insets in the figures 2 (a) - 2(e). It is observed that the magnitude of the real part of \nimpedance ( Z') decreases with the frequency, which is the normal behavior of ferrites43 and \nfollows Koop’s theory as discussed above. There is a sharp decrease in Z' at the lower \nfrequencies and eventually decreases towards high frequency and remain constant at very high \nfrequencies which indicate the dispersion of Z' at low frequency. This type of behavior observed \ndue to the polarization of valence states of cation  and space charge polarization. Impedance is  \n9 \n found to be non -dispersive at high frequency range (1 MHz -10 MHz). This happens  because the \ndipolar polariza tion and surface charge polarization fail to flip with respect to the fast changing \nof applied alternating electric field due to high frequency. Basically the impedance at very low \nfrequency is due to contribution from space charge polarization, dipolar po larization, atomic \npolarization and electronic polarization. As frequency increases, polarization due to dipolar and \nspace charge abruptly decreases and approach to zero at high frequency. At higher temperature, \ncharge carrier gets momentum, due to high th ermal energy, so the impedance dispersion is \nobserved at high frequency too. The signature of transition from high impedance to low \nimpedance value at a particular frequency as shown in figure 2 remains unaltered with the \nincreasing temperature. Hence this  signature could be leads to use the material  frequency filter or \nfrequency band selector at various temperatures. The  electric  dipol e polarization in the cobalt \nferrite can be understood by considering the electron ic interaction between cations.44 The \nelectronic exchange between Fe2+ and Fe3+ via oxygen causes the confined displacement of \nelectrons along the direction of electric field, which determines the polarization and relative \nimpedance behavior. As the resistance of grain boundary is very high, then  the electrons pile up \nat the interface which is the cause o f surface charge polarization. Hence, the applied voltage drop \nin the specimen could be mainly arise s across grain and grain boundary, causing space charge \naccumulation under the influenc e of elec tric field .45 \nThe magnitude of impedance is the order of mega ohm which is matched  well with the earlier \nreports .46 It has been observed that (figure 2) Z' increases with the increase in Cr doping \nconcentration. Generally the increase in impedance can be a ttributed to the grain size distribution \nin the oxides. However, the grain as well as crystalli te distribution in the present samples is very \nnarrow. Hence, the increase in Zˈ with the increase in Cr is due to the substitution effect. The  \n10 \n ionic size of Cr is smaller compared to that and Fe which creates lattice strain and hence \nmodification of dipole strength. The details of temperature dependence Z' are discussed in the \nlater section.  \nFigures 3(a) -3(e) show frequency variation of imaginary (i.e. ac loss) part of impedance ( Z'') as \na function of temperature. The logarithmic scale of frequency is taken in each graph in order to \nhave clarity of presentation. Each figure has an inset which depicts the frequency variation of Z'' \nfrom temperature 473 to 673 K.  A peak has been observed for all the samples in frequency \nvariation of Z'' spectrum, which can be seen from the figures 3(a) -3(e). Peaks are very broad and \nasymmetric in nat ure. The appearance of the peak represents the presence of electrical relaxation \nand the distribution of the relaxation times, which attributed to the asymmetrical broadening of \nthe peaks. The maximum loss on the  fmax position shifts to a higher frequency with the increase \nof temperature. It reveals that maximum loss part is temperature dependent. The amplitude of Z'' \nmaximum peak decreases with increase in temperature. The magnitude of the Z'' at fmax peak \nfrequency increases with the increase in Cr concen tration for a particular temperature, it is \nbecause of the presence of inhomogeneous atoms at lattice site. The value of a frequency \ncorresponding to Z'' maximum peak at a certain temperature is shown in table  II. The frequency \n2πf max related to Z''max could be modeled through the Arrhenius law as given in equation 4 where \nωo is the pre -exponential factor and EA is the activation energy. The activation energy was \ncalculated from the slope of the plot ln(ω max) verses 1000/T  and given in table V. \nIn this sect ion, the temperature dependence of impedance study is discussed. Figures 4 (a) -4(e) \nshows the temperature dependence of real part ( Z') of impedance as a function of frequency. \nFigures 4(a) -4(e) comprises of three categories, i.e ., i) Figures 4 (a1)-4(e1) ar e shown for the \nfrequencies 1097  Hz and 10722 Hz, ii) Figures 4 (a2)-4(e2) are shown for the different 6  \n11 \n frequencies in the range of 104761 to 3199267 Hz and iii) insets show enlarged version of the Z' \nvariation in the range of 500 to 873 K to represent the  clarity in peaks. It is observed that, the \nmagnitude of Z' increases with increase in Cr concentration. The similar trend has been observed \nby other research groups  in substituted cobalt ferrite .24 An increase in Cr concentration in CoFe 2-\nxCrXO4, causes a n increase in the impedance ( Z') because of strain introduced by inhomogeneous \natoms at lattice sites, as discussed in earlier sections. There are two peaks have been observed at \ndifferent temperatures. However, earlier reports on CoFe 2O4 exhibits only sin gle peak. It is worth \nnoting that earlier report for single peak 47 were for micrometer grain size  samples , however the \npresent samples are well dense nanocrystallites. Hence t his interesting observed feature has been \nstudied in detail . \n It is difficult to  find the exact peak position of the curve (figure 4) as the transition is very \nbroad. Hence the peak temperature was determined from the \nTdtdZ~' plot (figure not shown) . \nThe maxima were noted at\n0'dtdZ . The 1st peak trans ition temperature around 450 K (\nK100 450\n) is named a s TD and TM is called to the 2nd peak (2nd transition temperature) around \n600 K (\nK20 600 ).  Both the transition temperatures  TD and TM are en listed in the table III. (TD) \nfor each composition shifted towards higher temperature with the increase in frequency. This \nsignature is c onsistent with earlier reports24 which is the typical impedance behavior of the cobalt \nferrite. This is because at higher frequency, the dipoles lag behind th e fast changing alternating \napplied electric field. At higher frequency, polarization decreases and hence the impedance. But \ntemperature is also increasing with increasing frequency, henceforth to maintain the polarization \nwith rapid change in frequency, i ncreasing temperature play a very crucial role in order to \ncontribute energy to maintain this polarization. Eventually temperature in the form of energy  \n12 \n supplied to maintain polarization, and this result in the form of peak shift of Z' with the increase \nin frequency and temperature.  \n TD shifts more than 200 K by changing the frequency from 1 KHz to ~10 MHz, however, the \nsecond peak position  (TM) shifts ˂ 20 K, which is clear from figure 4 and enlisted in the table III. \nIt reveals that the second peak  almost  does not  respond to the frequency . is very less. \nParamagnetic to ferrimagnetic transition temperature for ~ 70 nm c obalt ferrite sample is around \n600 K48-49 which is close to the second observed peak in temperature verses Z' plot for the \npresent work . The magnetic ordering temperature ( TM) is lower  for nanocrystalline substituted \ncobalt ferrite compared to that of b ulk substituted cobalt ferrite .50 It is independent of applied \nelectric field frequency because the electric field does not respond to magnetic  dipole in the \ncobalt ferrites and hence a weak mag netoelectric coupling material, which has been almost \nestablished. The impedance at TM is changed due to only change in electrical resistive part of the \nsample which arises due to the presence of magnon. T he resistive part of impedance is frequency \nindependent. Hence, the transition temperature TM is independent of frequency. Earlier reports on \nthe impedance of cobalt ferrites are mostly in bulk sample where the magnetic transition \ntemper ature is very high.  Hence it  was not feasible to observe the para-ferrimagnetic transition \ntemperature . Although there are a few reports on nanocrystalline sample, but the impedance \nmeasurement has been  reported for a limited temperature  i.e. <500K. However, in the present \nreport the magnetic  transition is clearly observed. Hence the present study  opens the window to \nunderstand the different magnetic and electrical properties of cobalt ferrite. One can easily \nevidence the above characteristic by DC measurement where the secon d transition temperature \n(TM) will prominent ly compared to the 1st peak due to electron magnon scattering , and discussed \nin the later section.   \n13 \n The magnitude of  Z' decreases with the temperature above TM. This behavior accounts the \nsemiconducting nature of  the sample at high temperature. That is a negative temperature \ncoefficient of resistance (NTCR). This semiconduct ing behavior with increasing temperature \nleads to hopping of charge carrier and results in increase  of ac conductivity which is discussed in \nthe later section.  \nThe plots of temperature dispersion of the imaginary parts of impedance Z'' of CoFe 2-xCrxO4 \nfor x=0.0. 0.1, 0.2, 0.3 and 0.4 are shown in the figures 5(a) -5(e). Insets in figures 5(a1) -5(e1) \nshow the temperature dependent Z'' spectrum of  CoFe 2-xCrxO4 for x= 0.0. 0.1, 0.2, 0.3 and 0.4 in \nenlarge form to identify the transition temperature  TM. The magnitude of Z'' is high at low \ntemperature and gradually decreases towards high temperature and eventually Z'' became \nindependent of temperature . The magnitude of Z'' is decreasing with increasing frequency \n(because dipole lag behind increasing frequency) and temperature (because of semiconduct ing \nbehavior).  \n When an alternating electric field is applied the dipoles oscillates in its mean position . \nThis oscillation can be modeled to one of the electric oscillators such as the parallel RC or LCR \noscillator. A peak of maxima is observed when the frequency of the dipole oscillation is equal to \nfrequency of applied electric field, called the resonant f requency. The condition for the \nresonance is ωτ=1 , where time constant τ=RC . The amplitude of the oscillations will be small \nfor any frequency other than the resonant frequency. The electrical charge transport through \ngrain, grain boundary, partially elect rode interface modeled through consecutive parallel RC \ncircuit. The resonant frequency stands as ωr =1/RC . \nThe grain, grain boundary and electrode interface contribution can be observed individually \nfrom tot al impedance spectra .39 Fig 6 (a)-6(d) shows the Cole -Cole plots or complex impedance  \n14 \n spectra for samples CoFe 2-xCrxO4 for x=0.0. 0.1, 0.2 and 0.3 at selected temperatures. The Cole -\nCole plot gives the complete involvement of grain and grain boundaries. The plot exhibits single \nsemicircle which indicates  that the contribution of electrical conductivity mainly arises from the \ngrain, grain boundary and electrode interface. The semicircle obtained at low frequency \ncorresponds to grain boundary resistance while semicircle at high frequency correspon ds to grai n \nresistance .39 As it has been observed that there are two kinds of semicircles, one in low \nfrequency applied electric fields illustrates the grain boundary resistance and another in high \nfrequency applied electric field region indicates the role of grain resistance. The equivalent \ncircuits based on impedance data are shown in the figure 7. The parameter Rs, Rg, Cg, Rgb, Cgb, \nCPE and n  correspond to series resistance, grain resistance, grain capacitance, grain boundary \nresistance, grain boundary capacitance , constant phase element and exponential power n. All the \nparameter values are obtained by modeling the impedance data to an appropriate electrical circuit \noscillator. The electrical circuit models were fitted using the ZSimpWin software in which \nfrequency  is implicit in every curve. The values of electrical elements obtained from the analysis \nare enlisted in the table  IV. \nFrom the Cole -Cole plot, the semicircular arc depresses as the temperature increases and the \nintercepts of arc with the Z' axis shifted towards the origin with the increase of temperature. The \ndepression of an arc indicates that the dielectric relaxation deviates from the ideal Debye \nbehavior and the shifting of arc intercept towards the origin pointed that there is a decrease in \nbulk resi stances. As we observed from the table  IV, that Rg and Rgb decrease with the increase in \ntemperature, which once again validates the insulator or NTCR of semiconductor. It is also \nperceived that the grain boundary resistance increases with the increase in Cr concentration in  \n15 \n the cobalt ferrite. It implies that the Cr substitution enhances the barrier property not due to its \ninfluence against the flow of charge carriers.  \n \nAC Conductivity Analysis : \nPlots of variation of ac conductivity with temperature of com pound CoFe 2-xCrxO4 for x=0.0, 0.1, \n0.2. 0.3 and 0.4 are shown in the figures 8(a)-8(e). The temperature dependence of ac \nconductivity can be analyzed by emplo ying the Arrhenius equation45 \nTKE\no acBA\ne\n\n         or    \nkTE\no maxA\nexp\n                            (4) \nWhere σo is the electrical conductivity at infinite temperature. EA is the activation energy and KB \nis the Boltzmann constant. The activation energy is calculated at different frequencies of the \ncompounds CoFe 2-xCrxO4 for x=0.0, 0.1, 0.2, 0.3 and 0.4. The value of EA obtained at 533669 Hz \nfor all the compositions are enlisted in the table V. This indicates that the ion overcomes the \ndifferent barriers in the respective regions of conductivity. It is observed from the figure 8(a)-\n8(e) that the curve comprises of three parts. In region I, below 423 K conductivity was frequency \ndependent. The increase in frequency causes an increase in hopping of charge carriers. \nTherefore, increase in ac conductivity has been observed with the increasing frequency. It is \nconcluded that strong frequency dispersion exists in the region I (313 K to 423 K) and σac \nincreases with increase in temperature. The strong dispersion caused by polarization due to \nhopping of charge carriers at random sites of different barrier heights and separation. A \ndecreasing trend of activation energy has been noticed by the increase in Cr concentration. In \nregion II (423 K to 543 K), conductivity depends on both temperature and frequency. And \nconductivity in this region is a contri bution from short range oxygen vacancies. Followed to  \n16 \n region III (543 K to 673 K), electrical conductivity is independent of frequency and only depends \non temperature. The electrical conductivity in this range plays a role due to long range vacancy \nand the  creation  of a defect .51 Here the increasing fashion of activation energy was studied with \nthe increase in Cr concentration. It indicates that relatively more activation energy is required in \nthe form of thermal energy for the hopping of charge carriers du e to having  heterogeneous atoms \nat lattice site. The increase in electrical conductivity with increase in temperature at all the \nfrequencies was observed. It resembles the semiconduct ing behavior which represents a negative \ntemperature coefficient of resis tance (NTCR). The figure 8 represents the Arrhenius plot of the \nac conductivity at different frequencies for CoFe 2-xCrxO4 with x=0.0, 0.1, 0.2, 0.3 and 0.4 \ncompounds.  \nThe figures 9(c) – (e) depict the frequency dependence of ac conductivity of CoFe 2-xCrxO4 \ncompounds for x=0.0, 0.1, 0.2, 0.3 and 0.4 at different temperatures. The frequency dependence \nof ac conductivity curves shows the response of the material to the applied time varying field. \nThe nature of transport process in the material will be investi gated through these studies. The ac \nelectrical conductivity was  determined by the equation ; 41 \n) tan('o ac\n               (5) \n \nAt low frequency, more charge accumulation takes place at the electrode interface region due to \nspace charge polarizati on, called polarization region, which acts as a barrier in the conduction, so \nconductivity is low in this region. In the middle region, the conductivity is almost independent of \nfrequency called plateau region and equal to DC conductivity which is associat ed with drifting of \ncharge carriers. At high frequency, the conductivity increases with frequ ency, called dispersion \nregion .52 In this region frequency is very high, so space charges get dispersed rather than  \n17 \n relaxation, caused by localized charge carriers . The frequency region at which change of slope \ntakes place is called hopping frequency. The hopping frequency increases with the increase in \ntemperature.  \nThe behavior of the curve is showing an increase in ac conductivity with frequency. The curve \nconsist s of two parts: DC conductivity part, characterize by plateau behavior seen at low \nfrequencies. The second part is ac conductivity which increases at high frequencies. AC \nconductivity increase with the increase in the frequency for all the compositions. Th ere is a \ngradual rise in conductivity at low frequencies, whereas conductivity increases sharply at high \nfrequencies (figure 9). The cause of this conduction is grain boundary effect which act like a \nhindrance to the mobility of charge carriers. The presen ce of ionic part of the grain boundary is \nthe cause of conduction at high frequency. And this linear increase in AC conductivity with the \nincrease in frequency confirms the polaron type of conduction. It is cons istent with the earlier \nreport .53 The DC resi stivity has been studied to understand the polaron hopping mechanism \nwhich is discussed in the next section.  \nDC Resistivity Analysis : \nThe electrical behavior of CoFe 2-xCrxO4 compounds for x=0.0, 0.1, 0.2, 0.3, and 0.4 are \nanalyzed via measuring the DC resi stivity . Temperature variation of resistivity is shown in \nfigures 10(a)-10(e). It has been observed that resistivity decreases with increase in temperature \nand validates the behavior of semiconductor as negative temperature coefficient of resistance. It \nis observed that there are two anomalies in the temperature dependent DC resistivity curve. \nTransition temperatures, TD and TM were obtained from the temperature dependent DC resistivity \nmeasurement and t hese two temperatures are consistent with the peaks ob served in the Zˈ vs. \ntemperature plot which was discussed in the previous section (Figure 4 & table III).   \n18 \n Chromium substitution in CoFe 2O4 generally causes disorder in the system which may tend to \nlocalize the charge carriers at the doping site. The elect rical resistivity data were analyzed by \nemploying small polaron hoping (SPH) and Mott’s 3 -d variable range hoping model .54, 55 \nHowever, the lowest reliability factor was obtained from the analysis by employing Mott’s \nvariable range hopping (VRH) model whic h is given as;  \n41\no\no )TTexp( )T(\n                                 (6) \nHere ρo is the residue resistivity To is the Mott Characteristic temperature . The DC resistivity \nverses temperature plot s are shown  in figure s 14(a-e) along with the theoretical curve generated \nby the Mott VRH model . It is observed that in the whole temperature  range, there are four \nregions. However, only two regions could be analyzed by the VRH model. The resistivity at \nroom temperature (ρ300K) and, calculated residual resistivity ( ρo) and characteristic temperature \n(To) are enlisted in table VI. All the parame ters ( ρ300K, ρo and T0) increases with the increase in Cr \nconcentration , which is consisted with the impedance studies. Hence it is concluded that the \ntransport properties of Cr doped Cobalt ferrite could understood by the VRH model.  Also , it is \nworth noti ng that the hopping difference and hopping energy is different at two regions.  \nConclusion : \nChromium substituted nanocrystalline cobalt ferrite has been synthesized by the citrate precursor \nmethod. Samples are crystallize d to cubic crystal symmetry (spacegr oup    ̅ ). The \ntemperature variation of impedance spectroscopy reveals the two transitions in the material. First \ntransition temperature (TD) is observed at around 400K , which increases with the increase in the \nfrequency. Second transition temperature (TM) at around ~600K is identified as the para -\nferrimagnetic transition temperature. Both the transition temperatures are also identified from the \nprecise DC resistivity measurement. The above results open a window to understand the  \n19 \n electrical behavior of cob alt ferrite. The conduction mechanism of the chromium substituted \ncobalt ferrite followed the VRH model. The hopping distance below and above TD is different. \nThe impedance and DC resistivity on the material could be explained by the hopping mechanism \nbetw een Fe3+/Co3+/Cr3+ and Fe2+/Co2+/Cr2+ via oxygen.  \n The ac conductivity with temperature variation shows Arrhenius behavior. The \nconductivity is found to decrease with the increase chromium substitution. This result also \nsupports the observation of an increase in Rgb with the increase in Cr concentration. Rgb was \nobtained by modeling electrical equivalent circuit via Cole -Cole plot. The depression of an arc \nhas been observed in the Cole -Cole plot which shows the deviation of dielectric relaxation from \nthe ideal Debye behavior and the shifting of arc intercept towards the origin shows the decrease \nin bulk resistances and validates NTCR properties of semiconductor s. The activation energy ( EA) \nderived from the Arrhenius behavior obtained from  ac conductivity an d  ln(ω max) verses 1000/T  \nusing Z''max is found to be different because of different kinds of barrier in against the flow of \ncharge carriers. This study opens a window to understand the electrical transport properties in \ncobalt ferrite and its modification  by substitution for technological applications.  \nAcknowledgement  \n Authors are thankful to C ouncil of Scientitific Industrial Research , Department Of Science & \nTechnology  and D epartment of Atomic Energy ,  India vide sanction number 03/1183/10/EMR -\nII, SR/FTP/PS -103/2009 and 2011/20/37P/03/BRNS/007 respectively for financial assistance \nand also UGC -ref. No.: 4050/ (NET -JUNE 2013)  for JRF.  The authors also acknowledge IIT \nPatna for providing the working platform. Authors are thankful to Dr. Anup Keshri, Departmen t \nof Materials Sceince Engineering, IIT Patna for his help for hardness testing.  \n  \n20 \n Table I. Crystallite size and lattice parameter, density and V icker’s hardness of CoFe 2-xCrxO4 for \nx= 0, 0.1, 0.2, 0.3 & 0.4  samples.  \nSample  Crystallite \nsize (nm)  a = b = c ( Å) Density         \n( g/cm3) Vicker’s \nHardness (HV)  \nx=0.0  65 ± 1  8.377(5)  4.60 160 \nx=0.1  63 ± 1  8.376(2)  4.65 156 \nx=0.2  68 ± 1  8.373(1)  4.63 159 \nx=0.3  68 ± 1  8.373(9)  4.60 154 \nx=0.4  65 ± 1  8.376(2)  4.62 156 \nTable II. Peak frequency obtained from Z'' vs . frequency plot of CoFe 2-xCrxO4 for x= 0, 0.1, 0.2, \n0.3 & 0.4  samples . \nTemperature \n(K) Frequency (Hz)  \nx=0.0  x=0.1  x=0.2  x=0.3  x=0.4  \n323 576 568 112 572 95 \n373 9179  9101  3430  792 932 \n423 88831  75713  33516  6579  7742  \n473 632463  535391  200923  46415  6428 0 \n523 2738875  1215646  739072  278255  453487  \n573 3790914  2321528  1417470  5336]69  200923  \n623 3778943  1965804  739072  533669  236448  \n673 5213515  7753431  1963040  1023530  739072   \n21 \n  \nTable III. Transition temperatures obtained from impedance vs. temperature measu rement of \nCoFe 2-xCrxO4for x= 0, 0.1, 0.2, 0.3 & 0.4  samples . \n Transition temperatures (K)  \nFrequency(Hz)  x=0.0  x=0.1  x=0.2  x=0.3  x=0.4  \nTD TM TD TM TD TM TD TM TD TM \n1097  343 592 343 585 343 603 373 593 311 584 \n10722  383 592 383 604 412 632 433 593 333 595 \n104761  433 594 433 604 453 613 493 603 483 573 \n327454  473 604 463 604 493 613 553 603 553 603 \n533669  473 613 473 602 503 613 563 613 523 603 \n739072  493 613 483 604 523 603 573 613 533 613 \n1023531  553 623 503 595 533 623 573 623 523 623 \n3199267  543 623 553 615 553 623 583 623 593 623 \n \n \n \n \n \n \n \n  \n22 \n Table IV. Parameters of series resistance( Rs), grain resistance( Rg), grain capacitance ( Cg), grain \nboundary resistance ( Rgb), grain boundary capacitance ( Cgb), constant phase element ( CPE )  and \nn obtained from  the Cole -Cole plot and analysis of CoFe 2-xCrxO4 for x= 0, 0.1, 0.2, 0.3 & 0.4  \nsamples . CoFe 2O4 T(K)  Rs(Ω)  Cg(F) Rg(Ω) Cgb(F) Rgb(Ω) CPE  n \n323 4.615  1.16x10-9 6.537x105 3.39x10-10 9.354x105 4.46x10-8 0.37 \n373 5.42 1.06x10-9 5.206 x104 3.60x10-10 56610  2.91x10-7 0.3271  \n423 6.817  4.99x10-10 2119  6.45x10-10 7977  9.87 x10-8 0.4163  \n473 5.345  3.69x10-10 489.1  7.74x10-10 1288  2.41x10-7 0.01614  \n523 4.036  2.96x10-10 218.9  5.46x10-9 277.7  6.58x10-9 0.0964  \n573 9.0776  3.40x10-10 169.4  1.57x10-9 196.5  6.14x10-9 0.0164  CoFe 1.9Cr0.1O4 323 5.817  8.47x10-10 9.284 x105 2.72x10-10 9.88x105 2.27x10-8 0.4039  \n373 9.946  1.16x10-9 3.993 x104 7.02x10-10 7.02x104 2.068x10-7 0.3216  \n423 11.94  4.09x10-10 3754  7.13x10-10 7860  2.108x10-7 0.3874  \n473 13.05  3.38x10-10 939.4  1.57x10-9 4230  2.20x10-7 0.01234  \n523 13.4 9.90x10-11 753.7  2.76x10-10 801.7  2.29x10-9 0.1258  \n573 18.76  3.29x10-10 238.3  5.99x10-10 8.95 x108 4.50x10-9 0.1616  CoFe 1.8Cr0.2O4 323 9.778  5.45x10-9 10372  2.25x10-10 4.20x106 1.30x10-8 0.309  \n373 10.55  2.04 x10-8 9999  2.26x10-10 2.51x105 1.93x10-8 0.384  \n423 11.98  3.81x10-10 9346  5.77x10-10 2.16x104 1.29x10-6 0.1933  \n473 12.77  2.74x10-10 2820  1.84x10-9 1338  4.16x10-7 0.2669  \n523 11.63  1.25x10-9 25.41  2.04x10-10 971.7  4.59x10-5 0.6997   \n23 \n 573 18.43  1.95 x10-8 2.136  2.85x10-10 437 2.30x10-6 7.123  CoFe 1.7Cr0.3O4 323 14.07  1.65 x10-9 9.73x105 3.04x10-10 1.73x106 2.78x10-7 0.1739  \n373 14.92  1.15 x10-9 2.938 x104 3.44x10-10 7.15 x105 --- --- \n423 15.7 1.25 x10-9 4705  3.46x10-10 9.66 x104 6.44x10-7 0.1038  \n473 16.93  8.79x10-10 1779  4.14x10-10 6160  9.07x10-5 0.002877  \n523 17.54  1.99x10-10 1293  2.96x10-10 3346  4.70x10-7 0.0005934  \n573 13.11  2.89x10-10 1069  7.39x10-9 2138  2.10x10-6 7.36x10-11 CoFe 1.6Cr0.4O4 323 9.316  2.03x10-9 1.192 x106 3.04 x10-10 2.3 x106 4.11 x10-7 0.1493  \n373 14.9 4.33x10-10 1.924 x105 9.07 x10-10 2.64x105 - - \n423 15 1.32x10-9 2.28 x104 3.84x10-10 1.25x105 1.96x10-5 0.01866  \n473 15.95  3.72x10-10 5070  1.77x10-9 2923  3.90x10-6 0.003499  \n523 16.09  3.25x10-10 1205  2.01x10-9 2623  8.82x10-5 0.8349  \n573 12.95  3.29x10-10 23.92  5.44x10-7 68.41  5.49x10-6 0.3698  \n \n \n \n \n \n \n \n \n \n  \n24 \n Table V. Activat ion energy of CoFe 2-xCrxO4  for x= 0, 0.1, 0.2, 0.3 & 0.4  samples . \n \n \n \n \n \n \n \n \n \n \n \nTable VI. The value of resistivity and Mott characteristic temperature for different doping \nconcentratio n. \nSample  ρ300K(Ωcm)  Region I  Region II  \n ρo(Ωcm)  T0(x1010 K) ρo(Ωcm)  T0(x109 K) \nx=0.0  39788  4.19x10-32 1.47 3.75x10-10 1.60 \nx=0.1  42614  6.00x10-32 1.49 2.09x10-11 2.60  \nx=0.2  161485  1.53x10-33 1.62 2.12x10-19 2.40  \nx=0.3  2.23x107 4.50x10-33 1.71 1.08x10-25 7.30 \nx=0.4  2.35x107 6.77x10-33 1.85 2.79x10-27 10.04 \n \n           \n Activation Energy in eV (E A) taken at 533669 Hz \nfrom lnσ ac vs 1000/T plot.  Z'' Vs.   ω max data \nSample  Region \nI(below \n423 K)  Region II  \n(423 K to \n543K)  Region III  \n(543K to \n673 K)  \n EA(from log ω maxVs \n1000/T plot)  \nRegion I \n(323 K -\n523 K)  Region II \n(323 K -\n523 K)  \nx=0.0  0.29 0.54 0.15 0.62 0.12 \nx=0.1  0.34 0.50 0.28 0.57 0.32 \nx=0.2  0.33 0.55 0.37 0.61 0.10 \nx=0.3  0.26 0.48 0.61 0.46 0.24 \nx=0.4  0.39 0.20 0.69 0.61 0.24  \n25 \n      \n \n \n \n \n \n \n \n \n \n \n \n \n \n20 30 40 50 60 70 80CoFe1.7Cr0.3O4CoFe1.6Cr0.4O4\nCoFe1.8Cr0.2O4\nCoFe1.9Cr0.1O4(533)(440)(511)(422)(400)(311)\n  Intensity (A.U.)\n2(Degree )(220)CoFe2O4(222)(a)\n35.0 35.2 35.4 35.6 35.8 36.0\n2(Degree)\n   Intensity (A.U.) CoFe1.9Cr0.1O4\n CoFe1.8Cr0.2O4\n CoFe1.7Cr0.3O4\n CoFe1.6Cr0.4O4(311)(b)\nFIG. 1. (a)  X-ray diffraction (XRD) patterns of  \nCoFe 2-xCrxO4 for x=0.0, 0.1, 0.2, 0.3  and 0.4 \nsamples annealed at 1173 K and patterns are \nindexed to spacegroup in cubic \nsymmetry.  \nFIG. 1 . (b) Highest intensity peak (311) in \nthe XRD patterns of CoFe 2-xCrxO4 for \nx=0.1, 0.2, 0.3, 0.4 samples.   \n26 \n  \n \n \n \n \n  \n1001011021031041051061070200400600800100012001400\n1001011021031041051061070.00.51.01.5\n 323 K\n 373 K\n 423 K\n  Z' (K)\nFrequency (Hz)CoFe2O4(a)\n 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z' (K)\nFrequency (Hz)\n10010110210310410510610702004006008001000120014001600\n1001011021031041051061070246810\n323 K\n373 K\n423 K\n  Z' (K)\nFrequency (Hz)CoFe1.9Cr0.1O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z' (K)\nFrequency (Hz)(b)\n1001011021031041051061070200040006000800010000\n10010110210310410510610701234\nCoFe1.8Cr0.2O4 323 K  \n 373 K  \n 423 K\n  ' (K)\nFrequency (Hz) 473 K \n 523 K \n 573 K \n 623 K \n 673 K\n \n  Z' (K)\nFrequency (Hz)(C)\n1001011021031041051061070200040006000800010000120001400016000\n100101102103104105106107024681012\n323 K\n373 K\n423 K\n  Z' (K)\nFrequency (Hz)CoFe1.7Cr0.3O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z' (K)\nFrequency (Hz)(d) \n27 \n  \nFIG. 2. Frequency variation of the real part ( Z') of Impedance spectrum of CoFe 2-xCrxO4 for \n(a)x=0.0, (b)x=0.1, (c)x=0.2, (d)x=0.3 and  (e)x=0.4 at different  temperatures below 450 K. \nInsets depict the frequency variation of Z' at different temperatures within the range 473 K to \n673K.  \n \n   \n \n  \n \n \n100101102103104105106107040008000120001600020000\n10010110210310410510610702040\n323 K\n373 K\n423 K\n  Z' (K)\nFrequency (Hz)CoFe1.6Cr0.4O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z' (K)\nFrequency (Hz)(e)\n1001011021031041051061070100200300400500\n1001011021031041051061070.00.10.20.30.40.50.6\n(a) 323 K\n 373 K\n 423 K\n  Z'' (K)\nFrequency (Hz)CoFe2O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z'' (K)\nFrequency (Hz)\n1001011021031041051061070100200300400500\n1001011021031041051061070.00.10.20.30.40.50.6\n(b)323 K\n373 K\n423 K\n  Z'' (K)\nFrequency (Hz)CoFe1.9Cr0.1O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z'' (K)\nFrequency (Hz)\n10010110210310410510610702004006008001000\n1001011021031041051061070.00.51.01.5\n(c)323 K\n373 K\n423 K\n  Z'' (K)\nFrequency (Hz)CoFe1.8Cr0.2O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z'' (K)\nFrequency (Hz)\n1001011021031041051061070500100015002000\n1001011021031041051061070123456\n(d)Z'' (K)\n323 K\n373 K\n423 K\n  \nFrequency (Hz)CoFe1.7Cr0.3O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z'' (K)\nFrequency (Hz) \n28 \n  \n \nFIG. 3. Frequency variation of the imaginary part ( Z'') of Impedance spectrum of CoFe 2-xCrxO4 \nfor (a)x=0.0, (b)x= 0.1, (c)x=0.2, (d)x=0.3 and (e)x=0.4 at different temperatures. Insets show the \nfrequency variation of Z'' from temperature 473 K to 673 K.  \n \n   \n \n   \n \n \n10010110210310410510610705001000150020002500\n1001011021031041051061070.02.55.07.5\n(e)323 K\n373 K\n423 K\n  Z'' (K)\nFrequency (Hz)CoFe1.6Cr0.4O4 473 K\n 523 K\n 573 K\n 623 K\n 673 K\n \n  Z'' (K)\nFrequency (Hz)\n300 400 500 600 700 800020406080100120140\n500 600 700 8000.00.10.20.30.40.5\n1097 Hz\n10722 Hz\n  Z' (K)\nTemperature (K)CoFe2O4 (a1)Z' (K)\n  \nTemperature (K)\n300 400 500 600 700 8000.00.51.01.5  104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  Z' (K)\nTemperature (K)CoFe2O4 (a2)\n300 400 500 600 700 800050100150200\n500 600 700 8000.00.20.4\nTemperature (K)Z' (K)\n  Z' K\n  \nTemperature (K)CoFe1.9Cr0.1O4\n(b1)\n1097 Hz\n10722 Hz\n300 400 500 600 700 8000.00.51.01.52.02.5\n 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  Z' (K)\nTemperature (K)CoFe1.9Cr0.1O4 (b2) \n29 \n    \n \n  \n \n   \n \n \nFIG. 4. Temperature dependent real part ( Z') of Impedance for CoFe 2-xCrxO4 for (a)x=0.0 , \n(b)x=0.1, (c)x=0.2, (d)x=0.3 and (e)x=0.4. Insets are the temperature dependent Z' from 500 -800 \nK to distinguish the second peak( TM). \n300 400 500 600 700 800050100150200250\n500 600 700 80002004006008001000\n500 600 700 80002004006008001000\n  Z' ()\nTemperature (K)\n500 600 700 80002004006008001000\n  Z' ()\nTemperature (K)\n  \n1097 Hz\n10722 HzZ' (K)\nTemperature (K)CoFe1.8Cr0.2O4\n(c1)Z' (K)\nTemperature (K)\n  \n300 400 500 600 700 8000.00.51.01.52.02.53.0\n(c2) CoFe1.8Cr0.2O4\n 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\nTemperature (K)Z' (K)\n  \n300 400 500 600 700 800050100150200\n500 600 700 8000123\nTemperature (K)Z' (K)\n  \nTemperature (K)Z' (K)\n  \nCoFe 1.7Cr0.3O4(d1)\n1097 Hz\n10722 Hz\n300 400 500 600 700 8000.00.51.01.52.02.5\nTemperature (K)Z' (K) 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  \nCoFe1.7Cr0.3O4(d2)\n300 400 500 600 700 8000100200300400\n500 600 700 8000.00.51.01.52.02.5\nTemperature (K)Z' (K)\n  Z' (K) \n  \nTemperature (K)CoFe1.6Cr0.4O4\n(e1)\n1097 Hz\n10722 Hz\n300 400 500 600 700 8000.00.51.01.52.02.5\n 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  Z' (K)\nTemperature (K)CoFe1.6Cr0.4O4(e2) \n30 \n  \n \n \n   \n \n   \n \n    \n \n300 400 500 600 700 800050100150200250300350\n550 600 650 700 750 8000.00000.00020.00040.00060.0008\n  Z'' (K)\nTemperature (K)1097 Hz\n10722 Hz\n  Z'' (K)\nTemperature (K)CoFe 2O4\n(a1)\n300 400 500 600 700 800012345\n550 600 650 700 750 8000.000.050.10\n  Z'' (K)\n(a2) 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  Z'' (K)\nTemperature (K) CoFe 2O4\nTemperature (K)\n300 400 500 600 700 8000100200300400\n550 600 650 700 750 8000.0000.0010.0020.003\n  Z'' (K)\nTemperature (K)CoFe 1.9Cr0.1O4\n 1097 Hz\n 10722 Hz\n \n  Z'' (K)\nTemperature (K)(b1)\n300 400 500 600 700 800012345\n550 600 650 700 750 8000.000.020.040.060.080.10\n  Z'' (K)\nTemperature (K)(b2)CoFe 1.9Cr0.1O4Z'' (K)\nTemperature (K) 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  \n300 400 500 600 700 8000100200300400500\n550 600 650 700 750 8000.0000.0050.0100.0150.020\n  Z'' (K)\nTemperature (K)Temperature (K)Z'' (K)CoFe 1.8Cr0.2O4\n1097 Hz\n10722 Hz\n  \n(c1)\n400 500 600 700 80001234567\n600 650 700 750 8000.00.10.20.30.4\n  Z'' (K)\nTemperature (K)(c2)CoFe1.8Cr0.2O4\nTemperature (K)Z'' (K) 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n   \n31 \n  \n    \n \n \n   \n \n \nFIG. 5. Temperature dependent imaginary part ( Z'') of impedance of CoFe 2-xCrxO4 for (a)x=0.0, \n(b)x=0.1, (c)x=0.2, (d)x=0.3 and (e)x=0.4.  Insets are the temperature dependent Z'' spectrum of \nCoFe 2-xCrxO4 for x=0.0 -0.4 at different frequencies.  \n \n \n \n \n \n \n \n300 400 500 600 700 8000100200300400\n550 600 650 700 750 8000.000.010.020.030.04\n  Z'' (K)\nTemperature (K)CoFe1.7Cr0.3O4\n1097 Hz\n10722 Hz\n  Z'' (K)\nTemperature (K)(d1)\n300 400 500 600 700 8000123456\n550 600 650 700 750 8000.00.20.40.60.81.0\n  Z'' (K)\nTemperature (K)(d2)CoFe1.7Cr0.3O4\n 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n  Z'' (K)\nTemperature (K)\n300 400 500 600 700 800012345\n550 600 650 700 750 8000.0000.0050.0100.0150.020\n  Z'' (K)\nTemperature (K)CoFe 1.6Cr0.4O4\n 1097 Hz\n 10722 Hz\n \n  Z'' (K)\nTemperature (K)(e1)\n300 400 500 600 700 8000123456\n550 600 650 700 750 8000.00.20.40.60.81.01.2\n  Z'' (K)\nTemperature (K)(e2)CoFe 1.6Cr0.4O4Z'' (K)\nTemperature (K) 104761 Hz\n 327454 Hz\n 533669 Hz\n 739072 Hz\n 1023531 Hz\n 3199267 Hz\n   \n32 \n        \n \n         \n \nFIG. 6. (a) Cole -Cole plots or complex impedance spectra for sample CoFe 2O4 at different \ntemperatures.  \n \n   \n \n   \n \n \nFIG. 6. (b) Cole -Cole plots or complex impedance spectra for sample CoFe 1.9Cr0.1O4 at \ndifferent temperatures.  \n0 200 400 600 800 1000 12000100200300400500\nZ' (K) \n Z'' (K)323 K(a)\n0 20 40 60 800102030\nZ' (K) \n Z'' (K)373 K(b)\n0 20 40 60 800102030 \n Z'' ()\nZ' ()423 K(c)\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.10.20.30.40.5 \n Z'' (K)\nZ' (K)473 K(d)\n0.00 0.05 0.10 0.15 0.200.000.020.040.060.080.10\n523K \n Z'' (K)\nZ'' (K)(e)\n0.05 0.10 0.150.020.030.040.050.060.070.08 \n Z'' (K)\nZ' (K)573 K(f)\n0 500 1000 15000200400600 \n Z'' (K)\nZ' (K)323 K (a)\n0 20 40 60 80 100010203040 \n Z'' (K)\nZ' (K)373 K(b)\n0 2 4 6 8 1001234 \n Z'' (K)\nZ' (K) 423 K(c)\n0.0 0.5 1.00.00.10.20.30.40.50.6\n473 K \n Z'' (K)\nZ' (K)(d)\n0.05 0.10 0.15 0.200.0000.0250.0500.0750.100 \n Z'' (K)\nZ' (K)523 K(e)\n0.05 0.10 0.150.020.030.040.050.060.070.08 \n Z'' (K)\nZ' (K)573 K(f) \n33 \n       \n \n      \n \nFIG. 6. (c) Cole -Cole plots or complex impedance spectra for sample CoFe 1.8Cr0.2O4 at \ndifferent te mperatures.  \n \n   \n \n   \n \nFIG. 6. (d) Cole -Cole plots or complex impedance spectra for sample CoFe 1.7Cr0.3O4 at \ndifferent temperatures.  \n0 1000 2000 3000050010001500 \n Z'' (K)\nZ' (K)323 K(a)\n0 50 100 150 200020406080100 \n Z'' (K)\nZ' (K)373 K(b)\n0 5 10 15 20 25024681012 \n Z'' (K)\nZ' (K)423 K(c)\n0 1 2 3 40.00.51.01.5 \n Z'' () \nZ' (K)473 K(d)\n0.0 0.2 0.4 0.6 0.8 1.00.10.20.30.40.5 \n Z'' (K) \nZ' (K) 523 K(e)\n0.05 0.10 0.150.020.030.040.050.060.070.08 \n Z'' (K)\nZ' (K)573 K(f)\n0 200 400 600 800 1000050010001500 \n Z'' (K)\nZ' (K)323 K(a)\n0 200 400 6000100200300\nZ' (K) \n Z'' (K)373 K(b)\n0 20 40 60 80010203040\nZ'' (K) \n Z'' (K)423 K(c)\n0 2 4 6 8 10 120123456\nZ' (K) \n Z'' (K)473 K(d)\n0.0 0.5 1.0 1.5 2.00.00.20.40.60.81.0\nZ' (K) \n Z'' (K)523 K(e)\n0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.40.5\nZ' (K) \n Z'' (K)573 K(f) \n34 \n  \n \n \n \n \n \n \n \n \nFIG. 7. Equivalent Circuit model.  \n \n \n \n \n1.5 2.0 2.5 3.0-10-5\nRegion IRegion II(a)CoFe2O4\n 1097 Hz \n 10722 Hz\n 104761 Hz\n 327454 Hz\n 533669 Hz \n 739072 Hz\n 1023531 Hz \n 3199267  Hz\n   \n1000/T (K-1)lna.c() (m)-1Region III\n \n35 \n  \n \n \nFIG. 8. Temperature dependent of ac conductivity ( σac) of CoFe 2-xCrxO4 for (a) x=0.0, (b)x=0.1, \n(c)x=0.2, (d)x=0.3 and (e)x=0.4.  \n \n \n \n \n \n \n \n36 \n  \n \nFIG. 9. Frequency dependent conductivity of CoFe 2-xCrxO4 for (a) x=0.0, (b)x=0.1, (c)x=0.2, \n(d)x=0.3 and (e)x=0.4 at different temperatures.  \n \n   \n \n   \n \n \n37 \n  \n \nFIG. 10. Logarithmic DC resistivity (ln ρDC) versus Temperature [T(K)] -1/4 plot of DC resistivity. \nSolid lines represent the analyzed data point for VRH model.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n38 \n References:  \n \n \n1S. Verma, J. Chand and M. Singh, Adv. Mat. Lett. 4, 310 (2013) . \n2A. Mandal and C. 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Ravi , Materials Science and Engineering B 110, 46 (2004) . \n "    },    {        "title": "1611.01128v1.Room_temperature_multiferroism_in_polycrystalline_thin_films_of_gallium_ferrite.pdf",        "content": "1 \n Room temperature multiferroism in polycrystalline thin films of gallium \nferrite  \nMonali Mishra1, Amritendu Roy2, Ashish Garg3, Rajeev Gupta4,5 and Somdutta Mukherjee1,1 \n1 Colloids and Materials Chemistry Department, CSIR -IMMT Bhubaneswar -751013, India  \n2 Minerals, Metallurgical and Materials Engineering, Indian Institute of Technology, \nBhubaneswar -751007, India  \n 3 Materials Science and Engineering, Indian Institute of Technology, Kanpur -208016, India  \n4 Department of Physics, Indian Institute of Technology , Kanpur -208016, India  \n5 Materials Science Programme, Indian Institute of Technology, Kanpur -208016, India  \n  \nPACs: 77.84. -s, 77.80.Fm, 75.60. -d, 75.75. -c \nKeywords: polycrystalline, thin films, multiferroics, ferrimagnetism, ferroelectricity  \nAbstract  \nSol-gel deposited (010) textured polycrystalline thin films of gallium ferrite (GaFeO 3 or GFO) on \nn-Si(100) and  Pt/Si(111) substrates a re characterized for room temperature multiferroism. \nStructural characterization using X -ray diffraction and Raman spectros copy confirms formation \nof single phase with nano -sized crystallites. Temperature dependent magnetization study \ndemonstrates ferri to paramagnetic transition at ~300 K. Room temperature piezoresponse force \nmicroscopic analysis reveals local 180  phase swit ching of ferroelectric domains at very high \ncoercive field EC, ~ 1350 kV/cm consistent with recent experimental and first -principles studies. \nOur study opens up possibility of integrating polycrystalline GFO in novel room temperature \nmultiferroic devices.  \n \n \n \n \n                                                           \n \nCorresponding author: msomdutta@gmail.com  2 \n  \nMultiferroics have been recognized as next generation digital memory materials owing to \ntheir unique attributes such as lower energy consumption, fast read -write operation and \nextraordinary memory storage capability. 1, 2 Unfortunately, room temperat ure multiferroics are \nrare due to competing requirements of ferroelectricity and magnetism.3 The most extensively \nstudied room temperature multiferroic, bismuth ferrite (BiFeO 3)4 demonstrates weak/no net \nmagnetization at room temperature and has very weak magnetoelectric coupling limiting  its \napplication potential. Thus, search for newer multiferroics with appreciable room temperature \nproperties is a major challenge. Further, commercial device fabrication and design require \nintegration on to a standard sili con circuit such that the material should demonstrate sizable \nmagnetic as well as ferroelectric character along the growth direction. In other words, the \nmaterial should be grown in such a way that it yields maximum possible multiferroic properties \nalong t he growth direction. Thus, an alternative to B iFeO3 for prospective multiferroic memory \ndevice applications would be highly desirable. In this context, gallium ferrite is an interesting \nsystem  with exciting properties.5  \nGFO is a polar piezoelectric with near room temperature ferrimagnetism and \nmagnetoelectric coupling.6, 7 Further, ferri to paramagnetic transition temperature (T C) can be \ntailored to room temperature and above by tuning the Ga:Fe ratio slightly away from t he \nstoichiometry.8 The observed long range magnetic order in the A -type antiferromagnetic ground \nstate 9 could be attributed to  cation site disorder between similar sized Ga3+ and Fe3+ ions \nwherein some of the Fe3+ ions always occupy Ga sites and vice ver sa, even in stoichiometric \ncomposition. 6, 10 Early polarization vs. electric field (P -E) measurement on bulk polycrystalline \nand single crystalline samples showed leaky dielectric behavior with no/poor signature of \nferroelectric ity.11, 12 However, first -principles calculations predicted GFO to undergo \nferroelectric to paraelectric phase transition accompanied by a structural transition from polar \nPc2 1n to centrosymmetric Pnna  symmetry.13, 14 Our previous work on epitaxial thin film, in fact, \nshowed that po larization at room temperature could be switched by reversing  electric field within  \nnano -dimensional polar domains and thus established GFO as a room temperature ferroelectric14 \nand recently confirmed by a study showing saturated  ferroelectric P -E loops i n epitaxial films of \nGFO. 15 Ferroelectricity  in GFO , coupled with tunable ferrimagnetism at room temperature 3 \n opens up the possibility of adding a new member to the room temperature multiferroic family , \nexciting for device applications. However, use of high end deposition technique and expensive \nsingle crystalline oxide substrates put a serious question on the commercial efficacy of the \nmaterial in devices. Therefore, it is imperative to try and deposit GFO on commercial wafer \nmaterial us ing low cost deposition techniques. To the best of our knowledge, such an attempt on \nthis system has not been reported  so far. In this letter, for the first time, we report observation of \nnano -scale ferroelectricity as well as room temperature ferrimagneti sm in polycrystalline GFO \nthin films grown using a facile spin coating technique.  \nFilms were spin coated on cleaned n -Si(100) and Pt/TiO 2/SiO 2/Si(111) or Pt/Si(111) \nsubstrates using 0.1 M solution of the  precursor nitrates  (high purity gallium nitrate hydrate and \niron nona -hydrate ). Details of synthesis procedure could be found elsewhere.16 Crystal structure \nand phase purity of the films  were investigated using  X-Ray diffract ion Cu -K radiation . \nPiezoelectric and ferroelectric behaviors were examin ed using piezoresponce force  microscope \n(Asylum Research) and Radiant precision premier -II. Temperature dependent and isothermal \nmagneti zation  were studied  with a physical property measurement system ( Quantum Design).  \nFig. 1(a) shows room temperature XRD patterns of GFO films grown on Pt/Si(111) \n(GFO|| Pt/Si(111) ) and n -Si(100)  (GFO||n -Si(100)) substrates, respectively, over 2θ range , 18-\n80˚. XRD patterns were indexed using JCPDS card no. 76 -1005 confirm ing formation of single \nphase with orthorhombic Pc2 1n symmetry . One can notice that predominantly ( 0k0) reflections \nare present . This indicates  growth of (010) out -of-plane orientation that coincides  the \nspontaneous polarization direction of GFO .9, 14  Orthorhombic GFO  has lattice parameters a = \n0.872 nm, b = 0.937 nm and c = 0.507 nm  in bulk structure .7 For out -of-plane b-axis orientation, \nthe in -plane lattice parameter s of GFO (a2+c2) have a lattice mismatch of ~ 6.8% with twice of \nlattice parameter  of Si. This mismatch is quite large  compar ed to that of GFO grown epitaxially \non indium tin oxide  (0.7 %) or yttria stabilized zirconia  (1.6%) . 14  Due to such large mismatch, \na large tensile strain is exerted result ing in generation of  structural defec ts which in turn leads to \npolycrystallinity in the film . As a result, the out -of-plane lattice parameter , b ~ 9.34 Å is \ncompressed by only 0.3% compared to its bulk structure as determined from the XRD data. In \ncase of GFO|| Pt/Si(111), since the top Pt  layer is polycrystalline, heterogeneous nucleation \nwould be more favor ed at various defect sites such as grain boundaries and our FESEM study in 4 \n fact, as discussed below, demonstrates the grain size  is significantly less in case of \nGFO|| Pt/Si(111)  compar ed to GFO||n -Si(100) . We also calculated average crystallite size using \nDebye -Scherrer’s formula: 45 and 36 nm, for GFO|| n-Si(100)  and GFO|| Pt/Si (111) , respectively.  \nTo investigate the structural behavior, and strain  effect , samples were analyzed by Raman  \nspectroscopy  using  514 nm Ar+ laser as the excitation source. GFO, with Pc2 1n symmetry  has 8 \nformula   units per unit cell resulting in  a total of 117 Raman active modes at the zone center.7 \nFig. 1(b) shows unpolarized Raman spectra of GFO  films measured i n back scattering geometry  \nat room temperature . It is found that  Raman spectrum of GFO ||Pt/Si(111) is very similar to that \nobtained for bulk polycrystalline samples.8 The peaks are broad in complete agreement with the \nnanocrystalline  nature of the film. We observe a total of 19 modes over 100 to 850 cm-1 marked \nwith dotted vertical lines. However, t he spectrum of GFO|| n-Si(100) looks different compared to \nGFO|| Pt/Si(111).  Since the penetration depth of 514 nm is larger than the film thickness , intense \nRaman peak at ~520 cm-1 from Si substrate is visible and sample peaks are resolved  only after \nsubtracting this intense peak. Though, most of the GFO peaks are present, the spectrum is \ndominated by other peaks from Si such as at ~ 303 cm-1 and 620 cm-1.17, 18 Si peaks at 670 cm-1 \nappear as a shoulder to GFO peak at 692 cm-1.17, 18 GFO peak at 437 cm-1 has been suppressed by \na Si peak at 433 cm-1.17, 18 Since Raman peak positions of GFO|| Pt/Si(111) and GFO|| n-Si(100) \nare almost identical, we conclude that the strain levels at both the films are comparable . \n FE-SEM micrographs of the GFO thin films , GFO|| Pt/Si(111) and GFO|| n-Si(100) are \nshown in Fig.2 (a) and (b), respectively. The micrographs reveal a largely uniform deposition of \nGFO on both of the substrates with larger grains on n -Si(100) compared to on Pt/Si(111). \nAverage grain size of GFO|| Pt/Si(111) and GFO|| n-Si (100) are ~45 nm a nd 96 nm, respectively. \nCross sectional SEM of GFO ||Pt/Si(111) as shown in Fig. 2(c) demonstrates typical film \nthickness of ~200 nm. The energy dispersive X -ray spectrum analysis  (not shown here)  revealed  \nthat our films are slightly Fe rich with Ga to Fe r atio of 10:12. Surface topography analysis using \natomic force microscopy of GFO ||Pt/Si(111) as shown in, Fig. 2(d) reveals a RMS roughness of \n1.6 nm  bolstering the SEM observation that the films are largely uniform . \n Easy magnetization direction of GFO is along crystallographic c-axis6 that lies within our \nsample plane . Magnetization is studied as a function of temperature in presence of 500 Oe  field \nover 25 K to 320 K. Field cooled (FC) and zero field cooled (ZFC) in -plane magnetization with 5 \n temperature a re plotted in Fig. 3. It was found that GFO undergoes ferri -to-paramagnetic \ntransition  at ~ 300 and 302 K for GFO|| n-Si(100) and GFO|| Pt/Si(111), respectively. These are \nrather high compared to previously reported T C of epitaxial  film19 that closely matches with bulk \nGFO.6 The higher value of T C observed could be attributed to the reduced crystallite size of our \nfilm,20 suggesting  that the T C could be adjusted  by tailoring the synthesis conditions as well, in \naddition to sample compositi on. The maximum magnetization estimated from FC plots  ~ 60.1 \nemu/cc (or 0.33 B/Fe) and ~ 94.6 emu/cc (or 0.53 B/Fe) at 25 K for GFO|| Pt/Si(111) and \nGFO|| n-Si(100), respectively  are comparable to  the value  observed in epitaxial thin film.19 The \ninteres ting feature is that the FC and ZFC plots show bifurcation and ZFC magnetization goes \nnegative with decreasing temperature. Negative  magnetization is not unusual in ferrimagnetic \nmaterials and has also been observed in GFO nanoparticles with crystallite si ze ranges from 18 \nto 64 nm.21 However, for samples with larger grains , the ZFC magnetization is positive over the \nentire temperature range.21 GFO con tains  four types of cation sites: Fe1, Fe2, Ga2 octahedral \nsites and Ga1 tetrahedral sites  with  inherent site disorder .6 In low temperature synthesized \nGFO,6 site disorder predominantly affects octahedral sites :7  Fe3+ ions partially occupy \noctahedral Ga2 sites and Fe sites are partially occupied by Ga3+ ions.6, 9 Fe at Fe1 site couple \nanti-parallely  to Fe at Fe2 and Ga2 sites resulting in the observed ferrinmagnetic order.6, 7, 9 \nMagnetization at these different tetrahedral and octahedral sites may have different temperature \ndependence,22 Curie temperature (T C), coercive field21 etc. and the total magnetization is the sum \nof magnetization at all four sub -lattices.21, 23  The origin of negative magnetization observed in \nour films could be attributed to the variation in temperature dependence of magneti zation at \ntetrahedral and octahedral sites coupled to the surface spin structure which changes with \nsynthesis condition. However, in -depth studies are required to understand this effect in GFO \nnanostructures. Further, magnetization is plotted as a function  of magnetic field  (M-H) at 299K \nfor GFO ||Pt/Si(111) after subtracting the substrate contribution as shown in the inset of Fig. 3 . \nThe M -H loop is  saturated with saturation magnetization of M S ~ 5 emu/cc suggesting  our \npolycrystalline film is magnetic at r oom temperature. Th e magnitude of magnetization is \ncomparable to those  reported for BiFeO 3 polycrystalline films at room temperature.24, 25 \nHowever,  our previous study8 and recent report on epitaxial thin films26 suggest that room \ntemperature magnetization can be further enhanced by tailoring the Fe content in GFO.  6 \n  For electrical characterization and to explore the potential of GFO in MEMS device s, we \nchose GFO|| Pt/Si (111) . The film w as sputter coated with Au top electrodes of  0.2 mm diameter \nto form Pt/GFO/Au capacitors as shown schematically in Fig. 4(a). Fig. 4(b) shows room \ntemperature ferroelectric P -E hysteresis loops at 10 kHz under different electric fields. Form Fig. \n4(b) we find that P -E loops do not saturate even at  400 kV/cm and looks more like  leaky \ndielectrics,27 latter attributed to the conducting grain boundaries masking the intrinsic \nferroelectricity of the grains. First -principles calculations predicted that GFO possesses a very \nhigh activation energy ~  0.61 eV/ f. u. to switch from its centrosymmetric  to non -\ncentrosymmetric  structure which in turn, translates into requirement of a huge coercive field  \n(EC) to switch  polarization .14 In fact, a recent study on GFO epitaxial film reported a n EC of \n1400 kV/cm  to switch polarization in a b-axis oriented film.15 Due to large  leakage in our \npolycrystalline films, we could not apply such high field.   \n However , to appreciate  the switching behavior of ferroelectric domains, we analyzed the \nfilms using piezoresponse force microscopy (PFM). PFM , in addition to  visualization of domain \nstructure ,28 allows  to induce local polarization switching in individual grains obfuscating the \ngrain boundary contributions whose leakiness otherwise mask the intrinsic ferroelectricity.29 Fig. \n4(c) shows the PFM amplitude image of the as -grown film with domains of different polarity. \nWe chose a square region of 1 μm2 on the sample and sequentially applied -5V, +5V and -5V to \nstudy the effect of bias on the domain evolution. Fig. 4(d -f) show the out of plane PFM \namplitude images under different applied voltage s mentioned above over a scan area of 1.5 μm2.  \nUpon appl ying -5V, domains with bright shade (apparently with downward polarization) become \ndarker suggesting the change in polarizatio n direction. On the other hand application of +5V \ntransforms darker domains to comparatively brighter, indicating polarization  change from \nupward to downward. Further reversal of voltage  brings back the initial polarization state (Fig. \n4(f)).  Therefore, w e find that application of electric field of specific direction can switch the \npolarization direction in the film . However, it is apparent  that application of ±5 V cannot swap \nthe polarization state completely  since  selected area within the sample did not change  from \ncomplete dark to complete bright and vice -versa.  Above observation only suggests an \nincomplete polarization switching and indicat es requirement of larger EC. We in fact, chose a \nparticular domain and applied higher switching voltages and the e ffects are shown in switching \nspectroscopy data plotted in Fig. 5. Fig. 5(a) and (b) plot out -of-plane PFM amplitude and 7 \n corresponding phase as a function of switching potentials 10 V to 100 V. We find from the phase \nplot that upon sweeping voltage -10 to +10 V the phase changes about ~ 145o i.e. incomplete \nswitching and with increasing bias voltage phase difference increases. A ~180o phase change \noccurs  at an applied voltage of 100 V as shown in Fig. 5(b) indicating complete polarization \nreversal. An interesting feature of both Fig. 5(a) and 5(b) is that the loops are not centered at 0 V. \nThey are shifted in the negative voltage axis, instead. If we evaluate th e amplitude images (Fig. \n4(c-f)) carefully, similar effect could also be observed there. This is the signature of self -biasing \nin the film and similar effect was previously observed in other ferroelectric thin film s as well. 30 \nSuch self -biased behavior co uld originate from various defects in the film such as oxygen \nvacancies. The se positively charged defects can trap charges and hence generate a local inherent \nelectric field. To compensate this internal field larger amount of electric field is need ed to \ncomplete the domain switching process.  Due to the presence of downward internal field the \ncoercive field becomes -33 V and +21 V on negative and positive side, respectively at complete \npolarization switching that translates to an average EC of ~ 1350 kV/cm . Here we note that, this \nhigh EC is in complete agreement to our  first-principles study14 and comparable to the value \nreported recently on GFO epitaxial thin film determined  at macroscopic P -E hysteresis \nmeasurement.15  \nIn summary, we have, for the first t ime, synthesized polycrystalline yet oriented GFO \nthin films with orthorhombic Pc2 1n symmetry using low cost chemical solution deposition \ntechnique. The film growth direction coincides with the direction of spontaneous polarization. \nThough bulk electrical measurement demonstrates an unsaturated P-E hysteresis loop, nano -scale \nmeasurement clearly demonstrates switching of polarization states at a high EC of ~1350 kV/cm, \nin agreement with recent reports .15 14 Magnetization measurement reveals a ferri  to paramagnetic \ntransition at ~300 K.  Saturated r oom temperature ferrimagnetic loop is observed  with weak \nmagnetization. Magnetization value can further be improved by tuning Ga: Fe ratio.8, 31 Thus, \npolycrystalline GFO  thin films become a potential alternati ve to other competing systems such as \nBiFeO 3 for designing commercial multiferroic devices. However, the leakage behavior of the \nfilms needs to be improved significantly in order to realize room temperature bulk ferroelectric \nhysteresis loop. In this regar d, aliovalent doping at Fe3+ site by Mg2+ has been proved beneficial \nas reported in epitaxially grown thin film samples.31    8 \n This work was supported  by Department of Science and Technology, Govt. of India under the \nINSPIRE -Faculty Award Program through Gr ant No. IFA13/MS -03. S. Mukherjee and M. \nMishra thank Prof. B. K. Mishra, Director of IMMT Bhubaneswar, for providing support for this \nwork. AR thanks IIT Bhubaneswar for research initiation grant through Project No. SP059 and \nCentral Instrumentation Facil ity at IIT Bhubaneswar for characterization facilities.  \nReferences:  \n1. A. Roy, R. Gupta and A. Garg, Advances in Condensed Matter Physics 2012 , 12 (2012).  \n2. J. F. Scott, Nat Mater 6 (4), 256 -257 (2007).  \n3. N. A. Hill, The Journal of Physical Chemistry B 104 (29), 6694 -6709 (2000).  \n4. G. Catalan and J. F. Scott, Advanced Materials 21 (24), 2463 -2485 (2009).  \n5. A. Roy, S. Mukherjee, R. Gupta, R. Prasad and A. Garg, Ferroelectrics 473 (1), 154 -170 (2014).  \n6. T. Arima, D. Higashiyama, Y. Kaneko, J. P. He, T. Goto, S. Miyasaka, T. Kimura, K. Oikawa, T. \nKamiyama, R. Kumai and Y. Tokura, Physical Review B 70 (6), 064426 (2004).  \n7. S. Mukherjee, A. Garg and R. Gupta, Journal of Physics: Condensed Matter 23 (44), 44 5403 \n(2011).  \n8. S. Mukherjee, V. Ranjan, R. Gupta and A. Garg, Solid State Communications 152 (13), 1181 -1185 \n(2012).  \n9. A. Roy, S. Mukherjee, R. Gupta, S. Auluck, R. Prasad and A. Garg, Journal of Physics: Condensed \nMatter 23 (32), 325902 (2011).  \n10. A. Roy, R. Prasad, S. Auluck and A. Garg, Journal of Applied Physics 111 (4), 043915 (2012).  \n11. V. B. Naik and R. Mahendiran, Journal of Applied Physics 106 (12), 123910 (2009).  \n12. Z. H. Sun, S. Dai, Y. L. Zhou, L. Z. Cao and Z. H. Chen, Thin Solid Films 516 (21), 7433 -7436 (2008).  \n13. D. Stoeffler, Journal of Physics: Condensed Matter 24 (18), 185502 (2012).  \n14. S. Mukherjee, A. Roy, S. Auluck, R. Prasad, R. Gupta and A. Garg, Physical Review Letters 111 (8), \n087601 (2013).  \n15. S. Song, H. M. Jang, N. -S. Lee , J. Y. Son, R. Gupta, A. Garg, J. Ratanapreechachai and J. F. Scott, \nNPG Asia Mater 8, e242 (2016).  \n16. M. Somdutta, Amritendu Roy, in NMDATM 2016  (IIT Kanpur, India, 2016).  \n17. A. Hammouda, A. Canizarès, P. Simon, A. Boughalout and M. Kechouane, Vibratio nal \nSpectroscopy 62, 217 -221 (2012).  \n18. J. Kennedy, P. P. Murmu, J. Leveneur, A. Markwitz and J. Futter, Applied Surface Science 367, 52-\n58 (2016).  \n19. K. Sharma, V. Raghavendra Reddy, A. Gupta, R. J. Choudhary, D. M. Phase and V. Ganesan, \nApplied Physics  Letters 102 (21), 212401 (2013).  \n20. T. C. Han, T. Y. Chen and Y. C. Lee, Applied Physics Letters 103 (23), 232405 (2013).  \n21. S. Kavita, V. R. Reddy, G. Ajay, A. Banerjee and A. M. Awasthi, Journal of Physics: Condensed \nMatter 25 (7), 076002 (2013).  \n22. V. Raghavendra Reddy, K. Sharma, A. Gupta and A. Banerjee, Journal of Magnetism and \nMagnetic Materials 362, 97-103 (2014).  \n23. W. Kim, J. H. We, S. J. Kim and C. S. Kim, Journal of Applied Physics 101 (9), 09M515 (2007).  \n24. Y.-H. Lee, J. -M. Wu and C. -H. Lai, Applied Physics Letters 88 (4), 042903 (2006).  \n25. Z. Quan, W. Liu, H. Hu, S. Xu, B. Sebo, G. Fang, M. Li and X. Zhao, Journal of Applied Physics 104 \n(8), 084106 (2008).  9 \n 26. A. Thomasson, S. Cherifi, C. Lefevre, F. Roulland, B. Gautier, D. Albertini, C . Meny and N. Viart, \nJournal of Applied Physics 113 (21), 214101 (2013).  \n27. J. F. Scott, Journal of Physics: Condensed Matter 20 (2), 021001 (2008).  \n28. H. Yoo, C. Bae, M. Kim, S. Hong, K. No, Y. Kim and H. Shin, Applied Physics Letters 103 (2), \n022902 (2 013).  \n29. S. V. Kalinin and D. A. Bonnell, Physical Review B 65 (12), 125408 (2002).  \n30. P. Miao, Y. Zhao, N. Luo, D. Zhao, A. Chen, Z. Sun, M. Guo, M. Zhu, H. Zhang and Q. Li, Scientific \nReports 6, 19965 (2016).  \n31. C. Lefevre, R. H. Shin, J. H. Lee, S. H. Oh, F. Roulland, A. Thomasson, E. Autissier, C. Meny, W. Jo \nand N. Viart, Applied Physics Letters 100 (26), 262904 (2012).  \n \nFigure captions:  \nFig.1 (Color online) (a)  (a) Indexed X-ray diffraction patterns of (0k0) oriented GFO  films  \ngrown on Pt/ Si (111)  and n-Si (100) substrates. Presence of a weak GFO (221) peak is \nmarked.  A few unidentified peaks originating from the substrates are marked with \nasterisks.  (b) Raman spectra of GFO  films  grown on Pt/ Si (111)  and n-Si (100) substrates. \nRaman scattering peaks originating  from Si substrate are marked with arrows in Raman \nspectrum of GFO on n -Si substrate.  \nFig.2 (Color online) FE -SEM images of GFO films deposited on (a) Pt/Si (111) and (b) n -Si \n(100) substrates. (c) Cross -sectional SEM image of GFO films deposited on Pt/Si (111), (d) \n3-D AFM image of GFO film deposited on Pt/Si (111)  \nFig. 3 (Color online) FC and ZFC magnetization at 500 Oe as a function of temperature on GFO \nfilms deposited on Pt/Si(111) and n -Si(100)  substrates. Inset shows room temperature \nmagnetization vs.  magnetic field for GFO grown on Pt/Si(111) substrate.  \nFig. 4(a) (Color online) Schematic of MIM capacitor geometry used for ferroelectric \nmeasurement, (b) Room temperature ferroelectric hysteresi s loop of GFO film deposited \non Pt/Si (111) substrate measured at 10 kHz; (c) -(f) Out of plane PFM images of \nAu/GFO/Pt capacitors at applied ±5 V.  \nFig. 5 (Color online) Piezoelectric switching spectroscopy measurement on a particular \npiezoelectric domain: piezoelectric amplitude and phase hysteresis loops upon reversal of \nseveral bias voltages.  10 \n  \n \n \n \n \n \n \n \n \nFigures  \n  \nFig. 1  \n \n \n11 \n  \n \n \n \n \n \n \n \n Fig. 2  \n \n \n \n \n12 \n  \n \n \n \n \n \nFig. 3  \n \n \n \n \n \n \n \n \n \n \n \n \n \n13 \n  \n \n \n \n \n \nFig. 4  \n \n \n14 \n  \n \n \n \n \n \n \n  \n \n \nFig. 5  \n \n \n \n \n \n \n \n \n \n \n \n15 \n  \n \n "    },    {        "title": "1701.06468v1.Size_Effects_of_Ferroelectric_and_Magnetoelectric_Properties_of_Semi_ellipsoidal_Bismuth_Ferrite_Nanoparticles.pdf",        "content": "`To be submit ted to Journal of Alloys an d Co mpound s  \n \nSize Effects of Ferroelectric a nd Magnetoe lectric Properties of Semi-ellipsoidal \nBismuth Ferrite Nanopartic les  \n \nVictoria V. Khist1, Eugene A. Eliseev2, Maya D. Glinchuk2, Max im V. Silibin3, Dmitry V. \nKarpins ky41, and Anna N. Moro zovska52 \n \n1Institute of Magnetism , National Ac adem y of Sciences of Ukraine and Ministry of Education and \nScience of Ukraine, \n2 Institute for Problem s of Mate rials Science, National Academy of Sciences of Ukraine,  \n3, Krjijanovskogo, 03142 Kyiv, Ukraine, \n3 National Research University of  Electronic Technology “MIET”,  \nMoscow, Zelenograd, R ussia \n4 Scientif ic-Practical Ma terials Research Centre of NAS of Belarus, Minsk, Belarus  \n5Institute of Physics, National Acad emy of Sciences of Ukraine,  \n46, pr. Nauky, 03028 Kyiv, Ukraine \n \nBism uth ferri te (BiFeO 3) is one of t he most prom ising m ultiferroics with a sufficiently  high ferroelectric \n(FE) and anti ferro magneti c transition t emperature s, and magnetoelectric  (ME)  coupling coefficient at room \ntemperature, and thus it is highl y sensiti ve to the im pact of cross-influence of applied electric and magnetic \nfields. According t o the urgent dem ands of nan otechnolog y miniaturization  for ultra-hi gh densit y data \nstorage in advanced nonvolatile memor y cells, it is very important to reduce the sizes of m ultiferro ic \nnanoparticles in the self-assem bled array s without serious  deterioration of their properties. We study size \neffects of the  phase diagrams, FE and ME propertie s of se mi-ellipsoidal BiFeO 3 nano particles clam ped to a \nrigid cond uctive substrate. The spatial distributio n of the sp ontaneous p olarization v ector inside the \nnanoparticles, phase diagrams and paramagnetoelect ric (P ME) coefficient wer e calculated in the fram ework  \nof m odified Landau-Ginzburg-Devons hire (LGD) appro ach. Analy tical expressions wer e derived for  the \ndependences of the FE tra nsition tem perature, averag e polarization, linear dielectric susceptibility  and PME  \ncoefficient on the particle sizes for a general cas e of a semi-ellipsoidal nanoparticles with  three differe nt \nsemi-axes a, b and height c. The analy ses of the obta ined results leads to the conclusion that the size effe ct of \nthe phase diagram s, spontaneous polarization and PME coe fficient is rather  sensitive to the particle si zes \naspect ratio i n the polarization direction, and less sen sitive to the absolute value s of the sizes per se.  \n \nKeywords:  multiferroic; structural antiferrodistortion;  antiferromagnetic order; nanoparticle; semi-\nellipsoidal; nano-island  \n                                                 \n1 corresponding author,  e-mail: dmitry.karpinsky @gmail.co m\n2 corresponding author,  e-mail: anna.n.morozovska@g mail.co m\n 11. INTRODUCTION \n1.1. Multiferroic BiFeO 3 for fundamental studies and advanced applications \nMultiferroics, which are ferro ics with two or more long-range  order param eters, are ideal \nsystem s for  funda mental studies of the couplings among the ferroelectric polarization, structural \nantiferrodistortion, and antiferrom agnetic order param eters [1, 2, 3, 4, 5]. These couplings are in \nresponse of  unique phy sical properties of m ultiferro ics [6]. For instan ce, biquadratic and linear  \nmagnetoelectric ( ME) couplings lead to intriguing effect s, such  as a gian t magnetoelectric \ntunability  of multiferroic s [7]. Biquadratic coup ling of the s tructura l and polar and die lectric  orde r \nparam eters, introduced in Refs. [8, 9, 10], are responsible for the unus ual behavior of the dielectric, \npolar and other physical properties in ferroelastics − quantum paraelectrics. The linear-quadratic  \nparam agnetoelectric ( PME) effect should exist in the param agnetic phase of ferroics, below the \ntemperature of the paraelectric-t o-ferroelectric phase tr ansition, where the elec tric polarization is \nnon-zero. T his effect was observed in NiSO 4⋅6H2O [11], Mn-doped SrTiO 3 [12], Pb(Fe 1/2Nb1/2)O3 \n[13, 14, 15], and Pb(Fe 1/2Nb1/2)O3- PbTiO 3 solid so lution [ 16]. Note tha t PME ef fect can be \nexpected in m any nanosized ferroics, which becom es param agnetic due to the size-induced \ntransition from  the ferrom agnetic or antiferrom agnetic phase. \nBiFeO 3 is the one of the m ost interesting  multiferroics with a strong ferro electric \npolarization,  antiferromagnetism  at room  temperat ure as well as enhanced electro transpo rt at \ndomain walls [17, 18, 19, 20, 21, 22]. Bulk BiFeO 3 exhibits antiferrodistortive (AFD) order at \ntemperatures below 1200K; it is f erroelectric (FE)  with a large spontaneous polarization below \n1100 K and is antiferrom agnetic (A FM) below Neel temperature TN ≈ 650 K [23, 24]. The  \npronounced multiferro ic propert ies maintain in BiFeO 3 thin film s and het erostructures [25, 26, 27, \n28, 29]. Despite extensive experim ental and theoretical studies of  the physical properties of bulk \nBiFeO 3 and its thin film s [21 - 23, 30, 31, 32, 33, 34, 35, 36, 37], m any im portant issues concerning \nthe em ergence and theoretical background  of multiferroic polar, m agnetic and various  \nelectrophysical properties of BiFeO 3 nanoparticles rem ain almost unexplored [38, 39]. \n1.2. Multiferroelectric nano particles. State-of-art \nHowever according to the urgent dem ands of  nanotechnology m iniatur ization for ultra-high \ndensity of data storage in advanced nonvolatile m emory cells, it is very im portant to reduce the \nsizes of m ultiferroic nan oparticles in the self-assem bled arrays without serious  deterioration of their \npolar, m agnetic and ME properties. There are m any intriguing and encouraging exam ples of the \npolar and dielectric properties conservation, enhancement and modification in ferroelectric \nnanoparticles. In particular Yadlovker and Berger [40, 41, 42] present the un expected experim ental \nresults, which reveal the enhancem ent of polar pr operties of cylindrical na noparticles of Rochelle \nsalt. Frey and Payne [43], Zhao et al [44] and Erdem  et al [45] de monstra te the p ossibility to contro l \n 2the tem perature of the ferroelectric phase tran sition, the m agnitude and  position of the dielectric \nconstant m aximum for BaTiO 3 and PbTiO 3 nanopowders and nanoceram ics. The studies of KTa 1-\nхNbхO3 nanopowders [46] and nanograin ceram ics [47, 48, 49] discover the appearance of new \npolar phases, the shift of phase transition tem perature in comparison with bul k crystals. Strong size \neffects in SrBi 2Ta2O9 nanoparticles have been revealed by in situ Raman scattering by Yu et al [50] \nand by therm al analysis and Ra man spectroscopy by Ke  et al [51]. The list of experimental studie s \nshould be continued, m aking any new experim ental-and-theoretical study of ferroelectric  \nnanoparticles im portant for both fundam ental science and advanced applications.  \nIn particular, the surface and finite size eff ects i mpact on the phase diagram s, pol ar and  \nelectrophysical prope rties of BiFeO 3 nanoparticles are very poorly studied [38, 39]. Such study may \nbe very useful for science and a dvanced applications, b ecause the theo ry of finite size effects in \nnanoparticles allows one to establ ish the physical origin of the pol ar and other phy sical properties  \nanom alies, transition temperature and phase diagram s changes appear ed with the nan oparticle s sizes \ndecrease. In particular, using th e continual phenom enological appro ach Niepce [52], Huang et al \n[53, 54], Ma [55], Eliseev et al [56] and Morozovska et al [57, 58, 59, 60, 61] have shown, that the \nchanges of the transitio n tem peratures, the en hancem ent or weakening of  polar properties in \nspherical and cylindrical nanoparticles are conditi oned by the various physical m echanisms, such as \ncorrelation effect, depolarization fi eld, flexoelectricity, electrostric tion, surface tension and Vega rd-\ntype chem ical pressure.  \n1.3. Research motivation \nNanoparticles of (se mi)ellipsoidal shape can be  considered as the m odel objects to study size \neffects on physical properties of ferroic nano-islands. BiFeO 3 nano-islands and their self-assem bled \narrays can be form ed on anisot ropic substrates by di fferent low-dam age fabrication m ethods [62, \n63, 64]. The pa rticles typically have different in -plane axes due to the anisotropic therm al \nconductiv ity of substrate. Recent advances in the production technolo gy of ferroelectrics have \nresulted in a cost-effective synthesis of these nanoparticles, which are beginning to be used in \nfabrication of m icroactuators, m icrowave phase shif ters, infrared sen sors, transistor applications, \nenergy harv esting dev ices etc. A correlation m echan ism between the scaling factor, geom etry of the \nnanoparticles and their physical param eters, and related phenom ena vi z. spontaneous polarization, \nantiferromagnetic and a ntiferrod istortive order, widt h of the dom ain walls and the d omains stab ility \nis needed to be further inves tigated using both experim ental methods and theoretical m odeling. \nMost intriguing fundam ental issues  to be addressed include an es timation of the intrinsic lim it for \npolarization stability, m echanism  of dom ain wall motion, and polarization sw itching in nanoscale  \nvolum es [1- 6]. \n 3The analyses of the above state-of-art m otivated us to study theoreti cally the size effects  \ninfluence on FE, AFM and ME  properties of sem i-ellipsoidal BiFeO 3 nanoparticles in the  \nframework of the Landau-Ginzburg-Devonshire (LGD) approach, classical electrostatics and \nelasticity th eory. \n2. THEORY \n2.1. Problem statemen t \nIt is known that ferroelectricity is a c ooperative phenom enon associ ated with the dipole \nmom ents aligned on both short- and long-scale level. This alignm ent is characterized by a certain \ntrans ition te mperature ju stified by th e tem peratur e-dependen t forces which rela te to th e size ef fects, \ndimension of the m aterial, its structural hom ogeneity  etc. It is considered that the s ize effects are \nassocia ted either with  intrins ic (m ainly related  to the a tomic polariz ation ) or extrinsic (stres ses, \nmicrostructure, polarization sc reening etc.) factors [57-60]. \nLet us  consider f erroelectric nanop articles  in the form of sem i-elliptica l islands p recipitate d \non the rigid conducting substrate elec trode. The ellipsoid has different  values of sem i-axis length, a, \nb and c along X-, Y- and Z-axis  respectively. We denote bε and eε as th e dielectric pe rmittivity of \nferroelectric background and ex ternal m edia respectively. The one-com ponent ferroelectric \npolarization , directed along the crystallograp hic axis 3 inside the particle , that is parallel to the \ninterface z= 0 [Fig. 1 ].  ()rP\n \n \nac\nεbεe\nPt P\nImage in the metal \n1 \n3 z=0 \n \nFIG. 1. A semi-ellipsoidal uniform ly polarized ferro electri c nanoparticle is clam ped to a rigid conducting \nsubstrate electrode (e.g.,  Pt). The o ne-com ponent ferroelectric polarization  is directed along x-axes . \nSemi-ellipsoi d height is c and lateral se mi-axes are a and b. ()rP\n \nWe can assum e that in the crystallographic fr ame {1, 2, 3} the dependence of the in-plane \ncomponents of electric polarizatio n on the inner field electric Ei is linear ( )i i\nb E P1 0 1 1−εε=  and \n( )i i\nb E P2 0 2 1−εε= , where an is otropic back ground perm ittivity is relatively sm all, 10 [65], ≤εi\nb 0ε is \n 4a universal dielectric constant . Polarization com ponent 3 contai ns background and soft mode  \ncontributions, () () ( )i i\nb E EP E P3 0 3 3 3 1 , , −εε+ =r r . Electric displacem ent vector has the for m \n insid e the partic le an d  outside it;  is th e rela tive dielectric \nperm ittivity  of externa l media. Hereinaf ter the subscr ipt \" i\" corresponds to the electric field or  \npotential inside the particle, \" e\" – ou tside the pa rticle. P E D + εε=ii\nbi\n0ee eE D εε=0eε\n Inhom ogene ous spatial distribution of th e ferroelectric polarization component ( )3 3,E Pr  \ncan be determ ined from  the Landau-Ginzburg-Devonshire  (LGD) type equations inside a \nnanoparticle,  \n3 3 332\n335\n33\n3 3 2 E P QxxPg P P Pkl kli\nn mmn P P P = σ −∂∂∂− γ+ β+ α ,                         (1) \nwhere th e coefficient () ( )CT\nP P TT T − α= α)(, T is  the absolute te mperature,   is the Cu rie \ntemperature of the paraelectric-to-ferroel ectric phase transition. The param eters βP, and γP are \ncoefficients of LGD pot ential expa nsion on the polarization powers. , and are respe ctively \nelastic stress and electrostriction stress tensor. Fle xoelectric effect is regarded as sm all. Boundary \nconditions for the polarization P3 at the particle s urface S ar e regard ed to be natural,\n CT\nklσijklQ\n( )03 = ∂∂SPn .    \n Electric field  is defined via electric potential as iEi i x E ∂ϕ∂−= . For a ferroele ctric-\ndiele ctric, th e elec tric po tential ϕ can be found self-consistently from the Laplace equ ation outsid e \nthe nanoparticle  and Poisson equation inside it  00 =ϕ∆εεee\nkk\nj ib\nijxP\nxx ∂∂=∂∂ϕ∂εε2\n0 ,                                                     (2) \nε0=8.85 ×10−12 F/m the d ielectric permittivity  of vacuum ,  is background perm ittivity [66]. Free \ncharges are regarded ab sent inside the particle. b\nijε\nCorresponding electric b oundary con ditions  of poten tial continuity at the m echanically-free \nparticle su rface S, ( ) 0= ϕ−ϕSi e . The boundary condition for the norm al com ponents of electric \ndisplacem ents should  take into acco unt the surface screening produced  by e. g. am bient free ch arges \nat the particle surface S,  () 00 =⎟\n⎠⎞⎜\n⎝⎛\nλϕε+ −\nSi\ni e nD D , where λ is the surface screening length. The \npotential is constan t at the pa rticle-electrode interface, i.e.  00= ϕ=zi . Surface screening leads to the \ndecrease of extern al field inside the particle, as well as to th e decr ease of bare depolarization field \ncaused by the polarization gradient. \n 5 Within a phenom enological approach, linear and biquadratic ME couplings contribution to \nthe sys tem free energy are described  by the term s  and , respectively ( P is \npolarization and M is magnetization , and ij i j PMµ ijkli j k l PP MM ξ   \nijµ  and ijklξ    are corresponding tens ors of ME effects, \nrespectively) [67, 68, 69, 70, 71]. The PME coupli ng contribution is described by the term \nk j i ijk MMP ,,,η  [67, 72]. Let us use the phenom enological LGD-based m odel for the PME  \ncoefficients calculation [72, 73]. Assum ing that the magnetization M is linearly proportional to the \napplied m agnetic field H, \n ()HT MFMχ≈ , the PME coefficient η has t he form [72 ]: \n() ()()() ( )MP M FE S T T TP T ξ χ χ −= η2.                                 (3) \nHere the spontaneous polarization ()TPS  is th e averaged over the particle volum e spontaneous \npolarization  ()r3P  calculated  from  Eq.(1) at H=0 an d E=0. Functions  and ()TMχ ()TFEχ  are \naveraged ov er the p article volum e linear m agnetic susceptib ility and die lectric sus ceptibility in the \nferroelectric phase, resp ectiv ely. Ferroelectric su sceptibility can be calcu lated from  Eq.(1) using the \ndefinition  \n()\n033\n3=∂∂= χ\nEFEEPT .                                                              (4) \nApproxim ate expression for m agnetic susceptib ility is taken the sam e as in Ref .[72]: \n()() ()TP L TT\nS MP LMT\nMM 2 2 )(0\nξ+ ξ+θ− αµ= χ .                                (5) \nEquations (3) and (4) are valid in the ferro electric-antiferrom agnetic phase (with nonzero \nantiferrom agnetic long-range order param eter 0≠L ) as well as in  the f erroele ctric – para magnetic  \nphase without any magnetic order M=L=0. Param eters LMξ and  the biquadratic  \nmagnetoelectric coefficients that couple polari zation and m agnetic order param eters in the  \nmagnetoelectric energy MPξ\n(2 2 2\n21PL M GLP MP ME ξ+ ξ= ). It should be not ed, th at only two co efficients  in \nmagnetic energy, () ()2 2\n04 2 4 2\n2 4 2 4 2ML HM M MTL LTGLM M M L L\nMξ+ µ−β+α+β+α= , are \nassum ed to be dependent on tem perature, namely ()(NT\nM L TT T − α= α)() an d \n where  is the m agnetic Curie tem perature, and  is the Neel temperature.  () ( )θ− α= α T TT\nM M)(θNT\n \n 62.2. Analytical approximation base d on finite element modeling \n Using finite elem ent modeling (FE M) we num erically calc ulated the s patial dis tribution \n[Fig.2(a) ] and average electric field inside the par ticles. F erroelectric param eters of  BiFeO 3 are \nlisted in Table I . \n \nTable I . Param eters of BiFeO 3 used in our calculations  \nParameter SI units Value for BiFeO 3 \nSpontaneous polarization PS m /C2 1 \nElectrostriction coefficient Q12 m4/C2 −0.05 \nElectrostriction coefficient Q11 m4/C2 −0.1 \nBackground perm ittivity  εb dimensionless 10 \nAmbient per mittivity εe  dimensionless 1 \nGradient co efficient g 11 m3/F 10−10 \nLGD coefficient aS m2/F  10−4 \nLGD coefficient b J m5/C4 107 \nLGD coefficient a m/F  -107 (at 300 K) \nFerroelectric Curie tem peratu re Tc K 1100 \nTemperature coefficient aT m/(K F) 0.9×106 \nAntiferrom agnetic Neel tem perature  K 650 \nSurface screening leng th λ nm 10-3 to 102 \nUniversal d ielectric cons tant ε0  F/m  8.85×10−12 \n \n Num erical results  were approxim ated analy tically. Obtained elec tric field dependence on the \nsurface screening leng th λ has the f ollowing f orm: \n( )\n( )cba Rncban PEX\ndX,,,,\n0 ∞∞\n+λλ\nε−≈                                              (6) \nHere  is the “bare” depolarization fact or of the system  without scr eening charges (the case of the \nlimit ).  is the characteristic length - scale, prop ortion al to the size a of island along the  \npolar axis. Using Eq.(6) we can introdu ce th e effective dep olarization factor (∞n\n∞→λ R\n()\nXdX\ndPEcban0 ,, ε−= ) \ndependent o n the sem i-ellipso id geo metry  as  \n()( )\n( )cba Rncbancband,,,,,,\n∞∞\n+λλ=                                                (7) \nHigh accuracy of approxim ation (7) becom es evident from  Fig. 2(b) . \nThe f itting with Eq.(7)  allows us to obtain param eters  and  for a se t of island  sizes a, b \nand c , which are th e lengths of  ellipso id semi-axis. These sets w ere fitted with the Pa de-\napproxim ations of the following form : ∞n R\n 7()\n⎟⎟⎟⎟\n⎠⎞\n⎜⎜⎜⎜\n⎝⎛\n++ +ε+ε≈∞\na bbaca cc\na bbcban\ne b\n075.07.0,,\n2 22\n,                               (8) \n() ⎟\n⎠⎞⎜\n⎝⎛+ + ≈ca\nbaacbaR 25.0 19.0 62.0 ,, .                                              (9) \nNote that th e pre-facto r a bb\ne bε+ε in Eq.(8) is the exact expressi on for depolarization factor of \nelliptical cy linder with  sem i-axes a and b. High accura cy of approxim ations (8)-(9 ) beco mes \nevident fro m Fig. 2(b)-(d) . \n \n \n1 2 5 10  20 500.001 0.002  0.005 0.010 0.020 0.050  0.100 \nSemi- axis length c ( nm)  Bare depol arization factor  \nb increase  \n(c) 0.001 0.01 0.1 1 10 10010-510-40.0010.010.1\nScreening l ength λ (nm) Depolarization factor nd \nc=1 nmc=2 nmc=5 nmc=10 nm  c=50 nm  \nc=20 nm  \n1 2 5 10  20 501050\n2030\n15\nSemi-axis le ngth c (nm)  Characteristic size R   (nm) \nb=1 nm  \nb=2 nm  \nb=5 nm  \nb=10 nm\nb=50 nm  b=20 nm  \n(d)(b)(a) \n \nFIG. 2. (a) Distribution of the electric field co mponent Ex in the sy stem calculated numeric ally. (b) \nDependence of depolarization factor  on the screening lengt h for  different values of sem i-axis length c \n(num bers near the curves) and fixed value of a=10 nm b=20 n m. Dependences  of parameters  (c) and  \n(d) on the semi-axis length c for fixed values of a=10 nm and b=1, 2, 5, 10, 20, 50 nm  (numbers near the Xn\n∞n R\n 8curves). Rel ative dielectric perm ittivities 10=εb  and eε=1. Sym bols are results of FEM calculations, solid \ncurves represent fitting wit h Eq.(6) -  (9), respectively . \n \n Allowing for expres sions (6)-(9), th e transition tem perature to paraelectric phase ( )cbaTcr ,, \ncan be defined from  the condition 0\n0=ε+αdn and given by analytical expression  \n()( )\n0,,,,εα− =\nTd\nC crcbanTcbaT .                                                    (10) \nEquation (10) allows to write analytical expres sions for the average spontaneous polarization, and \nlinea r dielectric suscep tibility by con ventiona l form \n()()\n⎪⎩⎪⎨⎧\n>< −βα\n=\n. ,0, , ,,\ncrcr crT\nS\nTTTT TcbaTP ,                           (11) \n()()()\n() ()⎪⎪\n⎩⎪⎪\n⎨⎧\n>− α<− α= χ\n. ,,,1, ,,, 21\ncr\ncr Tcr\ncr T\nFE\nTTcbaTTTTTcbaTT .                        (12) \nAllowing for Eqs.(3), (5), (11) and (12) the anal ytical expression for PME coefficient acquires  the \nform:  \n()() ( )\n()( )\n⎪\n⎩⎪\n⎨⎧\n><\n− βαχ ξ−\n= η\n. ,0, ,\n,, 22\ncrcr\ncr TM MP\nTTTT\nTcbaTT\nT .                          (13) \nIt follows from the obtained f ormula (10) – (1 3), that the  main peculiar ities of  the ellip soidal \nnanoparticles originate from  depol arization field contribution. \n \n3. RESULT S AND DIS CUSSION \n3.1. Si ze effects of phase diagram s and average polariz ation  \nPhase diag rams of sem i-ellipso idal BiFeO 3 nanoparticles in coordina tes relative temperature CTT\n  \n- length of the part icle longer sem i-axis a are shown in Figs.3(a)  and 3(b) ( is bulk Curie \ntemperature). The boundary between paraelectric (PE) and ferroelectric (FE) pha ses (that is in fact \nthe c ritical tem perature  of the size-induced  phase trans ition CT\n( )cbaTcr ,, ) depends on the sem i-\nellipsoid s izes a, b a nd c (m ultiple s ize e ffect). The size ef fect m anifested itse lf in the \nferroele ctricity disapp earance at th e critical size ()cbacr, for whi ch  followed, by the  \nmonotonic increase of the transition tem perature with the size  increase a nd its f urther satura tion 0=crT\na\n 9to  for the sizes  nm. Different cu rves are calculated  for several values of sem i-axis b= \n3, 10, 30 and 100 nm and fixed particle height c. Figure 3(a)  correspon ds to the par ticles of  small \nheight c=10 nm , and Fig.3(b)  for c=100 nm . The critical size CT 100>>a\n()cbacr, monotonically decreases as \nwell as the phase boundary between PE and FE phase shifts to the right with b increase at the sa me \nc values [com pare different cu rves in Figs.3(a)  and 3(b)]. Moreover, the critical sizes ()cbacr, \ncalcu lated  at c=10 nm  are essen tially smaller tha n the s izes c alculated at c=100 nm  at the sam e b-\nvalues [compare th e curves calculated in Figs.3(a)  and 3(b)]. In num bers, the PE- FE transitio n \nexists  for all values of chosen sizes . At с = 10 nm , the critical s ize  varies in  the n arrow  \nrange (10 – 12) nm, and the curves calculated for different b values are very close to one another. \nAt с = 100 nm , the critical size (cbacr,)\n()cbacr, varies in the wider range (15 – 45) nm, and the curves \ncalculated f or different b values are well-separated from  one another. \n Hence the analysis  of Figs. 3(a)  and 3(b) allows us to co nclude that  the size effect of the \nphase diag rams is sensitive to the  value of  the particle as pect r atio in  the polarization direction, \n2abc , and less s ensitive to the absolute values of  the sizes pe r se. The sm aller is the r atio, th e \nsmaller is  the depolariza tion f ield and hence the higher is the tran sition te mperature  and the  smaller \nis the critical size. The resu lt seem s nontrivial a priory.  \n The spontaneous polarization dependence on the length of ellipsoid sem i-axis a calculated  \nfor different values of sem i-axis c = 10 nm  and 100 nm  and room  temperature are shown in Figs.  \n3(c) and 3(d), respect ively. The values of another sem i-axis b are chosen  the sam e as in Figs. 3(a)  \nand 3(b) [see different curves calcu lated  for b= 3, 10, 30 and 100 nm]. The polarization curves \ncalculated for different b values are very close to one another at с = 10 nm, and are well-separated \nfrom  one another at с = 100 nm . The spontaneous polarizati on appears at the critical size ()cbacr, \nand increases with the s ize a increas e. The polarization satura tes to the b ulk value ~ 1 C/m2 at sizes \nnm. Note that the po larization of  the particle s with heigh t c = 10 nm  saturates  essen tially \nfaster th an the one f or the partic les with c = 100 nm  [com pare curves s aturation in Figs. 3(c)  and \n3(d)]. 100>>a\n \n 1010 20 50 100 200 50010000.00.20.40.60.81.0\nSemi-axis a (nm)Temperature T/Tc \n310 PE\n30 100\n(b)FE\nc = 100 nm\n10 20 50 100  200 500 10000.0  0.2  0.4  0.6  0.8  1.0  \nSemi-axis a (nm)Temperature T/Tc \n(a) 3 30 100 PE \n10 FE\nc = 10 nm\n10 20 50 100 200 500 10000.0  0.2  0.4  0.6  0.8  1.0  Polariz ation Ps (C/m2) \n330 100 \n10 c = 10 nm\nT=300 K\nSemi-axis a (nm) (c) \n10 20 50 100200 50010000.00.20.40.60.81.0 Polarization Ps (C/m2) \n330 100 \n10\nc = 100 nm\nT=300 K \n(d) Semi-a xis a (nm) \n \nFIG. 3. Phase diagrams in coordinates tem perature – length of ellipsoid sem i-axis a calculat ed for different \nvalues of se mi-axis c = 10 nm and 100 n m (panels  (a) and (b) respe ctively ) and axis b= 3,  10, 30 and  100 nm \n(see nu mbers near the curves). The spontaneous polari zation dependence on the lengt h of ellipsoid sem i-axis \na calculated  at roo m temperature for di fferent values  of se mi-axis  c= 10 n m and 100 nm  (panels (c) and (d) \nrespectively ) and axis b= 3, 10,  30 an d 100 nm (see num bers near the curves). Screening length λ=1 nm, \nother param eters corresponding t o BiFeO 3 com pound  are listed in Table I . \n \n3.2. Si ze effect of param agnetoelectric coefficient \nThe dependences of  PME ef fect coef ficient on the length of  ellip soid se mi-axis a calculated  \nat room  temperatu re for diffe rent values of sem i-axis c = 10 nm  and 100 nm  are shown in Figs. 4(a)  \nand 4(b), respectively . The values of another sem i-axis b are chosen th e sam e as in the  previous \n 11figures  [see different cu rves calculated for b= 3, 10, 30 and 100 nm]. PME coefficient is norm alized \non its bu lk value. The  PME coef ficient is z ero at siz es ()cbaacr, <  because of spontaneous  \npolarization  disappearance, it appears at craa< and diverges at the critical size ()cbaacr, = , and \nthen it dec reases with th e size a increase. The PME coefficient satu rates to the bulk value at size s \nnm. The divergences at 100>>a ()cbaacr, =  dem onstrate the pos sibility  to ob tain g iant PME \neffect in BiFeO 3 nanoparticles in th e vicin ity of size-induc ed trans ition from  the FE phase to a PE \nphase. In p articula r the norm alized PME coeff icient is es sentially h igher than u nity f or siz es \n. The behavior of PME coefficient repr oduces the behavior of the dielectric  \nsusceptibility given by Eq.(12) in th e fram ework of our m odel. Note that the PME co efficient of the \nparticles with height c = 10 nm  saturates essen tially faster than the on e for the p articles  with c = 100 \nnm [com pare curves saturation in Figs. 4(a)  and 4(b)]. The PME coefficient curves  calcu lated for \ndifferent b values are very close to one another at с = 10 nm , and ar e well-separated from  one \nanother at с = 100 nm . () (cba acbacr cr , 2 , <≤ )\n The com parative analys es of the Figs.3(c)-(d)  and Figs.4(a)-(b)  approves our conclusion \nthat the size effect of the spont aneous polarization and PME coeffici ent is sensitive to the particle \naspect ratio in the polarization direction, 2abc , and le ss sensitive to the abso lute value s of the \nsizes. \n \n10 20 50 100  200  500012345\nSemi -axis a (nm) (b)PME coefficient  \n3\n1030 100 c = 100 nm\nT=300 K \n10 20 50 100 200 5000 1 2 3 4 5 \nSemi -axis a (nm) (a) PME coefficient  \n3 \n30 100 c = 10 nm\nT=300 K \n10 \n \nFIG. 4.  Normalized paramagnetoelect ric (PME) effect coefficient depende nce on length of ellipsoid semi-\naxis a calculated for different values o f semi-axis c= 10 n m and 100 nm  (pan els (a) and (b), respective ly). \nDifferent cur ves at each plot correspond to the different value of  the axis b= 3, 10 , 30 and 100 nm  (indicated \nby num bers near the curves). Screening length λ=1 nm , room temperature, other param eters correspondin g to \nBiFeO 3 compound  are listed in Table I . \n 12 \n4. CONCL USIONS \nWe have studied the size effects on the phase di agram s, ferroelectric and  magnetoelectric properties \nof sem i-ellipsoidal BiF eO3 nanoparticles clamped to a rigid conductive substrate. The spatial \ndistribution of the spontaneous polarization vector  inside the ferroelectric nanoparticles, phase \ndiagram s and param agnetoelectric coefficient were  calculated in the fram ework of the Landau -\nGinzburg-D evonshire approach, clas sical electrostatics and elasti city theory. Rather rigorous \nanalytical expressions were derived for the depende nces o f the ferroelectric transition tem perature, \naverage polarization, linear diel ectric susceptibility and param agnetoelectric coefficient on the  \nparticle sizes for a general case  of a sem i-ellipso idal n anoparticle s with three d ifferent sem i-axes  a, \nb and height  c. Due to the essential decrease  of depolarization field the in-plane orientation of the \nspontaneous polarization along the longer sem i-ellipsoidal axis a is ene rgetic ally p referable f or the \nnanoparticles of sm all height c<a. The analyses of the obtained resu lts leads to  the conclus ion that \nthe size effect on the phase diagrams, spontaneous  polarization and param agnetoelectric coefficient \nis rather sen sitive to th e particle s izes aspect ratio in th e polarization direction, 2abc , and less \nsensitive to the absolute values of the sizes per se. Th is opens the way to gove rn the properties by \nthe choice o f this ra tio value.  \n \nASKNOWLEDGMENTS. M.V.S. and D.V.K acknowledge  MK-1720.2017.8, BRFFI (# F16R-\n066) RFFI (# 16-58-00082). E. A.E. and A.N.M. acknowledge  the Center for Nanophase Materials \nSciences, w hich is a DOE Office of  Science User Facility, C NMS2016-061. \n \nAuthors contribution. A.N.M. generated the research idea, states the m athem atical problem  and \nwrote the m anuscrip t draft. V.V.K. with E.A.E. assis tance p erform ed num erical simulations. E.A.E. \nand A.N.M. derived analytical expressions. M. D.G. worked on the results inte rpretation and \nmanuscript im prove ment. M.V.S. and D.A.K. valu ably contribu ted to  the research ide a and \nmanuscript improvem ent. \n \nREFE RENCES \n                                                 \n[1] M. Fiebig, Revival of the magnetoelectric effect,  Journal of P hysics D: Applied Ph ysics, 3 8 (200 5) 123- \n152. \n[2] N. A. Spal din and M. 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Box MG-6, R-76900 Bucharest, Romania\ndSchool of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China\nePaul Scherrer Institute, Villigen, Switzerland\nAbstract\nKinetics of niobium and titanium carbide precipitates in iron has been simulated with cluster dynamics.\nThe simulations, carried out in austenite and ferrite for niobium carbides, respectively in austenite for\ntitanium carbide, were analyzed for dependency on temperature, solute concentration and initial cluster\ndistribution. The results are presented for di\u000berent temperatures and solute concentrations and compared\nto available experimental data. They show little impact of initial cluster distribution beyond a certain\nrelaxation time and that highly dilute alloys with only monomers present a signi\fcantly di\u000berent behavior\nthan less dilute alloys or alloys with di\u000berent initial cluster distribution.\nKeywords: cluster dynamics, precipitates, precipitation kinetics, titanium carbide, niobium carbide\n1. Introduction\nAddition of titanium and niobium to steels\nin metallurgy is conducive to titanium/niobium\ncarbide precipitates in solid solution due to their\ncombination with the carbon present in the steel.\nThis process limits the formation of chromium\ncarbides, thereby preventing intergranular\ncorrosion [1, 2]. Additionally, \fnely dispersed\ncarbide precipitates increase alloy strengths both\nat low and high temperature [2, 3]. Nuclear-\ngrade steels are required to meet higher, additional\nstandards, and the question of precipitate dynamics\nis raised with respect to them acting as point defect\nrecombination centers and sinks for helium, which\nreduces void swelling and helium embrittlement\n[4, 5, 6, 7, 8]. TiC/NbC precipitates tend to\nstabilize dislocation networks, and hence enhance\ncreep resistance [9, 10]. Titanium carbides are in\nparticular attractive in this respect, which led to\nthe development of 15-15Ti steel in the 1970's [11]\nfor nuclear reactor applications. This steel exhibits\n\u0003Corresponding authors at: Institute of Modern Physics,\nChinese Academy of Sciences, Lanzhou, China.\nE-mail addresses: nadezhda kv@bk.ru (N. Korepanova),\ngulong@impcas.ac.cn (L. Gu).excellent resistance to irradiation swelling and\ncreep and has been chosen as structural material\nfor several Generation-IV designs.\nTo simulate the precipitation behavior of carbides\nwe used cluster dynamics (CD), which is an e\u000bective\nmethod to predict microstructural evolution in a\nmaterial - with little computing overhead for long\ntime period simulations. In this method polyatomic\nclusters embedded in the solid solution exchange\nsolute atoms by absorption or emission. Time\nevolution of cluster distributions is computed from\ndi\u000berential equations coupled through monomer\nexchanges.\nIn the case of iron CD has yielded good results\nfor Cu precipitates in ferrite [12] and MnSiNi\nprecipitates in ferrite-martensitic steel [14].\nFor NbC and TiC precipitates in steel classical\nkinetic nucleation theory (CNT) has been used\n[15, 22, 16, 17], TiC precipitate modelling focusing\nmore on Precipitation-Temperature-Time (PTT)\ndiagrams, rather than on time-evolution of mean\nradius, volume fraction, and number density.\nThis paper is organized as follows: Section 2 gives\na brief description of the CD method; Section 3\npresents our simulation results for niobium carbides\nin austenite and ferrite and for titanium carbide in\naustenite; Section 3 presents a comparison of our\nPreprint submitted to Elsevier October 5, 2020arXiv:2005.10574v2  [cond-mat.mtrl-sci]  2 Oct 2020results to existing experimental data; and Section 4\nsummarises our study.\n2. Methodology\nIn cluster dynamics alloys are treated as binary\nsystems of an alloy matrix with clusters of solute\natoms. Clusters grow or shrink through the\nabsorption, respectively emission of solute atoms.\nTime evolution of solute clusters is dictated by a\nset of di\u000berential equations (1-2) which assume only\nmonomers to be mobile. Small cluster mobility\nmay result from common drift of monomers, an\nassumption reasonable in dilute alloys [18].\ndCn\ndt=\fn\u00001C1Cn\u00001\u0000(\u000bn+\fnC1)Cn+\n+\u000bn+1Cn+1; n6= 1;(1)\ndC1\ndt=\u00002\f1C1C1+\u000b2C2+\n+NmaxX\ni=2[\u000bi\u0000\fiC1]Ci;(2)\nwherenis the cluster size, Nmax the maximal\ncluster size, Cnthe uniform concentration of size n\nclusters,C1the concentration of monomers, \u000bnthe\nrate of monomer emission from size nclusters,\fn\nthe rate of monomer absorption by size nclusters,\nthe latter two calculated as:\n\fn= 4\u0019rnD=Vm\nat; (3)\n\u000bn=\fn\u00001Csol\neqexp\"\nA\u0002\n\u001bn2=3\u0000\u001b(n\u00001)2=3\u0003\nkT#\n;\n(4)\nwhereVm\natis the atomic volume of the alloy-\nmatrix,rnthe radius of size nclusters,Dthe\nthermal di\u000busion coe\u000ecient of solute atoms in the\nsystem,Athe geometrical factor, \u001bthe interfacial\nenergy between precipitates and matrix, Csol\neqthe\nequilibrium concentration of solute atoms in the\nsystem,Tthe temperature in degrees Kelvin, and\nkthe Boltzmann constant. The radius of size n\nclusters is:\nrn=\u00123nVp\nat\n4\u0019\u00131=3\n: (5)CD is a computationally e\u000ecient method,\nhowever with increasing Nmax&100 it can\nbecome CPU wise prohibitive. A traditional way\nto overcome this is to transform the di\u000berential\nequations into a Fokker-Planck partial di\u000berential\nequation [19, 20]:\n@Cn\n@t=\u0000@\n@n[(\fnC1\u0000\u000bn)Cn] +\n+1\n2@2\n@n2[(\fnC1+\u000bn)Cn];(6)\nThe discretization of the Fokker-Planck equation\nusing the central di\u000berence method brings Eq.6 into\nthe following form:\ndCnj\ndt=Cnj\u00001\nnj+1\u0000nj\u00001\u0002\n\u0002\u0014\n(\fnj\u00001C1\u0000\u000bnj\u00001) +\fnj\u00001C1+\u000bnj\u00001\nnj\u0000nj\u00001\u0015\n\u0000\n\u0000Cnj\nnj+1\u0000nj\u00001(\fnjC1+\u000bnj)\u0002\n\u0002\u00141\nnj+1\u0000nj+1\nnj\u0000nj\u00001\u0015\n+\n+Cnj+1\nnj+1\u0000nj\u00001\u0002\n\u0002\u0014\n\u0000(\fnj+1C1\u0000\u000bnj+1) +\fnj+1C1+\u000bnj+1\nnj+1\u0000nj\u0015\n;\n(7)\nand the evolution of monomer concentration to:\ndC1\ndt=\u00002\f1C1C1+\u000b2C2+NtrX\nj=2[\u000bj\u0000\fjC1]Cj+\n+NmaxX\nj>N tr[\u000bj\u0000\fjC1]Cjnj+1\u0000nj\u00001\n2:(8)\nwherenjis de\fned as follows:\n(\nnj=j;8j\u0014Ntr;\nnj=Ntr+1\u0000\u0015j\u0000Ntr\n1\u0000\u0015;8j >Ntr;(9)\nThe above system is reduced to the initial\ndi\u000berential equations for nj=j. This numerical\nscheme does not strictly conserve matter (as do\ndi\u000berential equations (1) and (2)), however under\ncarefully de\fned circumstances the looses are small\nand acceptable. To solve the above system we used\nin our study the ODEINT solver [21].\n2In this study we assume the di\u000busion coe\u000ecient\nof titanium/niobium carbide to be determined by\nthe most resistive element, i.e. - we used the\nTi and Nb di\u000busion coe\u000ecients in the simulation,\ncorrespondingly.\nReferences [34] and [16] cite the \\pipe-\ndi\u000busion\" e\u000bect for TiC/NbC precipitation\nkinetics, respectively a faster solute di\u000busion along\ndislocations than in the lattice in general. We\nincluded the e\u000bect of dislocations in the model, in\nEq.3, as a modi\fed e\u000bective di\u000busivity [16]:\nDeff=Ddisl\u0019R2\ncore\u001a+Dbulk(1\u0000\u0019R2\ncore\u001a);(10)\nwhereDdisl is the di\u000busion along dislocations\n(equal toDbulk\u000bdisl, with\u000bdisla correction factor\nde\fned according to [23] and presented in Table 2),\nDbulkthe bulk di\u000busivity, Rcorethe radius of the\ndislocation core, and \u001athe dislocation density.\nFig. 1 illustrates the e\u000bect of dislocations on\ndi\u000busivity and how this changes with temperature.\nThe \fgure shows the ratio of e\u000bective di\u000busivity to\nbulk di\u000busivity in austenite steel for several values\nof dislocation densities. The ratio increases with\ndislocation density increase and drops sharply with\nincreasing temperature.\n1011,m-2\n1012,m-2\n1013,m-2\n1014,m-2\n600 800 1000 1200 14000246810\nTemperature,oCDeff/Dbulk\nFigure 1: E\u000bect of dislocation on di\u000busivity - the ratio of\ne\u000bective di\u000busivity to bulk di\u000busivity in austenite steel for\nseveral values of dislocation densities.\nThe results of the simulation are the time-\nevolution of the mean radius, the \u0016 rvolume fraction,\nfv, and the number density of precipitates, Ntot-\ncalculated with the following equations:\n•Mean radius\n\u0016r=\u00123\u0016nVp\nat\n4\u0019\u00131=3\n; (11)whereVp\natis the atomic volume of precipitate,\nand \u0016nthe mean size of precipitate clusters:\n\u0016n=PNmax\njcutnjCnj\u0001njPNmax\njcutCnj\u0001nj; (12)\nwith \u0001nj=nj\u0000nj\u00001,jcuttaken such\nthatrjcut= 1nmfor TEM data (given the\nresolution limit in [24, 25], and rjcut= 0:5nm\nfor SANS data).\n•Volume fraction\nfv=Vp\nat\nVmat\natNmaxX\njcutnjCnj\u0001nj\u0002100%:(13)\n•Number density\nNtot(t) =1\nVmat\natNmaxX\njcutCnj(t)\u0001nj; (14)\nwithVmat\natthe atomic volume of the matrix.\nIn our study we used an initial cluster\ndistribution described by:\n8\n><\n>:C1=xC0;\nCn=(100\u0000x)C0\nnPM\nn=1\u0001n;2\u0014n\u0014M\nCn= 0; n>M(15)\nwhereC0is the concentration of the alloying\nelement in steel, xthe part of the alloying element\nin monomer form, Mthe maximal cluster size\nassumed to exist in the steel at moment t= 0.\nIn the next section, we show the dependence of CD\nresults on the initial state of the system.\n3. Results and Discussion\nIn this section, we present the results of our\nCD simulations for NbC and TiC in ferrite and\naustenite and compare them with experimental\ndata from literature. The parameters used in the\nsimulations are shown in Tables 1 and 2. Table 1\ndisplays the material parameters for TiC, NbC\nand iron matrix and Table 2 gives references to\nexperimental data and the conditions under which\nthey was obtained.\n3Table 1: Material parameters for Titanium and Niobium carbides and for the Iron matrix.\nParameter Symbol Units Value Reference\nTiC\nLattice parameter a nm 0.433 [26]\n\rFe\nDi\u000busion coe\u000ecient DTim2=s 0:15\u000110\u00004exp [\u0000251000=RT] [27]\nInterfacial energy \u001b J=m20.2 [28]\nsolubility limit log [ MC] 2 :97\u00006780=T [29]\nNbC\nLattice parameter a nm 0.445 [30]\n\rFe\nDi\u000busion coe\u000ecient DNbm2=s 0:75\u000110\u00004exp [\u0000264000=RT] [31]\nInterfacial energy \u001b J=m21:0058 \u00000:4493 \u000110\u00003T(Ko) [32]\nsolubility limit log [ MC] 2 :06\u00006770=T\n\u000bFe\nDi\u000busion coe\u000ecient DNbm2=s 1:27\u000110\u00005exp [\u0000224000=RT] [17]\nInterfacial energy \u001b J=m20.5 [17]\nsolubility limit log [ MC] 5 :43\u000010960=T [17]\nMatrix\nLattice parameter a\rFe nm 0.358\nLattice parameter a\u000bFe nm 0.287\nCorrection factort \u000bdisl(\rFe) 0 :643 exp(118700 =(R\u0003T)) [23]\nCorrection factor \u000bdisl(\u000bFe) 0 :0133 exp(115000 =(R\u0003T)) [23]\nTable 2: Experimental datasets and the concentrations and temperatures at which they were measured.\nReference Ti/Nb, wt% C, wt% Temperatures,oC Matrix Dislocation density, m\u00002\nTiC\n[25] 0.31 0.1 T=750 \rFe 6\u00011014(from [25])\n[33, 34] 0.1 0.05 T=925 \rFe 3:24\u00011013(calc. from [34])\n[28] 0.4 0.07 T=900 \rFe { (assumed 6 \u00011014)\n[36] 0.25 0.05 T=900 \rFe { (assumed 6 \u00011014)\nNbC\n[24] 0.031/0.095 0.1/0.1 T=900/950 \rFe { (assumed 1011)\n[35, 17] 0.040/0.079 0.0058/0.01 T=600/700/800 \u000bFe 2\u00011014(from [35, 17])\n3.1. Niobium Carbide\nFigures 2 and 3 show the dependence on the\ninitial cluster distribution for the time-evolution of\nprecipitate mean radius and number density. We\nassessed this in order to check the sensitivity of our\nsimulations to the initial state of the system. The\ninitial cluster distributions for our simulations are\ndescribed by Eq.15. Note in the \fgures that we\nindex the radius rMbyM, the maximal size of a\ncluster initially existing in the system. The cluster\ndistributions we used were both described by Eq.15,\nas well as arbitrary distributions: Poisson-like, or\nstep-function. The simulation results are however\nthe same for all distributions, warranting our use\nof Eq.15 described initial distributions throughout\nour study.\nFigure 2 shows the time-evolution of precipitate\nmean radius and number density in 4 distinct\ncases for which we varied the concentrations of\nmonomers and other clusters (see Eq.15), while\nkeepingrMconstant. Complementary, Figure 3presents the time-evolution of the mean radius\nand number density in relation to rM, for the\nsame correspoinding monomer concentrations. As\ncomparison Figure 3 shows the simulation only for\nmonomers.\nAs evident in Figures 2 { 3 the initial cluster\ndistributions play a role only in the initial\ndeparture time (with the notable exception of\nthe 0.031wt%Nb-steel simulation, with monomers\nand very small clusters). After 1000 s the e\u000bect\nof the initial cluster distribution washes out and\nall simulations become indistinguishable from one\nanother. The exception case of 0.031wt%Nb-\nsteel with monomers and very small clusters is\nshown in Fig.3. Its anomalous behaviour was also\nobserved for TiC in \rFe and NbC in \u000bFe. A\npossible explanation could be that in dilute alloys\nthere are few precipitation centers, which have the\nopportunity to grow faster than in the case of higher\nsolute concentrations. The amount of such clusters\nremains however small (see upper Fig.3). Had we\n41020102110221023Numberdensity, m-3\n100%monomers\n90%monomers\n80%monomers\n25%monomers\n0.1 1 10 100 1000 10402.×10-94.×10-96.×10-98.×10-91.×10-81.2×10-81.4×10-8\nTime,sMeandiameter,m\nFigure 2: Dependence of simulation results on initial cluster\ndistributions. T=950oC, C(wt%Nb)=0.095.\nintroduced higher clusters in the initial distribution,\nthe precipitate kinetic would have followed the\nstandard behaviour.\nComparing our simulations (Fig.4) with\nexperimental data for niobium carbide precipitates\nin austenite [24] we \fnd that in 0.031wt%Nb-steel\n90% of niobium should exist as monomers - because\nif we would accept the 100% assertion of [24], the\nsimulations would contradict the experimental\ndata. We assume1that the remaining 10% are\ndistributed in clusters with rM<1nm is invisible\nto TEM.\nFor niobium carbide precipitates in ferrite the\nsimulation results and experimental data [17, 35]\nare depicted in Fig.5. Our model with the\nset of parameters from Table 1 matches quite\nwell the experimental data of [17, 35], less to\nsome extent the volume fraction, predicting faster\nclustering of the precipitates than experimentally\nobserved. Fig.5 also shows better agreement with\nexperimental data at low temperatures versus high\n1The exact dislocation density for the steel used by\nHansen et al. [24] is not available. Hence, we adjusted this\nparameter to agree with the experimental data. The result\nshows the dislocation density of steel to be in the range\n1011\u00001012m\u00002. This concurs with the mention that the\nsteel was well annealed [24].temperatures. This could be due to the higher\nenergy available at high temperature that activates\nthe di\u000busion of small clusters along with monomers,\nwhereas in our model only the monomers are\nmobile.\n3.2. Titanium Carbide\nSimulation results for TiC precipitates in\naustenite and experimental data from [25, 28,\n33, 36] are presented in Figures 6 { 8. Fig.6\ndisplays the time evolution of mean particle\ndiameter - CD predicting a particle diameter /\nt1=3, regardless concentration, temperature, and\ndislocation density (except in 0.1wt%Ti-steel {\n/t1=2). Experimental data on the other hand\nexhibits two regions: an initial region, with mean\ndiameter is proportional to time exponent with\nfactor 0.5-1.0; and a secondary region with the time\nexponent having a very low factor, practically a\nplateau. Although in [36] the secondary region is\nconsidered/t1=3(re\recting the Ostwald ripening\nphenomenon controlled by bulk di\u000busion of Ti), the\nauthors believe the time exponent has a much lower\nfactor (likely due to a di\u000berent phenomenon). The\ntwo regions are also clearly observed in the size\ndistributions.\nThe size distributions from experimental data\n[36] are shown in Fig.7. Although our model\npredicts a smaller mean diameter, the overall\ndistribution shape of the sizes is strikingly similar\nto experimental one (up to 3610 s, after which the\ndistributions start to di\u000ber). Additionally, in the\ninserts in Fig. 7 the experimental and simulated\nsize distributions are plotted such that both have\nthe same mean diameter and maximal magnitude.\nThe comparison of size distributions relative to the\nexperiment [25] at 750oC(Fig.8) shows a similar\nchange in size distribution. The small di\u000berence\nobserved might suggest a competing mechanism\ncontroling the growth of TiC precipitates.\nIn [25] it was mentioned that the pinning\nof mobile dislocation a\u000bects TiC precipitate\nkinetics in the temperature range 650 { 900oC.\nHowever, CD applies only di\u000busion-controlled\ngrowth of precipitate clusters. To overcome\nthis obstacle, a model for time-evolution of\nmobile dislocation density should be introduced\nalongside with the dependency of di\u000busion on\nmobile dislocation density. The veri\fcation of\nthis suggestion is postponed to further study.\nNote that the data itself is very scarce for TiC\nprecipitations. We assess that there is a necessity\n51020102110221023Numberdensity, m-3\n100%monomers\n90%monomers,rM=1.25nm\n90%monomers,rM=1nm\n90%monomers,rM=0.5nm\n90%monomers,rM=0.25nm\n0.1 1 10 100 1000 10402.×10-94.×10-96.×10-98.×10-91.×10-81.2×10-81.4×10-8\nTime,sMeandiameter,m\n1091012101510181021Numberdensity, m-3\n100%monomers\n90%monomers,rM=1.25nm\n90%monomers,rM=1nm\n90%monomers,rM=0.6nm\n90%monomers,rM=0.25nm\n0.1 1 10 100 1000 10402.×10-94.×10-96.×10-98.×10-91.×10-81.2×10-81.4×10-8\nTime,sMeandiameter,m\nFigure 3: Dependence of simulation results on initial cluster distributions and radius rM. T=950oC, C(wt%Nb)=0.095 (left\ngraph), C(wt%Nb)=0.031(right graph).\n●●●\n■■■\n◆◆◆\n▲▲▲▲0.031Nb,900C\n0.031Nb,950C\n0.095Nb,900C\n0.095Nb,950C●0.031Nb,900C\n■0.031Nb,950C\n◆0.095Nb,900C\n▲0.095Nb,950C\n0.1 1 10 100 1000 10402.×10-94.×10-96.×10-98.×10-91.×10-81.2×10-81.4×10-8\nTime,sMeandiameter,m\nFigure 4: Comparison of simulation results with\nexperimental data for NbC precipitates in austenitic stainless\nsteel. The dots represent the experimental data of [24].\nfor more experimental and theoretical work on TiC\nprecipitates.\n4. Conclusion\nIn the present paper we applied cluster dynamics\nto model precipitation kinetics of niobium and\ntitanium carbides in steels. The kinetic of NbC\nprecipitates have been simulated for ferritic and\naustenitic iron matrices. Our simulation results\nare in agreement with experimental data. We\nhave analyzed the results for dependency on initialcluster distribution, where we considered various\ntypes of distributions and monomers concentration.\nThe analysis has shown that the initial distribution\nplays a role only in the initial-time range. After an\ninitialt-time all simulations follow the same pattern.\nThe analysis has also shown the \\special\" behavior\nof precipitates if only monomers are present in\nvery dilute alloys: the fast growth of mean particle\ndiameter, while number density remains small.\nWe have suggested that in dilute alloy fewer\nprecipitation centers are created and they have less\ncompetition, which allows those centers to grow\nfaster.\nFor TiC on the other hand, the simulation results\nand experimental data di\u000ber more pronounced.\nThis could be explained by another controlling\nmechanism besides di\u000busion. Such mechanism\ncould be mobile dislocation and its pining, which\nwas suggested in [25]. We believe, that more\nexperimental and theoretical work is needed\nto correctly model titanium carbide precipitates\nkinetics.\nAcknowledgments\nN. Korepanova is grateful for the CAS-TWAS\nPresident's Fellowship Programme for this doctoral\nfellowship (2016CTF004).\n6●●●●●●\n■■■■■■\n◆◆◆◆0.079Nb,600C\n0.079Nb,700C\n0.079Nb,800C\n●0.079Nb,600C\n■0.079Nb,700C\n◆0.079Nb,800C\n02.×10-94.×10-96.×10-98.×10-91.×10-8Meanradius,m\n●●●●●●■ ■ ■\n■\n■\n■◆◆\n◆\n◆\n1018101910201021102210231024Numberdensity, m-3\n●●●●●●\n■■■■■■■\n◆◆◆◆◆◆\n◆\n0.1 10 1000 1050.000.020.040.060.080.10\nTime,sVolumefraction, %\n●●●●●●\n■■■■\n◆◆◆◆0.040Nb,600C\n0.040Nb,700C\n0.040Nb,800C\n●0.040Nb,600C\n■0.040Nb,700C\n◆0.040Nb,800C\n02.×10-94.×10-96.×10-98.×10-91.×10-8Meanradius,m\n●\n●●●● ■■\n■\n■◆◆\n1018101910201021102210231024Numberdensity,m\n●●●●●●\n■■■■■■■\n◆◆◆◆◆\n0.1 10 1000 1050.000.020.040.060.080.10\nTime,sVolumefraction, %\nFigure 5: Simulation results for C(wt%Nb)=0.079 (left) and 0.040 (right). The dots represent experimental data of [35, 17].\nFor C(wt%Nb)=0.040 at T=800oCthe simulated number density and volume fraction of precipitates is too small and, therefore\ninvisible in the \fgures.\n0.31Ti, 750 C\n0.1Ti, 925 C\n0.4Ti, 900 C\n0.25Ti, 900 C\n0.31Ti, 750 C\n0.1Ti, 925 C\n0.4Ti, 900 C\n0.25Ti, 900 C\n0.1 10 1000 10510705.×10-81.×10-71.5×10-7\nTime, sMean diameter, m\n0.31Ti, 750 C\n0.1Ti, 925 C\n0.4Ti, 900 C\n0.25Ti, 900 C\n10 1000 1051071.×10-95.×10-91.×10-85.×10-81.×10-75.×10-7\nTime, sMean diameter, mt1/2\nt1/3\nt1/5\nt1/10\na) b)\nFigure 6: Comparison of simulation results with experimental data for TiC precipitates in austenitic stainless steel. The dots\nrepresent the experimental data of [25, 28, 33, 36].\n7calculation\nt=90sexperiment\n0 20 40 60 800.00.10.20.30.40.5\n0.00.10.20.30.40.5\nParticle diameter,nmRelative frequency\nDistribution, arb.units\ncalculation\nt=320sexperiment\n0 20 40 60 800.00.10.20.30.40.5\n0.00.10.20.30.40.5\nParticle diameter,nmRelative frequency\nDistribution, arb.units\ncalculation\nt=610sexperiment\n0 20 40 60 800.00.10.20.30.40.5\n0.00.10.20.30.40.5\nParticle diameter,nmRelative frequency\nDistribution, arb.units\ncalculation\nt=910sexperiment\n0 20 40 60 800.00.10.20.30.40.5\n0.00.10.20.30.40.5\nParticle diameter,nmRelative frequency\nDistribution, arb.units\ncalculation\nt=3610sexperiment\n0 20 40 60 800.00.10.20.30.40.5\n0.00.10.20.30.40.5\nParticle diameter,nmRelative frequency\nDistribution, arb.units\nFigure 7: Size distributions from experiment [36] and simulation for di\u000berent times at 900oC. The inserts inside the graphs\ndisplay the experimental and calculated distributions shifted such that both have the same mean particle diameter. The\ndot-dashed vertical line represents mean diameter.\ncalculation\nt=72sexperiment\n0 2 4 6 810 12 14050100150200250300\n0.0123456\nParticle diameter,nmNumber of precipitates\nDistribution, arb.units\ncalculation\nt=720sexperiment\n0 2 4 6 810 12 14050100150200250300\n0.00.20.40.680.811.2\nParticle diameter,nmNumber of precipitates\nDistribution, arb.units\ncalculation\nt=7200sexperiment\n0 2 4 6 810 12 14050100150200250300\n0.00.10.20.30.40.50.6\nParticle diameter,nmNumber of precipitates\nDistribution, arb.units\nFigure 8: Size distributions from experiment [25] and simulation for di\u000berent times at 750oC. The inserts inside the graphs\ndisplay the experimental and calculated distributions shifted such that both have the same mean particle diameter. The\ndot-dashed vertical line represents mean diameter.\nReferences\n[1] T. Thorvaldsson and G. L. Dunlop. Precipitation\nreactions in Ti-stabilized austenitic stainless steel.\nMetal Science , November:513{518, 1980.\n[2] Thomas Sourmail. Simultaneous Precipitation\nReactions in Creep-Resistant Austenitic Stainless\nSteels . PhD thesis, University of Cambridge, 2002.\n[3] Hao Jie Kong and Chain Tsuan Liu. A review on nano-\nscale precipitation in steels. Technologies , 6, 2018.\n[4] K. Herschbach, W. Schneider, and K. Ehrlich. E\u000bects\nof minor alloying elements upon swelling and in-pile\ncreep in model plain Fe-15Cr-15Ni stainless steels and\nin commercial DIN 1.4970 alloys. Journal of Nuclear\nMaterials , 203:233{248, 1993.\n[5] W. Kesternich and J. Rothaut. Reduction of helium\nembrittlement in stainless steel by \fnely dispersed\nTiC precipitates. Journal of Nuclear Materials , 103-\n104:845{852, 1981.\n[6] W. Kesternich. A possible solution of the problem of\nhelium embrittlement. Journal of Nuclear Materials ,\n127:153{160, 1985.\n[7] W. Kesternich and D. Meertens. Microstructuralevolution of a titanium-stabilized 15Cr-15Ni steel. Acta\nMetallurgia , 34(6):1071{1082, 1986.\n[8] Takayoshi Kimoto and Haruki Shiraishi. Dose\ndependence of void swelling and precipitation behavior\nin MC carbide dispersed austenitic Fe-Ni-Cr alloys.\nJournal of Nuclear Materials , 141-143:754{757, 1986.\n[9] Niels Cautaerts, Remi Delville, Wolfgang Dietz, and\nMarc Verwerft. Thermal creep properties of Ti-\nstabilized DIN 1.4970 (15-15Ti) austenitic stainless\nsteel pressurized cladding tubes. Journal of Nuclear\nMaterials , 493:154{167, 2017.\n[10] N. Cautaerts, R. Delville, E. Stergar, D. Schryvers,\nand M. Verwerft. Tailoring the TiC nanoprecipitate\npopulation and microstructure of titanium stabilized\naustenitic steels. Journal of Nuclear Materials ,\n507:177{187, 2018.\n[11] J. S\u0013 eran and M. Le Flem. Structural Materials for\nGeneration IV Nuclear Reactors , chapter Irradiation-\nresistant austenitic steels as core materials for\nGeneration IV nuclear reactors, page 285. Number\n106 in Woodhead Publishing Series in Energy. Elsevier,\n2016.\n[12] F. Christien and A. Barbu. Modelling of copper\nprecipitation in iron during thermal aging and\n8irradiation. Journal of Nuclear Materials , 324:90{96,\n2004.\n[13] Emmanuel Clouet, Alain Barbu, Ludovic La\u0013 e, and\nGeorges Martin. Precipitation kinetics of Al3Zr\nandAl3Scin aluminum alloys modeled with cluster\ndynamics. Acta Materialia , 53:2313{2325, 2005.\n[14] Jia-Hong Ke, Huibin Ke, G. Robert Odette, and\nDane Morgan. Cluster dynamics modeling of Mn-\nNi-Si precipitates in ferritic-martensitic steel under\nirradiation. Journal of Nuclear Materials , 498:83{88,\n2018.\n[15] Zhenqiang Wang, Qilong Yong, Xinjun Sun,\nZhigang Yang, Zhaodong Li, Chi Zhang, and\nYuqing Weng. An Analytical Model for the Kinetics of\nStrain-induced Precipitation in Titanium Micro-alloyed\nSteels. ISIJ International , 52:1661{1669, 2012.\n[16] B. Dutta, E. J. Palmiere, and C. M. Sellars. Modelling\nthe kinetics of strain induced precipitation in Nb\nmicroalloyed steels. Acta materialia , 49:785{794, 2001.\n[17] F. Perrard, Alexis Deschamps, and Philippe Maugis.\nModelling the precipitation of nbc on dislocations in\n\u000b-Fe. Acta Materialia , 55(4):1255{1266, 2007.\n[18] M.H. Mathon, A. Barbu, F. Dunstetter, F. Maury,\nN. Lorenzelli, and C.H. de Novion. Experimental study\nand modelling of copper precipitation under electron\nirradiation in dilute fecu binary alloys. Journal of\nNuclear Materials , 245:224{237, 1997.\n[19] N.M. Ghoniem and S. Sharafat A numerical solution\nto Fokker-Plank equation describing the evolution of\nthe interstitial loop microstructure during irradiation.\nJournal of Nuclear Materials , 92:121{135, 1980.\n[20] M.F. Wehner and W.G. Wolfer. Vacancy cluster\nevolution in metals under irradiation. Philosophical\nMagazine A , 52:189{205, 1985.\n[21] Karsten Ahnert and Mario Mulansky. Odeint {\nsolving ordinary di\u000berential equations in C++. AIP\nConference Proceeding , 1389:1586, 2011.\n[22] Zhenqiang Wang, Qilong Yong, Xinjun Sun, Zhigang\nYang, Zhaodong Li, Chi Zhang, , and Yuqing Weng.\nAn analytical model for the kinetics of strain-induced\nprecipitation in titanium micro-alloyed steels. ISIJ\nInternational , 52(9):1661{1669, 2012.\n[23] https://www.matcalc.at/wiki/doku.php?id=techpapers:\nprecipitation:di\u000busion.\n[24] S. S. Hansen, J. B. Vander Sande, and Morris\nCohen. Niobium carbonitride precipitation and\naustenite recrystallization in hot-rolled microalloyed\nsteels. Metallurgical Transactions A , 11A:387{402,\n1980.\n[25] W. Kesternich. Dislocation-controlled precipitation\nof TiC particles and their resistance to coarsening.\nPhilosophical Magazine A , 52(4):533{548, 1985.\n[26] Jae Hoon Jang, Chang-Hoon Lee, Yoon-Uk Heo, and\nDong-Woo Suh. Stability of (Ti,M)C (M = Nb, V,\nMo and W) carbide in steels using \frst-principles\ncalculations. Acta Materialia , 60:208{217, 2012.\n[27] Sheldon H. Moll and R. E. Ogilvie. Solubility\nand di\u000busion of titanium in iron. Transactions of\nthe American Institute of Mining and Metallurgical\nEngineers , 215:613{618, 1959.\n[28] Asa Gustafson. Coarsening of TiC in austenitic stainless\nsteel | experiments and simulations in comparison.\nMaterials Science and Engineering: A , 287(1):52 { 58,\n2000.\n[29] Lucla F. S. Dumitresc and Mats Hillert. Reassessmentofthe solubility of tic and tin in fe. ISIJ International ,\n39(1):84{90, 1999.\n[30] Heilong Zou and J.s. Kirkaldy. Carbonitride\nprecipitate growth in titanium/niobium microalloyed\nsteels. Metallurgical Transactions A , 22A:1511{1524,\n1991.\n[31] S. Kurokawa, J. E. Ruzzante, A. M. Hey, and\nF. Dyment. Di\u000busion of Nb in Fe and Fe alloys. Metal\nScience , 17:433, 1983.\n[32] Q.L. Yong, Y.F. Li, and Z.B. Sun. Theoretical\ncalculation of speci\fc interfacial energy of semicoherent\ninterface between microalloy carbonitrides and\naustenite. Acta Metallurgica Sinica , 2:153{156, 1989.\n[33] Zhenqiang Wang, Xinjun Sun, Zhigang Yang, Qilong\nYong, Chi Zhang, Zhaodong Li and Yuqing Weng.\nCarbide precipitation in austenite of a Ti{Mo-\ncontaining low-carbon steel during stress relaxation.\nMaterials Science and Engineering A , 573:84{91, 2013.\n[34] Zhenqiang Wang, Han Zhang, Chunhuan Guo, Zhe\nLeng, Zhigang Yang, Xinjun Sun, Chunfa Yao,\nZhengyan Zhang, and Fengchun Jiang. Evolution of (Ti,\nMo)c particles in austenite of a Ti{Mo-bearing steel.\nMaterials and Design ,109:361{366, 2016.\n[35] F. Perrard, A. Deschamps, F. Bley, P. Donnadieu,\nand P. Maugisb. A small-angle neutron scattering\nstudy of \fne-scale NbC precipitation kinetics in the \u000b-\nFe{Nb{C system. Journal of Applied Crystallography ,\n39:473{482, 2006.\n[36] W.J.Liu and J.J. Jonas. Ti(CN) Precipitation\nin Microalloyed Austenite during Stress Relaxation.\nMetallurgical Transactions A , 19A:1415{1424, 1988.\n9"    },    {        "title": "2004.07006v2.Synthesis_optimization_of_Zn_Mn_ferrites_for_magnetic_fluid_aplications.pdf",        "content": "   \n    Universidade de Aveiro   \nAno 201 9 Departamento de Física  \nAndré Filipe  \nCastanheira Horta  \n \n \n \n Otimização da s íntese de f errites para aplicações em \nfluidos magnéticos  \n \nSynthesis optimization of Zn -Mn ferrite s for magnetic \nfluid  aplication s \n  \n   \n \n \n \n \n \n \n \n  \n \n \n \n    \n   \n \n \n Universidade de Aveiro  \n2019 Departamento de Física  \nAndré Filipe  \nCastanheira Horta  \n Otimização da s íntese de f errites para aplicações em \nfluidos magnéticos  \n \nSynthesis optimization of Zn -Mn ferrite for magnetic \nfluids  aplication s  \n  \n \nDisserta ção apresentada à Universidade de Aveiro para cumprimento dos \nrequisitos necessários à obtenção do grau de Mestre em Engenharia Física, \nrealizada sob a orientação científica d o Dr. João Horta, investigador júnior do \nIFIMUP na Faculdade de Ciências da Unive rsidade do Porto, e co -orientação d o \nDr. João Amaral, investigador  auxiliar  do CICECO n a Universidade de Aveiro . \n \n   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n    \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n “In a world of magnets and mira cles”  \n-Pink Floyd in “High Hopes”     \n \n \n \n \n \n \n \n \n \n \n \n \n \n      \n \n \n  \n \no júri    \n \npresidente  Professor Auxiliar Dr. An tónio  Ferreira da  Cunha  \n Professor Auxiliar  do Departamento de Física  na Universidade de Aveiro  \n \norientador  Investigador  Júnior  Dr. João Filipe Horta Belo da Silva  \nInvestigador  Júnior  do IFIMUP  na Faculdade de Ciências da Universidade do Porto  \n \nco-orien tador   \nInvestigador Auxiliar Dr. João  Cunha de Sequeira  Amaral  \nInvestigador  Auxiliar  do CICECO na Universidade de Aveiro  \n  \n \narguente  Professor Auxiliar Dr. André Miguel Trindade Pereira  \n Professor Auxiliar  do IFIMUP  na Faculdade de Ciências da Universid ade do Porto  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n       \n \n \n \n  \n  \n  \nAcknowledgements  \n The delivery  of my master  thesis  is certainly one of the most important  mark s of \nmy life. This achievement  would have not been possible without the support  of \neveryone  who I have met in  this path.  To all of them I am eternally grate ful. \nTo my supervisor, Dr. João Horta, who embraced this project with me . Thank \nyou for the wise thesis proposal, thank you for all the support, motivation and  \nopportunities during this year, thank for the scolding  that made me better, thank \nyou for all you taught me . During this year, you were not only my supervisor but \nalso a teacher and a friend. Thank you.  \nTo my co -supervisor, Dr. João Amaral. When I first met you , you said was “do \nnot count with me, I’m very busy”, t urns out that I always could count with you. \nI’m thankful for all the advices, corrections and opportunities that you provided .  \nTo PhD student Farzin Mohseni and Dr. Carlos Amorim, with who I had the \npleasure to work. Thank for your expertise advices, pro mptitude and time spent  \nwith me.  To Prof. Nuno Silva, for the availability when using his magnetic \ninduction heating setup and for the useful lessons. To Prof. Vitor Amaral for the \ninterest in the proje ct and wise suggestions. T o Rosário Soares , XRD \ntechni cian, for helping in XRD matters . To Marta Ferro, TEM and SEM \ntechnician, for these techniques support and expertise.  \nI also have to  big thank s my family, my partner  and my friends. You were my \nsupport and strength during this year, the previous and the next. Thank you.   \n \n \n \n \n \n \n \n \n \n \n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n    \n \n \n \n \n  \npalavras -chave  \n Aquecimento Autorregulado, Ferrite de Mn -Zn, method de auto -combustão de sol -gel, \nmétodo hidrotermal, temperatura de Curie, temperatura de ordenamento magnético  \nresumo  \n \n Nanopartículas de ferrite Manganês -Zinco são cada vez  mais investiga das pelas  suas \npropriedades  desejadas  para uma vasta gama de  aplicações. Essas propriedades incluem  \ncontrolo nanométrico de  tamanho  de partícula , propriedades magnéticas ajustáve is, elevada \nmagnetização de saturação e baix a toxicidade, providenciando  estas ferrites com os \nrequerimentos necessários para tratamento d e cancro  por hipertermia magnética. Durante \nesta investigação , foram sintetizados  e caracterizados  pós de ferrite de M n-Zn, visando \notimizar as suas propriedades estruturais e magnéticas para futura aplicação num ferroflu ido. \nAmostras de Mn 1-xZnxFe2O4 (x=0; 0.5; 0.8; 1) foram sintetizad as pelos métodos de \nautocombustão  de sol -gel e  pelo método hidrotermal. Os pós sintetiz ados foram \ncaracterizados por XRD, SQUID, SEM, TEM e aquecimento por indução magnética. Os \ndifratogramas  de XRD das amostras  produzidas por hidrotermal apresentam a estrutura \ncristalina  de espinela com elevada percentagem de fase-única  (>88%). O refinament o de \nRietveld e a análise de Williamson -Hall revelam decréscimos no parâmetro de rede (8 .50 até \n8.46 Å) e no tamanho médio de cristalite (61 até 11 nm) com o aumento da razão Zn/Mn. As \nimagens de TEM revelam uma estreita distribuição de tamanhos e um decré scimo do \ntamanho  médio  de partícula (41 até 7 nm) com o aumento da razão Zn/Mn. Os resultados \nde SQUID mostram que o aumento de Zn resulta num decréscimo d e magnetização de \nsaturação (79 até 19 emu/g) e de magnetização remanente (5 até aproximadamente 0 \nemu/g). Notoriamente , as curvas M(T) revelaram um desvio na temperatu ra de ordenamento \nmagnético para mais baixas temperaturas com o aumento  de Zn , de ~556 ( estimado ) até \n~284 K. A experiência  de aquecimento por indução também revel ou um decréscimo na taxa \nde aquecimento com o aumento de Zn na ferrite.  \nNano -cristais de ferrite de Mn -Zn produzidos pelo método  hidrotermal apresentam melhor \ncristalinidade e propriedades magnéticas que as amostras de autocombustão de sol -gel. As \namostras sintetizadas pelo método  hidrotermal revelam dependência das suas propriedades \nestruturais e magnéticas com a razão Zn/Mn. A tem peratura de ordenamento magn ético \ndestas ferrites pode ser usada  como um mecanismo de aquecimento autorregulado , \nelevando estas ferrites para uma classe de materiais inteligentes.     \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n     \n \nkeywords  \n Self-Regulated Heating, Mn-Zn ferrite, sol -gel auto -combustion method, hydrot hermal \nmethod, Curie temperature, m agnetic ordering temperature  \nAbstract  \n \n \n \n Manganese -Zinc ferrite nanoparticles have been the subject of increasing research \ndue to their desired properties for a wide range of applications. These properties \ninclude nanometer particle size control, tunable magnetic properties, high \nsaturation magne tization and l ow toxicity, providing these ferrites with the \nnecessary requirements for cancer treatment via magnetic hyperthermia. During \nthis research, powders of Mn -Zn ferrite were synthe sized and characterized, aiming \nto optimize their structural and m agnetic proper ties for further application in a \nferrofluid.  \nSamples of Mn 1-xZnxFe2O4 (x=0; 0.5; 0.8; 1) were synthe sized via the sol -gel auto -\ncombustion and hydrothermal methods. Synthe sized powders were characterized \nby XRD, SQUID, SEM, TEM and magnetic induction heat ing techniques . The XRD \ndiffractograms of hydrothermally produced samples presented spinel crystal \nstructure with high single -phase percentage (>88%). Rietveld refinement and \nWilliamson -Hall analysis revealed  a decrease of lattice co nstant (8 .50 to 8 .46 Å)  \nand crystallite size (61 to 11 nm) with increase of Zn/Mn ratio. TEM images reveals \nnarrow particle size distributions and decrease of the mean particle size (41 to 7 \nnm) with the Zn/Mn ratio increase. SQUID results showed that the increase of Zn \nresults in a decrease of saturation magnetization (79 to 19 emu/g) and remnant \nmagnetization (5 to approximately 0 emu/g). More noticeably, the M(T) curves \npresent  a shift in the samples magnetic ordering temperature towards lower \ntemperatur es with the increase of  Zn content, from ~556 (estimated) to ~284 K. \nThe magnetic induction heating experiment also unveiled  a decrease in the heating \nrate with the increase of Zn in ferrite.  \nNanocrystals of Mn -Zn ferrite produced by hydrothermal method p resent better \ncrystalli nity and magnetic properties than the sol -gel auto -combustion samples. \nThe hydrothermally synthe sized samples revealed dependence of its structural and \nmagnetic properties with Mn/Zn ratio. The magnetic ordering temperature of these \nferrites can be used as  a self -controlled mechanism of heating, r aising these \nferrites to a class of smart materials . \n \n    \n \n \n \n André Horta  \nUnive rsidade de Aveiro   i Content:  \nContent:  ................................ ................................ ................................ ................................ ................ i \nList of Figures:  ................................ ................................ ................................ ................................ .... iii \nList of Tables:  ................................ ................................ ................................ ................................ .... vii \nList of Abbreviations:  ................................ ................................ ................................ ..........................  ix \nList of Nomenclatures:  ................................ ................................ ................................ ........................  xi \nSymbols list:  ................................ ................................ ................................ ................................ ..... xiii \nChapter 1:  Introdu ction  ................................ ................................ ................................ ...................  1 \n1.1 Motivation and Objectives  ................................ ................................ ................................ . 1 \n1.2 Thesis Layout  ................................ ................................ ................................ ....................  2 \nChapter 2:  Magne tic Behavior  ................................ ................................ ................................ ........  3 \n2.1 Magnetic Properties  ................................ ................................ ................................ ..........  3 \n2.1.1  Magnetization  ................................ ................................ ................................ ...............  3 \n2.1.2  Magnetic  Susceptibility  ................................ ................................ ................................ . 4 \n2.1.3  Exchange and Superexchange Interaction  ................................ ................................ .. 4 \n2.1.4  Magnetocrystalline Anisotropy  ................................ ................................ ......................  4 \n2.1.5  Dipolar Interactions  ................................ ................................ ................................ ....... 5 \n2.2 Macr o magnetism ................................ ................................ ................................ ..............  5 \n2.2.1  Diamagnetism  ................................ ................................ ................................ ...............  5 \n2.2.2  Paramagnetism  ................................ ................................ ................................ .............  6 \n2.2.3  Ferromagnetism  ................................ ................................ ................................ ............  6 \n2.2.3.1  Hysteresis Loops  ................................ ................................ ................................ . 7 \n2.2.3.2  Temperature Effect  ................................ ................................ ..............................  7 \n2.2.4  Antiferromagnetism ................................ ................................ ................................ ....... 8 \n2.2.5  Ferrimagnetism  ................................ ................................ ................................ .............  8 \n2.3 Nano Magne tism ................................ ................................ ................................ ...............  9 \n2.3.1  Single Domain  ................................ ................................ ................................ ..............  9 \n2.3.1.1  Size and Shape Effects  ................................ ................................ ......................  10 \n2.3.1.2  Temperature Effects  ................................ ................................ ..........................  10 \n2.4 Magnetic Induction Heating Mechanisms  ................................ ................................ ....... 11 \n2.4.1 Eddy Currents  ................................ ................................ ................................ .............  11 \n2.4.2  Hysteresis Power Loss  ................................ ................................ ...............................  12 \n2.4.3  Néel-Brown Relaxation  ................................ ................................ ...............................  12 \nChapte r 3: Mn-Zn Ferrite  ................................ ................................ ................................ ..............  13 \n3.1 Synthesis of Mn -Zn ferrite  ................................ ................................ ...............................  13 \n3.2 Crystal Structure  ................................ ................................ ................................ .............  14 \n3.1 Magnetic Structure  ................................ ................................ ................................ ..........  14 \nChapter 4:  Experimental Procedure  ................................ ................................ .............................  16 \n4.1 Sample Synthesis  ................................ ................................ ................................ ...........  16 \n4.1.1  Sol-Gel Auto -Combustion Method  ................................ ................................ ..............  16 \n4.1.1.1  Samples  ................................ ................................ ................................ .............  17 \n4.1.2  Hydrothermal Method  ................................ ................................ ................................ . 17 \n4.1.2.1  Samples  ................................ ................................ ................................ .............  18 \n4.2 Experimental Techniques  ................................ ................................ ...............................  18 \n4.2.1  X-Ray Diffraction (XRD) ................................ ................................ ..............................  18 \n4.2.1.1  Sampl e Preparation  ................................ ................................ ...........................  19 \n4.2.1.1  Phase Identification  ................................ ................................ ............................  19 \n4.2.1.1  Lattice Constant  ................................ ................................ ................................ . 19 \n4.2.1.1  Other Results from Rietveld Refinement  ................................ ...........................  20 \n4.2.1.2  Crystallite Size and Strain  ................................ ................................ ..................  20 \n4.2.1.3  Systematic Errors  ................................ ................................ ...............................  21 \n4.2.2  Scanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy (EDS)  . 22 \n4.2.2.1  Scanning Electron Microscopy (SEM)  ................................ ...............................  22 \n4.2.2.2  Sample Preparation  ................................ ................................ ...........................  22 \n4.2.2.3  Size Distribution Measurements  ................................ ................................ ........  23 \n4.2.3  Energy Dis persive Spectroscopy (EDS)  ................................ ................................ ..... 23 Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nii  Departamento de Física  4.2.3.1  Systematic Errors  ................................ ................................ ...............................  23 \n4.2.4  EDS Analysis  ................................ ................................ ................................ ..............  23 \n4.2.5  Transmission Electron Microscopy (TEM)  ................................ ................................ .. 24 \n4.2.5.1  Sample Preparation  ................................ ................................ ...........................  24 \n4.2.5. 2 Size Distribution  ................................ ................................ ................................ . 24 \n4.2.6  Superconducting Quantum Interference Device (SQUID)  ................................ ..........  24 \n4.2.6.1  Sample Preparation  ................................ ................................ ...........................  25 \n4.2.6.1 SQUID Data Analysis  ................................ ................................ .........................  26 \n4.2.6.2  Systematic Errors  ................................ ................................ ...............................  27 \n4.2.7  Magnetic Induction Heating  ................................ ................................ ........................  27 \n4.2.7.1  Sample Preparation  ................................ ................................ ...........................  28 \n4.2.7. 2 Data Analysis  ................................ ................................ ................................ ..... 28 \nChapter 5:  Results and Discussion  ................................ ................................ ..............................  29 \n5.1 Results of Sol -Gel Auto -combustion Method  ................................ ................................ .. 29 \n5.1.1  Structural Characterization  ................................ ................................ .........................  29 \n5.1.2  Morphological and Chemical Characterization  ................................ ...........................  30 \n5.1.3  Magnetic Characterization  ................................ ................................ ..........................  31 \n5.1.4  Particle Size Characterization  ................................ ................................ .....................  32 \n5.2 Discussion of Sol -Gel Auto -Combustion Method  ................................ ............................  32 \n5.3 Results Of Hydrothermal Method  ................................ ................................ ....................  33 \n5.3.1  Structural characterization  ................................ ................................ ..........................  33 \n5.3.2  Morphological and Chemical Characterization  ................................ ...........................  35 \n5.3.3  Magnetic Characterization  ................................ ................................ ..........................  35 \n5.3.4  Particle Size Characterization  ................................ ................................ .....................  39 \n5.3.5  Heat Generation  ................................ ................................ ................................ .........  39 \n5.4 Discussio n of Hydrothermal Samples  ................................ ................................ .............  40 \n5.4.1  Compositional Analysis  ................................ ................................ ...............................  43 \nChapter 6:  Conclusions  ................................ ................................ ................................ ................  44 \n6.1 Future Work ................................ ................................ ................................ .....................  45 \nReferences  ................................ ................................ ................................ ................................ ....... 46 \nAppendix A: Autoclave Time  ................................ ................................ ................................ ............  49 \nAppendix B: Magnetic Ordering Temperature  ................................ ................................ ..................  50 \n André Horta  \nUnive rsidade de Aveiro   iii List of Figures:  \nFigure 2.1 – Schematic repr esentation exchange interactions in a ferromagnet: a) direct exchange. \nb) superexchange.  ................................ ................................ ................................ ......................  4 \nFigure 2.2 - Diamagnetism; a) Field lines in diamagnetic material; b) magnetization  dependence of \nan external applied field.  ................................ ................................ ................................ ............  5 \nFigure 2.3 – Param agnetism; a) Schematic representation of magnetic dipoles in a paramagnet; b) \nMagnetization dependence with the applied field; c) Susceptibility and inverse susceptibility \ndependence with temperature.  ................................ ................................ ................................ ... 6 \nFigure 2.4 – a) Ferromagnetic unit cell. b) domain -wall dividing two antiparallel domains c) \nferromag netic domain observed by Kerr -effect microscope (author: C. V. Zureks) . d) \nferromagnetic M(H) curves: M S -saturation magnetization, M R - remant field, H C - coercive field.\n ................................ ................................ ................................ ................................ ....................  7 \nFigure 2.5 – Magnetic susceptibility and inverse susceptibility of a ferromagnetic material.  .............  8 \nFigure 2.6 – a) Unit cell of antiferr omagnetic moment s. b) Antiferromagnetic domains. c) \nMagnetization increases with external field. d) Susceptibility and inverse susceptibility as \nfunction of temperature, the maximum value is the Néel temperature.  ................................ ..... 8 \nFigure 2.7 – a) H C and M R dependence with the particle size and single domain regimes. b) chain of \nmagnetic nanop articles of Fe 3O4 found inside a magnetotactic  bacteria (draw by Marta Puebla).\n ................................ ................................ ................................ ................................ ....................  9 \nFigure 2.8 – Single domain representations. a) Stoner –Wohlfarth model. B EXT is the  externally \napplied magnetic field direction. M is the magnetization vector rotated φ from the applied field.  \nƟ is the angle between an easy axis and B EXT. b) and c) is the representation of atomic spin \ncanting at the nanoparticle surface.  ................................ ................................ .........................  10 \nFigure 2.9 – a) Dead layer increase with temperature increase . b) M(T) measurement of single \ndomains and T B distribution, f(T B) (point dashed). 𝐓𝐁 and T Irr are signed in the M(T) curve.  .. 11 \nFigure 3.1 – a) Normal spinel crystal structure of Mn 1-xZnxFe2O4. b) te trahedral (B) and octahedral \n(A) sites in Mn -Zn ferrite. Atoms and site captions are in the right side of each image. Vesta \nsoftware was used for unit cell 3D rendering.  ................................ ................................ ..........  14 \nFigure 3.2 –Magnetic structure of normal spinel Mn 1-xZnxFe2O4. Tetrahedral and octahedral magnetic \nmoments are represented by vector aligned antiparallelly in <111>. The unit cell schematic \ndisplays a vector length equal for tetrahedral and octa hedral sites, which is only true for \nMnFe 2O4. ................................ ................................ ................................ ................................ .. 15 \nFigure 4.1 – Sol-gel auto -combustion method - experimental scheme. ................................ ...........  16 \nFigure 4.2 – Hydrothermal method - experimental scheme.  ................................ ............................  17 \nFigure 4.3 – a) X-Ray diffractometer and components. b) Bragg -Brentano geometry: 1 – Cathodic \nrays’ tube; 2 – collimator; 3 and 4 – Slits; 5 - Sample holder; 6 – Goniometer; 7 – Beam knife; 8 \n– Slit; 9 – Collimator and Nickel filter; 10 – X-Ray photodetector.  ................................ ...........  18 \nFigure 4.4 – Bragg law: X -ray diffraction schematic. ................................ ................................ ........  18 \nFigure 4.5 – Rietveld refinement of: a) MnFe 2O4 (sol-gel) and b) Mn 0.2Zn0.8Fe2O4 (hydrothermal). \nAsterisks identifies Fe 3O4 phase. Red dots are for experimental data, black line is the Rietveld \nrefinement, blue line is the difference between data and fitting and green lines identify the peaks \nposition. C hi2 is the goodness of fitting.  ................................ ................................ ...................  19 \nFigure 4.6 – a) Peak fitting using Origin. b) Williamson -Hall linear fit.  ................................ .............  21 \nFigure 4.7 – (a) Diffractogram of LiB 6 with inset for K -α1 and K - α2 distinguished. (b) Linear fit of \nWilliamson -Hall equation.  ................................ ................................ ................................ .........  21 \nFigure 4.8 - a) Electrons flow within TEM, its comp onents and detectors. b) Different possible \ninteractions between the sample and the electron beam.  ................................ ........................  22 \nFigure 4 .9 – Particle counting using ImageJ. Black areas are the particles counted for size statistics.\n ................................ ................................ ................................ ................................ ..................  23 \nFigure 4.10 – EDS r esults for Mn 0.2Zn0.8Fe2O4. a) EDS spectrum – legend is: red for  O, green for Mn, \nlight blue for Zn and dark blue for Fe. b) EDS results table.  ................................ ....................  23 \nFigure 4.11 – a) TEM and its components. b) Sample -holder (Carbon grids) macro and micro scale \nc) Electrons’ path in TEM, components used: 1 – Electrons’ source; 2 – Condensing leans; 3 - \nSample; 4 – Objective lens; 5 – Objecti ve lens aperture; 6 – Intermediate lens; 7 – Projector \nlens; 8 – Fluorescent display.  ................................ ................................ ................................ ... 24 \nFigure 4.12 – a) Nanoparticle  count; b) Lorentz fit.  ................................ ................................ ..........  24 Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \niv  Departamento de Física  Figure 4.13 – (a) SQUID MPMS -3 (Quantum Design, Inc.). (b) Schematic of the SQUID \nsupercond ucting coils connected to a Josephson -junction.  ................................ .....................  25 \nFigure 4.14 – a) Curie -Weiss law linearization to obtain Ɵp. b) Determining Ɵp using the temperature \ndependence of the inverse susceptibility for ZnFe 2O4. ................................ ............................  26 \nFigure 4.15 – a) Magnetic i nduction heating experimental setup. 1) amplifier with a resonant circuit. \n2) Signal generator; 3) Hall sensor; 4) Inducti on core; 5) Fiber optics thermometer. b) inductor \ncore ................................ ................................ ................................ ................................ ...........  27 \nFigure 4.16 – Magnetic induction heating results as a function of time. a) Red line for sample. Blue \nline for sample -holder. b) Hall sensor Voltage p -p (V).  ................................ ............................  28 \nFigure 4.17 – Linear fit of the initial slope for MnFe 2O4 and ZnFe 2O4. ................................ .............  28 \nFigure 5.1 – a) XRD patterns of Mn 1-xZnxFe2O4. Miller indices of Mn -Zn ferrite spinel crystal structure \nare displayed. ZnO impurity peaks are represented by *. b) percentage of ZnO impurity as a \nfunction of Zn content, inset graph shows the evolution of ZnO peaks in the diffractogram.  .. 29 \nFigure 5.2 – a) Lattice c onstant as a function of the Zn content; b) Crystallite size as a function of Zn \ncontent. Green line is for synthesized samples. Black line is for bibliographic samples from A. \nDemir et al.  ................................ ................................ ................................ ...............................  30 \nFigure 5.3 – SEM images of samples: a) MnFe 2O4; b) Mn 0.2Zn0.8Fe2O4; c) ZnFe 2O4. .....................  30 \nFigure 5.4 – SEM images from literature: (a), (b) and (c), from A. Demir et al, sho wing the MnFe 2O4, \nMn 0.2Zn0.8Fe2O4 and ZnFe 2O4, respectively. Image (d), from R. Gimenes et al, with \nMn 0.8Zn0.2Fe2O4. ................................ ................................ ................................ .......................  30 \nFigure 5.5 – Metal ions ratio. a) Mn ratio; b) Zn ratio; c) Fe ratio. Green line is the sample ratio and \nblack line is the expected  ratio of the predicted stoichiometry.  ................................ ................  30 \nFigure 5.6 – M(H) curves at: 5K a), 300K b) and c) 380K for Mn1-xZnxFe2O4 (x  = 0; 0.2; 0.5; 0.8; \n1). Inset figure shows the hysteretic loops of the samples.  ................................ ......................  31 \nFigure 5.7 – Magnetic data obtaine d from M(H) curves at 300K. (a) Saturation Magnetization; (b) \nCoercive Field; (c) Remnant Magnetization. Green lines are for synthesized samples and black \nlines for the comparis on with samples of A. Demir et al.  ................................ .........................  31 \nFigure 5.8 – M(T)  curves measured at 100 Oe. a) Field cooled. b) Zero -field cool.  ........................  31 \nFigure 5.9 – TEM images for sol -gel auto -combustion samples. a) Zn 0.2Mn 0.8Fe2O4. b) ZnFe 2O4. . 32 \nFigure 5.10 – Size distribution of a) Zn 0.2Mn 0.8Fe2O4 and b) ZnFe 2O4.The distributi on was adjusted \nwith a Lorentz function.  ................................ ................................ ................................ .............  32 \nFigure 5.11  – XRD pattern of Mn 1-xZnxFe2O4 with x = 0; 0.5; 0.8; 1.Spinel phase peaks are signed by \nthe respective Miller indices, impurities peaks of Magnetite, Hematite -proto and Goethite are \nmarked with  open squares, circles and “x”s, respectively.  ................................ .......................  34 \nFigure 5.12 – Crystallographic parameters a) Percentage of spinel f errite phase. b) lattice constant. \nc) crystallite size.  ................................ ................................ ................................ ......................  34 \nFigure 5.13 – SEM images of all compositions: (a) MnFe 2O4; (b) Mn 0.5Zn0.5Fe2O4; (c) \nMn 0.2Zn0.8Fe2O4; (d) ZnFe 2O4. Poor definition in SEM images was o btained for intermediate \ncompositions (x = 0.5 and 0.8). For Mn and Zn ferrites, the image shows enough definition for \ncounting agglomerated nanoparticles.  ................................ ................................ .....................  35 \nFigure 5.14 – Metal ions relative quantity times 3 (number of metal ions) and comparison with the \nexpected values (Fe = 2; Mn = 1-x; Zn = x).  ................................ ................................ ............  35 \nFigure 5.15 – M(H) curves. From left to right column 5, 300  and 380 K respectivelly. ln the upper r ow \na full scan is shown (from -70000 to 70000 Oe), in the bottom row an amplification of the M(H) \ncurves is made for hysteresis loops analysis.  ................................ ................................ ..........  36 \nFigure 5.16 – Saturation magnetization (a), remnant magnetization (b) and coercive field (c) at 300K. \nGreen lines are for samples, black lines are samples from literature. Black open circles are for \nvalues which were not explic it in literature.  ................................ ................................ ..............  36 \nFigure 5.17 – M(T) measurements – FC and ZFC (a), FC (b) and ZFC (c). Range of temperature \nfrom 5  to 400K, except for Mn 0.5Zn0.5Fe2O4 which goes fr om 5 to 300 K.  ................................  36 \nFigure 5.18 – a) Normalized FC curve,  magnetic ord ering temperatures represented with arrows. b) \nmagnetic ordering temperatures obtained via Curie -Weiss law are represented in closed green \ncircles. Open circles are the values indirectly obtained.  ................................ ..........................  37 \nFigure 5.19 – a) Blocking temperature distributions. b) Ferromagnetic contribution of M(H) curves. c) \nParamagnetic contribution of M(H) curves.  ................................ ................................ ..............  38 \nFigure 5.20 – M/M S(H/T), at 300 K and 380 K, and Langevin  fit for a) ZnFe 2O4 b) Mn 0.2Zn0.8Fe2O4 \nand c) Mn 0.5Zn0.5Fe2O4. ................................ ................................ ................................ ............  38 André Horta  \nUnive rsidade de Aveiro   v Figure 5.21 – TEM images of hydrothermally synthesized Mn 1-xZnxFe2O4. Upper row images have a \n50 nm scale (except MnFe 2O4) and bottom row a 20nm scale. a) and d) is MnFe 2O4; b) and e) \nis Mn 0,2Zn0,8Fe2O4; c) and f) is ZnFe 2O4 ................................ ................................ ...................  39 \nFigure 5.22 – Size distributio n of Mn 1-xZnxFe2O4 x = 0; 0.8; 1 in TEM images. Mean grain size and \nstandard deviations for each sample. a) MnFe 2O4; b) Mn 0.2Zn0.8Fe2O4; c) ZnFe 2O4. ..............  39 \nFigure 5.23 – Magnetic induction heating results with an AC magnetic field of 25mT at a frequency \nof 364KHz. a) Temperature as a function of time for all compositions; b) Time derivative of \ntemperature composition dependen ce. ................................ ................................ ....................  40 \nFigure 5.24 – Magnetic induction heating results with a variable AC magnetic field amplitude at a \nfrequency of 364KHz. a) Temperature as a function of time for Mn 0.5Zn0.5Fe2O4; b) Heating rate \nquadratic dependence of the magnetic field amplitude.  ................................ ...........................  40 \nFigure 5.25 – (a) Iron metal ions ratio (EDS) and (b) % of impurity phase (XRD).  ..........................  40 \nFigure 5.26 – Crystallite size influence on the saturation magnetization, remnant magnetization and \ncoercive field. ................................ ................................ ................................ ............................  41 \nFigure 5.27 – Comparison of a) hysteretic loops area with b) heating rate.  ................................ .... 42 \nFigure 6.1 – Schematic of a self -pumping magnetic co oling dev ice by V. Chaudhary et al.  ...........  45 \nFigure A.0.1 – Percentage  of pure phase as a function of a) composition and b) autoclave time.  . 49 \nFigure A.2 – Mn and Zn ferrite dependency with autoclave time. a) D XRD(t) and a(t). b) M S(t), H C(t) \nand M R(t) c) ƟP(t). ................................ ................................ ................................ .....................  49 \nFigure A.0.1 – Temperature derivative of the FC curves for a) The red line is the F C measurement \nand the blac k line the respective temperature derivative.  ................................ ........................  50 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nvi  Departamento de Física   André Horta  \nUnive rsidade de Aveiro   vii List of Tables:  \nTable 4.1 – Available synthesis parameters and respective control. C ontrolled parameters are \nidentified by a range of used values. Free means that there was no control over the parameter.\n ................................ ................................ ................................ ................................ ..................  17 \nTable 4.2 – Crystal phases information exported from the Rietveld refine ment.  .............................  20 \nTable 5.1 – Synthesis methods summarized by H. Shokrollahi et al.  ................................ ..............  33 \nTable 5.2 - Estimation of single domain size, blocking temperature and anisotropic constant from the \nacquired magnetic and structural data. Asterisks means that the value was obtain from \nlinearization.  ................................ ................................ ................................ .............................  42 \nTable 5.3 – Estimation of single domain size and anisotropic constant for the blocked nanoparticle \nat higher temperature.  ................................ ................................ ................................ ..............  43 \nTable 6.1 – Properties of hydrothermall y prepared Mn 1-xZnxFe2O4 measured at 300 K. Asterisks \nmarks value that were not measured directly.  ................................ ................................ ..........  44 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nviii  Departamento de Física   André Horta  \nUnive rsidade de Aveiro   ix List of Abbreviation s: \ncgs - Centimeter -Gram -Second  \nS.I. – International System  \nXRD – X-Ray d iffractometer  \nTEM – Transmission electron microscop e \nSQUID – Superconducting Quantum Interference Device  \nSEM – Scanning electron microscop e \nFWHM – Full-width at Half M aximum  \nW-H – Williamson -Hall \nFC – Field Cool  \nZFC – Zero-Field Cool  \np-p – peak to peak  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nx  Departamento de Física   \n  André Horta  \nUnive rsidade de Aveiro   xi List of Nomenclatures:  \n𝑀 - Magnetization  \nMS – Saturation Magnetization  \nHC – Coercive Field  \nMR – Remnant Magnetization  \nTC – Curie temperature  \nTB – Blocking temperature   \n𝐻 – Applied magnetic field  \n𝑉 - Volume  \nT - Temperature  \n𝐾𝐸𝑓𝑓 – Anisotropy Constant  \nC – Curie Constant  \nTC – Curie Temperature  \nTN – Néel Temperature  \n𝐾𝐵 – Boltzman Constant  \nTIrr – Irreversibility Temperature  \nTB – Block Temperature  \nU - Entropy  \nQ - Heat  \nW – Work  \nD𝑋𝑅𝐷 – Crystallite size  \n𝑑𝑇𝐸𝑀 – Particle Size  \nPL – Power Loss  \nHA – Hysteresis Area  \nf - Frequency  \n \n \n \n \n \n \n \n \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nxii  Departamento de Física   \n \n \n \n \n \n \n \n \n \n \n André Horta  \nUnive rsidade de Aveiro   xiii List of Symbols:  \nµe – Bohr Magneton  \n𝑚𝑒 – Electron mass  \ne – Electron Charge  \nℎ - Plank constant  \nħ – Plank Constant/ 2𝜋 \n𝑆 – Spin Value  \n𝐿 – Orbital Momentum Value  \n𝐽 – Total Angular Momentum Value  \nge – Land é Electron g -factor   \nµ - Atomic spin  \nχ – Susceptibility  \n𝜏𝑁 – Néel Relaxation Time  \n𝜏𝐵 – Brownian Relaxation Time  \n𝜏𝑒𝑓𝑓- Relaxation Time  \n𝛿 – Inversion Parameter  \n𝜆 – Wavelength  \n𝑑 – Interplanar Distance  \nβ𝑇𝑂𝑇𝐴𝐿  - FWHM  \nβ𝑆𝐼𝑍𝐸 – DXRD contribution for FWHM  \nβ𝑆𝑇𝑅𝐴𝐼𝑁  – Strain contribution for FWHM  \n𝜂 - Strain  \nσ – Standard Deviation  \nµ - mean value  \nӨ𝑝 – Paramagnetic Curie Temperature  \nµ0 – Vacuum permeability  \n \n \n \n \n \n \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \nxiv  Departamento de Física   André Horta  \nUniversidade de Aveiro   1 Chapter 1:   INTRODUCTION  \nMagnetic materials attracted humanity ’s curiosity since ancient times. The characteristic distant \ninteraction between a magnet and a magnetic field still nowadays amaze s people who are not \nfamiliarized with magnetic phenomena. The first magnetic material discovered by Mankind was \nLodestone in Greece, 800 BC. Lodestone contains magnetite, Fe 3O4, which is a naturally magnetized \nmineral, that was  used to create the first co mpasses, [1]. Today, in the new era of nano research, \nthe magnetic materials are getting smaller and interesting magnetic pr operties are being studied and \napplied in novel applications such as medicine, higher efficiency devices or environmental aid.  \nEvery material reacts somehow when a magnetic field is applied to it. Some materials can have a \nmagnetic field of their own, thos e are magnets or ferromagnets. When a ferromagnet is heated, its \nmagnetic properties weaken until it completely loses its magnetiz ation. The temperature at which a \nferromagnet loses its magnetization is the Curie temperature.  \nThe main purpose of this maste r thesis is to tune the Curie temperature in nanoparticles of Mn -Zn \nferrite (Mn 1-xZnxFe2O4) with the objective of developing self -regulated  heating at the nanoscale.  \nThe interest in the Curie temperature tuning in nanoparticles of this material, is the vast range of \napplications emerging from it. The Mn -Zn ferrite is frequently mentioned for hyperthermia treatments, \ndue to its nano size, biocompatibility and Curie temperature close to room temperature, [2]. The self -\nregulated heating mechanism can also be used for self -pumping devices, energy harvesting, \ncontrolled melting, cookware, among many other applications, [3] [4] [5]. The advantage of preparing \nthese systems at the nanoscale is the possibly of scaling them to a larger scale, as a bulk material, \na ferrofluid, or even incorporated as a part of a composite . \n1.1 MOTIVATION  AND OBJECTIVES  \nThe engineering of magnetic  nanoparticles is a contemporary subject with applications in many areas \nof research. For this master thesis, the challenge is to synthe size magnetic nanoparticles capable of \nself-regulating their temperature in a magnetic induction heating experiment, via  Curie temperature \ntuning. The material of election was Mn -Zn ferrite due to the dependence of the Curie temperature \nwith the Zn/Mn ratio. In addition, the Curie temperature is expected to change from below room \ntemperature until high temperature with incr easing Mn content (~300oC). Another valuable feature of \nMn-Zn ferrite is its biocompatibility. In this point this ferrite outcomes other ferrites and hence signals \nthe Mn -Zn ferrite as a potential candidate for biomedical treatments, for example, cancer tr eatment \nvia hyperthermia , [6]. \nThe main objective of this master thesis is to synthe size magnetic nanoparticles of Mn -Zn ferrite, \nMn 1-xZnxFe2O4, capable of self-regulating their tempera ture for a further application in a ferrofluid. \nThis family of ferrites had never been synthe sized by this research group and, for this reason, the \ndevelopment of good quality samples, for further applications, is a priority. T his master thesis \nobjectives are the following:  \n❖ Synthesis optimization of Mn 1-xZnxFe2O4 nanoparticles with different compositions.  \n❖ Structural and magnetic characterization of the samples.  \n❖ Comprehension of the magnetic phenomena in this material and charact erization \ntechniques.  \n❖ Achieve ment of the self -regulated heating mechanism via Curie temperature.   \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n2  Departamento de Física  1.2 THESIS LAYOUT  \nThe present  report is divided  in 6 chapters . In “Chapter 1:  Introduction ”, the reader will find the  \nobjectives and motivations  of this master thesis work . In “Chapter 2: M agnetic behavior ”, an \nintroduction of the magnetic phenomena related with the synthe sized samples is discussed. Due to \nthe nano -size nature of the samples, this chapter starts with an increasing in scale until macroscale \nis ach ieved and then drops again to  the nanoscale magnetic behaviors. “Chapter 3: Mn -Zn ferrite ” is \nabout this remarkable  compound, it starts with the synthesis methods, passing to the crystallographic \nstructure and ends in magnetic structure. In “Chapter 4: Experimental Procedure ”, the syn thesis \nmethods (sol -gel auto -combustion and hydrothermal) are detailed, then the physical principles of the \ncharacterization techniques are explored. “Chapter 5: Results  and Discussion ”, starts with the results  \nand discussio n of the samples synthe sized by sol-gel auto -combustion method and ends in the \nresults and discussion of the hydrothermally synthe sized samples. This master thesis content ends \nin “Chapter 6: Conclusions ”, in this chapter the reader will  find some conclusi ons about the samples \nsynthe sized by both methods and suggestion for further work. After the conclusions the References \nare presented. In the last pages of this report a few Appendixes with complementary information are \nprovided.  André Horta  \nUniversidade de Aveiro   3 Chapter 2:   MAGNETIC BEHAVIOR  \nAll mat erials present a magnetic behavi or under the influence of a magnetic field, this behavior is \nexplained by the presence of electrons (charges) in all materials. Magnetic fields are created by \nmoving charges, like magnetic fields created by solenoids, or, as  an intrinsic property of elemen tary \nparticles, such as electrons, protons or neutrons - this quantum property is called spin. In \nferromagnetic materials, electrons are responsible for the magnetic field. For protons and neutrons \nthe magnetic moment is sev eral orders of magnitude lower t han for electrons, 2000 times lower, [1]. \nThe magnetic moment of an electron, µe, is 1.0012   µB, [7], and is related to the Bohr ma gneton, \nµB=−9.284×10−24 J/T (SI), by the relationship µe=geµB\nħ𝑆, where  ge is the Landé g -factor ≈2  and S \nthe spin value (ħ\n2). \nElectrons surround the atomic nucleus in discrete energy levels, N, each energy level has 2N+1 \norbital for electrons to oc cupy. E ach of these orbitals is classified with an L value, related with the \nangular orbital momentum. The electrons’ angular orbital momentum generates a magnetic field \nwhich interacts with the electrons’ spin. Concluding, every atom magnetic moment is de pendent  of \nthe total angular momentum, J, arising from the interaction between electrons spin and electrons \norbit, calculated as |L+S|,  …, |L-S|, for the ground state. The J value is proportional to the atomic \nmagnetic moment,  µ, equation 2.1, where g J is the Lan dé g-factor, µB is the Bohr magneton , [8]. \nµ= −gJµB\nħ𝐽      (2.1) \nTemperature interferes with the electrons’ d istribution, accordingly with Fermi -Dirac distribution. Any \ntemperature abov e 0K shifts the electrons’ ground state for higher energies, probabilistically. The \ntemperature plays a major role in the magnetic properties of all materials.  \nEvery material has e lectrons, as consequence, every material is affected by electromagnetic fiel ds. \nAt the ground state, electrons group up accordingly with the Hund -rules which determines which \nelectronic orbitals are first filled, the inner orbitals should be fully filed wi th paired electrons before \nouter -most orbitals. Paired electrons are found i n almost all elements and compounds. These pairs \nof electrons are responsible for magnetic repulsion (diamagnetic behavior). Unpaired electrons, in \nthe outer orbitals are responsib le for magnetic attraction.  \nWhen dealing with a molecule, the electromagnet ic interactions between atoms influence the spin \nvalue of each atom. The molecules magnetic properties are affected by the electromagnetic fields \narising from the atomic arrangemen t (hyperfine structure), magnetic  interactions (further explained), \ntemperat ure, external magnetic fields and others.  \nSection 2.1: Magnetic Properties is an introduction to the magnetic properties commonly found in all \nmaterials. Section 2.2: Macro Magneti sm is where the magnetic properties of the previous section \nare analyzed for  different macroscale magnetic behaviors. Section 2.3: Nano Magnetism, is directly \nrelated to the synthe sized samples and to ferromagnetic behavior at the nanoscale. Section 2.4 go es \nstraight to the objective of this work, understanding the magnetic induct ion heating mechanisms and \nheat generation  phenomena.  \n2.1 MAGNETIC PROPERTIES  \nThis section  intends to provide the reader with knowledge about fundamental magnetic properties. \nMagnetiza tion and magnetic susceptibility, when an external field is present. Magneti sm is a result \nof exchange interaction, superexchange interaction, magnetocrystalline anisotropy and dipolar \ninteractions. The magnetic properties are temperature dependent, as wil l be detailed in section  2.2. \n2.1.1  MAGNETIZATION  \nThe magnetization of a material  is described as “the magnetic dipole moment regarding either unit \nof volume or unit of mass”, [9]. The magnetization of a material is consequence of the alignment of \nthe atomic dipolar moments of a material  in the presence of a m agnetic field . Considering a finite Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n4  Departamento de Física  number  of non-interacting magnetic moments , n, each with a magnetic moment, µ, (mis) aligned with \nthe field direction by  an angle  Ɵ, the total magnetization of the group is giv en by equation 2.1, [10]. \n𝑀⃗⃗ =∫µ𝑛cos (Ɵ𝑛)𝑑𝑛𝑛\n0      (2.1) \nThe more aligned moments, the higher the magnetization, assuming the same volume or mass . The \nmagnetic order saturates for high fields when all magnetic moments  are aligned.  The magnetization \nunits are, usually,  normalized to the mass or volume  of the sample , emu/g or emu/ cm3 (cgs) . \n2.1.2  MAGNETIC SUSCEPTIBILITY  \nThe magnetic susceptibility is a prop erty of every material, since all materials react somehow when \na magnetic field is applied to it. A material with unpaired  electron is attracted towards a magnetic \nfield because its atomic dipolar moments align in the field direction, experiencing attracti on. Between \ntwo materials, the material with a higher magnetization for the same intensity of applied field, has a \nhigher magnetic susceptibility.  \nMagnetic susceptibility, χ, is defined as the amount of magnetization, 𝑀, induced in a material when \na magne tic field, 𝐻, is applied  to it, equation 2.2  \n𝑀⃗⃗ =χ𝐻⃗⃗                                                        (2.2) \nMagn etic susceptibility is measured with resource to magnetometers. Its measurement is done by \nchanging the intensity of an applied magnetic field while measuring the materials magnetization, \nM(H) measurements. These measurements are performed at constant temp erature. the magnetic \nsusceptibility is temperature dependent. The temperature effect will be individually detailed for each \nmagnetic behavior.  \nThe magnetic alignment  is field constant when considering non -interacting magnetic moments, like \nin paramagnetic  behavior. When dealing with interacting magnetic moments, for example in the \nferromagnetic state, the susceptibility  varies with temperature . \n2.1.3  EXCHANGE  AND SUPEREXCHANGE  INTERACTION  \nElectrons are indistinguishable particles, since all electrons present the  same fundamental properties \n(mass, charge and spin) , [9]. Electrons wave -function spread through space as time passes,  [11]. In \na molecule,  covalent bonding  causes the electron s’ wave -function to overlap between atoms, which \nmeans that electrons  have a non-zero probability of exchange its position with another electron of \nthe molecule.  \nExchange interaction  is a purely quantum -mechanical  phenomenon  as it affects electron s in an atom \nor in close -neighbor atoms . When the exchange interaction affects close -neighbor  atom s it is referred \nto as direct exchange. When th e exchange interaction is mediated through a non -magnetic atom is \ncalled superexchange. The exchange interaction  is the main cause of ferro - and antiferro -magnetism.  \nFigure 2.1 shows magnetic coupling s in a ferromagnetic material, direct exchange and \nsupe rexchange, both are mediated by the overlapping of the wave -functions , [1].  \n \n \n \n2.1.4  MAGNETOCRYSTALLINE ANISOTROPY  \n“The magnetocrystalline anisotropy is manifested by locking magnetic moments in certain \ncrystallographic directions”, by V. Sechoveský [1]. The magnetocrystalline anisotropy is a \nconsequence of magnetic alignment between neighbor atoms of a crystallograp hic plane. When the \nmagnetic alignment occurs between close -neighbor atoms, this direction is called an easy axis. If the \nalignment occurs be tween further atoms is called a hard axis. The magnetocrystalline anisotropy \ncauses the magnetic domains to be pref erentially aligned in specific crystallographic directions.  \na) \n b) \nFigure 2.1 – Schematic representation exchange interactions in a ferromagnet: a) direct exchange. \nb) superexchange.  André Horta  \nUniversidade de Aveiro   5 \nThe magnetocrystalline anisotropy is affected by the domain volume and the domain  shape. \nAssuming a spherical shape domain, t he larger the domain volume, the longer the chain of aligned \nmagnetic m oments, consequently , the larger the magnetocrystalline anisotropy , 𝐾𝑎(𝜃) given by \nequation 2.3, [12].  \n𝐸𝑎(𝜃)=𝐾𝐸𝑓𝑓𝑉𝑠𝑒𝑛2(𝜃)     (2.3) \nWhere 𝜃 is the angle between the magnetization vector and an eas y axis. The energy barrier \nbetween two easy axis corr espond to the product of the material anisotropic constant  and the domain \nvolume, 𝐾𝑉. The magnetocrystalline anisotropy has a maximum value when the magnetization is \nperpendicular to an easy direction, which corresponds to the highest potential energy of  the system.  \nA polycrystalline material is composed of many crystals, or magnetic domains, each crystal has its \nown preferred direction for alignment. The interaction between magnetic domains i s mainly dipolar.  \n2.1.5  DIPOLAR INTERACTIONS  \nThe dipolar interaction is the weakest of the interactions previously mentioned. This interaction is the \nclassical magnetic force between a magnet (dipole) and a magnetic field. The dipolar interaction \noccurs between magnetic domains or magnetic particles. Its strength depends on  the characteristics \nof the dipoles, such as magnetic moment, distance between dipoles and interaction angle.  \nThe dipolar interaction is responsible for the directional magnetization of a ferro magnet. If a magnetic \nfield is applied to a demagnetized ferrom agnet, it aligns its domains in the field direction, through \ndipolar interaction. After the field is removed, the ferromagnet might retain its magnetization (remnant \nmagnetization). In order to  return the magnetization to zero, an opposite magnetic field can be \napplied (coercive field). Both coercive field and remnant magnetization are a direct consequence of \nthe dipolar interaction between magnetic domains.  \n2.2 MACRO MAGNETISM  \nMacro magnetism section is dedicated to the different magnetic behaviors found in bu lk materials. \nThese behaviors  are categorized as  diamagnetic,  paramagnetic, ferromagnetic, antiferromagnetic  \nand ferrimagnetic . Other forms of magnetism will not be discussed, such as metamagne tism, spin \nglass and  molecular magnetism.  \n2.2.1  DIAMAGNETISM  \nDiamagne t materials have the interesting property of  repell ing an external  magnetic field, in \nopposition to all other s magnetic  behaviors.  Some examples of diamagnetic compounds are water, \ngraphite, si lver, mercury  and bismuth.  The magnetic susceptibility in diama gnets have a negative \nvalue , the materials with higher diamagnetic behavior are pyrolytic carbon ( χ = -23x10-5 cm3/g, [13]) \nand bismuth ( χ = -16.6x10-5), water has a susceptibility of -0,91x10-5, [14]. The stronge st diamagnetic \nmaterials are superconductors : they behave as ideal diamagnets , completely repelling magnetic \nfields. Superconductors have  a susceptibility of -1, [1], which enables th em to have a zero total \nmagnetic field within their volume . \nThe characteristic repulsion of diamagnets for an applied magnetic field is responsibility of paired \nelectrons. Due to compensated spins, the  electrons pair has a  null magnetic moment , thus, pai red \nelectrons  do not align in the magnetic field . Instead, the movement  of paired electrons create a  \nmagnetic field  with opposing polarization , experiencing repulsion from a magnetic field, Lenz law . \nEvery material has an intrinsic  diamagnetic behavior , since most  elem ents have  paired electrons in \ntheir fully filled orbitals.  \n \n \n \na) \n b) \nFigure 2.2 - Diamagnetism; a) Field lines in diamagnetic material;  b) magnetization dependence of an \nexternal applied field.  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n6  Departamento de Física  2.2.2  PARAMAGNETISM  \nParamagnetism is the most common type of magnetic behavior at room temperature. Some \nexamples of paramagnetic materials are oxygen, aluminum and titanium. Paramagnets  are weakly \nattract ed to magnetic fields and do not retain any magnetization at zero magnetic field.  \nParamagnetic material s have  unpaired electrons. Without an external magnetic field,  the electrons \nare randomly oriented , figure 2.3.a . The magnetization of  a paramagnet, without an applied magnetic \nfield, is 0 emu/g, as result of the randomly oriented atomic spins.  \nIn the presence of an external magnetic field, the atomic spins tend to align parallelly to the magnetic \nfield causing a weak attraction of the material towards the field source. Thus, in order to measure \nthese materials magnetization and magnetic susceptibility an external magnetic field must applied. \nThe magnetization follows a linear tendency with increasing  magnetic field, which is consequence  of \na nearly field independent magnetic susceptibility, figure 2.3.b). The magnetization of paramagnets \nreaches a maximum, saturation magnetization, value when all the atomic spins are aligned in the \nfield direction, al though this phenomenon is only observ able typically at high magneti c fields. When \nthe external magnetic field is removed, the material returns to the zero -magnetization state, as the \nthermal agitation randomly orientates the atomic spins.  \n  \n \n \n \n  \n \n \nThe magnetic susceptibility , is dependent of the applied magnetic field  and temperature , as \npresented  in section  2.1.2. This equation was formalized by  Pierre  Curie , [8], in the Curie law, \nequation 2.4. \nχ=M\nH= C\nT      (2.4) \nWhere χ is the  susceptibility, C is the Curie constant of the material, H the magnetic field and T the \ntemperature. By analyzing equation 2.4 it is noticeable that the susceptibility increases inversely \nproportional with the magne tic field  and decreases with increasing t emperature . Increasing \ntemperature decreases magnetization, as it suppresses the magnetic interactions . This equation is \nnot valid for high magnetic fields , due to the saturation  of the  magnetic domains or phase tra nsitions \ncaused by temperature.  \nParamagne tic materials at room temperature might not be  paramagnetic  at lower temperatures. \nParamagnetism is temperature dependent - the thermal energy is competing with the magnetic \nenergy by changing the atomic spins orien tation. At lower temperatures, paramagnet s at room \ntemperature might change its magnetic structure, this effect is  referred as phase transition . The \nphase transition from paramagnetism to ferro -/ferri-magnetism is said to occur at the  Curie \ntemperature.  For example, bulk Gadolinium is paramagneti c above 20oC (293.15 K),  while  below this \ntemperature it is paramagnetic.  The phase transition from paramagneti sm to antiferromagnetism \noccurs at  the Néel temperature. Manganese oxide, MnO 2, is paramagnetic  above  -157oC and \nbecomes antiferromagnetic  below  this temperature. Paramagnets which are paramagnetic at 0K are \ncalled ground -state paramagnets.   \n2.2.3  FERROMAGNETISM   \nFerromagnetism has a highlighted  place  between magnetic behaviors , due to the capacity of these \nmater ials to retain a ma gnetic field. Ferromagnetism is the most researched  type of magnetic \nbehavior and occur s at room temperature in elements such as Iron, Cobalt, Nickel and Rare-Earth \na) \n b) \n c) \nFigure 2.3 – Paramagnetism; a) Schematic r epresentation of mag netic dipoles in a paramagnet ; b) \nMagnetization dependence with the applied field ; c) Susceptibility and inverse susceptibility \ndependence with temperature.  André Horta  \nUniversidade de Aveiro   7 elements. It can have several  applications ranging from fridge magnets,  electric motors, co mpasses, \ntransformers and other devices, [8]. \nThe remnant magnetization arises from ferromagnets as conseque nce of magnetic doma ins \nalignment inside the material. A magnetic domain is typically associated with the  crystal lite of the \nferromagnet , which has its own preferred direction for magnetic alignment , as discussed previously \nin section 2.1.4 . The atomic spi ns alignment within a magnetic domain is  a result of preferred \norientation between atoms of a crystallographic plane  (magnetocrystalline anisotropy) . Figure 2. 4.a) \nis a representation of atomic spin arrangement in a  ferromagnetic unit cell . Figure 2. 4.b) is the \nrepresentation of two domain s, separated by  domain -wall, [8]. Figure 2. 4.c) is a n image  of magnetic \ndomains in a ferro magne tic material, obtained using a Kerr -effect microscope , [15].  \n \n \n \n \n \n \n \n2.2.3.1  HYSTERESIS LOOPS  \nWhen a ferr omagnet is demagnetized, its ferromagnetic domains are misaligned. If a magnetic field \nis applied to the ferromagnet, the magnetic domains tend to align paralle lly with the applied magnetic \nfield. When all the magnetic domains are pointed in the field dire ction the material reaches a point of \nmagnetic saturation, M S. When the applied field is removed, the aligned magnetic domains relax to \nmore stable orientation, however, some magnetization is retained in the applied field direction, \nremnant magnetization, MR. The remnant magnetization grants the ferromagnets with their own \nmagnetic field. This remnant magnetization is only lost by heating the ferromag net above the Curie \ntemperature or by applying an opposite magnetic field with enough intensity for the magn etization to \nbecome zero, called coercive field, H C. The dependence of the ferromagnet magnetization with an \napplied field is called a magnetic hyst eresis loop, figure 2.4.d).  \nThe hysteresis loops , obtained by M(H) measurements,  are consequence of magnetic  domain \naligning their magnetic moments in the presence of a magnetic field . This behavior can be described \nby three characteristic points of a hysteretic  curve: saturation magnetization, remnant magnetization \nand coercive field. Saturation magnetization occur s when all the  sample magnetic domains  are \naligned in the  field direction . Remnant magnetization is the sample remaining magnetization when \nthe magnetic field is removed . Coercive field is the  required field to take the sample magnetization \nto zero.  All these points are marked in figure 2. 4.d). \nThe inversion of magnetic domains , or domain walls movement,  causes the crystal lattice  to heat. \nThe hysteretic  loss can be used for heat generation via magnetic induction heating and will be \ndetailed in section  2.4. \n2.2.3.2  TEMPERATURE  EFFECT  \nThe temperature effect  on a material magnetization  is usually observed in  measurements of  \nmagnetization dependence  of the temperature, M(T) measurements . In ferromagnets, t he thermal \nenergy  (randomizing aligned spins)  competes  with the magnetic interactions  (spins alignment) . The \ntemperature which  cause s a material to alter its magnetic behavior from the ferromagnetic state to \nthe paramagnetic state is the Curie temperature , as referred in 2.2.2 . \nBelow the Curie temperature, in the ferromagnetic state, the temperature is not sufficiently high to \ncompete with the magnetic interactions mentioned in sections 2.1 .3 to 2.1.5 . While i ncreas ing the  \ntemperature , the overall magnetization weaken s as the thermal energy gradually overcomes the \na) \n b) \n c) \n d) \nFigure 2.4 – a) Ferromagnetic unit  cell. b) domain -wall dividing two antiparallel domains c) ferromagnetic \ndomain observed by Kerr -effect microscope  (author: C. V. Zureks) . d) ferromagnetic M(H) curves : M S -\nsaturation magnetization, M R - remant field, H C - coercive field.  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n8  Departamento de Física  \nFigure 2.5 – Magnetic susceptibility and inverse susceptibility of a ferromagnetic material.  \n \nFigure 2.6 – a) Unit cell of antiferromagnetic moments. b) Antiferromagnetic domains. c) Magnetization \nincreases with external field. d) Susceptibility and inverse susceptibility as function of temperature, the \nmaximum value is the Née l temperature.  magnetic exchange interaction energy, promoting the misalignment of atomic spins or magnetic \ndomains . Above the  Curie temperature , the thermal energy is higher that the magnet ic exchange \ninteractio ns and hence  while temperature increases the magnetic domai ns are destroyed  \n(paramagneti sm).  \nThe susceptibility, χ, of a ferromagnet in the paramagnetic state was formalized by Pierre Cu rrie and \nPierre -Ernest Weiss in the Curie -Weiss law, equation 2.5.  \n \nχ=C\nT−Ө𝑝    (2.5) \n                                                       \n \n \nWhere C is the material Curie con stant, T is the temperature and Ө𝑝 is the paramagnetic Curie \ntemperature, [1]. Figure 2. 5 shows the dependence of ferromagnetic magnetization with temperature , \nM(T) curve . Above the Curie temperature, the susceptibility decreases inversely proportion ally with \nthe increase of temperature. Measuring  the inverse susceptibility,  χ-1, a linear para magnetic behavior \nis revealed, being possible to estimate  Ө𝑝, the Curie temperature of the ferromagnetic material  by \nfitting its linear behavior as function of T . The Ө𝑝 differs from the Curie temperature since  it is a \nprediction  in the paramagnetic state. \n2.2.4  ANTIFERROMAGNETISM  \nAntiferromagnet ism is a magnetic ordering  occurring in materials where crystallographic planes have  \nanti-parallelly aligned  atomic  spins , figure 2. 6.a). A group of antiparallel spins  forms  an \nantiferromagnetic  domain, figure 2. 6.b), [16]. The anti -parallel alignment is a consequence of \nexchange , or supere xchange, interaction between atoms of a crystal lite. \nWhen a magnetic field is applied to the antiferromagnet, an increase in its magnetization is observed. \nThe positive susceptibility verified is due to an unequal number of antiparallel spins in the domains, \nwhich causes a weak magnetization oriented in the field direction. The dependency of magnetic \nsusceptibility with the magnetic field is linear until satu ration, [1], resembling the paramagnet s \nsusceptibility, figure 2. 6.c).  \n \n   \n  \n  \n \n \n \nFigure 2. 6.d), [1], shows the  susceptibility dependence of  temperature . The maximum value of 𝜕\n𝜕𝑇χ(T) \nis the Néel temperature, T N. At this point, t emperature induces a  magnetic phase transition : below \nTN the material is antiferromagnetic and above T N is paramagnetic.    \nExamples of antiferromagnetic material are hematite (Fe 2O3), chromium  and nickel oxide.  \n2.2.5  FERRIMAGNETISM   \nFerrimagnetism is the magnetic behavior present in most of the synthe sized samples of thi s report. \nFerrimagnetism was a term originally proposed by Néel to describe magnetic order  of ferrites , [1]. \nFerrites , and other ferrima gnets, are characterized for possessing ions with different magnetic \na) \n \nb) \n \nc) \n \nd) \n André Horta  \nUniversidade de Aveiro   9 moments in an antiferromagnetic arrangement.  These material s present a  spontaneous \nmagnetization  as consequence of the uncompensated magnetic  moments.  The dipolar interacti on of \nthe unequ al antiparallel magnetic moments often induces canting effects in the weaker magnetic ion.  \nThese materials are usually ceramic oxides, where the oxygen is the responsible for the \nantiferromagnetic alignment of different ions.  Ferrimagnetic a rrangement is a  consequence of \nsuperexchange interaction. The ferrimagnetic materials resembles ferromagnets, once the \nuncompensated magnetic spins create magnetic domains. Ferrimagnetic domains possess a \nspontaneous magnetization.  \nWhen a magnetic field is  applied to a f errimagnet its magnetic domains align in the field direction. \nWhen the magnetic field is removed, they retain magnetic moment due to dipolar interaction between \ndomains. The ferrimagnetic hysteresis loop is like the ferromagnetic hysteresis loop.  The \ntempe rature has a similar effect between ferrimagnet and ferromagnet. When a ferrimagnet heats \nabove the Curie temperature it becomes paramagnetic and lose s its magnetic order ing. When cooled \nbelow the Curie temperature, the ferrimagnet is demagn etized  due to t he misalignment of the \nmagnetic domain . \n2.3 NANO  MAGNETISM  \nMagnetic nanomaterials is an area that attracted many research  efforts lately,  due to its vast range \nof applications in the m ultiple fields , such as magnetic recording, [17], high-efficiency devices, \nbiomedicine  and other. During the last 60 years, improve ments  in synthesis methods and \ncharacterization techniques, allowed  control over  size, shape and magnetic properties of \nnanoparticles.  Also, h igh resolution characterization techniques allowed researchers to understand \nnanoparticles’ magnetic properties in detail.  \nDespite magnetic nanoparticles having  the same magnetic ordering  as the bulk material, the size \neffect has several implications in  the nanoparticles magnetic behavior. The large surface/volume \nratio, characteristic of nanoparticles , greatly impacts  their s tructural and magnetic properties . In \nparticular , MS, HC, MR and their temperat ure dependence is  affected by  their reduced size. Section \n2.3.1, refers to ferromagnetic nanoparticles , due to their similar ity with ferrimagnetic single domains .  \n \n \n \n \n \n \n \n2.3.1  SINGLE DOMAIN  \nFerromagnetic materials ar e composed of multiple domains, separated by domain walls. When the \nparticle size is small,  typically about 100nm , [18] , the formation of a domain wall is not energet ically \nfavorable . Thus, the entire particle becomes a magnetic domain, or, single domain, [19]. The \nferromagnetic single domain has all atomic spins aligned  in the same direction, which  confers  the \nparticl e a high value of  spontaneous magnetization . Furthermore, a single domain  has a high value \nof coercive field , since in order to rotate the particle magnetization the entire magnetic domain  \nrotate s. When a magnetic field is applied to a system of single dom ains, a n increase in magnetization \nis expected , as result of single domains alignment in the field direction.  Without an applied field the \nsystem present a low value of remnant magnetization, consequence of the  low magnitude  dipolar \ninteractions between pa rticles contributing to reduce the system magnetization.  \nSize \n \nHC and M R \nSuperparamagnetism  \n \nSingle Domain  \nMulti -domain  \na) \n \na) \nb) \n \nFigure 2.7 – a) H C and M R dependence with the particle size and single domain regimes. b) chain \nof magnetic nanopart icles of Fe 3O4 found inside a magnetotactic bacteria (draw by Marta Puebla).  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n10  Departamento de Física  \nFigure 2.8 – Single domain representations.  a) Stoner –Wohlfarth model . BEXT is the externally \napplied magnetic field direction. M is the magnetization vector rotated φ from the applied field .  Ɵ is \nthe angle between an easy axis and B EXT. b) and c) is the representation of atomic sp in canting at \nthe nanoparticle surface.  \n 2.3.1.1  SIZE AND SHAPE  EFFECTS  \nThe size and shape greatly af fect the magnetic properties of the single domain. The increase of the \nsingle domain volume increases the magnetic anisotropy, similarly with  what occurs with  the \nmagnetocrystalline anisotropy. The shape anisotropy of the single domain can lead to an increase \nof the magnetic anisotropy, affecting preferred orientation for the magnetization.  \nThe si ngle domain is usually  represented by the Stoner –Wohlfarth  model, figure 2. 8.a). In this model, \nthe magnetic anisotropy is consequence of the particle magnetization rotating with a phase diference  \nof the easy axis , φ-Ɵ. The angle between magnetic field and  particle easy -axis is Ɵ and with \nmagnetization  is φ. For a system of Stoner -Wohlfarth particles, when a magnetic field is applied, the \nsingle domains rotate in the field direction unt il all magnetic moments are aligned, saturati ng the \nmagnetization of the single domain s system . However, it  is frequently re ported that systems \ncomposed of single domains do not saturate, even at high fields. The cause of this is that the  atomic  \nsurface  spins are misaligned with the spins of the single  domain core . While the core spins are \nbounded by exchange interactions  to the surrounding spins,  the superficial spins are  only bounded  \nwith the core , thus, they are relatively free to align in other direction . This phenomenon is known as \ncanting and it i s a common cause for the  very high fields required to achieve complete  satura tion \nmagnetization , figure 2. 8.b) and c) , [20]. \n \n \n \n \n \n \n2.3.1.2  TEMPERATURE EFFECTS  \nThe effect of the temperature in a  single domain has similarities with the bulk material, if the bulk \nmaterial presents phase transitions , the s ingle domain might also present the  same phase  \ntransitions. The magnetic phase transition from ferromagnetic to paramagnetic,  Curie temperature,  \nalso occurs for  ferromagnetic  single domains. However, due to finite -size effects, the temperature \nfirstly affects the superficial spins , transiting them to  the paramagnetic state . The paramagnetic layer \nformed at the surface of the single domain is known as  dead layer,  figure 2.1 9.a). \nTemperature and time have curious effect s in magnetic nanoparticles . As ti me pas ses the particles’ \nmagnetization  can be flipped  to a different easy axis.  The time it takes for the  magnetization to be \nthermally fluctuate d is called  Néel relaxation time , equation  2.6. \n𝜏𝑁=𝜏0exp(𝐾 𝑉\n𝐾𝐵 𝑇)       (2.6) \nThe relaxation t ime is dependent of the energy barrier between easy axis  (anisotropy) , KV, and the \nthermal energy,  𝐾𝐵 𝑇. 𝜏0 is approximately 10-9 s, depend s on the chemical composition of the single \ndomain,  [20]. The larger the anisotrop y, or the lower the temperature, the longer the relaxation time.  \nThis equation explains why the magnetization of a bulk sample is not spontaneously change d by \ntemperature , due to its large volume,  the anisotropy is orders of magnitude larger t hat the therm al \nenergy.  When the  thermal energy is comparable with the anisotropy energy , the nanoparticle \nmagnetization rapidly fluctuate s between easy axis.  When measuring a system  of single domains , \nthe time window of the measurement is relevant in orde r to determine the system magnetic behavior.  \nIf the Néel relaxation time is  significantly  longer that the measurement window, the magnetization \nappears to be static during the measurement  - the single  domain is in  the blocked state. If the Néel \nrelaxation time is shorter that the time window, the magnetization flips during the measurement and , \nconsequently , the magnetometer will measure with equal probability a up (positive) or down \n(negative)  magnetiz ation  - the single domain is in the superparamagnetic st ate. The temperature \nwhich separates both states is the blocking temperature , TB. A system with a distribution of volumes \na) \n \nb) \n \nc) \n André Horta  \nUniversidade de Aveiro   11 \n \n \n \n \nFigure 2.9 – a) Dead layer increase with temperature increase. b) M(T) measurement of single \ndomains and T B distribution, f(T B) (point dashed). 〈𝐓𝐁〉 and T Irr are signed in the M(T) curve.  will simultaneously present a distribution of blocking temperatures , according ly with equation 2.6, 𝜏𝑁 \ndepends on volume.  \nThe thermal fluctuations in the superparamagnetic regime demote the system of any remnant  \nmagnetization  and coercive field . However, each particle is still characterized by a high value of \nmagnetization, which grants these systems a high susceptibility to the mag netic field , [18]. \nThe blocking temperature distribution is usually identified in magnetization versus  temperature \nmeasurements, M(T) measurements, 2. 9.b). The M(T) measurements are performed under two \nprotocols : cooling the sample with and without magnetic field, field cool (FC) and zer o-field cool \n(ZFC), respectively. The FC imposes that the single domai ns are blocked with a preferred alignment \nin the field direction.  ZFC, allows the nanoparticles to block  freely  oriented, thus,  decreas ing the total \nmagnetization of the system. Both measurements are performed during sample warming and with \nan external fie ld applied. When plotting both curves,  figure 29.b),  at the lowest measured \ntemperature, typically the FC and ZFC have different values of  magnetization, which is a \nconsequence of the nanoparticles being blocked in different magnetic arrangements. Increasi ng the \ntemperature will allow the ZFC single domains to be unblocked and align in the field direction, until \na maximum value of magnetizat ion is obtained. The magnetization drops due to the \nsuperparamagnetic behavior above the blocking temperature. The tem perature at which both curves \nmerge is known as irreversibility temperature, T Irr, and is the temperature where the particle with \nhigher b locking temperature is unblocked, [21]. For higher temperatures, both curves superimpose \ninto a single curve, due to all the single domains b eing above the blocking temperature, in the \nsuperparamagnetic state. Further increasing temperature will contribute fo r the superparamagnetic \nnanoparticles to transform  its magnetic phase from superparamagnetic to paramagnetic, at the Curie \ntemperature.  \n  \n \n \n \n \n2.4 MAGNETIC INDUCTION HEATING  MECHANISMS  \nThe magnetic induction heating mechanisms section  is dedicated to the heat generation mechanism s \nfound in magnetic materials, or  magnetic  nanoparticles. Magnetic i nduction  is a known ph enomenon \nsince Michael Faraday discovered it in 1831, [22]. Since then, magnetic induction heating has \nbecome more efficient and widely available as a heating mechanism. Nowadays, magnetic induction \nheating me chanisms are used in induction plates for cooking , biomedical treatments or in the \nmetallurgical industry for metal melting , [23], [24] [25]. In some cases , magnetic induction heating is \nan undesired effect, for example, through the heat generated in a transformer core or in wireless \nbattery chargers.  \nMagnetic in duction heating is based on the electromotive force experienced by a conductive material \nupon the  application of a magnetic field. In order to an electrical current to be induced in the material, \nthe intensity of the magnetic field should vary in time. For  this reason, most magnetic induction \nheating setups use oscillating magnetic fields with frequen cies in the kHz or MHz ranges.  \nHeating a ferromagnet via magnetic induction is  highly  efficien t due to the multiple heat  generation  \nmechanisms that these mater ials presents . Conductive materials generate  heat only through Joule \neffect,  ferromagnet s not onl y generate  heat via Joule effect but also by hysteretic loops. In \nnanoparticles form , ferromagnets  generate  heat through Néel and Brownian relaxation.  The foll owing \nsections  are dedicated to the heat generation  mechanism found in ferromagnetic materials.  \n2.4.1  EDDY CURRENTS  \nEddy currents are based on Faraday law of induction, which states that electrical charges are \naffected by the electromotive force (emf)  induced by  an alternating magnetic field. Furthermore, Lenz \na) \n \nb) \n \nb) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n12  Departamento de Física  law states that the alternating magnetic field induces current loops whose corresponding magnetic \nfield opposes the applied magnetic field changes - this behavior is similar with diamagnetic behavior. \nSince  electrons move in the metal surface, they decrease the magnetic field penetrating the material. \nThe electrons movement on the metal surface causes the metal to heat via Joule effect.  Nowadays, \nEddy currents are used in metallurgical industry for metal mel ting. Th is phenomenon  is also applied \nin magnetic breaks and to separate metallic materials from non-metallic , [26] . \n2.4.2  HYSTERE SIS POWER LOSS \nA ferromagnet will generate  heat at every hystere tic loop performed. The heat  source  is the \nmagnetization -demagnetization process , [27]. Heat  is release d as the domains  change their \nmagnetic orientation and /or domain  walls move along the ferromagnet.  \nThe first  law of thermodynamic s states that the total energy, U, of an isolated system is neither \ncreated nor destroyed. It is usually presented as ΔU = Q – W. Where ΔU is the change of the system \nentropy, Q is the energy given to the system and W is the work perfo rmed by the system. Fo r an \nadiabatic system, which do es not transfer energy out of the system, W = 0 . The increase of entropy \ncan be performed by demagnetizing the system. The energy  change  per hysteretic cycle is equal to \nthe hysteresis area,  present  equa tion 2. 7, [28].  \nΔU =−µ0∮𝑀𝑑𝐻     (2.7) \nThe total power loss in an induction heating experiment, considering only hysteretic losses is \nproportional to hysteresis area , HA, and to the frequency , f, of the alternating magnetic field, P L=HA.f, \n[29]. \n2.4.3  NÉEL-BROWN  RELAXATION  \nMagnetic induction heating mechanis ms occurring in single domain nanoparticles have a significant \ncontribution of both Néel relaxation and B rownian motion. Although the heat generation  occurs \ndifferently in both mechanisms,  they occur parallelly . Once  an external field is applied to  the sin gle \ndomain , its magnetization tries to follow the magnetic field, with a time lag, [29]. It is not energetically \nfavorable for the single domain to align  an individual  atomic moment , thus, for the magnetization to \nalign with the applied field, either  the entire nanoparticle rotate s, Brow n relax ation , or the materials’ \nmagnetization  rotate , Néel relaxati on.  \nNéel relaxation is the rotation of the single domain  magnetization , between easy  axis, leaving the \nnanoparticle  orientation  unaltered. The  heat generation  leads to an  increase  of the single d omain \ntemperature  which  is a consequence of the  magnetizatio n transpos ing the energy barrier that \nseparates two, or more, easy  axis. \nThe Brownian motion is a heating generation  mechanism that do not directly heat the nanoparticle. \nInstead, the  mechanical  rotation of the nanoparticle  heats its surrounding medium by Brownian \nmotion . \nBoth, Néel and Brownian relaxation , contribute  simultaneously  for the particle  temperature increase. \nThe effective  relaxation time , 𝜏𝑒𝑓𝑓, is expressed by  equation 2. 8. \n1\n𝜏𝑒𝑓𝑓 =1\n𝜏𝑁+1\n𝜏𝐵     (2.8) \nR.E. Rosensweig, [28], developed an numerical model for  the power generation  consequent of \nmagnetizatio n change in an AC magnetic field. The result is displayed in equation 2. 9. \n P = µ0πχ0𝐻02𝑓2π𝑓𝜏𝑒𝑓𝑓\n1+(2π𝑓𝜏𝑒𝑓𝑓)2    (2.9) \nWhere µ0 is the vacuum permeability ( 4π x 10-7 H/m), χ0 is the magnetic susceptibility, H0 is the \nexternal magnetic field amplitude and  𝑓 the external  magnetic field frequency . From the previous \nequation it is obtained  that the power generation  is proportional to the frequency and to the square \nof fie ld strength , [29]. \nIn the next chapter it is described the magnetism of the Mn -Zn ferrite.  André Horta  \nUniversidade de Aveiro   13 Chapter 3:   MN-ZN FERRITE  \nFerrites are Iron -Oxide based ceramics with a ferrimagnetic behavior.  The ferrite family arouse much \ninterest for being a magnetic material with high electrical resistivity, thus, reducing  the power loss \nfrom Eddy currents and presenting suitable magnetic properties for high -frequency applications.  The \nmain cause for their ferrimagnetic behavior is the Iron (III) cations present in its crystal structure along \nwith divalent cations. Ferrite s chemical formula is MFe 2O4, where M represents a divalent ion, or \nmultiple divalent ions. The choice of the divalent ion has a great imp act on the ferrite properties , as \nit can change its magnetic and crystallographic structure, as well as other intrinsi c properties . Ferrites \nare usually divided in two categories, hard ferrites, with high values of coercivity and soft ferrites, \nwith low va lues of coercivity.  \nAlthough the discovery  of ferrites is attributed to Y. Kato and T. Takey, [30], ferrites have been used \nby Mankind since the development of the  first compass es. Lodestone magnetic properties are \ncaused by the presence of magnetite . Magnetite (Fe3O4) is a ferrite , with two distinct Iron cations , \nFe2+ and Fe3+, and consequently its  chemical formula might also be written as Fe2+Fe23+O4. \nNowadays, ferrites are  spread through a vast range of application s due to their tunable magnetic \nproperties and low -cost production. De spite these materials were originally envisaged  for electronic  \napplications , today  they are used for  biomedicine, catalysts, pollution control, magnetic shielding , \namong many other applications, [31] [32]. \nMn-Zn ferrite , (Mn 1-xZnx)Fe 2O4, is a soft ferrite which attracted res earchers attention due to its \nversatility in multiple areas. This ferrite has been highly studied for biomedical applications , such as \ncancer treatment, via  hyperthermia, magnetic resonance imaging and drug -delivery agents, [33]. The \ntechnological advances in the medicine field derives from the remarkable properties of this ferrite, \nfor example,  biocompatibility , low toxicity , magnetic properties and  ease of  synthesis  in the \nnanoparticle form , [2].  \nFor this  master thesis, the int erest in Mn -Zn ferrite is the dependence of the structural and magnetic \nproperties with its composition. During the present chapter , synthesis methods, crystallograph y and \nmagnetic structure will be explored .  \n3.1 SYNTHESIS OF MN-ZN FERR ITE \nThe widely used  synthesis method for bulk ferrites is through  solid-state reaction route . This method \nrequires metal oxides (Fe 2O3, MnO 2, ZnO) and temperatures above 1000oC for the reaction to start, \n[34]. Stable  oxides  require high temperatures for their  reaction , which causes  this method to be  very \nexpensive. The synthesis of Mn -Zn ferrite in nanoparticle form is usually done via chemical methods , \nwhich can be performed close to room temperature, re ducing the production cost.  \nThere are multiple synthesi s methods for synthe sizing Mn-Zn ferrite as nanoparticle s. These \nsynthesis methods include:  hydrothermal, thermal decomposition, sol -gel auto -combustion, co -\nprecipitation, reverse -micelle , high energy ball milling , among others [35], [36]. During this work,  the \nelected synthesis methods were sol -gel auto -combustion and hydrothermal m ethods. A brief \ndescription of these methods is presented next and the experimental details of both methods are \ndetailed in “Chapter 4: Experimental Procedure ”. \nThe sol -gel auto -combustion met hod is  a chemical synthesis method that relies on the high \ntempe ratures generated by a self -sustained combustion to synthe size ceramic oxides. This method \nis frequently mentioned for Mn -Zn ferrite synthesis , as it  produc es quality samples in high quantities.  \nThe hydrothermal method is a chemical synthesis method. It r elies on the precipitation of Iron \nHydr oxides precursors through an acid -base reaction. These precursors are then subjected to \nconditions of high pressure and moderate temperature inside an autoclave , then  the ferrite  crystals \nare obtained . This synthesis method is frequently cited for the production  of high crystallinity Mn-Zn \nferrite.  Moreover, the use of chemical additives, control over temperature and the choice of the initial \nreagents allows the  control of  synthesis with varied morphologies, particles’ size distribution and \ndispersion levels [37], [38]. Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n14  Departamento de Física  \n3.2 CRYSTAL  STRUCTURE  \nThe Mn-Zn ferrite belongs to cubic  spinel  crystal structure (Fd -3m space group, 𝑂ℎ7 or 227),   [32], \nfigure 3.1 . The spinel crystal structure  is composed  of a face -centered cubic arrangement of oxyg en \nions, 32 oxygen atoms per unit ce ll. In the  interstitial  space, 8 tetrahedral  and 16 octahedral  sites are \noccupied by  transition metal cations . The cations  are required in order to maintain  this structure \nelectrically neutral. The tetrahedral site is frequently called A sit e, and the octa hedral site, B site. \nThus, the general spinel crystal structure molecular formula is represented by AB 2O4. respectively,  \nLiterature  frequently report s two types of spinel crystal s tructure, the normal spinel and the inverse \nspinel. In the normal spinel , the tetrahedral site is occupied by divalent cations (2+) and the \noctahedral site are occupied by trivalent cations (3+). In the inverse spinel, the tetrahedral sit e is \noccupied by t rivalent cation and the octahedral  site by divalent cations. However, the  most frequently \nconfiguration for ferrites is the mixed spinel crystal structure, this structure is characterized by a \nmixture of both normal and inverse spinel crystal structures. T hus, the molecular formula for spinel \nferrites must  be written as ( 𝑀1−𝛿2+𝐹𝑒𝛿3+)tetra[ 𝑀𝛿2+𝐹𝑒2−𝛿3+]octa𝑂42−, where 𝛿 represents the inversion \nparameter, [39]. For instance, in the normal spinel ( 𝛿=0) all the divalent cations are in the tetrahedral \nsite and all the trivalent cations are in the octahedral site. In the in verse spinel ( 𝛿=1), all divalent \ncations are in the octahedral site with half of the trivalent cations and the other half of the trivalent \ncations are in the tetrahedral site. For a matter of simplici ty, and since the crystal structure \ncharacterization te chnique (XRD) do es not allow such definition , the Mn -Zn ferrite molecular formula \nwill be represented as a normal spinel crystal structure, (M n1-xZnx)Fe 2O4, with the divalent cation \n(Mn2+ and Zn2+) in the tetrahedral site and all Iron  cations (Fe3+) in the  octahedral site.  Although it \nmust be highlight ed, that not only the synthe sized samples probably have a mixed spinel  crystal  \nstructure , but also the Manganese ions can be present in the divalent and t rivalent cation state s, \nMn2+ and Mn3+. All these parame ters are highly dependent of the synthesis method and synthesis \nconditions, [40]. \n \n \n  \n \n  \n \n \n \n \n3.1 MAGNETIC STRUCTURE  \nThe magnetic structure of Mn -Zn ferrite is the reason for it being highly studied, conferring  these \ncompounds  excellent  properties for ne w technological applications. Some of the magnetic properties \nthat arouse researchers’ in terest are the high saturation magnetization, high  magnetic susceptibility  \nat high -frequency magnetic fields , low coercivity, low losses through Eddy currents and hyst eresis \nloops, Curie temperature close to room temperature and the main reason, the tuning  of all these \nproperties by varying the Zn/Mn ratio.  \nIn order to understand the magnetic structure of this ferrite, is important to understand each  atom \nrole in the spinel crystal structure. Four  different  atoms are present in the Mn -Zn ferrite: Oxygen, \nIron, Manganese and Zinc.  \nThe Oxygen anion , O2-, has the fundamental role of bounding all the elements  together. Its electrons \nconfiguration is [He]2s22p6. Although Oxygen ions do not possess unpaired electrons, and \nconsequently no  corresponding  magnetic moment, they are responsible for the superexchange \ninteraction between metal cation s. Accordingly with P. J. Zaag, [41], Oxygen bounds through a single \nMn/Zn:  \nFe: \nO: \nFigure 3.1 – a) Normal  spinel crystal structure of  Mn 1-xZnxFe2O4. b) tetrahedral (B) and octa hedral \n(A) sites in Mn -Zn ferrite. Atoms and site captions are in the right side of each image. Vesta \nsoftware was used for unit cell 3D rende ring.   \nA site:  \nB site:  \na) \n \na) \nb) \n \nb) André Horta  \nUniversidade de Aveiro   15 p-orbital, which makes  180o between metal cations, thus , the shared electrons have opposite spins  \nand are responsible for the antiferromagnetic order.  \nThe Iron cation, Fe3+, has an electronic configuration of [Ar]3d5. Due to the high -spin d -orbital, it \ncontributes with magnetic moment for the material, with a value of 5  µB. The Manganese cation, Mn2+, \nhas an electronic configuration of [Ar]  3d5, same as Iron, co ntributing with approximately the same \nmagnetic moment, 5  µB. The Zn cation, Zn2+, has an electronic configuration of [Ar]3d10. With the \nvalence orbitals fully paired, this cation does not  contribute with magnetic moment . \nAs it might be predicted, the magnetic structure of Mn and Zn ferrite s strongly differ, consequence of  \nthe different magnetic moment of Mn and Zn cations.  An individual  analysis is present  in the next \nparagraphs.  \nThe Mn fe rrite, composed of magnetic Fe and Mn cations,  which are bounded antiferromagnetically \nthrough the oxygen anion , thus  present ing a ferrimagnetic behavior , figure 3.2.a) . The Fe-Fe \nmagnetic moments are aligned, the Mn -Mn magnetic moments are also aligned, h owever, the Fe -\nMn magnetic moments are antiparall elly aligned. This ferrite is expected to present a ferrimagnetic \nbehavior since the Fe cations are twice the amount of  Mn cations.  \nThe Zn ferrite,  which is  composed of magnetic Fe and non -magnetic Zn,  bound ed antiparallelly \nthrough Oxygen bonds. An enhanc ed overall magnetic moment  is expected, since  only parallelly \nalign ed Iron moments contribute for the materials’ magnetization. However,  this ferrite is barely \nmagnetic,  as discussed  by H. L. Anderson  et al, [42], the weak magnetic ordering is responsibility of \nthe inversion parameter as Fe3+ cations occupy tetrahedral sites . The  Fe magnetic moments in the \noctahedral site coupl e antiparallel. It is also reported that a low value of  saturat ion magnetization was \nobtained, which was caused by Fe located in tetrahedral site, due to the spinel inversion. The \nantiparallel coupling of Iron cations confers to this ferrite an antiferromag netic behavior . \nThe intermediate compositions have a ferrimag netic behavior due to the presence of Mn cations. An \nincrease of magnetization is expected while going  from the antiferromagnetic Zn ferrite to the \nferrimagnetic Mn ferrite. Besides, it is also reported by H. L. Anderson et al, that Mn ferrite have the \neasy axis along the <111> direction.  \n \n \n \n \n. \n \n \n \n \n \n \nThe Zn ferrite present s an antiferromagnetic behavior, with a respective Néel temperature, and the \nMn ferrite a ferrimagnetic behavior, with a respective Curie temperature. From this point on, when \nmentioni ng the full range of compositions of Mn 1-xZnxFe2O4, the temperature that  these compounds \nreturn/cease to have a magnetic order will be denominated “magnetic ordering temperature”.  \nThe different magnetic structures of Mn ferrite and Zn ferrite causes its ma gnetic properties to be \nhighly composition dependent. We aim to take adv antage of this dependency to observe the \nmagnetic ordering temperature varying with the ferrite composition, as observed by P. H. Nam, [43]  \nNext chapter, Experimental Procedure, explores the synthesis methods, characterization technique s \nand the data treatment methodology.  Figure 3.2 –Magnetic structure of  normal spinel Mn 1-xZnxFe2O4. Tetrahedral and octahedral magnetic \nmoments are represented by vector aligned antiparallelly in <111> . The unit cell schematic displays a \nvector length equal for tetrahedral and octahedral sites, which is only true for MnFe 2O4.   \nOtimização da síntese de ferrites para aplicações em fluidos magnéticos  \n16  Departamento de Física  Chapter 4:  EXPERIMENTAL PROCEDURE  \nThe “Experimental procedure” chapter includes details concerning the sample synthesis and the \ncharacterization techniques used, which are XRD, SEM, TEM, SQUID and magnetic induction \nheating.  \n4.1 SAMP LE SYNTHESIS   \n4.1.1  SOL-GEL AUTO-COMBUSTION METHOD  \nThe s ol-gel auto -combustion method is  a synthesis method, capable of producing different nano \npowders by an exothermic and auto -sustained reaction between metal salts and an organic \ncomponent. The simplified che mical reaction is written below:  \n2 Fe(NO 3)3 (aq) + (1 -x) Mn(NO 3)2 (aq) + x Zn(NO 3)2 (aq) + 2,2(2) C 6H8O7 (aq) +NH 4OH  (aq)   \n→ drying → auto -combustion trigger →  Mn 1-xZnxFe2O4 (s) + H 2O (g) + CO 2 (g) + NO x (g) + energy  \nAll nitrates were bought in Chem -Lab, Zn and Fe nitrate have a purity of 98+% and Mn nitrate 97+%, \ncitric acid was bought from Sigma -Aldrich and has a purity of 9 8+% and ammonium hydroxide at \n25% was bought in  Chem -Lab. The procedure starts by weighing  hydrated nitrates  salts of Fe, Mn \nand Zn,  followed by their dissolution in de -ionized water.  As presented by the chemical equation, the \nnumber of nitrates mols will influence the ferrite composition and its amount. Then, citric acid amount \nis weighed, dissolv ed and added in the solution. Citric acid has a fundamental role in this synthesis \nmethod, as it works as a reductant  for the salts (oxidants). However, there is not a consensus in the \nliterature about the amount of citric acid that should be added. Dong L imin et al,  [44], reported ferrite \nsynthesis with 3 mols of citric acid (1 mol of C6H8O7 per mol of metal ion), while A. Sutka et al, [45], \ndefended that the oxidation number must be equal  to the reduct ant number, in accordance with the \nvalence electrons of the compounds.  Both strategies were tried,  and the best results were analyzed .  \nThe aqueous solution of metal salts and organic component is called solution, or sol. The solution is \nstirred at 400 RPM at  a temperature between 60 and 70 oC. Then, ammonium hydroxide ( NH 4OH) \nis added to the solution dropwise until the solution ph is adjusted to 7. the pH influences the ignition \ntemperature and the fullness of the reaction, [46]. When NH 4OH  interacts with the solution, the \nhydroxide reacts with the acidic solution,  releasing heat and  leading to the evaporation  of water (in \nvapor form ), then ammoniu m nitrate (NH 4NO 3) precipitates together with other amorphous \nprecursors. During the pH adjustment , the sol was kept at a temperature below 70 oC (controlled by \na thermocouple). The sol was let dry in the hotplate at 120oC. Almost fully dehydrated, the sol ution \nis now a ge l, with the appearance  of a black mud . The gel is kept in the oven until fully dehydration \ntakes place.  \nThe auto -combustion is conducted in a hot -plate (inside a fume hood) at a temperature of 250oC. \nWhen ignition starts, ammonium nitrate  starts to decompose, generating heat and ions in the form of \na flame. The metallic ions are under the tempera ture conditions where ferrite synthesis is possible. \nThe combustion temperatures can vary from 600 to 1350oC, [45]. Mn -Zn ferrite is created as a \ncombustion ash. The ash is then m illed using a mortar until a fine powder is achieved.    \n \n2 Fe(NO 3)3.9H 2O \n(1-x) Mn(NO 3)2.4H 2O \nx Zn(NO 3)2.6H 2O \n \n2.2 C 6H8O7.H2O \n \n275 oC \n 60-70 oC \npH adjustment  \nwith ammonia:  \n \npH adjustment  \nReagents:  \n \nHot-Plate \nand Oven dry:  \n \nCombustion:  \n Samples:  \n \nSamples:  \n \nSamples:  \n \nSamples:  MnFe 2O4 \nMn 0,8Zn0,2Fe2O4 \nMn 0,5Zn0,5Fe2O4 \nMn 0,2Zn0,8Fe2O4 \nZnFe 2O4 \n \n❖ MnFe 2O4 \n❖ Mn 0,8Zn0,2Fe2O4 \n❖ Mn 0,5Zn0,5Fe2O4 \n❖ Mn 0,2Zn0,8Fe2O4 \n❖ ZnFe 2O4 \n \n❖ MnFe 2O4 \n❖ Mn 0,8Zn0,2Fe2O4 \n❖ Mn 0,5Zn0,5Fe2O4 \n❖ Mn 0,2Zn0,8Fe2O4 \n❖ ZnFe 2O4 \n60-70 oC \nFigure 4.1 – Sol-gel auto -combustion method - experimental scheme.  \n André Horta  \nUniversidade de Aveiro   17 4.1.1.1  SAMPLES  \nSamples of Mn -Zn ferrite,  Mn 1-xZnxFe2O4, of composition x = 0; 0 .2; 0.5; 0.8 and 1 were synthesized. \nIt was no ticed the presence of secondary phase of zinc oxide (ZnO) for Mn 0.2Zn0.8Fe2O4. A secondary \nphase is not desired because it could compromise  the main ferrite  properties . Efforts for its reduction \nwere made. Table 4.1 shows the synthesis parameters and how  these  parameters  were  manipulated.  \nParameter  Control  Parameter  Control  Parameter  Control  \nFerrite composition (x)  0 - 1 Citric acid amount  1; 2.2; 3 mol  pH adjustment  Fixed (7)  \nCombustion \ntemperature  Free Drying \ntemperature  0 - 120 Drying time  0 – 4 days  \nCombustion time  Free Combustion \natmos phere free Temperature \nset RT, C6H8O7, \nNH 4 \nTable 4.1 – Available  synthesis parameters  and respective control . Controlled parameters are \nidentified by a range of used value s. Free means that there was no control over the parameter .  \nMultiple testing for Mn 0.2Zn0,8Fe2O4 revealed that the samples that did not undergo the dehydration \nprocess presented higher single -phase nature (>88%). The results presented in the next chapter for \nsol-gel auto -combustion samples did not undergo  the oven dehydration process. The amount of citric \nacid used is 2.2 mol.  The temperature was increased before citric acid was added to the solution. \nAfter , samples were milled using a mortar to obtain a finer powder.  \n4.1.2  HYDROTHERMAL METHOD  \nHydrothermal method is a chemical method for fine powders synthesis. Its advantages over other \nmethods are the simplicity of the technique, control over the reaction and acquisition of high -quality \nnanoparticles, as electe d from I. Sharifi et al , [20] (table 5) . The simplified chemical equation is the \nfollowing:  \n2 Fe(NO 3)3 (aq) + (1 -x) Mn(NO 3)2 (aq) + x Zn(NO 3)2 (aq) + y NaOH  (aq) +  pressure +  \n+ temperature → (1-z-w) Mn 1-xZnxFe2O4 (s) + y NaNO 3 (aq) + z Fe2O3 (s) + w FeOOH (s)  \nThe experimental procedure follows the recipe presented in the article by Xin Li et al, [47]. It starts \nby calculating the stoichiometric masses of the nitrat es salt to obtain 4mmol of Mn -Zn ferrite. The \nsalts are dissolved in 80 mL of de -ionized water. The amount of NaOH, 1 .3877mg per 15mL,  these \nvalues were obtained in Xin Li et al article . NaOH initially defines the pH of the solution, settled to \n12, and its  exothermic contact with the w ater creates a series of compounds, some soluble in water, \nothers insoluble. See more about hydrothermal precursors  in appendix A: Autoclave time.  \nThe autoclave is kept in the oven for 6h at 180oC, the autoclave is filled wi th a maximum of 120mL \nand have a capacity of 150mL. This causes the solvent (water) to enter a supercritical fluid state. In \nthis state, liquid or gas are indistinguishable. The pressure inside the PTFE container should surpass \n3MPa , the maximum advice tem perature is below 2 20oC, [48]. This extreme conditions of controlled \npressure and temperature allows the solution ions to crystalize in Mn -Zn ferrite and with the proper \ntime, temperature and pH, minimize se condary phases . Figure 4.2 is a schematic o f the hydrothermal \nmethod.  \n \n2 Fe(NO 3)3.9H 2O \n(1-x) Mn(NO 3)2.4H 2O \nx Zn(NO 3)2.6H 2O \n \n1.3877g  NaOH  \n ❖ MnFe 2O4: \n \n \n \n❖ Mn 0,5Zn0,5Fe2O4 \n❖ Mn 0,2Zn0,8Fe2O4 \n❖ ZnFe 2O4: \n \n \n \n❖ Mn 0,5Zn0,5Fe2O4 \n❖ Mn 0,2Zn0,8Fe2O4 \n❖ ZnFe 2O4: \n \n❖ MnFe 2O4: \n \n \n Reagents:  \n \nReagents:  \n \nReagents:  \n \nReagents:  \nSamples:  \n \nSamples:  \n \nSamples:  \n \nSamples:  NaOH addition:  \n \nNaOH addition:  \n \nNaOH addition:  \n \nNaOH addition:  \nAutoclave for 6h at 180oC: \n60-70 oC \nFigure 4.2 – Hydrothermal method - experimental scheme.  \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n18  Departamento de Física  \nFigure 4.4 – Bragg  law: X-ray diffraction schematic.  4.1.2.1  SAMPLES  \nHydrothermal samples with Mn 1-xZnxFe2O4 (x = 0; 0 .5; 0.8; 1) compositions were prepared at \nconstant temperature (180oC), constant pH (12 ) and variable time in autoclave (0, 6 and 21 hours). \nSamples synthe sized for 6h presented the highest percentage of pure phase, for this reason, only \nthe results of these samples will be included in this thesis. Samples prepared by the autoclave \nprocedure  for 21h hours are not included in the core of this thesis but can be found in Appendix A: \nAutoclave Time   \n4.2 EXPERIMENTAL TECHNIQUES  \n4.2.1  X-RAY DIFFRACTION (XRD)  \nTo measure sample’s crystallinity and inspect the crystallographic phases present the X -ray \ndiffra ctometer Pan’Analytical EMPYREAN was used. Figure 4.3.a) is a picture of the diffractometer  \nand 5.2.b) , [49], a schematic of the Bragg -Brentano geometry .  \n \n \n \n \n \n  \n \n \nThe obtained  XRD diffraction patterns, were accomplished by directing an X -ray beam to the sample. \nSamples parallel atomic planes ( with associated  Miller indices) reflect the beam. The X -rays coming \nfrom parallel atomic planes interfere, due to X -ray wavelength being of the same scale as the \ndistance between atomic planes ( Angstroms). The diffraction pattern is the result of constructive and \ndestructive interference of x -ray waves coming from the sample. By scanning the spherical diffraction \npattern that surrounds the sample, maximum intensity peaks are displayed revealing interatomic \ndistances, d. The interatomic distance can be calculate d using Bragg law , equation 4.1 . Figure 4. 4 \npresents a schematic of the  Bragg law with two X-ray beams  diffracted in  parallel crystal lographic \nplanes.  \n \n          2 𝑑 𝑠𝑒𝑛(Ɵ)=𝑛 𝜆  (4.1) \n \n \n \nThe X -ray beam is generated in a Copper anode  when a high ener getic beam of electrons is directed \nagainst it, exciting the Cu atoms, which emit X -rays in k -α and k -β wavebands. Further in the \ndiffractometer, a nickel filter attenuates the k -β peak, whereas  kα emission remains. k-α emission is \ncompose d of k-α1 wavelen gth of 1.540598 Å and a less intense emission of k -α2 at 1.544418 Å \n(typically  k-α2/k-α1 ~0.5) . The sample rotates Ө and the photodetector also moves, Ө, radially, with \nthe sample. The diffractogram x -axis units are in 2Ө, which is the ang le between emitte d beam and \ndetected beam.  This configuration is known as Bragg -Brentano, figure 4. 3.b). Slits and beam knife \npurpose is to direct the beam to the sample, controlling the area of interaction and the beam area \nthat reaches the detector. The sample -holder rot ates around itself to ensure the maximum detection \nfrom all orientations  present in the analyzed powder . \na) \n \nb) \nFigure 4.3 – a) X-Ray diffractometer and components. b) Bragg -Brentano geometry: 1 – Cathodic \nrays’ tube; 2 – collimator; 3 and 4 – Slits; 5 - Sample holder; 6 – Goniometer; 7 – Beam knife; 8 – \nSlit; 9 – Collimator and Nickel filter; 10 – X-Ray phot odetector.  André Horta  \nUniversidade de Aveiro   19 The X -ray diffraction pattern gives different information about the crystal structure. Using Highscore \nPlus, Fullprof Suite and Origin  softwares, infor mation about the crystal phases, lattice constants and \ncrystallite sizes were obtained. The diffractogram analysis is explained below.  \n4.2.1.1  SAMPLE PREPARATION  \nThe powder was placed in a Silicon wafer (Si single crystal), which was cut in a spec ific crystal \ndirection to prevent the appearance of Silicon diffracted peaks  at 2Ө angles inferior to 100o.. The \nsamples are gently pressed against the wafer to ensure a smooth surface and then are placed below \nthe beam knife.  The powders should be fine, in order to ensure that the diffracted beam is a result of \nthe randomly orien ted atomic planes. The sample surface should be smooth for the diffraction to \noccur at the same height throughout the whole sample.  \n4.2.1.1  PHASE IDENTIFICATION  \nDifferent crystallographic structures were identified with resource to High Score Plus software. The \ndiffractogram is composed of several peaks, each peak corresponding to an interplanar distance, \nidentified by a Miller index. Every crystal structure present s a group of Miller indices. To identify the \nmaterials, present in the sample, High Score Plus compar es the sample diffractogram with a \ndatabase of diffractograms and rates their proximity by analyzing the similarity between Miller indices \nof different ato mic structures. The user is then allowed to manually select which atomic structural \nphases are presen t in the diffractogram.  \nThe percentage of phases was estimated by  Rietveld refinement with Fullprof Suit e software  \npackage , as it o ffers control over fitting and consequently better fitting results. Rietveld refinement \nrequires initial information of the  present phases in order to enable the fit, such as atomic positions, \nelements, space group, lattice constants and others. This infor mation is available in free data  bases , \nsuch as  Crystallographic Open Database , used in this work . The structural informati on, or .cif files, \nwere obtained and imported into Fullprof Suite in order to create a .pcr file for each phase. The .pcr \nfiles of al l phases are then merged in a single .pcr file. The final .pcr file is then loaded to Fullprof \nand fitted with the diffract ogram. The output is .sum file with crystallographic information. The \npercentage of phases is automatically calculated based on the s cale of each diffractogram. Figure \n4.5.a shows the Rietveld refinement  of single -phase  Mn Ferrite  from sol -gel auto -combust ion \nmethod . Figure 4.5.b) shows the Rietveld refinement of Mn 0.2Zn0.8Fe2O4 of hydrothermal method, the \nrefinement of both spinel crystal structure and impurity (hematite).  \n \n \n \n \n \n \n \n4.2.1.1  LATTICE CONSTANT  \nThe lattice constant was also fitted using Fullprof suit e software. The lattice constant is calculated \nwith resource to the interplanar distances of a unit cell, the interplanar distance is related to the \npeaks’ positions in the 2 θ-axis, as previously mentioned. Changes of interplanar distances are visible \nas sh ifts in the peak position (2 Ɵ). Full Prof Suit e software uses the atomic positions and space group \ninformation to calculate the unit cell lattice constant . \nThe lattice constant is sensitive to different phenomena, such as synthesis conditions, pressure, \ntemperature, interatomic interactions and nanoparticles size . Also, nanoparticles differ from the bulk \nmaterial as the ratio surface/volume is larger in nanoparticles, being more likely to suffer strains due \n* \n* \n* \n * \n* \n* \na) \n \nb) \nFigure 4.5 – Rietveld refinement  of: a) MnFe 2O4 (sol-gel) and b) Mn 0.2Zn0.8Fe2O4 (hydrothermal). Asterisks \nidentifies Fe 3O4 phase. Red dots are for experimental data, black line is the Ri etveld refinement, blue line is the \ndifference between data and fitting and green lines identify the peaks position. Chi2 is the goodness of fitting.  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n20  Departamento de Física  to the pressure of  the surrounding medium . Lattice constant in bulk samples also change when \npressure is applied, however, a bulk material lattice constant does  not change as much as the same \nmaterial in nanoparticle form , [50]. \n4.2.1.1  OTHER RESULTS FROM RIETVELD REFINEMENT  \nThe main results obtained from  Rietveld refinement are the phases quantification, their lattice \nconstants and atomic positions . However , the Rietveld refinement from Fullprof Suite software \noutputs other fitted  parameters . Some of these parameters  are identified in table 4. 2. This info rmation \nis available in the .sum file exported after the Rietveld refinement.  \nAtomic Parameters:  Cell Parameters:  Fitting parameters:  Others:  \nAtomic Position  a, b, c  scale factor  Volumic mass  \nOccupancy  alpha,  beta, gama  Peaks position and \nFWHM  Temperatur e effects  \n  unit cell volume  Sharpest peaks : position \nand FWHM  \nTable 4.2 – Crystal phases information exported from the Rietveld refinement.  \nThe Rieveld refinement quality is identified for each phase and fo r the complete diffractogram by \nRWP, R exp and χ2 (Chi squared), [51]. Several iterations with different fitting parameters were \nperformed in order to decrease these values, increasing the fit quality.  \n4.2.1.2  CRYSTALLITE SIZE AND STRAIN  \nCrystallite size is defined as the mean size of the single crystals th at compose the sample. A single \ncrystal is the coherent length throughout which the unit cell repeats periodically with the same \norientation. When an X -ray beam interacts with parallel a tomic planes it creates an interference \npattern. If an atomic plane re peats itself throughout a long distance, the more X -rays reflected from \nthat plane will interfere constructively, and an increase in the intensity and narrowing of the peak is \nexpected.  \nIt was G.K. Williamson and W.  H. Hall who merged the Scherrer equatio n (for crystallite size) and \nStokes and Wilson equation (for strain), creating the Williamson -Hall equation , [52]. The Williamson -\nHall equation  (H-W) relates the experimental band broadening of the peaks with the crystallite s ize \nand strain as displayed in equation 4.1,  [53]. \nβ𝑇𝑂𝑇𝐴𝐿= β𝑆𝐼𝑍𝐸+β𝑆𝑇𝑅𝐴𝐼𝑁=𝐾.𝜆\nd𝑋𝑅𝐷.cos (Ө) + 4.𝜂.tan (Ө)   (4.1) \nWhere βTOTAL is the experimental FWHM, β𝑆𝐼𝑍𝐸 and β𝑆𝑇𝑅𝐴𝐼𝑁  are the FWHM contribution of crystallite \nsize and strain, respectively. K is the Debye -Scherr constant (~0 .94 for  spherical nanoparticles), 𝜆 is \nx-rays wavelength, d𝑋𝑅𝐷 is the crystallite size, 𝜂 is the crystal strain and Ө is the angle (in radians) of \nthe peak.  For this analysis Origin software was used for peak fitting and calculations. Peaks are fitted \nusing a pseudo -Voigt function. The fit automatically outputs the FWHM, or β𝑇𝑂𝑇𝐴𝐿 , and the peak \nposition, 2 Ө. Figure 4.6.a is the fitting of a p eak with a pseudo -voigt function. This procedure is \nrepeated for a characteristic set of peaks (typically th e most intense). Hence, the obtained fitting \nresults are plotted according with the Williamson -Hall equation by making 𝑦= β𝑇𝑂𝑇𝐴𝐿.cos(Ө)\n𝐾.𝜆, and 𝑥=\n 4.sen(Ө)\n𝐾.𝜆. In this plot, slope is m = 𝜂 and the interception is b = 1\nd𝑋𝑅𝐷.  Figure 4.6.b presents the \nlinearization using W -H equation for Mn Ferrite.  \n \n \n \n André Horta  \nUniversidade de Aveiro   21 \na) \n \na) \nb) \n \nb) \nFigure 4.6 – a) Peak fitt ing using Origin. b) Williamson -Hall linear fit.   \n \n \n \n \n \n4.2.1.3  SYSTEMATIC ERRORS  \nFor Williamson -Hall analysis, fit ting was performed  with Lorentz , Gaussian  and pseudo -Voigt \nfunction s. The one which presented a better value of R2 was the pseudo -Voigt function . All the fitted \npeaks presented a R2 value greater than 90%.  \nThe Williamson -Hall analysis is based on the linearization described in section  4.2.1.2.  As in every \nlinearization the value R2 is  and has an erratic contribu tion for the calculated properties, this means \nthat the crystallite size ha s an error associated with the linearization. When linearized in Origin, it \npresents the errors associated with the interception value, the error in crystallite size was calculated \nas presented in equation 4.2 . \n𝛥𝐷𝑋𝑅𝐷=𝜕\n𝜕𝑏𝐷𝑋𝑅𝐷 .𝛥𝑏=1\n𝑏2 .𝛥𝑏    (4.2) \nCrystallite size  values , dXRD ± 𝛥d𝑋𝑅𝐷, are presented in the crystallite size plots . \nAn amorphous/nanocrystalline material has very small crystall ites which results  in ve ry broadened  \npeaks, sometimes just very spread humps. Amorphous phases are not identified by XRD, usually \ncontributing as  background for the diffraction pattern . \n4.2.1.3.1  Experimental Line Broadening  \nThe experimental line broadening is an equipment limitation that affects the Williamson -Hall analysis. \nThe experimental line broadening can be experienced while making XRD in a monocrystal. If the \ncrystal size is orders of magnitude larger than the wavelength, β𝑇𝑂𝑇𝐴𝐿  should approach zero (a Dirac -\ndelta function), h owever this is not observable as the diffraction pattern peaks will always have a \nFHWM greater than zero. This phenomenon occur s for multiple reasons, for example due to the \ndetector limited ape rture or atoms not being in their equilibrium position (therma l kinetics, for \nexample).  \nThe experimental line broadening was calculated for a monocrystal of Lanthanum Hexaboride \n(LaB 6). Being a monocrystal, the FWHM of the peaks should give a Dirac -delta  function. The analysis \nof this crystal is important to understand de error of β𝑇𝑂𝑇𝐴𝐿  and the maximum coherence length \ndetectable by this diffractometer. The peak fitting for this s ample is present in figure 4. 7. \n \n \n \n \n \n \n \n \nFrom the Williamson -Hall analysis performed on this diffractogram, it was obtained the maximum \nsize of cry stallite detectable by this diffractometer: 120.62 ± 4.55 nm. The peaks are broadened as \nthe angle increases: from 15 to 80 degrees the FWHM value var ies from 0.00952º to 0.0059º (in 2 Ɵ). \na) \n \na) \nb) \nFigure 4.7 – (a) Diffractogram of LiB 6 with inset for K -α1 and K - α2 distinguished. (b) Linear fit of \nWilliamson -Hall equation.  \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n22  Departamento de Física  4.2.2  SCANNING ELECTRON MICROSCOPY (SEM)  AND ENERGY DISPERSIVE SPECTROSCOP Y (EDS)  \nScanning Electron Microscopy (SEM) and Energy Dispersive Spectroscopy Scanning electron \nmicroscope (SEM) is a technique  that relies on electro magnetically accelerated electrons and their \ninelastic collision with the sample for imaging processing. E nergy dispersive spectroscopy (EDS) is \nbased on radiation emission from the sample, when excited with high energy electrons.  During this \nwork, the SEM  equipment available, and capable of analyzing magnetic nanoparticles, was Hitachi \nS-4100 system, coupled with an EDS sensor. The manual of a close model, S -4800, can be f ound \nin [54]. \n4.2.2.1  SCANNING ELECTRON MICROSCOPY (SEM)  \nThe scanning electron microscope (SEM) is a complex tool for nano and mic ro-scale imaging. It \nconsists o n the detection of electrons emitted from a sample when interacting with an electron beam. \nThe SEM consists o f a vacuum  chamber containing  an electron gun, a series of electromagnetic \nlens, apertures  and detectors . The high e nergy electrons are produced in the electron gun and \naccelerated by a potential difference. The high -speed electrons are then selected in an aperture and \ngo through a series of condensing lens and apertures for beam focusing. The beam is finally \ndeflected by the deflecting lenses, in order to scan the sample, while the beam intensity is controlled \nby the last aperture.  Sample rotation and tilting motors  allows sample observation from various \nangles. It is schematically represented in figure 4.8.a.  \nWhen th e electrons reach the sampl e, figure 4.8.b), different interactions take place within the depth \nof penetration. Electrons will inelastically collide with the sample lattice, which will originate different \nemissions: backscattered electrons, secondary elect rons, Auger electrons, char acteristic x -ray, and \nothers.  Image formation occurs when the sample is scanned. At each sample point, a quantity of \nbackscattered and secondary electrons is detected , [54]. The final image pixel intensity is defined by \nthe number of detected electrons.  \n \n \n \n \n \n \n \n \n \n4.2.2.2  SAMPLE PREPARATION  \nThe magnetic powders were held by  carbon tape,  on top of  a silicon wafer, and then coated with \ncarbon. Carbon coating is performed b y putting the samples in a vacuum chamber and sublimating \na carbon wire by Joule effect. The carbon coating is necessary due to the insulator nature of the \npowders. If samples were  not coated with carbon, they would accumulate charge when interacting \nwith the electron beam, deflecting the electron beam and saturating the image brightness to a limit \nwhere no particles would be  distinguishable.  \nThe samples were then attached to SEM pre -chamber to be subjected to vacuum and introduced in \nSEM main chamber for visualization.  \nElectron Beam  \nAuger Electron s  \nCharacteristic X -Ray \nCathodoluminescence  \nSecondary Electrons  \nBackscatered Electrons  \nContinuos X -Ray \nSample  \n \n Inelastic Scattering  \nIncoherent Elastic  \nElastic Scattering  \nTransmitte d Electrons  \nVacuum Pump  \nElectron Gun  \nAnode  \nAperture  \nConden ser Lens  \nAperture  \nCondenser Lens  \nObjective Lens  \nDeflection Coils  \nAperture  \nBackscattered Electrons Detector  \nX-Rays detector  \nSecondary Electrons Detector  \nSample  \na) \n \na) \nb) \nFigure 4.8 - a) Electrons flow within TEM, its components and detectors . b) Different possible \ninteraction s between the sample and the electron beam.  André Horta  \nUniversidade de Aveiro   23 \n1 2 3 4 5 6 7 8 9 10\nkeV0.00.20.40.60.81.01.21.4 cps/eV\n  O   Mn   Mn \n  Fe   Fe \n  Zn   Zn 4.2.2.3  SIZE DISTRIBUTION MEASUREMENTS  \nThe nanoparticles were imaged  using SEM, the ImageJ software was employed to measure the \nparticle area. F igure 4.9.a depicts the image before counting and figure 4.9.b after counting, where \nonly particles wit h a clear limit were measured . The area values and number of particles were then \ntreated using Origin software.  The particle area data was converted into  radius (assuming spherical \nshape) and then diameter.  \n \n \n \n \n \n4.2.3  ENERGY DISPERSIVE SPECTROSCOPY (EDS)  \nEnergy  dispersive spectrometer (EDS) is a sensor attached to SEM  capable of distinguish different \nX-ray energies . As previous mentioned, once the electron beam interacts with the sample, different \ntypes of emission can occur. The EDS sensor receives x -ray atomic  characteristic emissions from \nthe sample and group them accordingly with their energy.  \nX-ray characterist ic emission is created by each atom when a high energy electron interacts with it. \nThe electron beam excite s electrons from the inner electronic leve ls. Consequently, electrons from \nouter orbitals occupy the ir place , releasing energy in the form of x-rays,  with an amount of energy \ndependent of the energy between energy levels . The released radiation is element -specific and \nhence enables the indexing of  the obtained spectrum peaks with respective element .  \nEDS results i n a spectrum, yield photons versus energy (keV), as represented in figure 4.10.a). \nDatabases of x -ray characteristic emission for every element allows the microscope software to \nperform  a fitting of the experimental spectrum whose outcome is the atomic percentage fracti ons \npresent in the sample, as exemplified in figure 4.10.b).  \n \n \n \n \n \n \n4.2.3.1  SYSTEMATIC ERRORS  \nFor lighter elements like Oxygen, the EDS presents a considerable error. For this re ason, when \nanalyzing the samples composition, it was assumed that the Oxygen was fully integrated in ferrite.  \n4.2.4  EDS  ANALYSIS  \nFor stoichiometry estimation, it was assumed that the Oxygen has 4 elements per molecule and the \nsum of all metal ions results in a t otal of 3 elements, accordingly with ferrite metal/Oxygen ratio: Mn 1-\nxZnxFe2O4. The composition of the sample was then calculated by normalizing the metal ions and \nmultiplying them by 3. Equation 4.3 shows the stoichiometry calculus, per element, for every  sample.   \n#𝑀𝑛=3%𝑀𝑛\n%𝐹𝑒+%𝑀𝑛+%𝑍𝑛 ; #𝐹𝑒=3%𝐹𝑒\n%𝐹𝑒+%𝑀𝑛+%𝑍𝑛  ; #𝐹𝑒=3%𝐹𝑒\n%𝐹𝑒+%𝑀𝑛+%𝑍𝑛  (4.3) \nMissing elements, or excess, are represented by the difference between the measured fraction of \nelements and the expected fraction of elements.  \nEl AN  Series  unn. C norm. C Atom. C Error (1 Sigma) \n               [wt.%]  [wt.%]  [at.%]          [wt.%]  \n-----------------------------------------------------  \nO  8  K-series  19,65   26,17   56,22            3,49  \nMn 25 K-series   3,01    4,01    2,51            0,14  \nFe 26 K-series  38,21   50,91   3 1,33            1,05  \nZn 30 K-series  14,20   18,91    9,94            0,46  \n-----------------------------------------------------  \n        Total:  75,06  100,00  100,00  \n1 2 3 4 5 6 7 8 9 10\nkeV0.00.20.40.60.81.01.21.4 cps/eV\n  O   Mn   Mn \n  Fe   Fe \n  Zn   Zn \na) \n \na) \nb) Figure 4.9 – Particle counting using ImageJ. Black areas are the particles counted for size statistics.  \na) \n \na) \nb) \nFigure 4.10 – EDS results for Mn 0.2Zn0.8Fe2O4. a) EDS spe ctrum – legend is: red for O, green for \nMn, light blue for Zn and dark blue for Fe. b) EDS results table.  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n24  Departamento de Física  4.2.5  TRANSMISSION ELECTRON MICROSCOPY (TEM)  \nTransmission electron microscope (TEM) is a device that relies on e lectromagnetic accelerated \nelectrons being transmitted through a sample for imaging it. TEM is a powerful device for \nnanoparticles imaging d ue to its high resolution, in some cases (high -resolution TEM), capable of \nimaging the atomic lattice of the sample,  [55] \nThe working principle of TEM is the following: electrons are created in a hot -filament (electron gun) \nand accelerated by anodes in sample direction, then a group of condensing lenses focus and direct \nparallelly the be am towards the sample.  The h igh-energy electrons , that reach the sample, have a \nshort wavelength , which allows them to diffract in the atomic lattice , thus creat ing a reciprocal -space \nimage or a real -space image . Changing the focus of the condensing lenses  the user can change \nbetween a reciprocal -space or a real -space image,  [56]. During this work, the TEM equipment \navailable, and capable of magnetic nanoparticles visualizat ion was H9000 Hitachi , figure 4.11.  \n \n \n \n \n \n \n \n \n4.2.5.1  SAMPLE PREPARATION  \nThe preparation of nanoparticles for TEM starts with the dilution of a small fraction of powder in \nethanol, then the prepared solution goes to an ultrasonic  bath to favor particle dispers ion in ethanol \nand prevent particle agglomeration. When the powders are sufficiently diluted a TEM support grid is \ndive inside the solution and is let to dry. The used TEM  uses Copper , or Carbon , grids as support for \nthe nano particles . The grids must be co nductive in order to prevent electrical charging of the sample. \nThe grids were  mounted in TEM support and attached to the microscope.  \n4.2.5.2  SIZE DISTRIBUTION   \nNanoparticles’ area was acquired using  ImageJ software . This software outputs the count s and the \narea of a manually surrounded nanoparticles. These results allow ed the estimation of the \nnanoparticles’ diameter which is then displayed in a histogram  of coun ts as a function of particle size, \nfigure 4.12. The mean grain size, 〈𝑑𝑇𝐸𝑀〉 is obtained by adjusting the size distribution to a Lorentz \nfunction. The standard deviation, σ, is obtained using the equation 4.4, where µ is the mean size and \nxi is the size o f each nanoparticle.  \n \n   \nσ= √1\n𝑁 ∑(𝑥𝑛−µ)2 𝑛\n0  (4.4) \n \n \n4.2.6  SUPERCONDUCTING QUANTUM INTERFERENCE DEVICE (SQUID)  \nThe S uperconducting Quantum Interference Device (SQUID) is a highly sensitive magnetometer with \na Josephson junction which allows the measurement of magnetic fluxes down to the 10-17 Wbm . Its \na) \n b) \n c) \na) \n b) Figure 4.11 – a) TEM and its components . b) Sample -holder (Carbon grids) macro and micro \nscale  c) Electrons’ path in TEM, components used: 1 – Electrons’ source; 2 – Condensing leans; 3 \n- Sample; 4 – Objective lens; 5 – Objective lens aperture; 6 – Intermedi ate lens; 7 – Projector lens; \n8 – Fluorescent display.  \nFigure 4.12 – a) Nanoparticle count; b) Lorentz fit.  André Horta  \nUniversidade de Aveiro   25 incomp arable sensitivity makes it the election magnetometer for a vast range of applications, from \nmeasurements of materials magnetic properties to 3D mapping of the brain.  \nThe model of the used SQUID is a MPMS -3 (Quantum Design, Inc.),  figure 4.13.a), with a \nmagnetization sensibility of 5x10-8 emu and capable of producing magnetic fields up to  7T, [57]. \nFigure 4.13.b) is a schematic of the main cha mber, with the 4 superconducting coils presented, \nconnected to a Joseph -junction,  [58]. \n \n \n \n \n \n \n \nA SQUID is a complex and versatile magnetometer capable of working in a vast range of \ntemperatures, fields and materials. It is c omposed of many different components, sensors and four \nmain coils. Superconducting w ires are connected to the induction coils and to the Josephson -\njunctions. A DC current creates a magnetic field, magnetizing the sample. The sample is measured \nby the Josep hson -junctions as it oscillates in the vertical axis of the magnetometer. While movi ng, \nthe sample inducts a magnetic field flux which is detected by the Josephson -junctions. A series of \nmeasurements, and sequences, can be pre -programmed using the magnetom eter software. Inside \nthe program sequence it is possible to make measurements while  changing variables. For the \npurpose of this work, temperature, T, and applied field, H, are the variables and sample \nmagnetization, M, is outputted by the magnetometer as M(T) and M(H).  \nThe Josephson -junction working principle is based on the Josephson ef fect. A Josephson -junction \nis made of 2 superconducting layers separated by a thin insulator layer. Josephson predicted \nmathematically how the supercurrent phase changes wh en tunneling through the insulator layer, \n[59]. The Josep hson -junction presents an impedance and capacitance, thus it resonates at some \nfrequ encies. The flux change generated by the vibrating samples is acquired by the Josephson -\njunction as a phase difference between the junction sides, resulting in a destructiv e or constructive \ninterference of spin -waves. This extremely sensitive device is the  reason why SQUID magnetometer \nis capable of distinguishing magnetic fluxes of the magnetic flux quanta order (h/2e), [60].  \nThe function of the nitrogen and helium reservoirs is to cool the Jose phson -junction and the induction \ncoils (keeping them in a superconducting state) and to control the sample temperature, the \nmagnetometer is also provided with resistors for heating. The vacuum pump is connected to the main \nchamber and to the antechamber. T he main chamber is always kept in vacuum, the antechamber \nmust be in vacuum before passing the sample to the main chamber. The vacuum pump is an \nimportant component of the magnetometer f or various reasons: the magnetometer is composed of \nsmall tubes for co oling liquids delivery, the tubes are so thin that they can easily clog if a solid \ncompound freezes in the tube. Also, at liquid helium temperature (boiling point at ~4,4 K) some \ngaseous  compounds can crystalize, for example oxygen, that in the solid state  presents a magnetic \ntransition from antiferromagnet to ferromagnet at 43 K, [61]- oxygen impurity can contribute with a \nparasite signal.  \nA series of measurements, and  sequences, can be pre -programmed using the magnetometer \nsoftware. The used sequences consisted in measurement of the M(H) and M(T) curves of Mn -Zn \nferrite.  \n4.2.6.1  SAMPLE PREPARATION  \nPowder samples are encapsulated and weighed.  Due to their ferromagnetic behavior they are h eld \ninto a sample holder of brass with Kapton tape - both have a paramagnetic behavior with low \nsusceptibility in order not to interfere with the sample signal. The sample holder is kept in the Figure 4.13 – (a) SQUID MPMS -3 (Quantum Design, Inc.). (b) Schematic of the SQUID \nsuperconducting coils connected to a Josephson -junction.  \nPickup Coil  \n \n \na) \n b) \nJosephson Junction  \n \n \n \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n26  Departamento de Física  antechamber until vacuum leve ls of main chamber and antechamber are equal. The sample is then \nmoved to the main chamber and centered between coils with resource to a small applied field and a \nM(z) curve. Once centered, oscillation amplitude and frequency must be adjusted to the magnet ic \nbehavior of the sample in order not to saturate the detector. Then the  measuring sequenc e is \nselected and the  measurement  can start . Samples measurement sequence starts with  M(T) curves \nand then M(H) curves.  \n4.2.6.1  SQUID  DATA ANALYSIS  \nThe SQUID was used for ob taining M(H)  curves  at 5, 300 and 380 K  and M(T) curves between 5 and \n380K, with an applied  field of 100  Oe. Both data sets give information about the magnetic properties \nof the samples at different temperatures. The extracted information from M(H) curves was saturation \nand remnant magnetization, coercive field, hysteresis area and magnetic susc eptibility. The magnetic \nordering temperature  and blocking temperatures were obtained from the M(T) curves (when visible) . \nAll magnetization (in emu) were normalized  to the sample mass in order to obtain the magnetization \nper mass (emu/g).  \nThe M(H) curves  give information on how the materials magnetization change with the applied field. \nThese curves are taken by starting the magnetic field at 70000 Oe , measurements are made while \ndecreasing  the field until -70000 Oe  and inversing  it to 70000 Oe . The data a cquisition is logarithmic \nclose to 0 T in order to have more points in this region and hence decrease the error on the coercive \nfield and remn ant mag netization estimations close to zero -field. The linear magnetic susceptibility , χ, \nis calculated by lineari zing of M(H) slope for high fields . this constant is related with paramagnetic  \nand antiferromagnetic states, as other misalignment of spins .  \nThe measured  M(T) curves reveal  how the materials magnetization change with the evolution of \ntemperature. Two M(T)  curves were taken, the zero -field cool (ZFC) and field cool (FC). For the ZFC \nmeasurement the sample is cooled until 5 K without a magnetic field ap plied. In this procedure the \nsample is cooled down without any preferred direction for the magnetic particl es. Then, a field of 100 \nOe is applied to the sample and measurements starts , the time between measurements is \napproximately 11 s . FC measurements st art at room temperature, when a field of 100 Oe is applied \nto the sample, followed by the sample cooling do wn. Measurements start when the sample achieves \n5/6 K. The measurements are taken while increasing the temperature, maintaining the 100 Oe \nmagnetic f ield). FC curve must be acquired after the ZFC, as if it was the opposite the samples might \nkeep a preferre d magnetic orientation  which is not desired for the zero -field cool measurements.  \nThe magnetic ordering  temperature was estimated by the linearization of Curie -Weiss law at the \nparamagnetic state . Figure 4.14 shows the used linearization and the H/M(T) p lot. From the \nlinearization it is possible to obtain a close value of the Curie temperature,  the calculated value is Ɵp, \nas assigned in the graph.   \nχ=C\nT−Ɵ𝑃    χ(T−Ɵ𝑃)=C  \n T=𝐶\nχ+Ɵ𝑃 \n𝑦= 𝑇 ; 𝑥=1\nχ=𝐻\n𝑀 \n𝑚=𝐶 ; 𝑏=Ɵ𝑃 \n \n \nThe mean blocking temperature was calculated with resource to the  maximum of the  temperature \nderivative of the difference bet ween FC and ZFC magnetization curves, [53], 𝜕(𝐹𝐶−𝑍𝐹𝐶)\n𝜕𝑇. The different \nmagnetization of the FC -ZFC curves is due to the blocked states.  The derivative have a maximum in \nthe mean blocking temperature, 〈𝑇𝑏〉, and the maximum temperature  of the distri bution agrees  with \nthe irreversibility temperature, [10]. Figure 4.14 – a) Curie-Weiss law  linearization to obtain Ɵp. b) Determining  Ɵp using the temperature \ndependence of  the inverse susceptibility  for ZnFe 2O4. \na) \n b) André Horta  \nUniversidade de Aveiro   27 The susceptibility, χP, was obtained from the M(H) curves by linearizing the hysteresis loop at high \nfield intensity , in the regime where the magnetization is almost saturated. By subtracting the \nsusceptibility from M(H) curves, it is possible to decompose the ferrimagnetic h ysteresis loops in a \nferromagnetic loop and a paramagnetic line.  \n4.2.6.2  SYSTEMATIC ERRORS  \nThe SQUID magnetometer is a very sensitive device, that fits the data points while acquiring, for this \nreason the measurements are reliable. However, the machine is calibrat ed for specific shape and \nsize, as the s ample shape influences the materials magnetization and the flux lines passing through \nthe Josephson -junction. This said, a systematic error was detected when samples magnetization \nwas measured with resource to 2 diff erent measuring modes, DC and VSM. The s ample \nmagnetization must remain the same independently of the measuring mode, using this principle a \ncorrection was proposed by  [62]. All data were corrected accordingly with personal communication \nof the correction.  \n4.2.7  MAGNETIC INDUCTION HEATING  \nMagnetic induction heating is a technique that measures the temperature change of a sample while \nsubmitting it to an AC magnetic field. Heating a sample with an AC magnetic field is a phen omenon \nbased on the sample hysteresis lo ops loss, i.e. the inversion of the sample magnetic moment \nreleases energy in the form of heat. Consecutive inversions of the sample magnetization will \ngenerate  energy in heat form which is read by a temperature sens or. Magnetic hysteretic loss is the \nmain energy generation  mechanism, whereas other magnetic  loss mechanism s were previous \nmentioned in section  2.5. \nThe magnetic induction setup was developed by Dr. Nuno João Silva from “CICECO – Aveiro \nInstitute of M aterials” , it is homemade and has an extensive apparatus . Figure 4.15.a) displays  a \nschematic of the system  and figure 4.1 5.b) presents the inductor coil.   \n \n \n \n \n \n \n \n \n \nThe system is composed of a magnetic inductor core in a ring shape with an aperture wh ere the \nsample and reference are placed. The aperture, which is seen in figure 4.1 5.b), has an important \nrole, as the magnetic field lines between this space are parallel , this ensures that, both sample and \nreference, are subjected to the same intensity of  magnetic field.  \nThe current flow starts in an AC generator, goes through an operational amplifier, then into a \ncapacitors system in order to achieve resonance. The descr ibed system has the function of \nincreasing the current intensity passing by the coil, which is directed to the sample by the inductor \ncore. The magnetic field generated by the magnetic core is controlled by the electric current that \nflows in the coil strap ped around the core. For a better control of the magnetic flux, a Hall -sensor is \nattac hed to the wire that turns around the coil, measuring the potential caused by an  inducted flux. \nFor temperature measurements, the magnetic induction heating system has tw o temperature \nsensors, both are composed of an optical fiber with a silic a end.  Both temperatures (sample and \nreference) as a function of time are recorded . It is important that the reference (empty)  sample holder \nSample__  \nReference  \n1) \n 2) \n3) \n 4) \n5) \na) \n b) \nFigure 4.15 – a) Magnet ic induction heating experimental setup. 1) amplifier with a resonant circuit. \n2) Signal generator; 3) Hall sensor; 4) Induction core; 5) Fiber optics therm ometer. b) inductor core  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n28  Departamento de Física  \na) \n b) has a temperature sensor of its own,  because, the AC magnetic field will not only heat the sample \nbut also the induction core . The inductor core temperature, T(t), is subtracted from the sample signal.   \n4.2.7.1  SAMPLE PREPARATION  \nMn-Zn ferrite powder s are p laced into an organic, non-metallic and non -magnetic sample holder. The \nreference sample holder remains empty. Due to the polymeric nature of the sample holder the \nmaximum temperature reached was 80oC. The frequency of the generator is then set into resonance, \nat 364 kHz, and th e magnetic field can be controlled using the system software (a matlab GUI for \ncontrolling the AC generator w as used).  \nBoth sample holders are placed in the induction core aperture, with a temperature sensor inside each \nof them. The measurements can now st art. \n4.2.7.2  DATA ANALYSIS  \nThe raw data of a magnetic induction heating experiment is shown in figure 4.16. The data  file a \nmatlab figure file . To extract  the data, a matlab script was developed . \n \n \n \n \n \nThe difference in the initial temperature is associated with differences in both sensors and it does \nnot compromise the experiment, as the analysis is based on the sample  initial slope. The blue line \nshows the reference sample holder increasing the temperature - it happens due to the increasing \ninduction core temperature.  \nFor plotting the sample heating rate, dT/dt, the reference curve was subtracted from the sample \ncurve, as this is representative of how much sample heated more than the reference. Then, the initial \ntemperature was added to the difference to show the real temperature values achieved by the sample \nunder induction heating mechanism. The heating rate is the n obtained by linearizing the initial slope \nof the magnetic induction heating results, as shown in figure 4.17  \n \n \n  \n  \n \n \n \nSAR (Specific Absorption Rate)  values are calculated with resource to  equation  4.5, [2]: \n𝑆𝐴𝑅=𝐶.∆𝑇\n∆𝑡.1\n𝑚𝑠𝑎𝑚𝑝𝑙𝑒= 𝑐.∆𝑇\n∆𝑡     (4.5) \nWhere C is th e heat capacity of the medium, ∆𝑇\n∆𝑡 is the rate change of the temperature (given by the \ninitial slope of the magnetic induction heating graph) and 𝑚𝑠𝑎𝑚𝑝𝑙𝑒  is the sample mass. Heat capacity \nis usually associated with the medium where t he part icles are suspended, however, the experiment \nwas made only with powder, for this reason, the heat capacity, C, must be calculated with resource \nto the specific heat, c, the relationship is c= 𝐶\n𝑚. The value of specific heat for Mn -Zn ferrite is 750  J \nKg-1 K-1, accordingly to [63] and [64]. \nFigure 4.17 – Linear fit of the initial slope for MnFe 2O4 and ZnFe 2O4. \nFigure 4.16 – Magn etic induction heating results as a function of time. a) Red line for sample. Blue \nline for sample -holder. b) Hall sensor Voltage p -p (V).  \na) \n b) André Horta  \nUniversidade de Aveiro   29 Chapter 5:  RESULTS  AND DISCUSSION  \nThe results of sol-gel auto -combustion  method  and hydrothermal method were independently \nanalyzed by the charac terization techniques detailed i n the previous chapter . After results of sol-gel \nauto-combustion method a brief discussion is  displayed . Follows th e results of hydrothermally \nsynthesized samples , which  are presented  in higher  detail  due to the su perior qua lity of this samples . \nThe discussion  of hydrothermal ly prepared  samples  is presented  after its results . The discussion on \nthe hydrothermal samples intends to correlate results and finishes with a general review  of the Mn-\nZn ferrite  nanoparticles.  \n5.1 RESULTS O F SOL-GEL AUTO-COMBUSTION METHOD  \nSol-gel auto -combustion samples were ch aracterized via XRD, SEM, SQUID and TEM.  Analy zed \nsamples have the composition of Mn 1-xZnxFe2O4 (x= 0; 0 .2; 0.5; 0.8; 1)  and were not  dehydra ted. \nMost results are compared with the res ults from “ Magnetic and Optical Properties of Mn 1-xZnxFe2O4 \nNanoparticles ” of A. Demir et al , [65]. This article is referenced  due to  the similarities in experimental \nprocedur es, full composition range (x = 0  to 1), similar characterization techniques and coherent \nresults.  \n5.1.1  STRUCTURAL CHARACTERIZATION  \nX-ray diffraction was the technique used to explore the crystallographic order and distinguish \ncrystallographic phases. Figure 6.1 shows the XRD patterns for all the  compositions.   \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \nThe diffractograms displayed in figure 5.1.a present the peaks of the spinel crystal structure, space \ngroup fd -3m belonging to Mn -Zn ferrite, and the respective Miller indices are indexed.  The seconda ry \nphase, Zincite (ZnO) , peaks are marked with *. The percentage of ZnO is composition dependent, \nonce the samples with higher Zn/Mn ratio present higher percentage of impurity phase, figure5.1.b).  \nZnFe 2O4 is the composition which presents the highest perc entage of impurity phas e (32%). \nMagnetite (Fe 3O4) can be present in the samples, but it is not distinguishable from Mn -Zn ferrite \ndiffractogram , since it belongs to the same space -group and has the same peaks positions.    \nA shift of the main peak of Zn ferrite for higher angle s in comparison with Mn ferrite was noted, which \npredicts a smaller lattice constant for Zn ferrite, figure 5.2.a). The lattice constant was obtained via \nRietveld refinement. Figure 5.2 is a comparison between the obtained lattic e parameters and \ncrystallit e sizes with those obtained by A. Demir et al,  [65]. Figure 5.1 – a) XRD patterns of Mn 1-xZnxFe2O4. Miller indices of Mn -Zn ferrite spinel crystal structure \nare displayed . ZnO impurity peaks are represented by *. b) percentage of ZnO impurity as a function of \nZn content, inset graph show s the evolution of ZnO peaks in the diffractogram.  \n(220) \n(422)  \n(400)  \n (422)  \n(511) \n(440) \n* \n* \n* \n* \n(111) \n* \n * \n * \n*  \na) \n b) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n30  Departamento de Física  \n \n \n \n \n \n \n \nLattice parameter is in rough agreement with A. Demir et al, varying from 8 .47 to 8 .41 Å with the \nincrease of Zinc content. The crystallite sizes for samples with higher Mn content are more displaced \nfrom A. Demir et al values than ferrites with higher Zn content.  \nZincite (ZnO) lattice constant was obtained from the Rietveld refinement for both samples with more \nimpurity phase (x = 0 .8 and 1).   ZnO has an hexagonal crystal structure, space -group: P6 3mc, lattice \nconstant values for a(=b) of  3.250 Å and c of 5 .207 Å, in  [66]. The Rietveld refinement val ues for \nZnO lattice constant were a(b) = 3.25 and 3.25 Å, and c = 5.21 and 5.27 Å, respectively for x=0.8 \nand 1 samples . Lattice constant s for both compositions agree with bibli ographic values,  [66], [67] \n5.1.2  MORPHOLOGICAL AND CHEMICAL CHARACTERIZATION  \nA selection of the  best SEM images obtained for sol -gel a uto-combustion samples are shown in \nfigure 5.3. \n \n \n \n \n \n \nSEM images cannot distinguish individual nanoparticles even with the maximum amplification \nachievable (x20k , scale of 1 .5 µm). All images present agglomerated particles. The morphology of \nthe powders i s similar between compositions, it is also similar with the  samples reported by A. Demir \nand R. Gimenes et al . \n \n \n \n \n \n \n \nThe EDS results allowed to estimate  the metal ions ratio in the samples,  figure 5.5.  \n \n \n \n \nFigure 5.3 – SEM images of samples : a) MnFe 2O4; b) Mn 0.2Zn0.8Fe2O4; c) ZnFe 2O4. \nFigure 5.5 – Metal ions ratio. a) Mn ratio; b)  Zn ratio; c) Fe ratio. Green line is the sample ratio and  \nblack line is the expected ratio of the predicted st oichiometry.  Figure 5.2 – a) Lattice constant as a function of the Zn content; b) Crystallite size as a function of Zn \ncontent. Green line is for synthesized samples. Black line is for bibliographic samples from A. Demir et al.  \nFigure 5.4 – SEM images from literature: (a), (b) and (c), from A. Demir et al, showing the MnFe 2O4, \nMn 0.2Zn0.8Fe2O4 and ZnFe 2O4, respectively. Image (d), from R. Gimenes et al, with Mn 0.8Zn0.2Fe2O4. \na) \n b) \na) \n b) \n c) \nb) \n a) \n c) \n d) \na) \n b) \n c) André Horta  \nUniversidade de Aveiro   31 \nBy the analysis of figure 5.5 it’s visible that the largest deviation from the expected values occurs for \nZnFe 2O4. Zinc in excess (twice the expected) and Iron  lacking  (50%) might be a consequence of  \nexcess of Zinc and lack of Iro n during the synthesis, contributing for the formation of  ZnO impurit y \nand lack of ferrite phase. Mn is easily incorporated by the samples, with almost no deviation from the \nexpected values. The calculated ratios by EDS strongly correlate with the percenta ge of zincite \nimpurity calculated via XRD . \n5.1.3  MAGNETIC CHARACTERIZATION  \nThe powders were analyzed in SQUID magnetometer. The powders magnetization was measured \nas a function of applied field (M(H) curves) at different temperatures , figure 5.6. The magnetization \nas a function of temperature at 100 Oe  (M(T) curves)  is presen ted in figure 5.8.  \n \n \n \n \n \n \n \n \nThe M(H) loops are expected to present an increase of saturation magnetization from Zinc to \nManganese compositions, accordingly with A. Demir et al,  [65]. This is expected due to the magnetic \nmoment of Mn ion being superior than Zn ion. Mn ferrite has the highest M S at 300 and 380 Kelvin  \nbut, at 5K , Mn 0.5Zn0.5Fe2O4 saturation  magnetization surpass es MnFe 2O4. MS values vary from 77 .8 \nto 43 .8 emu/g with the increase of Zn/Mn ratio, at ro om temperature , figure 5.7.a) . \n \n \n \n \n \n \n \n \nValues of coercive field, H C, and remnant magnetization, M R, do not present the expected tendency \nas can be seen in figure 5.7 .b) and c),  against Demir et al obtained values.  Remnant magnetization \ngenerally seems t o follow the tendency  observed in literature,  [65] [68], however, the composition \nx=0.5 disrupts the decreasing behavior.  \n \n \n \n \n \n \nThe FC curves show a decrease of magnetization with the increase of temperature, which is \nexpected for ferrimagnetic materials. In accordance with the literature on sol -gel auto -combustion \nFigure 5.6 – M(H) curves  at: 5K a), 300K  b) and c) 380K  for Mn1 -xZnxFe2O4 (x = 0; 0.2; 0.5; 0.8; 1). \nInset figure shows the hyste retic loops of the samples.  \nFigure 5.8 – M(T)  curves measured at 100 Oe. a) Field cooled. b) Zero -field cool.  \nFigure 5.7 – Magnetic data obtained from M(H) curves at 300K. (a) Saturation Magnetization; (b) \nCoercive Field; (c) Remnant Magnetization. Green lines are for synthesized samples and black lines for \nthe comparison with samples of A. Demir et al.  \na) \n b) \n c) \na) \nb) \n c) \na) \n b) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n32  Departamento de Física  samples, the magnetic ordering temperature  occurs only for high er temperatures (above the \nmeasurement range, 400K) unless thermal treatments, encapsulation or combustion in alternative \nmediums is performed , [67], [69], [70], [71], [72]. This result is expected but not desired for the \npurpose of this thesis. Then, s ol-gel auto -combustion samples were analyzed via TEM.   \n5.1.4  PARTICLE SIZE CHARACTERIZATION  \nTwo samples were selected  to be analyzed by TEM microscope : Zn0.2Mn 0.8Fe2O4 due to low quantity \nof secondary phase and ZnFe 2O4 due to high quantity of secondary phase. In Figure 5.9, two  \nobtained images for both samples  are displayed . \n \n \n \n \n \n \nBoth images show agglomerated nanoparticles. For the first image, Zn0.2Mn 0.8Fe2O4, it is noticeable \nthat the image is blurry, probably due incomplete evaporation of the solvent (eth anol). Whereas in \nthe second image, ZnFe 2O4, is was not possible to distinguish single nanoparticles for higher \namplification, and a higher amplification ha d to be used. T he wide size  particle distribution is clear , \nalso the larger particle size, which can  be consequence of counting agglomerated nanoparticles . \nParticle counting and diameter analysis confirms the previous sentence, figure 5.10. \n \n \n \n \n \n \n \nFrom the particle counting,  it is visible the mean size is ~10nm for Mn 0.2Zn0.8Fe2O4 and ~6 4nm for \nZnFe 2O4. The particle size  distribution  for ZnFe 2O4 is wider  (σ ~ 92 nm) that its mean size. \nMn 0.2Zn0.8Fe2O4 size distribution it’s of the same scale of the mean size (σ ~ 10 nm). Such a wide \nsize distribution leads to a consequential w ide distribution of the mag netic properties (Tc, Msat, etc..) \nand ultimately impacts in their magnetic hysteretic losses because of their strong dependence on the \nparticle siz e. Consequently,  the correct assessment of these important magnetic parameters \nbecomes impractical or nearly  impossible.  Difficulties  controlling the particle size will difficult the \ncontrol of all magnetic and structural properties and therefore all the further work.  \n5.2 DISCUSSION OF SOL-GEL AUTO-COMBUSTION METHOD  \nThe s ol-gel auto -combustion method is a reliable  synthesis method to produce oxide samples in \nlarge amounts. This method requires high control over the combustion in order to obtain reproducible \nsamples. The combustion is dependent of multiple synthesis paramet ers which difficult the synthesis.  \nThe XRD results show the expected spinel crystal structure of the Mn -Zn ferrite. Impurity phase of \nZincite increases in percentage as the content of Zn increases in the ferrite, the Zn ferrite has 32% \nof ZnO. Lattice cons tant is in rough agreement with bibliograph ic values. The crystallite size diverged \nfrom the expected values towards larger crystallite sizes.  \nThe EDS results showed a good incorporation of the metal ions in the samples, except for ZnFe 2O4. \nThe SEM images present a morphology similar with literatur e. \nFigure 5.9 – TEM images for sol -gel auto -combustion samples. a) Zn0.2Mn 0.8Fe2O4. b) ZnFe 2O4. \nFigure 5.10 – Size distribution of a)  Zn0.2Mn 0.8Fe2O4 and b) ZnFe 2O4.The distribution was adjusted with a \nLorentz function.  \n〈𝑑𝑇𝐸𝑀〉 ≈ 64.42 nm  \nσ = 92.45 nm  \n〈𝑑𝑇𝐸𝑀〉 ≈ 10.21 nm  \nσ = 10.33 nm  \na) \n b) \na) \n b) André Horta  \nUniversidade de Aveiro   33 \nM(H) curves displays saturation magnetization in agreement with the bibliographic behavior, \nhowever, coercivity and remnant magnetization are below the expected values probably due to \nequipment limitations. M(T ) curves did not present the magnetic order ing temperature in the \nmeasurement range, 5 -400K. TEM analysis was employed and the particles have a wide size \ndistribution.  \nTEM revealed that the synthe sized nanoparticles have a  wide size distribution. This resu lt \ndemystifies why the magnetic characteriz ation do not present the expected results. In particular the \nmost important result for this work purpose, the magnetic ordering temperature dependence with \nZinc content, is not clearly definable in the measured FC  curves. Also, a wide size distribution wou ld \nbe a problem in a magnetic induction heating experiment, nanoparticles of different size in the sample \nwould heat up by different mechanisms and achieve different temperatures. Furthermore, the wide \nsize distri bution  also constitutes a disadvantage for biomedical applications. For instance, in \nhyperthermia applications, a narrow size distribution is required to ensure that the functionalization \nof the nanoparticles occurs similarly for every particle.  \nThe cause  of the undesired results might be the unco ntrolled combustion temperature, atmosphere \nand the multi -step procedure which enables room for mistakes. If the objective was simply the \nsynthesis of Mn -Zn ferrite, it would have been accomplished and the samples  have good enough \nproperties for a ferroflu id or a bulk application. For the main objective of this work,  self-regulated \nheating via Curie temperature, the synthe sized samples via sol -gel auto -combustion method are not \nsuitable.  \nThe undesired results obtained via sol -gel auto -combustion method lea d to the change of the \nsynthesis method. I . Sharifi et al, [20], published a paper with a review on various synthesis methods \nfor Mn -Zn ferrite, where co -precipitat ion, thermal decomposition, microemulsion and hydrother mal \nmethods are compared (table 5 of the paper resume them), from the description of these different \nsynthesis methods hydrothermal was the elected as the alternative to sol -gel auto -combustion \nmethod,  as it a simple and inexpensive technique which enables  a narrower particle size distribution \nand a finer control over particle shape.  \n \n \n \n \n \n \n5.3 RESULTS OF HYDROTHERMAL METHOD  \nHydrothermal method is adopted due to the controlled environment where nanoparticles are grown. \nThe analyzed samples were synthesized fol lowing the autoclave procedure for 6 hours at a \ntemperature of 180oC. The chosen compositions of the produced samples were x= 0; 0 .5; 0.8; 1; \nMn 1-xZnxFe2O4, similarly to sol -gel method . Hydrothermal method results are in great accordance \nwith the samples o f P. H. Nam et al,  [43], in “Effect of zinc on struct ure, optical and magnetic \nproperties and magnetic heating efficiency of Mn 1-xZnxFe2O4 nanoparticles”.  \n5.3.1  STRUCTURAL CHARACTERIZATION  \nThe crystal structure of hydrothermally synthe sized samples was investigated using the X -ray \ndiffraction pattern of the powder s. XRD patterns are presented in figure 5.11 for the compositions of \nMn 1-xZnxFe2O4 x = 0; 0 .5; 0.8; 1. \n \n \n Table 5.1 – Synthesis methods summarized by H. Shokrollahi et al.  Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n34  Departamento de Física  \nFigure 5.11 – XRD pattern of Mn 1-xZnxFe2O4 with x = 0; 0.5; 0.8; 1.Spinel phase peaks are sig ned by \nthe respective Miller indices, impurities peaks of Magnetite, Hematite -proto and Goethite are marked \nwith open squares, circles and “x”s, respectively.  \n(220) \n(422)  \n(400)  \n (422)  \n(511) \n(440) \no \nx \n x \nx \no \n(111) \n□ \n□ \n□ \n□ \n□ \n□ \no \n o \n o \no – Hematite -proto (Fe1,9H0,06O3)  \n□ - Magnetite (Fe 3O4) \nx – Goethite (FeOOH)   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nRietveld refinement of the presented samples outcome that the single phase spinel structure fraction \nis above 88% for all samples, a s plotted in figure 5.11.a. Impurities of Magnetite (Fe 3O4), Hematite -\nproto (Fe 1.9H0.06O3), and Goethite (FeOOH) are present in every sample at low quantities (below \n12%);  MnFe 2O4 sample presents mostly goethite,  whereas all the other samples seem to have both \nHematite -proto and Goethite. The  previously mentioned  Magnetite  phase  is indistinguishable of \nspinel crystal structure and for this reason it is not easily distinguishable using XRD .  \n \n \n \n \n \n \n \n              \nThe lattice constant of these ferrites, refine d by Rietveld refinement method in Fullprof Suite software, \nreveals a shrinkage of the lattice constant from 8 .5 to 8 .46 Å with the increase of Zinc content in \nferrite, as shown in figure 6.11.b. The cause of the lattice constant shrinkage is the smaller ionic \nradius of Zn2+ (0.74 Å) in com parison with Mn2+ (0.82 Å), [73], [74] . \nThe cr ystallite size  varies from 61 to 11 nm with the increase of Zinc content in ferrite. Zinc decrease  \nresults are obtained by , [43], [68]. The explanati on of this phenomenon was given by  [75] and [68], \nwhich attributes the larger crystallite size for Mn ferrite as responsibility of Mn ions forming 2+ and \n3+ cations, which means that th ey have a chance to  be incorporated in octahedral and tetrahedral \nvacancies, while the Zinc ions only form Zn2+ cations and the probability of being incorporated is \nsmaller, thus, Zn ferrite has a smaller probability of creating particles larger than Mn fe rrite, ,. \nBased on the lattice parameter, the unit cell volume was calculated, V = a3, since it’s a cubic cell. \nThe ferrite density was also calculated based on the mass and site occupancy of the elements \ncomposing the ferrite: Zinc (10.9x10-25 kg), Mangan ese (9.12x10-26 kg), Iron  (9.27x10-26 kg) and \nOxygen  (1.33x10-26 kg). The site occupancy of  unit cell is 8 for Zn or Mn, 16 for Fe and 32 for O. The \ncalculations showed  that the ferrite density increases from 4300 kg/m3 to 4580 kg/m3 with the \nincrease of Zn/Mn r atio, which is in agreement with the trend of the unit cell volume decrease with \nFigure 5.12 – Crystallographic parameters a) Pe rcentage of spinel ferrite phase. b) lattice constant. c) \ncrystallite size.  \na) \n b) \n c) André Horta  \nUniversidade de Aveiro   35 Zinc content, and the fact Zinc is heavier than Manganese. A similar trend was ob served in the \nliterature  [76]. \n5.3.2  MORPHOLOGICAL AND CHEMICAL CHARACTERIZATION  \nMorphology and size distribution of Mn 1-xZnxFe2O4 nanoparticles  were studied with resource to SEM . \nThe characterized samples have x = 0; 0 .5; 0.8 and 1. Image 6.12 present the obtained ima ges for \ndifferent samples.  \nX \n \n \n \n \n \n \nSEM images of hydrothermally prepared samples present agglomerated nanoparticles, as the \nmicroscope amplification is not high enough to distinguish single nanoparticles.  For the Mn to count \nand estimate the particles diam eter (with resource to ImageJ softw are). The Mn ferrite has a mean \nagglomerate size between 100 and 125 nm, comparing with the crystallite size, around 60 nm, \nsuggest a mean particle size composed of 2 crystallites.  For other compositions, the definition  of the \nimages is not high enough to distinguish agglomerates . \n   EDS analysis was carried out as explained in the experimental procedure chapter . The ratio \nof each metal ion is compared with the expected stoichiometry of the ferrite. Figure 5.14 presents \nthe results of EDS analysis for all metal ions with respect to the  expected values.  \n \n \n \n \n  \n \n \n \nThe EDS results show the deviations on the fraction of elements in the samples. In the Zinc element \nplot (6.14.a)), larger deviations are found in the samples  with higher content of Zinc (up to 15% of \ndeviation) . Manganese plot i s in good agreement with the expected quantity of Manganese. Iron \nmetal ratio reveals an excess of Iron for all samples, in particular samples with higher Zn/Mn ratio \npresent higher excess o f Iron.  \nThe samples with larger deviations from the expected are t hose with higher Zn content. Still, all the \ndeviations are inferior to 15% of change in elements relative quantity.  \n5.3.3  MAGNETIC CHARACTERIZATION  \nThe hydrothe rmally prepared samples were analyze d via SQUID magnetometer with M(H) loop s \nmeasured  at 5, 300 and 380 K, figure 5.15. The dependence of their magnetization with the increase \nof temperature in FC -ZFC curves  is present in  figure 5.17. Samples of Mn 1-xZnxFe2O4 (x = 0; 0 .5; 0.8; \n1) were analyz ed. \n \nFigure 5.13 – SEM images of all compositio ns: (a) MnFe 2O4; (b) Mn 0.5Zn0.5Fe2O4; (c) Mn 0.2Zn0.8Fe2O4; (d) \nZnFe 2O4. Poor definition in SEM images was obtained for intermediate compositions (x = 0.5 and 0.8). For \nMn and Zn ferrites, the image shows enough definition for counting agglomerated nanoparticles.  \nFigure 5.14 – Metal ions relative quantity times 3 (number of metal ions) and comparison with the \nexpected values (Fe = 2; Mn = 1 -x; Zn = x).  b) a) c) d) \na) \n b) \n c) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n36  Departamento de Física  \nFigure 5.16 – Saturation magnetization (a), remnant magnetization (b) and coercive field (c) at 300K. \nGreen lines are for samples, black lines are samples from literature. Black open circles are for values \nwhich were not explicit in literature.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nDifferent types of magnetic information can be extracted from the  M(H)  curves, as explained  in \n“Chap ter 4: Experimental Procedure ”. Figure 6.16 show the evolution of the magnetic parameters \nMS, M R and HC, with the increase of Z n conte nt ratio, at room temperature .  \n \n  \n \n \n \n \n \n \nFigure 5.16 shows a decrease in saturation magnetization, from 79 to 19 emu/g, and remnant \nmagnetization, from 5 to 0 emu/g, with the increase of Zn/Mn ratio. The obtained values are in good \nagreement with P.H. Nam et al, [43], and were complemented by  [77]. Coercive field does not present \nthe bibliographic reported behavior, in which H C is almost  zero for ZnFe 2O4 sample , [78]. The \nobtained values for the coercive field might be due to the remnant currents in the SQUID \nsuperconducting coil, since the samples with x = 0 .5 and 0 .8 appear to be superparamagnetic \n(theoretically having no coercivity) and x= 1 is in the paramagnetic regime, having no hysteresis and \nno coercive field. \nFor a better understanding of the depen dence of magnetization with temperature  (M(T) curves ) figure \n5.17.a) shows the FC (field cool ed) and ZFC (zero -field cool) measurements.  \n \n \n \n \n \n \nFigure 5.17 – M(T) measurements – FC and ZFC (a), FC (b) and ZFC (c). Range of temperature \nfrom 5 to 400K, except for Mn 0.5Zn0.5Fe2O4 which goes from 5 to 300 K.  \nFigure 5.15 – M(H) curves. From left to right column 5, 30 0 and 380 K respectivelly. ln the upper \nrow a full scan is shown (from -70000 to 70000 Oe), in the botto m row an amplification of the M(H) \ncurves is made for hysteresis loops analysis.  \n \na) \n b) \n c) \na) \n b) \n c) André Horta  \nUniversidade de Aveiro   37 The Mn -Zn ferrite presents a strong dependence of the magnetization with temperature. In a general \nanalysis, all  FC magnetization decrease with the increase of temperature, this is related to the \ntransition from ferromagnetic to paramagnetic phase. At higher temperatures more nanoparticles are \nin the paramagnetic state, thus reducing the sample magnetization.  The magnetic ordering  \ntemperature dependence with ferrite composition is the basic property to develop a s elf-regula ted \nheater with tuning capacity . The magnetic ordering  temperature decreasing with the increase of \nZn/Mn is observable in the FC measurements, the  magnetization drops at lower temperatures when \nZn content is higher in ferrite.   \nAnalyzing the FC curves, it is noticeable that the intermediat e composition (x=0,5) has a higher \nmagnetization than the Mn ferrite, for temperatures below 275 K. Above this t emperature, Mn ferrite \nhas the highest magnetization.  Also, by the FC curves it is possible to estimate the magnetic ordering \ntemperature, rely ing on Ɵp, as explained in “Chapter 4: Experimental Results ”. A decrease of the \nmagnetic ordering temperature with the increase of Zinc content, from ~610 to 250 K  is noticeable in \nthese curves. The MnFe 2O4 Curie temperature value is from P. H. Nam et al. Going t hrough the FC \ncurve from higher to lower temperature, an increase in magnetization is clear as the temperature \ndrops. This occurs as the paramagnetic particles (above the Curie temperature) undergo a \nparamagnetic -ferrimagnetic transition. The magnet ization  increases until it reaches a maximum and \nthen drops , the drop of magnetization  might be  related to inter -particle interaction , [10]. The blocking \nof less anisotropic nanoparticles is dependent of th e orien tation of previously blocked nanoparticles.  \nThe ZFC measurement starts by cooling the samples without an applied magnetic field. At high \ntemperatures, the superparamagnetic nanoparticles change their magnetization rapidly as they \npossess more therma l energ y than anisotropic energy, with relaxation time, . While cooling without \nfield, the nanoparticles are free to rearrange their magnetization in order to decrease the total \nmagnetization of the system. When cooled below the blocking temperature, the nanopar ticle \nmagnetization is said blocked, which means t hat the thermal energy takes  longer to change the \nnanoparticles’ magnetization and the  system stabilizes with a low magnetization value. When the \nmeasurement starts, a magnetic field of 100 Oe is applied an d the temperature is increased. The \nincreasing of temperature will allow the smaller nanoparticles (with low block temperature) to unblock \nand thermally change their magnetization in the applied magnetic field direction. This is the cause \nfor the magnetiza tion increasing until a maximum.  \nAn approximate value for  magnetic ordering  temperature is obtained by analyzing the  paramagnetic \nphase of the  FC curve with the Curie -Weiss law, as explained in “Chapter 4: Experimental \nProcedures ”. Figure 5.1 8.a) is the  normalization of the FC curves, for com parison improvement. \nMn 0.5Zn0.5Fe2O4 Curie temperature was estimated by extrapolation of th e normalized FC curve  for \nthis sample .  \n \n \n \n \n \n \n \n \n \nFigure 5.1 8.b) sho ws the decrease of the magnetic ordering  temperature as t he Zn/Mn ratio  \nincreases and compared with P.H Nam et al and Y. Xuan et al, [43] and [79]. These authors introduce \na linear dependency of the Curie temperature with the composition, which was used  to extrapolate \nthe Curie temperature  of MnFe 2O4, 556  K. This value was not considered, for further analysis the \nCurie temperature of MnFe 2O4 is considered as above 400 K.  \nFigure 5.18 – a) Normalized FC curve, magnetic ordering temperatures represented with arrows. \nb) magnetic ordering temperatures obtained via Curie -Weiss law are represented in closed green \ncircles. Open circles are the values indirectly obtained.  \na) \n b) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n38  Departamento de Física  \nAnalysis of the blocking temperature, ferromagnetic M(H) curves  at 300K  and susceptibility are \npresent in figure 5.19.   \n \n \n \n \n \n \n \n \nThe blocking temperature distribution is present in figure 5.19.a). This figure has an inset for a clearer \nobservation of the T b distribution near 0 Kelvin. By the analysis of this figure it is possible t o see that \nthe narrowest T b distribution happens for the Zn ferrite, with a mean blocking temperature of 11K. \nFor Mn ferrite, the blocking temperature it  is not in the 5-400K  range. Mn 0.2Zn0.8Fe2O4 and \nMn 0.5Zn0.5Fe2O4 have a wide T b distribution and the 〈𝑇𝑏〉 is hard to precise, for this reason it was not \nassumed.  \nFigure 5.19.b) shows the hysteresis loops subtracted by  the linear paramagnetic component . This \nfigure represents the ferromagnetic component of the synthe sized samples and will be used to \ndetermine the superparamagnetic behavior of the nanoparticles.  \nFigure 5.19.c) shows the susceptibility  of the M(H) curves. I t is visible that the Zn ferrite has the \nhighest  contribution , at 300 K , which is expected , since  the Zn ferrite has  the lowe st magneti c \nordering temperature . Mn ferrite has the lowest paramagnetic contribution, which is also expected \ndue to the ferr imagnetic contribution of the Mn ions. For the intermediary composition s, \nMn 0.5Zn0.5Fe2O4 has a higher susceptibility  than the Mn 0.2Zn0.8Fe2O4. \nTo clarify if any sample is in the superpa ramagnetic regime, the M/Ms(H/T) was plot ted for 300 and \n380 K . Accordingly with [10], the M/Ms(H/T) plot at different temperatures and above the irreversibili ty \ntemperature should superimpose into a universal  Langevin curve. This review article  also refer s that \nin order to obtain a better Langevin fit , the susceptibility should be subtracted from the M(H) curves, \nwhich is equivalent to M(H) curves  of figure 5.1 9.b). The cause for t he susceptibility  above the \nirreversibility temperatur e might be caused by  the buildup of paramagnetic moments on the partic les’ \nsurface. The obtained  results are present in figure 5. 20 along with the Langevin fit performed by \nOrigin.  \n \n \n \n \n \n \n \n \nIt was verified that the ZnFe 2O4 sample, figure 5. 20.a), superimpose into a universal Langevin curve \nat 300 and 380 K, thus revealed to be in the superparamagnetic state. The low initial susceptibility \nof these curves is because of the measurement  temperature, almost all nanoparticles are above the \nmagnetic  ordering temperature. Both samples, Mn 0.2Zn0.8Fe2O4 and Mn 0.5Zn0.5Fe2O4, do not \nsuperimpose into a universal Langevin curve, thus, they are not in the superparamagnetic state.  Figure 5.20 – M/M S(H/T), at 300 K and 380 K, and Langevin fit for a) ZnFe 2O4 b) Mn 0.2Zn0.8Fe2O4 \nand c)  Mn 0.5Zn0.5Fe2O4. \nFigu re 5.19 – a) Blocking temperature distributions. b) Ferromagnetic contribution of M(H) curves. \nc) Paramagnetic contribution of M(H) curves.  \na) \n b) \n c) \na) \n b) \n c) André Horta  \nUniversidade de Aveiro   39 \n〈𝑑𝑇𝐸𝑀〉 ≈ 40.6 nm  \nσ = 23.44 nm \n \n〈𝑑𝑇𝐸𝑀〉 ≈ 8.01 nm  \nσ = 2.66 nm  \n〈𝑑𝑇𝐸𝑀〉 ≈ 6.9 nm  \nσ = 2.18 nm \n 5.3.4  PARTICLE SIZE CHARACTERIZATION  \nHydrothermally prepared samples of Mn 1-xZnxFe2O4 of composition x = 0; 0.8 and 1 were analyzed \nvia TEM. Multiple amplifications were used for inter -composition analysis  and higher amplifications \nwere used to construct the nanoparticles size  distribution. Figure s 5.21 are  a selection of the obtained \nimages for the mentioned composition s. \n \n \n \n \n \n \n \n \n \n \n \n \nA brief analysis to the  TEM images of the synthe sized samples reveal the tendency previously \nobserved in crystallite size by XRD: a decrease in  grain size with the increase of Zn/Mn ratio.  All \nimages present agglomerated nanoparticles, which is expected due to their magnetic behavior. \nZnFe2O4 nanoparticles are more disperse than the other samples which might be a consequence \nof the higher parama gnetic -like behavior associated with this sample  \nTEM images of Mn Fe2O4 revealed that the  nanoparticles form agglomerates  in a flower arrangement. \nIn the higher amplification i mage, it is distinguishable a cubic  shape nanoparticle. The higher \namplification image also reveals that the flower shape agglomerates are formed of spherical  \nnanoparticles.  Irregular  particle shapes  were  also found. The images of higher amplification were  \nused to determine the size distribution of the samples, figure 5.22.  \n \n \n \n \n \n \n \n \n \nParticles counting revealed  a decrease in mean  grain  size with increase of Zn/Mn ratio , from 40 .6 \nnm to 6 .9 nm, MnFe 2O4 to ZnFe 2O4. The standard deviation is also reduced with the shrinking of the \ndistribution.  \n5.3.5  HEAT GENERATION  \nThe hydrothermal samples an alyzed in the magnetic induction heating setup were measured  without \nthe use of a solvent,  i.e. the  dry ferrite powders  were measured . For this reason, the following \nFigure 5.21 – TEM images of hydrothermally synthesized  Mn 1-xZnxFe2O4. Upper row images have a \n50 nm scale (except MnFe 2O4) and bottom row a 20nm scale. a) and d) is MnFe 2O4; b) and e) is \nMn 0,2Zn0,8Fe2O4; c) and f) is ZnFe 2O4 \nFigure 5.22 – Size distribution of Mn 1-xZnxFe2O4 x = 0; 0.8; 1 in TEM images. Mean grain size and \nstandard deviations for each sample. a) MnFe 2O4; b) Mn 0.2Zn0.8Fe2O4; c) ZnFe 2O4. \n \na) \n b) \n c) \nd) \n e) \n f) \na) \n b) \n c) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n40  Departamento de Física  \nmeasurements are introductory to further studies, as they are i ncomparable with literature , where \ntypically , the induction heating performance is measured for NPs immersed in a liquid solvent.  \n \n \n \n \n \n \n \n \nFigure 5.23.a shows the magnetic induction heating results and figure 5.23.b the dT/dt extracted \nfrom the initial slope of the magnetic indu ction heating curves.  From these images it is visible that  \nthe heating rate decreases with the increase of Zn content in ferrit e. \n \n  \n  \n \n \n \n \n \nFigure 5.2 4.a) shows the temperature increasing of Mn 0.5Zn0.5Fe2O4  with different applied magnetic \nfield. Figure 5. 24.b) proves the linear dependency of the heating rate with the squared applied  field, \nas predicted by  R.E. Rosensweig, [28], equation 2.9.The linearization has a slope of 6.06x10-4, an \ninterception of -0.02 and R2 of 0.998. \n5.4 DISCUSSION  OF HYDROTHERMAL SAMPLES  \nLet us start by looking at the samples  from a structural point of view. All synthe sized samples a re \nnano -powders of Mn 1-xZxFe2O4, spinel ferrite almost single phase, with impurities percentage below \n12%. The most common impurity found by XRD is hematite -proto, which is present in every \ncomposition except MnFe 2O4 which has a secondary phase of goethite . The presence of impurities \nis also suggested by EDS: the exc ess of iron found by EDS is closely related with the XRD impurities \n(mainly iron oxides) - the comparison can be found in figure  5.25. \n \n \n \n \n \n \n \nImpurity phases of are found in low percentage (bel ow 12%) in the samples. Due to the residual \nphases of goethite and hematite and the fact that both impurities have an antiferromagnetic nature, \ntheir interferen ce in the magnetic analysis is not considerable. Additionally, Hematite has the Néel Figure 5.23 – Magnetic induction heat ing results with an A C magnetic field of 25mT at a frequency of \n364KHz. a) Temperature as a function of time for all compositions; b) Time derivative of temperature \ncomposition dependence.  \nFigure 5.24 – Magnetic induction heating results with a variable AC magnetic field amplitude at a \nfrequency of 364KHz. a) Temperature as a function of time for Mn 0.5Zn0.5Fe2O4; b) Heating rate \nquadratic dependence of the magnetic field amplitude.  \nFigure 5.25 – (a) Iron metal ions ratio  (EDS) and (b) % of impurity phase (XRD).  \na) \n b) \na) \n b) \na) \n b) André Horta  \nUniversidade de Aveiro   41 transition below 955 K and remains antiferromagnetic at 6 K, [80]. Goethite is also antiferromagnetic \nwith a Néel temperature around 400 K and remains antiferromagnetic at temperatures lower than 5 \nK, [81]. Magnetite was not identified in magnetic analysis , since its saturation magnet ization \nincreases as crystallite size increases, [82]. Similarly with Mn ferrite in [83]. Thus, it is \nindistinguishable from Mn ferrite and it is not present in Zn ferri te, or it would increase its M S value.  \nThe magnetic properties of the nanoparticle are mainly dependent of the composition, however, the \nnanoparticles’ size als o as a significant role. The particle size is roughly calculated via Williamson -\nHall equation, w hen assuming every particle as a single crystal. Though this is not always true, it is \na good approximation due to the nanoparticles scale. The most reliable si ze measurement of the \nnanoparticles was performed via TEM, which revealed a constant size distri bution decreasing with \nthe increase of Zn/Mn ratio and reasonable standard deviations. Also, it revealed a particle size \nsmaller than crystallite size, which is  contradictory at first, but if the errors associated with the \nmeasurement techniques are consid ered, the values are not disagreeing. A crystallite size of the \nsame order of the particle size (or superior) suggests that most nanoparticles are single crysta l. \nFor structural and magnetic discussion, the crystallite size was used, due to the same com positions \nbeing analyzed  via SQUID and XRD. The crystallite/grain size has a great influence in the magnetic \nproperties of these ferrites, a s discussed in section  2.3.1. Crystallite size, saturation magnetization, \nremnant magnetization and coercive field a re displayed in figure 5.2.6  \n \n \n \n \n \n \n \nAs previously mentioned, the decrease of the crystallite size with the increase of Zn/Mn ratio is due \nto the lower probability of incorporation for Zn2+ than to Mn2+/3+. Zinc ions do no t contribute for the \nferrite magn etization as they do not have unpaired electrons, furthermore, they contribute for an \nantiparallel alignment of Iron magnetic moment, thus, causing all magnetic parameters to decrease \nwith the Zn increase.  \nThe main cause f or saturation magnetization to de crease is the incorporation of Zinc ions. However, \ncrystallite size also has a relevant role: larger  crystallite sizes , means smaller surface/volume ratio \nand consequently less spin canting contribution, which leads to a higher value of M S. Z. X. Tang et \nal showed that the increase of crystallite size favors an increase of saturation magnetization for \nMnFe 2O4, in [83]. This correlates with larger crystallite size found in synthe sized MnFe 2O4 sample \nwith the value of saturation magnetization being superior to the ones found by P. H. Nam et al.   \nThe remnant ma gnetization decreasing is mainly dependent from the Zn2+/Mn2+ ratio increasing. The \nMR decreasing is also affected by the crystallite si ze decreasing, as shown  by Y. Su et al, [84], and \nM. H. Mahmoud, [85]. The obtained values for the synthe sized samples closely agree with P. H. Nam \net al samples, however, the small values of remnant magnetization can easily be affected by the \nremnant field present in the s uperconducting coils.  \nThe coercive field is composition and grain size dependent. The composition dependence of coercive  \nfield can be observed in P. H. Nam et al samples. The relationship between grain size and coercivity \nwas studied by M. Vopsaroiu, [86], for FeCo films, the larger the grain size, the higher the coercivity. \nIn Mn -Zn ferrite, the increase of Zinc decreases the crystallite size, thus, a general decrease in \ncoercive f ield is expected. In order to understand the reason why this is not observed we should look \nat the samples from an indiv idual perspective. MnFe 2O4 has the largest crystallite and the higher \ncoercivity, which agrees with the previous citations. Mn 0.2Zn0.8Fe2O4 and ZnFe 2O4 present an \nambiguous comparison: the higher Zn composition the smaller the expected coercivity, however the \ncrystallite size is larger for ZnFe 2O4 having a coercive field similar with Mn 0.2Zn0.8Fe2O4. The \nintermediate composition, Mn 0.5Zn0.5Fe2O4, presents the higher deviations from the other Figure 5.26 – Crystallite size influence on the saturation magnetizatio n, remnant magnetization and \ncoercive field.  \na) \n b) \n c) d) Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n42  Departamento de Física  \nsynthe sized samples but a smaller deviation from the reference val ues. The reason for this might be \nthe measurement in a different SQUID equipment, also, the M(H) was measured at a  maximum \napplied field of 5T (in comparison with 7T of all other samples). A possible cause for the coercive \nfield of Mn 0.5Zn0.5Fe2O4 to have a smaller value might be caused by a lower value of remnant field in \nthe SQUID superconducting coils  \n \n \n \n \n \n \n \nThe magnetic induction heating experiment shows an initial heating rate decreasing with the increase \nof Zn/Mn ratio. The heating rate main contri butions are the hysteretic loops loss and Néel relaxation. \nThe experiment was made without a liquid medium, which means that the Brownian motion do not \ncontribute for the nanoparticles heating. The higher heating rate belongs to MnF e2O4, which has the \nlarger hysteresis area. Mn 0.5Zn0.5Fe2O4 and Mn 0.2Zn0.8Fe2O4 have close values of heating rate, the \nhysteresis of both samples is considerably small, thus the observable heating can occur via hysteric \nlosses and Néel relaxation. The heat ing rate for Zn ferrite is close to zero as this ferrite is completely \nparamagnetic at room temperature.   \nCrystallite size, blocking temperature and anisotropic constant are also related, accordingly with \nequation 5.1, [10]. \n𝑇𝐵=𝐾𝐸𝑓𝑓𝑉\nln(𝜏𝑀\n𝜏0 )𝐾𝑏      (5.1) \nThis time  interval of recorded data is 11 s, 𝜏𝑀,  where ln(𝜏𝑀\n𝜏0 )=23.12. Where K Eff is anisotropic \nconstant for Mn ferrite as found in [53], 1.24 kJ/m3, and K Eff for Zn ferrite, 46  kJ/m3 in [87], the \nintermediary composition’s K Eff was obtained by linear interpolation between these two values. The \nanisotropic constant is highly dependent on the size, shape, temperature and number of magnetic \nmoments of a nanoparticle . In the mentioned articl es both ferrites have a similar size. From the above \nequation several correlations were tried: for the volume calculus, the mean blocking temperature, \n〈𝑇𝐵〉, was used. For the T B calculus the crystallite volume was used. The K Eff was calculated with \nresourc e to both D XRD and 〈𝑇𝐵〉. For the conversion of crystallite size (D XRD) to volume and from \nvolume to calculated size (D CALC) were performed assuming spherical nanoparticles. The results are \npresented in Table 5.1. \nComposit ion \n (x=Zn)  DXRD  \n(nm) 〈TB〉FC/ZFC  \n(K) KEff_BIBILIO   \n(J/m3)                  DCALC  \n(nm) 〈TB〉CALC  \n(K) KEff_CALC  \n (J/m3) \n1 10.50 15.21 4600   12.6 8.7 7997  \n0.8 8.05 11.85 3932*   12.2 3.4 13817  \n0.5 11.70 12.21 2916*   13.7 7.7 4640  \n0 61.31 >400  1240   >58.1 469.0 >1057  \nTable 5.2 - Estimation of single domain size, blocking temperature and anisotropic constant from \nthe acquired magnetic and structural data. Asterisks means that the value was obtain from \nlinearization.  \nThe c alculus of the single domain  size, D CALC, with resource to the mean blocking temperature \npresents agreeing results, corroborating the XRD estimation and following the same tendency with \nZn content. The estimation of the mean blocking temperature with resou rce to the crystallite size \nresults: the largest deviation is for Mn 0.2Zn0.8Fe2O4 (280%), the lowest deviation is for Mn 0.5Zn0.5Fe2O4 Figure 5.27 – Comparison of a) hysteretic loops area with b) heating rate.  \na) \n b) André Horta  \nUniversidade de Aveiro   43 (72%). It was possible to estimate the blocking temperature of MnFe 2O4 (433K) which is above the \nFC-ZFC measurement range,  however, the single domain size was not performed due to the \ninexistence of a mean blocking temperature in FC -ZFC curves. In resume, the overall results are in \nrough agreement with the measured values.  \nTIrr was used for calculating the largest single doma in size. Also, using the D XRD value, it was possible \nto estimate the anisotropic constant of the highest temperature blocked nanoparticles, table 5.2. \nComposition  \n (x=Zn)  DXRD   \n(nm) 𝑇𝐼𝑟𝑟 (K) KEff_BIBLIO  \n (J/m3)         Dmax_CALC \n(nm) KEff_MAX_CALC  \n (J/m3) \n1 10.50 23.8 4600   14.7 12513  \n0.8 8.05 76 3932 *  22.8 88618  \n0.5 11.70 196 2916 *  34.5 74833  \n0 61.31 >400  1240   >58.0 >1057  \nTable 5.3 – Estimat ion of single domain size and anisotropic constant for the blocked nanoparticle \nat higher temperature.  \nThe observed increase tendency of the calculated domain size with the increase of Mn content in \nferrite is on agreement with the crystallite size tendenc y estimated by Willia mson -Hall analysis and \nTEM analysis.  \nThe anisotropic constant estimated with resource to the crystallite size and the irreversibility \ntemperature does not follow the linear tendency assumed. The calculus of K Eff depends mostly on \nthe n anoparticles’ composi tion, yet, the size and shape and of the nanoparticles, also plays a role, \nas explained in section  2.3.1. The hydrothermally synthesized nanoparticles have a relatively narrow \nsize distribution and its shape varies between spherical an d cubic.  \n5.4.1  COMPOSITIONAL ANALYSIS  \nMnFe 2O4 is ferrimagnetic, the Mn2+/3+ magnetic moment aligns with the iron lattice inside the ferrite, \ncreating a ferrimagnetic single domain. Mn ions have a canted spin due to the magnetic moment \nbeing weaker than iron mag netic mome nt, this canting categorizes the Mn ferrite as a ferrimagnet. \nMn ferrite has the highest saturation and remnant magnetization, coercivity, Curie and Block \ntemperatures and the highest heating rate of the Mn -Zn ferrite family.  Structurally, it has  the large st \ncrystallite size and lattice constant, with smaller density.  \nFerrimagnetism and paramagnetism at room temperature, occur simultaneously for Mn 0.5Zn0.5Fe2O4 \nand  Mn 0.2Zn0.8Fe2O4. The ferrimagnetic behavior of these samples comes from the Iron la ttice and \nthe Mn sublattice. The Zinc does not contribute with an individual magnetic moment,  however, it \ninfluences Fe spins by shrinking the lattice. For both ferrites, figure 5.20 shows that the M/Msat(H/T) \ncurves do not collapse into a universal Langev in curve, which means that these ferrites are not in \nthe superparamagnetic state at room -temperature.  \nSuperparamagnetism and paramagnetism occurs for ZnFe 2O4 at room temperature. The Zn ions in \nthe spinel structure do not contribute with magnetic moment fo r the ferr ite, instead they contribute \nfor the shrinkage of the crystallite. In this ferrite only the antiparallel Iron atoms contribute with \nmagnetic moment, thus, ZnFe 2O4 is antiferromagnetic below the Néel temperature. From the 5K \nM(H) curve, it was obs erved that  the magnetization does not saturate even at 70000 Oe, this might \nindicate spin canting at the surface of the nanoparticles. At room temperature, and at higher \ntemperatures, these nanoparticles are above their magnetic ordering temperature, which  makes \nmost of them paramagnetic. The nanoparticles that still contribute with magnetic moment are in the \nsuperparamagnetic state.  \n \n Otimização da síntese de ferrites para aplicações em fluidos magnéticos  \n44  Departamento de Física  Chapter 6:  CONCLUSION S  \nDuring this work nanoparticles of Mn -Zn ferrite were synthe sized by two different methods: sol -gel \nauto-combus tion method and hydrothermal method. Despite both methods are suitable for the \nnanoparticle synthesis, the hydrothermally prepared samples present better crystallinity and \nmagnetic properties than the sol -gel auto -combustion samples.  \nThe hydrothermally syn thesized samples revealed dependence of all structural and magnetic \nproperties with the Zn/Mn ratio . The XRD revealed the expected spinel crystal structure with high \nsingle -phase percentage (>88%). Lattice constant, determined by Rietveld refinement, decre ases \nfrom 8 .50 to 8 .46 Å from Mn ferrite to Zn ferrite. The Williamson -Hall analysis presents the crystallite \nsize decreasing from 61 to 11 nm, with the increase of Zn in ferrite. SEM images present \nagglome rated nanoparticles. TEM images showed  mean partic le size  varying, from 41 to 7 nm , with \nthe Zn/Mn ratio increase . Much narrower particle size distributions were obtained in comparison with \nthe samples of sol -gel auto -combustion method. SQUID results showe d that the increase of Zn \ncontent in ferrite decre ases saturation magnetization (79 to 19 emu/g ) and  remnant magnetization  \n(5 to approximately 0 emu/g ). The M(T) curves , FC -ZFC,  revealed that the mean blocking \ntemperature and the irreversibility temperatur e also decrease with the increase of Zn/Mn ratio.  From \nthe magnetic induction heating experiment , higher heating rate s were obtained for higher Mn content. \nTable 6.1 resumes the structural and magnetic properties of the hydrothermally synthe sized Mn -Zn \nferrite  measured at 300K . \n Composition \n(x=Zn)  1 0.8 0.5 0 \nXRD  Pure Phase (%)  93.1 88.56 93.65 95.07 \na (Å)  8.46 8.44 8.46 8.50 \nDXRD (nm)  10.50 8.05 11.70 61.31 \nDensity (kg/m3) *4581  *4562  *4478  *4301  \nTEM  〈𝑑𝑇𝐸𝑀〉 (nm)  6.9 8.01 - 40.6 \nSQUID  MS (emu/g)  15.87  23.42  39.6 77.07  \nHc (Oe)  29.84  29.88  11.13  41.85  \nMR (emu/g)  0.01 0.02 0.67 4.23 \nƟP (K) 284 340 420 *556 \n〈TB〉FC/ZFC (K) 15.2 11.9 12.2 - \n𝑇𝐼𝑟𝑟 (K) 23.8 76 196.9  >400  \nMagnetic Induc tion Heating  𝜕𝑇\n𝜕𝑡 (K/s) 18 255 285 912 \nTable 6.1 – Properties of hydrothermally prepared Mn 1-xZnxFe2O4 measured at 300 K . Asterisks \nmarks value that were not measured  directly . \nThe hydrothermally synthe sized nanoparticl es of Mn 1-xZnxFe2O4, revealed a  decreas ing of the \nmagnetic ordering temperature , from ~556 K  (estimated) to ~284 K , with x value increasing . \nMoreover,  the decrease of the heating rate might also be correlated with the magnetic ordering \ntemperature, the inc rease of Zn /Mn ratio leads to a lower ƟP and a lower heating rate . Although it \nwas not proved in a magnetic induction heating experiment, in principle, t he magnetic ordering \ntemperature of the synthe sized nanoparticles can be used as a self -regulated mechanism of heating .  André Horta  \nUniversidade de Aveiro   45 Figure 6.1 – Schematic of a se lf-pumping magnetic cooling device by V. Chaudhary et al . 6.1 FUTURE WORK \nFuture w ork guidelines are the optimization of the synthe sized ferrites by hydrothermal method. For \ninstance, increasing the  single -phase percentage by finding the optimum time and temperature for \nthe autoclave procedure. Organic solvents can be added during the s ynthesis in order to increase \nthe nanoparticles dispersion, [37]. The initial reagents affect the nanopar ticles size and shape, [38], \nthus, trying different co mpounds might be interesting to study the size and shape effect on the \nnanoparticles’ magnetic properties.  \nAlthough the  tuning of the  magnetic ordering temperature  was shown , the self -regulated induction \nheating mechanism was not proven during the inductio n heating experiment , probably due to the \nnon-adiabaticity of the system. A system with higher thermal  insulation could be used for this \npurpose. In order to obtain a more precise measurement of the magnetic ordering temperature, an \nexperiment of AC magnet ic susceptibility as a function of temperature could also be performed, [88]. \nAnother idea is to use a polymeric matrix to fix the nanoparticles during the  magnetic induction \nheating measurement, preventing them to rotate. This would mitigate the Brownian relaxation and \nonly the Néel rela xation contribution would be measured, [89].  \nAn interesting future application for the self-regulated heating nanoparticles  is a self -pumping \nmagnetic  cooling  device, as the one presented by V.  Chaudhary et al, [3]. For this application, the \nnanoparticles should be in the ferroflui d form. The fluid is in a close circuit between a heat load and \na heat sink. The magnetic fluid is attracted , by a magnet, to a heat load, where it will heat above its \nCurie temperature. Once above the Curie temperature, the fluid cease to be attracted to the magnet  \nand cold fluid take s its place. Thus , creating a self -pumping fluid, or a negative viscosity  fluid. The \nhot ferrofluid is then driven to the heat sink, where it will cool down and regain its magnetic properties. \nThis system not only allow the he at transport between heat load and heat sink but  can also be used \nfor generating ene rgy. The self -propelled magnetic fluid can be use d to induce an electromotive force \nin a coil , generating a potential difference, or even, by adding turbines in the fluid p ath it will force \nthe turbines to rotate. It works as transducer f rom mechanical to electric energy.  \n \n \n \n \n \n \n \nAnother future work guideline is the application of the synthe sized nanoparticles in other s tructural \nforms, such as, functionalized nanoparticles , ferrofluids, or bulk materials. 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Alloys Compd. , vol. 663, pp. 77 –81, 2016.  \n[89] R. Dannert, H. H. Winter, R. Sanctuary, and J. Baller, Rheol. Acta , vol. 56, no. 7 –8, pp. 615 –\n622, 2017.  André Horta  \nUniversidade de Aveiro   49 \nFigure A. 2 – Mn and Zn  ferrite dependency with autoclave time. a) D XRD(t) and a(t). b) M S(t), H C(t) \nand M R(t) c) ƟP(t). \nAPPENDIX A: AUTOCLAVE TIME \nThe hydrothermal results presented during the present report belong to samples that went the \nautoclave procedure for 6 hours. Other samples were synthe sized with different autoclave procedure \ntimes, 0 and 21 hours , in order to understand the synthesis evolution. Rietveld refinement analysis \nwas employed to determine the spinel -phase percentage.  \n \n \n \n \n´ \n \n \nAs noticeable from figure A.1, almost pure  spinel -phase samples are acquired from both, 6 and 21h. \nFigure A. 1.b) shows the spinel -phase percentage as function of the autoclave time. An increasing of \npure-phase percentage from 0 to 6 hours is reported for Mn and Zn ferrites. From 6 to 21h the pure -\nphase perc entage declines.  \nSamples without the autoclave procedure (0 hours) are very small, which is revealed in XRD \ndiffractogram by the broa dened peaks and have low percentage of the spinel -phase. The Rietveld \nrefinement results  of these samples , revealed  that Zinc ferrite along with a  secondary phase of \ngoethite , (FeOOH).  The Mn ferrite at 0h revealed the absence of spinel  crystal structure, instead it \nshowed Mn-doped goethite (Mn 0,18Fe0,82OOH)  and Mn(III) oxide ( MnO3 ). For higher autoclave times \n(21h) the pure -phase percentage decreases, comparing with 6h samples. The Rietveld Refinement \nrevealed iron hydroxides of hematite , Fe 1,9O2,7(OH) 0,3, for all samples  and goethit e for Mn ferrite .  \nThese results indicate that autoclave -time plays a major role in single -phas e spinel crystal structure  \nformation . Iron Hydroxides are formed before the autoclave procedure and , with temperature and  \npressur e, the metal cations  are incorporated in the spinel crystal structure .  \nThe evolution of the properties of Mn and Zn ferrite  with autoclave time are present in figure A.2. The \n0 hours samples were removed due to high impurity phase . \n    \n  \n \n \n \n \n \nFrom figure A.2.a it is observable that when increasing the autoclave time, the crystallite size \nincrea ses and the lattice constant dec reases. From figure A.2.b) It is shown the superior magnetic \nproperties of 6 hours samples  over 21 hours , even with longer autoclave times . Longer time in \nautoclave decreases M S, HC and M R for both Mn and Zn ferrite. Figure A.2.c) shows that the ƟP value \nof Zn ferrite do not change for longer time in the autoclave procedure.  \nThe presented figure shows minor changes between 6 and 21 hours samples, for this reason they \nwere not  included in t he thesis body. Its noticeable a slight increase of crystallite size  and  remnant \nmagnetization and decrease in lattice constant, saturation magnetization and coercivity. The ƟP \nvalue remains unaltered.      \nFigure A. 0.1 – Percentage of pure phase as a function of a) composition and b) autoclave time.  \nOtimização da síntese de ferrites para aplicações em fluidos magnéticos  \n50  Departamento de Física  APPENDIX B: MAGNETIC ORDERING TEMPERATURE  \nThe ma gnetic ordering temperature , presente d in the body text , was determined using the Curie -\nWeiss law, as detailed in Chapter 4: Experimental Procedure.  It was suggested that the temperature \nderivative of the magnetization , 𝜕𝑀\n𝜕𝑇, could also be used to calc ulate the magnetic ordering \ntemperature. Thus, the magnetic ordering temperature would be defined as the temperature at which \nthe FC curve presents a maximum curvature. The  Curie temperature was tested with resource to this \nmethod, the results are present in figure A.1.  \n \n \n \n \n \n \n \n \n \n \n \n \n \nThrough this method the magnetic ordering temperature would present a lower value of those obtain \nvia the Curie -Weiss law  for the paramagnetic state . Furthermore, the magnetic ordering \ntemperature s calculated via 𝜕𝑀\n𝜕𝑇 are inconsistent with the bibliographic values present in figure 5.19 . \nThis method also relies in the range of temperatures where the nanoparticles are under the \nunblocking procedure , thus, the unblocking of the magnetic domains will influence t he maximum of \nthe derivative. For all these  reasons, these  results were not included in the body text.  \n \n \n \n  Figure A. 0.1 – Temperature derivative of the FC curves for a) The red line is the FC \nmeasurement and the blac k line the respective temperature derivative.  \n"    },    {        "title": "2009.03750v1.On_the_assessment_of_an_optimized_method_to_determine_the_number_of_turns_and_the_air_gap_length_in_ferrite_core_low_frequency_current_biased_inductors.pdf",        "content": "arXiv:2009.03750v1  [physics.app-ph]  8 Sep 20201\nOn the assessment of an optimized method to\ndetermine the number of turns and the air gap\nlength in ferrite-core low-frequency-current biased\ninductors\nAndr´ es Vazquez Sieber and M´ onica Romero\nAbstract —This paper presents a ﬁrst assessment of a design\nmethod [1] aiming at the minimization of the number of turns\nNand the air gap length gin ferrite-core based low-frequency-\ncurrent biased AC ﬁlter inductors. Several design cases are\ncarried on a speciﬁc model of Power Module (PM) core, made\nof distinct ferrite materials and having different kinds of air gap\narrangements The correspondingly obtained design results are\nﬁrstly compared with the classic approach by linearization of\nthe magnetic curve to calculate Nand the use of a fringing\nfactor to determine g. Next, a reﬁned design approach of\nspecifying the inductance roll-off at the peak current and i ts\npotential limitations are discussed with respect to [1]. Fi nally,\nthe behaviour of inductors designed according to [1] operat ed\nbeyond their design speciﬁcations is analyzed.\nI. I NTRODUCTION\nFERRITE core based inductors are being increasingly em-\nployed into high-frequency high-power converters. Their\nlow loss ﬁgures at such frequencies, their vast availabilit y\nin shapes, sizes and materials already tailored for speciﬁc\napplications as well as their mature and well-known technol -\nogy mainly justify that trend. To design a ferrite-based low -\nfrequency-biased AC ﬁlter inductor is more challenging tha n\na pure AC inductor since in the former case, the required in-\nductance to block relatively high-frequency currents, has to be\nmaintained even when usually a higher level of relatively lo w-\nfrequency currents is superimposed. Complying the inducto r\nwith those constrains, [1] presents a design method that fur ther\ndetermines the minimum number of turns to be winded and\nthe optimum air gap length to be made on a given model of\nungapped ferrite core. This is a very convenient strategy to\nminimize the copper power loss and simultaneously to reduce\nthe intra-winding stray capacitance. It is therefore neces sary\na) to assess the results yield by [1] through comparisons wit h\nother design approaches and b) to establish its beneﬁts and\npotential limitations.\nAlthough the design procedure of [1] can be applied to\nan inductor based on any kind of ferrite core, it is clearly\noriented to high-power inductors. Any reduction in the numb er\nof winding turns translates into important savings in cost a nd\nthermal stress as well as the provision of a minimum, precise\nA. Vazquez Sieber and M. Romero are with Departamento de Elec tr´ onica,\nFacultad de Ciencias Exactas, Ingenier´ ıa y Agrimensura, U niversidad Na-\ncional de Rosario, Rosario, Santa Fe, 2000 Argentina e-mail :{avazquez,\nmromero}@fceia.unr.edu.ar.1 4.75 8.5 12.25 16020406080\ni^\nLF [A]N (turns)a)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 1004080120160\ni^\nLF [A]N (turns)b)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1602.557.510\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1002.557.510\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\nFig. 1.{Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant Lˆrev:\na) 0.5mH, b) 2mH. N27 ferrite material, qg= 1.\n1 4.75 8.5 12.25 16020406080\ni^\nLF [A]N (turns)a)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 1004080120160\ni^\nLF [A]N (turns)b)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1601234\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1001234\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\nFig. 2.{Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant Lˆrev:\na) 0.5mH, b) 2mH. N27 ferrite material, qg= 2.\nand accurate air gap length minimizes the heating and radiat ion\neffects of fringing ﬂuxes while it facilitates the ﬁnal indu ctance\nadjustment during manufacturing. Accordingly, the assess ment\nof [1] should be ﬁrstly performed on inductors having core\nshapes well-suited to that power levels and so this paper is a\nﬁrst attempt in that direction. Among the many existing ferr ite\ncores designed to handle relatively high currents, the Powe r\nModule (PM) core [2] is an attractive shape because a) it\nprovides a balanced trade-off between magnetic shielding a nd\npower dissipation capabilities, b) it can be conveniently p otted2\ninto a metallic heatsink to reduce the thermal resistance, c ) its\ncircular coil former offers a lower mean turn length than in a\nrectangular E core for the same cross-sectional area and d) i t\nis easier to wind and to adjust the air gap compared to toroids .\nHence, the assessment carried out in this paper focalizes on\nthis type of core, but due to space restrictions, it is limite d\nto a particular PM core model, the TDK-EPCOS PM 62/49\n[2]. However, it is examined in combination with two differe nt\nferrite material, N27 and N87, as well as having two differen t\nair gap arrangements: a) a single air gap located in the centr al\nleg or b) three air gaps of equal length each one located in\nthe respective leg of the core. The detailed deﬁnitions and\ndatasheet parameters required to apply [1] to this particul ar\nmodel or to any other size/brand of PM ferrite cores are found\nin [3]. Likewise, datasheet and model parameters related to\nferrite material N27 and N87 needed by [1] are available in\n[4].\nThis paper is organized as follows. In section II, the funda-\nmentals of [1] are revisited and the general setup of the sim-\nulations are presented. In section III, the design results y ields\nby [1] are compared with that obtained from the traditional\nmethod by linearization of the ferrite magnetization curve plus\nthe use of a fringing factor to determine the air gap. In secti on\nIV, a conceptual critique is posed on a reﬁned method which\nrequires to specify the initial inductance and its roll-off at peak\ncurrent. In section V, inductors designed according to [1] a re\ndriven beyond their original speciﬁcations and their behav iours\nare then analyzed. In section VI conclusions are presented.\nFor the sake of completeness, deﬁnitions of variables and\nparameters belonging to the models and methods of [1], which\nare referred all along this paper are summarized in Appendix\nA.\nII. F UNDAMENTALS AND SETUP OF THE DESIGN METHOD\nIn this section the key concepts and requirements of the\nmethod developed in [1] are refreshed as well as the gen-\neral setup and some particular implementation details are\nexplained.\nThe goal of the design method presented in [1] is the\nobtention of an inductor with the minimum number of turns\nNmin and the minimum nominal air gap length with its\ntolerance g=gopt±∆g, for a target reversible inductance\nLˆrevat given low-frequency peak current ˆiLFand a core\ntemperature distribution Tc. The adoption of goptis optimum\nin the sense that a) the maximum possible manufacturing\ntolerance ∆gis ensured, for the resulting Nmin and b) if the\nactualg∈[gopt−∆g,gopt+∆g]then the actual Lˆrevis at\nleast the target one. This requires the initial selection of a) an\nungapped core model to get its dimensions and the properties\nof the ferrite material and b) the winding wire gauge, from\nwhich are obtained estimations of wire temperature Twand\nTc. The following constraints should be initially speciﬁed: a )\nthe maximum number of winding turns allowed ( N∗\nhigh), b) the\nnumber of air gaps used ( qg) and c) the per-unit manufacturing\nprecision limit for the air gap length ( Tolg). Additionally, an\nunreachable lowest limit of Nmin,N∗\nlowshould be selected in\norder to ultimately ﬁnd Nmin∈(N∗\nlow,N∗\nhigh]. Inductors arespeciﬁed as qg= 1 for being based on a core having a single\nair gap in the central leg while as qg= 2 for having the two\nhalves of the PM core separated by an spacer, conforming so\nanother gap divided in each of the two external legs [3].\nAlthough [1] is suitable to design inductors subjected to\na certain type of non-uniform temperature distribution in\nthe core volume, all inductors designed in this paper are\nbased on a ferrite PM core model operated at a uniform\nTc=/vector100oCunless otherwise noted, since the other methods\nto be compared with only allow the hypothesis of uniform\ncore temperature. Tolerance Tolgis here set to 10% for the\nsake of algorithm convergence in the case of qg= 1 but it\nis disregarded elsewhere in order to obtain the best possibl e\nNmin. In all the simulation, ∆ΨHFis related to ∆iHFas\n∆iHFLˆrev≈∆ΨHF=1\n6000Vs\nAccordingly, the target values of Lˆrevare selected in such a\nway that∆iHF\nˆiLF≪1to be so placed in a small-signal scenario\nin which Lˆ∆is very close to Lˆrev[4]. This is required for a\nfurther experimental assessment of Lˆrevby measuring Lˆ∆.\nIII. C OMPARISON : LINEARIZATION OF THE\nMAGNETIZATION CURVE AND USE OF THE FRINGING\nFACTOR\nIn this section, the design method presented in [1] is\ncompared with the widely used method explained in [5]. It\nessentially considers Li=La=Lˆ∆=Lˆrevas long as in any\npart of the core, the absolute peak induction ˆBLF+∆BHF\n2≤\nBmax.Bmax≈0.35Tis commonly set regardless the kind of\nferrite material while a core temperature Tc= 100oCis also\nusually assumed uniform in all parts of the core.\nThe number of turns Nis simply given by\nN=ceil/parenleftBigg\nLˆrevˆiLF\nB∗maxAmin/parenrightBigg\nB∗\nmax=Bmax\n1+1\n2∆iHF\nˆiLF\nwhereAminis the minimum core cross-sectional area. In PM\ncores,Ac1stands for the cross-sectional area of the central leg\n[3] which coincides with Amin [7]. As was noted before, in\nall simulations it holds∆iHF\nˆiLF≪1and soB∗\nmax≈Bmax.\nBeing the PM a three-legged core, to obtain the gap length\ngwhenqg= 1, [5] avoids solving the nonlinear implicit\nequations [6]\ng=µ0AeN2\nLˆrevF−le\nµi(1)\nF= 1+g√Ac1ln/parenleftbigg2h1\ng/parenrightbigg\n(2)\nby using the approach explained next. Fis the so-called\nfringing factor; µ0is the vacuum permeability; Aeandleare\nthe effective core area and core length respectively. h1is the\nheight of the core central leg, deﬁned for PM cores in [3].\nIn Equation (1), the approximation F≈(F∗−1)F∗+ 1 is\nthen used, where F∗comes from Equation (2) when g=g∗.\nThe ideal air gap g∗is obtained from Equation (1) by making3\n1 4.75 8.5 12.25 16020406080\ni^\nLF [A]N (turns)a)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 1004080120160\ni^\nLF [A]N (turns)b)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1602.557.510\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1002.557.510\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\nFig. 3.{Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant Lˆrev:\na) 0.5mH, b) 2mH. N87 ferrite material, qg= 1.\n1 4.75 8.5 12.25 16020406080\ni^\nLF [A]N (turns)a)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 1004080120160\ni^\nLF [A]N (turns)b)\n  \n0255075100\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1601234\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1001234\ni^\nLF [A]g [mm]\n  \n020406080\n|gopt−g|/g [%]gopt\ng\nFig. 4.{Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant Lˆrev:\na) 0.5mH, b) 2mH. N87 ferrite material, qg= 2.\nF= 1. Note that [5] does not have provisions for the case\nqg= 2, where there are two air gaps with equal gbut different\ncross-sectional areas: the central leg area Ac1and the external\nlegs combined area Ac5[3]. To obtain better design results in\nthose cases, this paper proposes the use of\nF=1\n1\nF1+1\nF5(3)\nas a natural extension to the fringing factor Fof Equation (2).\nThe fringing factors of each air gap, F1andF5are identical to\nEquation (2) except for F5in which Ac1is replaced by Ac5.\nIn the following simulation, the resulting solutions\n{Nmin,gopt}using the design algorithm of [1] are compared\nagainst{N,g}of [5], for the same design speciﬁcations. A\nﬁrst set of comparisons (Figures 1-4) are made as a function o f\nthe target ˆiLFfor a constant target Lˆrev: 0.5mH in Subﬁgures\na); 2mH in Subﬁgures b). A second set of comparisons\n(Figures 5-6) are made as a function of the target ˆiLFfor a\nconstant target∆iHF\nˆiLF: 1% in Subﬁgures a); 5% in Subﬁgures\nb). A third set of comparisons (Figures 7-8) are made as a\nfunction of the target∆iHF\nˆiLFfor a constant target ˆiLF: 4A in\nSubﬁgures a); 16A in Subﬁgures b). In all the upper parts of\nSubﬁgures a) and b), the dashed blue curves are references to\nthe maximum number of turns Nmax that could be allocated\ninside the coil former for kuJ= 1.5A/mm2, wherekuis the1 4.75 8.5 12.25 1604080120160\ni^\nLF [A]N (turns)a)\n  \n020406080\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 10010203040\ni^\nLF [A]N (turns)b)\n  \n020406080\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1602468\ni^\nLF [A]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1000.1250.250.3750.5\ni^\nLF [A]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\nFig. 5. {Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant\n∆iHF\nˆiLF: a)1%, b)5%. N27 ferrite material, qg= 2.\n1 4.75 8.5 12.25 1604080120160\ni^\nLF [A]N (turns)a)\n  \n020406080\n|Nmin−N|/N [%]Nmin\nN\n0.5 2.875 5.25 7.625 10010203040\ni^\nLF [A]N (turns)b)\n  \n020406080\n|Nmin−N|/N [%]Nmin\nN\n1 4.75 8.5 12.25 1602468\ni^\nLF [A]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\n0.5 2.875 5.25 7.625 1000.1250.250.3750.5\ni^\nLF [A]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\nFig. 6. {Nmin,gopt}and{N,g}as a function of ˆiLF, for a constant\n∆iHF\nˆiLF: a)1%, b)5%. N87 ferrite material, qg= 2.\nwinding utilization factor and Jis the current density allowed\nin the wires. In all the lower parts of Subﬁgures a) and b), the\ndashed blue curves are gopt±∆g.\nThe results show that Nmintend to be higher than Nwhen\nkuJis set relatively low, for example to purposely obtain low\nstray capacitance and/or low loss inductors. In these cases ,\nNminshould be a more accurate solution than Nbecause [5]\ndoes not consider RcaandRcˆrevto be dissimilar and to be\ncomparable to Rgg, for determining the magnetic ﬂux. When\nthe number of turns is even lower, the impact of Rgocan also\nfurther increase Nminwith respect to N. For the same reasons,\nthe difference between Nmin andNtends to disappear when\nkuJincreases since Rggdominates then. Moreover, Nmin\ntends to be lower than Nfor relatively large kuJlimits which\nenhances efﬁciency. goptis clearly smaller than gin most\ncases.\nIf the ferrite material N27 is replaced by the higher quality\nN87 one, Nmintends to get closer to NsinceRcˆrevis lower\nand varies less from Rcias the target ˆiLFincreases. For the\nsame design speciﬁcations, the use of an N87 material leads\nto a lower Nmin than with an N27 material, which reduces\nthe winding loss. However, [5] yields the same Nregardless\nthe ferrite material employed.\nIt can be noted that when N≈Nmin, the trend is g > gopt.4\n1 3.25 5.5 7.75 1004080120160\n∆ iHF / i^\nLF [%]N (turns)a)\n  \n010203040\n|Nmin−N|/N [%]Nmin\nN\n1 3.25 5.5 7.75 1004080120160\n∆ iHF / i^\nLF [%]N (turns)b)\n  \n010203040\n|Nmin−N|/N [%]Nmin\nN\n1 3.25 5.5 7.75 1000.250.50.751\n∆ iHF / i^\nLF [%]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\n1 3.25 5.5 7.75 1001.252.53.755\n∆ iHF / i^\nLF [%]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\nFig. 7.{Nmin,gopt}and{N,g}as a function of∆iHF\nˆiLF, for a constant\nˆiLF: a)4A, b)16A. N27 ferrite material, qg= 2.\nTo explain this, consider the arithmetic difference betwee n\nthe achievable1\nLˆrevfor a target ˆiLFand the target1\nLˆrev,\nas a function of ˆΨLFfor a given N. That is referred to as\nf(ˆΨLF,N)[1] and it is plotted in green lines in Figure 9\nfor increasing values of N. Wherever f(•)>0, the current\nLˆrevis lower than the target one, and vice versa. Since [5]\nconsiders µˆrev=µaits corresponding f(•)follows the ideal\ndashed cyan line of Figure 9. As it usually crosses the x-axis\nat a lower ˆΨLFthan in the case of the corresponding f(•)\nof [1], the gap required in the former case would be larger\nto maintain that lower ﬂux. It is worth mentioning that [1]\nadjustsgoptto place ˆΨLFin between the two points where\nF(•)crosses the x-axis so a bounded manufacturing tolerance\nin the actual gcan still maintain the actual Lˆrevat least equal\nto the target one. On the contrary, [5] would place ˆΨLFjust\nin the limit and thus any minimum increase of the gap length\nwould immediately decrease the actual Lˆrevfrom the target\nvalue. Likewise, [5] overestimates µˆrevwhereg≈goptsince\nthereN < N min. This is evident from Figure 9 because [5]\nassumesµˆrevto be always equal to µibut in reality, µˆrev< µi\nat the point where ˆΨLFcurrently operates.\nAs a result of those issues and according to [1]\nLˆrev=N2\nRcˆrev+Rg(g)(4)\n{N,g}would yield an actual Lˆrevlower than the correspond-\ningly predicted by {Nmin,gopt}wherever are concurrently\ng > goptandN < N min.\nIV. C OMPARISON : INITIAL INDUCTANCE AND ROLL -OFF\nAT PEAK CURRENT\nIn this section, the design method described in [1] is\ncompared with the one proposed in [8]. It requires to initial ly\nset a target Liand adopt an arbitrary roll-off, RO, to getLˆrev\nat the targeted ˆiLF.\nOnce the ungapped core model is selected\nµe=Lˆrev\n(1−RO)N2le\nµ0Ae(5)\nRO=Li−Lˆrev\nLi1 3.25 5.5 7.75 1004080120160\n∆ iHF / i^\nLF [%]N (turns)a)\n  \n010203040\n|Nmin−N|/N [%]Nmin\nN\n1 3.25 5.5 7.75 1004080120160\n∆ iHF / i^\nLF [%]N (turns)b)\n  \n010203040\n|Nmin−N|/N [%]Nmin\nN\n1 3.25 5.5 7.75 1000.250.50.751\n∆ iHF / i^\nLF [%]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\n1 3.25 5.5 7.75 1001.252.53.755\n∆ iHF / i^\nLF [%]g [mm]\n  \n010203040\n|gopt−g|/g [%]gopt\ng\nFig. 8.{Nmin,gopt}and{N,g}as a function of∆iHF\nˆiLF, for a constant\nˆiLF: a)4A, b)16A. N87 ferrite material, qg= 2.\n0 0.2 0.4 0.6 0.8 1−10000100020003000400050006000\nΨLF^/Ψsatf(ΨLF^,N)\n  \nµ^\nrev=µa0 0.2 0.4 0.6 0.8 1\n0100020003000400050006000\nµ\n  \nµ^\nrev\nµa\nN\nFig. 9.µa,µˆrevinAmin andf(ˆΨLF,N)as a function of ˆΨLFnormalized\nagainst its saturation value Ψs[1]\nwhereµeis the effective permeability of the derived gapped\ncore to be ﬁnally employed. Nandµeare obtained simulta-\nneously solving\n0 =1−RO\nLˆrevN2−ˆiLF\nAminˆBLF(µˆrev)N−le\nµ0µiAe(6)\nµˆrev=1\nRO\n1−RO1\nµe+1\nµi\nalong with Equation (5). The explicit function for µˆrev/parenleftBig\nˆBLF/parenrightBig\ndeﬁned in [4], which is also used by the inductance model of\n[1], has to be numerically inverted to ﬁnally obtain ˆBLF(µˆrev)\nin Equation (6). Note that if the resulting µeneeds to be ad-\njusted to the nearest commercial off-the-shelf value avail able,\nthenNis recalculated, which may end altering LˆrevorˆiLF.\nOtherwise, a customized air gap of length\ng=/parenleftbigg1\nµe−1\nµi/parenrightbigg\nle\nis here proposed, which is only valid for relatively small ai r\ngaps and qg= 1. Customized gapped core with qg= 2, cannot\nbe directly handled by this design approach.5\n1 4.75 8.5 12.25 1600.250.50.751\ni^\nLF [A]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n0.5 2.875 5.25 7.625 1001234\ni^\nLF [A]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 10. Li,LˆrevandRO as a function of ˆiLF, for a constant Lˆrev: a)\n0.5mH, b) 2mH. N27 ferrite material, qg= 2.\n1 4.75 8.5 12.25 1600.250.50.751\ni^\nLF [A]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n0.5 2.875 5.25 7.625 1001234\ni^\nLF [A]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 11. Li,LˆrevandRO as a function of ˆiLF, for a constant Lˆrev: a)\n0.5mH, b) 2mH. N87 ferrite material, qg= 2.\nThe main purpose of this section is not to compare ab-\nsolute results of both design methods, like was done in the\nprevious section. It is rather oriented to conceptually dis cuss\nthe rationale of [8]. To this end, LiandLˆrevare calculated\nusing [1] alike in the previous section. The resulting RO are\nthen examined. Figures 10-13 show in solid lines the resulti ng\nLiandLˆrev using nominal parameters {Nmin,gopt,AL}\nas a function of the target ˆiLF, whereALis the nominal\ninductance factor of the ungapped core. In Figures 10-11,\nconstant target Lˆrev: 0.5mH in subﬁgure a); 2mH in subﬁgure\nb) are used. In Figures 12-13, constant∆iHF\nˆiLF:1%in subﬁgure\na);5%in subﬁgure b) are employed. In Figures 14-15, Li\nandLˆrevare obtained as a function of∆iHF\nˆiLFfor constant\nˆiLF: 4A in subﬁgure a); 16A in subﬁgure b). For all ﬁgures,\nTc= 25oCin cyan lines. Upper and lower dashed blue and\ncyan lines correspond to the limit variations on Liwhen it\nis calculated with parameters {Nmin,gopt−∆g,ALmax}and\n{Nmin,gopt+∆g,ALmin}, respectively. ALmax andALmin\nare the tolerance limits of AL. The red dashed lines are the\nlimits on Lˆrevcalculated under the same extreme parameters\nas before, which certainly coincide with the target Lˆrev. The\nresulting associated RO is then obtained with values of Li\nandLˆrevfrom the solid lines in blue and red respectively.\nTheRO tend to be smaller where ferrite material N871 4.75 8.5 12.25 1605101520\ni^\nLF [A]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n0.5 2.875 5.25 7.625 1002.557.510\ni^\nLF [A]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 12. Li,LˆrevandRO as a function of ˆiLF, for a constant∆iHF\nˆiLF: a)\n1%, b)5%. N27 ferrite material, qg= 2.\n1 4.75 8.5 12.25 1605101520\ni^\nLF [A]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n0.5 2.875 5.25 7.625 1002.557.510\ni^\nLF [A]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 13. Li,LˆrevandRO as a function of ˆiLF, for a constant∆iHF\nˆiLF: a)\n1%, b)5%. N87 ferrite material, qg= 2.\nrather than N27 is employed which indicates that the former\none would yield inductors with more stable inductance as\nthe existing peak current increases from zero to the rated\nˆiLF. For both materials however, inductors rated at higher\nˆiLForLˆrevalso tend to have an smaller RO since the air\ngap (linear) mandates over the core reluctance (non-linear )\nin those situations. In light of these simulations it can be\nconcluded that, if the inductor is to be designed with the\nminimum number of turns in mind, the resulting ROis highly\ndependent on the target Lˆrev,ˆiLFand the ferrite material\nselected. However, the actual RO can only be determined\nonce the design is over. Thus, an arbitrary adoption of RO\nat the beginning of the design process, as the approach in [8]\ndemands, it would yield in most cases a sub optimal design.\nV. B EHAVIOUR OF INDUCTORS BEYOND SPECIFICATIONS\nFigure 16 shows the evolution of Lˆrev, as the current value\nofˆiLFthrough the inductor increases from 0A up to a 10%\nmore than the rated current ˆiLF= 6A, for a design target Lˆrev\nequal to2mH,1mH and0.5mH. While the current ˆiLFis\nlower than the design target ˆiLF, the upper and lower dashed\nlines correspond to the limit situations {gopt−∆g,ALmax}\nand{gopt+∆g,ALmin}respectively; after that, their relative\npositions invert. The solid line between them is the resulti ng6\n1 3.25 5.5 7.75 1001234\n∆ iHF / i^\nLF [%]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n1 3.25 5.5 7.75 1000.250.50.751\n∆ iHF / i^\nLF [%]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 14. Li,LˆrevandRO as a function of∆iHF\nˆiLF, for a constant ˆiLF: a)\n4A, b)16A. N27 ferrite material, qg= 2.\n1 3.25 5.5 7.75 1001234\n∆ iHF / i^\nLF [%]L [mH]a)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\n1 3.25 5.5 7.75 1000.250.50.751\n∆ iHF / i^\nLF [%]L [mH]b)\n  \n05101520\n|L^rev−Li|/Li [%]Li\nLi @25oC\nL^\nrev\nFig. 15. Li,LˆrevandRO as a function of∆iHF\nˆiLF, for a constant ˆiLF: a)\n4A, b)16A. N87 ferrite material, qg= 2.\nnominal inductance with {gopt,AL}. As it is guaranteed, the\ndashed lines converge to the design target LˆrevwhenˆiLF\nmatches the design target ˆiLFand, as is expected, the more\ngapproaches to gopt+ ∆g, the more robust is the inductor\nwhen the design target ˆiLFis surpassed. This is achieved at\nthe expense of obtaining an inductance that, though higher\nthan the design target Lˆrev, is lower than the nominal while\nthe current ˆiLFis below the design target ˆiLF. The previous\nanalysis was done under the assumption of a uniform and\nconstant core temperature Tc=/vector100oC. Figure 17-a) shows\nthe behaviour of LˆrevwhenTcvaries from /vector100oCdown to\n/vector25oC. The plane parallel to the x−yaxes is the design target\ninductance Lˆrev= 0.5mH. The lower curved surface is the\nlower limit of inductance, which touches the plane of target\ninductance only at the design point {ˆiLF= 8A,Tc=/vector100oC}.\nThe upper surface is the nominal inductance, always well\nabove the other two surfaces. Although in this particular de sign\nmade atTc=/vector100oCnone of the curved surfaces trespasses\nthe plane of target inductance while the current Tcdecreases\nfrom that design temperature, this cannot be ensured to be\nso for any design. The behaviour of another inductor based\non the same core model with the same Lˆrevbut rated at\nˆiLF= 4Ais depicted in Figure 17-b). It reveals that the\nminimum value of inductance can take values below the design\ntarget as Tcapproaches /vector25oC. That may or may not be a\nproblem but it suggests doing another design attempt at the0 1 2 3 4 5 6 700.511.522.5\ni^\nLF [A]L^rev [mH]\nFig. 16.Lˆrevas a function of ˆiLF, varying from 0Aup to a10% more than\nthe design target ˆiLF= 6A, for three different values of target inductance\n204060801000\n2\n4\n6\n80.490.50.510.520.530.540.550.560.570.58\ni^\nLF [A]a)\nTc [oC]L^rev [mH]\n204060801000\n1\n2\n3\n40.480.50.520.540.560.580.60.620.640.66\ni^\nLF [A]b)\nTc [oC]L^rev [mH]\nFig. 17. Lˆrevas a function of current ˆiLFandTc. Design {Lˆrev,Tc}=\n{0.5mH,/vector100oC}; designˆiLFequal to a) 8A and b) 4A.\nlowest temperature of interest and then to test whether that\ninductor complies with the required inductance all through the\ntemperature span.\nVI. C ONCLUSION\nThe comparative results obtained from the simulation on\nthe PM core TDK-EPCOS 62/49 have shown that the in-\nductance model and the design method developed in [1] has\nclear advantages over the other common design approaches\nhere examined. Being grounded on a sufﬁciently detailed\ninductance model, the optimized design method brings to\nthe designer much more conﬁdence in obtaining an inductor\nwith a guaranteed inductance at the design speciﬁcations of\npeak current and core temperature. Also it proves that the\ninevitable manufacturing tolerances in the air gap length, if\nare kept below a certain limit, are not going to degrade the\nspeciﬁcations of the designed inductor. Although [1] requi res a\nmore complex modelling of the inductor and more number of\nsteps are devoted in ﬁnding Nminandgopt, all core parameters\nare readily available from the manufacturer’s datasheet wh ile\nthe model and the design strategy can be easily implemented\nunder any scientiﬁc programming language, such as Matlab7\nor Scilab. Once the inductance model is numerically imple-\nmented, it can also be conveniently employed for a given\ninductor, designed according to any methodology, to verify\nits inductance evolution under typical or limiting operati onal\nscenarios as well as predict behaviours beyond speciﬁcatio ns.\nFuture works are expected in assessing inductors based on\nother core shapes and performing experimental measurement s\nof inductance over prototype inductors designed with [1].\nAPPENDIX A\nREFERENCES TO VARIABLES AND PARAMETERS\nTABLE I\nLIST OF VARIABLES AND PARAMETERS\nSymbol Deﬁnition Ref.\nˆBLF Low-frequency magnetic induction at ˆΨLF [4]\n∆BHF High-frequency incremental magnetic induction [4]\nˆiLF Low-frequency peak current [4] [1]\n∆iHF High-frequency incremental inductor current [4] [1]\nLi Initial inductance [4] [1]\nLa Amplitude inductance at ˆiLF [4] [1]\nLˆ∆Incremental inductance at ˆiLF [4] [1]\nLˆrev Reversible inductance at ˆiLF [4] [1]\nµi Initial permeability [4]\nµa Amplitude permeability at ˆBLF [4]\nµˆrev Reversible permeability at ˆBLF [4]\nRci Initial core reluctance [4]\nRca Amplitude core reluctance at ˆΨLF [4]\nRcˆrevReversible core reluctance at ˆΨLF [4]\nRgg Air gap reluctance due to length g [1]\nRgo Residual air gap reluctance [1]\nˆΨLF Low-frequency magnetic ﬂux at ˆiLF [4] [1]\n∆ΨHF High-frequency incremental magnetic ﬂux [4] [1]\nTc Vector of core temperature distribution [1]\nIn Table I a list of variables and parameters referred in this\npaper, which are employed by the inductance model and the\ndesign method of [1], is presented. In each cited reference,\nthe deﬁnition of the parameter is given and explained into an\nappropriate context.\nACKNOWLEDGMENT\nThe ﬁrst author wants to thank Dr. Hernan Haimovich for\nhis guidance and constructive suggestions.\nREFERENCES\n[1] A. Vazquez Sieber, M. Romero and H. Haimovich, Optimum Joint Air\nGap Length and Number of Turns for Ferrite-core Low-frequen cy-current\nBiased AC Filter Inductors , IEEE Trans. Power Electron. Submitted,\n2020.\n[2] TDK Electronics, Ferrites and accessories, PM 62/49, Core\nand accessories , May 2017. [Online] Available: www.tdk-\nelectronics.tdk.com/inf/80/db/fer/pm 6249.pdf.\n[3] A. Vazquez Sieber and M. Romero, Power Module (PM) core-speciﬁc\nparameters for a detailed design-oriented inductor model . Congreso\nArgentino de Control Autom´ atico AADECA, Buenos Aires, Arg entina.\nSubmitted, 2020. [Online]. Available: http://arxiv.org/ abs/2008.13659.\n[4] A. Vazquez Sieber and M. Romero, A collection of deﬁnitions and\nfundamentals for a design-oriented inductor model . Congreso Argentino\nde Control Autom´ atico AADECA, Buenos Aires, Argentina. Su bmitted,\n2020. [Online]. Available: http://arxiv.org/abs/2008.1 3634.\n[5] A. Van den Bossche and V . C. Valchev, Inductors and Transformers for\nPower Electronics , 1st ed. Boca Raton, USA: Taylor & Francis, 2005.[6] C. Wm. T. McLyman, Transformer and Inductor Design Handbook ,\n3rd ed. New York, USA: Marcel-Dekker, 2004.\n[7]PM-cores made of magnetic oxides and associated parts - Dime nsions ,\nIEC 61247 edition 1.0b., International Electrotechnical C ommission,\nGeneva, Switzerland, 1995.\n[8] M. Esguerra, DC-Bias Speciﬁcations for Gapped Ferrite\nCores , Power Electronics Technology, Oct. 2003. [Online]\nAvailable: www.researchgate.net/publication/23542588 8DC-\nBias Speciﬁcations ofgapped ferrites."    },    {        "title": "1701.02041v2.Nonlocal_magnon_spin_transport_in_NiFe__2_O__4__thin_films.pdf",        "content": "Nonlocal magnon spin transport in NiFe 2O4thin ﬁlms\nJ. Shan,1,a)P . Bougiatioti,2L. Liang,3G. Reiss,2T. Kuschel,1and B. J. van Wees1\n1)Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG Groningen,\nThe Netherlands\n2)Center for Spinelectronic Materials and Devices, Department of Physics, Bielefeld University, Universit ¨atsstraße 25,\n33615 Bielefeld, Germany\n3)Device Physics of Complex Materials, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4,\n9747 AG Groningen, The Netherlands\n(Dated: 18 October 2021)\nWe report magnon spin transport in nickel ferrite (NiFe 2O4, NFO)/ platinum (Pt) bilayer systems at room temperature.\nA nonlocal geometry is employed, where the magnons are excited by the spin Hall effect or by the Joule heating induced\nspin Seebeck effect at the Pt injector, and detected at a certain distance away by the inverse spin Hall effect at the Pt\ndetector. The dependence of the nonlocal magnon spin signals as a function of the magnetic ﬁeld is closely related to\nthe NFO magnetization behavior. In contrast, we observe that the magnetoresistance measured locally at the Pt injector\ndoes not show a clear relation with the average NFO magnetization. We obtain a magnon spin relaxation length of 3.1\n\u00060.2\u0016m in the investigated NFO samples.\nThe transport of spin information is one of the most exten-\nsively studied topics in the ﬁeld of spintronics.1,2Spin current,\na ﬂow of angular momentum, is a non-conserved quantity that\nis mostly transported diffusively in various material systems,\nregardless of the carrier being conduction electrons or quasi-\nparticles such as magnons.3In traditional metallic systems4\nand 2D materials such as graphene,5a nonlocal spin valve ge-\nometry is usually applied to study the spin diffusion phenom-\nena and their relevant length scales.\nVery recently, it was shown that thermal magnons with typ-\nical frequencies of around kBT=h can be excited and detected\npurely electrically in Pt/yttrium iron garnet (YIG) systems, by\nalso employing a nonlocal geometry where the injector and\ndetector are both Pt strips, spaced at a certain distance.3,6–9An\nelectric current through the injector excites non-equilibrium\nmagnons both electrically via the spin Hall effect (SHE)10,11\nand thermally via the spin Seebeck effect (SSE),12–14and\nthey are detected nonlocally via the inverse spin Hall effect\n(ISHE).15At room temperature and below,16a magnon relax-\nation length \u0015mof typically around 10 \u0016m is observed, for\nboth electrically and thermally generated magnons indepen-\ndent from the YIG thickness.17\nAn open question is whether the nonlocal effects can be\nalso observed in other magnetic materials, such as ferrites, be-\ning ferrimagnetic at room temperature with a relatively large\nbandgap. Two local effects have been studied in Pt/ferrite\nsystems so far: the ﬁrst is the spin Hall magnetoresistance\n(SMR),18–21which results from the simultaneous action of\nSHE and ISHE in the Pt layer, while the magnetization in the\nmagnetic substrate modiﬁes the spin accumulation at the in-\nterface and hence the Pt resistance. SMR has been reported in\nPt/NiFe 2O4(NFO), Pt/Fe 3O4and Pt/CoFe 2O4systems.20,22–24\nSecond is the SSE, one of the central topics in the ﬁeld of\nspin caloritronics,25which is the excitation of magnon cur-\nrents when exerting a temperature gradient on the magnetic\nmaterial. Previously, SSE has been observed in ferrites and\na)j.shan@rug.nlother magnetic spinels.26–32However, the nonlocal transport\nof magnon spin has not yet been explored in ferrite systems.\nIn this study, we focus on the NFO thin ﬁlm systems\nwhich can be prepared by co-sputtering,33whereby a typi-\ncal bandgap of 1.49 eV and a resistivity of 40 \n\u0001m can be\nobtained at room temperature.34The electrical properties of\nthe NFO ﬁlms can be further tuned by temperature26or oxy-\ngen contents.35The employed NFO thin ﬁlms were grown by\nultra high vacuum reactive dc magnetron co-sputtering in a\npure oxygen atmosphere of 2\u000210\u00003mbar, with the depo-\nsition rate of 0:12˚A/s. The substrate is MgAl 2O4(MAO), a\nnonmagnetic spinel which is known to have a lattice mismatch\nto NFO as small as 1.3 %. It was heated up to 610\u000eC during\ndeposition and kept rotating to ensure a homogeneous growth.\nThe crystallinity of the NFO/MAO sample was investigated\nby x-ray diffraction, conﬁrming a (001) orientation for both\nNFO layer and MAO substrate. The thickness of the NFO\nlayer was determined by x-ray reﬂectivity to be 44.0 \u00060.5 nm.\nThe sample was characterized by a superconducting quantum\ninterference device (SQUID) to obtain its magnetic behavior.\nIt is known that in an inverse spinel magnetic thin ﬁlm with\n(001) orientation, a four-fold magnetic anisotropy is expected\nin-plane, with two magnetic easy axes aligned perpendicular\nto each other.26,27Figure 1(a) plots the NFO magnetization\nwhen an in-plane magnetic ﬁeld is applied along one of the\nmagnetic hard axes, showing a coercive ﬁeld of around 0.2 T.\nTo study the magnon spin transport in the NFO, two Pt\nstrips, parallel to each other and separated by a center-to-\ncenter distance d, were patterned by e-beam lithography and\ngrown on the NFO layer by dc sputtering. The Pt strips are\nall oriented along one of the magnetic hard axes. The lengths\nof the Pt strips are typically 10 \u0016m and the widths range from\n100 nm to 1 \u0016m. Two series of samples were fabricated, with\nthe Pt thickness of 2 nm (series A) and 7 nm (series B). Due\nto the difference in thickness, the Pt resistivities of the two se-\nries turn out to be quite different, where \u001aA=(0.9 - 2.4) \u000210\u00006\n\n\u0001m and\u001aB=3.5\u000210\u00007\n\u0001m, respectively, which is within a\nfactor of two in line with literature.20,36,37As a ﬁnal step, the\nPt strips were connected to Ti (5 nm)/Au (50 nm) contacts.\nA lock-in detection technique was employed in the elec-arXiv:1701.02041v2  [cond-mat.mtrl-sci]  13 Apr 20172\nB\nNFO (b)\nMAO α\nVNL+\n-\nVL+-\nI (a)M / Ms\nB (T)d\nhard axis\nhard axis-2 -1 0 1 2-1.0-0.50.00.51.0\nFIG. 1. (a) In-plane magnetization curve obtained by SQUID mea-\nsurements. A diamagnetic linear background has been subtracted,\nwhere the slope is determined from the high-ﬁeld regime up to B=\n7 T. The whole curve is subsequently normalized to the saturation\nmagnetization MsatB=7 T. The coercive ﬁeld is around 0.2 T.\n(b) Schematic representation of the device geometry and measure-\nment conﬁguration. Two Pt strips, one serves as the injector and the\nother as the detector, were sputtered onto the NFO surface, separated\nby a center-to-center distance d. The local voltage VLat the injector\nand nonlocal voltage VNLat the detector can be measured simultane-\nously. The magnetic ﬁeld is applied in the plane by an angle \u000b. All\nmeasurements are performed at room temperature.\ntrical measurements. A low-frequency ( \u001813 Hz) ac current,\nwith an rms value I0(typicallyI0= 100\u0016A), was sent\nthrough the Pt injector as input, while two output voltages\ncan be monitored simultaneously: the local voltage VLat the\nsame strip, and the nonlocal voltage VNLat the Pt detector, as\nshown in Fig. 1(b). Both VLareVNLare separated into the\nﬁrst (V1f) and second ( V2f) harmonic signals by the lock-in\nampliﬁers, which probes the linear and quadratic effects, re-\nspectively. The mathematical expressions are V1f=I0\u0001R1f\nandV2f=1p\n2I2\n0\u0001R2f, whereR1f(R2f) is the ﬁrst (second)-\norder response coefﬁcient.38,39Hence, for the local detection,\nR1f\nLrepresents the Pt strip resistance, as well as its magnetore-\nsistance, and R2f\nLshows the local SSE that was induced by\nJoule heating.40,41The transport behavior of magnons can be\nfound in the nonlocal detection, where R1f\nNLdenotes the signal\ndue to the magnons that are injected electrically via the SHE,\nandR2f\nNLillustrates the nonlocal signals of the thermally gen-\nerated magnons.3,16,17,42All measurements were performed in\nvacuum at room temperature.\nFigure 2 shows the experimental results obtained by ro-\ntating the sample in-plane, under a certain magnetic ﬁeld\nstrengthB. The nonlocal results are shown in the left panel\nwhile the local results are plotted in the right panel as a com-\nparison. One typical measurement curve of the nonlocal ge-\nometry in its ﬁrst order response is shown in Fig. 2(a), where\nd=1.5\u0016m. The applied in-plane magnetic ﬁeld, B=3 T,\nis large enough to align the NFO magnetization Mduring the\nfull rotation. The measured data exhibits a sinusoidal behav-\nior with a period of 180\u000e, the same as observed in Pt/YIG\nsystems.3,16,17,42In the injector, as a result of the SHE, a\nspin accumulation \u0016sbuilds up at the Pt/NFO interface, with\nits orientation always transverse to the electric current. The\nmagnon excitation is activated when the projection of \u0016son\ntheMis nonzero. The excited magnons become maximal\nwhen\u0016sis collinear with M, and vanish when they are per-pendicular to each other. Hence, the injection efﬁciency is\ngoverned by sin(\u000b), and the same holds for the reciprocal pro-\ncess at the detector, in total yielding a sin2(\u000b)dependence.\nWe further investigate the amplitude of this signal, \u0001REI, as\na function of the magnetic ﬁeld B, as shown in Fig. 2(b). Each\ndatapoint that is extracted by ﬁtting the corresponding angular\nsweep data to a sin2(\u000b)curve, represents the amplitude of the\noscillation. It can be seen that \u0001REIincreases rapidly from 0\nto\u00061T, and grows slowly as Bbecomes larger. Two other de-\nvices withd= 10\u0016m and 12\u0016m, show the same dependence\ndespite with different signal amplitudes. This dependence is\nin accordance with the NFO magnetization curve shown in\nFig. 1(a). In the non-saturated situation, the local Mis not\noriented along the external magnetic ﬁeld Bas a result of do-\nmain formation. When \u000b=\u000690\u000e, the projection factor of \u0016s\nonMis equal to 1 for the saturated case and becomes smaller\nthan 1 for the non-saturated case. Similarly, when \u000b= 0\u000e,\nthe projection factor for the saturated case is 0, but becomes\nnonzero for the non-saturated case. In this way, the difference\nbetween a parallel and perpendicularly applied ﬁeld decreases\nwhenBbecomes smaller and Mgets more unsaturated.\nSimultaneously we recorded the local signals. Figure 2(c)\nshows a typical ﬁrst-order response under B= 3 T, exhibiting a\nmagnetoresistance behavior, and Fig. 2(d) shows the MR am-\nplitude as a function of the magnetic ﬁeld. In the SMR sce-\nnario, \u0001RMRshould depend on Minstead of on B, as the key\ningredient in the SMR theory is the interaction between \u0016s\nandM. Surprisingly, our results show that \u0001RMRkeeps in-\ncreasing with a larger B, even when above the saturation ﬁeld\nof NFO. This behavior can be alternatively explained by the\nrecently reported Hanle magnetoresistance (HMR),43which\nis an instrinsic property of metallic thin ﬁlms with large spin-\norbit coupling and depends only on Binstead ofM. The MR\nratio we obtained is in the same order of magnitude as reported\nin Ref.43. However, it is not yet clear why we do not observe\nthe SMR feature on top of HMR.\nThe different dependences between the \u0001REIand\u0001RMR\nas a function of Brule out the possibility of any charge cur-\nrent leakage from the injector to the detector, in which case\nthe nonlocal signal would mimic the local magnetoresistance\nbehavior. Moreover, the ratios of the resistance changes com-\npared to the backgrounds differ by two orders of magnitude\nfor the local and nonlocal responses, further eliminating this\nscenario.?In addition, the nonlocal signals were also investi-\ngated at different lock-in excitation frequencies, and the \u0001REI\nkeeps almost unvaried with no systematic dependence on fre-\nquency, implying that the \u0001REIis not affected by any capac-\nitive coupling. Therefore, we can conclude that the \u0001REIwe\nmeasured is indeed due to magnon spin transport in NFO.\nThe second-order local responses which are due to ther-\nmally generated magnons are shown in the lower right panel\nof Fig. 2, detected in a nonlocal (left) or a local method (right).\nBoth signals show a sin(\u000b)behavior as a function of \u000b, gov-\nerned by the ISHE at the detector. Their amplitudes, \u0001RTG\nand\u0001RSSE, mainly follow the evolution of M, in accordance\nwith previous studies in the Pt/NFO system26,45and other\nPt/ferrite systems27,28. However, the rise of the thermal sig-\nnals is less sharp than that of Maround the coercive ﬁeld, for3\n-90 -60 -30 0 30 60 900.00.10.20.30.40.5 RL  - R L0 (Ω)\nangle (deg)-0.1∆ RMR\n∆ RMR (Ω)\n-1 0 1 2 3 4 5 6 70.00.20.40.60.81.01.2 \nmagnetic /f_i eld (T)-90 -60 -30 0 30 60 900510 \n ∆ REIRNL - R NL0 (mΩ)\nangle (deg)\n∆ REI (mΩ)(c)\nmagnetic /f_i eld (T)(d) (a) (b)d = 1.5 /uni03BCm, B = 3 T 1f 1flocal MR, B = 3 T 1f 1fNonlocal Local\n-90 -60 -30 0 30 60 90-400-2000200400 \nangle (deg)local SSE, B = 3 T RL  - R L0 (V/A2)2f 2f\n-1 0 1 2 3 4 5 6 7-300-1500150300450600\nmagnetic /f_i eld (T)∆ RSSE\n∆ RSSE (V/A2)(g) (h)\n∆ RTGRNL   (V/A2)\n-1 0 1 2 3 4 5 6 7-1.0-0.50.00.51.01.5  d = 0.3 /uni03BCm∆ RTG (V/A2)(e) (f)\nmagnetic /f_i eld (T)-90 -60 -30 0 30 60 90-1.5-1.0-0.50.00.51.01.5 d = 0.3 /uni03BCm , B = 3 T \nangle (deg)Electrical\n(1st harm.)Thermal\n(2nd harm.)\n2f-1 0 1 2 3 4 5 6 7051015d = 1.5 /uni03BCm1.5\n1.0\n0.00.5×10-4\nMR ratio\nFIG. 2. Comparison of both the electrical and thermal effects between nonlocal and local geometries under angle sweep, measured with\ndifferent magnetic ﬁelds. (a) The ﬁrst harmonic nonlocal signal with Pt spacing d= 1.5\u0016m while sweeping \u000b, measured at B= 3 T. The\nbackground resistance RNL0is -4.733 \n. The red curve shows a sin2(\u000b)ﬁt to the data. \u0001REIis deﬁned as the amplitude of the electrically\ninjected, nonlocally detected magnon signal. (b) The dependence of \u0001REIas a function of the magnetic ﬁeld at d=1.5\u0016m. (c) Local MR\nmeasurement at B= 3 T. The background resistance RL0is 8056 \n. The red curve shows a sin2(\u000b)ﬁt to the data. \u0001RMRis deﬁned as the\namplitude of the local MR signal. (d) The dependence of \u0001RMRas a function of the magnetic ﬁeld. Right axis indicates the MR ratio, which\nis\u0001RMR=8056 \n . (e) The nonlocal detection of the thermally generated magnons with Pt spacing d= 0.3\u0016m,B= 3 T. The red curve is a\nsin(\u000b)ﬁt. Its amplitude, \u0001RTG, depends on the magnetic ﬁeld as shown in (f). (g) The angular dependence of the local SSE measured at B=\n3 T. The subtracted background is -21.4 kV/A2. The red curve shows a sin(\u000b)ﬁt to the data. \u0001RSSEis deﬁned as the amplitude of the local\nSSE signal. (h) The dependence of \u0001RSSEas a function of the magnetic ﬁeld. Data in (e), (f) are from sample series B and the rest are from\nseries A.\nreasons that are not yet clear to us. The sign of the local SSE\nresults shows to be the same as in Pt/YIG systems.46\nExperimentally we deﬁned the polarities of the local and\nnonlocal voltages to be opposite in the measurement scheme\n(see Fig. 1(b)). Hence, the same shape in Figs. 2(e) and (g) in-\ndicates that the actual signs of the local and nonlocal SSE sig-\nnals are opposite. This is similar to the observation in Pt/YIG\nsystems, where at closer spacings the sign of the nonlocal SSE\nsignals are the same as the local one, but at further dthe sign\nis reversed.3,17However, to determine the exact sign-reversal\ndistance in this sample and how it evolves on the NFO thick-\nness, requires further study and is beyond the scope of this\npaper.\nNote that Figs. 2(e)(f) are obtained from sample series B.\nDue to the large resistivities of the Pt strips in sample se-\nries A and hence a limited electric current that can be sent,\nthe second-harmonic signals in the nonlocal detection, which\nscale withI2\n0, are below the noise level. We can, however,\ndetect them in series B. The local behaviors for both series\nare very similar as a function of \u000bandB, with the amplitude\n\u0001RSSEaround 5 times larger in sample series B. However,\n\u0001REIin series B is observed to be much smaller compared to\nseries A, which can be attributed to the thicker Pt ﬁlms and\nlower resistivity. Only for the shortest distance, where d=\n300 nm, we obtained a \u0001REIresponse beyond the noise ﬂoor,\nshowing the same magnetic ﬁeld dependence as series A.\nTo further study the relation between the observed signals\nand the NFO magnetization, we also performed magnetic ﬁeldsweep measurements at two speciﬁc angles, \u000b= -90\u000eand\n\u000b= 0\u000e, as shown in Fig. 3. In principle, this measurement\nwould yield the same information as obtained from the angu-\nlar sweep measurements, as the differences between \u000b= -90\u000e\nand 0\u000ecorrespond to the signal amplitudes extracted from the\nsinusoidal curves in Fig. 2. However, in the angular sweep\nexperiments,Mrotates in the plane, and hence the effects\nrelated to the magnetization hysteresis cannot be directly ob-\nserved. In comparison, ﬁeld-sweep measurements allow to re-\nsolve these features. Note that \u000b= 0\u000eand -90\u000ecorrespond to\nthe two equivalent in-plane magnetic hard axes. In both cases,\nthe behavior of Mcan be described by the M\u0000Bcurve in\nFig. 1(a).\nThe ﬁeld-sweep results are shown in Fig. 3. Similar as in\nFig. 2, the local magnetoresistance do not show any features\nrelated to the NFO magnetization curve, which would be pro-\nduced by the SMR. In contrast, both the local and nonlocal\nSSE signals show the typical hysteresis behaviors, with the\ncoercive ﬁelds being very close to the ones extracted from the\nM\u0000Bhysteresis loop.\nOne interesting observation is the electrically injected\nmagnon transport signal under the ﬁeld sweep, as shown in\nFig. 3(a). The peaks and dips for \u000b=-90\u000eand 0\u000e, occurring\nat the coercive ﬁelds, correspond to the situation where the\nnet magnetization in the ﬁeld direction is zero. In this case,\nthe thermally generated magnon signals vanish to zero, as ex-\npected, but interestingly the electrically injected magnon sig-\nnals show half of its maximum signal amplitude. Consider-4\n-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0-20246810121416  \n-8 -6 -4 -2 0 2 4 6 8-0.20.00.20.40.60.81.01.21.41.6  \nα=-90oα= 0oRL  - R L0 (Ω)\nmagnetic /f_i eld (T)d = 1.5 µm \nmagnetic /f_i eld (T)RNL - R NL0 (mΩ)(b)\n(d)1f 1flocal MR1f 1f\nα= 0oα=-90o\n-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.52.0-0.6-0.4-0.20.00.20.40.6 d = 0.8 µm \nmagnetic /f_i eld (T)RNL  - R NL0 (V/A2)(c)\n-2.0-1.5-1.0-0.5 0.0 0.5 1.0 1.5 2.0-400-2000200400 \nmagnetic /f_i eld (T)local SSE(a)\nRL  - R L0 (V/A2)2f 2f2f 2f\nα= 0oα=-90oseries Aseries A\nseries A series Bα=-90o\nα= 0o\nFIG. 3. Magnetic ﬁeld sweep results for (a) the nonlocal signal by\nelectrical injection, (b) the local MR, (c) the nonlocal signal by ther-\nmal generation and (d) the local SSE at \u000b= -90\u000eand\u000b= 0\u000e. The\nresults in (a), (b) and (d) are obtained from sample series A and (c)\nis from sample series B.\ning that multiple domains can form with the magnetizations\naligned along both of the magnetic easy axes in this material\naround the coercive ﬁelds, our results hence suggest the trans-\nport of magnons in a multi-domain state.\nTo estimate \u0015min the NFO sample, we performed a\ndistance-dependent study of the nonlocal signals. In Fig. 4,\nwe plot the thermally generated nonlocal signals as a func-\ntion ofdwhenMis saturated by the ﬁeld. Due to the\nmore complicated behavior for the short- dregime,17we only\nﬁt the data exponentially where d>1\u0016m. This yields a\n\u0015mof 3.1 \u00060.2\u0016m in the investigated NFO sample. This\nresult is supported by the electrically injected magnon sig-\nnals from series A obtained at B= 7 T, which can be ﬁt-\nted satisfactorily with the same \u0015m, by applying \u0001RNL(d) =\nC=\u0015m\u0001exp(d=\u0015m)=(1\u0000exp(2d=\u0015m))3(see inset of Fig. 4).\nGiven that the Gilbert damping coefﬁcient \u000bof an NFO thin\nﬁlm is 3.5 \u000210\u00003,47around one order of magnitude higher\nthan a typical \u000bof YIG thin ﬁlms, a reduction of \u0015mof NFO\ncompared to YIG is expected, as observed in our experiments.\nIn conclusion, we have experimentally observed the trans-\nport of both electrically and thermally excited magnons in\nNFO thin ﬁlms. The nonlocal signals of both exciting meth-\nods are directly related to the average NFO in-plane magne-\ntization, while the local MR is not, showing that the nonlocal\nresults are more sensitive to the NFO magnetization or domain\ntexture. Our results also suggest that the study of magnon\nspin transport can be extended to other materials such as ferri-\nmagnetic spinel ferrites, not only limited to YIG, showing the\nubiquitous nature of the exchange magnon spin diffusion.\nWe would like to acknowledge M. de Roosz, H. Adema,\nT. Schouten and J. G. Holstein for technical assistance. This\nwork is part of the research program of the Foundation for\nFundamental Research on Matter (FOM), and DFG Prior-\n ∆ RTG (V/A2) \ndistance (µm)0 2 4 6 8 10 12 14 160.0010.010.11\n0 5 10 150.1110\ndistance (µm)∆ REI (mΩ)FIG. 4. The thermally generated nonlocal signal response R TGas\na function of d, plotted in logarithmic scale. Red dashed line is an\nexponential decay ﬁt Aexp(\u0000d=\u0015m), withAbeing ad-independent\ncoefﬁcient, yielding a \u0015mof 3.1 \u00060.2\u0016m. The results are obtained\nfrom series B. 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In this work we explore the tunneling conductance across  the spin-filter material \nCoFe 2O4 interfaced with Au electrodes , a geometry which provides nearly perfect lattice matching  at the \nCoFe 2O4/Au(001)  interface . Using density functional theory calculations we demonstrate  that interface \nstates play a decisive role in controlling the transport spin polarization  in this tunnel junction . For a \nrealistic CoFe 2O4 barrier thickness , we predict a  tunneling spin polarization of about -60%. We show that \nthis value is lower than what is expected based solely on considerations of the spin -polarized band \nstructure  of CoFe 2O4, and therefore that these interface states can play a detrimental role.  We argue this is \na rather general feature of ferrimagnetic ferrites and could  make an important impact on spin -filter \ntunneling applications.   \nPACS numbers: 72.25. -b, 75.47.Lx , 73.40.Gk  \n \nIn the last few decades spintronics has been  one of the \nmost active  fields in condensed matter physics,  mostly \nbecause of its vast potential for device applications.1 The \ncornerstone of spintronics is the generation, injection and \ntransport of spin -polarized current  (SPC) . The \nconvention al approach of manipulating SPC  is based on \nmagnetic tunnel junctions (MTJ s) in which two \nferromagnetic electrodes are separated by a non -magnetic \ninsulating barrier. In MTJ s the tunneling current depends \non the  relative  magnetization orienta tion of the electrodes, \neffect  known as tunneling magnetoresistance (TMR).2 An \nalternative approach is to use spin-filter tunneling  where  a \nferro (ferri) magnet  is used as a barrier in a tunnel junction  \nwith non -magnetic electrodes .3 Spin-filter  tunneling relies  \non different probabilities for electrons with opposite spin to \nbe transmitted through a spin-dependent energy barrier of \nthe ferro(ferri)magnetic insulator.  The spin -dependence of \nthe energy barrier is due to the excha nge splitting of the \nband structure , which leads to  the conduction band \nminimum  (CBM)  and/or the valence band maximum \n(VBM)  lying  at different energies for majority - and \nminority -spin electrons. The tunneling transmission \ndepends exponentially on the barrier height, therefore \ntunneling conductance is expected to be spin -dependent .  \nDespite some promising e arly experiments  on Eu \nchalcogenides , such as  EuS4 and EuSe5 and EuO6, \ndemonstrating  the potential of spin-filter tunneling using \nthe Tedrow -Meservey technique7, practical application s are \nlimited d ue to their low  Curie temperature s. For that \nreason , the focus  recently has shifted to  the spinel -based  \nmaterials , such as CoFe 2O4,8,9 NiFe 2O4,10 NiMn 2O4,11 \nBiMnO 3,12 CoCr 2O4,13 and MnCr 2O413, which exhibit \nmuch higher Curie temperatures . \nThe theoretical understanding of the spin -filter tunneling has been largely based on the free -electron \nmodel3,14 and more recently on the analysis of the complex \nband structure.15,16 In the  former,  the spin -filter efficiency \nis entirely determined by the spin -dependent barrier height \nin the ferromagnetic insulator . The latter approach  takes \ninto account the realistic electronic structure of the bulk \nmaterial , in particular the orbital character  and symmetry \nof the complex bands . Both approaches  work , at best , in \nthe limit of large barrier thickness, thereby  neglect ing any  \npossible effects of the electrode/barrier interface s. In \nparticular, the presence of localized interface states are \nknown to play a decisive role in spin -dependent \ntunneling.17,18 This question  has yet to be  addressed for \nspin-filter system s.  \nWe employ  here density functional theory (DFT) \ncalculations to explore spin filtering in a prototype  \nAu/CoFe 2O4/Au (001) tunnel junction . CoFe 2O4 (CFO) has \na much narrower minority -spin band gap9, and hence \nstrong spin filtering with a large negative spin polarization \nis expected for large  thickness of CFO. We demonstrate , \nhowever,  that majority -spin states present at the \nCoFe 2O4/Au interface can produce a sizable contribution to \nthe tunneling conductance  for reasonable barrier \nthicknesses (i.e. ~2 nm), thereby reduc ing the spin \npolarization anticipated from the complex band structure of \nbulk CFO alone . We demonstrate that these inter face states \noriginate from native surface states of CFO. W e argue that \nsuch interface states are a rather general feature of \nferrimagnetic ferrites and will have  an important imp act on  \nspin-filter tunneling .  \nWe perform DFT  calculations using the Quantum \nEspresso (QE) package.19 We use the generalized gradient \napproximation  (GGA)  according to the Perde w-Burke -\nErnzerhof (PBE) formulation20 with energy cutoff of 5 00 \neV for the  plane -wave expansion and  k-point sampling of 2 \n 6×6×4  (bulk CoFe 2O4) and 4×4×1 (heteros tructure) for the \nself-consistent  calculations. Tunneling  transmission  \nthrough  a CoFe 2O4 (CFO)  barrier separating  two semi -\ninfinite leads of Au is calculated using the wave function -\nmatching  formalism implemented for  plane waves and \npseudopotentials  in the QE package.21,22 All calculations \nare performed with Hubbard U correction,23 which  is \nnecessary to accurately describe the insulating electronic \nstructure of CFO .24 We set U = 3 eV and J = 0 eV for the \nd-orbitals of both Fe and Co, in accordance with a recent  \ntheoretical study.16 Analysis of the complex band structure \nis achieved by constructing Wannier orbitals from the  \nGGA+ U band structure of bulk CFO25 and using standard \ntight-binding techniques thereafter.  \nCFO  is a ferrimagnetic insulator with a b ulk Curie \ntemperature of 796 K. 9 The oxygen atoms form a face-\ncentered cubic (FCC) sublattice, with cation atoms \ndistributed over tetrahedral ly and octahedrally coordinated \nsites. CFO has an inverse spinel structure with  Fd¯3m \nsymmetry  with 56 atoms per cell. Fe atoms occupy all of \nthe tetrahedral  whereas  the octahedral sites are randomly \noccupied by  Co and Fe .24 For a manageable computational \ncell we arrange the Co and Fe atoms on the octahedral sites \nin order to increase the symmetry of the cell . This  allows a  \nreduction  in the size of the  unit cell to a tetragonal cell of \n28 atoms with space group Imma . In this geometry the \ncalculated lattice parameters for the CFO are a = 5.91 Å \nand c/a=1.41.  \nThe ground state of CFO is ferrimagnetic, where \nmagnetic moments on octahedral sites  are aligned parallel \nto one another, but  antiparallel to the magnetic moments of \nFe atoms at tetrahedral sites. The magnetic moments \nprojected on individual atomic sites  are 2.5 µB for Co, 4.0 \nµB for Fe at octahedral sites, and –3.9 µ B for Fe at \ntetrahedral sites. There a re also induced magnetic moments \non O atoms:  0.05 µ B per O in the CoO 2 plane s and 0.15 µ B \nper O in the FeO 2 plane s. The total magnetic moment of \nCFO is 3 .0 µB per formula  unit, consistent with the \nexpected formal electron ic configuration s of the transitio n-\nmetal cations  (Co2+ and Fe3+ both in their high -spin \nconfigurations ) and the ferrimagnetic alignment.  \nFig. 1 (b) shows the calculated local densities of states \n(LDOS) for the bulk CFO. We fin d that a band gap is \nabout 0.8  eV, determined by minority -spin states, \nconsistent with previous DFT+ U calculations of CFO \nwhich report  band gaps in the range of 0.5 to 1 eV .24,26,16  \nThe exchange splitting of the CBM  is Δex = 0.9 eV , \nconsistent with previously reported values in the range of \n0.5 to 1.2 eV .24,26,16 The VBM is predominantly composed \nof Co (hybridized with O) states, while the CBM in both \nspin channels  are composed of Fe states. Thus, Δex is \nalmost entirely due to the splitting between the Fe states on \nthe octahedral and tetrahedral sites , in agreement with \nrecently published data.16  \nFig. 1 (a) shows an Au/CFO/Au supercell used in our \ncalculations. We construct the supercell  by lattice matching \n(001) oriented fcc Au with bulk CFO , leading to a  tensile strain on the Au  of less than 1%. We assume  a CoO 2 \ntermination of the CFO (001) layer and place interfacial \nAu atop O atoms. The supercell contains 8 formula units of \nCFO plus an additional monolayer (ML) of Co 2O4 to \nensure symmetric interface termination, resulting in non -\nstoichiometry of the CFO barrier. The structure is then \nfully optimized with constrained in -plane lattice parameter \nof bulk CFO, a = 5.91 Å.  \n \n \nFig. 1:  (color online) (a) S tructural model for the Au/CFO/Au \ntunnel junction. The magnetic moment direction in each layer is \nindicated by the  arrow s. LDOS  of (b) bulk CFO , (c) interfacial \nand (d) middle CFO layers o f the Au/CFO/Au tunnel junction,  \nand (e) the surface layer of stand -alone (001) CFO slab . Green \nline – octahedral Co, red line – octahedral Fe, blue line – \ntetrahedral Fe, yellow line – O, black line – total LDOS. The \nmajority - and minority -spin LDOS are displayed in the upper and \nlower panels, respectively. The vertical dashed lines denote the \nFermi energy.   \n \nThe calculated LDOS  for the Au/CFO/Au tunnel \njunction is shown in Fig.1 for interfacial (Fig. 1 (c)) and \nmiddle (Fig. 1 (d)) CFO layers. While the LDOS for the \nmiddle CFO  layer  closely  resembles that of  bulk (compare \nFig. 1 (b) and (d )), the interface LDOS  exhibits  different  \nbehavior.27 As is evident from Fig. 1(c), interface states \nappear within  the band gap of CFO for the majority -spin \nelectrons, with a peak near the Fermi energy  (EF). \nThe CFO/Au interface states  originate from native  \nCFO (001) surface states , as confirmed from a separate \ncalculation  of a stand -alone CFO (001)  slab with the same \nstructure as in the supercell . The surface  LDOS  of this slab \n(Fig. 1(e)) displays surface  states  in the bulk gap  for \nmajority - but not for minority -spins . Details of the surface \nstates are shown in Fig. 2. In Fig. 2(a -c) the k||-resolved \nLDOS is plotted in the two-dimensional Brillouin zone  \n(2D BZ) for the surface  atomic  layer  in the CFO (001)  \n1680816\n-4 -2 0 2 41680816\n-4 -2 0 2 4(e) (d)(c)LDOS (states/eV) total      Co\n Fe oct   O\n Fe tet\n(b)\n  \nE - EF (eV)  E - EF (eV)3 \n slab, calculated for the majority -spin at different energies  \n(see the  figure caption) . Fig. 2 (d) shows the majority -spin \ndensity for the (001) CFO slab calculated by  integrating \nthe surface layer LDOS from EF to EF + 0.4 eV.  The \nmajority -spin surface  states  mostly consist of O-px, O-py, \nand Co -dxy orbitals , as shown on Fig. 2(d ) and confirmed \nby additional calculations of the orbital ly-resolved  k||-\ndistribution (see Supplementary Materials ). These states \noriginate from the fact that, at the surface, the Co atoms \nlose their octahedral coordination due to one “missing” O \natom at the apex.  The octahedral crystal field  in the bulk \nsplits the  Co d-states into  a low energy t2g and higher  \nenergy  eg manifold. Absence  of the apex O atom at the \nsurface further splits the eg states which make up the \nmajority -spin VBM, lowering the dz2 states and raising the \ndxy states.28 The higher crystal field of the dxy orbitals leads \nto the formation of the surface states. As seen in Fig. 2(d), \nthe structure consists of relatively  well-separated parallel \nchains of CoO 2 oriented along the y-direction, leading to \nlarger dispersion along y than x and therefore giving rise to \nthe two -fold rotational symmetry seen in Fig. 2(a -c). \n \n \nFig. 2 : (color online) (a-c) k||-resolved majority -spin LDOS  \n(arbi trary units) at  (a) EF, (b) EF + 0.2 eV and (c) EF + 0.4 eV for \nthe surface atomic layer in the stand -alone (001) CFO slab. (d) \nIntegrated LDOS, Δ n, in real space for the surface  layer of the \n(001) CFO slab  from EF to EF + 0.4 eV.  Color  indicates the \ndensity on a plane cutting through the surface Co atoms and the \nshaded surfaces correspond to a constant -density Δ n = 0.05 Å-3. \n \nThese majority -spin surface states survive at the \nAu/CFO interface , as can be seen in the  k||-resolved LDOS  \nfor the interfacial CFO layer , plotted in Fig. 3 . As seen \nfrom Fig. 3  (a), these states exhibit the same distinct stripe -\nlike features originating from  the CFO  surface states \n(compare Fig. 2 (c) and Fig. 3(a)). We note that the \ninterface states shown of  Fig. 3(a) are calculated at  EF of \nthe Au/CFO/Au tunnel junction , which is shifted by about \n0.25 eV away from the VBM  in the (001) CFO slab. The \nsurface states shown on the Fig. 2(c) are plotted at EF + 0.4 eV. The small energy difference is due to the sli ghtly \ndifferent nature of the LDOS at EF for the CFO surface and \ninterface.29 These majority -spin interface states of CFO in \nAu/CFO/Au tunnel junction have a significant effect on the \ntunneling conductance with CFO, as confirmed below.  \n \n \nFig. 3 : (color online) k||-resolved majority -spin LDOS  (arbitrary \nunits) at the Fermi energy  for (a) interfacial and (b) middle CFO \nlayers in Au/CFO/Au tunnel junction.  \n \nWe calculate  tunneling conductance by taking the \nAu/CFO/Au supercell  as a scattering region and attaching \nit on both sides to semi -infinite  FCC Au leads. The \ncalculations are performed  at zero bias using a uniform 60 \n× 60  k-point mesh in the 2D BZ . The calculated \nconductance per unit cell area is  G↑ = 0.11×10-4 e2/h for \nmajority - and G↓ = 0.40×10-4 e2/h for minority -spin \nchannels , respectively. The spin polarization of the \ntunneling current is P = (G↑ – G↓)/(G↑ + G↓) = -57% . The \nnegative sig n of P is consistent with the expectation \nfollowing from the lower minority -spin band gap \ncompared to the majority -spin band gap and is in \nagreement with  experimental results for fully epitaxial \njunctions  with CFO as a tunneling barrier. 9 \nFigs. 4(a -b) show the k||- and spin -resolved \nconductance of the Au/CFO/ Au tunnel junction.  The \nmajority -spin conductance, Fig. 4 (a), can be explained by \ncorrelating it with  the k||- resolved LDOS shown in Fig. 3 \n(b). The interface states seen in Fig. 3 (a) as stripes for the \ninterfacial CFO layer strongly decay away from the \ninterface, however , even in the middle of the CFO barrier \nlayer  they do not completely vanish (Fig. 3 (b)). Moreover, \ncomparison of  Fig. 4 (a) with Fig. 3  (b) indicates  a clear \ncorrelati on between the k||- resolved  conductance and \nLDOS profiles, both exhibiting  maxima  in the same area of \nthe 2D BZ. The transmission distribution bears little \nresemblance, however, to the distribution of lowest decay \nrates for majority spins, Fig. 4 (c), as d etermined by the \ncomplex band structure calculations. We conclude, \ntherefore,  that the tunneling conductance of majority -spin \nelectrons is , in fact,  dominated by the interface  states and \ntherefore cannot be deduced by consideration of the \ncomplex band structure of CFO alone.   \nThe conductance profile for the minority -spins, on the \nother hand,  is very reminiscent of the distribution of  \n(b)0510LDOS\n(a)n (Å-3)\nCo\n(c)\n-1 0 1-101\n(d)\n kx (/a) ky (/a)\nO OO O10-1\n10-2\n10-3\n10-4\nxy\n-1 0 1-101\n(b)\n kx (/a) ky (/a)0 6 0.00 0.05\n(a)4 \n evanescent states in the band gap of CFO. Fi gure 4(d) \nshows the lowest decay rates of the minority -spin \nevanesce nt states in the band gap of CFO, where we see a \nclose resemblance between the conductance (Fig. 4(b)) and \nthe decay rate distribution (Fig. 4(d)) for the minority -spin. \nFinally, we notice  that both majority - and minority -spin \nchannels demonstrate minimal conductance at the  ¯Γ point \n(Fig. 4 (a -b)), somewhat inconsistent with the distribution \nof the decay rates (Fig. 4 (c -d)) and with recently published \nresults.16 This is due to the  mismatch of the band \nsymmetries for both majority and minority spin channels \nof Au and CFO, calculated for kx = ky = 0, along the [001] \ndirection. In particular, for Au along this direction there is \nonly one band crossing Fermi level  having  Δ1 symmetry, \ni.e. with orbital contributions s, pz, and dz2. None  of these \norbital characters belong to the slowest dec aying  bands  \nnear the  VBM of CFO , being primarily of  py and dxy orbital \ncharacter  for majority -spin and dx2-y2 and py for minority -\nspin.   \n \n \nFig. 4: (color online). k||-resolved transmission  at EF for (a) \nmajority - and (b) minority -spin channels  of the  Au/CFO/Au  \ntunnel junction . Lowest decay rate, κ, of the (c) majority - and (d) \nminority -spin evanesce nt states  of bulk CFO  as a function of k|| \nin the 2D BZ  at VBM +  0.4 eV. \n \nThe contribution of the majority -spin interface states is \ndetrimental to the net spin -polarization of the tunneling \nconductance. To see this, we return to the simpler \ndescription of the spin -filter effect based solel y on the \ncomplex band structure where we assume featureless \nelectrodes and perfect interface transmission functions. In \nthis case the conductance  for each s pin-channel  is \ndetermined by G ∝ ∫e-2κ(k||)td2k|| where t is the thickness of \nthe barrier, κ (k||) is the calculated lowest decay rate at  EF \nand k|| and the integral is over the entire 2D BZ.  Using t = \n1.9 nm and EF = VBM + 0.4 eV,  we find a spin -\npolarization of P = -80%. This is significantly larger than \nwhat is found from our full transport calculations, where \ninterface states dom inate the majority spin channel.  The predicted effect of interface states on spin -\npolarized tunneling is not limited to the particular \ngeometry of the tunnel junction considered above . We find  \nthat a terminating layer  of the CFO (001) with a mixture of  \nFe and Co, as well as  a purely  FeO 2 terminating layer , both \nalso lead to majority -spin interface  states which produce  \nsimilar  detrimental  effect s on spin -polarized tunneling . \nOne could  expect a different behavior for Fe at tetrahedral \nsites comprising  the interface;  we find , however,  that this \ntermination is unstable .      \nIn summary , we have  shown that  the spin polarization \nof the tunneling conductance in  Au/CoFe 2O4/Au (001)  \ntunnel junction  is strongly affected by majority -spin \ninterface states, leading to a reduction in spin -polarization \nas compared to expectations based on the spin -polarized \nband -gap alone.  Interface states are a general feature of the \nferrimagnetic ferrites that are used as spin -filter barriers. \nThus, t he predicted effect has important implications for \nthe design of spin -filter tunnel junctions, where the \ninterface states need to be a voided to exploit the unspoiled \nspin filtering anticipated from the band structure of the \nbulk material.  \nThis research  was supported by the NSF (Grant No. \nEPS-1010674)  and the Nebraska Research Initiative.  The \nwork  at the University of Puerto Rico was supp orted by \nNSF Grant s DMR -1105474 and EPS -1010094 . \nComputations were performed at the University of \nNebraska,  Holland Computing Cente r. \n \n† pavel.lukashev@unl.edu  \n* tsymbal@unl.edu  \n                                                           \n1.  Handbook of Spin Transport and Magnetism , eds. E. Y. \nTsymbal and I. Žutić (CRC press, Boca Raton, FL, 2011), 808 \npp.  \n2.  E. Y. Tsymbal, O. N. Mryasov, and P. R. LeClair, J. Phys.: \nCond. 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By symmetry  DOS are slightly different at right and left CFO \ninterfaces, but the difference is small and ignored throughout \nthe text.  \n28.  Here x and y to refer to the in plane lattice vectors . These  are \nrotated by 45° with respect to the usual coordinate system used  \nto describe octahedral ly coordinated transition metals . \nTherefore, in our notation,  dxy belongs to the eg manifold, with \nlobes along the directions of O nearest neighbors  as seen in Fig. \n2(d). \n29.  We confirmed the presence of the CFO surface states by an \nadditional calculation for  a Au/CFO/Au tunnel junction with \ndifferent interface termina tion. In particular,  similar stripe -like \nfeatures are present for k||- resolved DOS of the interfacial CFO \nlayer if the interface layer of CFO consists of both Co and Fe. 1 \n Interface  states in CoFe 2O4 spin-filter tunnel junctions : supplementary materials  \nPavel  V. Lukashev ,1† J. D. Burton,1 Alexander Smogunov,2 Julian P. Velev ,3 and Evgeny Y. Tsymbal1* \n1Department of Physics and Astronomy  & Nebraska Center for Materials and Nanoscience,  \nUniversity of Nebraska, Lincoln, Nebraska 68588 , USA  \n2 CEA , Institut Rayonnement Matière de Saclay , SPCSI, F -91191 Gif -sur-Yvette  Cedex, France  \n3Department of Physics, Institute for Functional Nanomaterials, University of Puerto Rico, San Juan,  \nPuerto Rico 00931, USA  \n \nOrbita l resolved surface states in CoFe 2O4 \n \nFig. S1(a-c) shows the  orbital -resolve d contributions \nto the majority -spin surface states calculated at EF + 0.3 \neV for the (001) CoFe 2O4 (CFO ) slab (EF – Fermi \nenergy) . We see that the majority -spin surface  states  \nmostly consist of O-px, O-py, and Co -dxy orbitals. All other \ncontributions are negligibly small and are not shown here.  \n \n \n \nFig. S1: (color online) k||-resolved majority -spin DOS  (arbi trary \nunits) of (001) CFO slab calculated at EF + 0.3 eV for O -px (a), \nO-py (b), and Co -dxy (c) orbitals.  \n \nComplex band structure of CFO  \n \n  \nFig. S2: (color online) Compl ex band structure of CFO  in the Γ \n→ Z direction  for majority ( left panel ) and minority ( right \npanel ) spin . The middle panel shows real bands  for the same \ndirection .  \n Fig. S2 shows the calculated spin -dependent compl ex \nband structure of CFO for k|| = 0 in the Γ → Z direction . \nThe complex bands (left and right panels) are connected \nto the real bands (middle panel) and inherit their \nsymmetry properties. The curvature for complex and real \nbands is the same at the connect ing points due to the \nanalytic pro perties of the energy dispersion function , \nE(kz). For detailed discussion of the complex band \nstructure ’s significance for the spin -filter materials, see \nRef. [1].  \n \n                                                           \n1.  P. V. Lukashev, A. L. Wysocki, J. P. Velev , M. van \nSchilfgaarde, S. S. Jaswal, K. D. Belashchenko, and E. Y. \nTsymbal, Phys. Rev. B 85, 224414 (2012).  \n-2-1012\n0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 up\n dn\n[0 0 kz] Z\n   (2/c) E - VBM (eV) \n   (2/c)\n "    },    {        "title": "0704.2456v1.Classical_Heisenberg_Hamiltonian_Solution_of_Oriented_Spinel_Ferrimagnetic_Thin_Films.pdf",        "content": " 1Classical Heisenberg Hamiltonian Solution of Oriented Spinel Ferrimagnetic Thin \nFilms \nP. Samarasekara \nDepartment of Physics, University of Ruhuna, Matara, Sri Lanka. \nAbstract \n         The classical Heisenberg Hamiltonian was solved for oriented spinel thin and thick \ncubic ferrites. The dipole matrix of complicated cubic cell could be simplified into the form of dipole Matrix of simple cubic cells. This study was confined only to the highly oriented thin films of ferrite. The variation of total energy of Nickel ferrite thin films with angle and number of layers was investigated. Also the change of energy with stress induced anisotropy for Nickel ferrite films with N=5 and 1000 has been studied. Films \nwith the magnetic moments ratio 1.86 can be easily oriented in θ=90\n0 direction when N is \ngreater than 400. Although this simulation was performed only for \n5 5 ,0 ,10 ,100)4(\n1)2(\n1= === = =∑ ∑\n= =\nω ωωω ω ωmN\nm s out inmN\nmD\nandK H HDJ as an example, these \nequations can be applied for any value of ω ωωωωω)4(\n1)2(\n1, , , ,mN\nm s out inmN\nmD\nandK H HDJ∑ ∑\n= =. \nKeywords: Ferrites, Heisenberg Hamiltonian, thin films, anisotropy, spin \nPACS numbers: 75.10.Hk, 75.30.Gw. 75.70.-i  \n1. Introduction: \n        For the first time dipole matrix and the total energy of the cubic spinel ferrites were \ncalculated using classical model of Heisenberg Hamiltonian in detail. All the relevant  2energy terms such as spin exchange energy, dipole energy, second and fourth order \nanisotropy terms, interaction with magnetic field and stress induced anisotropy in Heisenberg Hamiltonian were taken into account. The spin exchange interaction energy and dipole interaction have been calculated only between two nearest spin layers and within same spin plane. These equations derived here can be applied for spinel ferrites such as Fe\n3O4, NiFe 2O4 and ZnFe 2O4 only. But these equations can not be applied for \nferrites such as Lithium ferrite.          The structure of spinel ferrites with the position of octahedral and tetrahedral sites is given in detail in some early report \n1-5. Although there are many filled and vacant \noctahedral and tetrahedral sites in cubic spinel cell 1, only the occupied octahedral and \ntetrahedral sited were used for the calculation in this report. Only few previous reports could be found on the theoretical works of ferrites \n6-9. The solution of Heisenberg ferrites \nconsist of spin exchange interaction term only has been found earlier by means of the retarded Green function equations \n6.     \n \n2. The model \n        The Hamiltonian in Heisenberg model can be written as following for a film. \nH= - ∑∑ ∑ ∑ − − − +\n≠ nmm mz\nmz\nm\nnmmnn mn mnm\nmnn m\nn m S D S D\nrSr rS\nrSSSSJ\nm m\n,4 )4( 2 )2(\n5 3) ( ) ( )).)(.(3 .( .λ λ ωrr rr rrrr\n \n             ∑∑ − −\nmmm s m SinK SH θ2 ..rr\n       (1) \n                Here θ is the angle between local magnetization (M) and the stress. Within a \nsingle domain, M is parallel to the spin.  If stress is applied normal to the film plane, then \nθm is the angle between the normal to the film plane and the local spin. Here the last term  3indicates the change of magnetic energy under the influence of a stress. K s depends on the \nproduct of magnetostriction coefficient (λs) and the stress ( σ). K s can be positive or \nnegative depending on the type of stress whether it is compressive or tensile. Integer m \nand n denote the indices of planes, and they vary from 1 to N for a film with N number of \nlayers. First, second, third and fourth terms represent the spin exchange interaction, magnetic dipole interaction, second order anisotropy and fourth order anisotropy, \nrespectively. Here \niSr\n is a spin vector at pointirr in layeriλ. Therefore the ground state \nenergy will be calculated per spin with Z-axis normal to film plane. Hr\n is the external \nmagnetic field with the effective magnetic moment µ of the spins incorporated.  \n                   The spin exchange interaction energy is negative and positive for parallel and \nantiparallel spins, respectively. Hence spin exchange interaction energy constant (J) is positive and negative for parallel (Ferromagnetic) and antiparallel (ferrites or antiferromagnetic) spin arrangements, respectively. Similarly dipole interaction energy is positive and negative for parallel and antiparallel spin arrangements. In this report, the spin structure of AFe\n2O4 spinel ferrite cell described by Kurt et al will be used 1. In this \nconsidered spinel ferrite cubic cell, all spins in one spin layer are produced by either Iron or other metal (A) ions, but spins in two consecutive spin layers are produced by Iron and metal (A) ions alternatively. Therefore, all the spins in one layer are parallel, and spins in consecutive layers are antiparallel. Although dipole interaction energy and J are positive within one spin layer, both of them are negative between two nearest spin layers.  Only the interaction between two nearest layers has been considered for these calculations.  \n \n  43. Results and discussion \n                The length of one side of the cubic cell was taken as a. The spin exchange or \ndipole interaction between two spins with separation less than a was taken into account. Eight spin layers with separation a/8 in the cubic unit cell were considered as given in Kurt et al \n1. The number (Z) of metal (A) and Iron ions with separation less than a in \neach layer and in between two nearest spin layers is given in table 3. The spin with \nmagnitude s will be given as s(0, sin θµ, cosθµ). The dipole interaction energy between \ntwo spins is given by \nj ij i SrWS Er r\n).(.ω=                     (2) \nHere \n\n\n\n\n− − −−− −− −−\n=\n222\n3\n313 33 3133 3 31\n1)(\nz zy zxyz y yxxz xy x\nr rr rrrr r rrrr rr r\nrrW                         (3) \nand 32\n0\n4aπµµω=  \n                    The spins of A and Fe ions are given as 1 and p, respectively. For an \nexample, the ratio between spins in Nickel ferrite (NiFe 2O4) can be given as p=2.5. The \nmatrix elements calculated within each in spin layer and in between two nearest spin layers are also given in table 3. A film with (001) orientation of spinel ferrite cell has been considered. As an example, the calculations of some matrix elements are given below. In layer one (bottom layer of spinel cell), five metal ions occupy A\n1(0, 0, 0), A 2(1, \n0, 0), A 3(0, 1, 0), A 4(1, 1, 0) and A 5(0.5, 0.5, 0) sites 1. Because the interactions between \nspins with separation less than a have been considered, only the A 5A1, A 5A2, A 5A3 and \nA5A4 spin interactions have been taken into account. For these spin interactions, \nindividual and total matrix elements are given in table 1. The spins of Fe ions in the  5second spin layer located at 0.125 above the bottom layer of spinel cell occupy Fe 1(0.125, \n0.625, 0.125), Fe 2(0.375, 0.875, 0.125), Fe 3(0.625, 0.125, 0.125) and Fe 4(0.875, 0.375, \n0.125) sites 1. Because the separations between following interactions are less than a, \nthe spin interactions between A 3Fe1, A3Fe2, A2Fe3, A2Fe4, A5Fe1, A5Fe2, A5Fe3 and A 5Fe4 \nwere considered for this calculations. The individual and total dipole matrix elements of these interactions are given in table 2. Similarly the total dipole matrix elements calculated for other spin layers are given in table3.                  The spin exchange interaction energy for nearest spins in one layer and spins between nearest layers with separation less than a can be given as below for one unit cell using the nearest spin neighbors given in table 3. J is assumed to be a constant for all spin layers throughout the whole film. E\nEx unit cell = -10J+80Jp-22Jp2 \nFor a thin film with thickness Na (height of N cubic cells), \nTotal spin exchange interaction energy=EEx\nTotal=2J(-5N+40Np-11p2N-4p)             (4)   \nFor all A type spins given in table 3, the dipole interaction energy can be given as \nfollowing by using equation 2. \n()\n\n\n\n\n\n\n\n−− =\nνν µ µ\nθθ θθω\ncossin0\n1 0002100021\n) 2842.28( cos sin0 AdipoleE    \nHere 28.2842 is the addition of W 33 matrix elements of all A type rows given in table 3. \nAlso this dipole matrix is similar to that of a highly symmetric cubic cell 10. Within one \nferrite unit cell, all the spins are either parallel or antiparallel to each other due to the  6super exchange interaction between spins. Therefore, the angle θµ=θν=θ within unit cell \nwill deduce above equation to following equation. \n)2cos43\n41( 2842.28 θ ω + −=AdipoleE    \nSimilarly for all Fe layers and A-Fe nearest layer interactions, \n)2cos43\n41(2572θ ω + −= p E Bdipole \n)2cos43\n41( 3464.576 θ ω + =p E ABdipole \nThe dipole interaction energy of a unit cell= ABdipole\nBdipole\nAdipole\nunitcelldipoleE E E E + + =  \nIf the film is highly oriented the angle θ remains same throughout the whole film. \n For a thin film of thickness Na, \nTotal dipole interaction energy= unitcelldipole\nTotaldipoleNE E =  \n) 3464.576 257 2842.28)(2cos43\n41(2p p N E Totaldipole+− − += θ ω                (5) \nThis equation is valid only for large N. The 5th term of equation 1 can be given as, \n∑ − + =\nmout in m p H HN SH ) 1)( cos sin (4 . θ θrr\n                                    (6) \nTherefore, from equation 1, 4, 5 and 6 the total energy can be given as, \n)4 11 40 5(2)(2p Np Np N J E −−+−=θ    \n      ) 3464.576 257 2842.28)(2cos43\n41(2p p N +− − ++ θ ω         \n     )2sin cos sin )( 1(4 cos cos)4(\n114 )2( 2θ θ θ θ θs out in mN\nmN\nmm K H Hp N D D + + −− − − ∑∑\n==  (7)   \nUsing 0 90 ,0 ,1000=∂∂== =pEand HJ\ninθω for minimum energy,  7Np19.086.1−=  \nThe graph between p and N is given in figure 1. The number of layers corresponding for \npreferred perpendicular orientation can be found for different ferrites using this graph. \nFor ferrites with p=1.86, the films with N>400 can be easily oriented in θ=900 direction.   \nFor Nickel ferrite, p=2.5  \n)2cos31( 415.48)10 25.26(2)( θ ω θ + −− = N N J E  \n         ) 2sin cos sin (6 cos cos)4(\n114 )2( 2θ θ θ θ θs out in mN\nmN\nmm K H HN D D + + + − − ∑∑\n==        (8) \nWhen 5 5 ,0 ,10 ,100)4(\n1)2(\n1= === = =∑ ∑\n= =\nω ωωω ω ωmN\nm s out inmN\nmD\nandK H HDJ \nθ θ θ θωθ2sin 30 cos5 cos10 2cos 25.145 2000 6. 5201)(4 2N N NE+ − − −− =  \nThe 3-D graph of ωθ)(E versus N and θ is given in figure 2. When the number of layers \nincreases, the energy gradually increases with some sinusoidal variation. But for one \nvalue of N, the energy remains constant.  \nWhen N=5 (thin) and ωsK is a variable from equation 8, \nθωθ θωθ2sin 30 cos5 cos6. 1462 247343)(4 2 sK E+ − − =  \nHere the other constants given above have been used. The 3-D plot of ωθ)(E versus θ and \nωsK is given in figure 3. Because the energy indicates minimums at some stress values,  8the value of stress corresponding to different orientations with minimum energy can be \nobtained using this graph.    \nWhen N=1000 (thick) and ωsK is a variable, \nθωθ θωθ2sin 6000 cos5 cos 290500 5344830)(4 2 sK E+ − − =  \nThe 3-D plot of ωθ)(E versus θ and ωsK for N=1000 is given in figure 4. According to \ngraph 3 and 4, the stress corresponding to minimum energy increases with number layers \nat lower angles. But at higher angles, this stress corresponding to minimum energy does not depend on number of layers. Also according to these two graphs, the maximum energy increases with the number of layers.   For Fe\n3O4 and ZnFe 2O4, the ratio p=1.25 and 1, respectively. Above simulation can be \nrepeated for theses ferrites too by using equation 7. \n4. Conclusion \n           The dipole matrix of this complicated spinel cubic cell could be simplified into the form of dipole Matrix of simple cubic cells. Films with the magnetic moments ratio 1.86 \ncan be easily oriented in \nθ=900 direction when N is greater than 400. The total energy of \nNickel ferrite thin films gradually increases with number of layers for the values used in \nthis simulation. Also the energy indicates some minimum values at some stress values implying that the film can be easily oriented in some directions under the influence of some particular applied stress. Also this stress corresponding to minimum energy varies  9with number of layers at lower angles. This simulation can be similarly applied for any \nvalue of ω ωωωωω)4(\n1)2(\n1, , , ,mN\nm s out inmN\nmD\nandK H HDJ∑ ∑\n= =. \n \n  10References: \n1. Kurt E. Sickafus, John M. Wills and Norman W. Grimes, J. Am.Ceram. Soc. 82(12) ,   \n    3279 (1999) 2. I.S. Ahmed Farag, M.A. Ahmed, S.M. Hammad and A.M. Moustafa Egypt, J. Sol.      \n24(2) , 215 (2001) \n3. V. Kahlenberg, C.S.J. Shaw and J.B. Parise, Am.Mineralogist 86, 1477 (2001) \n4. I.S. Ahmed Farag, M.A. Ahmed, S.M. Hammad and A.M. Moustafa, Cryst. Res.      Technol. \n36, 85 (2001) \n5. Z. John Zhang, Zhong L. Wang, Bryan C. Chakoumakos and Jin S. Yin, J. Am. Chem.    Soc. \nI20, 1800, (1998) \n6. Ze-Nong Ding, D.L. Lin and Libin Lin, Chinese J. Phys. 31(3) , 431 (1993) \n7. D. H. Hung, I. Harada and O. Nagai, Phys. Lett. A53, 157 (1975) \n8. H. Zheng and D.L. Lin, Phys Rev. B37, 9615 (1988) \n9. S.T. Dai and Z.Y. Li, Phys. Lett. A146 , 50 (1990) \n10. K.D. Usadel and A. Hucht: Phys. Rev. B 66, 024419-1 (2002) \n \n \n  11Figure and Table Captions \nTable 1. The individual and total dipole matrix elements of the bottom layer of spinel cell \nTable2. The individual and total dipole matrix elements for the interaction between first               and second layer of spinel cell Table 3. The number of nearest neighbors and matrix elements of dipole tensor for each                layer and two nearest layers Figure 1. Graph between p and N at minimum energy for perpendicular orientation \nFigure 2. 3-D graph of \nωθ)(E versus N and θ for Nickel ferrite \nFigure 3. 3-D plot of ωθ)(E versus θ and ωsK for Nickel ferrite with N=5 \nFigure 4. 3-D plot of ωθ)(E versus θ and ωsK for Nickel ferrite with N=1000  \n \n     \n \n \n \n \n \n  12 \n \n \n \n W11 W 12=W 21W13=W 31 W22W23=W 32 W33 \nA5A1 -1.41421  -4.24264 0 -1.41421 0 2.828427 \nA3A5 -1.41421 4.242641 0 -1.41421 0 2.828427 \nA2A5 -1.41421 4.242641 0 -1.41421 0 2.828427 \nA4A5 -1.41421 -4.24264 0 -1.41421 0 2.828427 \nTotal -5.65685 0 0 -5.65685 0 11.31371 \n \nTable 1 \n \n \n  13 \n \n W11 W 12=W 21 W13=W 31 W22 W23=W 32 W33 \nA2Fe3 -20.4131 11.48235 11.48235 10.20653 -3.82745 10.20653 \nA2Fe4 10.20653 11.48235 3.82745 -20.4131 -11.4823 10.20653 \nA5Fe3 10.20653 11.48235 -3.82745 -20.4131 11.48235 10.20653 \nA5Fe4 -20.4131 11.48235 -11.4823 10.20653 3.82745 10.20653 \nA3Fe1 10.20653 11.48235 -3.82745 -20.4131 11.48235 10.20653 \nA3Fe2 -20.4131 11.48235 -11.4823 10.20653 3.82745 10.20653 \nA5Fe1 -20.4131 11.48235 11.48235 10.20653 -3.82745 10.20653 \nA5Fe2 10.20653 11.48235 3.82745 -20.4131 -11.4823 10.20653 \nTotal -40.8261 91.8588 0 -40.8261 0 81.65226 \n \n \nTable 2 \n \n \n \n \n  14 \nLayer Z W 11 W 12=W 21W13=W 31W22 W 23=W 32 W 33 \n1 Metal 4 -5.65685 0 0 -5.65685  0 11.31371  \n1 and 2 8 -40.8261  91.8588  0 -40.8261  0 81.65226  \n2 Fe 6 -27.4797  -55.754  0 -27.4797  0 54.9594  \n2 and 3 8 -26.9009  37.81957  0 -26.9009  0 53.80186  \n3 Metal 1 -1.41421  -4.24264  0 -1.41421  0 2.828427  \n3 and 4 8 -26.9009  -37.8196  0 -26.9009  0 53.80186  \n4 Fe 5 -36.7696  110.3087  0 -36.7696  0 73.53911  \n4 and 5 16 -49.4587  -78.9235  0 -49.4587  0 98.91731  \n5 Metal 4 -5.65685  0 0 -5.65685  0 11.31371  \n5 and 6 16 -49.4587  78.92348  0 -49.4587  0 98.91731  \n6 Fe 5 -36.7696  -110.309  0 -36.7696  0 73.53911  \n6 and 7 8 -26.9009  37.81957  0 -26.9009  0 53.80186  \n7 Metal 1 -1.41421  4.242641  0 -1.41421  0 2.828427  \n7 and 8 8 -26.9009  -37.8196  0 -26.9009  0 53.80186  \n8 Fe 6 -27.4797  55.75403  0 -27.4797  0 54.9594  \n8 and 9 8 -40.8261  -91.8588  0 -40.8261  0 81.65226  \n  \n \nTable 3 \n \n \n \n \n \n \n \n  15 \n \n \n \n \n \n \n \nFigure 1 \n \n \n \n  16 \n \n \n \n \n \n \n \nFigure 2 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \n \n \n \n \n \n \nFigure 3 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \n \n \n \n \n \n \n \n \nFigure 4 \n \n \n \n "    },    {        "title": "1803.09315v1.Analytical_modeling_of_demagnetizing_effect_in_magnetoelectric_ferrite_PZT_ferrite_trilayers_taking_into_account_a_mechanical_coupling.pdf",        "content": "1 \n Analytical modeling of demagnetizing effect in magnetoelectric \nFerrite/ PZT/Ferrite trilayers  taking into account a mechanical coupling . \nV. Loyau, A.  Aubert , M. LoBue, and F. Mazaleyrat  \nSATIE UMR 8029 CNRS, ENS Cachan, Université Paris -Saclay, 61, avenue du président \nWilson, 94235 Cachan Cedex, France.  \nKeywords  : Magnetoelectric  composite; Layered magnetic structure; Demagnetizing effect ; \nMagnetostriction  \nArticle  in Journal of Magnetism and Magnetic Materials 426  · November 2016  \nDOI: 10.1016/j.jmmm.2016.11.125  \n \nAbstract.  In this paper, we investigate  the demagnetizing  effect in ferrite/PZT/ferrite \nmagnetoelectric (ME) trilayer composites  consist ing of  commercial PZT discs bonded by  \nepoxy layer s to Ni-Co-Zn ferrite discs  made by a reactive Spark P lasma Sintering (SPS) \ntechnique . ME voltage coefficients  (transversal mode)  were measured on ferrite/PZT/ferrite \ntrilayer ME samples with  different thicknesses or phase volume ratio  in order to highlight the \ninfluence of the magnetic f ield penetration governed by the se geometrical parameters. \nExperimental ME coefficients and voltages were compared  to analytical calculation s using a \nquasi -static model . Theoretical d emagnetizing factors of two magnetic discs that interact \ntogether i n parallel magnetic structures we re derived from a n analytical calculation based on a  \nsuperposition method. These factors were  introduced in  ME voltage  calculation s which take  \naccount of the demagnetizing effect . To fit the experimental results, a mechanical coupling \nfactor was also introduced in the theoretical formula. This reflects the differential strain that  \nexist s in the ferrite and PZT layers due to shear effect s near the edge of the ME samples  and \nwithin the bonding epoxy layers . From this study, an optimization  in magnitud e of the ME \nvoltage  is obtained . Lastly, a n analytical calculation of demagnetizing effect was conducted \nfor layered ME composites containing higher numbers of alternated layers (𝑛≥5). The \nadvantage of such a structure is then discussed.   \nI. Introduction.  \nMagnetoelectric (ME) composites using the product -property concept are particularly suitable \nfor smart sensors fabrication ( e.g. magnetic field or current sensors1-5). The product -property \neffect is obtained w hen piezoelectric and magnetostrictive phases are mechanically coupled to \neach other.  At the present date, layered ME composites have high interest because they \nproduce the best ME performances. Bilayers of piezoelectric and magnetostrictive materials \nare the simplest layered composites but these structures exhibit low ME effects. In order to \nachieve high ME responses, some authors6,7 have focused their studies on co -sintered ME \nsamples containing a high number of alternated PZT/ferrite thin layers. Among the different \nstructures of laye red composites, the trilayer , consisting of a piezoelec tric layer sandwiched \nbetween  two magnetostrictive layers , achieve s a good balance between ease of fabrication and \nperformances8,9. In a recent paper4, we have shown that a piezoelectric layer stressed on its 2 \n two faces by two ferrite layers (ME trilayer) has much better mechanical coupling in \ncomparison with a bilayer configuration where  the piezoelectric layer is stressed only on one \nface. Furthermore,  the symmetric configuration of a trilayer sample avoids any flexural strain \nthat can reduce the ME response. The shape demagneti zation is an another important \nparameter that affect the ME response4, 10-13. In the same way, the ME response is increased \nusing  a trilayer configuration because the magnetic field penetration is impro ved within two \nseparate ferrite  layers in comparison with a single ferrite layer of the same  total thickness12. \nHowever,  the calculation of demagnetizing factor s is more complicate when two (or more) \nseparated magnetic layers interact together , thus it is difficult to predict and optimize the ME \nresponse in such a geometry . There are few  publications on this subject in the literature12,13,14, \nand to the best of ou r knowledge, only one concerns a demagnetizing factor calculation using \nan analytical method12. In this paper, we propose an alternative method to calculate the \ndemagnetizing factor of two magnetic disc s in a parallel configuration. This analytical \ncalculation is based on a superposition principle and it is valid for a wide range of material \npermeabil ities. Moreover, the model is extended to stacked configuration s including more \nthan two magnetic layers.  The aim of this work is to quantify and optimize  the magnetic field \npenetration within the ferrites layers in order to maximize the ME response of the ME trilayer. \nObviously, the field penetration is better when two magnetic layers are thin and far from each \nother but at the same time,  the mechanical coupling between the magnetic and piezoelectric \nphases is reduced. So in addition, we have taken into account a mechanical coupling that vary \naccording to the volume ratio of the two phases.  We have studied  the ME response  of ME \ntrilayers theoretically and experimentally in terms of demagnetizing  effect and mechanical \ncoupling , and we have investigated the influence of geometrical parameters (thicknesses of \nPZT and ferrite layers) in order to reach the optimum ME performance  of a layered ME  \ncomposite .   \nII. Theoretical basis.  \nLet us consider a trilayer ME sample made by sandwiching a PZT disc between two ferrite \ndiscs,  where the direction (3) (z axis) corresponds to the cylindrical axis of symmetry.  \nUsually, the ME effect is measured by applying a small external AC field 𝐻1𝑎 (in direction \n(1)) superimposed to an external DC bias field. The transversal coupling coefficient 𝛼31 is \nthen obtained by measuring the induced electrical field  𝐸3 in the dir ection (3) . According to \nthis method, t he theoretical coupling coefficient  (transversal mode)  is given by4:  \n𝛼31=𝐸3\n𝐻1𝑎=−𝜂𝑑31𝑒\n𝜀33𝑇 [𝑠11𝐸+𝑠12𝐸+𝜂𝛾(𝑠11𝐻+𝑠12𝐻)]−2(𝑑31𝑒)2× (𝑑11𝑚+𝑑12𝑚)\n1+𝑁𝜒                                                       (1) \nwhere, for the piezoelectric material,  𝑠11𝐸 and 𝑠12𝐸 are zero field compliances , 𝑑31𝑒 is the \npiezoelectric coefficient , and 𝜀33𝑇 is the zero stress permittivity ; and for the magnetic material , \n𝑠11𝐻 and 𝑠12𝐻 are zero field compliances, 𝑑12𝑚 and 𝑑11𝑚 are intrinsic piezomagnetic coefficients,  𝜒 \nis zero stress dynamic susc eptibility, and  𝑁 is the radial magnetometric demagnetizing factor. \n𝛾=𝑉𝑝/𝑉𝑓 is the volume ratio of PZT with respect to ferrite material. The mechanical \ncoupling factor4 𝜂=〈𝑆1𝑒〉/〈𝑆1𝑚〉 (in average) , takes into account the differential strain between \nthe PZT layer (average strain: 〈𝑆1  𝑒〉) and the ferrite layers (average strain: 〈𝑆1𝑚〉). The ME 3 \n response curve is mainly shaped by the right hand term in Eq. (1) because the internal DC \nfield 𝐻𝐷𝐶 sets the value of the intrinsic piezomagnetic coefficients and the dynamic \nsusceptibility. Usually, ME curves are plotted against the external app lied DC field 𝐻𝐷𝐶𝑎, and \ndue to the demagnetizing effect , the ME curves are shifted along the  𝐻𝐷𝐶𝑎 axis because  the \nlink between the internal and external DC field is:  \n𝐻⃗⃗ 𝐷𝐶=1\n1+𝑁.𝜒𝐷𝐶  𝐻⃗⃗ 𝐷𝐶𝑎                                                                                                                 (2) \nwhere  𝜒𝐷𝐶 is the static susceptibility. It must be noted that the radial magnetometric \ndemagnetizing factor N is for two parallel ferrite layers configuration .  \n The left hand term in Eq. (1) is mainly dependent upon the mechanical properties of the PZT \nand the ferrite material s, and the mechanical structure of the ME composite. In case of a \ntrilayer sample, the PZT disc is stressed on both faces. The propagation of the longitudinal \nstrain from the two PZT/ferrite interfaces to inner PZT layer depends on the thickness to \ndiameter ratio of the ME sample, and the relative  thicknesses o f the PZT and ferrite layers. \nCons equently,  the mechanical coupling factor  𝜂 is affected by these parameters.  \nSummarizing , the ME response is affected by the dimensions  (thicknesses and diameter)  of \nthe layers in the following way: (i) a strong AC field penetration, and consequently a high \nlevel M E response are obtained for layer  configurations producing low demagnetizing effects; \n(ii) the demagnetizing factor N sets the optimal working point 𝐻𝐷𝐶𝑎 for which the ME response \nis maximum for a given sample;  (iii) the mechanical coupling factor is reduced  when the \nstrain distribu tion is strongly non -uniform across the ME sample.  In the next parts of the \npaper, we will quantify the influences of the geometries ( layer sizes) on the ME responses.  \nIII. Experimental aspect . \nA. ME samples fabrication  \nTrilayer ME samples were made by bonding a commercial PZT disc with two ferrite discs of \nthe same composition ((Ni 0.973Co0.027)0.875Zn0.125Fe2O4) chosen for its high piezomagnetic \nproperties4. The ferrite material  was made by a reactive Spark Plasma S intering (SPS) \ntechnique (see ref. 4). After the SPS stage, all ferrite discs (2  mm in thickness and 10  mm in \ndiameter) were annealed in air at 1000  °C during 1 hour for full re -oxidation. Then the ferrite \ndiscs were sliced  by means of diamond saw  (Struers  Secotom -10) and grinded (using silicon \ncarbide pap ers) to reduce their thickness  to appropriate values between 0.165  mm and 1  mm. \nIn the same way, PZT  discs (Ferroperm, Pz27 , poled along the thickness ) with 1  mm or 0.5  \nmm in thickness  and 10  mm in diameter  were machined and grinded to reduce their \nthicknesses from 0.75  mm to 0.25  mm.  Lastly, two ferrite discs with the same thickness were \npasted (using conductive silver epoxy Epotek E4110) on each faces of the PZT disc.  Two sets \nof sample s were prepared . First, ME samples with different thicknesses (0.75, 1.05, 1.5, 2.25, \nand 3  mm) but with the same PZT/ferrite volume ratio ( 𝛾=0.5), and s econdly, ME samples \nwith the same total thickness (1.5  mm) but with different PZT/ferrite volume ratio ( 𝛾=\n0.2,0.5,1,2, and 3.5).  This relatively simple  method of ME samples fabrication  process  \nallow s however to  obtain  very accurate c haracterization results. Indeed , all the PZT discs 4 \n were poled (with the same optimal electric field) before the fabrication  of the ME samples, so, \nall the ME sampl es have optimal and reproducible  piezoelectric properties. On the other hand, \nseveral  authors6,7,9 have already conducted studies on co -sintering PZT/ferrites multilayer. \nMoreover , in those cases, the co -sintering stage occurs before the poling stage of the PZT \nlayers, and thus the poling process is influenced by the ferrite materia l for the following \nreason : (i) the electric field is applied through the ferrite layers which have v ery low relative \npermittivity (~10) in comparison t o the PZT material (~1000 -2000);  (ii) if the fer rite material \nhas high resistivity  (so no electrical current exists) , most of the applied  field is absorbed by \nthe ferrite layers. Thus, the poling field within the PZT material may be  suboptimal and it \nmay depend  on the volume ratio of PZT and ferrite. As a consequence, the piezoelectric \nproperties of the PZT layer  and thus the ME properties  can vary from a ME sample to an \nanother. This problem is av oided  when the PZT material is poled before the fabrication of the \nME samples.  \nB. ME measurement results  and discussion  \nThe experimental transversal ME coefficient 𝛼31 is derived from the voltage V measured \nacross the electrode s of the PZT layer (direction (3)) when a small external AC magnetic field \n𝐻1𝑎 (1 mT in our case) is applied in the direction (1): 𝛼31=𝑉/𝑡𝑝/ 𝐻1𝑎, where 𝑡𝑝 is the \nthickness of the PZT layer. The measurements are repeated for different external stat ic field \n𝐻𝐷𝐶𝑎 (applied in direction (1)) defining the working points ( 0<𝐻𝐷𝐶𝑎<6×104𝐴/𝑚). To \navoid any resonance phenomena, the AC magnetic field has been kept low  frequency (80  Hz). \nIn a first experiment, ME coefficients were measured on trilayer samples with always the \nsame PZT /ferrite  volume ratio : 𝛾=0.5. For a given sample, each layer s have the same \nthickness t, and consequently, the total thickness of a sample is 3t. Experiments were \nconducted on samples with total thickness between 0.75  mm and 3 mm, and the results are \ngiven in Fig. 1. The increase  in the sample thickness (for a given PZT /ferrite  volume ratio ) \nleads  to a decrease  in the ME peak amplitude and in a upshift in the peak s position s. These \neffects are due to the  increase of the  demagnetizing factor. In fact, the demagnetizing effect \nreduces the AC field penetration in a ratio 1/(1+𝑁.𝜒) and the amplitude of the ME peak is \naffected accordingly. Furthermore , the DC field penetration is  also diminished in a ratio \n1/(1+𝑁.𝜒𝐷𝐶) and thus, the maximum s of the  intrinsic piezomagnetic coefficients 𝑑11𝑚 and  \n𝑑12𝑚 are shifted (by the factor 1+𝑁.𝜒𝐷𝐶 ) to higher external DC field s. It must be noted that \nthere  is a magnetic coupling between the two ferrite layers and in this case, the demagnetizing \nfactor  N is higher than the one obtained for a single ferrite layer with the same dimensions.  In \nFig. 2, the measured ME peak coefficients (circle symbols) and the measured  1/(𝐻𝐷𝐶𝑎)𝑚𝑎𝑥 \nfunction (square symbols) are plotted versus the thickness t. For better comparison s, the \namplitude of the curves were normalized  (with respect to the value  given by the thicker ME \nsample ). It appears that the two curves are similar, exhibiting the  same behavior concerning \nthe demagnetizing effect. It means that in our case, the static permeability  𝜒𝐷𝐶 and the \nreversible permeability 𝜒 may have a similar  value. Using a Vibrating Sample Magnetometer \n(VSM) technique, the static permeability  𝜒𝐷𝐶=𝑀/𝐻 and the differential permeability \n𝜒𝑑𝑖𝑓𝑓=𝑑𝑀/𝑑𝐻 were measured on a spherical sample of the ferrite material. At the internal 5 \n field for which the ME coefficients are maximum we obtain:  𝜒𝐷𝐶=84 and 𝜒~𝜒𝑑𝑖𝑓𝑓=94, \nwhich agrees with the previous assumption.  \nThe demagnet izing effect is not the only  parameter affecting the M E coefficient 𝛼31. In Eq. \n(1), it can be seen that the left hand term is a function of the mechanical coupling 𝜂. We can \nsuppose that this mechanical coupling is improved when the thickness to diameter ratio t/d is \ndecreased , thus improving the ME coefficient . On the other hand, as a  basic principle, the \nintrinsic piezomagnetic coefficients are assumed to be unaffected by the mechanical coupling  \nand then, free of the influence of the ratio t/d. Thus, (𝐻𝐷𝐶𝑎)𝑚𝑎𝑥 , the field at maximum  ME \ncoefficients is free from such an influence . Nevertheless, the two curves  plotted in Fig. 2 \nmatch well  and the 1/(𝐻𝐷𝐶𝑎)𝑚𝑎𝑥 curve is known to be  unaffected  by the ratio t/d. As a \nconsequence, the ME coefficient 𝛼31, and then the mechanical coupling 𝜂, is free from such \nan influence . So, another important finding from the previous measurements is that the \nmechanical coupling  𝜂 seems to be independent of the thickness to diameter ratio t/d (in the \nrange  0.025≤𝑡/𝑑≤0.1).  \nThe voltage gain, 𝑉=𝛼31.𝑡𝑝, is an important parameter for a ME sample used in a real \napplication (a current sensor for example). In our case, when subjected to a 1mT external AC \nfield, the 0.75  mm thick ME sample produces 0.19  V and the 3  mm thick ME sample \nproduces 0.28  V. So when the thickness  𝑡𝑝 of the piezoelectric layer is increased by a factor 4, \nthe voltage is increased by a factor 1.5 only. The demagnetizing effect explains this \ndiscrepancy: the increase in the ferrite layers thicknesses diminishes the magnetic field \npenetr ation. So, increasing the piezoelectric thickness and at the same time decreasing the \nferrite layers thicknesses is the way to obtain high voltage gains in trilayer ME sample s. To \nverify this point, t rilayer  ME samples were fabricated, all with the same total thickness (1.5  \nmm), but with various PZT/ferrite volume ratio  (𝛾=0.2,0.5,1,2, and 3.5).  The measured \nME voltages are plotted in Fig. 3 for the ME samples subjected to a n external AC field  of \n1mT . It appears that t he ME peak voltages increases continuously with the increase of the \nvolume ratio 𝛾.  The maximum voltage (0.47  V) is obtained for the thicker PZT layer \n(𝑡𝑝=1.17 𝑚𝑚). This is due to a combination of two causes. First, for a given electric  field E, \nthe voltage is proportional to the thickness 𝑡𝑝 because  𝑉=𝐸×𝑡𝑝, so increasing 𝑡𝑝 will \nincrease V. Secondly,  the demagnetizing factor is low because the ferrite layers are thin \n(𝑡𝑓=0.16 𝑚𝑚 each) and they are far from each other (distance: 𝑡𝑝=1.17 𝑚𝑚), so their \nmagnetic interactions are weak. On the other hand, this phenomenon is a little bit counter \nbalanced by a weaker mechanical coupling coefficient 𝜂 at high PZT /ferrite  volume ratio  𝛾. \nFrom those experiments, it would appear that the demagnetizing effect is one of the most \nimportant phenomenon th at influences the ME voltages and  the ME coefficients.   \nIV. Analytical modeling.  \nA. C alculation of magnetometric demagnetizing factor s for two parallel ferrite discs  \nThe calculation of the  magnetometric demagnet izing factor of a single disc is  not trivial. But \nChen et al. have published several paper s on the subject  where useful tables of demagnetizing \nfactors for a single disc are given15,16. From those data, demagnetizing factors of two ferrites 6 \n discs in parallel configurations can be derived. The method of calculation is based on a one -\ndimension superposition method. The calculation is restricted to small thickness  t to diameter \nd ratio (in practice: 𝑡/𝑑≤10). In this case, the magnetic material s are assumed \n(approximately ) homogeneously magnetized even for magnetization states far from the \nsaturation.  \nLet us consider two ferrite discs with the same diameter d and thickness t in a parallel \nconfiguration. For simplicity, the distance between the two discs is t. As an approximation, \nwe suppose that the two ferrite discs are homogeneously magnetized with the same value 𝑀⃗⃗ , \nparallel to the direction 𝑥  (which is the consequence of  an external magnetic field applied in \nthe direction 𝑥 ). The magnetic structure can be divided in to 3 cells: the cells (1) and (3) are \nthe bottom and top ferrite discs, respectively, and the cell (2) is the vacuum  between the two \ndiscs where 𝑀⃗⃗ =0⃗  (see Fig. 4). For example, t he bottom magnetized ferrite  (cell (1)) creates \na magnetic field in all the space.  This magnetic field inside the cell (1) is called the \ndemagnetizing field whereas the magnetic field created outside the cell (1) is usually called \nthe interaction field. Here, we have chosen to name all those created fields  (inside and \noutside) , “dipolar  field”  because they have the same origin . The total dipolar  field 𝐻⃗⃗ 1𝑑 within \nthe bottom ferrite disc ( cell (1)) is the sum of two contribution  (in average) : \n 𝐻⃗⃗ 1𝑑=𝐻⃗⃗ 1,1𝑑+𝐻⃗⃗ 3,1𝑑 (3)                                                                                                           \nwhere 𝐻⃗⃗ 1,1𝑑 is the dipolar  field created by the magnetized cell (1) and acting on it , and 𝐻⃗⃗ 3,1𝑑 is \nthe dipolar field created by the magnetized cell (3) and acting on the cell (1). In the same way, \nthe total dipolar  field 𝐻⃗⃗ 3𝑑 within the top ferrite disc ( cell (3)) is: \n 𝐻⃗⃗ 3𝑑=𝐻⃗⃗ 3,3𝑑+𝐻⃗⃗ 1,3𝑑                                                                                                                     (4) \nwhere 𝐻⃗⃗ 3,3𝑑 is the dipolar  field created by the magnetized cell (3) and acting on it, and 𝐻⃗⃗ 1,3𝑑 is \nthe dipolar field created by the cell (3) and acting on the cell (1). Due to symmetry of the \nproblem, 𝐻⃗⃗ 1𝑑=𝐻⃗⃗ 3𝑑 and so 𝐻⃗⃗ 1,1𝑑=𝐻⃗⃗ 3,3𝑑 and 𝐻⃗⃗ 3,1𝑑=𝐻⃗⃗ 1,3𝑑.  \nThe global demagnetizing factor N of the two ferrite discs  interact ing in a parallel \nconfiguration  (structure given in Fig. 4) can be defined as:  \n𝐻⃗⃗ 1𝑑=𝐻⃗⃗ 3𝑑=𝐻⃗⃗ 3,3𝑑+𝐻⃗⃗ 1,3𝑑=−𝑁.𝑀⃗⃗                                                                                            (5)                                                                                                         \nwhere 𝐻⃗⃗ 3,3𝑑 is the dipolar field of a single magnetized disc. In this case, some tables of \ncalculated demagnetizing coefficient are available in the literature15,16 and 𝐻⃗⃗ 3,3𝑑 can be simply \ndetermin ed from:  \n 𝐻⃗⃗ 3,3𝑑=−𝑁𝑡,𝑑.𝑀⃗⃗                                                                                                                 (6) \nwhere 𝑁𝑡,𝑑 is the demagnetizing coefficient of a single disc with thickness t and diameter d. \nOn the other hand, the determination of 𝐻⃗⃗ 1,3𝑑 is not as direct as 𝐻⃗⃗ 3,3𝑑. Nevertheless , using a \nsuperposition method,  the demagnetizing factor N of a magnetic structure consisting in two \nparallel disc can be derived from  a sum of demagnetizing factors of single discs  with different \nthicknesses . 7 \n Consider now a single ferrite disc with a diameter d and a thickness 3t uniformly magnetized \nin the direction 𝑥  at the same value 𝑀⃗⃗  (see Fig 5(a)) . This disc can be divided into three cells, \neach with the same thickness t. In each cell, we define, 𝐻⃗⃗ 𝑖,𝑗𝑑, the dipolar field generated by the \ncell (i) and acting on the  cell (j). So, we obtain the total dipolar field 𝐻⃗⃗ 1𝑑 within the cell  (1): \n𝐻⃗⃗ 1𝑑=𝐻⃗⃗ 1,1𝑑+𝐻⃗⃗ 2,1𝑑+𝐻⃗⃗ 3,1𝑑                                                                                                          (7) \nIn the same way, the total dipolar field  𝐻⃗⃗ 2𝑑 within the cell  (2) is given by:  \n𝐻⃗⃗ 2𝑑=𝐻⃗⃗ 1,2𝑑+𝐻⃗⃗ 2,2𝑑+𝐻⃗⃗ 3,2𝑑                                                                                                          (8) \nAnd 𝐻⃗⃗ 3𝑑 within the cell  (3) is given by:  \n𝐻⃗⃗ 3𝑑=𝐻⃗⃗ 1,3𝑑+𝐻⃗⃗ 2,3𝑑+𝐻⃗⃗ 3,3𝑑                                                                                                          (9) \nDue to the geometry and the symmetry of the problem, 𝐻⃗⃗ 𝑖,𝑗𝑑=𝐻⃗⃗ 𝑗,𝑖𝑑  if 𝑖≠𝑗 and if 𝑖=𝑗 and \n𝐻⃗⃗ 1,1𝑑=𝐻⃗⃗ 2,2𝑑=𝐻⃗⃗ 3,3𝑑 . Thus,  the mean value of the global dipolar  field within a single disc with \na thickness 3t can be written : \n〈𝐻⃗⃗ 123𝑑〉=(𝐻⃗⃗ 1𝑑+𝐻⃗⃗ 2𝑑+𝐻⃗⃗ 3𝑑)/3=𝐻⃗⃗ 3,3𝑑+2\n3𝐻⃗⃗ 1,3𝑑+4\n3𝐻⃗⃗ 2,3𝑑=−𝑁3𝑡,𝑑.𝑀⃗⃗                                    (10)                                 \nwhere 𝑁3𝑡,𝑑 is the demagnetizing factor of a single ferrite disc uniformly magnetized with a \nthickness 3t and a diameter d. \nFor a single ferrite disc with a diameter d and a thickness 2t uniformly magnetized and \ndivided into two cells (region (2) and (3)) with the same thickness t (see Fig. 5(b)), the mean \nvalue of the global dipolar  field is:  \n〈𝐻⃗⃗ 23𝑑〉=(𝐻⃗⃗ 3,3𝑑+𝐻⃗⃗ 2,3𝑑+𝐻⃗⃗ 2,2𝑑+𝐻⃗⃗ 3,2𝑑)/2=𝐻⃗⃗ 3,3𝑑+𝐻⃗⃗ 2,3𝑑=−𝑁2𝑡,𝑑.𝑀⃗⃗                                     (11)    \nwhere 𝑁2𝑡,𝑑 is the demagnetizing factor of a single ferrite disc uniformly magnetized with \nthickness 2t and diameter d. \nLastly, considering a single ferrite disc with a diameter d and a thickness t uniformly \nmagnetized ( cell (3)) (see Fig. 5(c)), the mean value of the global dipolar  field is:  \n〈𝐻⃗⃗ 3𝑑〉=𝐻⃗⃗ 3,3𝑑=−𝑁𝑡,𝑑.𝑀⃗⃗                                                                                                         (12) \nwhere 𝑁𝑡,𝑑 is the demagnetizing factor of a single ferrite disc uniformly magnetized with \nthickness  t and diameter d. \nCombining Eqs. (10), (11), (12), leads to: \n𝐻⃗⃗ 1,3𝑑=−(3\n2𝑁3𝑡,𝑑−2.𝑁2𝑡,𝑑+1\n2𝑁𝑡,𝑑).𝑀⃗⃗                                                                                 (13)    \nCombining Eqs. (5), (6), and (13), we o btain the dipolar  field within two ferrite disc s, each \nwith thickness t and diameter d, separated by a distance t, and interacting in a parallel \nconfiguration : 8 \n 𝐻⃗⃗ 1𝑑=𝐻⃗⃗ 3𝑑=𝐻⃗⃗ 3,3𝑑+𝐻⃗⃗ 1,3𝑑=−(3\n2𝑁3𝑡,𝑑−2.𝑁2𝑡,𝑑+3\n2𝑁𝑡,𝑑).𝑀⃗⃗ =−𝑁.𝑀⃗⃗                                 (14)    \nwhere 𝑁 is the magnetometric demagnetizing factor for such a magnetic structure:  \n𝑁=3\n2𝑁3𝑡,𝑑−2.𝑁2𝑡,𝑑+3\n2𝑁𝑡,𝑑                                                                                               (15) \nWhen the two parallel ferrite disc s are spaced with a distance e different from the thickness t \nof a disc, using the previous method, the calculation of the  magnetometric demagnetizing \nfactor leads to:  \n𝑁=𝑁𝑡,𝑑−(1+𝑒\n𝑡).𝑁𝑒+𝑡,𝑑+(1+𝑒\n2𝑡).𝑁𝑒+2𝑡,𝑑+𝑒\n2𝑡.𝑁𝑒,𝑑                                                      (16) \nwhere 𝑁𝑒+𝑡,𝑑, and 𝑁𝑒+2𝑡,𝑑 are demagnetizing factor s for single ferrite discs with thicknesses \n(𝑒+𝑡) and (𝑒+2.𝑡) respectively , and diameter d.  Eq. (16)  is similar to the one calculated \nby Liverts et al.12. for two parallel rectangular ferromagnetic prisms.  A useful formula for the \ncalculation of the radial demagnetizing factor 𝑁𝑡,𝑑 of a single disc is given in Appendix A. \nB. Estimation of the mechanical coupling behavior  \nIn a stacked ME sample, the mechanical coupling is due to the shear stress that propagates \nfrom the ferrite layers to the PZT layer through the interfaces . The induced strain field results \nfrom the equilibrium between shear and extensional stresse s in each laye rs. The ME response \nis obtained when a small alternative strain field (produced by the alternative magnetic field) is \nsuperimposed with a DC strain field  (produced by the bias magnetic field) . Consequently, the \nstrain distribution is inhomogeneous in a ME sample and the exact alternative strain f ield can \nbe accurately predicted only by numerical method s17 (Finite Element Method for example).  \nNevertheless, in some special cases , the mechanical coupling could be estimated as following.  \nWhen the PZT layer thickness 𝑡𝑝 is very small compared to the total thickness  of the ferrite \nlayers 𝑡𝑓, a far -field strain is obtained in the PZT layer (because 𝑡𝑝≪𝑑) and the ferrite layers \nare almost mechanically free. Thus , the strain fields are almost homogeneous and equal  in \neach layer and the relative differential strain is close to zero:  \n〈𝑆𝑓〉−〈𝑆𝑝〉\n〈𝑆𝑓〉~0                                                                                                                              (17) \nOn the other hand, when the ferrite layers thickness is very small compared to the PZT one  \n(𝑡𝑓≪𝑡𝑝), the strain field is almost homogeneous in the ferrite layers (far -field \napproximation) and merges with the PZT/ferrite interfaces strain  〈𝑆𝑝,𝑓〉. The problem is now \nresumed to that of a PZT disc stressed on its both faces. In this case, the relative differential \nstrain have a finite value a, (0<𝑎<1) which depends o n the mechanical properties and \ndimensions of the PZT layer:  \n〈𝑆𝑓〉−〈𝑆𝑝〉\n〈𝑆𝑓〉~〈𝑆𝑝,𝑓〉−〈𝑆𝑝〉\n〈𝑆𝑓〉~𝑎                                                                                                          (18) 9 \n These two extr eme situations suggest that (〈𝑆𝑓〉−〈𝑆𝑝〉)/〈𝑆𝑓〉 is a function of 𝑡𝑝/(𝑡𝑝+𝑡𝑓) \nrather than 𝑡𝑝/𝑡𝑓, because when 𝑡𝑓≪𝑡𝑝, the ratio 𝑡𝑝/𝑡𝑓 tends toward s infinity, which \ncontradicts Eq. (18). Since the thickness to diameter ratio of each layer  of our ME samples are \nless than 10  %, we can assume a first order approximation in the mechanical coupling \nbehavior, and then:  \n〈𝑆𝑓〉−〈𝑆𝑝〉\n〈𝑆𝑓〉~𝑎.𝑡𝑝\n𝑡𝑝+𝑡𝑓                                                                                                                    (19) \nin the range 0<𝑡𝑝/(𝑡𝑝+𝑡𝑓)<1. From the previous equation, we deduce an approximated  \nmechanical coupling factor 𝜂: \n𝜂=〈𝑆𝑝〉\n〈𝑆𝑓〉=−〈𝑆𝑓〉−〈𝑆𝑝〉\n〈𝑆𝑓〉+1 ~−𝑎.𝑡𝑝\n𝑡𝑝+𝑡𝑓+1                                                                           (20)         \nThe elastic bonding layers (silve r epoxy  in our case ) absorb a part of the stress at the \nPZT/ferrite interface s13 and the  result  is a down shift of the mechanical coupling factor:  \n𝜂 ~−𝑎.𝑡𝑝\n𝑡𝑝+𝑡𝑓+𝑏                                                                                                                 (21)   \nwhere 𝑏<1 is the coupling factor of the bonding layers (obtained when 𝑡𝑝≪𝑡𝑓). The value s \nof a and b in Eq. (21) are un known and are therefore fitting  parameters.                                                            \nC. Application to an a nalytical modeling of the ME effect in trilayer ME samples  \nTo calculate the ME response of trilayer samples, the theoretical demagnetizing factor N of \ntwo parallel ferrite discs must be derived from the previous theory. First, radial \nmagnetometric demagnetizing factors for a single ferrite disc were interpolated  (𝜒=94) from \ndata published by Chen et al.15,16 (circle symbols in Fig. 6). These results were fitted using a \npolynomial function of degree 3 (for 0.01≤𝑡/𝑑≤0.3): \n𝑁𝑡,𝑑≈𝐴+𝐵×(𝑡/𝑑)+𝐶×(𝑡/𝑑)2+𝐷×(𝑡/𝑑)3                                                             (22) \nwhere the polynomial coefficients are: A=0.0027, B=1.014,  C=-2.087, D=2.313.  The fitted \nresult is plotted in dashed line in Fig. 6. Then, an analytical approximation of the \ndemagnetizing factor N of two ferrite discs in parallel configuration (with thickness t each) \nseparated by a distance t is calculated using Eq s. (15). To verify our analytical calculation for \ntwo parallel discs, (solid line in Fig. 6), we have solved the same magnetic problem by a \nnumerical method based on a Finite Element Method (FEM) software ( ANSY S Maxwell ). \nThe numerical results plotted in Fig. 6 (square symbols) validate the analytical method  of \ndemagnetizing calculation  developed  in this paper. To verify the relationship between the ME \nresponse and the demagnetizing effect, the theoretical field reduction ratios 1/(1+𝑁𝜒) were \ncalculated, normalized, and plotted in Fig.  2 (triangle symbols) for the five ME samples. \nThere is a good agreement between the theory and the experiment  that confirm the influence \nof the demagnetizing effect,  except for the thinnest sample ( 𝑡=0.25 𝑚𝑚). In this case, the \nexperimental result is 18  % over the theoretical one. It suggests that the magnetic properties \nare different for this sample: we may suppose that the AC susceptibility  is lower than  94 and 10 \n then, the field penetration is improved , leading to a stronger  ME response . In this study, since \nthe PZT /ferrite  volume ratio  𝛾 is maintained constant in the ME samples, the mechanical \ncoupling factor 𝜂 seems to have a constant value ( 𝜂~0.7). On the other hand, the ME voltages \npresented in Fig. 3 are obtained for ME samples whose 𝛾 ratios vary in a large proportion  \n(0.2≤𝛾≤3.5), involving  a large variation in the mechanical coupling factor 𝜂. The peak \nME voltages of these samples  were theoretically calculated using Eq. (1), by including both \nthe demagnetizing effect (using Eq. (16)) and the mechanical coupling . Materials  properties \nused for the calculation are summarized in Table I. It must be noted that 𝑑11𝑚 is the intrinsic \npiezomagnetic coefficient  and 𝑑12𝑚~−𝑑11𝑚/2 for polycrystalline Ni -Co-Zn ferrites . First, we \nhave plotted the theoretical peak ME v oltage s (dotted and dashed line s in Fig. 7), taking into \naccount constant mechanical coupling factor s 𝜂=1 and 𝜂=0.71 repectivly  (see Ref . 4 for \nthe choice of this value). We see that the theory do not match with the experiments when the \nvalue of the mechanical coupling factor is assumed constant . Then, according  to Eq. (21), we \nhave introduced a factor 𝜂 that decreases linearly against the  increase of the PZT volume ratio \n𝑉𝑝/(𝑉𝑝+𝑉𝑓). The obtained theoretical ME voltage, plotted in solid line in Fig. 7 shows a n \nimproved  agreement between theory and measurement s, especially for the ME samples with a \nthick PZT layer.  The linear curve 𝜂 that permits to fit the experimental ME voltage s is plotted \nin Fig. 8 in dashed line. In the 𝑉𝑝/(𝑉𝑝+𝑉𝑓)~0 extrapolated area, the mechanical coupling \nfactor is lower than 100%,  which means that the strains are not perfectly transmitted through \nthe glue layers, and a fraction ( 20%)  is absorbed. The analytical demagnetizing factor N used \nin the ME voltage calculation and plotted in Fig.  8 is nearly linear  as confirmed by the FEM \nsoftware calculations (square symbols).  Since all samples have the same total thickness (1.5 \nmm), the ME coefficient normalized with respect to the total thickness (1.5 mm) is given on \nthe right vertical axis in Fig. 7. Experimentally, the ME coefficient reach 𝛼31=0.4 𝑉/𝐴 for \nthe optimal ME sample ( 𝛾=0.35). Nevertheless, even for this optimized sample, due to the \ndemagnetizing effect, the internal magnetic field reaches only 30 % of the external applied \nfield (see Fig . 9). This low value is due to the high dynamic magnetic susceptibility ( 𝜒=94) \nthat appear s in the term 1/(1+𝑁𝜒). \nD. Numerical calculation of the mechanical coupling factor.  \nIn the previous part, it was shown that the coupling factor 𝜂 is strongly affected by the epoxy \nbonding layers, and even in the best case, more than 20% o f the strain is absorbed by th ose \nlayers.  In fact, there are mechanical properties mismatches between the ceramic materials \n(Young modulus:  𝐸=59 𝐺𝑃𝑎, for the PZT and 𝐸=154 𝐺𝑃𝑎 for the ferrite) and an epoxy \nresin (𝐸=3−10 𝐺𝑃𝑎, depending on material references), leading to a strong shear strain  \nfield within the bonding layers13. Some authors18 have shown that the mechanical properties \nof the epoxy layers have high influence on the ME response of a ME device. Furthermore, a \ncoupling coefficient18 must be introduced to model an interface detachment between ceramic \nlayers (PZT and ferrite) and the epoxy layers. In order to evaluate the influence of the bondin g \nlayers on the mechanical coupling factor 𝜂, the strain field within the  magnetoelectric \nstructure was modeled using a 3 -dimensional Finite Element Method (ANSYS software). The \nsimulated mechanical structure consists in two ferrite discs bonded by two epoxy layers on \nboth face of the PZT disc. The thickness of each epoxy layer (almost 30µm) was measured by 11 \n means of an optical microscope. This relatively high thickness can be explained by the size of \nthe silver particles constituting the conductive filler. The manufacture r (Epoteck) indicates \nparticle  sizes lower than 48µm, which can explains the thickness of the bounding layer s. The \nmechanical properties were  experimentally ob tained from ultrasonic velocity measurement s in \na sample  of Epotek E4110  (pulse -echo method4). The results are: Young modulus, 𝐸=\n8 𝐺𝑃𝑎, and Poisson ratio, 𝜈=0.35 (estimation).  \nFor a given AC magnetic field excitation, the strain field was calculated in each layers (using \nFEM method) and lastly, the theoretical mean strain ratio in the PZT and ferrite layer, \n𝜂=〈𝑆1𝑒〉/〈𝑆1𝑚〉, was obtained.  The FEM calculations were done for each of  the five samples \nof the study. These theoretical results are given in Fig. 10, in comparison with the \nexperimental coupling factor used to  fit the data  (solid line) . First, we studied a structure \nwhere the epoxy layers are perfectly mechanically coupled to the PZT and ferrite layers. This \nmeans that there is no sliding at the  PZT/epoxy and ferrite/epoxy interfaces , or in other words, \nthe coefficient of friction is 𝑘=1 at the interface (square symbols in Fig. 10). It is seen that \nthe behavior is almost li near with the PZT volume ratio (as predicted by Eq. 21), with a slope \nclose to the experimental curve (solid line), but with a bias overestimation. To overcome  this \nsystematic error , the same structure was  simulated, but we have introduced a coefficient of \nfriction modeling a sliding produced by local interface detachments or cracks within the \nepoxy layers  (square symbols). The value 𝑘=0.25 was chosen to fit the experimental \ncoupling factor 𝜂. Lastly, for  comparison, we have studied the case of an assumed structure, \nwhere the ferrite discs are perfectly clamped ( 𝑘=1, no sli ding)  on both faces of the PZT disc \nwithout intermediate layers of epoxy resin  (triangle symbols) . In this case, the coupling factor \nreflects only the shear strain effect within the PZT and ferrite layers.  This mechanical study \nshows that the differential strain between the PZT and ferrite layers is mainly due to the shear \nstrain within the epoxy layers and the sliding at the PZT/epoxy and ferrite/epoxy interfaces. \nThe shear strain along the thickness of  the PZT layer plays a secondary role.  \nE. Analytical calculation of demagnetizing effect for multilayer ed ME samples  \nIn the previous part, we have demonstrated that  a low demagnetizing effect, so a high ME \nvoltage is obtained in a ME trilayer when the two magnetic layers are thin and far from each \nother. By increasing the number of layers ( 𝑛>3) at a given PZT/ferrite volume ratio  and at a \ngiven  total thickness , we m ay assume a reduction of the global demagnetizing effect because \nthe magnetic layers are thinner. But in the other hand, in this case, the magnetic layer s are \ncloser to each other and this tends to counterbalance the previous effect , and i t is difficult to  \nsay which phenomenon dominates. To answer this question, u sing the calculation met hod \npresented in section IV. A.  (with the same restrictions) , we have derived an analytical formula \ngiving the global demagnetizing factor of a multilayered ME sample.  This formula is an \nextrapolation of Eq. (16) which gives the demagnetizing factor of two magnetic discs (Fig. 4). \nIn this simple structure, the demagnetizing factor of a layer is the sum of two contributions. \nThe first one, 𝑁𝑡,𝑑 is the influence of a given layer on itself. The second one, that we  call \nℕ𝑡,𝑒,𝑑, is the influence from the other layer (with thickness t and diameter d) situated at a \ndistance  e, where:  12 \n ℕ𝑡,𝑒,𝑑=−(1+𝑒\n𝑡).𝑁𝑒+𝑡,𝑑+(1+𝑒\n2𝑡).𝑁𝑒+2𝑡,𝑑+𝑒\n2𝑡.𝑁𝑒,𝑑                                                        (23) \nConsider  now a stack of n alternated ferrite layers (thickness  t each) and PZT  layers \n(thickness e each),  with a ferrite layer  at the top and bottom of the stack , so that n is an odd \nnumber  (see Fig 11). Then, t he radial demagnetizing factor 𝑁𝑡,𝑑𝑝 of a ferrite layer numbered p \n(odd number  between 1 and n), is the sum of  the 𝑁𝑡,𝑑 term (the influence of the layer p on \nitself) and all the ℕ𝑡,𝑥,𝑑 terms produced by each of the other layers  distant of 𝑥, where \n𝑥=𝑒,2𝑒+𝑡,3𝑒+2𝑡,… ,(𝑛−𝑝\n2)𝑒+(𝑛−𝑝\n2−1)𝑡, for the magnetic layers above the layer p, \nand where 𝑥=𝑒,2𝑒+𝑡,3𝑒+2𝑡,… ,(𝑝−1\n2)𝑒+(𝑝−1\n2−1)𝑡, for the magnetic layers below \nthe layer p. Then, the radial demagnetizing factor 𝑁𝑡,𝑑𝑝 of a ferrite layer numbered p can be \nwritten as:  \n𝑁𝑡,𝑑𝑝=∑ℕ𝑡,(𝑘𝑒+(𝑘−1)𝑡),𝑑+(𝑝−1)/2\n𝑘=1∑ℕ𝑡,(𝑘𝑒+(𝑘−1)𝑡),𝑑+𝑁𝑡,𝑑 (𝑛−𝑝)/2\n𝑘=1                                      (24) \nReplacing the ℕ𝑡,(𝑘𝑒+(𝑘−1)𝑡),𝑑 term in Eq.  (24) by its expression given in Eq.  (23), we obtain:  \n𝑁𝑡,𝑑𝑝=∑\n[    𝑘𝑒+(𝑘+1)𝑡\n2𝑡 𝑁 𝑘𝑒+(𝑘+1)𝑡,𝑑\n−𝑘(𝑡+𝑒)\n𝑡 𝑁𝑘(𝑡+𝑒),𝑑\n+𝑘𝑒+(𝑘−1)𝑡\n2𝑡 𝑁𝑘𝑒+(𝑘−1)𝑡,𝑑]    𝑝−1\n2\n𝑘=1 \n+∑\n[    𝑘𝑒+(𝑘+1)𝑡\n2𝑡 𝑁 𝑘𝑒+(𝑘+1)𝑡,𝑑\n−𝑘(𝑡+𝑒)\n𝑡 𝑁𝑘(𝑡+𝑒),𝑑\n+𝑘𝑒+(𝑘−1)𝑡\n2𝑡 𝑁𝑘𝑒+(𝑘−1)𝑡,𝑑]    𝑛−𝑝\n2\n𝑘=1 \n+ 𝑁𝑡,𝑑                                                                        \n  \n \n \n \n                \n    \n     \n \n \n \n       (25)                \nwhere t  is the thickness of each ferrite layer, and e is the thickness of each PZT layer ; 𝑁𝑥,𝑑 is \nthe radial demagnetizing factor of a single magnetic disc with diameter d and thickness x. The \nglobal demagnetizing factor 〈𝑁〉 of the st ack is deduced from the average  over all magnetic \nlayers:  \n〈𝑁〉=1\n(𝑛+1)/2 ∑𝑁𝑡,𝑑𝑝=2𝑘+1(𝑛−1)/2\n𝑘=0             \n  (26) \n \nwhere (𝑛+1)/2  is the quantity  of magnetic layers.   \nUsing Eq. (25) and (26), the global demagnetizing factor was calculated for a number of \nlayers between 3 and 21 at different values of PZT/ferrite volume ratio 𝑉𝑝/𝑉𝑓 for a ME sample \nof 1.5  mm total thickness  and 10  mm diameter . The theoretical calculations  are plotted in Fig. \n12. The result shows  that, for a given PZT/ferrite volume ratio, the global demagnetizing \nfactor is almost independent of the number of magnetic layers in the ME stack. So, in theory, \nfrom a demagnetizing effect point of view, the ME voltage cannot be enhanced by increasing \nthe numb er of layers. On the other hand, in terms of mechanical coupling effect, we can \nexpect a better strain uniformity for  structures comprising large  numbers of layers, which 13 \n means that the mechanical coupling factor approaches its maximum value (we may assume  \n𝜂~0.8 at best ).  In this later case, the ME voltage can be improved by 20  %. Fig. 13 shows the \ntheoretical profile of the demagnetizing factor through the thickness for three different ME \nstacks (𝑛=7,11, and 21)  at a given PZT/ferrite volume ratio ( 𝑉𝑝/𝑉𝑓=3.5). Obviously, \ndemagnetizing factor s are  higher for magnetic layers close to the centre of the stack (where \nthe influences of all the magnetic layers are stronger), and lower for the external layers. \nHowever, the inhomogeneity is lower than 17% w hatever the number of layers, which \nvalidates the use of a global demagnetizing factor 〈𝑁〉 averaged over the whole stack.  \nV. Conclusion.  \nWe have proposed an analytical model suitable for the calculation of the demagnetizing field \nin ME trilayers where two m agnetic discs interact together in a parallel configuration. This \nanalytical model is based on a superposition method, involving the demagnetizing factors of  \nsingle magnetic disc s. A Finite Element Method was used to solve numerically the magnetic \nproblem, permitting to validate the analytical approach.  The analytical demagnetizing factors \nwere introduced into a model predicting the ME voltage coefficient. It appears that the \ntheoretical calculations fit  well the experimental ME respons es when a mechanical coupling \nfactor  depending on the PZT/ferrite volume ratio  is introduced. This work has revealed that a \nmaximum ME voltage is reached for ME samples with high PZT volume ratios. In this case, \nthe low mechanical coupling between the PZT and ferrite layers is counter balanced by a \nbetter magnetic field penetration  because the two ferrite layers are thin and are relatively far \nfrom each other . This implies that t he diameter and the thickness  of the PZT and ferrites \nlayers are the geometrical parameters that affect the ME response through the demagnetizing \neffect and the mechanical coupling.  The superposition method that we used to calculate the \ndemagnetizing factor of a trilayer ME sample  has been  extended to multilayered ME samples. \nThe important finding is that an increase of the number of layers (at a given sample thickness \nand PZT/ferrite volume ratio) do not change the magnetic field penetration, and the ME \nvoltage remain s unchang ed from this point of view. Summarizing , the ME trilayer structure \ncombines the adv antages of high ME performance and ease of fabrication.   \n \n \n \n \n \n \n \n 14 \n Appendix  A: magnetometric radial demagnetizing factor calculation for a \nsingle disc.  \nThis appendix is an additional part of Sec. IV.  A formula for the calculation of the \nmagnetometric radial demagnetizing factor of a single magnetic disc is given below. The \nvalidity domain is: susceptibility 𝜒 between 1 and 190, and thickness to diameter ratio t/d \nbetween 0.01 and 0.5, which satisfy most of the layered ME sample s with cylindrical \ngeometries.  This formula is derived from the works of Chen et al.15,16. \n𝑁𝑡,𝑑(𝜒)=(𝐶11𝜒4+𝐶12𝜒3+𝐶13𝜒2+𝐶14𝜒+𝐶15).(𝑡\n𝑑)4\n \n+(𝐶21𝜒4+𝐶22𝜒3+𝐶23𝜒2+𝐶24𝜒+𝐶25).(𝑡\n𝑑)3\n \n+(𝐶31𝜒4+𝐶32𝜒3+𝐶33𝜒2+𝐶34𝜒+𝐶35).(𝑡\n𝑑)2\n \n+(𝐶41𝜒4+𝐶42𝜒3+𝐶43𝜒2+𝐶44𝜒+𝐶45).(𝑡\n𝑑) \n+(𝐶51𝜒4+𝐶52𝜒3+𝐶53𝜒2+𝐶54𝜒+𝐶55) \nWhere the 𝐶𝑖𝑗 coefficients are:  \n𝐶11=−1.27×10−8;  𝐶12=5.69×10−6; 𝐶13=−8.79×10−4;  \n𝐶14=5.42×10−2;     𝐶15=−2.88; \n𝐶21=1.41×10−8;  𝐶22=−6.29×10−6; 𝐶23=9.67×10−4;  \n𝐶24=−5.89×10−2;     𝐶25=4.15; \n𝐶31=−5.42×10−9;  𝐶32=2.41×10−6; 𝐶33=−3.67×10−4;  \n𝐶34=2.20×10−2;     𝐶35=−2.55; \n𝐶41=9.78×10−10;  𝐶42=−4.33×10−7; 𝐶43=6.55×10−5;  \n𝐶44=−3.90×10−3;     𝐶45=1.08; \n𝐶51=2.57×10−11;  𝐶52=−1.23×10−8; 𝐶53=2.10×10−6;  \n𝐶54=−1.55×10−4;     𝐶55=7.00×10−3; \n \n \n 15 \n References.  \n1S. Dong, J. G. Bai, J. Zhai, J.F. Li, G.Q. Lu, D. 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Silva et al. , ACS Applied Materials a nd Interfaces, Vol. 5, Issue 21, pp 10912 -10919 \n(2013).  \n19 http://www.ferroperm -piezo.com/  \n \n 16 \n  \n \n \n \n \n \n \n \n \n \n \n 𝑑31𝑒 \n(pC/N)  𝑑11𝑚 \n(nm/A)  𝑠11𝐸 or 𝑠11𝐻 \n(m2/N) 𝑠12𝐸 or 𝑠12𝐻 \n(m2/N) µT or 𝜀33𝑇 \n(in relative)  \nPz27 -170  17×10−12 −6.6×10−12 1800  \nferrite   -9.5 6.47×10−12 −1.84×10−12 95 \n \nTABLE 1  : Material properties for Pz27 (cited from  Ferroperm19), and Ni -Co-Zn ferrite  (Ref. \n4). Note that 𝑑11𝑚 is the intri nsic piezomagnetic coefficient.                                      \n \n \n \n \n \n \n \n \n \n \n 17 \n  \n \n \n \n \n \n \n \n \n \nFIG. 1.  Magnetoelectric coefficient s for trilayer sample with various thicknesses. t is the \nthickness of a ferrite layer  (or a PZT layer) . The total thickness of a sample is 3 t. \n \n \n \n \n \n \n \n \n \nFIG. 2.  Circles: normalized measured ME peak coefficient s. Squares: normalized measured \n1/(𝐻𝐷𝐶𝑎)𝑚𝑎𝑥 function.  Triangles: normalized theoretical ME peak coefficient s. t is the \nthickness of a ferrite layer  (or a PZT layer ). The total thickness of a sample is 3 t. \n \n \n \n \n18 \n  \n \n \n \n \n \n \n \n \n \nFIG.  3.  Magnetoelectric voltage s for trilayer sample s with various PZT/ferrite volume ratio \n𝛾=𝑉𝑝/𝑉𝑓. All the samples have the same total thickness (1.5 mm).  The amplitude of the \nexternal AC field is 1  mT. \n \n \n \n \n \n \n \n \n  \n \nFIG 4. Sketches of a magnetic structure consisting in two ferrite discs uniformly magnetized \n(cells (1) and (3)) in a parallel configuration separated by a layer of vacuum (cell (2) in \ndashed line).  \n \n \n \n𝑀⃗⃗  \n𝑀⃗⃗  \n𝐻⃗⃗ 3,3𝑑 \n𝐻⃗⃗ 1,2𝑑 \n𝐻⃗⃗ 1,1𝑑 \n𝐻⃗⃗ 1,3𝑑 \n𝐻⃗⃗ 3,1𝑑 \n𝐻⃗⃗ 3,2𝑑 \n 𝑀⃗⃗ =0⃗  \n1 \n2 \n3 \nt \nt \nt \nd \n𝑥  \n𝑦  \n𝑧  \n19 \n  \n \n \n \n \n  \n \n \n \n \n \n \n                                                                                      \n \n \n \n \n \n \n                                                   \n \nFIG 5. Sketches of  three different  magnetic stru ctures; (a)  a single magnetic disc divided into \nthree cells with thickness t each; (b)  a single magne tic disc divided into two cells; (c)  a single \nmagnetic disc corresponding to a unique cell.  \n \n \n \n \n𝑥  \n𝑦  \n𝑧  \n𝑀⃗⃗  \n𝑀⃗⃗  \n𝐻⃗⃗ 3,3𝑑 \n𝐻⃗⃗ 1,2𝑑 \n𝐻⃗⃗ 1,1𝑑 \n𝐻⃗⃗ 1,3𝑑 \n𝐻⃗⃗ 3,1𝑑 \n𝐻⃗⃗ 3,2𝑑 \n1 \n2 \n3 \nt \nt \nt \nd \n𝑀⃗⃗  \n𝐻⃗⃗ 2,3𝑑 \n𝐻⃗⃗ 2,1𝑑 \n𝐻⃗⃗ 2,2𝑑 \n(a) \n𝑀⃗⃗  \n𝐻⃗⃗ 3,3𝑑 \n𝐻⃗⃗ 3,2𝑑 \n2 \n3 \nt \nt \nd \n𝑀⃗⃗  \n𝐻⃗⃗ 2,3𝑑 \n𝐻⃗⃗ 2,2𝑑 \n(b) \n𝑀⃗⃗  \n 𝐻⃗⃗ 3,3𝑑 \n3 \nt \nd \n(c) 20 \n  \n \n \n \n \n \n \n \n \nFIG 6. Theoretical  magnetometric demagnetizing factor . Circle symbols:  single ferrite di sc \n(data derived from Chen et al. ). Dashed line: polynomial fit. Solid line: demagnetizing factor \nfor two parallel discs (distance: t) deduced from data of a single ferrite disc. Square symbols: \nnumerical calculation  for two parallel discs. In all cases, 𝜒=94. \n \n \n \n \n \n \n \n \n \nFIG 7. Theoretical ME peak voltage s as a function of the PZT/ferrite volume ratio 𝛾. Dashed \nand dotted lines: the mechanical coupling factor 𝜂 is maintained constant . Dotted line: 𝜂=1 \nand d ashed line : 𝜂=0.71. Solid line: the mechanical coupling factor decrease linearly with \nthe increasing thickness of the PZT layer . Square symbols: experimental peak ME voltages \nextracted from Fig. 3. The corresponding ME coefficient normalized with respect to the total \nthickness (1.5 mm) of the ME samples is given on the right vertical axis.   \n \n \n21 \n  \n \n \n \n \n \n \n \n \nFIG 8. Mechanical coupling factor 𝜂 (dashed line)  and magnetometric demagnetizing factor N \n(solid line) used for the analytical  calculation  of the ME voltage s and plotted against the PZT \nvolume ratio 𝑉𝑝/(𝑉𝑝+𝑉𝑓). Square symbols: numerical calculation (FEM software)  of \ndemagnetizing factors ( 𝜒=94) for comparison . \n \n \n \n \n \n \n \n \n \n \n \nFIG 9. Theoretical r atio between the internal magnetic field and the external field as a \nfunction of the PZT/ferrite volume ratio.  \n \n \n \n22 \n  \n \n \n \n \n \n \n \n \nFIG 10 . Mechanical coupling factor 𝜂 versus PZT volume ratio. Solid line: linear function \nthat permits to fit the experimental ME voltages. Symbols: FEM calculation results. Square \nsymbols: structure with epoxy layers perfectly coupled ( 𝑘=1) to the PZT and ferrite layers. \nStar symbols: structure with sliding at the interfaces ( 𝑘=0.25). Triangle symbols: structure \nassuming ferrite and PZT layers perfectly coupled (without intermediate layers) . Dotted lines \nare linear interpolations.  \n \n \n \n \n \n \n \n \n \n \n \nFIG 11 . Sketches  of a multilayered  structure consisting in (n+1)/2 ferrite discs uniformly \nmagnetized (cells in grey) in a parallel configuration separated by (n -1)/2 PZT discs (cells in \ndashed line).  \n \n1 \n2 \np \np-1 \np-2 \np+1 \np+2 \nn-1 \nn \nt \ne \nd \n23 \n  \n \n \n \n \n \n \n \n \n \nFIG 12 . Theoretical g lobal demagnetizing factor , as a function of the quantity  of layers in the \nME stack , plotted for three different PZT/ferrite volume ratio s. Circle symbols: 𝑉𝑝/𝑉𝑓=3.5; \ntriangle symbols: 𝑉𝑝/𝑉𝑓=2; diamond symbols: 𝑉𝑝/𝑉𝑓=1. Star symbols: numerical FEM \ncalculation ( 𝑉𝑝/𝑉𝑓=2) for comparison. In each case s, the total thickness of a ME sample is \n1.5 mm and the diameter is 10  mm. \n \n \n \n \n \n \n \n \n \n \n \nFIG 13 . Theoretical p rofile of the demagnetizing factor within ME stack s for a volume ratio  \n𝑉𝑝/𝑉𝑓=3.5. Circle symbols: 21 layers in the stack ; triangle symbols: 11 layers in the stack; \ndiamond symbols: 7 layers in the stack. In each case s, the total thickness of a ME sample is \n1.5 mm and the diamete r is 10  mm. \n24 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nView publication statsView publication stats"    },    {        "title": "1011.1867v1.Calculation_of_the_substitutional_fraction_of_ion_implanted_He_in_an_Fe_target.pdf",        "content": "arXiv:1011.1867v1  [cond-mat.mtrl-sci]  8 Nov 2010Calculation of the substitutional fraction of ion-implant ed He in an Fe target\nPaul Erhart and Jaime Marian\nLawrence Livermore National Laboratory, Livermore, CA 945 51\nAbstract\nIon-implantation is a useful technique to study irradiatio n damage in nuclear materials. To study He effects in\nnuclear fusion conditions, He is co-implanted with damage i ons to reproduce the correct He/dpa ratios in the desired\nor available depth range. However, the short-term fate of th ese He ions, i.e.over the time scales of their own collisional\nphase, has not been yet unequivocally established. Here we p resent an atomistic study of the short-term evolution\nof He implantation in an Fe substrate to approximate the cond itions encountered in dual ion-implantation studies in\nferritic materials. Speciﬁcally, we calculate the fractio n of He atoms that end up in substitutional sites shortly afte r\nimplantation, i.e.before they contribute to long-term miscrostructural evol ution. We ﬁnd that fractions of at most 3%\nshould be expected for most implantation studies. In additi on, we carry out an exhaustive calculation of interstitial\nHe migration energy barriers in the vicinity of matrix vacan cies and ﬁnd that they vary from approximately 20 to 60\nmeV depending on the separation and orientation of the He-va cancy pair.\nKeywords: fusion materials, helium, ion implantation\n1. Introduction\nThis paper tries to answer a simple yet important ques-\ntion in He-implantation studies: Do ion-implanted He\natoms end up as interstitial or as substitutional particles\nin the target matrix? The difference is critical because\nof the large diffusivity difference between both forms of\nHe: interstitial He (i-He) diffuses extremely fast, sam-\npling large portions of the conﬁgurational space quickly,\nreadily ﬁnding other defects or microstructural features.\nConversely, substitutional He (s-He), while energeticall y\nmore stable, is immobile, necessitating migration of other\npoint defects before it can move. This can happen either\nby reacting with a self-interstitial atom (SIA) that recom-\nbines with the vacancy and knocks the He back to an in-\nterstitial site, or by correlated lattice exchange reactio ns\nwith a vacancy in nearest-neighbor positions. Either way,\nthe diffusivity of s-He is still several orders of magnitude\nlower than that of i-He.\nDespite the important implications of these mecha-\nnisms on the subsequent microstructural evolution, at\npresent most researchers consider that 100% of the im-\nplanted He is interstitial [1, 2, 3] and can only become\nsubstitutional by ﬁnding an isolated vacancy in an un-\ncorrelated fashion via long-range diffusion. The question\nwe ask here is whether this is true for all He atoms or\nwhether some of them can become substitutional as part\nof their own implantation process prior to uncorrelated\ndiffusion taking place.\nHere we present a computational study involving thebinary collision approximation (BCA), molecular dynam-\nics (MD), and kinetic Monte Carlo (kMC) simulations.\nThe BCA is used to simulate the penetration of He beams\nof various energies into Fe targets, and to obtain en-\nergy distributions of Fe recoils due to He impact. MD\nis then used to simulate He thermalization after the pri-\nmary knock-on event in its own collisional environment,\nand to ascertain whether He atoms create stable Frenkel\npairs that can result in correlated recombination. Finally ,\nwe use kMC to calculate the fraction of freely-migrating\nHe atoms from those that do create defects but do not\nﬁnd the vacancy during MD time scales. From these sim-\nulations, we ﬁnd that nearly 3% of the He atoms end up\nin subsitutional sites. While this number appears small,\nit nonetheless leads to dramatic differences in the mi-\ncrostructural evolution of the material, as will be shown\nin future studies.\n2. Results\n2.1. Calculation of recoil distributions and He energies\nThe He energy range of interest for fusion materials\nlies between 3.5 and 0.33 MeV , corresponding to the en-\nergy of αparticles emitted from fusion reactions and\nthose produced via (n, α) transmutation reactions in Fe1.\nIn addition, ion beam experiments typically use energies\nin this range to achieve penetrations of a few microns,\nso it is useful to have an intermediate energy for refer-\nence. Thus, we ﬁrst calculate the Fe recoil distribution\nPreprint submitted to Journal of Nuclear Materials March 3, 20180.00.20.40.60.81.0\n101102103104Cumulative Fe recoil distribution\nRecoil energy (eV)0.33 MeV\n1.7 MeV\n3.5 MeV\nFigure 1: Cumulative Fe recoil distribution for He-ion beam irradia-\ntions with recoil energies of 0.33, 1.7 and 3.5 MeV incident e nergy. The\nthreshold displacement energy is 25 eV .\nTable 1: SRIM parameters for the three He-beam energies cons idered.\nHe ion energy (MeV) 0.33 1.7 3.5\nDepth ( µm) 0.7 2.6 6.0\n% energy to recoils 0.43 0.11 0.07\nAverage recoil energy (eV) 194 211 222\nMaximum recoil energy (keV) 66 134 315\nfor three He-ion energies, namely 0.33, 1.7 and 3.5 MeV ,\nusing SRIM [5]. The cumulative recoil energy distribu-\ntions in each case are given in Fig. 1, where a threshold\ndisplacement energy of 25 eV was used. As the ﬁgure\nshows, the three recoil distributions are almost identical .\nThis is because He ions only create recoils when they\nhave slowed down to a few keV , without much participa-\ntion from their higher-energy histories, which as shown\nin Tab. 1 only contribute to penetration and maximum re-\ncoil energy. It is more informative to compare the average\nrecoil energies, which, in contrast, differ only by a few eV .\nIn all cases, the energy expended in recoils amounts to\nless than 0.5% of the total ion energy.\nThese results suggest that in the energy range relevant\nto fusion materials the actual ion energy is irrelevant for\ndamage purposes. Therefore, we take the 1.7 MeV spec-\ntrum shown in Fig. 2 as representative of all He energies\nand proceed to simulate the effect of these recoils on lat-\ntice damage. We note that SRIM does not capture chan-\nneling, which may have some impact on the ﬁnal results.\n2.2. Molecular dynamics simulations of He impact in Fe\nNext we study the fate of He ions at the end of their\ncollision trajectories, when they collide with the last of\ntheir recoils before thermalizing in the host lattice. We\nassume that these recoils are ejected with energies consis-\ntent with the recoil spectrum obtained using SRIM (see\nFig. 2), as the probability for high energy recoils, pro-\nduced early in the He collision sequences, is very small.10−510−410−310−2\n101102103Relative number of events\nEnergy (eV)25 eVSRIM\nsampling\nFigure 2: Spectrum of Fe recoils sampled by MD simulations in com-\nparison with spectrum for 1.7 MeV He obtained using SRIM.\nThe objective is then to investigate whether He ions can\nbecome substitutional by interacting with vacancies of\ntheir own creation, rather than by long-range diffusion.\nFor this, we perform MD simulations of He collisions in\nFe and analyze the ﬁnal conﬁgurations.\nSimulations were carried out using the massively par-\nallel MD code lammps [6]. The simulation cell consisted\nof 38×38×38 conventional body-centered cubic (BCC)\nunit cells corresponding to 109,744 Fe atoms. All simula-\ntions were carried out at a temperature of 700 K, which is\nrepresentative of fusion conditions. Atomic interactions\nwere modeled using the Fe-He interatomic potential of\ndeveloped by Juslin and Nordlund [7], which builds on\nthe Fe potential by Mendelev et al. [8] and gives an equi-\nlibrium lattice constant of a0=2.871 Å at 700 K. The col-\nlision simulations were done with He as an energetic ion\nin an otherwise perfect Fe lattice. The He atoms was\nassigned kinetic energies that produce a Fe recoil dis-\ntribution consistent with the SRIM data in Fig. 2. We\nsample the Fe recoil velocity vFedirectly from the SRIM\nspectrum and, assuming purely elastic collisions, assign\na velocity vHeto the He atom\nvHe=/parenleftbiggmHe\nmFe−1/parenrightbigg\nvFe. (1)\nwhere the mass ratio is mHe/mFe≈0.077 and the velocity\nvector is directed at the Fe PKA.\nThe atoms that were initially within a radius of 18.65 a0\nof the center of the simulation cell were considered the\ncore region while all other atoms were assigned to the\nedge region. The latter were coupled to a Nosé-Hoover\nthermostat at 700 K, while the remainder of the system\nevolved in time according to the microcanonical ensam-\nble. The equations of motion were integrated until ei-\nther the He atom thermalized (kinetic energy less than\n0.1 eV) or escaped the core region. The ﬁnal conﬁgura-\ntion was relaxed using conjugate gradient minimization\n2 0 1 2 3 4 5 6 7 8 9\n 25 50 75 100 125 150 175Number of Frenkel pairs\nRecoil energy (eV)\nFigure 3: Average number of stable Frenkel pairs created dur ing the\nMD simulations of He collision cascades in Fe. The bars indic ate the\nstandard deviation of the number of defects created.\nand defects were identiﬁed by analyzing the occupancies\nof the Wigner-Seitz cells of the initial perfect BCC lattice .\nIn total, we simulated 5,000 events to extract sufﬁcient\nstatistics.\nOut of the total 5,000 events simulated, 2,668 (53.4%)\nwere terminated when the He atom exited the core re-\ngion. They are interpreted as cases in which the He equi-\nlibrates in a region “far” away from the PKA, such that\nthe probability for the He to recombine with defects cre-\nated in the original cascade is very small. Accordingly\nthese events are counted as i-He in the total atom tally.\nThe remaining 2,332 events were analyzed to obtain the\nnumber and distribution of irradation induced point de-\nfects. In 25 cases the helium ended up in a substitutional\nsite corresponding to 1.07% of the “thermalized” cases\nand 0.5% of the total number of events. This occurred\ntypically within 4 to 6 ps of simulated time. Of the re-\nmaining events, 673 cases (13.5%) resulted in He ther-\nmalization but no Frenkel pairs, further adding to the\ni-He tally.\nThe remaining 1,634 cases (32.7%) deserve special at-\ntention. This group includes those He atoms that have\nthermalized within the core region and have created sta-\nble Frenkel pairs but have not become substitutional dur-\ning the MD simulation. The number of stable point de-\nfects created by the He collisions as a function of PKA\nenergy is shown in Fig. 3. Although in principle, He can\nalso be trapped by SIAs [9], we did not observe any in-\nstances where this occurred. SIAs created during the cas-\ncade either diffused away from the core region or were\nsituated too far from the ﬁnal position of the He atom.\nAnalysis of the resulting conﬁgurations enables us to\ndetermine the spatial correlation of He interstitials with\nvacancies as shown in Fig. 4. The overall distribution has\na mean of approximately 30 Å, with lower energy con-\ntributions slightly shifted to shorter separations. In any 0 10 20 30 40 50 60 70 80\n 0 10 20 30 40 50 60Fraction of events (×10−3)\nSeparation between He and vacancy (Å)all events\n25 to 40 eV\n40 to 150 eV\nFigure 4: Pair correlation of He interstitials and Fe vacanc ies obtained\nfrom 1,634 cases of He impact in a Fe lattice. Comparison of th e relative\ncontributions of different energy windows shows a slight sh ift toward\nsmaller distances for smaller energies.\ncase, to determine the fate of these He ions conclusively,\none needs to “age” these conﬁgurations further using a\ntechnique capable of probing longer time scales. This is\nakin to calculating the fraction of freely migrating defects\nin high-energy cascade simulations [10]. To this end, we\ncarry out kinetic Monte Carlo (kMC) simulations of He-\nvacancy reactions according to the distribution given in\nFig. 4.\n2.3. Kinetic Monte Carlo simulations of He-vacancy reactio ns\nThe kMC simulations consisted of a single vacancy lo-\ncated at the center of a reaction sphere and a randomly-\noriented He atom separated from the vacancy by a dis-\ntance sampled from the distribution shown in Fig. 4. We\nhave used the Green’s function Monte Carlo method [11]\nin a continuum Fe medium with two spherical particles\nrepresenting the He atom and the vacancy. The sum of\nthe radii of the vacancy and He atom was set equal to\nthe third-nearest neighbor distance in the BCC lattice,\nr=a0√\n2 to be consistent with the He-V binding en-\nergy calculations performed in the Appendix. Other au-\nthors have suggested that binding occurs up to the ﬁfth\nnearest neighbor distance [12]. No further correlation\nbetween the He atom and the vacancy is assumed, i.e.\njumps toward and away from the central vacancy are\nsampled with equal probability. The critical parameters\nfor the kMC simulations are the temperature, the diffu-\nsivities, and the size of the simulation box. The tempera-\nture was 700 K, the same as in the MD simulations. With\nrespect to the diffusivities, we neglect vacancy diffusion ,\nas its diffusion coefﬁcient is known to be several orders\nof magnitude lower than that of He, even at these tem-\nperatures. The diffusion coefﬁcient of He in BCC Fe was\ntaken from Terentyev et al. [12], who used the same in-\nteratomic potential as in the present work and obtained\n3Table 2: Summary of results from MD and kMC simulations of He i mpacts in BCC Fe. For the data obtained from kMC simulations t he third\ncolumn states the number of occurences equivalent to the num ber of events treated by MD simulations. The actual number of events simulated by\nkMC was 106(see text for details).\nEvent Number of occurences Relative fraction\ni-He He escaped core region during cascade (MD) 2668 53.3%\nno stable defects created during cascade (MD) 673 13.5%\nHe escaped core region after thermalization (kMC) 1513 30.3 %\ntotal 97.1%\ns-He He trapped during cascade (MD) 25 0.5%\nHe trapped after thermalization (kMC) (MD) 121 2.4%\ntotal 2.9%\nDHe=5.1×10−3exp(−76/kT)cm2s−1(migration en-\nergy in meV). Finally, also for consistency with the MD\nsimulations, we considered a spherical region with ra-\ndius equal to 18.65 a0.\nIf over the course of a simulation the He atom escaped\nthe spherical region, it was added to the i-He count,\nwhereas, if it reacted with the vacancy in the center of\nthe simulation sphere, it was tallied as a s-He. After\n106events, we calculated the probability for He-vacancy\nrecombination under these conditions to be 7.4%. The\nprobability of He escape in an inﬁnite medium has a\nknown analytical solution given by [13]\np=1−2r\nd(2)\nwhere dis the initial separation distance. This formula\nyields an average reaction probability of approximately\n12%, which is higher than the kMC value because it is\nnot limited to a ﬁnite reaction volume.\nWhen prorated to the number of cases that consti-\ntute the He-vacancy pair correlation in Fig. 4, this re-\nsult, added to the 0.5% computed directly from the MD\nsimulations, results in a total fraction of s-He of 2.9% in\nion implanted BCC Fe. The various contributions to this\nnumber are summarized in Tab. 2 for clarity.\n3. Discussion and conclusions\nUnderstanding how implanted or transmutation He\nis partitioned between i-He and s-He is paramount be-\ncause of the different characteristics of both species in th e\nBCC lattice. A comprehensive review on the role of He\nin Fe has been recently published [14], where the basic\nenergetics are given and a review of the existing litera-\nture is provided. All studies agree that s-He is energeti-\ncally more favorable with a lower formation energy and a\nstrong He-vacancy binding energy but cannot move un-\nless aided by other point defects. For its part, i-He dif-\nfuses very fast in all three dimensions, rapidly probing\nvast regions of conﬁguration space and ﬁnding sinks orother defects very efﬁciently. These different behaviors\nhave important implications in terms of the long term\nmicrostructural evolution. For example, He in solution in\nthe BCC lattice is known to stabilize vacancy clusters pro-\nduced directly in high-energy cascades. In terms of its ef-\nfect on direct damage production, however, the evidence\nreported in the literature is contradicting. On the one\nhand, some researchers using the Fe–He Wilson-Johnson\npotential [15] have found that high-energy cascades in\nBCC Fe doped with small concentrations of s-He result\nin higher numbers of vacancy clusters than in pure Fe\n[16]. They also found these clusters to be generally larger\nin size. On the other hand, using the Juslin-Nordlund\npotential —employed here— Lucas and Schäublin have\nfound that it is i-He in solution that causes larger clus-\nter sizes and number densities to appear [17]. In fact,\nthey observe that s-He reduces the number of stable de-\nfects with respect to pure Fe. These workers also per-\nformed a systematic Fe–He potential comparison, not-\ning that the Juslin potential gives results that are overall\nin better agreement with DFT calculations [18]. In any\ncase, these results show the importance of determining\nthe correct partition of implanted He, something typi-\ncally neglected in most rate theory studies, where He is\ngenerally inserted as i-He (although some notable excep-\ntions exist [19]).\nNext we discuss the validity and limitations of our\napproach. This work hinges on the fact that Fe recoil\nspectra from He ions with a wide range of incident ener-\ngies are almost identical. Further, assuming that recoils\nproduce isolated cascades, data extracted from ion beam\nexperiments can be used to infer the behavior of αparti-\ncles created in (n,α)reactions. Then, the only difference\nbetween (n,α)reactions and He-ion irradiations is that\nthe former are created homogeneously within the matrix,\nwhereas ions penetrate a short distance into the material\n(c.f. Table 1), but up to the cooling-down phase of the\ncascade, damage is produced in an identical fashion in\nboth instances.\nWith regard to the interatomic potentials used, we\n4have already mentioned the studies in Refs. [17, 14].\nStewart et al. [20] have recently carried out an exhaustive\ncomparison of Fe–He and He–He potentials available in\nthe literature. These authors noted that the Fe–He poten-\ntial is strongly dependent on the matrix (Fe–Fe) to which\nit is coupled, and that the potentials used in the present\nwork produce little clustering. In addition, Yang et al.\n[21] and Pu et al. [22] have shown that different poten-\ntials can have a noticeable inﬂuence on vacancy cluster\nformation in displacement cascades. All of these effects\nshould again be mitigated at high temperatures, yet in\nlight of these results some variability in the ﬁnal fraction\nof s-He can be expected.\nAnother, limitation is the assumption of uncorrelated\ndiffusion in the kMC calculations. Presumably, the mi-\ngration energy of an i-He varies as a function of its\nproximity to a vacancy from the value in the bulk (here\nEm=58 meV, see Fig. A.5) to zero in the two intersti-\ntial sites closest to a vacancy (leading to spontaneous re-\ncombination). In this work, however, we have neglected\nthis dependency, for two main reasons. First, because\nat the simulation temperature, the i-He diffusion barri-\ners, which are typically on the order of 40 to 60 meV (see\nTab. A.3), are well below kinetic energy ﬂuctuations, as\nshown in the analysis provided in the Appendix. Sec-\nond, this correlated effect is partially captured by adjust -\ning the interaction distance, which was set to 5thnearest\nneighbor distance in this work. Indeed, other (lattice)\nkMC calculations of He and He-vacancy complex diffu-\nsion have also disregarded this effect for similar reasons\n[23].\nIn conclusion, we have performed a computational\nstudy of He implantation in BCC Fe and have calculated\nthe fraction of implanted He that becomes substitutional\nduring its own collisional phase. We ﬁnd that a fraction\nof approximately 3% is reasonable for the energy range\nof interest to fusion materials. Damage accumulation cal-\nculations should take this datum into account, although\nmore calculations are needed to accuratelt quantify the\nimpact on long-term microstructural evolution.\nAcknowledgments\nThis work was performed under the auspices of the\nU.S. Department of Energy by Lawrence Livermore Na-\ntional Laboratory under Contract DE-AC52-07NA27344.\nWe acknowledge support from the Laboratory Directed\nResearch and Development Program under project 09-SI-\n003 and allocations of computer time at the National En-\nergy Research Scientiﬁc Computing Center, which is sup-\nported by the Ofﬁce of Science of the U.S. Department of\nEnergy under Contract No. DE-AC02-05CH11231.He (tet)\nHe (oct)\nFe\nAB\nC ED127 meV\n58 meV\nK L\nFigure A.5: Conventional unit cell of the body-centered cub ic lattice\nillustrating the locations of tetrahedral and octahedral i nterstitial sites,\nand He migration barriers in the defect-free iron. Note that for clarity\nonly a subset of the interstitial sites is shown.\nNotes\n1The transmutation reaction56Fe(n, α)53Cr results in an excess\nmass of me=mFe+mn−(mCr+mα) = 55.9349375 +1.0086692 −\n(52.9406494 +4.0026032 ) = 0.0003541 amu. This is equivalent to E=\nmec2=0.33 MeV, although some variability in the form of a relative ly\nnarrow energy spectrum is to be expected depending on local c ondi-\ntions such as orientation, atomic vicinity, etc. (Source of particle rest\nmasses: NIST [4])\nAppendix A. Helium interstitial migration in the\nvicinity of an iron vacancy\nTo supplement the kMC simulations conducated as\npart of the present work, we carried out a systematic\nstudy of He interstitialimigration in the vicinity of a va-\ncancy. Calculations of migration barriers for He intersti-\ntials in defect-free iron and in the vicinity of a vacancy\nwere carried out using the drag method implemented by\nthe authors in the MD code lammps . All barrier calcu-\nlations were carried out at the zero-K lattice constant of\n2.855 Å using 16 ×16×16 supercells based on the con-\nventional BCC unit cell.\nThe formation energies of interstitial helium in tetrahe-\ndral and octahedral sites are 4.39 eV and 4.52 eV , respec-\ntively. In the ideal lattice the coordinates of the tetrahe-\ndral and octahedral interstitial are (0,1\n4,1\n2)and(0, 0,1\n2)\nin units of the lattice constant. As can be deduced from\nFig. A.5, there are 12 tetrahedral and 6 octahedral sites in\nthe conventional BCC unit cell. Figure A.5 also summa-\nrizes the results for the migration barriers of tetrahedral\nHe interstitials in defect-free iron. We obtain a barrier\nof 58 meV for migration along /angbracketleft110/angbracketrightin agreement with\nRef. [7]. For migration along /angbracketleft100/angbracketrightthe calculations yield\na value of 127 meV , which precisely corresponds to the\nenegry difference between the tetrahedral and octahedral\nsites as the latter coincides with the saddle point.\n5Table A.3: Migration barriers for He interstitials in the vi cinity of a\nvacancy. xini: initial position in fractional coordinates with respect t o\nthe vacancy located at the origin, Nini: initial neighbor shell, xf in: ﬁnal\nposition, Nf in: ﬁnal neighbor shell, ∆Ei−f=Ef−Ei: energy difference\n(meV), ∆Eb: migration barrier (meV).\nxini Nini xf in Nf in ∆Ei−f∆Eb\n11\n21\n43 13\n41\n24 −9 31\n11\n41\n23 0 45\n5\n41\n20 5 85 123\n0 0 0 S −1945 16\n11\n23\n44 −9 31\n0 0 0 S −1945 23\n13\n41\n24 11\n23\n44 0 17\n11\n21\n43 9 40\n5\n411\n26 69 120\n3\n411\n24 0 4\n11\n41\n23 9 40\n15\n41\n26 69 120\n5\n41\n20 53\n23\n40 8 −20 45\n3\n21\n40 7 12 78\n11\n21\n43−85 38\n11\n2-1\n43−85 38\n7\n41\n20 13 −18 55\n0 0 0 S −2030 59\n5\n411\n263\n213\n410 2 63\n3\n211\n49 1 56\n15\n41\n26 0 56\n13\n41\n24−69 51\n7\n411\n214 7 58\n3\n411\n24−69 51\n3\n21\n40 77\n41\n20 13 −30 35\n5\n41\n20 5 −12 66\n3\n201\n47 0 58\n3\n20 -1\n47 0 58\n3\n23\n40 8 −32 40\n3\n2-1\n40 7 0 125\nBefore we address the migration of He in the vicinity\nof a vacancy, we ﬁrst study the effect of the vacancy strain\nﬁeld on the energetics of He interstitials. To this end, we\nconstructed all crystallographically distinct He-vacanc y\npairs that can occur within a radius of four lattice con-\nstants, which yields 78 distinct pairs.\nThe binding energy for a He-vacancy pair is deﬁned as\n∆Eb=∆Ef(He−V)−∆Ef(He)−∆Ef(V), (A.1)\nwhere ∆Ef(He−V),∆Ef(He) =4.391 eV, and ∆Ef(V) =\n1.721 eV are the formation energies of the He-vacancy\npair, the isolated He interstitial as well as the isolated\nvacancy. Negative and positive values of ∆Ebindicate\nattraction and repulsion of the He-vacancy pair, respec-−80−60−40−20 0 20 40\n1.0 1.5 2.0 2.5 3.0 3.5 4.0Binding energy (meV)\nHe−vacancy separation in units of lattice constant\nFigure A.6: Binding energy of He-vacancy pairs as a function of sepa-\nration after relaxation.\ntively. The formation energy is given by\n∆Ef=Edef−Ndef\nNidEid, (A.2)\nwhere Eiand Niare the total energy and the number of\nFe atoms in conﬁguration i. Here, we have quietly set\nthe chemical potential of He to zero which has a mini-\nmal effect on the formation energies and no effect on the\nbinding energy.\nThe binding energy is shown as a function of the He-\nvacancy separation in Fig. A.6. Note that the two nearest\nHe-vacancy pairs have been omitted in Fig. A.6, since\nthey spontaneously recombine with the vacancy leading\nto a strongly negative binding energy of −2.01 eV . We\nﬁnd notable variations in the binding energy at least up\nto a separation of about 2.2 lattice constants. It is fur-\nthermore noteworthy that the variation in the binding\nenegry is not monotonic with the distance but exhibits\npronounced oscillations that to some extent can be corre-\nlated with different crystallographic directions.\nWe can now consider the different possibilities for He\ninterstitials migrating in the vicinity of a vacancy. From\nFig. A.5, we can deduce that from any tetrahedral site\nthere are six possible jumps, four of which are along\n/angbracketleft110/angbracketrightand two of which are along /angbracketleft100/angbracketright. More speciﬁ-\ncally,\n1. tet→tet:(0,1\n4,1\n2)+( 0,+1\n4,+1\n4)→(0,1\n2,3\n4)\n2. tet→tet:(0,1\n4,1\n2)+( 0,+1\n4,−1\n4)→(0,1\n2,1\n4)\n3. tet→tet:(0,1\n4,1\n2)+(+1\n4,−1\n4, 0)→(1\n4, 0,1\n2)\n4. tet→tet:(0,1\n4,1\n2)+(−1\n4,−1\n4, 0)→(−1\n4, 0,1\n2)\n5. via oct: (0,1\n4,1\n2)+( 0,+1\n2, 0)→(0,3\n4,1\n2)\n6. via oct: (0,1\n4,1\n2)+( 0,−1\n2, 0)→(0,−1\n4,1\n2)\nThe barriers that are obtained for these paths start-\ning from different initial He-vacancy arrangements are\nsummarized in Table A.3. We ﬁnd that while there is a\n6great variability of the values, in general migration bar-\nriers range from about 20 to 50 meV for shells 3 and 4,\nand from 40 to 60 meV for shells 5 to 7, relatively quickly\nclosing in on the bulk value of 58 meV . Using these data,\nwe rationalized (see Sec. 3) that a good choice for the\ninteraction radius entering the kMC simulations is ﬁve\nlattice constants.\nReferences\n[1] M. H. Yoo and L. K. Mansur, J. Nucl. Mater. 85&86 (1979) 571.\n[2] N. M. Ghoniem, S. Sharafat, J. M. Williams, and L. K. Mansu r,J.\nNucl. Mater. 117(1983) 96.\n[3] K. C. Russell, Metall. & Mat. Trans. A 39(2008) 956.\n[4] http://www.nist.gov/physlab/data/comp.cfm\n[5] J. F. Ziegler, J. P . Biersack, and U. Littmark, The Stopping and Range\nof Ions in Solids , vol. 1 of series: “Stopping and Ranges of Ions in\nMatter” (Pergamon Press, New York, 1984).\n[6] S. J. Plimpton, J. Comp. Phys. 117(1995) 1.\n[7] N. Juslin and K. Nordlund, J. Nucl. Mater. 382(2008) 143.\n[8] M. I. Mendelev, S. Han, D. J. Srolovitz, G. J. Ackland, D. Y . Sun,\nand M. Asta, Phil. Mag. 83(2003) 3977.\n[9] L. Ventelon, B. D. Wirth, and C. Domain, J. Nucl. Mater. 351(2006)\n119.\n[10] N. Soneda and T. Diaz de la Rubia, Phil. Mag. A 78(1998) 995.\n[11] M. H. Kalos, Phys. Rev. 128(1962) 1791.\n[12] D Terentyev, N. Juslin, K. Nordlund, and N. Sandberg, J Appl Phys\n105(2009) 103509.\n[13] T. S. Hudson, S. L. Dudarev, M.-J. Caturla, and A. P . Sutt on,Phil.\nMag. 85(2005) 661.\n[14] M. Samaras, Materials Today 12(2009) 46.\n[15] W. D. Wilson, and R. A. Johnson, in Interatomic Potentials and the\nSimulation of Lattice Defects , edited by P . C. Gehlen, J. R. Beeler, and\nR. J. Jaffee (Plenum Press, New York, 1972) p. 375.\n[16] L. Yang, X. T. Zu, H. Y. Xiao, F. Gao, H. L. Heinisch, R. J. K urtz,\nand K. Z. Liu, Appl. Phys. Lett. 88(2006) 091915.\n[17] G. Lucas and R. Schäublin, J. Phys. Condens. Matter 20415206.\n[18] C.-C. Fu and F. Willaime, J. Nucl. Mater. 367(2007) 244.\n[19] Y. Katoh, R. E. Stoller, Y. Kohno snd A. Kohyama, J. Nucl. Mater.\n210(1994) 290.\n[20] D. M. Stewart, Yu. N. Osetsky, R. E. Stoller, S. I. Golubo v, T. Selet-\nskaia, and P . J. Kamenski, Phil. Mag. 90(2010) 935.\n[21] L. Yang, X. T. Zu, Z. G. Wang, H. T. Yang, F. Gao, H. L. Heini sch,\nand R. J. Kurtz, J. Appl. Phys. 103(2008) 063528.\n[22] J. Pu, L. Yang, F. Gao, H. L. Heinisch, R. J. Kurtz, and X. T . Zu,\nNucl. Ins. & Meth. Phys. Res. B266(2008) 3993.\n[23] B. D. Wirth and E. M. Bringa, Phys. Scripta T108 (2004) 80.\n7"    },    {        "title": "1404.2965v1.Thermal_effect_on_magnetic_parameters_of_high_coercivity_cobalt_ferrite.pdf",        "content": "Thermal effect on magnetic parameters of high -\ncoercivity cobalt ferrite  \nChagas, E. F.1, Ponce , A. S. 1, Prado, R. J.1, Silva, G. M. 1, J. Bettini3 and Baggio-\nSaitovitch, E.2 \n1Instituto de Física , Universidade Federal de Mato Grosso, 78060- 900, Cuiabá- MT, \nBrazil  2Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150 Urca. Rio de Janeiro, \nBrazil.  \n3Laboratório Nacional de Nanotecnologia, Centro Nacional de Pesquisa em Energia e \nMateriais, 13083- 970, Campinas, Brazil  \nPhone number: 55 65 3615 8747  \nFax: 55 65 3615 8730  \nEmail address: efchagas@fisica.ufmt.br  \n \nAbstract  \nWe prepared very high -coercivity cobalt ferrite nanoparticles using short -time high -energy \nmechanical milling. After the milling process , the coercivity of the nanoparticles reach ed 3.75 \nkOe - a value almost five times higher than  that obtained for the non-milled sample. We \nperformed a thermal treatment on the milled sample at 300, 400, and 600 oC for 30  and 180 \nmins , and studied the changes in the magnetic parameters due to the thermal treatment usin g \nthe hysteresis curves, William son-Hall analysis, and transmission electron microscopy. The \nthermal treatment at 600 oC causes a  decrease in the microstructural strain and density of \nstructural defe cts resulting in a significant  decrease in coercivity.  Furthermore, this thermal \ntreatment increases the size of the nanoparticles and, as a consequence, there is a substantial \nincrease in the saturation magnetization. The coercivity and the saturation ma gnetization are \nless affected by the thermal treatment at 300 and 400 oC. \n \nIntroduction  \nThe hard magnetic compound CoFe 2O4 (cobalt ferrite) presents interesting \ncharacteristics , such as chemical stability, electrical insulation, high magnet -elastic effect [1], thermal chemical reduction [2 -4], moderate saturation magnetization (M S) \nand hard coercivity (H C). Due to these characteristics, the cobalt ferrite is a promising \nmaterial for several technological applications , such as high density magnetic storage \n[5], electronic devices, biomedical applications [6,7], and permanent magnets [8].  \nTo permanent magnets applications two quantities  are fundamental : coercivity and \nremanence (M R). Both compose the figure of merit in a hard  magnetic material, the \nquantity  energy product (BH) max, that gives an idea of the amount of energy that can be \nstored in the material. Several works report tuning of the coercivity using different \nmethods: thermal annealing [9], capping [10] and mechanica l milling treatment [11,12] \nof the grains. Liu et al. increased the H C of cobalt ferrite, from 1.23 to 5.1 kOe, with a \nrelatively small decrease in M S due the decrease of the grain size. They used a brief \n(1.5h) mechanical milling process on relatively lar ge particles (average grain size of \n240nm). In a previous work [12], using mechanical milling, we reached an increase of \nup to 4.2 kOe in the  coercivity of cobalt ferrite nanoparticles with an average grain size \nof about 23 nm. \nTo increase the saturation m agnetization and remanence is more complicated, but \nthe exchange spring effect can be used to increase the saturation magnetization of some nanomaterials [4,13- 15]. The exchange spring effect was observed  for the  first time in \n1989 by Coehoorn et al. [16] , and explained in 1991 by E. F. Kneller and R. Hawig \n[13], whor argue that, under  certain conditions, hard and soft magnetic materials may \npresent exchange coupling when in nanocomposite form. In this case, the high anisotropy of the hard material gives to nanocomposite a high coercivity , and high \nsaturation magnetization of the soft material gives the high M\nS, substant ially increasing \nthe product (BH) max when  compared with any one of the individual phases of the \nnanocomposite.  \nCobalt  ferrite is a promising material for obtaining optimized exchange -spring \nmagnets [4,15] due its charact eristic of thermal chemical reduction , cited above . This \nproperty allows the transformation of CoFe 2O4 into CoFe 2 (a soft ferromagnetic \nmaterial with high M S value of about 230 emu/g [17]) at  moderate/high temperatures , \nand controlling the ratio of CoFe 2 (in the nacomposite CoFe 2O4/CoFe 2) must be \noptimizing the quantity (BH) max [15]. All processes utilized to obtain the nanocomposite \nCoFe 2O4/CoFe 2 employ  thermal treatment [4,15]. Is reasonable to suppose that more hard is the precursor material (CoFe 2O4) better  the nanocomposite will be for permanent \nmagnet application . \nTo investigate the thermal effect on magnetic parameters , we prepared high-\ncoercivity cobalt ferrite using  mechanical milling and annealing  it at various \ntemperatures for different periods of time.  \n \nExperimental Procedure  \nThe cobalt ferrite nanopowder (sample CF0) was synthesized by a pH -controlled \nnitrate -glycine gel -combustion process [18,19]. High- purity (99.9%) raw compounds \nwere used. The synthesis process was adjusted to obtain 3 g of the final product. \n Cobalt nitrate and  iron nitrate (VETEC, Brazil) were dissolved in 450 ml of \ndistilled water in a ratio corresponding to the selected final composition. Glycine (VETEC, Brazil) was added in a proportion of one and half moles per mole of metal \natom s, and the pH of the solution was adjusted with ammonium hydroxide (25%, \nMerck, Germany)  in the range of 3 to 7. The pH was tuned as close ly as possible to 7, \ntaking care to avoid  precipitation. The resulting solution was concentrated by \nevaporat ion using a hot plate at 300°C until a viscous gel was obtained. This hot gel \nfinally burnt out as a result of a vigorous exothermic reaction. The system remained homogeneous throughout  the entire  process and no precipitation was observed. Finally, \nthe as -reacted material was calcined in air at 700°C for 2 h in order to remove the \norganic residues.  \nIn this work, CF0 is the sample as synthesized and CF is the same sample after \nmilling , but without any thermal treatment. The sample CF300_180 was thermally \ntreated at 300 C (first number) for 180 minutes (second number). The naming of the \nother samples followed the same label ing rule.  \nThe mechanical processing used to increase the coercivi ty of the samples was described \nin detail in a previous work [12]. A Spex 8000 high- energy mechanical ball miller, \nequipped with 6 mm diameter zirconia balls, was used for 1.5 h for all samples with \nball/sample mass ratio of about 1/7.  \nThe morphology and p article size distribution of the samples were examined by direct observation via transmission electron microscopy (TEM) using a JEOL -2100 \napparatus installed at LNNano / LNLS – Campinas – Brazil, working at 200 kV. \nThe crystalline phases of the cobalt ferr ite nanoparticles were identified by X -ray \npowder diffraction (XPD) patterns, obtained on a Shimadzu XRD -6000 diffractometer \ninstalled at the “ Laboratório Multiusuário de Técnicas Analíticas ” (LAMUTA / UFMT \n– Cuiabá - MT – Brazil). It was equipped with gra phite monochromator and \nconventional Cu tube (0.154178 nm), working at 1.2 kW (40 kV, 30 mA). Bragg-\nBrentano geometry was used. For the Williamson -Hall analysis [20 -22], the \ninstrumental broadening of the apparatus was determined using a Y 2O3 diffraction \npattern as standard.  \nMagnetic measurements were carried out using a vibrating sample \nmagnetometer (VSM) model VersaLab Quantum Design, installed at CBPF, Rio de \nJaneiro -RJ – Brazil. Experiments were done at room temperature and using a magnetic \nfield up to 2.7 T. \n \nResults and discussions  \nThe XPD patterns obtained from samples CF, CF600_180, CF600_30, \nCF400_30, CF300_180, and CF300_30 (see figure 1) confirm that all samples are \nCoFe 2O4 with the expected inverse spinel structure. Furthermore, these measurements \nindicate the absence of any other phases or contamination after milling and thermal \ntreatments.  \n  \nFigure 1  - XPD patterns of the  samples CF, CF600_180, CF600_30, CF400_30, \nCF300_180, and CF300_30. \n \nThe magnetic behaviour was evaluated via measurements of  the hysteresis loops \nobtained by VSM at room temperature. A substantial increase in  coercivity was \nobtained for the milled sample CF (H C=3.75kOe) compared with the non-milled sample \n(0.76kOe). This H C value is close to that obtained in a previous work (4.2 kOe) [12].  \nTo facilitate the analysis, we summarized in T able 1 the results of H C, M S, and \nMR for all samples. Analysing these three parameters, one can note that there are two \nclear tendencies: first is the decrease of the H C, and second is the increase of the M S and \nMR with the increase in temperature and time of the thermal treatment.  \nTable 1 -  Magnetic parameters  \nSample  HC (kOe)  MS* (emu/g)  MR (emu/g)  \nCF0 0.76 - - \nCF 3.75 57 33.0 \nCF300_30  3.46 57 32.8 \nCF300_180  3.39 58 34.0 \nCF400_30  3.00 62 36.7 \nCF600_30  2.24 66 39.6 \nCF600_180  1.93 70 42.0 \n*value of magnetization obtained at magnetic field equal to 2.7 T. \n \nFor clarity , we show in Figure 2 the hysteresis curve for only three samples: CF, \nCF300_30, and CF600_180. Despite the tendency  for the  HC to decrease in the samples \ntreated at 300oC, the decrease was only about 8 % and 10% in  samples treated for 30 \nmin and 180 min respectively , and no significant change in M S and M R was observed. \nHowever, in the sample treated at 600 oC for 180 min, H C decreas ed almost 50% , and \nsaw a significant and simultaneous increase in M S. The samples CF600_30 and \nCF400_30 also followed the tendencies of decreasing H C and increasing M S when \nincreasing the temperature of the thermal treatment.  \n \n \nFigure 2 – Hysteresis loop of cobalt ferrite nanocomposites CF, CF300_30 and \nCF600_180. \nTo explain these results  it is necessary first to understand the cause of the increase in \ncoercivity in the milled cobalt ferrite. This increase was studied in a previo us work [12] \nand by Liu et al. [11], and  two possible factors  were identified : the increase of the stress \nanisotropy (due the microstructural strain); and the increase of the density of structural \ndefects that causes  the increase of the centres of  domains wall pinning . Therefore , we \nassociate the decrease in coercivity  following the thermal treatment observed in this \nwork with  decreases in the stress anisotropy and density of structural defects.   \nThe thermal treatment causes the increase of the nanoparticle  grains and \nconsequently decreases the ratio between surface and bulk atoms in the material. The \nsurface magnetic atoms , due the magnetic disorder caused by finite -size behaviour, \ncontribute a smaller portion to the macroscopic magnetic moment when compared to the bulk atoms [23,24]. We associate this effect with the increase  in the M\nS observed to \nsamples treated at 400 and 600oC. The same effect , but in an opposite direction (i.e., a \ndecrease in the M S caused by the decrease of the nanoparticle grains) was  observed by \nLiu and Ding [11] to milled cobalt ferrite.  \nTo confirm our explanations of  the decrease in H C and the increase in M S when \nincreasing the temperatures of thermal treatment, a Williamson -Hall analysis of the \nXPD data and TEM measurements were performed.  The Williamson- Hall analysis in \nFigure 3 shows that the broadening of the diffraction peaks after milling evidences both \ngrain size reduction and an increase in microstructural strain.  \n \nFigure 3 – Williamson -Hall analysis of cobalt ferrite nanoco mposite milled and \nannealed at various temperatures for different periods of time. \nTo facilitate the analysis of the influence of structural parameters on the \nmagnetic behaviour of the samples, we included in Table III results obtained from the \nWilliamson -Hall analysis  together with coercivity and saturation magnetization . \nTable 2 -  Results obtained from Williamson -Hall analysis together with coercivity and \nsaturation magnetization.  \nSample  Average crystallite \nsize (nm)  MS (emu/g)  Strain \n(%) HC (kOe)  \nCF 24 57 0.92 3.75 \nCF300_30  28 57 0.98 3.46 \nCF300_180  25 58 0.95 3.39 \nCF400_30  30 62 0.92 3.00 \nCF600_30  55 66 0.73 2.24 \nCF600_180  51 70 0.57 1.93 \n \nOne can see that the Williamson -Hall analysis is in agreement with our \nassumptions and magnetic measurements. For both samples treated at 300oC there is no  \nincrease in the mean crystallite size or the strain , nor is  there a significant decrease in \nthe H C or increase in the M S (see Table 1). In the sample CF400_30, the small increase \nin crystallite size agrees with the small increase in M S, and the decrease in the strain is \nconsistent with the decrease in H C. The results of the Williamson -Hall analysis on the \nsamples treated at 600oC are also consistent with the magnetic measurements.  \nThe TEM images al so support our explanations for the decrease of H C and the \nincrease of M S, and are consistent with the Williamson -Hall analysis. In Figure 4 we \nshow the TEM images of the more strained samples: CF0; CF300_30 and CF300_180. One can note contrasts in the nanoparticle images that are characteristic of strained \nmaterial [11]. \nMany regular dislocations ( see Figure 4 ) in the nanopartic les were also \nobserved . These structural defects are moiré fringes  [11,12]  caused by the dislocation of \ndifferent crystalline planes  (see Figure 4E) . \n  \nFigure 4  – TEM images of ( A and B)  sample CF, (C) CF300_180, (D) \nCF300_30, (E) regular  and irregular squematic pictures of moiré  fringes . The circled \nareas show  some  moiré  fringes in the nanoparticles. \n \nSome struct ural defects (regular dislocations) were observed in the images \nobtained from sample  CF400_30, Figure 5 (A). For images of the samples treated at \n600oC (see F igure 5  B, C , and D) , we did not observe  any dislocation as was observe d \nin other samples. Analysis of size distribution indicate s that there is an increase in the \nmean  size when comparing the sample CF0 with the  sample CF600_180, which  is \ncompatible with our explanation. \n \nFigure 5  - TEM images of ( A) sample CF400_30, ( B) CF600_180, (C  and D ) \nCF600_30. The circled area shows  the moiré  fringes in the nanoparticle.  \n \nConclusion  \n \nThe magnetic coercivity  of the milled cobalt ferrite nanoparticles, as expected, is \naffected by the thermal treatment. This behaviour is associated with the decrease of \nmicrostructural strain and the density of structural defects, as confirmed by Williamson -\nHall analysis and TEM  measurements.  \nThe increase in M S observed in  the samples treated mainly at 600oC is associated \nwith the increase of the nanoparticle grains. This assumption is in agreement with the \nWilliamson -Hall analysis.  \nTo Summa rize, high -coercivity  cobalt ferrite induced by mechanical milling \nprocess is affected by thermal treatment. For temperatures up to 300oC the coercivity is \nslightly affected, but the result depends on the duration of the treatment. Also, there is \nnot a significant increase in mean size of the nanoparticles grains. Thermal treatment at \ntemperatures equal  to or higher than 600oC is not recommended, even  for a short \nduration, because a significant decrease in coercivity and the increase of the mean nanoparticles size occurs.  \n \nAckn owledgments  \nThis work was supported by the following Brazilian funding agencies CAPES (project \n#2233/2008) and FAPEMAT (project # 850109/2009). \n \nReferences  \n[1] - Nlebedim I C, Snyder J E, Moses A J and Jiles D C 2010 J. Magn. Magn. Mater. 322  \n3938.  \n[2] - Scheffe J R, Allendorf M D, Coker E N, Jacobs B W, McDaniel A H, Weimer A W, 2011 \nChem. Mater.  23 2030.  \n[3] - Cabral F A O, Machado F L A, Araújo J H, Soares J M, Rodrigues A R, Araújo A, 2008 \nIEEE Trans. Magn.  44 4235. \n[4] - Leite G C P, Chagas E F, P ereira R, Prado R J, Terezo A J , Alzamora M and Baggio-\nSaitovitch E 2012 J. Magn. Magn. Mat. 324 2711.  \n[5] - Pillai V and Shah D O 1996 J. Magn. Magn. Mater. 163 243.  \n[6] - Colombo M, Carregal -Romero S, Casula M F, Gutiérrez L, Morales M P, Bohm I B, \nHeverhagen J T, Prosperi D and Parak W J 2012 Chem. Soc. Rev.  41 4306.  \n[7] - Torres T E, Roca A G, Morales M P, Ibarra A, Marquina C, Ibarra M R and Goya G F  \n2010 J. Phys.: Conference Series, 200  072101.  \n[8] - Wang Y C, Ding J, Yi J B, Liu B H, Yu T and S hen Z X 2004 Appl. Phys. Lett. 84  2596.  \n[9] - Chiua W S, Radiman S, Abd- Shukor R, Abdullah M H and Khiew P S 2008 J. Alloys \nCompounds 459 291.  [10] -  Limaye M V, Singh S B, Date S K, Kothari D, Reddy V R, Gupta A, Sathe V, Choudhary  \nR J and Kulkarni S K  2009 J. Phys. Chem. B  113 9070. \n[11] - Liu B H and Ding J 2006 J. Phys. Chem. B  88 042506.  \n[12] - Ponce A S,  Chagas, E F  ; Prado, R J, Fernandes, C H M, Terezo, A J and Baggio-\nSaitovitch, E. 2013 J. Magn. Magn. Mat. 344 182.  \n[13] - Kneller E F and Hawig R 1991 IEEE Trans. Magn. 27 3588.  \n[14] - Debangsu Roy and Anil Kumar P S 2009 J. Appl. Phys. 106  073902.  \n[15] - Soares J M, Cabral F A O, de Araújo J H and Machado F L A  2011 Appl. Phys. Lett.  98 \n072502.  \n[16] - Coehoorn R, de Mooij D B and de Waard C 1989 J. Magn. Magn. Mat.. 80 101. \n[17] - Mohan M, Chandra V and Manoharan S S  2008 Curr. Sci. India, 94 473.  \n[18] - Ibiapino A L, Figueiredo L P, Lascalea G E,  Prado  R J 2013 Quim. Nova  36 762.  \n[19] - Zimicz  M G, Larrondo S A, Prado  R J, Lamas D G 2012 In. J. Hydrogen Energ. 37 \n14881.  \n[20] - Williamson G K and Hall W H 1953 Acta Metall.   1 22.  \n[21] - Mote V D, Purushotham Y and Dole B N 2012 J. Theor Appl. Phys. 6 6. \n[22] - Khorsand Zak A, Abd. Majid W H, Abrishami M E and Yousefi R 2011 Solid State Sci.  \n13 251.  \n[23] - Rajendran M, Pullar R C, Bhattacharya A K, Das D, Chintalapudi S N and Majumdar C \nK 2001 J. Magn. Magn. Mat. 232 71. \n[24] - Franco A, e Silva F C and Zapf V S 2012 J. Appl. Phys.  111 07B530.  \n "    },    {        "title": "0906.4979v1.Relaxation_Mechanism_for_Ordered_Magnetic_Materials.pdf",        "content": "arXiv:0906.4979v1  [cond-mat.mtrl-sci]  26 Jun 2009Relaxation Mechanism for Ordered Magnetic Materials\nC. Vittoria and S.D. Yoon\nCenter for Microwave Magnetic Materials and Integrated Cir cuits\nECE Department, Northeastern University, Boston MA. 02115 USA\nA. Widom\nPhysics Department, Northeastern University, Boston MA. 0 2115 USA\nWe have formulated a relaxation mechanism for ferrites and f erromagnetic metals whereby the\ncoupling between the magnetic motion and lattice is based pu rely on continuum arguments con-\ncerning magnetostriction. This theoretical approach cont rasts with previous mechanisms based\non microscopic formulations of spin-phonon interactions e mploying a discrete lattice. Our model\nexplains for the ﬁrst time the scaling of the intrinsic FMR li newidth with frequency, and1\nMtemper-\nature dependence and the anisotropic nature of magnetic rel axation in ordered magnetic materials,\nwhereMis the magnetization. Without introducing adjustable para meters our model is in reason-\nable quantitative agreement with experimental measuremen ts of the intrinsic magnetic resonance\nlinewidths of important class of ordered magnetic material s, insulator or metals.\nPACS numbers: 76.50.+g\nINTRODUCTION\nSince the discovery of magnetic resonance, the physics\ncommunity has been fascinated with possible mecha-\nnisms to explain the absorption linewidth or the relax-\nationtimeinmagneticmaterials. Itwasandstillisavery\nchallengingproblem. Magneticrelaxationissoimportant\ntounderstandbecauseitaﬀectsanumberoftechnologies,\nincludingcomputer,microwave,electronics,nanotechnol-\nogy, medical, etc.. Ultimately, the physical limitation of\nany technology which incorporates magnetic materials of\nany size, shape and combinations thereof comes down to\nprecise knowledge of the relaxation time of the magnetic\nmaterialbeing utilized. Thebackgroundofvariouscalcu-\nlationsorformulationsofmagneticrelaxationforthepast\nsixty years or so can be summarized brieﬂy as follows:\n(i) The relaxation times in paramagnetic materials [1]\nis characterized by two parameters, T1andT2, wherein\nT−1\n2describes the magnetic resonance linewidth and T1\ndescribes the time taken for the external magnetic ﬁeld\nZeemann energy density −Hext·Mto relax into thermal\nequilibrium. These times have been modeled in terms\nof various coupling schemes, i.e. spin-spin and/or spin-\nlattice interactions [2]. Since the coupling between spins\nisrelativelyweak,asitshouldbeinaparamagneticmate-\nrial, the coupling to the lattice involvesdiscrete spin sites\nrather than a collective cluster of spins. As such, param-\nagnetic coupling is necessarilymicroscopic in nature. For\nexample, a microscopic coupling scheme was formulated\n[3] whereby a spin Hamiltonian was modulated by the\nlattice motion. Variants to this approach have been very\nsuccessful in explaining relaxation in paramagnetic ma-\nterials. (ii) The magnetic relaxation of ferrimagnetic or\nferromagnetic resonance (FMR) linewidth is character-\nized by the Gilbert parameter [4] α, or equivalently by\nLandau-Lifshitz parameter [5] λL. A distinguishing fea-tureofthe collectivecoherentmagneticmomentsinFMR\nis that the magnitude of the magnetization, M=|M|re-\nmainsﬁxedwhichrequiresamagneticresonanceequation\nof the simple form\ndM\ndt=γM×Htot=γM×(H+H′),(1)\nwherein the gryomagnetic ratio γ=ge/2mc. Thetotal\nmagnetic intensity Htothas a thermodynamic part H\ndetermined by the energy per unit volume u,\ndu=Tds+H·dM, (2)\nand a dissipative part H′determined by the Gilbert lin-\near operator ˜ α,\nH′=/bracketleftbigg1\nγM/bracketrightbigg\n˜α·dM\ndt. (3)\nEqs.(1) and (3) imply that all components of the magne-\ntization must relax simultaneously in a way which con-\nserves the magnitude of the magnetization. Much of the\nsuccessful microscopic approaches or formulations uti-\nlized in paramagnetic materials were transferred over to\nmodels [6] which attempted to explain Eqs.(1) and (3).\nIn some sense this presented a contradiction or paradox\nwhich was conveniently ignored. As it is well known that\ncollective excitations in a ferri or ferromagnetic state can\nbe adequately described in classical continuum termi-\nnologies, although microscopic descriptions remain per-\nhapsmoreaccurate[7]. Toourknowledgeveryfeworany\nmicroscopicmodelshavebeensuccessfulinexplainingthe\norigin of Eq.(3). For example, much attention was given\nin the seventies to explain the FMR linewidth in YIG\n(Y3Fe5O12), since its linewidth was the narrowest ever\nmeasured in a ferrimagnetic material [8]. Clearly, there\nwas less to explain, and perhaps spin-lattice interactions2\ncould be treated at discrete spin sites as in paramagnetic\nmaterials. These calculations[8]containedmanyapprox-\nimations and predicted an FMR linewidth about 1/10 to\n1/100 of the measured linewidth. We believe that this\nis the best agreement between theory and experiment on\nrelaxation in an ordered magnetic material. The pur-\npose of this work is to improve upon the predictability\nof a theoretical model not only on a given material but\nin general for any ordered magnetic materials without\nrestoring to any approximations and assumptions.\nWe have adopted a conventional continuum magneto-\nmechanical description of the magnetic and elastic states\nof the ferri or ferromagnetic crystal [9, 12]. The ad-\nvantage of this description is that the microscopic spin-\nlattice coupling need not be formulated, since it has al-\nready been included in the continuum model which has\nbeenprovedtobeexperimentallycorrect. Weintroducea\nthermodynamic argument stating that the heat exchange\nbetween the magnetic and elastic systems must be the\nsame. Assuch, Eq.(3) maybe directlyrelatedtotheelas-\ntic sound wave relaxation time and the coupling strength\nbetween the magnetic and elastic systems. Speciﬁcally,\nwe will show that αis proportional to the square of the\nmagnetostriction constant. i.e. λ2and inversely propor-\ntional to γMτwherein τthe elastic relaxation time. In\naddition, the model predicts that ˜ αcannot be presumed\nto be a scalar as it has been done in the past; i.e. ˜ α\nis predicted to be anisotropic a second rank tensor in a\nsingle crystal material.\nIt is clear that one needs an interaction between\nphonons and electron spins to account for Gilbert damp-\ning parameter α. Suhl [10] and more recently Hickey\nandMoodera[11]haveconsideredsuchcouplingschemes.\nThe Gilbert damping parameter can be thought of as a\ntransport coeﬃcient in much the same way as conduc-\ntivity and/or viscosity are transport coeﬃcients. Such\ntransportcoeﬃcientsdescribeheatingprocessesbywhich\notherwise long lived modes are damped. One can in\nfact relate the Gilbert damping parameter to conduc-\ntivity and/or viscosity. For metallic ferromagnetic mate-\nrials, conductivity as well as electron viscosity produces\nconsiderable amount of magnetic damping via eddy cur-\nrent heating. For magnetic insulators it is the viscos-\nity which determines the magnetic damping. As it is\nwell known, conductivity and viscosity can be non zero\neven in zero frequency limit. Hence, the implied Gilbert\ndamping parameter is also non zero at zero frequency. In\nSuhl and Hickey and Moodera’s papers they ﬁnd, in the\nlimit of zero frequency and zero wave number, that the\nreal part of αis zero. This limiting case suggests that\nthey have not included the zero frequency transport co-\neﬃcients consistently in their theory. In our derivation\nthe expected result at zero frequency occur naturally in\nour formalism. In general, we believe the very nature of\ndiscreteness (as in paramagnetic materials) gives rise to\nrelatively long magnetic relaxation times. However, themagnetic relaxation time of a coherent collection of spins\n(as in FMR) implies shorter relaxation times, since it in-\nvolvescollectiveacousticwavesinthe interactionscheme.\nOur present theoretical treatment takes this into account\nvia the continuum magneto-mechanics.\nTHEORETICAL MODEL\nFrom Eq.(3), it is evident that the heating rate per\nunit volume due to the dissipative magnetic intensity H′\nobeys\n˙Q=dM\ndt·H′\n˙Q=/bracketleftbigg1\nγM/bracketrightbiggdM\ndt·˜α·dM\ndt\n˙Q=M\nγ˙Niαij˙Njwherein N=M\nM, (4)\nand ˜αis a second rank tensor\n˜α=\nαxxαxyαxz\nαyxαyyαyz\nαzxαzyαzz\n. (5)\nThe crystaldisplacement uyields in elasticity theory [13]\nthe strain tensor\neij=1\n2(∂iuj+∂jui) (6)\nIn virtue of the magneto-elastic eﬀect [14], a chang-\ning magnetization dM/dtwill produce a changing strain\nde/dt. In detail, in terms of third rank magneto-elastic\ntensor Λ ijklone ﬁnds\neij= ΛijklNkNl,\n˙eij= 2λijklNk˙Nl. (7)\nFinally, the fourth rank crystal viscosity tensor, ηijkl\ndtermines the heating rate per unit volume due to the\ntime dependent strain\n˙Q=˙eijηijkl˙ekl. (8)\nEmploying Eqs.(7) and (8) and comparing the result to\nEq.(4) yields the central result of our model.\nFor any crystal symmetry the Gilbert damping tensor\ndue to magnetostriction coupling is rigorously given by\nαij=/bracketleftbigg4γ\nM/bracketrightbigg\n(ΛnmpiNp)ηnmrl(ΛrlqjNq).(9)\nThe following properties of the Gilbert damping tensor\nEq.(9) are worthy of note: (i) The Gilbert damping ten-\nsor ˜αis inversely proportionalto the magnetization mag-\nnitudeM. (ii) The Gilbert damping tensor ˜ αis propor-\ntional to the squares of the magnetostriction tensor el-\nements. (iii) The tensor nature of ˜ αdictates that the3\nmagnetic relaxation is anisotropic . To a suﬃcient degree\nof accuracy, one may employ an average of the form\nα=1\n3tr{˜α}=/bracketleftbiggαxx+αyy+αzz\n3/bracketrightbigg\n(10)\ndeﬁning a scalar function α. (iv) The crystal viscosity\ntensorηnmrlmay be employed to describe the acoustic\nwave damping [15]. For a mode label a, e.g. a longitudi-\nnal (a=L) or a transverse ( a=T) mode, the acoustic\nabsorption coeﬃcient at frequency ωis given by [15]\nτ−1\na=ω2ηa\n2ρv2a, (11)\nwherein vais the acoustic mode velocity and ρis the\nmass density. Finally, for a cubic crystal, there are only\ntwoindependent magneto-elastic coeﬃcients which may\nbe deﬁned\nΛxxxx=3\n2λ100and Λ xyxy=3\n2λ111 (12)\nwherein the Cuachy three index magneto-elastic coeﬃ-\ncients are λijk.\nCOMPARISON WITH EXPERIMENT\nThe Gilbert damping factor αmay be deduced from\nthe measurement of the intrinsic FMR linewidth. How-\never, the measurement of the intrinsic linewidth is, in-\ndeed, very diﬃcult. The reason for this conclusion is\nthat there are too many extrinsic eﬀects that inﬂuence\nthe measurement. For example, in ferromagnetic metals\nlike Ni, Co and Fe the intrinsic linewidth contribution to\nthe total linewidth measurement [16, 17] may be between\n10% and 30%. The rest of the linewidth [18] may be due\nto exchange-conductivity eﬀects.\nHowever, there may be other contributions, such as\nmagnetostatic excitations, surface roughness, volume de-\nfects [19], crystal quality, interfaces [20], size, etc.. Sim-\nilar conclusions apply to ferrites except there are no\nexchange-conductivity eﬀects [18]. Thus, the reader\nshould be mindful that when we quote or cite an intrinsic\nvalue of the linewidth it represents a maximum value for\nthere can be some hidden extrinsic contributions in an\nexperiment. However, we have relied on data well estab-\nlished over the years. The criteria that we have adopted\nin choosing an ensemble of intrinsic linewidth measure-\nments are the ones exhibiting the narrowest linewidth\never measured in single crystal materials. In addition,\nwe required full knowledge of their elastic, magnetic and\nelectrical properties [16, 17, 18, 21]. The objective is not\nto introduce any adjustable parameters.\nThe experimental value of Gilbert damping parameter\nαexpmay be deduced from the FMR linewidth ∆ Hatfrequency fas\nαexp=√\n3\n2/parenleftbiggγ∆H\n2πf/parenrightbigg\n. (13)\nThefactor√\n3/2assumesLorentzianlineshapeoftheres-\nonance absorption curve. The theoretical Gilbert damp-\ning parameter αthvalue is expressed in terms of known\n[17] parameters so that there are no adjustable parame-\nters in our comparison to experiments, as shown in TA-\nBLE I. The theoretical prediction for the Gilbert damp-\ning paramter is that\nαth=36ργ\nMτ/bracketleftbiggλ2\n100\nq2\nL+λ2\n111\nq2\nT/bracketrightbigg\n, (14)\nwhereinρis the mass density, qT≈vTM\n2γAis the trans-\nverse acoustic propagation constant, qLis the longitudi-\nnal acoustic propagation constant, v Tis the transverse\nsound velocity, Ais the exchange stiﬀness constant, λ100\nandλ111are magnetostriction constants for a cubic crys-\ntal magnetic material. The transverse acoustic propaga-\ntion constant, was approximatedon the basisthat the re-\nlaxationprocessconservedenergyandwavevector. Since\nthe acoustic frequency is ﬁxed in the process the longi-\ntudinal propagation constant may be also calculated to\nbeqL=qT(vT/vL) for magnetic materials, wherein v L\nis the longitudinal sound wave velocity.\nIn FIG.1, we plot the experimental and theoretical val-\nues Gilbert damping constants as given by Eqs.(13) and\n(14). We note that the agreement between theory and\nexperiment is remarkable in view of the fact that any\nof the cited parameters could diﬀer from the ones listed\nin TABLE I by as much as 20-30%. For example, the\nlinewidth reported in TABLE I may not be on the same\nsample where the elastic or magnetic parameters were\ncited. In a few cases we needed to extrapolate the value\nofA, since there was no published value. In FIG.1, we\ndid not present data on the ferromagnetic metals for lack\nof conﬁdence on the linewidth data. For example, mag-\nnetostatic mode excitations have a deleterious eﬀect on\nthe dependence of the FMR linewidth on size. Most, if\nnotall, previousFMRlinewidthmeasurementshavebeen\nperformed on slabs, wiskers, etc.. whcih can indeed sup-\nportmagnetostaticmode excitations. Additional compli-\ncations arise as a result of exchange-conductivity excita-\ntions in the linewidth data. Nevertheless, the agreement\nbetween theory and experiment is quite satisfctory.\nCONCLUSION\nQualitative and quantitatively our model is in agree-\nment with experimental observations of the intrisic FMR\nlinewidth reported over the years. Speciaﬁcally, experi-\nmentallythe mostimportant characteristicsofthe intrin-\nsic FMR linewidth, ∆ H, measured on ordered magnetic4\nTABLE I: Calculated and measured Gilbert damping ( α) parameters\nqT λ100λ111 M A ∆H f τ α thαexp\nMaterials (10−6cm−1) (10−6) (10−6) (G/4 π) (10−6erg/cm) (Oe) (GHz) (10−13sec) (10−5) (10−5)\nY3Fe5O12a3.8 1.25 2.8 139 0.40 0.33 9.53 4.4 5.56 9.0\nY3Fe4GaO12a1.46 −1−1 36 0.28 3.0 9.53 4.4 51 76\nLi0.5Fe2.5O4b8.6 −8 + 0 310 0.40 2.0 9.50 1.5 26 50\nNiFe2O4b7.49 −63−26 270 0.40 35 24.0 710 26 350\nMgFe2O4b9.30 −10−1 90 0.1 2.3 4.9 1.5 120 120\nMnFe 2O4b6.6 −30−5 220 0.4 238 9.2 1.5 930 1040\nBaFe12O19c9.6 15 350 0.4 6 55 1.5 18 26\nNid6.3 −46 25 484 0.75 102 9.53 1.8 770 2600\nFed8.75 20 −20 1690 1.9 9 9.53 1.8 30 220\nCod5.1 80 1400 2.78 15 9.53 1.8 530 370\naGarnetsbSpinelscHexagonal FerritedFerromagnetic Materials\n(Note: Longitudinal acoustic wave constant is qL= (vT/vL)qT)\n/s49/s48/s45/s53\n/s49/s48/s45/s52\n/s49/s48/s45/s51\n/s49/s48/s45/s50\n/s49/s48/s45/s49/s49/s48/s45/s53/s49/s48/s45/s52/s49/s48/s45/s51/s49/s48/s45/s50/s49/s48/s45/s49/s101/s120/s112\n/s116/s104\nFIG. 1: Shown are the experimental and theoretical values of\nthe Gilbert damping constants as given by Eqs.(13) and (14).\nmaterials (metal or insulator) for the past ﬁfty years are\nthat ∆Hscales with frequency and1\nM[16, 22, 23]. In-\ndeed, these arethe predictions of our theory. In addition,\n∆Hscales with the magnetostriction constant squared,\nsee FIG.1. FIG.1 was plotted in a logarithmic scale only\nto be able to include all ofthe datain TABLE I. Another\nprediction of our theoretical work is that the Gilbert\ndamping parameter ˜ αis not simply a scalar parame-\nter but a tensor quantity. This implies that the FMR\nlinewidth is intrinsically anisotropic in single crystals of\nferri-ferromagnetic materials. There was much contro-\nversy in the seventies about whether or not the intrin-\nsic linewidth should be anisotropic or not. Poor qualityof samples seemed to have incited the controversy. Im-\nproved or more accurate angular linewidth data [18, 19]\nsupports the notion of an anisotropic linewidth in or-\ndered magnetic materials in agreement with our model.\nIn summary, we believe that the comparison between\ntheory and experiment is very encouraging in terms of\ncontinuing this continuum approach to explain intrinsic\nlinewidths in ordered magnetic materials.\nAcknowledgement\nWe wish to thank to Prof. V.G. Harris and A. Geiler for\nstimulus discussions about magnetic materials and their\nrelaxation.\n[1] F. Bloch, Z. Physik, 74, 295 (1932).\n[2] H.B. Callen, Fluctuation, Relaxation, and Resonance in\nMagnetic Systems , Editor D. ter Haar, p 176, Oliver &\nBoyd Ltd., London (1962).\n[3] R. Orbach, Proc. Phys. Soc., 77, 821 (1961).\n[4] T.L. Gilbert, IEEE Trans. Mag., 40, 3443 (2004).\n[5] L.D. Landau and E.M. Lifshitz, Phys. Z. Sowjet., 8, 153\n(1935).\n[6] C. Kittel, J. Phys. Soc. Japan Suppl., 17(B1), 396\n(1962).\n[7] M. Sparks, Ferromagnetic Relaxation Theory , McGraw-\nHill Book Co., New York, (1970).\n[8] R.C. LeCraw and E.G. Spenser, J. Phys. Soc. Japan,\nSuppl.,17(B1), 401 (1962).\n[9] S. Chikazumi, Physics of Magnetism John Wiley and\nSons, Inc., New York (1964).\n[10] H. Suhl, IEEE Trans. Mag., 34, 1834 (1998).\n[11] M. C. Hickey and J. S. Moodera, Phys. Rev. Letts., 102,\n137601 (2009).\n[12] L.D. Landau, L.P. Pitaevskii, and E.M. Lifshitz, Elec-\ntrodynamics of Continuous Media , Elsevier Butterworth-\nHeinemann, Oxford, (2000).5\n[13] L.D. Landau and E.M. Lifshitz, Theory of Elasticity ,\nButterworth-Heinemann, Oxford, (1986).\n[14] L.D. Landau, L.P. Pitaevskii, and E.M. Lifshitz, op. cit.\npp 144-146.\n[15] L.D. Landau and E.M. Lifshitz, op. cit., Chapt. V.\n[16] S.M.BhagatandP.Lubitz, Phys.Rev., B 10, 179(1974).\n[17] G.C. Bailey and C. Vittoria, Phys. Letts., 37A, 3, 261\n(1972).\n[18] W.S.AmentandG.T.Rado, Phys.Rev., 97, 1558(1955).\n[19] A.M.Clogston, H.Suhl, L.R.WalkerandP.W.Anderson,\nJ. Phys. Chem. Solids, 1, 129 (1956).\n[20] C. Vittoria and J.H. Schelleng, Phys. Rev., B 16, 4020\n(1977).[21] Landolt-Bornstein. Numerical Data and Functional Rela-\ntionships in Science and Technology , Editors K.-H. Hell-\nwege and A.M. Hellwege, Magnetic and Other Prop-\nerties of Oxide and Related Compounds ,4part-a,b\n(Spring-Verag, Berlin, Heidelberg, New York, 1970); 12\npart-a,b,c (Spring-Verag, Berlin, Heidelberg, New York,\n1980).\n[22] P.Roschmann, IEEETrans.Mag., 11, 1247(1975); IEEE\nTrans. Mag., 20, 1213 (1984).\n[23] C. Vittoria, P. Lubitz, P. Hansen, and W. Tolksdorf, J.\nAppl. Phys., 57, 3699 (1985)."    },    {        "title": "2007.00602v1.Impact_of_V_substitution_on_the_physical_properties_of_Ni_Zn_Co_ferrites__structural__magnetic__dielectric_and_electrical_properties.pdf",        "content": "Impact of V substitution on the physical properties of Ni -Zn-Co ferrites: \nstructural, magnetic, dielectric and electrical properties  \n    \nM. D. Hossain1, A. T. M. K. Jamil1, M. R. Hasan2, M. A. Ali3, I. N. Esha4, M. A. Hakim5, M. N. \nI. Khan2* \n \n1Department o f Physics, Dhaka University of Engineering and Technology (DUET), Gazipur, \nBangladesh  \n2Materials Science Division, Atomic Energy Center, Dhaka 1000, Bangladesh  \n3Department of Physics, Chittagong University of Engineering and Technology (CUET), \nChattogram 4 349, Bangladesh  \n4Department of Physics, University of Dhaka, Dhaka 1000, Bangladesh  \n5Department of Glass and Ceramic Engineering, Bangladesh University of Engineering and \nTechnology (BUET), Dhaka 1000, Bangladesh  \n \n*Corresponding author´s e -mail address:   ni_khan77@yahoo.com  (M.N.I. Khan)  \n \nAbstract  \nWe have investigated the Vanadium - (V) substituted Ni -Zn-Co ferrites where the samples were \nprepared using solid -state reaction technique. The impact of V5+ substitution on the structural, \nmagnetic, dielectric and  electrical properties of Ni -Zn-Co ferrites has been studied. XRD \nanalysis confirmed the formation of a single -phase cubic spinel structure. The lattice constants \nhave been calculated both theoretically and experimentally along with other structural \nparame ters such as bulk density, X -ray density and porosity. The FESEM images are taken to \nstudy the surface morphology. FTIR measurement is also performed which confirms spinel \nstructure formation. The saturation magnetization ( Ms), coercive field ( Hc) and Bohr  magneton \n(µB) were calculated from the obtained M -H loops. The temperature dependent permeability is \nstudied to obtain the Curie temperature. Frequency and composition dependence of permeability \nwas also analyzed. Dielectric behavior and ac resistivity ar e also subjected to investigate the \nfrequency dependency. An inverse relationship was observed between the composition \ndependence of dielectric constant and ac resistivity. The obtained results such as the electrical \nresistivity, dielectric constants and m agnetic properties suggest the appropriateness of the studied \nferrites in microwave device applications.  \nKeywords: Ni-Zn-Co ferrites, structural properties, cation distribution, magnetic properties, \npermeability, dielectric and electrical properties.  Intro duction  \nFerrites, influential ceramic materials are composed by combining, firing and blending a huge \nportion of Fe 2O3 (iron (III) oxide) with a small amount of at least one metallic element, for \nexample, manganese, zinc, nickel, barium, cobalt, etc. [1].   Ferrites mainly used in three sectors \nof electronics: power applications, low -level applications and Electro -Magnetic Interference \n(EMI) suppression. Therefore, a constant challenge is being posted to improve their \ncharacteristics for ever -increasing dema nds in home communication appliance, computer, \nelectrical and other technical fields [2]. The interests in this oxide emerge from their versatile \napplicability from relatively high to microwave frequency region. Among various ferrites, soft \nferrites being highly resistive at high frequencies are used extensively in electronic applications \nsuch as transformer cores, inductors, antenna rod, microwave devices, computer memory chip, \nmagnetic recording media, etc. after their first commercial introduction [3]. M ostly these \nelectrical and magnetic properties depend on the method of preparation, preparative parameters, \npreparative conditions, particle size, nature of dopants etc. [4]. Hence, many researchers \nbestowed their time and efforts on various ferrites with many dopants to enhance its electric and \nmagnetic characteristics.  \nThe important electronic properties make the spinel Ni -Zn-Co ferrites prominent [5]. A good \ncombination of its magnetic and electric properties as well as it low -cost aspect makes them \npotential candidates for application in high -frequency purpose [6]. These ferrites have high \nsaturation magnetization, low eddy current loss, high resistivity, high permeability, high Curie \ntemperature [7 -10]. It is technologically sound due to its prospective  use in targeted microwave \ndevices, sensors, catalysis, magnetic recording applications and drug delivery systems [11 -15]. \nOwing to the mentioned interest many researchers have paid their attention to the Ni -Zn-Co \nferrites. Mallapur et al.  have investigate d the structural and electrical properties of the spinel \nferrite system of Ni -Co-Zn [16]. Electric properties of nanocrystalline Ni -Co–Zn ferrites have \nbeen reported by Ghodake et al [17]. Spectral studies such as room temperature Mossbauer, X -\nray and infr ared IR spectra of Ni -Co–Zn ferrites have been carried out by Amer et al.  [18]. \nKnyazev et al.  carried out a study on the structural and magnetic properties of Ni -Co–Zn ferrites \n[19]. Besides, Mohit et al.  [20], Stergiou et al.  [21], Ghodake  et al.  [22], H assan et al.  [23], \nMattei et al.  [24], Chen et al.  [25], etc. have also investigated the Ni -Co–Zn ferrites. Moreover, \nreports on the different ions substituted -Ni-Zn-Co ferrites are also available e.g., Ren et al.  [26] have performed a study on La3+ substi tuted Ni -Zn-Co ferrites. Saini et al.  [27] have investigated \nIn substituted Ni -Co-Zn ferrites. Y and La Substituted Ni -Co-Zn ferrites have been investigated \nby Stergiou et al.  [28], Gd and La -doped Ni -Zn-Co ferrites are studied by Zhou et al.  [29] \nBaO‐dope d Ni-Zn-Co magneto ‐dielectric ferrites have been studied by Zheng et al [30]. We have \nalso studied the Gd substituted Ni -Zn-Co ferrites [31].  \nThe dependency of the physical properties of ferrites on the distribution of cation over \ntetrahedral ( A) and octah edral ( B) sites open the way of changing their properties by introducing \ndifferent ions into these two sites [32 -34]. Therefore, alteration of physical properties by the \nincorporation of a small amount of V into Ni -Zn-Co is also expected. Reports on the ad dition or \nsubstitution of vanadium into different ferrites systems are available [35, 36].   Korkmaz et al.  \n[37] noted a decline of saturation magnetization due to V substitution in NiFe 2−xVxO4 (x ≤ 0.3) \nnanoparticles system. Magnetic properties of V substituted Li -Zn-Ti ferrites have been \ninvestigated by Maisnam et al.  [38]. They reported decrease in Curie temperature ( Tc) and \nsaturation magnetization with an increase in V contents. Slim ani et al.  [39] have investigated how \nmagnetic and optical properties NiFe 2−xVxO4 (0.0≤x≤0.3) NPs  are influenced by varying calcination \ntemperature. Heiba et al.  [40] studied the V substituted Co ferrites. M. Kaiser also reported the \neffect of V substi tution on the magnetic and dielectric properties of Ni -Zn-Cu ferrites [41]. The \ndecrease of saturation magnetization owing to the presence of V with the low melting \ntemperature which causes the formation of the liquid phase, accelerating the grain growth \nprocess at low sintering temperature [42, 43]. Based on the available report, a considerable \nalteration of properties of Ni -Zn-Co ferrites is expected through V substitution. Therefore, the \nobjective of the current study is to investigate the influence of V  substitution on the electrical and \nmagnetic properties of Ni -Zn-Co ferrites.  \nEXPERIMENTAL DETAILS  \n \nConventional double sintering method was used to prepare V5+ substituted Ni -Zn-Co ferrite \n[Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ]. The following operatio n are used to prepare the desired \nsamples the details of which can be found elsewhere [31]. However, the sample preparation \nprocedure is shown in Fig. 1.  \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1:  The sample preparation procedure of Ni 0.7Zn0.2Co0.1Fe2-xVxO4(0 ≤ x ≤ 0.12) fe rrites.  \n \nThe synthesized samples have been studied by X -ray diffraction (XRD) using Philips X'pert \nPRO X -ray diffractometer (PW3040) with Cu -Kα radiation (λ=1.5405 Å), Field Effect Scanning \nElectron Micrographs (FESEM) (JEOL JSM -7600F), Fourier transform i nfrared spectroscopy \n(FTIR) (PerkinElmer FT -IR Spectrometer). A Wayne Kerr precision impedance analyzer \n(6500B) [Frequency range: 1kHz –120 MHz, drive voltage of 0.5 V] was also used to study the \ndielectric and permeability properties. The magnetic properti es were obtained by a physical \nproperties measurement system (PPMS), Quantum Design Dyna Cool at ambient conditions.  \n \nResults and discussion  \n \nStructural properties  \n \nFig. 2 illustrates the X -ray diffraction (XRD) patterns of Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) \nferrites sintered at 1100 °C. The diffraction peaks at different planes (111), (220), (311), (222), \n(400), (422), (511) and (440) confirmed the cubic spinel structures (JCPDS #08 -0234) of the \nsynthesized samples [44]. The diffraction peaks b ecome broaden with the increase of the \nconcentration of V doped which indicates the decrease of grain size [Table 1] with the increase Oxide of raw \nmaterials (Fe 2O3, \nNiO, ZnO, CoO \nand V 2O5) Weighing by \ndifferent mole  \npercentage   \nDry mixing \nby milling,  \n6 hrs \n \nPressing to \ndesired shape   \nMilling,  \n2 hrs  Pre-sintering  \n(800 °C, 4 hrs)  \nSintering  \n(1100 °C, 4 hrs)   \nFinished \nProduct  of V contents. It is occurred because of the smaller ionic radius of V5+ ion than the ionic radius \nof Fe3+ ion.  \n15 30 45 60 75(440)(511)(422)(400)(222)(311)(220)Intensity (a.u)\ndegree )(111)(a)\nx=0.00x=0.02x=0.05x=0.07x=0.10x=0.12\n35.0 35.5 36.0(311)Intensity (a.u)\ndegree )(b)\n0.00 0.04 0.08 0.128.328.368.408.448.48\n atheo\n aexpLattice constant,a(A) \nV-content (x)(c)\n \nFig.2: (a)  X-ray diffraction patterns (b)Maximum pick broaden and  (c)Lattice constant( a) \ndependence of V -content of Ni 0.7Zn0.2Co0.1Fe2-xVxO4(0 ≤ x ≤ 0.12) ferrites.  \n \nThe experimental lattice constant ( a) for all the samples are calculated using the formula [45]:  \n                                          aexp = d hkl\n2 2 2l k h                                                               (1) \nwhere, h, k and l are the Miller indices of the crystal planes. The calculated values are given in \nTable 1. The obtained value of aexp for x = 0.00 is 8.362 Å and the reported value is 8.3719 Å  \n[26].  Our obtained  value is lower than the reported va lue, might be due to different synthesis \ncondition. The change of lattice constant ( a) with V content is shown in Fig.2(c). At first aexp \ndecreases with the increase of V contents and then increases for x = 0.12. The trend in the \nvariation lattice constant  is observed to have a good agreement with earlier report [46] of V -\nsubstituted Ni –Zn ferrites. The decrease of the lattice constant is owing to the differences of \nionic radii (Shannon 6 -coordination numbers) of V5+ions (0.59 Å) and Fe3+ions (0.645 Å) [47,48 ].  \nThe lattice constant of studied samples Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) has also been \ncalculated to compare with the experimental one. The lattice constant can be calculated \ntheoretically using the following equation [2]:  \n                                           \n  0 0 3\n338Rr Rr aB A th  \n \nwhere, R 0, rA, rB are th e ionic radii of oxygen (1.32 Å) [49], A- and B-sites, respectively. The r A   \nand r B can be calculated by the following equations:  \n) ( ) (3 2 \n  Fer C ZnrCreAF AZn A\n and \n)].( ) ( ) ( ) ( [215 3 2 2        VrC FerC CorC NirC rBV BFe BCo BNi B  \n [50, 51].  (2)  The calculated X -ray density, bulk density and porosity for th e studied samples \nNi0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) were  represented in Table 1. The equations used can be \nfound elsewhere [31]. A decreasing trend of X -ray density and the bulk density with increasing \nV-content till x = 0.12 is noted while an inverse relation is followed by the porosity. Th is trend \ncan be explained on the basis of atomic mass of V5+ (50.94 amu) and F3+ (55.84 amu). Here,  V \nwith lower atomic mass is substituted for Fe ions with comparatively higher atomic mass. \nWhereas, for x = 0.12 the increase in density was observed due t o the increase of  lattice \nconstant. Moreover, the b is found to be lower than the x, because the pores are considered in \nthe calculations of bulk density which are absent in the x -ray density calculations [52].  \n \nFig. 3 demonstrates FESEM images from whe re the changes in the microstructure owing to the \nV substitution can be observed. In addition, the average grain size for considered compositions \nhas been estimated [shown in Table 1] by linear intercepting method and a decreasing trend with \nV contents is also observed except  x = 0.12 which is similar to the variation of density and \nporosity with V contents. The changes in the average grains size, density and porosity can be \nunderstood from the FESEM images. The change in average grains size is also related  to the \ndifference in between ionic radii of V5+ (0.59 Å) ions and Fe3+ ions (0.645 Å).  \n \n \n  \n  \n \n \n  \n  \n \nFig. 3:  The FESEM images of Ni 0.7Zn0.2Co0.1Fe2-xVxO4ferrites.x=0.02  x=0.05  X=0.00  \nx=0.12  x=0.10  x=0.07  Table 1:  Cation distribution (tetrahedral A -site and octahedral B -site), Ionic radii (r Afor A -site and r Bfor B -site), Lattice constant (a th \nfor theoretical and a exp for experimental value), X -ray density ( x), Bulk density ( b) and Porosity (P) of Ni 0.7Zn0.2Co0.1Fe2-xVxO4(0 ≤ x \n≤ 0.12) ferrites.  \nV \ncontent  \n(x) Cation Distribution  \n \n Ionic radii  Lattice \nconstant  X-ray \ndensity  \n \nρx \n \n(g/cm3) Bulk \ndensity  \n \nρb \n \n(g/cm3) Grain \nSize \nD \n \n(μm)  Porosity  \n \n \nP \n(%) Bond \nLength (Å)  Hoping \nLength (Å)  \nA-site \n \n  \nB-site \n \n  \nrA \n \n(Å)  \nrB \n \n(Å)  \nath \n \n(Å)  \naexp \n \n(Å) \n  \nA-\nsite  \nB-\nsite  \nA-\nsite  \nB-site \n0.00 \nA Fe Zn ) (3\n8.02\n2.0  \n  2\n43\n2.12\n1.02\n7.0 O Fe CoNiB\n 0.664 0.666  8.349   \n8.362  \n 5.35 5.05 6.87 5.71 1.810  2.091  3.621  2.956  \n0.02 \nA Fe Zn ) (3\n8.02\n2.0 \n  2\n45\n02.03\n18.12\n1.02\n7.0 O V Fe CoNiB 0.664  0.665  8.348  8.361  5.51 4.26 \n 4.43 22.68  1.810  2.090  3.620  2.956  \n0.05 \nA Fe Zn ) (3\n8.02\n2.0 \n  2\n45\n05.03\n15.12\n1.02\n7.0 O V Fe CoNiB  0.664  0.664  8.346 8.354  5.49 4.19 \n 3.73 23.68  1.808  2.089  3.617  2.954  \n0.07 \nA Fe Zn ) (3\n8.02\n2.0 \n   2\n45\n07.03\n13.12\n1.02\n7.0 O V Fe CoNiB 0.664  0.664  8.345  8.353  5.48 4.11 3.70 25.0 1.808  2.088  3.617  2.953  \n0.10  \nA FeZn ) (3\n8.02\n2.0\n \n  2\n45\n1.03\n1.12\n1.02\n7.0 O VFe CoNiB 0.664  0.663  8.343  8.350  5.46 4.08 2.58 25.27  1.807  2.088  3.616  2.952  \n0.12 \nA Fe Zn ) (3\n8.02\n2.0 \n  2\n45\n12.03\n08.12\n1.02\n7.0 O V Fe CoNiB 0.664  0.662  8.341  8.361  5.48 4.43 3.98 19.16  1.810  2.090  3.620  2.956  FTIR analysis  \n \nFTIR Spectroscopy is an important technique to study the completion of the solid -state reaction \nand inspect the presence of deformation in the spinel ferrites due to substitutions of ions [53, 54]. \nIn case of ferrit es, there are two characteristic peaks in the FTIR spectra. The first peak at low \nfrequency is associated M -O (Metal -Oxide) stretching vibrations at B -sites while the peak at \nhigher frequency side results from M -O stretching vibrations at A -sites [55]. Fig . 4 displays the \nFTIR spectra of V substituted Ni -Zn-Co ferrite compositions in which the peaks are observed at \nexpected positions, confirmed the completion of solid state reaction [56]. It has two major \nabsorption bands in the range of 365 to 800 cm-1. Th e band's position for the studied \ncompositions is tabulated in Table 2. The bands at higher frequency (  υ1 = 590 cm-1) is for \nstretching vibrations of the M -O clusters at the tetrahedral site. Lower frequency bands (  υ2 = \n365cm-1) are assigned to the stret ching mode of the M -O bond in the octahedral sites [57].   \n800 700 600 500 40020406080Transmittance(%)\nWavenumber (cm-1) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12\n \n \nFig. 4:  FTIR spectra of Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites.  \n \nTable 2: The vibrational frequencies of two prominent IR bands corresponding to tetrahedral \nand octahedral sites of Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites.  \n \nV-content ( x) υ1(cm-1) υ2(cm-1) \n0.00 578.49  372.19  \n0.02 590.09  379.98  \n0.05 583.69  381.18  \n0.07 583.69  370.99  \n0.10 579.69  373.58  \n0.12 574.70  364.59  \n \nMagnetic properties  \n \nFig. 5 s hows the magnetic hysteresis (M -H) loops of Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) \nferrites. The parameters such as saturation magnetization ( Ms), coercivity ( Hc), magnetic \nremanence ( Mr), etc were obtained from the M -H loops and listed in Table 3. The na rrow hysteresis loops assure the soft magnetic nature of all the samples [58]. In the present study for \nun-doped Ni -Zn-Co ferrites, the value of saturation magnetization is 71.6 emu/g. It is observed to \ndecrease due to V substitution because of having non -magnetic nature. The V5+ ions have a \nliking to occupy B-sites. Although, the magnetic moment of ferrites is dominated by A -B \ninteraction but contribution from A -A interaction and B -B interaction is noticeable. The net \nmagnetic moment is given by the equati on: M = M BMA, where MB represents the total magnetic \nmoment at B -sites and the MA indicates the total magnetic moment at A -sites. The total magnetic \nmoment of B -sites is lowered by replacing Fe ions by V substitution and it caused the reduction \nof magneti c moment of ferrites system. Therefore, the lowering of saturation magnetization due \nto V substitution is reasonable. However, the Ms is observed to increase for x = 0.12, might be \nowing to the increased grain size at  x = 0.12 because of the proportional r elationship between \nmagnetization and grains size [59]. The values of Mr also follow the similar trend with V \ncontents. Another important parameter, known as Bohr magneton ( μB) is also calculated using \nthe relation: \n , Mʹ is the molecular weigh t. The calculated values of μB also follow the \ntrend of Ms with V contents.  \n-10000 -5000 05000 10000-75-50-250255075Ms(emu/g)\n \nApplied field(Oe) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(a)\n \nFig.5:  The variation of saturation magnetization dependence on (a) applied field and (b) V \ncontent( x) for Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites.  \n \nTable 3 : Saturation magnetization ( Ms), Magnetic remanence ( Mr), Resonance frequency ( fr) \ncoercivity ( Hc), Magnetic moment (µB), Curie Tem. ( Tc), Real permeability ( µ') and Relative \nquality factor (RQF) for Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites.  \n \nV \ncontent  \n(x) Saturation \nMagnetization  \nMs(emu/g)  Mr \n(emu/g)  fr \n(MHz)  Coercivity  \nHc (Oe) nB in µ B \n Curie \nTem.  \nTc (0C) µ' RQFx103 \nTheo  Exp. \n0.00 71.6 22.08  60.97  277.2   3.70 3.21 478 253 5.11 \n0.02 28.24  9.42 56.92  20.39   3.66 2.78 476 237 27.5 0.05 25.16  7.96 59.00  27.36   3.60 2.36 474 226 67.6 \n0.07 22.95  7.46 59.00  29.5  3.56 1.99 458 213 164.3  \n0.10 18.98  6.22 49.91  20.26  3.50 1.86 436 184 203.2  \n0.12 21.55  5.36 61.20  15.64  3.46 2.15 465 202 220.9  \n \nStudy of te mperature dependent permeability: Curie temperature  \nThe magnetic properties such as saturation magnetization, permeability etc. are very sensitive to \ntemperature. One of the characteristic parameters, Curie temperature ( Tc) can be obtained from \nthe tempera ture dependent permeability. The permeability remains almost constant up to certain \ntemperature and after increasing slightly by exhibiting a peak, known as Hopkinson’s peak, it \nfinally drops sharply to be zero. The temperature at which the permeability be comes zero is \nknown as Curie temperature ( Tc), the temperature at which completely disorderliness of atomic \nmoments took place and the ferrimagnetic materials converted to paramagnetic.  The temperature \ndependent initial permeability was measured and shown for different composition in Fig. 6(a). \nThe mentioned features are also observed for our obtained data and the calculated Curie \ntemperature is shown in Fig. 6 (b).   \n100 200 300 400 500 600100150200250300 Real part of permeability,   \nTemperature (oC) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(a)\n0.00 0.04 0.08 0.12400450500550600 Curie temperature (oC)\nV content (x)(b)\n \n \nFig. 6:  Variation of ( a) permeabilit y with temperature and ( b) V content ( x) dependence curie \ntemperature for Ni 0.7Zn0.2Co0.1Fe2-xVxO4(0 ≤ x ≤ 0.12) ferrites.  \n \nThe measured Tc is noted to decline with the V substitution. The transition temperature  attributed \nfrom the distribution of cation over A- and B-site and value of exchange coupling constant ( J) \n[60]. The JAB attributed from intera ction between ions on A- and B-sites is much stronger than the \nJAA and JBB attributed from interaction between ions of same sites ( A or B ). The replacement of \nFe3+ ions by V5+ ions, decrease the magnetic ions in the B -sites causing a decrease in the streng th \nof exchange coupling  constant JAB and thus decrease the Tc values.    \n Frequency dependence of real permeability and Relative Quality Factor  \n \nFig. 7 (a) shows the real part of permeability ( µ') as a function of frequency. The toroid shaped \nsamples were  prepared for this characterization . The value of \n remains almost constant up to ~ \n30 MHz, a noticeable increase is noted at which the \n became maximum and then fall sharply \nto certain low values. The steady value is important for ma ny applications such as in transformer \nas a broadband pulse and in video recording system as read -write heads (wide band) [61] . \n \nIn different circumstances, a significant peak is exhibited by \n  [figure is not shown] at the \nfrequency where sharp  decline of \n  is taken place. This phenomenon is termed as ferrimagnetic \nresonance [62] and the prepared compositions are found to follow the Snoek’s limit [63].  \n103104105106107108 100150200250300'\nFrequency,f(Hz) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(a)\n0.00 0.04 0.08 0.12150200250300'\nV content (x)(b)\n104105106107108109 -2-10123\n1031041051061071080200040006000 x = 0.00RQF 105\nFrequency,f(Hz) 0.00\n 0.02\n 0.05\n 0.07\n 0.10\n 0.12(c)\n \nFig. 7: (a) Frequency dependence of initial permeability ( '), (b)V-content ( x) dependence of \nreal permeability and ( c) Frequency dependent relative quality factor (RQF) of \nNi0.7Zn0.2Co0.1Fe2-xVxOs4(0 ≤ x ≤ 0.12) ferrites.  \n \nFig. 7(b) illustrates the composition dependence of \n which revealed that the initial \npermeability decreased gradually with V contents up to x = 0.10 and then increased slightly for x \n= 0.12. The permeability of ferrite s depends on various factors like grain size, density, porosity, \nsaturation magnetization, anisotropy, etc. [64]. A good correlation is observed among \npermeability, average grain size [Table -1] and saturation magnetization [Fig.5(b)]. The Globus -\nDuplex rel ation \n12\nμ\nKDMs, (where, Ms, D and K1 represent the saturation magnetization, average \ngrain size and anisotropic constant, respectively) can be used to explore the variation of µ' with \nV contents. The above relation exhibits that the initial p ermeability is proportional to Ms and D \nwhere Ms and D are also found to decrease with V contents up to x = 0.10 and then increases for \nx = 0.12.  The relative quality factor (RQF) measures the performance of a material for use in filter. Fig. 7 \n(c) shows the usual behavior of RQF versus frequency for different V contents in which a \nprominent peak is observed for each composition [31, 65, 66]. The exhibition of peak is followed \nby very low value both in the low frequency and in the high frequency regions. T he declination \nof RQF corresponds to the frequency (~30 MHz) wherein the \n  (loss component) started to \nincrease sharply. Moreover, the value of RQF is observed to increase owing to V substitution \nwith the maximum for x = 0.12. The values of RQF  are given in Table 3.  \n \nDielectric behavior  \n \nA very common dielectric behavior of ferrites is that the εʹ and εʹʹ decrease with frequency but \nexhibit three different responses in three ranges of frequencies: (i) a sharp decrease with low \nfrequency in ԑʹ and ԑʹʹ (ԑʹʹ is not shown), (ii) a slow decrease within mid frequency range in ԑʹ \nand ԑʹʹ and (iii) final ly become almost frequency independent at high frequencies [31,65,67,68]. \nComparatively slower decrease in ԑʹ and ԑʹʹ is observed in the mid -frequencies region where the \norientational polarization is mainly responsible for dielectric constant. The dielectr ic constant \nbecomes constant at the very high frequency region where the contribution comes from the \natomic and electronic polarization [69]. The frequency dependence of dielectric constant ( ԑʹ) and \ndielectric loss factor ( tanδ = ε\"/ʹ) of Ni 0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites are shown in \nFig. 8 (a and b). The dielectric properties can be explained by considering the studied ferrites as \ncomposed of two layers: grains of high conductivity which are separated by grain boundary of \nhigh resistivity [70-72]. The hopping mechanism is responsible of dielectric mechanism [73, 74]. \nThe charges produced through electron hopping between Fe3+ and Fe2+ are responsible for \nelectrical conduction in ferrites. The motion of these charges are supposed to be hinder ed at the \ngrain boundaries owing to their activity in the low frequency region and accumulated at the \ninterface causing space charge polarization. Although, the grains are active at the high frequency \nregion but electron hopping cannot follow the frequency  of applied external electric that causes a \nreduction of the charges produced by hopping between Fe3+ and Fe2+ [75]. Therefore, the \ndecrease of ԑʹ and ԑʹʹ with increasing frequency is expected. From Fig. 8 (a), it is obvious that the \ndielectric constant is  decreased due to V substitution which is explained later.  \n Fig. 8 (b) demonstrates the unusual behavior of dielectric loss (tanδ) as a function frequency. The \ncurves tanδ vs frequency exhibit a maximum at a particular frequency but different with V \nconce ntration variation. This peak is observed to shift to the high frequency side owing to the \nincreasing V substitution. The similar abnormal behavior (of tanδ) has also been reported for V \nsubstituted Ni -Zn-Cu ferrites [41], Sn substituted Ni -Zn ferrites [65 ] and Y substituted Mg -Zn \n[68]. This type of maximum in tanδ usually occurred when the jumping frequency of electron \nhopping becomes approximately equal to that of the externally applied ac electric field  [76] \n \n       \n103104105106107108010203040 103\nFrequency,f(Hz) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(a)\n103104105106107108015304560\nFrequency,f(Hz) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(b)tan  \nFig.8:   Frequency dependence of ( a) dielectric constant and ( b) dielectric loss of \nNi0.7Zn0.2Co0.1Fe2-xVxO4(0 ≤ x ≤ 0.12) ferrites.  \n \nFrequency dependence of AC resistivity  \n \nThe frequency dependent ac resistivity ( ρac) is shown in Fig. 9(a) i n the frequency range 1 kHz -\n120 MHz.  Again, the normal behavior in the frequency dependence of ac resistivity is observed \nfor each sample. The resistivity is observed to decrease with increasing frequency and then \nbecomes invariant at high frequency [65] . Fig. 9 (a) also demonstrates that the ac resistivity is \nnoted to increase with V contents as shown in Fig. 9 (b). It is seen from Fig. 8 (a) and 9 that the \ndependence of dielectric constant and ac resistivity on V contents follows the opposite trend that  \nis normal trends in ferrites. The dependence of dielectric constant or ac resistivity on V contents \ncan be explained by assuming two mechanisms. Firstly: The V is substituted for Fe3+ ions at the \nB-sites that lead to the decrease of electronic hopping bet ween Fe3+ and Fe2+ occurs at B -sites \ndue to reduced Fe3+ ions. Thus, the decrement of electronic conductivity attributed from the \nelectronic hopping mechanism is expected. Such type of reducing electronic conductivity is also \nreported by means of Fe3+ ions replacement with ions that have a tendency to occupy the B -sites [65]. Secondly: The decrement of grain sizes with V contents also contributed to the enhanced \nresistivity. The FESEM micrographs shows that the average grains size is observed to be \ndecrease d [Table 1] with V contents that leads to the increased in number of grain boundaries, \nthe more grain boundaries results more insulating barriers in the way of charge carriers [69]. \nConsequently, the decrease of grain size also revealed the reduction of th e conductive area. [77 ] \nHence, the electrical resistivity is expected to be increased owing to V substitutions as shown in \nFig. 9 (b).  \n     \n103104105106107108015304560ac107(-cm)\nFrequency,f(Hz) x=0.00\n x=0.02\n x=0.05\n x=0.07\n x=0.10\n x=0.12(a)\n0.00 0.04 0.08 0.120204060 ac107(-cm)\nV content (x)(b)  \nFig. 9: (a) Frequency dependence  and ( b) V content( x) depende nce of ac resistivity of \nNi0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites.  \n \n \nConclusion  \nNi0.7Zn0.2Co0.1Fe2-xVxO4 (0 ≤ x ≤ 0.12) ferrites have been synthesized by conventional ceramic \ntechnique. The decrease of lattice constant with V content is observed. A good correlation is \nobserved among lattice consta nt, density and porosity for different V contents. Average grains \nsize is found to decrease with V contents. The formation of spinel cubic ferrites is confirmed \nfrom the obtained values of vibrational frequency υ 1 (in the range: 574 to 590 cm-1) and υ 2 (in the \nrange: 364 to 382 cm-1). The soft ferromagnetic nature is observed from hysteresis loops with \nlow values of coercive field. The lowering of Ms (from 71.6 emu/g to 18 – 28 emu/g) is due to \nnon-magnetic V substitution. The Curie temperature obtained fro m temperature dependent \npermeability is observed to decrease owing to V substitution by means of decreasing the strength \nof exchange coupling constant. The study of the frequency dependent permeability exhibits a \ngood correlation among the permeability, av erage grain size and saturation magnetization. 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Shavit1 \n \n1 Microwave Magnetic Laboratory, \nDepartment of Electrical a nd Computer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \n2 Department of Electrical and E lectronics Engineering, Shamoon College of \nEngineering, Beer Sheva, Israel \n \nJanuary 17, 2017 \n \nAbstract \n \nMagnetic-dipolar modes (MDMs) in a quasi-2D ferrite disk are mi crowave energy-\neigenstate oscillations with topol ogically distinct structures of rotating fields and \nunidirectional power-flow circulat ions. At the first glance, th is might seem to violate \nthe law of conservation of an angular momentum, since the micro wave structure with \nan embedded ferrite sample is mechanically fixed. However, an a ngular momentum is \nseen to be conserved if topological properties of electromagnet ic fields in the entire \nmicrowave structure are taken in to account. In this paper we sh ow that due to the \ntopological action of the azimuth ally unidirectional transport o f  e n e r g y  i n  a  M D M -\nresonance ferrite sample there exists the opposite topological reaction on a metal screen \nplaced near this sample. We ca ll this effect topological Lenz’s  effect. The topological \nLenz's law is applied to opposite topological charges: one in a  ferrite sample and \nanother on a metal screen. The MDM-originated near fields – the  magnetoelectric (ME) \nfields – induce helical surface el ectric currents and effective  charges on a metal. The \nfields formed by these currents and charges will oppose their c ause. \n \nI. INTRODUCTION \n \nIn dielectric microcavities, when sizes are comparable with a w avelength inside a \ndielectric material, the rotati on modifies the cavity resonance s. The electromagnetic fields \nin a rotating resonant microcavity are subject to the Maxwell e quations generalized to a \nnoninertial frame of reference in  uniform rotation. Due to a no n-electromagnetic torque, \nthe path length of clockwise (CW) propagating light is differen t from that of counter-\nclockwise (CCW) propagating light. As a result, one has phase d ifference or frequency \nsplitting between counter propagating beams in a loop. This is the Sagnac effect in optical \nmicrocavities [1, 2]. The phase and frequency difference betwee n CW and counter-\nclockwise CCW propagating beams are proportional to the angular  velocity. In the case \nof the two-dimensional (2D) disk dielectric resonator rotating at angular velocity , the \nresonances can be obtained by solvi ng the following stationary wave equation [2]: \n 2\n                                    22\n22\n22 21120ik n krr r r c        ,                       (1) \n \nwhere  is an electric field component. The solutions for ,r is given as \n ,imrf r e , where m is an integer. When the disk  cavity is rotating, the wave \nfunction is the rotating wave ()im\nmmJKre, where 22 22( )mK nk k cm  . The frequency \ndifference between the counter-p ropagating waves is equal to 22( )mn . For a \ngiven direction of rotation, the C W and CCW waves inside a diel ectric cavity experience \ndifferent refraction index n. Their azimuthal numbers are m.     \n      In the case of effects in rotating magnetic samples, very  specific properties of \ngeometrical phases become evident. In Ref. [3] it was shown tha t the sample rotation \ninduces frequency splitting in nuc lear-quadrupole-resonance (NQ R) spectra. For the \nrotation frequency much less th an the NQR characteristic freque ncy, the observed \nsplitting in this magnetic resonance experiment can be interpre ted in two ways: ( a) as a \nmanifestation of Berry’s phase, associated with an adiabaticall y changing Hamiltonian, \nand ( b) as a result of a fictitious magnetic field, associated with a  rotating-frame \ntransformation. The main standpoin t is that the dynamical phase  evolution in a magnetic \nstructure is unaffected by the rotational motion and any rotati onal effects must arise from \nBerry’s phase. Another studies on the rotational effects in mag netic samples are devoted \nto rotational Doppler effect observed in ferromagnetic-resonanc e (FMR) rotating \nmagnetic samples. Frequencies of the FMR are typically in the g igahertz range, which is \nfar above achievable angular vel ocities of mechanical rotation of macroscopic magnets. \nIn [4, 5] it was discussed that a rotating ferromagnetic nanopa rticle (having a very small \nmoment of inertia) can be a resona nt receiver of the electromag netic waves at the \nfrequency of the FMR. If such a receiver is rotating mechanical ly at an angular velocity \n, one has splitting of the FMR frequency. The frequency of the wave perceived by the \nreceiver equals ()n\nFMR  ,  where FMR  is the frequency of the ferromagnetic \nresonance, n\nI, I is the moment of inertia of a nanoparticle, and 0. 1, 2, 3,...n  \nThe experimental results for the rotational motion of solid fer romagnetic nanoparticles \nconfined inside polymeric cavitie s give evidence for the quanti zation [5].  \n      While in the above studies, specific wave effects are obs erved due to mechanical \nrotation of samples, there exists evidence for azimuthally unid irectional wave propagation \nin mechanically fixed magnetic samples. The multiresonance spli tting in the FMR spectra \nwas observed in mechanically non-rotating macroscopic quasi-2D ferrite disks with \nmagnetic-dipolar-mode (MDM) (or magnetostatic-wave (MS-wave)) o scillations. Such \noscillations in a ferrite disk are rotating eigenmodes with azi muthally unidirectional \ntransport of energy [6 – 11]. In the MDM ferrite disks, the Ber ry’s phase is generated from \nthe broken dynamical symmetry [6]. It was shown that when a fer rite disk is loaded by a \ndielectric sample, the ferrite is under the reaction torque and  the magnetization motion is \ncharacterized by a fictiti ous magnetic field [10, 11]. \n     MDM oscillations in small (with sizes much less than the m icrowave radiation \nwavelength) ferrite spheres excited by external microwave field s were experimentally \nobserved, for the first time,  by White and Solt in 1956 [12]. A fterward, experiments with 3\nsmall ferrite disks revealed uni que spectra of s uch oscillation s. While in a case of a small \nferrite sphere one observes only a few and very wide absorption  p e a k s  o f  M D M  \noscillations, for a small quasi-2D ferrite disk there is a mult iresonance (atomic-like) \nspectrum with very sharp resonance peaks [13 – 15]. Analyticall y, it was shown [16, 17] \nthat, contrary to spherical geometry of a ferrite particle anal yzed in Ref. [18], the quasi-\n2D geometry of a ferrite disk gives the Hilbert-space energy-st ate selection rules for MDM \nspectra. MDM oscillations in a qua si-2D ferrite disk are macros copically quantized states. \nLong range dipole–dipole correlation in position of electron sp ins in a ferromagnetic \nsample can be treated in terms of  collective excitations of the  system as a whole. When a \nferrite-disk resonator with an energy spectrum is subjected to a weak action by an external \nmicrowave field, its energy levels do not change (change slight ly, to be more precise). \nHowever, due to such a weak external action, the entire structu re of precessing elementary \nmagnetic dipoles acquires an orbital angular momentum. The MS-p otential wave function \nin a ferrite disk with the  disk axis oriented along z axis, is written as [6 – 11]: \n \n                                                    ()(, )Cz r ,                                                      (2) \n                                          \nwhere  is a dimensionless membrane function, r and  are in-plane coordinates of a \ncylindrical coordinate system, ()z  is a dimensionless function of the MS-potential \ndistribution along z axis, and C is a dimensional amplitude coefficient. Being the energy-\neigenstate oscillations, the MDMs in a ferrite disk also are ch aracterized by topologically \ndistinct structures of the fields. This becomes evident from th e boundary condition on a \nlateral surface of a ferrite disk of radius , written for a membrane wave function [6 – \n11]: \n \n                                     0a\nrr rirr                \n  .                           (3)  \n \nHere,  and a are, respectively, diagonal and off-diagonal components of the  \npermeability tensor  .  One can compare Eq. (3) with Eq. (1). In both cases, there are \nthe terms with the first-order derivative of the wave function with respect to the azimuth \ncoordinate. In a case of the MS-potential wave solutions, one c an distinguish the time \ndirection (given by the direc tion of the magnetization precessi on and correlated with a \nsign of a) and the azimuth rotation direction (given by a sign of \n\n). For a given sign \nof a parameter a, there are different  MS-potential wave functions, () and (), \ncorresponding to the positive and negative directions of the ph ase variations with respect \nto a given direction of azimuth coordinates, when 02 . So a function  is not a \nsingle-valued function. I t changes a sign when  is rotated by 2 [6]. \n     Similar to the Sagnac effect in optical microcavities, MDM  oscillations in a quasi-2D \nferrite disk are described by th e Bessel-function azimuthally r otating waves. There are \nmicrowave-frequency rotating fiel d configurations with power-fl ow vortices. Such fields \nwith the azimuthally unidirectional energy transport around the  disk axis are observed \nboth inside and outside a ferrite sample. For a given direction  of a bias magnetic field, the 4\npower-flow vortices are the same in the vacuum near-field regio n above and below the \ndisk. Fig. 1 shows schematically such power flow rotations. The  fact of the presence of \nthe power flow circulations in the planes inside the ferrite an d in vacuum above and below \nthe disk arises a question on mechanical stability of the ferri te sample. Really, the disk \nbody should sense the impact of the torque due to power-flow ci rculations and so should \nrotate mechanically. However, in all of the studies of MDM osci llations in a ferrite disk \n[6 – 1, 13 – 15], the entire microwave structure with an embedd ed ferrite sample, is \nmechanically fixed.  \n     The field orbital rotation (and the power-flow circulation s related to this field rotation) \nare due to topological effects in MDM oscillations. The fields are orbitally driven to \nsatisfy properly the boundary conditions on a surface of a ferr ite disk. Because of path-\ndependent interference, we observe topological-phase effects in  the near-field region of \nthe ferrite disk [6 – 11]. Neverthe less, the entire system (the  microwave structure with an \nembedded ferrite disk) should be i ntegrated and so, the system should be under the \nangular-momentum-balance conditi ons. In other words, there shou ld exist conditions for \nneutrality of the vortex topologi cal charges. Such the angular momentum balance can be \nrealized only when the opposite power flow circulations  are created on metal walls of the \nmicrowave structure, in which the ferrite disk is embedded (see  Fig. 2). It means that at \nthe MDM resonances in a ferrite  disk, specific  (topologically d istingushed) conductivity \nelectric currents should be induced on the metal parts of a mic rowave structure. The entire \nmicrowave structure, external to a ferrite disk, should “work o ut” the situation to give the \npossibility for integration of t he entire Maxwell-equation syst em. The recoil from the \nmetal parts of the microwave structure is not similar to the cl assical Lenz’s law. It takes \nplace via topological effects on a  metal surface: creation of e ffective charges and surface \nchiral currents and recoil ME fields. There is the “topological  r e c o i l ” .  T h e  c o u p l i n g  \nbetween an electrically neutral M DM ferrite disk and an electri cally neutral metal objects \nshould involve the electromagne tic-induction and Coulomb intera ction through virtual \nphotons mediation. Moreover, there is the discrete-state proces s related to the quantization \nproperties of MDM oscillations, in which quantized virtual fiel d excitations appear in \nvacuum. \n     Why there is a topological nat ure of the rotating fields i n a MDM ferrite disk? A ferrite \nis a magnetic dielectric with low losses. This may allow for el ectromagnetic waves to \npenetrate the ferrite and results  in an effective interaction b etween the electromagnetic \nwaves and magnetization within the ferrite. Such an interaction  can demonstrate \ninteresting physical features. Because of gyrotropy, electromag netic waves incident on a \nferrite-dielectric interface have  reflection symmetry breaking.  In Ref. [19], different cases \nof reflection of electromagnetic waves from a ferrite-dielectri c interface were studied. It \nwas shown that in a certain range of the material parameters, a  frequency region, and a \nregion of a bias magnetic field, the phase shift on total refle ction of electromagnetic waves \nfrom a lossless ferrite is nonreciprocal and magnetically tunab le. The reflection of \nmicrowaves at a dielectric-ferrite interface from the view poin t of ray and energy \npropagation shows [20, 21] that fo r a given direction of a bias  magnetic field one has \nunidirectional energy transport along the interface. Thus, on a  lateral surface of a normally \nmagnetized ferrite disk, one has a n azimuthally nonreciprocal p h a s e  s h i f t  o f  \nelectromagnetic waves reflected from a ferrite (see Fig.3). The  MDM oscillations in a \nferrite-disk sample give evidence for quantized effects of inte raction between the 5\nelectromagnetic waves and magnetization within the ferrite. In this case a nonreciprocal \nphase at the orbit circulation on a lateral surface is quantize s as well.  \n     In the present paper, we anal yze numerically the interacti on of an MDM ferrite disk \nwith a metal screen. We show that to the topological action of the azimuthally \nunidirectional transport of ene rgy in a MDM-resonance ferrite s ample there exists the \nopposite topological reaction on a metal screen placed near thi s sample. We call this effect \ntopological Lenz’s effect. The topological Lenz's law is applie d to opposite topological \ncharges: one in a ferrite sample and another on a metal screen.  The MDM-originated near \nfields – the magnetoelectric (ME) fields – induce helical surfa ce electric currents and \neffective charges on a metal. The ME fields formed by these cur rents and charges will \noppose their cause.  \nII. RESULTS \n \nCompared to our previous nume rical studies [ 7, 9 – 11, 22, 23] of MDM oscillations in a \nferrite disk placed in standard microwave rectangular waveguide s, in this work we use \nextremely thin rectangular wave guide. In such a structure, the waveguide metal walls are \nsituated very close to the ferrite-disk surface. We also study a microwave microstrip \nstructure with an embedded MDM ferrite disk. In the structures under consideration we \nare able to consider the angular-momentum balance conditions se parately from the \nbalance for a linear momentum of propagating waves. \n     In the analysis, we use the same disk parameters as in Ref s. [7, 9 – 11, 22, 23]: The \nyttrium iron garnet (YIG) disk has a diameter of \n23  m m  and the disk thickness is t \n= 0.05 mm; the disk is normally magnetized by a bias magnetic f ield H0 = 4900 Oe; the \nsaturation magnetization of the ferrite is 4π Ms = 1880 G. For better understanding the \nfield structures, we assume in our numerical studies that a fer rite disk has very small \nlosses: The linewidth of a ferrite is H= 0.1 Oe. A ferrite disk is placed inside a thin \nTE10-mode rectangular waveguide symmetrically to its walls so that the disk axis is \nperpendicular to a wide wall of a  waveguide. The waveguide wall s are made of a perfect \nelectric conductor (PEC). A rectangular waveguide cross-section  sizes are a = 22.86 mm, \nb = 1 mm. Fig. 4 shows a thin rectangular waveguide with an embe dded quasi-2D ferrite \ndisk. The entire microwave structure (a ferrite disk and a wave guide) can be considered \nas a quasi-2D structure. \n     Figure 5 shows the module of t he reflection and transmissi on coefficients (the S 11 and \nS21 scattering-matrix parameters, respectively). The resonance peak s are designated in \naccordance with the mode classification used in Ref. [17]. This  classification shows the \nnumber of variations of the MS- potential function in a ferrite disk with respect to azimuth \nand radial coordinates. In the present analysis we use the 1st and 2nd radial modes, i. e. the \nmodes with one and two radial variations of the MS-potential fu nction, respectively. For \nboth these modes, the azimuth number is equal to 1. Fig. 6 show s the Poynting vector \ndistributions for the 1st radial mode on the upper plane o f a ferrite disk for two direc tions \nof the electromagnetic wave propagation in a rectangular wavegu ide and at the same \ndirection of a bias magnetic field. In a view along zdirection, one clearly sees the CCW \npower-flow vortex. We will classify this vortex by the topologi cal charge 1 Q . On a \nmetal screen (the upper wall of a waveguide), we have the CW po wer-flow vortex, which \ncan be classified by the topological charge 1 Q (see Fig. 7).  The view is along z\ndirection on an upper waveguide wall from the outside metal reg ion. A vacuum gap 6\nbetween the upper plane of a ferr ite disk and the upper wall of  a waveguide is sufficiently \nnarrow. On vacuum planes inside such a gap, we can see the effe ct of counteraction of the \ntwo topological charges. On a cer tain vacuum plane inside this gap, the condition of \n“topological neutrality” should ex ist. The power-flow distribut ion on a vacuum plane 20 \num above the upper plane of a ferrite disk in Fig. 8, shows tha t the vortex structure, \noriginated from a ferrite dis k, becomes much emasculated.  \n     The fields in vacuum near a MDM ferrite disk – the ME fiel ds – are determined by the \nmagnetization distributions. The vector field of RF magnetizati on m in a ferrite disk has \ntwo parts: the potential ( 0m ) and curl ( 0m) ones. While the magnetic \ncomponent of the ME field is originated from the potential part  of the magnetization, the   \nelectric component of the ME field is originated from the curl part of the magnetization \n[10]. An analytical expression for magnetization m is given in Ref. [24]. For the known \ndisk parameter, we calculated analytically the magnetization fo r the 1st radial mode. The \ndistribution shown in Fig. 9 give s evidence for the spin and or bital angular momentums \nof the magnetization field. In t he orbitally driven distributio n of m, one can clearly \nobserve the regions of magnetization vortices. At the same time , it is evident that the fields \nin vacuum near a metal wall are related to the electric current  distributions on a metal \nsurface. Fig. 10 shows the orbita lly driven electric current di stributions on the upper wall \nof a waveguide for the 1st rad ia l m o d e.  In  th i s case,  th e re  ex is t th e r e g io n s o f  cu r ren t  \nvortices. For the present study, it is worth noting that the in -plane rotating distributions in \nFigs. 9 and 10 are very similar topologically one to another.  \n     The structures of the fields of a MDM ferrite disk were st udied numerically and \nanalytically in Refs. [7, 9 – 11, 22, 23, 25]. The Poynting-vec tor vortices of such fields on \na disk plane are formed by the m agnetic-field components normal  to the disk surface and \nthe electric-field components tangential to the disk surface: (ferrite) (ferrite)\ntnEH\n. Because of \nthe presence of surface electric charges and electric currents induced on a closely situated \nmetal wall by the fields originated from MDM oscillations, the Poynting-vector vortices \nof on a screen are formed by the electric-field components norm al to the metal surface \nand the magnetic-field components tangential to the metal surfa ce: (metal) (metal)\nntEH\n. The \nfield structures shown in Fig.11, can explain qualitatively why  the power-flow circulations \non a plane of a ferrite disk a nd on a metal screen are opposite ly directed.  \n     The topological nature of the observed effect becomes clea r when one analyzes more \nin details the electric and magnetic fields on a metal wall. Fi g. 12 shows the magnetic field \ndistribution for the 1st radial mode on the upper wall of a waveguide. One can see spec ific  \nregions of surface topological magnetic charges (STMCs). Simila r STMCs were observed \nin previous studies when a MDM ferrite disk was placed inside a  standard (“thick”) \nwaveguide [9, 22]. The STMCs are points of divergence and conve rgence of a 2D \nmagnetic field (or a surface magnetic flux densitySB\n) on a waveguide wal l. As is evident \nfrom Fig. 12, one has nonzero outward (inward) flows of the vec tor field SB\n through a \nclosed flat loop  lying on the metal wall and surrounding the points of divergen ce or \nconvergence: 0SSBn d\u0000\n . Here Sn is a normal vector to contour , lying on a metal \nsurface. At the same time, it is clear, however, that 0SSB\n, since there  are zero \nmagnetic fields at the points of divergence or convergence. Suc h topological singularities 7\non the metal waveguide wall show unusual properties. One can se e that, for the region \nbounded by the circle , no planar variant of the divergence theorem takes place. It i s \nworth noting also that the STMC s are observed exactly at the sa me places on a metal wall \nwhere the centers of the electri c current vortices are situated  (see Fig. 10). \n     Distributions of the magnetic-  and electric-field componen ts, on the metal surface, are \nalso orbitally driven with the same angular velocity as other d istributions (such as \nmagnetization in a ferrite and surface magnetic current on a wa ll). Importantly, both these, \nelectric and magnetic, fields have their maximums (minimums) ex actly at the same \nazimuth coordinates. This result in appearance very interesting  topological structures on \na metal wall originated from the MDM resonances in the disk. Th e fields on an upper \nmetal wall for the 1st radial mode, shown in Fig. 13, give evidence for clearly disti nguished \nregions of the surface electric and topical-magnetic charges. T he “color” picture shows a \ndistribution of a normal component of the electric field (metal)\nnE\n while the “arrow” picture \nshows a distribution of a tangential component of the magnetic field field (metal)\ntH\n. The \ntime phase of 255t corresponds to the case when the electric and topical-magnetic  \ndipoles are perpendicular to the waveguide axis. The distributi ons in Fig. 13 show that on \na metal wall there is a “glued pair” of 2D rotating dipoles: th e electric and topological \nmagnetic ones. To a certain extent (in our case, there is a 2D,  topological, and rotating \nstructure) we have an analog of  a Tellegen particle [26, 27].  \n     On a surface of a MDM ferrite disk, there are both the reg ions of the orbitally driven \nnormal magnetic field nB\n and normal electric field nE\n. Importantly, the maximums \n(minimums) of the rotating fields nE\n and nB\n are situated at the same places on a disk \nplane [7, 10, 25]. This is illust rated in Fig. 14 for a certain  time phase t. This field \nstructure is projected on a metal wall closely placed to a ferr ite disk. Together with surface \nelectric charges S induced on a metal wall by the electric field nE\n, there also the \nFaraday-law eddy currents induc ed on a metal surface by the tim e-derivative of a normal \ncomponent of the rotating MDM magnetic field nB\n. Evidently, the induced surface \nelectric current Sj\n should have two components. There are the linear currents aris ing from \nthe continuity equation for surface electric charges, for which  0SSj\n, and the \nFaraday-law eddy currents, for which 0SSj\n. As a result, we have the sources of the \nlinear-current and eddy-current c omponents situated at the same  place on a metal wall. \nThus, the form of a surface electric current is not a closed li ne. It should be a spiral. This \nform of a surface electric current can be traced on the picture s in Fig. 10. More clearly, \nthe current lines, modeled by the spirals, are shown in Fig. 15 . There are both the right-\nhanded and left-handed flat spirals.     The effect of the angular-momentum opposite topological rea ction on a metal screen \nfor the 2\nnd radial mode in a “thin” waveguide with a ferrite sample is sho wn in Ref. [24]. \nAlso, the angular-momentum-balance condition has been proven in  a microwave \nmicrostrip structure with an e mbedded MDM ferrite disk [24]. \n  \nIII. DISCUSSION AND CONCLUSION \n 8\nLong-range magnetic-dipolar interactions in confined magnetic s tructures are not in the \nscope of classical electromagnetic problems and, at the same ti me, have properties \nessentially different from the effects of exchange ferromagneti sm. The MDM spectral \nproperties in confined magnetic structures are based on postula tes about physical meaning \nof the magnetostatic (MS) potential function (,)rt as a complex scalar wave function, \nwhich presumes long-range (on the  s c a l e s  m u c h  b i g g e r  t h a n  t h e  e xchange-interaction \ns c a l e s )  p h a s e  c o h e r e n c e .  T h e  M D M s  i n  a  q u a s i - 2 D  f e r r i t e  d i s k  a r e characterized by \nenergy-eigenstate orthogonality r elations. Due to topological s tates on a lateral surface of \na ferrite disk, these modes have orbital angular momentums. In a vacuum subwavelength \nregion abutting to a MDM ferrite disk, one can observe the quan tized-state power-flow \nvortices. In such a region, a c oupling between the time-varying  electric and magnetic \nfields is different from such a coupling in regular electromagn etic fields. These specific \nnear fields, originated from MDM oscillations, we term magnetoe lectric (ME) fields. The \nME field solutions give evidence for spontaneous symmetry break i n g  a t  t h e  r e s o n a n t  \nstates of MDM oscillations [10]. As a source of the ME field, t here is the pseudoscalar \nparameter of magnetization helicity. This parameter, appearing since the magnetization \nm in a ferrite disk has two parts:  the potential and curl ones, is calculated as \n \n                                        * *() ( )Im1RemeVm m j jc             ,                         (4)  \n \nwhere ()mj im and ()ejc m   are, respectively, the ma gnetic and electric current \ndensities in a ferrite medium, c is the light velocity in vacuum [10, 27, 28]. For the MS-\npotential wave function in a ferrite disk represented by Eq. (2 ), the pseudoscalar parameter \nV was derived analytically in [24]. This parameter, analytically  calculated for the 1st radial \nmode (with normalization of t he mode amplitude coefficient C [25]), is shown in Fig. 16. \nThe pseudoscalar parameter V gives evidence for the presence of two coupled and \nmutually parallel currents – the electric and magnetic ones – i n a localized region of a \nmicrowave structure. This is the cause of appearance of flat-sp iral currents and a “glued \npair” of 2D rotating dipoles on a metal wall.       In a microwave structure with an embedded ferrite disk, an  orbital angular momentum, \nrelated to the power-flow circul ation, must be conserved in the  process. Thus, if power-\nflow circulation is pushed in one  direction in a ferrite disk, then the power-flow circulation \non metal walls to be pushed in t he other direction by the same torque at the same time. In \nthe present paper, we analyzed numerically the interaction of a n MDM ferrite disk with a \nmetal screen. Our numerical studies give evidence for the fact that to the topological action \nof the azimuthally unidirectiona l transport of energy in a MDM- resonance ferrite sample \nthere exists the opposite topolog ical reaction on a metal scree n placed near this sample. \nWe call this effect topological Len z’s effect. The topological Lenz's law is applied to \nopposite topological charges: one in a ferrite sample and anoth er on a metal screen. The \nMDM-originated near fields – the magnetoelectric (ME) fields – induce helical surface \nelectric currents and effective charges on a metal. The ME fiel ds formed by these currents \nand charges will oppose their cause. As we have shown, the in-p lane magnetization \ndistribution in a ferrite disk and the surface current distribu tion on a metal wall are very \nsimilar topologically one to another. These ME sources, in a fe rrite and on a metal wall, \nresult in appearance of the ME field in the vacuum gaps. To cha racterize the ME properties 9\nof the field, we use distribu tion of the normalized helicity fa ctor in a vacuum region. The \nnormalized helicity fac tor, calculated as \n  \n                                                        *Re\ncosEH\nEH\n\n ,                                                  (5)  \n \ngives evidence to the fact that the electric and magnetic field s in vacuum are not mutually \nperpendicular [10, 22, 23, 25]. Th e normalized helicity factor distribution on the xz cross-\nsectional plane of the vacuum gap, numerically calculated for t he fields of the 1st radial \nmode, is shown in Fig. 17. One can see that a maximal angle bet ween the electric and \nmagnetic fields in vacuum is about 70. \n     In the present study, we used microwave waveguiding struct ures with the metal walls \nsituated very close to the ferrite-disk surfaces. The vacuum ga ps between the metal and \nferrite are much less than a diameter of a MDM ferrite disk and  thus the entire microwave \nstructure (a ferrite disk and a waveguide) can be considered as  a quasi-2D structure. In the \nshown distributions, the regions of the power-flow circulations  in a ferrite disk and on \nmetal wall are, in fact, opposite one another and the angular-m omentum balance does not \ndepend, actually, on the directio n of the electromagnetic wave propagation in a \nwaveguide. In this case, we have exchange of two types of virtu al photons: ( a) the static \nelectric force and ( b) the electromagnetic induction. The observed properties rely o n ME \nvirtual photons to act as the mediator.          However, the fields in a ferr ite disk rotate at microwave  frequencies and situation \nbecomes more complicated when the vacuum region scale is about the disk diameter or \nmore and the finite speed of wave propagation in vacuum – the r etardation effects – should \nbe taken into consideration. This means that for a brief period  the total angular  momentum \nof the two topological charges (one in a ferrite, another on a metal) is not conserved, \nimplying that the difference should be accounted for by an angu lar momentum in the fields \nin the vacuum space. The magnetization dynamics have an impact on the phenomena \nconnected with fluctuation energy in vacuum. Because of MDM res onances, this angular \nmomentum is quantized. As the rot ational symmetry is broken in this case, the Casimir \ntorque [29 – 33] arises because the Casimir energy now depends on the angle between the \ndirections of the magnetization vectors in a ferrite and electr ic-current vectors on a metal \nwall. A vacuum-induced Casimir torque allows for torque transmi ssion between the ferrite \ndisk and metal wall avoiding any di rect contact between them.  \n \nReference \n \n[1] E. J. Post, Rev. Mod. Phys. 39, 475 (1967). \n[2] S. Sunada and T. Harayama, Phys. Rev. A 74, 021801(R) (2006). \n[3] R. Tycko, Phys. Rev. Lett. 58, 2281 (1987). \n[4] S. Lendínez, E. M. Chudnovsky, and J. Tejada, Phys. Rev. B 82, 174418 (2010). \n[5] J. Tejada, R. D. Zysler, E. Molins, and E. M. Chudnovsky, P hys. Rev. Lett. 104, \n027202 (2010). \n[6] E. O. Kamenetskii, J. Phys. A: Math. Theor. 40, 6539 (2007). \n[7] M. Sigalov, E. O. Kamenetskii, and R. Shavit, J. Phys.: Con dens. Matter 21, 016003 \n(2009). 10\n[8] E. O. Kamenetskii, J. Phys.: Condens. Matter 22, 486005 (2010). \n[9] E. O. Kamenetskii, R. Joffe, R. Shavit, Phys. Rev. A 84, 023836 (2011). \n[10] E. O. Kamenetskii, R. Jo ffe, R. Shavit, Phys. Rev. E 87, 023201 (2013). \n[11] R. Joffe, E. O. Kamenetskii , and R. Shavit, J. Appl. Phys.  113, 063912 (2013). \n[12] R. L. White, I. H . Solt Jr, Phys. Rev. 104, 56 (1956). \n[13] J. F. Dillon Jr, J. Appl. Phys. 31, 1605 (1960). \n[14] T. Yukawa, K. Abe, J. Appl. Phys. 45, 3146 (1974). \n[15] E. O. Kamenetskii, A. K. S aha, I. Awai, Phys. Lett. A 332, 303 (2004). \n[16] E. O. Kamenetskii, Phys. Rev. E 63, 066612 (2001). \n[17] E. O. Kamenetskii, M. Sigalov, R. Shavit, J. Phys.: Conden s. Matter 17, 2211 (2005). \n[18] L. R. Walker, Phys. Rev. 105, 390 (1957). \n[19] N. C. Srivastava, J. Appl. Phys. 49, 3181 (1978). \n[20] S. S. Gupta and N. C. S rivastava, Phys. Rev. B 19, 5403 (1979);  \n[21] B. H. Renard, J. Opt. Soc. Am. 54, 1190 (1964). \n[22] M. Berezin, E. O. Kamenet skii, and R. Shavit, J. Opt. 14, 125602 (2012). \n[23] M. Berezin, E. O. Kamenetsk ii, and R. Shavit, Phys. Rev. E  89, 023207 (2014). \n[24] See Supplementary Information.  [25] R. Joffe, R. Shavit, and E. O. Kamenetskii, J. Magn. Magn.  Mater. 392, 6 (2015). \n[26] In 1948 Tellegen [Philips Res. Rep. 3, 81 (1948)] suggested that an assembly of the \nlined up electric-magnetic dipol e twins can construct a new typ e of an electromagnetic \nmaterial. Until now, however, the problem of realization of a s tructural element of \nsuch a material – the Tellegen particle – is a subject of stron g discussions (see e. g. \n[27] and references therein).  \n[27] E. O. Kamenetskii, M. Sig alov, and R. Shavit, J. Appl. Phy s. 105, 013537 (2009). \n[28] J. D. Jackson, Classical Electrodynamics , 2\nnd ed. (Wiley, New York, 1975). \n[29] C.-G. Shao, A.-H . Tong, and J. Luo, Phys. Rev. A 72, 022102 (2005). \n[30] J. N. Munday, D. Iannuzzi, F. Capasso, New J. Phys. 8, 244 (2006). \n[31] R. B. Rodrigues, P. A. Maia Neto, A. Lambrecht, and S. Rey naud, Europhys. Lett. \n76, 822 (2006). \n[32] R. Guérout, C. Genet, A. Lambrecht, and S. Reynaud, Europh ys. Lett. 111, 44001 \n(2015). \n[33] D. A. T. Somers a nd J. N. Munday, Phys. Rev. A 91, 032520 (2015). \n \n \n   \n \n   \n \n    \n 11\n \nFig. 1. Power flow rotat ions. For a given di rection of a bias m agnetic field, the power-\nf low  circula tions ar e th e sam e inside a f errite and in the vacu um near-field regions \nabove and below the disk.   \n \n \nFig. 2. Angular momentum balance conditions: In a view along z axis, there are opposite \npower flow circulations on a surface of a ferrite disk and on a  metal surface. At the \nsame time, in a view along y axis, both power flow circulatio ns are clockwise rotations.  \n   \n \nFig. 3. The topological nature of the rotating fields in a MDM ferrite disk. On a lateral \nsurface of a normally magnetized ferrite disk, one has an azimu thally nonreciprocal \nphase shift of electromagnetic wav es reflected from a ferrite. \n  \n12\n \nFig. 4. A thin rectangular waveguide with an embedded quasi-2D ferrite disk. An insert \nshows the disk position inside a waveguide.  \n \n                                                                     (a) \n \n                                                                         ( b) \n Fig. 5. The MDM spectra in a waveguide. ( a) Reflection coefficient; ( b) transmission \ncoefficient.  \n13\n \n                                  ( a)                                                                      ( b)  \nFig. 6. Power-flow vortex for the 1st radial mode on the upper plane of the disk in a \nview along zdirection. ( a) Electromagnetic wave propagation in a waveguide from \nport 1 to port 2  (1  2) , ( b) wave propagation fro m port 2 to port 1  (2 1). A black \narrow clarifies the power-flow direction. A bias magnetic field  is directed along z \ndirection. The vortex is classif ied by the topological charge 1 Q.  \n \n    \n                                  ( a)                                                                  (b)  \nFig. 7. Power-flow vortex for the 1st radial mode on the upper wall of a waveguide. ( a) \nWave propagation 1  2, (b) wave propagation 2 1. A black arrow clarifies the \npower-flow direction. A bias ma gnetic field is directed along z direction. The view \nis along zdirection on an upper waveguide w all from the outside metal reg ion. The \nvortex is classified by t he topological charge 1 Q.  \n \n \n                                  ( a)                                                                    (b)  \nFig. 8. Power-flow distribution 1st radial mode on a vacuum plane 20 um above the \nupper plane of a ferrite disk. ( a) Wave propagation 1  2 , (b) wave propagation 2 1. \nA bias magnetic field is directed along z direction. The view is along z direction. \nThe vortex structure with the charge 1 Q is much emasculated.  \n \n \n14\n \n                            ( a)                                        ( b)                                         ( c) \nFig. 9. The magnetization distributions in a ferrite disk for t he 1st radial mode. Wave \npropagation 1  2. A bias magnetic field is directed along z direction. ( a) Time \nphase 0t, (b) 90t, (c) 180t. The regions of magnetization vortices are \nclearly observed. \n     \n                                     ( a)                                                                   (b) \n \n                                                                      ( c) \nFig. 10. Surface electric current for the 1st radial mode on the upper wall of a waveguide. \nWave propagation 1  2. A bias magnetic field is directed along z direction. ( a) \nTime phase 0t, (b) 90t, (c) 180t. One can see the current vortices. \n \n \n                             ( a)                                                                          ( b) \n \n15\nFig 11. The fields on lines of t he power flow circulations. ( a) On the ferrite-disk plane, \n(b) on the metal-wall plane. The angular-momentum vectors  (ferrite) (ferrite)\ntn rE H  and \n(metal) (metal)\nnt rE Hare oppositely directed. \n \n \n                                      ( a)                                                                 (b)  \n \n \n                                                                         ( c) \nFig. 12. Magnetic field distribution for the 1st radial mode on the upper wall of a \nwaveguide. Wave propagation 1  2. A bias magnetic field is directed along z \ndirection. ( a) Time phase 0t, (b) 90t, (c) 180t. \n \n \n \nFig. 13. The orbitally driven fields on an upper metal wall for  the 1st radial mode. The \n“color” picture shows a distribut ion of a normal component of t he electric field (metal)\nnE\n \nw h i l e  t h e  “ a r r o w ”  p i c t u r e  s h o w s  a distribution of a tangential component of the \nmagnetic field field (metal)\ntH\n. The regions of the surface electric and topical-magnetic \n16\ncharges are clearly disti nguished. The time phase of 255t corresponds to the case \nwhen the electric and topical-magnetic dipoles are perpendicula r to the waveguide axis.  \n                           \n                                       ( a)                                                                 (b)  \nFig. 14. The magnitudes of the normal electric field nE\n (a) and normal magnetic field \nnB\n (b) on a surface of a MDM ferrite disk at a certain time phase t. The fields are \norbitally driven and mutually synchronized. The maximums (minim ums) of the rotating \nfields nE\n and nB\n are situated at the same places on a disk plane.  \n \n \n \nFig. 15. Surface electric current for the 1st radial mode on the upper wall of a waveguide \nshown schematically as the ri ght-handed and left-handed flat sp irals (red lines). \n17\n \n                             ( a)                                                                 (b)  \n \n(c) \nFig. 16. Pseudoscalar parameter *ImVm m      on the upper plane of a ferrite disk \ntheoretically calculated for the 1st radial mode. ( a) On the upper plane of a ferrite disk at \na bias field directed along z axis; ( b) on the upper plane of a ferrite disk at a bias field \ndirected along - z axis; (c) on the xz cross-sectional plane of a ferrite disk at a bias field \ndirected along z axis. \n \n \nFig. 17. The normalized helicity factor for the fields on the xz cross-sectional plane of \nthe vacuum gaps.         \n18\nSupplementary information for “Azimuthally unidirectional \ntransport of energy in magnetoel ectric fields: Topological Lenz ’s \neffect” \n \nR. Joffe 1,2, E. O. Kamenetskii 1, and R. Shavit 1 \n \n1 Microwave Magnetic Laboratory, Department of Electrical and Co mputer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \n2 Department of Electrical and El ectronics Engineering, Shamoon C ollege of Engineering, \nBeer Sheva, Israel   January 17, 2017   \nIn the supplementary material we show the effect of the angular -momentum opposite \ntopological reaction on a metal screen for the 2\nnd radial MDM in a “thin” waveguide with \na ferrite sample. We prove numerically the angular-momentum-bal ance condition in a \nmicrowave microstrip structure with an embedded MDM ferrite dis k. Also, we derive and \ncalculate analytically a pseudos calar parameter of magnetizatio n helicity. \n \nA. The 2nd radial MDM in a “thin” wave guide with a ferrite disk \n \nPower-flow vortices in a ferrite disk are observed for every MD M in the spectrum and so \nthere should exist neutrality for  the vortex topological charge s for all the modes. We \nillustrate here the angular momentum balance for the 2nd radial MDM in a “thin” \nwaveguide with a ferrite sample. Fig. 1 shows the power-flow vo rtices for the 2nd radial \nmode on the upper plane of the disk. One sees the two CCW and o ne CW power-flow \nvortices, which can be classified, successively, by three topol ogical charges: 1, 1, 1 . \nThe angular momentum balance ca n be realized only when the oppo site power flow \ncirculations are created on metal parts of a microwave waveguid e. Fig. 2 shows the power-\nflow vortices for the 2nd radial mode on the upper metal wall for two directions of the \nelectromagnetic wave propagation in a rectangular waveguide and  at the same direction \nof a bias magnetic field. The vortices are observed from the ou tside region above a \nwaveguide along z direction. One sees that now there are two CW and one CCW powe r-\nflow vortices. In succession, th e vortices are classified by th e topological charges:\n1, 1, 1 . In a vacuum gap between the upper plane of a ferrite disk and  the upper wall \nof a waveguide we have counteraction of the two topological cha rges. In Fig. 3, we can \nsee that on a vacuum plane 20 um above the upper plane of a fer rite disk the vortex \nstructure, originated from a ferrite disk, becomes emasculated.  In comparison with the \nsituation shown in Fig. 8 in the  article, this emasculation is not so evident since the fields \nof the 2nd radial mode are concentrated more close to the disk plane and so the metal \ninfluence on this level in vacuum is not so strong. \n  19\nB. The 1st radial MDM in a microwave m icrostrip structure with  \n \nFor more illustration of the angular momentum balance in microw ave waveguides with \nan embedded MDM ferrite disk, we analyze also the situation wit h a microwave microstrip \nstructure. The structure is shown in Fig. 4. There is the struc ture combined with two \nmicrostrip lines. The disk is placed between these lines on a s urface of a dielectric \nsubstrate. The thickness of a  dielectric substrate is 1.5 mm. \n     We consider the angular-momentum-balance conditions for th e 1st radial MDM. Fig 5. \nshows the power-flow distribution on a bottom plane of the disk  when a bias magnetic \nfield is directed along z direction.. In a view along z direction, there is the CCW \npower-flow circulation. We classi fy this vorte x by the topologi cal charge 1 Q . In Fig. \n6, one can see the power-flow distribution on a ground plane of  a microstrip structure at \nthe same direction of a bias magnetic field. In this case, in a  view along z direction one \nobserves the CW power-flow circ ulation. This vortex is classifi ed by the topological \ncharge 1 Q . Importantly, for a given direction of a bias magnetic field, the power-flow \ncirculation on a metal region co rresponding to the projection o f the ferrite-disk area, is the \nsame for two opposite directions of the wave propagation in the  entire microwave \nstructure.       Likewise to the surface electric current on the upper wall  of a waveguide shown in the \nArticle, the electric current d istribution on a ground plane of  a microstrip structure is \norbitally driven and has the regions of current vortices. This current, shown in Fig. 7 for \na certain time phase, is topologi cally similar to the magnetiza tion distributions in a ferrite \ndisk.  \nC. Pseudoscalar parameter of magnetization helicity \n \nMDM oscillations are describe d by a MS-potential wave function,  which is related to the \nRF magnetic field as \nH\n [S1 – S3]. The MS-potential wave function in a ferrite \ndisk with the disk axis oriented along z axis, is written as: ()(, )Cz r , where  is \na dimensionless membrane function, r and  are in-plane coordina tes of a cylindrical \ncoordinate system, ()z  is a dimensionless function of the MS-potential distribution \nalong z a x i s ,  a n d  C is a dimensional amplitude coefficient. The magnetization m i s  \ndefined as m , where  is the magnetic susceptibility  tensor. For a ferrite disk \nmagnetized along z axis we have [S2, S3] \n \n                                                     0\n0\n00 0a\nai\nmi\n\n  \n .                                      (S1)                          \n \n     Assuming that MDMs are the fields rotating with an azimuth  number  ie , \nwe obtain \n                                         20\n      0\nˆ ˆ 0( )\n00 0a\na\na ai\nmi C z r irr r r\n                                (S2) \n \nThe parameter of magnetization helicity is defined as the produ ct *mm . Since \nmagnetization vector m has only in-plane components, we are interested in only in-pla ne \ncomponents of the vector m as well: \n \n,()ˆˆ ˆˆa r\narm m zmrC i rzz z r r r r\n                                 .   (S3) \n \nAs a result, we obtain   \n                 \n*2 ()2( )a\nazmmi C zzr rr r                  .                (S4) \n \nWe can see that the parameter of magnetization helicity is pure ly an imaginary quantity: \n  \n                                                *ImVm m      .                                                     (S5) \n \nThis parameter, appearing since the magnetization m in a ferrite disk has two parts: the \npotential and curl ones, can be also represented as \n \n                                             *() ( ) 1RemeVj jc    \n,                                                (S6)  \n \nwhere ()mj im and ()ej cm   are, respectively, the magnetic and electric current \ndensities in a ferrite medium, c is the light velocity in vacuum [S4 – S6]. This parameter, \nanalytically calculated for the 1st radial mode, is shown in Fig. 16 of the article. \n \n--------------- ---------- \n[S1] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media  (Pergamon \nPress, Oxford, 1960).  \n[S2] A. G. Gurevich and G. A. Melkov, Magnetization Osci llations and Waves  (CRC \nPress, 1996). \n[S3] D. D. Stancil, Theory of Magnetostatic Waves  (Springer-Verlag , New York, 1993). \n[S4] E. O. Kamenetskii, R. Jo ffe, R. Shavit, Phys. Rev. E 87, 023201 (2013). \n[S5] E. O. Kamenetskii, M. Sigal ov, and R. Shavit, J. Appl. Phy s. 105, 013537 (2009). \n[S6] J. D. Jackson, Classical Electrodynamics , 2nd ed. (Wiley, New York, 1975). \n \n 21\n \n \n \nFig. S1. Power-flow vortices for the 2nd radial mode on the uppe r plane of the disk. \nElectromagnetic wave propagation in a waveguide from port 1 to port 2  (1  2).  Black \narrows clarify the power-flow directions. A bias magnetic field  is directed along z \ndirection. The vortices are classi fied by three topological cha rges. In succession, there  \nare the topological charges 1, 1, 1 .  \n \n \n                                  ( a)                                                                  (b) \nFig. S2. Power-flow vortices for the 2nd radial mode on the uppe r wall of a waveguide. \n(a) Wave propagation 1  2, (b) wave propagation 2 1. Black arrows clarify the \npower-flow directions. A bias m agnetic field is directed along z direction. The view \nis along z direction on an upper waveguide wall from the outside metal re gion. The \nvortices are classified by three topological charges. In succes s i o n ,  t h e r e  a r e  t h e  \ntopological charges 1, 1, 1 .  \n \n \nFig. S3. The 2nd radial mode. 20 um above the upper plane of the disk. 1  2. Black \narrows clarify the power-flow directions.  \n22\n \n \n \nFig. S4. A microstrip structure with an embedded quasi-2D ferri te disk. \n \n \nFig. S5. Power-flow vortex for the 1st radial mode. Bottom plane of the disk. A black \narrow clarifies the power-flow direction. A bias magnetic field  is directed along z \ndirection. The view is along z direction. The vortex is classified by the topological \ncharge 1 Q .  \n \n \n (a)                                                                   (b) \nFig. S6. Power-flow vortex for the 1st radial mode on a ground plane of a microstrip \nstructure. A black arrow clarifies the power-flow direction. A bias magnetic field is \ndirected along z direction. The view is along z direction. The vortex is classified \nby the topological charge 1 Q . (a) Electromagnetic wave propagation in a \nwaveguide from port 1 to port 2  (1  2) , (b) wave propagation from port 2 to port 1  \n(2 1).  \n  \n23\n \n \n \nFig. S7:  Surface electric current for the 1st radial mode on a ground plane of a microstrip \nstructure for a certain time phase.    \n \n \n \n"    },    {        "title": "1606.01698v1.Band_gap_tuning_and_orbital_mediated_electron_phonon_coupling_in__HoFe__1_x_Cr_xO_3_.pdf",        "content": "1 \n Band  gap tuning and orbital mediated electron – phonon coupling  \n in HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  \n \nGanesh Kotnana1 and S. Narayana Jammalamadaka1*  \n \n1Magnetic Materials and Device Physics Laboratory, Department of Physics, Indian Institute \nof Technology Hyderabad , Hyderabad, India – 502 205.  \n*Corresponding author: surya@iith.ac.in   \nAbstract:  \nWe report on the evidenced orbital mediated electron – phonon coupling and band gap tuning \nin HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds. From the room temperature Raman scattering, it is \napparent  that the electron -phonon coupling is sensitive to the presence of both the Fe and Cr \nat the B-site. Essentially, a n A g like local oxygen breathing mode is activated due to the \ncharge tr ansfer between Fe3+ - Cr3+ at around 670 cm-1, this observation is explained on the \nbasis of Franck -Condon (FC) mechanism. Optical absorption studies infer that there exists a \ndirect band gap in the HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds. Decrease  in band gap until x = \n0.5 is ascribed to t he broadening of the oxygen p - orbitals as  a result of the induced spin \ndisorder due to Fe3+ and Cr3+ at B-site. In contrast, the increase  in band gap above x = 0.5 is \nexplained on the basis of the reduction in the available unoccupied d - orbitals of Fe3+ at the \nconduction band.  We believe that above results would be helpful for the development of the \noptoelectronic devices based on the ortho -ferrites.  \n \nKey words:  Charge transfer,  electron phonon coupling, optical absorption, band gap tuning  \nPACS: 63., 63.20. -e, 63.20.kd,  71.35.Cc, 71.20.Nr  \n \n 2 \n Introduction : \nThe b and gap engineering may play a significant  role in the future spin -photonic and ultra -\nviolet (UV) photonic devices such as  laser diodes, solar blind UV photo -detectors light -\nemitting diodes (LED), and transparent electronic devices.  In order to fulfil the above \nrequirement, there is a quest for devel opment of  the new materials and engineer the band gap \nselectively based on the requirement.  In search of the new materials, materials based on the \nrare earth ortho -ferrites R FeO3 (R - rare earth metal and the Fe - transition metal) have \nappealed continued experimental and theoretical interest as a result of their impressive \nmagnetic and structural changes1-3. Technologically these ortho – ferrites have gained much \nattention in solid oxide fuel cells4, magneto optic devices5, gas sensors6 and for the detection \nof ozone in monitoring environment7. On top of that these compounds also have much \npotential for the photocatalytic activity8 and the multiferroi c behaviour9. \nElectron – lattice dynamics are very important in the rare earth -transition metal oxides to \nunderstand the phenomenon of the colossal magnetoresistance10. On the other hand, \nexploring the correlation between the structural, magnetic and orbital ordering is very much \nessential to understand  the charge transfer mechanism which is base for the electron – phonon \ncoupling11. It has been believed that in the cubic perovskites, due to the distortion of the \nlattice , symmetry of the structure lowers and this leads to the appearance of the Raman -active \nphonons12. The influence of the electronic configuration and orbital ordering on the Raman \nspectra of these distorted perovskites has been addressed by Allen et. al.13.  Local oxygen \nbreathing mode in the Raman scattering has been observed in the distorted perovskite  \ncompound based on the mixed Fe and Cr ions at B – site. Such intriguing phenomenon has  \nbeen explained on the basis of the Franck -Condon picture following a photon induced \ntransfer of an electron from Fe to adjacent Cr ion11. Similar behaviour has been observed on \nthe other compound La1−ySryMn1−xMxO3 (M=Cr , Co, Cu, Zn, Sc or Ga)14. To our knowledge, 3 \n until now the electron - phonon  coupling  has been observed only in the Jahn – Teller active \ncompounds. However, i t would be very interesting if we can observ e the electron - phonon \ncoupling in the Jahn – Teller inactive compound . For this purpose we choose the HoFeO 3 \nwhich is a Jahn – Teller inactive and  may become Jahn – Teller active  by substituting Cr. We \nanticipate that electron transfer mechanism can takes place from the Fe site to the Cr site  \nwhen the incident  photon energy equals to the ground state energy gap between Fe3+ and Cr3+ \n15.  \nInterestingly, HoFeO 3 has been believed to possess the canted G -type antiferromagnet ism and \na potential candidate for the ultrafast recording16 with a magnetic ordering temperature \naround 641 K17. On the other hand, the rare earth ortho -chromites of the formula RCrO 3 (R = \nHo, Er, Yb, Lu, Y) have been believed to exhibit the multiferroicity with a canted \nantiferromagnetic behaviour  in the temperature range of 113-140 K (T N) and a dielectric \ntransition in the temperature range of 472 - 516 K18. In contrary, ortho -ferrites  and ortho -\nchromites  have been believed to possess a p – type semiconducting behaviour19, which may \nbe very much useful  in developing the optoelectronic devices. However, to the best of  our \nknowledge, detailed information on the band gap values is not available in the literature \nneither on the HoFeO 3 nor on the HoCrO 3. In addition, it would also be of great interest to \nmonitor the change in the band gap  value  by applying the chemical pressure either at the Fe \nsite or the Cr site. To achieve our goal, in the present work, Cr is chosen to apply the \nchemical pre ssure at the Fe-site in HoFeO 3. Essentially, the fa ct that both the Fe and Cr are \nhaving same ionic state (in 3+ valence state) and the smaller ionic radius of the Cr3+ compared \nwith the Fe3+ may influence the optical properties by occupying the octahedral Fe3+ site.  \nAs x – rays and the Raman scattering are very much sensitive for the structural changes and \nthe local environment in a n unit cell respectively, one can get precise information about the \natomic positions and information about the Raman active modes. Hence, the first aim of our 4 \n manuscript is to study the interplay between the electron and lattice dynamics using the \nRaman scattering in the HoFe 1-xCrxO3 [x = 0, 0.25, 0.5, 0.75 and 1]. Secondly, we also would \nlike to understand how the band gap value varies  by applying the chemical pressure at Fe site.  \nExperimental  Details : \nPolycrystalline compounds  of HoFe 1-xCrxO3 (0 ≤ x ≤ 1) were prepared by the conven tional \nsolid state reaction method. High purity oxide powders of Ho 2O3, Fe 2O3, and Cr 2O3 (purity > \n99.9%) (Sigma -Aldrich chemicals India) were used as starting raw materials and were mixed \ntogether in stoichiometric ratios. The mixture thus obtained was th oroughly and repeatedly \nground in the isopropanol alcohol using an agate mortar and pestle to ensure the \nhomogeneity. Pellets were prepared using t he resultant powder and sintered sequentially at \n1000oC for 12 h, 1200oC for 12 h and 1250oC for 24 h  respectively . The phase purity or the \nstructural analysis was carried out at room temperature using the powder x - ray diffraction \n(XRD)  (Panalytical X -ray diffractometer ) with Cu K α rad a     (    1   0   ) and with a \nstep size of 0.0170 in the wide range of the Bragg angles 2 θ (200 - 800). Raman spectra was \nmeasured at room temperature using a Laser Micro Raman spectrometer (Bruker, Senterra) \nwith an excitation source of 535 nm and with a power of 10 mW.  Optical absorbance of the \nsamples were measured at room temperature using the Perkin Elmer Lambda 1050 UV -Vis-\nNIR spectro photo meter  in the wavelength range of 200 – 800 nm.  \nResults and Discussion : \nPhase purity of the HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds  is confirmed with the room \ntemperature XRD . Intense reflections that are present in Fig. 1 are allowed reflections for a \nGdFeO 3 type disordered pervoskite structure described by the orthorhombic with a space \ngroup of the Pbnm. We do not see any impurity phase apart from the parent phase within the \ndetectable limits of the XRD. In order to get more insights about the structural aspects, we 5 \n also have performed the Rietveld refinement using the General  Structural Analysis Sys tem \n(GSAS)20. Information extracted from the refinement is depicted in the Table. 1.  From th e \nRietveld refinement data, small χ2 values of all the compounds  infer  that there exists  a good \nagreement between the observed and the calculated diffraction patterns. As it is evident from \nthe Fig. 2(a) that extracted lattice parameter values from the refinement indicates a tendency \ntowards decrement, which is consistent with the fact that the ionic radius of the Fe3+ (0     \n )     ar e r than that of the Cr3+ (0  1   )21. It is worth noting that the change in the position \nand the shape of the diffraction peaks with Cr concentration is minimal, hinting that there \nexists no structural transformation as a result of the Cr dopants. This mismatch in ionic radii \ncan leads to a distortion of the lattice and such distortion can be quantified using the \nG  d chm d ’     era ce fac  r  (GTF) . In general, the GTF can be defined as  \n                                                      \n2( )AO\nBOrrt\nrr\n                                                (1) \nWhere rA, rB, and rO are the radii of Ho3+, Fe3+/Cr3+, and O2- respectively , where\n1B Fe Crr x r xr  \n. Calculated GTF for all the compounds in the present investigation are \nfound to be in the range of 0.848 – 0.868, the range for a compound to be in an orthorhombic \nstructure. The variation of the GTF with respect to the Cr composition is depicted in Fig. \n2(b). Upon closer  observation, the GTF is found to increase as a function of the chromium \ncontent, hinting that the increase in stability and tendency towards the cubic structure upon \nCr doping. We have also calculated the average tilt angle < φ>  f FeO 6 octahedral around the \npseudo cubic [111] direction using the geometric rel ation that has been proposed by O’Keefe  \nand Hyde22 and the two super exchange angles θ1 = Fe (Cr) -O1-Fe (Cr), θ2 = Fe (Cr) -O2-Fe \n(Cr).  Both the above angles Fe (Cr) -O1-Fe (Cr) and Fe (Cr) -O2-Fe (Cr) are extracted from the \nstructural refinement. From Fig. 2(b) it is evident that average tilt angle < φ>  f the FeO 6 \noctahedral decreases with an  increase in Cr content. It is apparent  that the tolerance factor 6 \n increases whereas  the tilt angle diminishes, hinting  that the internal stresses as a result of the \nchemical pressure  (via Cr3+ doping) leads to a distortion of the lattice .   \nTable 1:  Lattice parameters, cell volumes, selected bond lengths, bond angles from Rietveld \nrefinement of HoFe 1-xCrxO3. \nCompounds                 HoFeO 3           HoFe 0.75Cr0.25O3       HoFe 0.5Cr0.5O3        HoFe 0.25Cr0.75O3            HoCrO 3 \nSpace group                   Pbnm                    Pbnm                          Pbnm                      Pbnm                           Pbnm  \nLattice Parameters  \na (Ao)                             5.28339(6)        5.27487(6)                  5.26645(6)              5.25658(7)                  5.25009(8)  \nb (Ao)                             5.59100(6)        5.57407(6)                  5.55568(6)              5.53709(7)                  5.51635(8)  \nc (Ao)                             7.60986(8)         7.59292(8)                  7.57590(8)              7.55790(10)                7.54373(11)  \nCell Volume (Ao3)           224.7909          223.  2507                   221.6610                 219.9814                     218.4764  \nSelected bong angles (o) \nFe(Cr) – O1 – Fe(Cr)      144.2(5)            144.0(5)                      145.3(4)                  145.8(4)                      146.2(4)  \nFe(Cr) – O2 – Fe(Cr)      144.7(4)           145.5(4)                      144.45(33)                145.47(31)                  146.13(34)  \nSelected bond lengths (Ao) \nFe(Cr) – O1                    1.9990(30)        1.9959(28)                  1.9840(23)               1.9767(21)                  1.9708(23)  \nFe(Cr) – O2                    2.037(9)            2.033(8)                      2.035(8)                   2.015(6)                      2.005(7)  \n \n \nNow we discuss the results pertinent to the Raman scattering which essentially gives the local \nstructure, shift and distortion of the modes as a result of the chemical doping (here Cr3+ \ndoping). In addition, we also would like to correlate our structural information with the \nRaman data that we obtained. For this purpose the room temperature Raman spectroscopy \nwas used on the HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds  to understand  the aforesaid properties. 7 \n It has been observed that i n an ideal perovskite  (ABO 3), the B -site transition metal cation \nlocates at the centre  of the oxygen octahedral and A -site cation locat es at the corners of the \ncube23. However, due to the displacement of the crystallographic sites from the ideal cubic \npositions, most perovskites show the symmetry breaking which results the appearance of the \nRaman active modes in the Raman spectra. The rare earth o rtho-ferrite HoFeO 3 is an \northor hombically d istorted perovskite  with a space group  of D2h16 (Pbnm ). The irreducible \nrepresentations corresponding to the phonon modes at the Brillouin zone center24 can be \ndefined as follows   \n                           7Ag + 7B1g + 5B2g + 5B3g + 8Au + 8B1u + 10B2u + 10B3u  \nHere, Ag, B1g, B2g, B3g are the Raman active mode species, B 1u, B2u, B3u are the infrared mode \nspecies and A u is the i nactive mode. Among them the modes which are above 300 cm-1 are \nrelated to the vibrations of oxygen  and the modes below the wave number 300 cm-1 are \nassociated with the rare earth ions25. However , the Raman vibrational modes corresponding \nto an orthorhombic structure are: A g + B 1g and 2B 2g + 2B 3g, which are symmetric and \nantisymmetric  modes respectively; In contrast, Ag + 2B 1g + B 3g,  2Ag + 2B 2g + B 1g + B 3g,  \nand 3Ag + B 2g + 3B 1g + B 2g are associated with the bending modes, rotation and tilt mode of \nthe octahedral and for the changes in the ra re earth movements respectively26. Fig.  3 shows \nresults pertinent  to the Raman Spectra of HoFe 1-xCrxO3 (0 ≤ x ≤ 1) samples recorded at room \ntemperature and with the wavenumber in the range of 50 to 800 cm-1. Peaks with the high \nintensities are evident at 109, 137, 158, 337, 423 and 494 cm-1 which is normal for a typical \northoferrite24. Apart from aforesaid mode s, a mode at 670 cm-1 prevailed in the compounds \nwith the Cr3+ ion. However, the peak which is evident at 670 cm-1 consists very less intensity \nin the parent  HoFeO 3 and HoCrO 3. Basically, such peak picks intensity only if we have the \ncombination of both the Fe and Cr at the B site. Such intriguing phenomenon may be \ncorrelated to the orbital mediated electron – phonon coupling like in case of LaFe 1-xCrxO311. 8 \n The Raman active modes  of the samples are designated according to the method proposed by \nGupta27 et. al ,. The appearance of an A g like mode with observable intens ity at high \nfrequency of around  670 cm-1 can be attributed to an in -phase stretching (breathing) mode of \noxygen in the close vicinity of the substituted Cr3+ ion. Essentially the oxygen breathing \nmode is  activated by the charge transfer between the Fe3+ and Cr3+ through an orbital \nmediated electron -phonon coupling mechanism11.  \nIn HoFe 1-xCrxO3 compounds, the ground state electron configuration of Fe3+ does not support \norbital mediated electron -phonon coupling due to the lack of strongly interacting half -filled e g \nlevels in both the Fe3+ (d5) and Cr3+ (d3). To facilitate such an orbital mediated  electron -\nphonon coupling, the electronic configuration of Fe3+ must contain partially filled e g orbital \n(like Fe4+) such that d4 electron of the Fe4+ ion can move to the upper e g levels of the Cr2+ to \ncreate electronic excitation as shown in LaMnO 328. Fig. 4 explains the mechanism for the \norbital mediated electron -phonon coupling. The left part of Fig. 4(a) shows electronic states \nof Fe3+ and Cr3+ ions. νg,0, νg,1, νg,2, … , ν g,n a d ν e,0, νe,1, νe,2, …, ν e,n represents the vibrational \nstates of the Fe3+ and the Cr3+ respectively.  Right part of the Fig. 4(a) reveals electronic states \nof the Fe4+ and Cr2+ ions.   Arrows on both Fig. 4(a) and 4(b) represents various transitions \nbetween Fe3+Cr3+ and Fe4+  Cr2+ respectively.  \nIt has been reported that the  photon mediated charge transfer can takes place between the \nFe3+ and Cr3+ ions upon irradiation with a laser of wavelength 535 nm11, 15.  In this process, \nthe overlap between the d-orbitals of Cr2+ and the p-orbitals of the oxygen couples through a \nlattice  distortion, causing a self -trapping motion. Evidently, t his motion increases the lifetime \nof excited Cr2+ electronic ground state long enough to interact with the intrinsic phonon \nmode.  Essentially, d uring the charge transfer  mechanism  (CT) , when the photon energy \nequals to the CT energy gap b etween the two transition metal ions  Fe3+ and Cr3+, electron s in \nthe Fe3+ excite  to the Cr3+ ion and leaves them in a strongly coupl ed d4-d4 configuration with 9 \n the half-filled bands . The change in the charge density of e g orbital of transition metal \nsurrounded by the oxygen octahedral activates  a breathing distortion of O 6 around the \ntransition metal cation which appears in the Raman spectrum at around 670 cm-1. This \nconfiguration is Ja hn-Teller active , hence, that it leads to a volume preserving lattice \ndistortion  (δ), involves a stretching of Fe (Cr) -O bonds along z - direction and a compression \nin x-y plane . As a result of the Jahn - Teller effect, an electron in e g orbital collapse into the \nlower energy state which produces a potential minimum. This minimum potential t raps the \nelectron in that orbit al (self-trapping) and increases its  life time in the excited Cr2+ state long \nenough for it to inte ract with intrinsic phonon mod e/lattice distort ion. In the perturbed state, \nindeed there exists a contraction of the oxygen octahedral surrounding to the Fe4+ ion and \nleads to an expansion in the adjacent octahedron surrounding to Cr2+ ion as shown in Fig . \n4(b). The oxygen lattice relaxes back to its unperturbed state when the electron transfers back \nto Fe3+ state. In this fashion , the charge transfer of an electron from Fe3+ to the Cr3+ ion \nactivates an oxygen breathing mode of A g symmetry through orbital mediated electron -lattice \ncoupling.   \nThe incre ase in the intensity of the peak at around 670 cm-1 is observed only in doped \nsamples as shown in Fig.  5 (a). This effect can be related to the increase in the degree of \ndisorder, which can be supported by an increase in the tolerance factor as a function of the Cr \ncontent. This is a striking coincidence between our structural and Raman studies. As the Cr \ncontent incr eases, there would indeed be  an increase in the degree of disorder, which may \nenhance the interaction between the lattice distortion and the  charges transferred between \nFe3+ - Cr3+. Eventually as a result of this there would be an enhancement in the electr on-\nphonon coupling which leads to increase in the intensity of the peak at 670 cm-1. The \nbroadening of the peak at 670 cm-1 with an increase in the amount of the Cr3+ ions at the Fe3+ \nsite can correlate with the  structural disorder. This is confirmed by the observed change in the 10 \n lattice parameters by the substitution  of the Cr3+ ions at the Fe site. The observed shift in the \nwave number towards the higher values ( Blue shift) in the doped compounds  shown in Fig \n5(b) can correlate to the compressive  strain produced in the compounds  by the incorporation \nof the Cr3+ ion at the Fe  site. This effect is supported by the change in Fe (Cr) – O bond \nlengths as well as FeO 6 octahedral tilt angle with respect to the chromium content at Fe site \nas shown in the Table 1. \nThe optical absorbance of the HoFe 1-xCrxO3 (0 ≤ x ≤ 1) c mp u d  wa  rec rded a  r  m \ntemperature and is  shown in Fig. 6.  Indeed there exists a direct band gap and the value of gap \nis determined using the Tauc’  equation29. Essentially this equation relates the optical \nabsorption coefficient  (α), photon energy  (\nhv) and  the energy gap \ngE  as given below  \n12\ng hv hv E\n \n Optical band gaps of the HoFe 1-xCrxO3 (0 ≤ x ≤ 1) c mp u d  are  b a  ed u     the above \nequation and by extrapolating the linear region of the curve to the zero in the \n2hv vs. \nhv\ngraph  as shown in Fig. 7 (a). The direct band gap value for HoFeO 3 and HoCrO 3 is calculated \nas 2.07 eV and 3.26 eV respectively.  From the Fig.  7(b) and  for the compounds with the \ncombination of Fe and Cr, it is evident  that the band gap  decreases with Cr content, and \nreached a minimum value of 1.94 eV at x = 0.5 . Further increase in Cr content results in \nincrease of the band gap value and  reached a maximum value of 3. 26 eV at x = 1. In order to \nexplain the observed band gap variation we would like to propose a possible mechanism \nusing energy diagram of HoFeO 3, HoCrO 3 and HoFe 1-xCrxO3. Fig. 8 (a) s hows the energy \ndiagram of HoFeO 3 with the band gap value of 2.07 eV (b) the e nergy diagram of HoCrO 3 \nwith the bad gap val ue of 3.26 eV and (c) probable energy diagram for HoFe 1-xCrxO3 for \nwhich the band gap varies between 2.07 – 3. 26 eV. The variation of the band gap with Cr \nconcentration could be due to a complex interplay between the Fe3+ and the Cr3+ electronic 11 \n levels me diated by oxygen through superexchange interaction.  From the Fig. 8(c) i t is evident \nthat when x < 0.5, the valance band maxima (VBM) and conduction band minima (CBM) \nshifts to higher energy  (dark red colour) . The shift in VBM may be explained on  the basis of \nthe hybridization of d - orbitals of Fe and Cr with  p - orbitals of  oxygen in the valance band. \nEssentially, the ferrimagnetically15  coupled Fe3+ and Cr3+ induce a spin disorder  on oxygen  \nwhich can en hance the broadening of oxygen p - orbitals  and valance band edges of Fe3+ and \nCr3+ 30, 31, hinting a smaller band gap until x = 0.5. When x > 0.5, the width of available un \noccupied d - orbitals  of Fe3+ at the conduction band reduces, which can leads to a shift of the \nconduction band minima to higher energies  (as shown in fig 8(c)  (purple colour) ). As a result, \nthe band gap in HoFe 1-xCrxO3 increases above x = 0.5 and reaches a maximum value of 3.26 \neV at x = 1. From the above results, indeed it is possible to tune the band gap in rare earth \northo -ferrites and the other compounds with a similar structure by controlling the Fe/Cr ratio.  \nIn summary, we have explored the orbital mediated electron – phonon coupling mechanism \nin the compounds HoFe 1-xCrxO3 (0 ≤ x ≤ 1) . There is a striking coincidence between our \nstructural and Raman studies. Raman studies infer that , A g like symmetric oxygen breathing \nmode at around 670 cm-1 in the compounds with both the Fe and Cr. The decrease in optical \nband gap is ascribed to the ind uced spin disorder due to the Fe3+ and the Cr3+ on oxygen, \nwhich can leads to broadening of the oxygen p-orbitals. On the other hand, increase in band \ngap value explained on the basis of th e reduction in the available un occupied d-orbitals  of \nFe3+ at the conduction band. 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C  117, 25504 (2013 ). \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 15 \n Figure captions  \nFig. 1:  Powder x -ray diffraction patterns of HoFe 1-xCrxO3 (0 ≤ x ≤ 1) . It is evident that all the \ncompounds are formed in single phase.  \nFig. 2:  (a) Variation of lattice parameter with Cr composition. (b) Variation of tolerance \nfactor (circle symbol) and FeO 6 average tilt angle (square symbol) with Cr composition.  \nFig. 3: Room temperature Raman spectra of HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds with an \nexcitation of 535 nm.  \nFig. 4:  (a) Franck -Condon (FC) mechanism for Jahn -Te  er ac  ve per v k  e   ν g,0, νg,1, νg,2, \n… , ν g,n a d ν e,0, νe,1, νe,2, …, ν e,n represents the vibrational states of Fe3+ and Cr3+ respectively.  \nFor FC mechanism to happen for a vibrational mode, the virtual state \nrv of Raman process \nmust coincide with any vibrational   a e  f e ec r   ca  y exc  ed   a e  δ   d ca e   a   ce \ndistortion due to Jahn - Teller effect as a result of charge transfer mechanism (b) Octahedral \nsites of Fe4+ and Cr2+ respectively. Dotted arrow in the figure indicates charge transfer \nmechanism an d lattice relaxation.    \nFig. 5: (a) Intensity variation of A g peak and (b) wave number shift of A g peak for the HoFe 1-\nxCrxO3 (0 ≤ x ≤ 1)  compounds.  \nFig. 6: Absorption spectra of HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds.  \nFig. 7:  (a) Tauc’  p        de erm  e  he ba d  ap va ue   f HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  \ncompounds. (b) Variation of the band gap with respect to Cr composition.   \nFig. 8:  (a) Shows the energy diagram of HoFeO 3 (b) Energy diagram of HoCrO 3 (c) probable \nenergy diagra m for HoFe 1-xCrxO3. It is evident from the frame (c) that when x < 0.5, the \nvalance band maxima (VBM) and conduction band minima (CBM) shifts to higher energy  \n(dark red color) . However, the shift in VBM is due to strong hybridization of d orbitals of Fe \n& Cr with p - orbitals of oxygen  in valance band .  When x > 0.5, band gap is dominated by \nunoccupied d - orbitals of Cr in conduction band  which leads to increase in band gap (purple \ncolor) .  \n \n \n \n \n 16 \n  \n \n \nFig. 1:  Powder x -ray diffraction patterns of HoFe 1-xCrxO3 (0 ≤ x ≤ 1) . It is evident that all the \ncompounds are formed in single phase.  \n17 \n  \n \n \n \n \nFig. 2:  (a) Variation of lattice parameter with Cr composition. (b) Variation of tolerance \nfactor (circle symbol) and FeO 6 average tilt angle (square symbol) with Cr com position.  \n \n \n \n0.8480.8520.8560.8600.864 Tolerance factorHoFe1-xCrxO3\n20.620.821.021.221.421.621.8FeO6 average tilt angle (deg)\n0.0 0.2 0.4 0.6 0.8 1.05.06.07.08.09.0(b)\n  \nCr composition (x)abcLattice parameter (Å)(a)18 \n  \n  \n \n \n \n \n \nFig. 3: Room temperature Raman spectra of HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds with an \nexcitation of 535 nm.  \n \n \n \n100 200 300 400 500 600 700 800\n Ag AgB3gx =1\nx =0.75\nx =0.50\nx =0.25 \n x =0\nWave number (cm-1) \n  \n  \n \nB1g\nAgAgB3g  \n \nB1gIntensity (arb.units)HoFe1-xCrxO319 \n  \n \nFig. 4:  (a) Franck-Condon (FC) mechanism for Ja hn-Teller active perovskites. νg,0, νg,1, νg,2, \n… , ν g,n a d ν e,0, νe,1, νe,2, …, ν e,n represents the vibrational states of Fe3+ and Cr3+ respectively.  \nFor FC mechanism to happen for a vibrational mode, the virtual state \nrv of Raman process \nmust coincide with any vibrational state of electronically excited state.  δ   d ca e   a   ce \ndistortion due to Jahn - Teller effect as a result of charge transfer mechanism (b) Octahedral \nsites of Fe4+ and Cr2+ respectively. Dotted arrow in the figure indicates charge transfer \nmechanism and lattice relaxation.    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n20 \n  \n \n \n \n \nFig. 5: (a) Intensity variation of A g peak and (b) wave number shift of A g peak for the HoFe 1-\nxCrxO3 (0 ≤ x ≤ 1)  compounds.  \n \n \n \n \n  Normalized Intensity (arb. units)\n x=0\n x=0.25\n x=0.5\n x=0.75\n 1Ag peak\n600 630 660 690 720 7500200040006000800010000(b)\n  Intensity (arb. units) \nWave number (cm-1)HoFe1-xCrxO3(a)21 \n  \n  \n \nFig. 6: Absorption spectra of HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  compounds . \n \n \n \n \n \n \n \n1 2 3 4 50123456\nx = 1\nx = 0.75\nx = 0.50\nx = 0.25\n  Absorbance (arb.units)\nEnergy (eV)x = 0HoFe1-xCrxO322 \n  \nFig. 7:  (a) Tauc’  plots to determine the band gap values of HoFe 1-xCrxO3 (0 ≤ x ≤ 1)  \ncompounds . (b) Variation of the band gap with respect to Cr composition.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n1.5 2.0 2.5 3.0 3.5 4.0h (eV)\n  (h)2\nEnergy (h) (eV) x=0\n x=0.25\n x=0.5\n x=0.75\n x=1\n(a)1.8 2.0 2.2\n   \n \n0.0 0.2 0.4 0.6 0.8 1.01234\n   \nCr composition (x)Band gap (eV)(b)23 \n  \n \nFig. 8:  (a) Shows the energy diagram of HoFeO 3 (b) Energy diagram of HoCrO 3 (c) probable \nenergy diagram for HoFe 1-xCrxO3. It is evident from the frame (c) that when x < 0.5, the \nvalance band maxima (VBM) and conduction band minima (CBM) shifts to higher energy  \n(dark red color) . However, the shift in VBM is due to strong hybridiza tion of d orbit als of Fe \n& Cr with p - orbitals of oxygen  in valance band .  When x > 0.5, band gap is dominated by \nunoccupied d - orbitals of Cr in conduction band  which leads to increase in band gap (purple \ncolor) .  \n \n \n \n \n"    },    {        "title": "0909.4979v4.Strain_induced_isosymmetric_phase_transition_in_BiFeO3.pdf",        "content": "arXiv:0909.4979v4  [cond-mat.mtrl-sci]  10 Feb 2010Strain-induced isosymmetricphase transitioninBiFeO 3\nAlison J. Hatt and Nicola A. Spaldin\nMaterials Department, University of California, Santa Bar bara, CA 93106-5050\nClaude Ederer\nSchool of Physics, Trinity College, Dublin 2, Ireland\n(Dated: July23, 2021)\nWe calculate the effect of epitaxial strain on the structure and properties of multiferroic bismuth ferrite,\nBiFeO3, using ﬁrst-principles density functional theory. We inve stigate epitaxial strain corresponding to an\n(001)-oriented substrate and ﬁnd that, while small strain c auses only quantitative changes in behavior from the\nbulkmaterial, compressive strains of greater than 4% induc e an isosymmetric phase transitionaccompanied by\na dramatic change in structure. In striking contrast to the b ulk rhombohedral perovskite, the highly strained\nstructure has a c/aratio of∼1.3 and ﬁve-coordinated Fe atoms. We predict a rotation of po larization from\n[111] (bulk) to nearly [001], accompanied by an increase in m agnitude of ∼50%, and a suppression of the\nmagnetic ordering temperature. Our calculations indicate critical strain values at which the two phases might\nbe expected to coexist and shed light on recent experimental observation of a morphotropic phase boundary in\nstrainedBiFeO 3.\nI. INTRODUCTION\nBiFeO3(BFO)hasbeenwidelystudiedforitsroomtemper-\naturemultiferroicproperties,inwhichtheelectricpolar ization\nis coupledto antiferromagneticorder,allowing for manipu la-\ntionofmagnetismbyappliedelectricﬁeldsandviceversa1–5.\nIn its bulk form, BFO occurs in the R3cspace group6, but it\nis oftenstudied in the formofthin ﬁlms whereit is subject to\nan epitaxial constraint. Such constraints impose coherenc y\nand strain that in general distort the bulk structure and/or\nstabilize phases not present in the bulk material7–9. Indeed,\nseveral groups have reported stabilization of a nearly tetr ag-\nonal phase in highly strained BFO ﬁlms grown on LaAlO 3\nsubstrates with a giant c/aratio close to 1.310,11. A simi-\nlar result was found by earlier theoretical studies that con -\nstrained BFO to tetragonal P4mmsymmetry, thus prevent-\ningtherotationaldistortionsfoundinthe R3cbulkphase12,13.\nWithinthisconstraintthetheoreticalgroundstateofBFO h as\na massively reduced in-plane lattice parameter of 3.67 ˚A and\na “super-tetragonal” unit cell with c/a=1.27. This P4mm-\nconstrained structure has an enhanced polarization roughl y\n1.5timesthatofthebulksinglecrystal,aremarkableincre ase\nthathasalso beenreportedexperimentally,thoughnotwide ly\nreproduced14. Lisenkov et al.recently used model calcula-\ntionstosuggestthatthe P4mmphasecanalsobestabilizedby\nelectricﬁelds,andtheyidentiﬁedanadditionalphaseinte rme-\ndiatetothestrained-bulkphaseand P4mminducedbyapplica-\ntion of a [00 ¯1] electric ﬁeld15. The most recent experimental\nresults, from B´ ea et al.and Zeches et al., have resoundingly\nconﬁrmed the existence of a second phase with giant c/ain\nBFO thin ﬁlms grown on (001) LaAlO 3but suggest a mono-\nclinically distorted space group Cc10,11. In contrast to earlier\nexperimentalstudies,B´ ea etal.onlyfoundamodestenhance-\nment of the polarization. Intriguingly,Zeches et al.observed\nthecoexistenceofthehighstrain“super-tetragonal”phas eand\na low strain bulk-like phase with defect- and dislocation-f ree\ninterfaces between phase domains, in spite of a large differ -\nenceinout-of-planelattice parameters.\nIn this work we use ab initio calculations to investigatethe effect of epitaxial strain on BFO without the additional\nimposed symmetry constraints used in earlier ﬁrst-princip les\nstudies. We address the most widely used (001)-orientedcu-\nbic substrates and cover a range of strains encompassing all\nexperimentally accessible states. Our calculations revea l a\nstrain-inducedphase transitionat ∼4.5% compressivestrain,\nconsistent with recent experimental reports10,11and in addi-\ntion show that the transition is isosymmetric . We reported\nthesebasicresultsinRef. 11,andinthepresentworkwepro-\nvide the complete theoretical background and analysis. Our\nanalysisof the isosymmetric behaviorin the transition reg ion\nsuggests an explanation for several experimental results r e-\nported in Ref. 11; namely, the coexistence of bulk-like and\nsuper-tetragonalphasesinﬁlmsgrownonLaAlO 3substrates,\nand the reversible movement of domain walls in these same\nﬁlmsbyelectricﬁeld. Inaddition,ourcalculationsofthem ag-\nnetic properties of the super-tetragonal phase point to mag -\nnetic behavior that is distinct from the rhombohedral phase .\nFinally, in the low strain regime, we conﬁrm that epitaxial\nstrain has only a small quantitativeeffect on the propertie sof\nrhombohedralBFO12.\n1 1.1 1.2 1.3\nc/a-36-35.6-35.2E (eV)a = 3.65\na = 3.71 \na = 3.81\n3.65 3.7 3.75 3.8 3.85 3.9\na (Å)00.050.10.15E (eV)6 5432 1strain (%)\nP4mm\nCc, β=90ο\nCc, β relaxed\nFIG.1:Left:Totalenergy per formula unit as afunction of c/aratio\nfor three constrained aparameters. Right:Energy per formula unit\nrelative to bulk BFO for the ground state structures as a func tion\nofaforP4mm, un-relaxed monoclinic angle ( β=90◦), and relaxed\nmonoclinic angle.2\nII. COMPUTATIONALMETHOD\nWe perform density functional calculations using the local\nspin density approximation plus Hubbard U(LSDA+U) ap-\nproach as implemented in the software package VASP16. We\nuse an effective Uof 2 eV, which has been shown to give\na good description of bulk properties of BFO1. We use the\nprojector augmented wave method17, the default VASP po-\ntentials (Bi d, Fepv, O), a 5 ×5×5 Monkhorst-Pack k-point\nmesh, and a 500 eV energycutoff. Spin-orbit couplingis not\nincluded in these calculations and unless otherwise noted w e\nimpose the bulk G-type antiferromagneticorder. Electric p o-\nlarizationiscalculatedusingthe Berryphasemethod18,19.\nTo address the effect of epitaxial strain, we use a 10 atom\nunit cell with lattice vectors /vectora1=(a,a,0),/vectora2=(Δ,a+Δ,c),\nand/vectora3= (a+Δ,Δ,c). This unit cell can accommodate the\nalternating rotations of the FeO 6octahedra found in the bulk\nR3cstructure, while enforcing the formation of a square lat-\ntice within the x-yplane, corresponding to the epitaxial con-\nstraint imposed by a (001) oriented cubic substrate with in-\nplanelattice constant a. Furthermore,it allowsusto relaxthe\nout-of-planelattice parameter canda possiblemonoclinictilt\nβof the pseudo-cubicperovskiteunit cell (tan β=c/Δ). For\na=c=3.89˚AandΔ=0oneobtainsthelatticevectorsofthe\nrelaxedR3cstructureofbulkBFO, albeitwiththerhombohe-\ndral angle ﬁxed to 60◦. Note that since the relaxed value of\nthisanglefor Ueff=2eVis59.99◦(seeRef.1)thisisbarelya\nconstraint. For computationalsimplicity we ﬁrst consider the\ncaseΔ=0,i.e. nomonoclinicdistortionoftheperovskiteunit\ncell (β=90◦); thisconstraintisrelaxedlater.\nIII. RESULTSAND DISCUSSION\nA. Energetics and Structure\nWe start by examining how the energy varies with out-of-\nplane lattice parameter for strained BFO. For in-plane latt ice\nparameters corresponding to compressive strains up to 6.2%\nwe varyc/aand, for each c/avalue, we relax all internalco-\nordinatesuntiltheHellman-Feynmanforcesarenolargerth an\n1meV/˚Aonanyatom. Theinternalcoordinatesareinitialized\naccordingtotherelaxedbulk R3cstructure,resultinginspace\ngroup symmetry Cc. Fig. 1(left) shows the resulting total en-\nergy as a function of c/afor three representative cases. For\na=3.81˚A, corresponding to a moderate compressive strain\nof 2.06 % relative to the LSDA bulk value (3.89 ˚A), a sin-\ngle energyminimumis observedat c/a∼1.05,i.e. relatively\nclosetounity. For a=3.65˚A,i.e. 6.17%compressivestrain,\nweagainﬁndasingleenergyminimum,butthistimeatavery\nlargec/aof almost 1.3. Incontrast, foran intermediatevalue\nofa=3.71˚A (4.63 % compressive strain), the energy curve\nis almost ﬂat between c/a∼1.1 and 1.3. Our calculations\nindicate the presence of two minima at this intermediate a, a\nglobal minimum at large c/a≈1.25 and a local minimum at\nsmallerc/a≈1.15. Furthercalculationswouldberequiredto\nfullyresolvetheenergycurveinthisregion.\nWe thenuse polynomialﬁts to extractthe valueof c/athat606264cell volume (Å3)\n2.02.53.0Fe-O dist. (Å)in-plane Fe-O\nout-of-plane Fe-O\n3.65 3.7 3.75 3.8 3.85 3.9\na (Å)050100150200P (µC/cm2)\ntotal\nin-plane\nout-of-plane1.01.11.21.3c/a-6 -5 -4-3-2 -1strain (%)\n(a)\n(b)\n(c)\n(d)\nFIG.2: Structural parameters asfunctions of latticeparam eter/strain\nrelative to the bulk LSDA+ Ulattice parameter. (a) c/aof the pseu-\ndocubic cell; (b) volume per formula unit; (c) Fe-O bond leng ths for\nin-plane and out-of-plane bonds; (d) total polarization an d compo-\nnents lying in-plane and out-of-plane. All values are for co nstrained\nmonoclinic angle, β=90◦.\nminimizes the total energy for ﬁxed a, and relax all atoms\nagain at the obtained c/aratio, still maintaining β=90◦.\nThe total energyof the resulting structures is shown in Fig. 1\n(right), relative to the energy of bulk R3c. The correspond-\ning evolution of various structural parameters( c/aratio, unit\ncell volume, Fe-O distances) and of the electric polarizati on\nas a function of strain is shown in Fig. 2. All of these quan-\ntities exhibit a sharp discontinuity between a=3.71˚A and\na=3.73˚A, i.e. 4-4.5% strain, where also the total energy\ndepictedin Fig. 1(right)hasa kink,indicativeofa ﬁrst-or der\nphasetransition.\nFinally,we performadditionaltotalenergycalculationsf or\nβ/negationslash=90◦, determine the optimal βfor each value of afrom a\npolynomialﬁt,andagainrelaxatompositionswithinthisne w\nunitcell. For a<3.72˚Aweﬁndamonoclinicdistortionofthe\nunitcell of ∼1◦to 2◦that reducesthe energyby 10meV/f.u.\ncomparedto β=90◦,asshowninFig.1. Theresultingatomic\nconﬁguration represents only a minor structural change fro m\ntheβ=90◦case and we thus neglect this distortion in all fur-\nthercalculations,onlyreportingresultsfor β=90◦.\nWestartourdiscussionbyﬁrstcharacterizingthestructur es\nfor two extreme cases representative of the small and large\nstrainregimes, a=3.85˚Aanda=3.65˚A,∼1%and6%com-\npressivestrain,respectively. At1%straintherelaxedstr ucture3\nFIG. 3: Isosymmetric phases of CcBFO:Left:tetragonal-like T-\nphase;Right:rhombohedral-like R-phase.\nclosely resembles the rhombohedral R3cbulk phase, but the\nepitaxial constraint causes a monoclinic distortion that l ow-\ners the symmetry from R3ctoCc. The corresponding struc-\nture is depicted in Fig. 3 (right). Following the notation in -\ntroduced in Ref. 11, we call this the ‘ R’ phase to emphasize\nits similarity to the rhombohedral parent phase. The intern al\nstructure remains similar to the unstrained bulk, with octa he-\ndrally coordinated Fe, a ferroelectric distortion consist ing of\nionic displacements along the pseudocubic (PS) [111] axis,\nandantiferrodistortiverotationsoftheFeO 6octahedraaround\n[111]PS. The [001] PScomponentofthe antiferrodistortivero-\ntation increases and the [110] PScomponent decreases as we\nshrink the in-plane lattice constant, but no qualitative ch ange\noccurs for compressive strains up to 4%. The electric po-\nlarization, constrained by symmetry to lie within the mono-\nclinic glide plane, is almost entirely along [111] PS, rotating\nslightly towards [001] PSas compressive strain is increased\n(seeFig.2d). Ourresultsareconsistentwiththegrowingbo dy\nofliteratureonBFOundermoderatecompressivestrain11,20,21\nAt 6% strain, the ground state resembles the P4mmstruc-\nturedescribedpreviouslyfortetragonally-constrainedB FO22.\nUsing the notationfrom Zeches et al.11, we refer to the high-\nstrain structure as the ‘ T’ phase for its similarity to tetrag-\nonalP4mm. The Fe undergoes large displacement towards\none of the apical oxygens(see Fig. 2c), resulting in a “super -\ntetragonal” structure with ﬁve-coordinated Fe, similar to the\ncoordination of the transition metals in perovskites PbVO 3\nand BiCoO 323,24(see Fig. 3). P4mmsymmetry is broken\nby antiferrodistortive tilting of the square-pyramidal ox ygen\ncages (analogous to octahedral tilt) around [110] PSby 5.1◦.\nInterestingly, the Tphase does not exhibit the [001] PSro-\ntations that have been associated with compressive strain i n\nother perovskites25,26. This is likely associated with the\nchange in Fe coordination, which encourages displacement\nalong[001] PSinresponsetodecreasedlatticeparameterrather\nthanrotationoftheoxygencages.\nAscanbeseeninFig.1(right),the Tphaseissigniﬁcantly\nlower in energy than P4mm. Furthermore, it follows from\nFig. 1 that at about 5% strain the energy reduction obtained\nby allowing the tiltings (going from P4mmtoCcatβ=90◦)\nis even larger than the subsequent reductionfrom relaxing β.\nTheseresultsindicatethattheformationofatetragonal P4mmphase in BFO ﬁlms grown on SrTiO 3substrates, as reported\ne.g. in Ref. 14, ishighlyunlikely.\nThe polarization vector is within the glide plane, as in\nthe low strain regime, but is now strongly rotated to be al-\nmost entirely out-of-plane, along [001] PS(Fig. 2d). In ad-\ndition, the magnitude (relative to a Pm¯3mreference struc-\nture)increasesfrom 96 µC/cm2in the unstrainedbulk to 150\nµC/cm2. This is in contrast to the modest enhancement re-\nportedinRef.10,wherearemnantpolarizationof75 µC/cm2\nis reported for Mn-doped ﬁlms on LaAlO 3substrates, com-\npared to 60 µC/cm2in bulk27. The discrepancy in enhance-\nment might be a result of the 5% Mn doping used to reduce\ncurrent leakage but it could also indicate incomplete switc h-\ning of the polarization. This is reasonable, as we expect the\nlarge distortion accompanying the polarization to be assoc i-\natedwith a largecoercivity. Reorientationof the out-of-p lane\npolarizationwouldrequireasigniﬁcantchangeinbondingf or\nthe5-coordinatedstructureanditispossiblethatthenatu reof\nswitching is quite differentin the super-tetragonalphase than\ninthebulk.\nWe observe that the total polarization /vectorPis nearly constant\nwithin the low strain regime (1%–2%) but that the out-of-\nplane projection grows at the expense of the in-plane projec -\ntion (Fig. 2d). This is in part associated with the growing\nc/a, but when we correct for that we ﬁnd that /vectorPis rotated\ntoward the [001] direction by several degrees at -1% strain.\nThis rotation in the low strain regime is qualitatively simi lar\nto that reported in Ref. 28, although we ﬁnd a much smaller\nenhancementofthetotal polarization. Therotationacross the\ntransition from RtoTis much larger: at a=3.73˚A the an-\ngle between /vectorPand [001] is 31.52◦; ata=3.71˚A it is 9.90◦.\nWe suspect that this dramatic rotation is responsible for th e\nrecentlydemonstratedabilitytoshiftthephaseboundaryw ith\nan electric ﬁeld in BFO ﬁlms with mixed TandRphases11.\nAnelectricﬁeldappliedalong[001]willfavoranout-of-pl ane\npolarization, thus growing the Tphase at the expense of R.\nThe propertiesthat are responsible for the side-by-side co ex-\nistence of TandRdomains, and for the reversible nature of\nthisdomainshifting,areaddressedlater.\nB. Octahedral Modes\nTo furtherquantifythe phasetransition, we decomposethe\nstructuraldistortionrelativeto a P4/mmmreferencestructure\ninto symmetry irreducible modes using the ISODISPLACE\nsoftware29. We ﬁndthelargestchangesasafunctionofstrain\nin the antiferrodistortive A4−andA5−modes (see Fig. 4).\nThese modes correspond to alternating rotations of the FeO 6\noctahedra around the [001] PSand [110] PSaxes, respectively,\nand are therefore a measure of the octahedral rotations and\ntilts,similartotheanalysispresentedinpreviouswork25,26. In\nfact,thedependenceofrotationandtiltangles(usingthet erms\nas deﬁned in Ref. 25) on lattice parameter are qualitatively\nidentical to those of the corresponding symmetry adapted\nmodes. As can be seen from Fig. 4, the transition between\nTandRphases can be characterized by the evolution of the\nA4−andA5−modesasa functionoflattice parameter,which4\nagain exhibit pronounced discontinuities around a=3.71˚A.\nInparticular,theA 4−modevanishesinthe Tphase. Thissug-\ngests that the isosymmetric phase transition is related to t he\nstabilityofthecorrespondingphononmode,whichisunstab le\nfor larger in-plane lattice parameterswhereas it becomess ta-\nbleforlargecompressivein-planestrain. Thephasetransi tion\nthusseemstobeofsimilarmicroscopicoriginasconvention al\nsoftmodetransitions,albeitwithnoresultingchangeinsp ace\ngroupsymmetry.\nC. MagneticCoupling\nFinally, we calculate the relative energies of likely mag-\nnetic orderings in the Tphase. We double the unit cell to\nallowA-andC-typeorderingsandre-relaxtheatomposition s\nfor each ﬁxed magnetic order. We do not optimize the vol-\nume for the different conﬁgurations but maintain that of the\nG-type unit cell. For the Rphase, G-type antiferromagnetic\nordering is the most stable magnetic conﬁguration across th e\nwhole strain range, consistent with bulk. In contrast, we ﬁn d\nthat in the Tphase, G-typeand C-type are nearlydegenerate,\nwithC-typebeingslightlylowerinenergy;forexample,at6 %\nstrain, the differenceis 6 meV per formulaunit30. In both G-\nandC-type,neighboringFe momentswithin(001) PSareanti-\nferromagnetically aligned, but whereas for G-type neighbo r-\ningmomentsarealsoantialignedintheout-of-planedirect ion,\nthey are ferromagnetically aligned for C-type. A-type orde r-\ning, with parallel orientationof all magnetic momentswith in\nthesame(001) PSplanebutalternatingorderintheperpendic-\nulardirection,isstronglyunfavorable.\nTo further quantify this, we map the calculated total ener-\ngies onto a nearest neighbor Heisenberg model where E=\n−1\n2ΣijJijSiSj, and calculate the magnetic coupling constants\nJijforSi=±5\n2. We distinguish between in-plane coupling,\nJin, and out-of-plane coupling Jout. For the T-phase at 6%\nstrain, we ﬁnd Jin=−10 meV and Jout=0.48 meV, whereas\nfor theR-phase at 1% strain, Jin=−9.8 meV and Jout=\n−7.6 meV. This shows that the increased distance between\nFe atoms along [001] PSin the high strain regime strongly re-\nducesthemagneticcouplingstrengthinthatdirection,lea ding\nto the very similar energiesof C- and G-type magnetic order.\nTheweakmagneticcouplingbetweenadjacent(001) PSplanes\nislikelytosigniﬁcantlyreducethemagneticorderingtemp er-\nature,inspiteofthestrongcouplingwithinindividual(00 1)PS\nplanes.\nIV. SUMMARYANDDISCUSSION\nInsummary,ourcalculationsoftheeffectofstrainonBFO\nrevealtwodistinctstructuresinthehighandlowstrainreg ime\nwith a discontinuous ﬁrst-order transition between them at\n∼4.5 % compressive strain. We note that, as a result of the\nconstraints imposed by coherence and epitaxy, both phases\nhave the same space group symmetry; such phase transi-\ntions are known are isosymmetric and are necessarily ﬁrst3.65 3.70 3.75 3.80 3.85 3.90\na (Å)00.20.40.60.8normalized mode amplitude6 5 432 1strain (%)\nA4-\nA5-\nFIG. 4: Displacement mode amplitudes for dominant antiferr odis-\ntortive modes. A 4−and A5−correspond to octahedral rotation and\ntilt,respectively.\norder31. Isosymmetric transitions have also been demon-\nstrated in pressure-induced transitions, as in Fe 0.47NbS2and\nα-PbF2,32,33orbytemperaturechanges,asin β-YbV4O8and\nthe fulleride Sm 2.75C60,34,35. Large volume changes appear\nto be characteristic. For example the pressure-induced spi n-\nstate transition in cerium is accompaniedby a 16.5% volume\nchange. IndeedBFO also undergoesa 9% change in volume\nfromR-toT-phase(seeFig.2b). Toourknowledgethisisthe\nﬁrstexampleintheliteratureofastrain-inducedisosymme tric\nphasetransition. TherecentworkbyLisenkov etal.15showed\nthat isosymmetricphasetransitionsin BiFeO 3ﬁlms mayalso\nbeinducedbyelectricﬁelds.\nWhile such isosymmetry might enable the coexistence of\nthe two phases recently reported in strained thin ﬁlms11, we\nsuggestthatthelowbarrierbetween RandTphase,asshown\nin Fig. 1, is likely also a requirement. An Rphase ﬁlm with\nlarge compressive strain near the transition region can low er\nits energy signiﬁcantly by relaxing to a slightly larger lat tice\nparameter. Thislatticeexpansionwouldnormallycreatema s-\nsive dislocationswithin the crystal, but becauseof the unu su-\nallyﬂatenergysurfacenearthetransitionregion,itcanbe ac-\ncommodatedby the simultaneousdevelopmentof Tdomains\nwith decreased lattice parameter, incurring a relatively s mall\nenergypenalty. Thismodelforcoexistencein BFO thin ﬁlms\ncan provide guidelinesfor identifying other systems with c o-\nexistingphases,namelythepresenceofanisosymmetrictra n-\nsition with a low energy barrier between phases. We also\nsuggest that the isosymmetric nature of the transition faci li-\ntatesthereversiblemovementoftheboundarybetween Rand\nTphases with an electric ﬁeld11, as it obviates the need to\nchange symmetry at the transition. The practical and techno -\nlogical implications of phase coexistence and morphotropi c\nphaseboundary-likebehaviorarebeingactivelyexploreda nd\nwe hope that the criteria suggested by our calculations will\nprovidedirectionforfuturestudies.\nAcknowledgments\nWe gratefully acknowledge support from the following:\nNSF Award Nos. DMR-0820404 and NIRT-0609377 (NAS\nand AJH) and Science Foundation Ireland through Contract5\nNo. SFI-07/YI2/I1051(CE). Travel support was providedby\nthe International Center for Materials Research through th e\nIMI Program of the NSF under Award No. DMR04-09848\n(AJH). Computational resources used include the SGI Altix\n[Cobalt] system and the TeraGridLinuxCluster [Mercury]at\nthe National Center for Supercomputing Applications underGrant No. DMR-0940420; CNSI Computer Facilities at UC\nSanta Barbara under NSF Grant No. CHE-0321368.; and fa-\ncilities provided by the Trinity Centre for High Performanc e\nComputing. We thank R.J. Zeches, M.D. Rossell, L.W Mar-\ntin,andR.Rameshforhelpfuldiscussions.\n1J. B. Neaton, C. Ederer, U. V. Waghmare, N. A. Spaldin, and\nK. M. Rabe,Phys. Rev. B 71, 014113 (2005).\n2C.EdererandN.A.Spaldin,Phys.Rev.B 71, 060401(R) (2005).\n3J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale,\nB.Liu,D.Viehland,V.Vaithyanathan,D.G.Schlom,U.V.Wag h-\nmare, et al.,Science 299, 1719 (2003).\n4T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A. 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Kippenberg1,z\n1Institute of Physics, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, Lausanne 1015, Switzerland\n2Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom\n(Dated: June 2, 2017)\nDevices that achieve nonreciprocal microwave transmission are ubiquitous in radar and radio-\nfrequency communication systems, and commonly rely on magnetically biased ferrite materials.\nSuch devices are also indispensable in the readout chains of superconducting quantum circuits\nas they protect sensitive quantum systems from the noise emitted by readout electronics. Since\nferrite-based nonreciprocal devices are bulky, lossy, and require large magnetic \felds, there has been\nsigni\fcant interest in magnetic-\feld-free on-chip alternatives, such as those recently implemented\nusing Josephson junctions. Here we realise recon\fgurable nonreciprocal transmission between two\nmicrowave modes using purely optomechanical interactions in a superconducting electromechanical\ncircuit. We analyse the transmission as well as the noise properties of this nonreciprocal circuit.\nThe scheme relies on the interference in two mechanical modes that mediate coupling between\nmicrowave cavities. Finally, we show how quantum-limited circulators can be realized with the\nsame principle. The technology can be built on-chip without any external magnetic \feld, and\nis hence fully compatible with superconducting quantum circuits. All-optomechanically-mediated\nnonreciprocity demonstrated here can also be extended to implement directional ampli\fers, and it\nforms the basis towards realising topological states of light and sound.\nNonreciprocal devices, such as isolators, circulators,\nand directional ampli\fers, exhibit altered transmission\ncharacteristics if the input and output channels are in-\nterchanged. They are essential to several applications\nin signal processing and communication, as they protect\ndevices from interfering signals [1]. At the heart of any\nsuch device lies an element breaking Lorentz reciprocal\nsymmetry for electromagnetic sources [2, 3]. Such el-\nements have included ferrite materials [4{6], magneto-\noptical materials [7{10], optical nonlinearities [11{13],\ntemporal modulation [14{19], chiral atomic states [20],\nand physical rotation [21]. Typically, a commercial non-\nreciprocal microwave apparatus exploits ferrite materi-\nals and magnetic \felds, which leads to a propagation-\ndirection-dependent phase shift for di\u000berent \feld polar-\nizations. A signi\fcant drawback of such devices is that\nthey are ill-suited for sensitive superconducting circuits,\nsince their strong magnetic \felds are disruptive and re-\nquire heavy shielding. In recent years, the major ad-\nvances in quantum superconducting circuits [22], that\nrequire isolation from noise emanating from readout elec-\ntronics, have led to a signi\fcant interest in nonreciprocal\ndevices operating at the microwave frequencies that dis-\npense with magnetic \felds and can be integrated on-chip.\nAs an alternative to ferrite-based nonreciprocal tech-\nnologies, several approaches have been pursued towards\nnonreciprocal microwave chip-scale devices. Firstly, the\nmodulation in time of the parametric couplings between\nmodes of a network can simulate rotation about an axis,\ncreating an arti\fcial magnetic \feld [14, 18, 23, 24] ren-\n\u0003These authors contributed equally to this work\nyalexey.feofanov@ep\r.ch\nztobias.kippenberg@ep\r.chdering the system nonreciprocal with respect to the ports.\nSecondly, phase matching of a parametric interaction can\nlead to nonreciprocity, since the signal only interacts\nwith the pump when copropagating with it and not in\nthe opposite direction. This causes travelling-wave am-\npli\fcation to be directional [24{27]. Phase-matching-\ninduced nonreciprocity can also occur in optomechanical\nsystems [28, 29], where parity considerations for the in-\nteracting spatial modes apply [30{32]. Finally, interfer-\nence in parametrically coupled multi-mode systems can\nbe used. In these systems nonreciprocity arises due to\ninterference between multiple coupling pathways along\nwith dissipation in ancillary modes [33]. Here dissipa-\ntion is a key resource to break reciprocity, as it forms\na \row of energy always leaving the system, even as in-\nput and output are interchanged. It has therefore been\nviewed as reservoir engineering [34]. Following this ap-\nproach, nonreciprocity has recently been demonstrated\nin Josephson-junctions-based microwave circuits [35, 36]\nand in a photonic-crystal-based optomechanical circuit\n[37]. These realisations and theoretical proposals to\nachieve nonreciprocity in multi-mode systems rely on a\ndirect, coherent coupling between the electromagnetic in-\nput and output modes.\nHere, in contrast, we describe a scheme to attain recon-\n\fgurable nonreciprocal transmission without a need for\nany direct coherent coupling between input and output\nmodes, using purely optomechanical interactions [28, 29].\nThis scheme neither requires cavity-cavity interactions\nnor phonon-phonon coupling, which are necessary for the\nrecently demonstrated optomechanical nonreciprocity in\nthe optical domain [37]. Two paths of transmission be-\ntween the microwave modes are established, through two\ndistinct mechanical modes. Interference between those\npaths with di\u000bering phases forms the basis of the nonre-arXiv:1612.08223v3  [quant-ph]  1 Jun 20172\nA\nB\nC\nFIG. 1. Optomechanical nonreciprocal transmission\nvia interference of two asymmetric dissipative cou-\npling pathways. A . Two microwave modes ^ a1and ^a2are\ncoupled via two mechanical modes ^b1and^b2through optome-\nchanical frequency conversion (as given by the coupling con-\nstantsg11;g21;g12;g22). Nonreciprocity is based on the inter-\nference between the two optomechanical (conversion) path-\nwaysg11;g21andg12;g22, in the presence of a suitably chosen\nphase di\u000berence \u001ebetween the coupling constants as well as\nthe deliberate introduction of an asymmetry in the pathways.\nB-C. The symmetry between the pathways can be broken by\no\u000b-setting the optomechanical transmission windows through\neach mechanical mode (dashed lines in dark and light green)\nby a frequency di\u000berence 2 \u000e. Each single pathway, in the\nabsence of the other mode, is described by eq. (2). In the\nforward direction ( B), the two paths interfere constructively,\nallowing transmission and a \fnite scattering matrix element\nS21on resonance with the \frst microwave cavity. In contrast,\nin the backward direction ( C), the paths interfere destruc-\ntively, such that S12\u00190, thereby isolating port 1 from port 2\non resonance with the second microwave cavity. The isolation\nbandwidth is determined by the intrinsic dissipation rate of\nthe mechanical modes.\nciprocal process. In fact, due to the \fnite quality factor\nof the intermediary mechanical modes, both conversion\npaths between the electromagnetic modes are partly dis-\nsipative in nature. Nonreciprocity is in this case only\npossible by breaking the symmetry between the two dis-\nsipative coupling pathways. We describe the mechanism\nin detail below, shedding some light on the essential in-\ngredients for nonreciprocity using this approach.\nWe \frst theoretically model our system to reveal how\nnonreciprocity arises. We consider two microwave modes\n(described by their annihilation operators ^ a1, ^a2) hav-\ning resonance frequencies !c;1,!c;2and dissipation rates\n\u00141,\u00142, which are coupled to two mechanical modes (de-\nscribed by the annihilation operators ^b1,^b2) having res-\nonance frequencies \n 1, \n2and dissipation rates \u0000 m;1,\n\u0000m;2(\fg. 1A). The radiation-pressure-type optomechan-ical interaction has the form [28, 29] g0;ij^ay\ni^ai(^bj+^by\nj)\n(in units where \u0016 h= 1), where g0;ijdesignates the vac-\nuum optomechanical coupling strength of the ithmi-\ncrowave mode to the jthmechanical mode. Four mi-\ncrowave tones are applied, close to each of the two lower\nsidebands of the two microwave modes, with detunings\nof \u0001 11= \u0001 21=\u0000\n1\u0000\u000eand \u0001 12= \u0001 22=\u0000\n2+\u000e\n(\fg. 2C). We linearise the Hamiltonian, neglect counter-\nrotating terms, and write it in a rotating frame with re-\nspect to the mode frequencies\nH=\u0000\u000e^by\n1^b1+\u000e^by\n2^b2+g11(^a1^by\n1+^ay\n1^b1)+g21(^a2^by\n1+^ay\n2^b1)\n+g12(^a1^by\n2+ ^ay\n1^b2) +g22(ei\u001e^a2^by\n2+e\u0000i\u001e^ay\n2^b2) (1)\nwhere ^aiand^bjare rede\fned to be the quantum \ruc-\ntuations around the linearised mean \felds. Here gij=\ng0;ijpnijare the \feld-enhanced optomechanical coupling\nstrengths, where nijis the contribution to the mean in-\ntracavity photon number due to the drive with detun-\ning \u0001ij. Although in principle each coupling is complex,\nwithout loss of generality we can take all to be real except\nthe one between ^ a2and^b2with a complex phase \u001e.\nWe start by considering frequency conversion through\na single mechanical mode. Neglecting the noise terms,\nthe \feld exiting the cavity ^ a2is given by ^ a2;out=\nS21^a1;in+S22^a2;in, which de\fnes the scattering matrix\nSij. For a single mechanical pathway, setting g12=g22=\n0 and\u000e= 0, the scattering matrix between input and\noutput mode becomes\nS21(!) =r\u0014ex;1\u0014ex;2\n\u00141\u00142pC11C21\u0000m;1\n\u0000eff;1\n2\u0000i!; (2)\nwhere\u0014ex;1,\u0014ex;2denote the external coupling rates\nof the microwave modes to the feedline, and the (mul-\ntiphoton) cooperativity for each mode pair is de\fned\nasCij= 4g2\nij=(\u0014i\u0000m;j). Conversion occurs within the\nmodi\fed mechanical response over an increased band-\nwidth \u0000 e\u000b;1= \u0000 m;1(1 +C11+C21). This scenario,\nwhere a mechanical oscillator mediates frequency conver-\nsion between electromagnetic modes, has recently been\ndemonstrated [38] with a microwave optomechanical cir-\ncuits [39], and moreover used to create a bidirectional link\nbetween a microwave and an optical mode [40]. Optimal\nconversion, limited by internal losses in the microwave\ncavities, reaches at resonance jS21j2\nmax=\u0014ex;1\u0014ex;2\n\u00141\u00142in the\nlimit of large cooperativities C11=C21\u001d1.\nWe next describe nonreciprocal transmission of the full\nsystem with both mechanical modes. We consider the\nratio of transmission amplitudes given by\nS12(!)\nS21(!)=g11\u001f1(!)g21+g12\u001f2(!)g22e+i\u001e\ng11\u001f1(!)g21+g12\u001f2(!)g22e\u0000i\u001e(3)\nwith the mechanical susceptibilities de\fned as \u001f\u00001\n1(!) =\n\u0000m;1=2\u0000i(\u000e+!) and\u001f\u00001\n2(!) = \u0000 m;2=2+i(\u000e\u0000!). Con-\nversion is nonreciprocal if the above expression has an\nmagnitude that di\u000bers from 1. If S21andS12di\u000ber only3\nA B\nC DIn\nOut\n2 µm\n/uni21261+δ/uni21262–δ\n/uni21261+δ/uni21262–δ\nωs,2 ωs,1 ωc,1 ωc,2eiϕp\n/uni21261\n2π=6.5MHz/uni21262\n2π=10.9MHz(0,1) (0,2)\nFIG. 2. Implementation of a superconducting microwave circuit optomechanical device for nonreciprocity. A . A\nsuperconducting circuit featuring two electromagnetic modes in the microwave domain is capacitively coupled to a mechanical\nelement and inductively coupled to a microstrip feedline. The end of the feedline is grounded and the circuit is measured\nin re\rection. B. Scanning electron micrograph of the drum-head-type vacuum gap capacitor (gap distance below 50 nm),\nmade from aluminium on a sapphire substrate. C. Frequency domain schematic of the microwave pump setup to achieve\nnonreciprocal mode conversion. Microwave pumps (red bars) are placed at the lower motional sidebands - corresponding to\nthe two mechanical modes - of both microwave resonances (dashed purple lines). The pumps are detuned from the exact\nsideband condition by \u0006\u000e= 2\u0019\u000118 kHz, creating two optomechanically induced transparency windows detuned by 2 \u000efrom\nthe microwave resonance frequencies. The phase \u001epof one the pumps is tuned. The propagation of an incoming signal (grey\nbar) in the forward and backward direction depends on this phase and nonreciprocal microwave transmission can be achieved.\nD. Finite-element simulation of the fundamental (0 ;1) and second order radially symmetric (0 ;2) mechanical modes, which\nare exploited as intermediary dissipative modes to achieve nonreciprocal microwave conversion.\nby a phase, it can be eliminated by a rede\fnition of either\n^a1or ^a2[24, 33]. Upon a change in conversion direction,\nthe phase\u001eof the coherent coupling (between the mi-\ncrowave and mechanical mode) is conjugated, while the\ncomplex phase associated with the response of the dissi-\npative mechanical modes remains unchanged. Physically,\nscattering from 1 !2 is related to scattering from 2 !1\nvia time-reversal, which conjugates phases due to coher-\nent evolution of the system. Dissipation is untouched\nby such an operation and thus remains invariant. In-\ndeed, the mechanical dissipation is an essential ingredi-\nent for the nonreciprocity to arise in this system, but not\nsu\u000ecient on its own. In fact, if we align the frequency\nconversion windows corresponding to the two mechanical\nmodes by setting \u000e= 0, the system becomes reciprocal\non resonance ( != 0), since there is no longer any phase\ndi\u000berence between numerator and denominator. This sit-\nuation corresponds to two symmetric pathways resulting\nfrom purely dissipative couplings; they can interfere only\nin a reciprocal way.\nWe study the conditions for isolation, when backward\ntransmission S12vanishes while forward transmission S21\nis non-zero. A \fnite o\u000bset 2 \u000ebetween the mechanical\nconversion windows causes an intrinsic phase shift for a\nsignal on resonance ( != 0) travelling one path compared\nto the other, as it falls either on the red or the blue side of\neach mechanical resonance. The coupling phase \u001eis then\nadjusted to cancel propagation in the backward directionS12(\fg. 1C), by cancelling the two terms in the numera-\ntor of eq. (3). In general, there is always a frequency !for\nwhichjg11\u001f1(!)g21j=jg12\u001f2(!)g22j, such that the phase\n\u001ecan be tuned to cancel transmission in one direction.\nSpeci\fcally, for two mechanical modes with identical de-\ncay rates (\u0000 m;1= \u0000 m;2= \u0000 m) and symmetric couplings\n(g11g21=g12g22), we \fnd that transmission from ports\n2 to 1 vanishes on resonance if\n\u0000m\n2\u000e= tan\u001e\n2: (4)\nThe corresponding terms of the denominator will have a\ndi\u000berent relative phase, and the signal will add construc-\ntively instead, in the forward direction (\fg. 1B). The de-\nvice thus acts as an isolator from ^ a1to ^a2, realised with-\nout relying on Josephson-junctions [35, 36]. We now\ndescribe the conditions to minimise insertion loss of the\nisolator in the forward direction. Still considering the\nsymmetric case, the cooperativity is set to be the same\nfor all modes (Cij=C). For a given separation \u000e, trans-\nmission on resonance ( != 0) in the isolating direction\nhas the maximum\njS21j2\nmax=\u0014ex;1\u0014ex;2\n\u00141\u00142\u0012\n1\u00001\n2C\u0013\n(5)\nfor a cooperativity C= 1=2 + 2\u000e2=\u00002\nm. As in the case for\na single mechanical pathway in eq. (2), for large coopera-4\nA B C\nD\n E\nFIG. 3. Experimental demonstration of nonreciprocity. A -C. Power transmission between modes 1 and 2 as a function\nof probe detuning, shown in both directions for pump phases \u001ep=\u00000:8\u0019;0;0:8\u0019radians. Isolation of more than 20 dB in the\nforward ( C) and backward ( A) directions is demonstrated, as well as reciprocal behaviour ( B).D. The ratio of transmission\njS21=S12j2, representing a measure of nonreciprocity, is shown as a function of pump phase \u001epand probe detuning. Two regions\nof nonreciprocity develop, with isolation in each direction. The system is recon\fgurable as the direction of isolation can be\nswapped by taking \u001ep!\u0000\u001ep.E. Theoretical ratio of transmission from eq. (3), calculated with independently estimated\nexperimental parameters. The theoretical model includes e\u000bectively lowered cooperativities for the mechanical mode ^b1due to\ncross-damping (optomechanical damping of the lower frequency mechanical mode by the pump on the sideband of the higher\nfrequency mechanical mode) acting as an extra loss channel.\ntivity the isolator can reach an insertion loss only limited\nby the internal losses of the microwave cavities.\nThe unusual and essential role of dissipation in this\nnonreciprocal scheme is also apparent in the analysis of\nthe bandwidth of the isolation. Although the frequency\nconversion through a single mechanical mode has a band-\nwidth \u0000 e\u000b;j(c.f. eq. (2)), caused by the optomechanical\ndamping of the pumps on the lower sidebands, the non-\nreciprocal bandwidth is set by the intrinsic mechanical\ndamping rates. Examination of eq. (3) reveals that non-\nreciprocity originates from the interference of two me-\nchanical susceptibilities of widths \u0000 m;j. One can conclude\nthat the intrinsic mechanical dissipation, which takes en-\nergy out of the system regardless of the transmission di-\nrection, is an essential ingredient for the nonreciprocal\nbehaviour reported here, as discussed previously [33, 34].\nIn contrast, optomechanical damping works symmetri-\ncally between input and output modes. By increasing\nthe coupling rates, using higher pump powers, the overall\nconversion bandwidth increases, while the nonreciprocal\nbandwidth stays unchanged.\nWe experimentally realise this nonreciprocal scheme\nusing a superconducting circuit optomechanical sys-\ntem in which mechanical motion is capacitively coupled\nto a multi-mode microwave circuit [39]. The circuit,schematically shown in \fg. 2A, supports two electro-\nmagnetic modes with resonance frequencies ( !c;1;!c;2) =\n2\u0019\u0001(4:1;5:2) GHz and energy decay rates ( \u00141;\u00142) =\n2\u0019\u0001(0:2;3:4) MHz, both of them coupled to the\nsame vacuum-gap capacitor. We utilise the fun-\ndamental and second order radially symmetric (0 ;2)\nmodes of the capacitor's mechanically compliant top\nplate [41] (cf. \fg. 2B and D) with resonance frequencies\n(\n1;\n2) = 2\u0019\u0001(6:5;10:9) MHz, intrinsic energy decay\nrates (\u0000 m;1;\u0000m;2) = 2\u0019\u0001(30;10) Hz and optomechanical\nvacuum coupling strengths ( g0;11;g0;12) = 2\u0019\u0001(91;12)\nHz, respectively (with g0;11\u0019g0;21andg0;12\u0019g0;22,\ni.e. the two microwave cavities are symmetrically cou-\npled to the mechanical modes). The device is placed at\nthe mixing chamber of a dilution refrigerator at 200 mK\nand all four incoming pump tones are heavily \fltered and\nattenuated to eliminate Johnson and phase noise (details\nare published elsewhere [42]). We establish a parametric\ncoupling between the two electromagnetic and the two\nmechanical modes by introducing four microwave pumps\nwith frequencies slightly detuned from the lower motional\nsidebands of the resonances, as shown in \fg. 2C and as\ndiscussed above. An injected probe signal !s1(s2) around\nthe lower (higher) frequency microwave mode is then\nmeasured in re\rection using a vector network analyser.5\nA\nB\nC\nD\nE\nFS21\nS12\nFIG. 4. Asymmetric noise emission of the nonreciprocal circuit. The noise emission is mainly due to mechanical\nthermal noise, that is converted through two paths to the microwave modes. The resulting interference creates a di\u000berent noise\npattern in the forward ( A-C ) and the backward ( D-F) directions when the circuit is tuned as an isolator from mode ^ a1to\n^a2.A, D . The two possible paths for the noise are shown for each mechanical mode. For ^b2, the direct path (orange) and\nthe indirect path going through mode ^b1(yellow) are highlighted. B, E . Each path on its own would result in a wide noise\nspectrum that is equally divided between the two microwave cavities (dashed yellow and orange lines). When both paths are\navailable, however, the noise interferes di\u000berently in each direction. In the backward direction ( E), a sharp interference peak\nappears, of much larger amplitude than the broad base. The theoretical curves (on an arbitrary logarithmic scale) are shown\nfor the symmetric case (\u0000 m;1= \u0000m;2) and for the single mode ^b2. Note that for the mode ^b1, the shape of the asymmetric peak\nin the backward noise would be the mirror image. C, F . Measured output spectra of modes ^ a2(C) and ^a1(F), calibrated to\nshow the photon \rux leaving the circuit. Because cross-damping provides extra cooling for the mode ^b1, the thermal noise of\n^b2is expected to dominate.\nFrequency conversion in both directions, jS21(!)j2and\njS12(!)j2, are measured and compared in \fg. 3A-C. The\npowers of the four pumps are chosen such that the asso-\nciated individual cooperativities are given by C11= 520,\nC21= 450,C12= 1350 andC22= 1280. The detun-\ning from the lower motional sidebands is set to \u000e=\n2\u0019\u000118 kHz. By pumping both cavities on the lower\nsideband associated with the same mechanical mode, a\nsignal injected on resonance with one of the modes will\nbe frequency converted to the other mode. This process\ncan add negligible noise, when operating with su\u000eciently\nhigh cooperativity, as demonstrated recently [38]. In the\nexperiment, the four drive tones are all phase-locked and\nthe phase of one tone \u001epis varied continuously from \u0000\u0019\nto\u0019. The pump phase is linked to the coupling phase \u001e\nby a constant o\u000bset, in our case \u001ep\u0019\u001e+\u0019. Between the\ntwo transmission peaks corresponding to each mechani-\ncal mode, a region of nonreciprocity develops, depending\non the relative phase \u001ep.\nThe amount of reciprocity that occurs in this process is\nquanti\fed and measured by the ratio of forward to back-\nward conversionjS21=S12j2. Figure 3D shows this quan-\ntity as a function of probe detuning and the relative pump\nphase. Isolation of more than 20 dB is demonstrated ineach direction in a recon\fgurable manner, i.e. the direc-\ntion of isolation can be switched by taking \u001ep!\u0000\u001ep,\nas expected from eq. (4). The ideal theoretical model,\nwhich takes into account \u0000 m;16= \u0000 m;2, predicts that the\nbandwidth of the region of nonreciprocity is commensu-\nrate with the arithmetic average of the bare mechani-\ncal dissipation rates, \u00182\u0019\u000120 Hz. However, given the\nsigni\fcantly larger coupling strength of the fundamental\nmechanical mode compared to the second order mode,\nand that\u00142=\n1;2is not negligible, the pump detuned by\n\n2\u0000\u000efrom the microwave mode ^ a2introduces consid-\nerable cross-damping (i.e. resolved sideband cooling) for\nthe fundamental mode. This cross-damping, measured\nseparately to be \u0000(cross)\nm;1\u00192\u0019\u000120 kHz at the relevant\npump powers, widens the bandwidth of nonreciprocal be-\nhaviour by over two orders of magnitude and e\u000bectively\ncools the mechanical oscillator. It also acts as loss in\nthe frequency conversion process and therefore e\u000bectively\nlowers the cooperativities to ( C11;C21)\u0019(0:78;0:68).\nThis lowered cooperativity accounts for the overall \u001810\ndB loss in the forward direction. This limitation can be\novercome in a future design by increasing the sideband\nresolution with decreased \u0014ior utilising the fundamental\nmodes of two distinct mechanical elements with similar6\n3g13\ng31 g22g21g11 g12\n,out3\n,in3,in\n,out,in ,out\n1\n2A\nB\nC\nD\nE\nFIG. 5. Proposal for a microwave optomechanical cir-\nculator. A . With a third microwave mode ^ a3coupled to the\nsame two mechanical oscillators, circulation can be achieved\nbetween the three microwave cavities. The circuit now in-\nvolves two independent loops, with two phases \u001e1and\u001e2\nthat can be tuned with the phases associated with g21and\ng11, respectively. B-C. The theoretical transmission in the\ncirculating direction (counter-clockwise) and the opposite di-\nrection (clockwise) are shown for two values of the coopera-\ntivityC. The isolation bandwidth scales with Cand is only\nlimited by the energy decay rates of the microwave modes.\nExperimentally realistic parameters are chosen with overcou-\npled cavities of energy decay rates \u00141=\u00142=\u00143= 2\u0019\u0001200 kHz\nand \u0000 m;1= \u0000 m;2= 2\u0019\u0001100 Hz. D-E. Noise emission spec-\ntra for the same two cooperativities, for \u0016 nm;1= \u0016nm;2= 800.\nNote that for the circulator the noise is symmetric for all the\ncavities, and that it decreases with increasing cooperativity.\ncoupling strengths. To compare the experiment to theory\nwe use a model that takes into account the cross-damping\nand an increased e\u000bective mechanical dissipation of the\nfundamental mode. The model is compared to the exper-\nimental data in \fg. 3E, showing good qualitative agree-\nment.\nFrom a technological standpoint, it does not su\u000ece\nfor an isolator to have the required transmission proper-\nties; since its purpose is to protect the input from anynoise propagating in the backward direction, the isola-\ntor's own noise emission is relevant. We therefore return\nto the theoretical description of the ideal symmetric case\nand derive the noise properties expected from the de-\nvice, in the limit of overcoupled cavities ( \u0014ex;i\u0019\u0014i).\nIn the forward direction and on resonance, the emitted\nnoise amounts to Nfw(0) = 1=2 + (\u0016nm;1+ \u0016nm;2)=(4C),\nwhere \u0016nm;jis the thermal occupation of each mechan-\nical mode. In the limit of low insertion loss and large\ncooperativity, the added noise becomes negligible in the\nforward direction. More relevant for the purpose of us-\ning an isolator to protect sensitive quantum apparatus\nis the noise emitted in the backward direction, given by\nNbw(0) = 1=2 + (\u0016nm;1+ \u0016nm;2)=2. Here the noise is di-\nrectly commensurate with the occupation of the mechan-\nics which can be of hundreds of quanta even at cryogenic\nmillikelvin temperatures, due to the low mechanical fre-\nquencies. This is a direct consequence of isolation with-\nout re\rection, since it prevents \ructuations from either\ncavity to emerge in the backward direction. In order to\npreserve the commutation relations of the bosonic out-\nput modes, the \ructuations consequently have to origi-\nnate from the mechanical modes. A practical low-noise\ndesign therefore requires a scheme to externally cool the\nmechanical modes, e.g. via sideband cooling using an\nadditional auxiliary microwave mode.\nThe origin of this noise asymmetry can be understood\nas noise interference. The thermal \ructuations of one\nmechanical oscillator are converted to microwave noise in\neach cavity through two paths, illustrated in \fg. 4A, D: a\ndirect (orange) and an indirect (yellow) link. Each path-\nway, on its own and with the same coupling strength,\nwould result in symmetric noise that decreases in mag-\nnitude with increasing cooperativity. When both are\npresent, however, the noise interferes with itself di\u000ber-\nently in each direction (cf. SI). In the forward direction,\nthe noise interferes destructively (\fg. 4B) leading to low\nadded noise, but in the backward direction a sharp in-\nterference peak arises (\fg. 4E) with \fnite noise in the\nnonreciprocal bandwidth even in the high-cooperativity\nlimit. In an intuitive picture, the circuit acts as a circu-\nlator that routes noise from the output port to the me-\nchanical thermal bath and in turn the mechanical noise to\nthe input port. We demonstrate experimentally the noise\nasymmetry by detecting the output spectra at each mi-\ncrowave mode while the device isolates the mode ^ a1from\n^a2by more than 25 dB (\fg. 4C, F). The cooperativities\nare here set to (C11;C21;C12;C22) = (20:0;14:2;106;89)\nwith a cross-damping \u0000(cross)\nm;1\u00192\u0019\u00012:6 kHz, in order\nto optimise the circuit for a lower insertion loss and in-\ncrease the noise visibility. As there is additional cooling\nfrom the o\u000b-resonant pump on mode ^b1, we expect noise\nfrom ^b2to dominate. There exists a way to circumvent\nthe mechanical noise entirely: introducing one extra mi-\ncrowave mode ^ a3, we can realise a circulator, where in-\nstead of mechanical \ructuations, the \ructuations from\nthe third microwave mode emerge in the backward direc-\ntion. The scheme is illustrated in \fg. 5A. As before, the7\ntwo mechanical modes are used to create two interfer-\ning pathways, now between the three microwave cavities.\nSince there are now two independent loops, two phases\nmatter; we choose the phases associated to the couplings\ng11andg21and set them respectively to \u001e1= 2\u0019=3\nand\u001e2=\u00002\u0019=3. With the mechanical detunings set\nto\u000ei=p\n3\n2(C+1\n3)\u0000m;i, the system then becomes a cir-\nculator that routes the input of port ^ a1to ^a2, ^a2to ^a3\nand so on. Critically and in contrast to above, counter-\npropagating signals are not dissipated in the mechanical\noscillators, but directed to the other port, with two ad-\nvantages. First, the bandwidth of nonreciprocity is not\nlimited to the mechanical dissipation rate, but instead\nincreases withCuntil reaching the ultimate limit given\nby the cavity linewidth (see \fg. 5B and C). Second, the\nmechanical noise emission is symmetrically spread be-\ntween the three modes, and over the wide conversion\nbandwidth (see \fg. 5D and E). In the large cooperativity\nlimit, the nonreciprocal process becomes quantum lim-\nited, irrespective of the temperature of the mechanical\nthermal baths.\nIn conclusion, we described and experimentally demon-\nstrated a new scheme for recon\fgurable nonreciprocal\ntransmission in the microwave domain using a supercon-\nducting optomechanical circuit. This scheme is based\npurely on optomechanical couplings, thus it alleviates\nthe need for coherent microwave cavity-cavity (or di-\nrect phonon-phonon) interactions, and signi\fcantly fa-\ncilitates the experimental realisation, in contrast to re-\ncently used approaches of optomechanical nonreciprocity\nin the optical domain [37]. Nonreciprocity arises due tointerference in the two mechanical modes, which medi-\nate the microwave cavity-cavity coupling. This interfer-\nence also manifests itself in the asymmetric noise output\nof the circuit. This scheme can be readily extended to\nimplement quantum-limited phase-preserving and phase-\nsensitive directional ampli\fers [43]. Moreover, an ad-\nditional microwave mode enables quantum-limited mi-\ncrowave circulators on-chip with large bandwidth, limited\nonly by the energy decay rate of the microwave modes.\nFinally, the presented scheme can be generalised to an\narray, and thus can form the basis to create topological\nphases of light and sound [44] or topologically protected\nchiral amplifying states [45] in arrays of electromechan-\nical circuits, without requiring cavity-cavity or phonon-\nphonon mode hopping interactions.\nACKNOWLEDGMENTS\nThis work was supported by the SNF, the NCCR\nQuantum Science and Technology (QSIT), and the EU\nHorizon 2020 research and innovation programme under\ngrant agreement No 732894 (FET Proactive HOT). 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Phys. 82, 1155 (2010).9\nSupplementary Information\nAppendix A: Theoretical background: Hamiltonian, scattering matrix, nonreciprocity\nIn this section we derive the e\u000bective Hamiltonian relevant for our system, calculate the input-output scattering\nmatrix for the electromagnetic modes, and discuss the conditions for obtaining nonreciprocal microwave transmission.\nWe consider two mechanical degrees of freedom whose positions parametrically modulate the frequencies of two\nelectromagnetic modes via radiation-pressure coupling [28]. The Hamiltonian describing this situation is given by\n(\u0016h= 1)\n^H=2X\ni=1\u0010\n!c;i^ay\ni^ai+ \ni^by\ni^bi\u0011\n+^Hint+^Hdrive; (A1)\nwhere ^a1and ^a2are the annihilation operators associated with the two electromagnetic modes with frequencies !c;1\nand!c;2, and ^b1and^b2are those for the two mechanical modes with mechanical frequencies \n 1and \n 2, respectively.\nRadiation-pressure coupling between the microwave and mechanical modes is described by the interaction Hamiltonian\n[28]\n^Hint=\u00002X\nj=12X\nk=1g0;jk^ay\nj^aj(^bk+^by\nk); (A2)\nwithg0;jkthe vacuum optomechanical coupling strength between electromagnetic mode jand mechanical mode k\nand where we neglect cross coupling terms /^ay\ni^aj, which is a good approximation for spectrally distinct modes\nj!c;1\u0000!c;2j\u001d\ni[46, 47].\nIn the experiment, both cavity modes are driven with two microwave tones each. These four tones are close to the\nlower mechanical sidebands, but the ones driving the mechanical sidebands at frequency \n 1are slightly detuned to the\nred, whereas the ones driving the sidebands at frequency \n 2are slightly detuned to the blue from the lower sideband.\nThat is, the detuning of the four drives are \u0001 jk=!jk\u0000!c;jwith \u0001 11= \u0001 21=\u0000\n1\u0000\u000eand \u0001 12= \u0001 22=\u0000\n2+\u000e.\nWe separate mean and \ructuations in the microwave \felds and move to a frame rotating at the cavity frequencies\n^aj=e\u0000i!c;jt \n(\u000e^aj) +2X\nk=1\u000bjke\u0000i\u0001jkt!\n(A3)\nwhere\u000bjkis the coherent state amplitude due to the microwave drive with detuning \u0001 jkwithj;k= 1;2 and (\u000e^aj)\ndescribe the \ructuations of the two microwave modes j= 1;2. We then linearize the Hamiltonian by approximating\n^ay\nj^aj\u0019\u0010\n\u000e^ay\nj\u0011 2X\nk=1\u000bjke\u0000i\u0001jkt!\n+ H:c: (A4)\nTo obtain a time-independent Hamiltonian we will assume that the system is in the resolved-sideband limit with\nrespect to both mechanical modes, i.e. \n 1;\n2\u001d\u00141;\u00142, and that the two mechanical modes are well separated in\nfrequency, i.e.j\n1\u0000\n2j\u001d\u0000m;1;\u0000m;2. Moving into a rotating frame with respect to the free evolution of the microwave\nmodes, and keeping only non-rotating terms, we obtain the e\u000bective Hamiltonian describing our system, which is given\nas equation (1) in the main manuscript,\nH=\u0000\u000e^by\n1^b1+\u000e^by\n2^b2+g11(^a1^by\n1+ ^ay\n1^b1) +g21(^a2^by\n1+ ^ay\n2^b1) +g12(^a1^by\n2+ ^ay\n1^b2) +g22(ei\u001e^a2^by\n2+e\u0000i\u001e^ay\n2^b2):(A5)\nHere,gjk=g0;jkj\u000bjkjare the optomechanical coupling strengths enhanced by the mean intracavity photon numbers\nnjk=j\u000bjkj2due to the drive at frequency !jkand where we have renamed ( \u000e^aj)!^ajfor notational convenience.\nWithout loss of generality, the phase of all but one coupling constant gjkcan be chosen real. Here, we take all of\nthem real and write out the phase \u001eexplicitly which is varied in our experiment.\nFrom the Hamiltonian (A5) we derive the equations of motion for our system which can be written in matrix form\nas [48, 49]\n_u=Mu+Luin (A6)10\nwith u = (^a1;^a2;^b1;^b2)T,uin = (^a1,in;^a2,in;^a(0)\n1,in;^a(0)\n2,in;^b1,in;^b2,in)Tand uout =\n(^a1,out;^a2,out;^a(0)\n1,out;^a(0)\n2,out;^b1,out;^b2,out)T, where ^ai;in/out are the input-output modes of the external microwave\nfeedline and ^ a(0)\ni;in/outare those corresponding to internal dissipation.\nThe matrix Mreads\nM=0\nBB@\u0000\u00141\n20\u0000ig11\u0000ig12\n0\u0000\u00142\n2\u0000ig21\u0000ig22e\u0000i\u001e\n\u0000ig11\u0000ig21 +i\u000e\u0000\u0000m,1\n20\n\u0000ig12\u0000ig22e+i\u001e0\u0000i\u000e\u0000\u0000m,2\n21\nCCA(A7)\nwhere the cavity dissipation rates are the sum of external and internal dissipation rates, i.e. \u00141=\u0014ex,1+\u00140,1and\n\u00142=\u0014ex,2+\u00140,2, and the matrix Lreads\nL=0\nBB@p\u0014ex,1 0p\u00140,1 0 0 0\n0p\u0014ex,2 0p\u00140,2 0 0\n0 0 0 0p\n\u0000m,1 0\n0 0 0 0 0p\n\u0000m,21\nCCA: (A8)\nUsing the input-output relations for a one-sided cavity [48, 49]\nuout=uin\u0000LTu (A9)\nwe can solve the input-output problem in the Fourier domain\nuout(!) =S(!)uin(!) (A10)\nwith the scattering matrix\nS(!) =16\u00026+LT[+i!14\u00024+M]\u00001L: (A11)\nEliminating the mechanical degrees of freedom from the equations of motion (A6) we obtain\n\u0012\u00141\n2\u0000i!+g2\n11\u001f1(!) +g2\n12\u001f2(!)g11\u001f1(!)g21+g12\u001f2(!)g22e+i\u001e\ng11\u001f1(!)g21+g12\u001f2(!)g22e\u0000i\u001e\u00142\n2\u0000i!+g2\n21\u001f1(!) +g2\n22\u001f2(!)\u0013\u0012\n^a1\n^a2\u0013\n= p\u0014ex;1^ain,1+p\u00140;1^a(0)\nin,1\u0000ig11\u001f1(!)p\n\u0000m;1^b1;in\u0000ig12\u001f2(!)p\n\u0000m;2^b2;inp\u0014ex;2^ain,2+p\u00140;2^a(0)\nin,2\u0000ig21\u001f1(!)p\n\u0000m;1^b1;in\u0000ig22\u001f2(!)e\u0000i\u001ep\n\u0000m;2^b2;in!\n(A12)\nwhere we introduced the mechanical susceptibilities \u001f\u00001\n1(!) = \u0000 m;1=2\u0000i(\u000e+!) and\u001f\u00001\n2(!) = \u0000 m;2=2 +i(\u000e\u0000!).\nInverting the matrix in equation (A12) and exploiting the input-output relation (A9), we obtain equation (3) of the\nmain manuscript\nS12(!)\nS21(!)=g11\u001f1(!)g21+g12\u001f2(!)g22e+i\u001e\ng11\u001f1(!)g21+g12\u001f2(!)g22e\u0000i\u001e: (A13)\nNote that the expressions in the nominator and denominator in (A13) are equal to the matrix elements coupling\nthe two electromagnetic modes in (A12) which are the sum of the two (complex) amplitudes for the two dissipative\noptomechanical pathways. Equation (A13) is used to generate Fig. 3 E of the main text, with all the parameters\n(\u0000m;j;gij) independently measured.\nFor identical mechanical decay rates \u0000 m;1= \u0000m;2= \u0000mand identical cooperativities C=Cij=4g2\nij\n\u0014i\u0000m;jwe \fnd that\nthe transmission 2 !1 vanishes on resonance != 0, i.e.S12= 0, if\n\u0000m\n2\u000e= tan\u001e\n2: (A14)\nFor a given \u000e, maximal transmission in the opposite direction 1 !2 is then obtained for C=1\n2+2\u000e2\n\u00002mand given by\njS21j2\nmax=\u0014ex;1\u0014ex;2\n\u00141\u001424\u000e2\n\u00002m+ 4\u000e2=\u0014ex;1\u0014ex;2\n\u00141\u00142\u0012\n1\u00001\n2C\u0013\n: (A15)11\nA B C\nFIG. A.1. Microwave transmission of the nonreciprocal electromechanical device in each direction for di\u000berent values of the\ncooperativityC, derived from eq. (A11) for the case of symmetric mechanical modes (\u0000 m;1= \u0000m;2= \u0000m). The detuning \u000eand\nthe phase\u001eare set for maximal transmission according to eq. (A15). As the cooperativity is increased, the overall bandwidth\nof the frequency conversion increases to \u0000 e\u000b, however the bandwidth of nonreciprocal transmission stays constant and is on\nthe order of the intrinsic mechanical damping rate \u0000 m. This illustrates the fact that the intrinsic dissipation of the mechanical\noscillator is the underlying resource for the nonreciprocity.\nWe see that for \u000e\u001d\u0000mthe optimal cooperativity C!1 andjS21(0)j2!1. Thus, we see that in this limit the\nelectromagnetic scattering matrix of our system becomes that of an ideal isolator, i.e. S11=S12=S22= 0 and\njS21j= 1.\nThe full scattering matrix Sijof (A11) is used in \fg. A.1 to show optimal transmission in each direction for the\nsymmetric case, with di\u000berent values of the cooperativity. As the cooperativity increases, the overall bandwidth\nof conversion increases to \u0000 e\u000b, but the nonreciprocal bandwidth stays constant. This can be seen in the ratio\nS12(!)=S21(!) in (A13) that depends only on the bare mechanical susceptibilities \u001f1(!) and\u001f2(!).\nFor unequal decay rates \u0000 m;16= \u0000m;2, but equal e\u000bective decay rates of the mechanical modes \u0000 e\u000b;j= \u0000m;j(1+C1j+\nC2j), nonreciprocity is obtained for\u0000+\n2\u000e= tan\u001e\n2o\u000b-resonance at a frequency !=\u0000+\u0000\u0000\n4\u000ewhere \u0000\u0006=1\n2(\u0000m;1\u0006\u0000m;2).\nFor unequal decay rates \u0000 m;16= \u0000 m;2, but matched cooperativities Cjk=C, we \fnd nonreciprocity for\u0000+\n2\u000e= tan\u001e\n2,\nbut at!=\u0000\u0000\u0000\u000e\n\u0000+.\nAppendix B: Theoretical background: Noise analysis of the device\nIn this section we analyse the noise properties of the nonreciprocal electromechanical device. We assume the bosonic\ninput noise operators obey\nh^a1,in(t)^ay\n1,in(t0)i=\u000e(t\u0000t0) (B1)\nh^a2,in(t)^ay\n2,in(t0)i=\u000e(t\u0000t0) (B2)\nh^a(0)\n1,in(t)^a(0)y\n1,in(t0)i=\u000e(t\u0000t0) (B3)\nh^a(0)\n2,in(t)^a(0)y\n2,in(t0)i=\u000e(t\u0000t0) (B4)\nh^b1,in(t)^by\n1,in(t0)i= (\u0016nm,1+ 1)\u000e(t\u0000t0) (B5)\nh^b2,in(t)^by\n2,in(t0)i= (\u0016nm,2+ 1)\u000e(t\u0000t0); (B6)\ni.e. the baths of the microwave modes are assumed to be at zero temperature whereas the mechanical modes have a\n\fnite thermal occupation \u0016 nm;1and \u0016nm;2, respectively.\nThe symmetrised output noise spectra [49] are determined by the scattering matrix of the device (A11) as well as\nthe noise properties of the microwave and mechanical baths (B1)-(B6). Explicitly, we \fnd that the cavity output\nspectra are given by\n\u0016S1,out(!) =1\n2Z1\n\u00001dtei!th^ay\n1,out(t)^a1,out(0) + ^a1,out(0)^ay\n1,out(t)i\n=1\n2\u0002\njS11(\u0000!)j2+jS12(\u0000!)j2+jS13(\u0000!)j2+jS14(\u0000!)j2\u0003\n+jS15(\u0000!)j2(\u0016nm,1+1\n2) +jS16(\u0000!)j2(\u0016nm,2+1\n2)\n(B7)12\nand\n\u0016S2,out(!) =1\n2Z1\n\u00001dtei!th^ay\n2,out(t)^a2,out(0) + ^a2,out(0)^ay\n2,out(t)i\n=1\n2\u0002\njS21(\u0000!)j2+jS22(\u0000!)j2+jS23(\u0000!)j2+jS24(\u0000!)j2\u0003\n+jS25(\u0000!)j2(\u0016nm,1+1\n2) +jS26(\u0000!)j2(\u0016nm,2+1\n2):\n(B8)\nIn the limit of overcoupled cavities \u0014ex;i\u0019\u0014iand for the optimal phase \u001eand detuning \u000e, the noise emitted in the\nbackward direction 2 !1 on resonance != 0 is\nNbw=\u0016S1,out(0) =jS11j2\u00011\n2+jS12j2\u00011\n2+jS15j2\u0001\u0012\n\u0016nm,1+1\n2\u0013\n+jS16j2\u0001\u0012\n\u0016nm,2+1\n2\u0013\n= 0\u00011\n2+ 0\u00011\n2+1\n2\u0001\u0012\n\u0016nm,1+1\n2\u0013\n+1\n2\u0001\u0012\n\u0016nm,2+1\n2\u0013\n=1\n2+\u0016nm,1+ \u0016nm,2\n2;(B9)\ni.e. in the backward direction the noise of the device is dominated by the noise emitted by the mechanical oscillators.\nThe noise emitted in the forward direction 1 !2 on resonance != 0 is\nNfw=\u0016S2,out(0) =jS21j2\u00011\n2+jS22j2\u00011\n2+jS25j2\u0001\u0012\n\u0016nm,1+1\n2\u0013\n+jS26j2\u0001\u0012\n\u0016nm,2+1\n2\u0013\n=\u0012\n1\u00001\n2C\u0013\n\u00011\n2+ 0\u00011\n2+1\n4C\u0001\u0012\n\u0016nm,1+1\n2\u0013\n+1\n4C\u0001\u0012\n\u0016nm,2+1\n2\u0013\n=1\n2+\u0016nm,1+ \u0016nm,2\n4C;(B10)\ni.e. in the forward direction the noise contribution from the mechanical oscillators vanishes at large cooperativity\nC\u001d 1. Therefore, intriguingly, the noise emitted on resonance by the nonreciprocal device is not symmetric in the\nforward and backward directions.\nAppendix C: Theoretical background: Noise interference as origin of asymmetric noise emission\nIn the previous section we concluded that the circuit emits more noise in the backward direction as compared to\nthe forward direction. This is also corroborated by the experimental data, shown in Fig. 4 in the main text. In the\nfollowing, in order to understand the di\u000berent noise performance in the forward and backward direction, we consider\ntwo di\u000berent points of view. First, we derive the scattering amplitude from one mechanical resonator to one cavity,\neliminating the other two modes. In this picture, the imbalance can be understood as an interference of the two paths\nthe noise can take in the circuit, analogously to the interference in the microwave transmission. Second, we eliminate\nthe mechanical resonators, but taking their input noise into account. This leads to the same scattering matrix for\nthe microwaves as discussed in the main text, but the mechanical noise appears as additional, e\u000bective noise input\noperators for the cavities. In the second formulation we can therefore use our knowledge of the microwave scattering\nmatrix to deduce properties of the noise scattering.\nLet us \frst consider the scattering from a mechanical resonator to cavities 1 and 2. Since in the experiment mechan-\nical resonator 1 is strongly cross-damped due to o\u000b-resonant couplings, the noise emitted stems almost exclusively\nfrom resonator 2. If we are interested in the noise scattering from mechanical resonator 2 to cavity 2, we can eliminate\nthe other two modes and drop their input noise. In frequency space, their equations of motion are\n\u0012\u001f\u00001\nc;1(!)\u0000ig11\n\u0000ig\u0003\n11\u001f\u00001\n1(!)\u0013\u0012^a1\n^b1\u0013\n=\u0012\nig12^b2\nig\u0003\n21^a2\u0013\n+ noises: (C1)\nWe drop the noise terms and solve for ^ a1;^b1\n\u0012^a1(!)\n^b1(!)\u0013\n=1\n\u001f\u00001\n1(!)\u001f\u00001\nc;1(!) +jg11j2\u0012\u001f\u00001\n1(!)ig11\nig\u0003\n11\u001f\u00001\nc;1(!)\u0013\u0012\nig12^b2(!)\nig\u0003\n21^a2(!)\u0013\n\u0011\u001f^a1^b1(!)\u0012\nig12\nig\u0003\n21\u0013\u0012^b2(!)\n^a2(!)\u0013\n;(C2)13\nwhere we de\fned the cavity susceptibility \u001f\u00001\nc;i(!) =\u0014i=2\u0000i!and the susceptibility of the coupled system of modes\n^a1;^b1,\u001f^a1^b1(!). We turn to the other two modes, the ones that we are actually interested in. For those, we have a\nsimilar equation, which can be obtained from interchanging 1 $2\n\u0012\u001f\u00001\n2(!)\u0000ig\u0003\n22\n\u0000ig22\u001f\u00001\nc;2(!)\u0013\u0012^b2(!)\n^a2(!)\u0013\n=\u0012\nig\u0003\n12\nig21\u0013\u0012^a1(!)\n^b1(!)\u0013\n+\u0012p\n\u0000m;2^b2;in(!)p\u00142^a2;in(!)\u0013\n: (C3)\nEliminating the modes ^ a1;^b1with eq. (C2), we arrive at\n\u0012p\n\u0000m;2^b2;in(!)p\u00142^a2;in(!)\u0013\n=2\n664\u0012\u001f\u00001\n2(!)\u0000ig\u0003\n22\n\u0000ig22\u001f\u00001\nc;2(!)\u0013\n\u0000\u0012\nig\u0003\n12\nig21\u0013\u0012\u001f\u00001\n1(!)ig11\nig\u0003\n11\u001f\u00001\nc;1(!)\u0013\u0012\nig12\nig\u0003\n21\u0013\n\u001f\u00001\n1(!)\u001f\u00001\nc;1(!) +jg11j23\n775\u0012^b2(!)\n^a2(!)\u0013\n\u0011\u0014\n\u001f\u00001\n^b2^a2(!)\u0000\u0012\nig\u0003\n12\nig21\u0013\n\u001f^a1^b1(!)\u0012\nig12\nig\u0003\n21\u0013\u0015\u0012^b2(!)\n^a2(!)\u0013\n:(C4)\nIn the second line, we have formulated the equation in terms of the susceptibilities of the two subsystems (^ a1;^b1) and\n(^a2;^b2). This equation is a bit complicated, but we note that the coupling between ^ a2and^b2is\niT^a2^b2(!) =\u0000ig22\"\n1\u0000e\u0000i\u001epC12C21=(C22C11)\n1 + (\u001fc;1(!)\u001f1(!)jg11j2)\u00001#\n: (C5)\nAnalogously, changing the indices referring to the cavity, we obtain the coupling between ^ a1and^b2\niT^a1^b2(!) =\u0000ig12\"\n1\u0000e+i\u001epC11C22=(C12C21)\n1 + (\u001fc;2(!)\u001f1(!)jg21j2)\u00001#\n; (C6)\nThe coupling phase \u001epappears as the relative phase between indirect and direct coupling path, as for the microwave\nsignal transmission. Equations (C5) and (C6) demonstrate that the transmission of noise from the mechanical res-\nonators to the microwave cavities is subject to interference, which ultimately leads to the di\u000berence in noise emitted\nin the forward versus the backward direction.\nIn a second picture, we can also understand the mechanical noise interference in terms of the nonreciprocity in the\nscattering matrix for the microwave modes. In order to do so, we solve the equations of motion for the mechanical\nresonators (given in eq. (A6)), which leads to\n^bj(!) =\u001fj(!)\"\niX\nig\u0003\nij^ai(!) +p\n\u0000m;j^bj;in(!)#\n: (C7)\nWe obtain equations that only relate the cavities\n\u0012\u001f\u00001\nc;1(!) +iT11(!)iT12(!)\niT21(!)\u001f\u00001\nc;2(!) +iT22(!)\u0013\u0012\n^a1(!)\n^a2(!)\u0013\n=i\u0012\ng11g12\ng21g22\u0013\u0012p\n\u0000m;1\u001f1(!)^b1;in(!)p\n\u0000m;2\u001f2(!)^b2;in(!)\u0013\n+\u0012p\u00141^a1;in(!)p\u00142^a2;in(!)\u0013\n;(C8)\nwhere\niTij(!)\u0011\u0000iX\nk\u001fk(!)gikg\u0003\njk: (C9)\nWe can think of mechanical noise as coloured and correlated noise in the optical inputs. That is, consider the\nreplacement\n\u0012p\u00141^c1;in(!)p\u00142^c2;in(!)\u0013\n\u0011i\u0012\ng11g12\ng21g22\u0013\u0012p\n\u0000m;1\u001f1(!)^b1;in(!)p\n\u0000m;2\u001f2(!)^b2;in(!)\u0013\n: (C10)\nThe e\u000bective noise ^ ci;inis both colouredh^cy\n1;in(!)^c1;in(!0)i6=\u000e(!+!0)\u0016n1,e\u000band correlatedh^cy\n1;in(!)^c2;in(!0)i6= 0.\nUsing the input-output relation ^ aout= ^ain\u0000p\u0014^a, the cavity output is given by\n\u0012\n^a1;out(!)\n^a2;out(!)\u0013\n=S(!)\u0012\n^a1;in(!)\n^a2;in(!)\u0013\n+ [S(!)\u000012]\u0012\n^c1;in(!)\n^c2;in(!)\u0013\n; (C11)14\nwhere in the last step we have identi\fed the 2-by-2 optical scattering matrix S(!) that relates the cavity inputs to\nthe outputs ^ ai;out(!) =P\njSij(!)^aj;in(!). The fact that eq. (C11) contains mechanical noise as well, but can be\nwritten entirely in terms of the optical scattering matrix constitutes the central result here. Since the two e\u000bective\ninput noises ^ ci;inare coloured and correlated, they can interfere.\nMost importantly, we can consider what happens when the circuit is impedance matched to the signal and perfectly\nisolating. We choose the detunings \u000e1= \u0000m;1\u000e=2;\u000e2=\u0000\u0000m;2\u000e=2, for some dimensionless parameter \u000e. For simplicity,\nlet us choose all cooperativities to be equal C=Cij. For\u000e2= 2C\u00001 (impedance matching), the optical scattering\nmatrix of the isolator is (up to some irrelevant phase)\nS(0) =\u00120 0p\n1\u00001=(2C) 0\u0013\n\u0011T\u0012\n0 0\n1 0\u0013\n: (C12)\nThe cavity output on resonance is\n\u0012\n^a1;out\n^a2;out\u0013\n=T\u0012\n0\n^a1;in\u0013\n\u0000ip\n2C\u0012\nei\u001ep 1\n1\u0000Tei\u001ep1\u0000T\u0013\u0012^b1;in(0)\n^b2;in(0)\u0013\n: (C13)\nAsC!1 ,T!1 and\u001ep= arccos(1\u00001=C)!0, such that the second cavity does not receive any noise, which\nis due to an interference of ^ c1;inwith ^c2;in. In the backward direction, no interference can take place, since cavity\n2 is isolated from cavity 1. As a consequence, the number of noise quanta emerging from cavity 1 on resonance is\nNbw= (\u0016nm;1+ \u0016nm;2+ 1)=2.\nAppendix D: Theoretical background: optomechanical circulator\nWe consider three microwave modes (described by their annihilation operators ^ a1, ^a2, ^a3) with resonance frequencies\n!c;1,!c;2,!c;3and dissipation rates \u00141,\u00142,\u00143. These three microwave modes are coupled to two mechanical modes\n(described by the annihilation operators ^b1,^b2) with resonance frequencies \n 1, \n2and dissipation rates \u0000 m;1and\n\u0000m;2. The optomechanical coupling strengths gijare taken to be real and we write out three phases ( \u001e1,\u001e2,\u001e3).\nEach loop has a relevant phase which is a linear combination of those previous phases. The three cavities are\ndriven with two microwave tones each. These six tones are close to the lower motional sidebands, with detunings of\n\u000111= \u0001 21= \u0001 31=\u0000\n +\u000e1and \u0001 12= \u0001 22= \u0001 32=\u0000\n2+\u000e2;\u000e1and\u000e2are to be determined. The cooperativities\nare matchedC=Cij=4gij\n\u0014i\u0000j.\nThe linearised Hamiltonian, describing the system, in a frame rotating with the cavity frequencies and keeping non-\nrotating terms is given by\n^H=\u000e1^by\n1^b1+\u000e2^by\n2^b2\n+g11(^ay\n1^b1ei\u001e1+ ^a1^by\n1e\u0000i\u001e1) +g12(^ay\n1^b2+ ^a1^by\n2)\n+g21(^ay\n2^b1ei\u001e2+ ^a2^by\n1e\u0000i\u001e2) +g22(^ay\n2^b2+ ^a2^by\n2)\n+g31(^ay\n3^b1ei\u001e3+ ^a3^by\n1e\u0000i\u001e3) +g32(^ay\n3^b2+ ^a3^by\n2):(D1)\nFrom the Hamiltonian (D1), we derive the equations of motion for our system in their matrix form\n(A6) with u= (^a1;^a2;^a3;^b1;^b2)T,uin= (^a1;in;^a2;in;^a3;in;^a(0)\n1;in;^a(0)\n2;in;^a(0)\n3;in;^b1;in;^b2;in)Tand uout =\n(^a1;out;^a2;out;^a3;out;^a(0)\n1;out;^a(0)\n2;out;^a(0)\n3;out;^b1;out;^b2;out)T. The Mmatrix reads\nM=0\nBBBB@\u0000\u00141\n20 0 \u0000ig11ei\u001e1\u0000ig12\n0\u0000\u00142\n20\u0000ig21ei\u001e2\u0000ig22\n0 0 \u0000\u00143\n2\u0000ig31ei\u001e3\u0000ig32\n\u0000ig11e\u0000i\u001e1\u0000ig21e\u0000i\u001e2\u0000ig31e\u0000i\u001e3\u0000\u0000m,1\n2+i\u000e1 0\n\u0000ig12\u0000ig22\u0000ig32 0\u0000\u0000m,2\n2+i\u000e21\nCCCCA; (D2)\nwhere the cavity dissipation rates are the sum of the external and internal dissipation rates, i.e. \u0014i=\u0014ex;i+\u00140;i, and\ntheLmatrix reads\nL=0\nBBBB@p\u0014ex,1 0 0p\u00140;10 0 0 0\n0p\u0014ex,2 0 0p\u00140;20 0 0\n0 0p\u0014ex,3 0 0p\u00140;3 0 0\n0 0 0 0 0 0p\n\u0000m,1 0\n0 0 0 0 0 0 0p\n\u0000m,21\nCCCCA: (D3)15\nUsing the input-output relation (A9) and the matrix form of the equations of motion (A6), we can compute the\nscattering matrix S(!) similarly to (A11).\nWe arbitrarily choose to suppress the propagation in the clockwise direction, i.e. S12(0) = 0,S23(0) = 0, and\nS31(0) = 0. To enforce this suppression to occur on resonance ( != 0) ,\u000e1scales with \u0000 m,1and\u000e2with \u0000 m,2, i.e.\n\u000e1=\u000b\u0000m,1and\u000e2=\f\u0000m,2.\nThe set of equations corresponding to S12(0) =S23(0) =S31(0) = 0 is given by\n8\n><\n>:\u00002i\u000b\u00002i\fei(\u001e1\u0000\u001e2)\u0000C(1\u0000ei(\u001e1\u0000\u001e3)\u0000ei(\u001e3\u0000\u001e2)+ei(\u001e1\u0000\u001e2))\u0000(1 +ei(\u001e1\u0000\u001e2)) = 0\n\u00002i\u000b\u00002i\fei(\u001e2\u0000\u001e3)\u0000C(1\u0000ei(\u001e2\u0000\u001e1)\u0000ei(\u001e1\u0000\u001e3)+ei(\u001e2\u0000\u001e3))\u0000(1 +ei(\u001e2\u0000\u001e3)) = 0\n\u00002i\u000b\u00002i\fei(\u001e3\u0000\u001e1)\u0000C(1\u0000ei(\u001e3\u0000\u001e2)\u0000ei(\u001e2\u0000\u001e1)+ei(\u001e3\u0000\u001e1))\u0000(1 +ei(\u001e3\u0000\u001e1)) = 0(D4)\nAnalysing this set of equations, we see that only two phases are independent. Setting \u001e1= 2\u0019=3,\u001e2=\u00002\u0019=3 and\n\u001e3= 0 leads to a set of fully degenerated equations.\nS13(0) =S32(0) =S21(0) = 0 if\n2p\n3\f\u00003C\u00001 +i\u0010\n4\u000b\u00002\f+ 3p\n3C+p\n3\u0011\n= 0: (D5)\nSolving the previous equation (D5) with respect to the cooperativity Cgives\nC=2\fp\n3\u00001\n3and\u000b=\u0000\f: (D6)\nIf\u000b6=\u0000\f,Ccontains an imaginary part leading to complex coupling strengths but we assumed them to be real.\nAlso,Chas to be positive (such that gij2R) and non-zero (else gij= 0). The lower bound for \fis given by\n\f >1\n2p\n3: (D7)\nLet us express the transmission in the counter clockwise direction ( jS13j2,jS32j2andjS21j2) on resonance as a\nfunction of the cooperativity C\njS13j2=\u0014ex;1\u0014ex,3\n\u00141\u001431\n(1 +1\n3C)2;\njS32j2=\u0014ex,3\u0014ex,2\n\u00143\u001421\n(1 +1\n3C)2;\njS21j2=\u0014ex,2\u0014ex,1\n\u00142\u001411\n(1 +1\n3C)2:(D8)\nWe \fnd that, in the case of overcoupled cavities \u0014i\u0019\u0014ex;i, the transmission approaches unity with increasing\ncooperativity.\nThe symmetrised output noise spectra is computed as in B. In the limit of overcoupled cavities, \u0014i\u0019\u0014ex;i, the\nnoise emitted on resonance at each port is given by\nN=1\n2+3C\n(3C+ 1)2(\u0016nm,1+ \u0016nm,2) (D9)\nwith 0<C<min (\u0014i=\u0000j). In the limit of large cooperativity, the noise contribution from the mechanical oscillators\nis entirely suppressed, leaving only vacuum noise of half a quantum."    },    {        "title": "2012.03164v1.Magnetless_Circulators_Based_on_Synthetic_Angular_Momentum_Bias__Recent_Advances_and_Applications.pdf",        "content": "Submitted to the IEEE Antennas and Propagation Magazine  \n 1 \n \nAbstract— In this paper, we discuss recent progress in \nmagnet -free non -reciprocal structures b ased on a synthetic form of \nangular momentum  bias imparted via spatiotemporal modu lation. \nWe discuss how such components can support metrics of \nperformance comparable with traditional magnetic -biased ferrite \ndevices, while at the same time offering distinct advantages in \nterms of reduced size, weight, and cost due to the elimination of \nmagnetic bias. We further provide an outlook on potential \napplications and future directions based on these components, \nranging from wireless full -duplex communications to metasurfaces  \nand topological insulators . \nIndex Terms — Angular -momentum, n on-recipr ocity, \nmagnet -free, spatio temporal modulation, metamaterials.  \nI. INTRODUCTION  \nHE demand for multimedia -rich applications and services \nhas been dramatically increasing over the last few decades. \nAs a result, the sub -6 GHz electromagnetic spectrum, where \nmost o f deployed communication systems operate, has become \nsaturated with unprecedented data traffic leaving no more \nbandwidth for further growth in the transmission  rate. In order \nto overcome thi s problem, full‐duplex radios have been \nproposed. In such systems, data streams are transmitted in the \nuplink and downlink simultaneously on the same carrier \nfrequency [1]-[9], which doubles the spectral efficiency \ncompared to half -duplex time - and frequency -division systems. \nThe key challenge in full -duplexing, however, is that it requires \nconsiderably large self -interference cancellation (SIC) between \nthe transmitter ( Tx) and receiver (Rx) of every communication \nnode in order to prevent the strong Tx signal from saturating the \nRx frontend modules. For instance, WiFi signals are transmitted \nat an average power of +20 dBm and the noise floor is around \n–90 dBm, hence SIC i s required to be as high as +110 dB. For \n \n Manuscript received April  6, 2020. This work wa s supported by a \nQualcomm Innovation Fellowship , an IEEE Mi crowave Theory and \nTechniques Society Graduate Fellowship,  the Air Force Office of Scientific \nResearch,  the Defense Advanced Research Projects Agency,  Silicon Audio, the \nSimons Foundation, and the National Science Foundation. (Corresponding \nauthor: Andrea Alù.) \nA. Kord is with the Department of Electrical Engineering, Columbia \nUniversity, New York, NY 10027, USA.  \nA. Alù is with the Department of Electrical and Computer Engineering,  \nUniversity of Texas at Austin, Austin, TX 78712 USA, and also with the  \nAdvan ced Science Research Center, City University of New York, New York,  \nNY 10031 , USA . (e-mail: aalu@gc.cuny.edu)  \n Color versions of one or more of the figures in this paper are available online \nat http://ieeexplore.ieee.org.  many years, this level of SIC was impossible to attain, therefore \nfull-duplex radios were presumed irrelevant in practice. \nRecently, several works challenged this long -held assumption \nand demonstrated that full -duplexing can, in fact, be achieved \nby using a combination of radio -frequency (RF) [3]-[5], analog \n[6]-[7], and digital techniques [8]-[9]. The basic idea in digital \nand analog approaches is to generate a weighted version of the \nTx signal that mimics i ts echo into the receiver and subtract it \nfrom the Rx signal before and after the analog -to-digital \nconverter. Since the strength contrast between the Tx and Rx \nsignals can be billions in magnitude, the subtraction must be \nperformed extremely accurately, o therwise it may end up adding \nmore interference. But the presence of non -idealities such as \nnoise, non -linearity , and dynamic variation of the antenna input \nimpedance, make this process a non -trivial task. Therefore, \nanalog and digital SIC approaches must be preceded  by another \nlayer of isolation at the RF frontend in order to relax their design \nrequirements and mitigate their sensitivity to inevitable errors. \nFortunately, circulators can play this role [10]-[64]. Not only \nthey can achieve low insertion loss, thus overcoming the \nfundamental limit of electrical -balance duplexers, but they also \nallow the Tx and Rx nodes to share a single antenna thus making \nfull-duplexing  more attractive compared to multi -input \nmulti -output (MIMO) radios.  \n For decades, circulators have been built almost exclusively \nthrough magnetic biasing of rare -earth ferrites [10]-[19]. These \nmagnetic devices can achieve excellent performance in terms of \nisolation, power handling, bandwidth of operation , and insertion \nloss, yet they typically result in bulky, non -integrable , and \nexpensive devices that may not be sui ted for various \ncommercial systems. In order to overcome these problems, \nmagnet -free implementations of circulators based on active \nsolid -state devices  or nonlinear materials with geometric \nasymmetries have been pursued [20]-[29]. Despite significant \nresearch efforts over many years, these alternative solutions \ncontinued to suffer from a fundamentally poor noise figure and \nlimited power handling, hence the appeal of a fu lly passive \nmaterial that can provide non -reciprocity kept favoring \nmagnetic approaches. Recently, however, it was shown that \nparametric circuits can overcome these limitations and achieve \nexcellent performance together with low -cost and small -size. \nFor ex ample,  [31] introduced the concept of spatiotemporal \nmodulation angular -momentum (STM -AM) biasing to \nsynthesize cyclic -symmetric magnetless circulators. This was \nthen followed by numerous works extending these c oncepts to Ahmed Kord, Student Member, IEEE , and Andrea Alù, Fellow , IEEE  Magnetless Circulators Based on Synthetic  \nAngular -Momentum  Bias: Recent Advances \nand Applications  \nT Submitted to the IEEE Antennas and Propagation Magazine  \n 2 \nrealize various magnet -free components, including circulators, \ngyrators, and isolators [31]-[55]. In particular, [33]-[36] refined \nthe STM -AM concept by finding the necessary conditions to \nideally mimic magnetic biased cavities, thus enabling the \ndevelopment of many new circuits with unprecedented \nperformance. In this art icle, we review recent advances in \nSTM -AM devices and provide an outlook on their possible \nimpact on technology and future research directions. We also \ndiscuss the unusual properties of two-dimensional  metasurfaces \nconsisting of arrays of such elements, w hich may be engineered \nto behave as magnetized ferrites or topological insulators \n[66]-[85]. \nII. OPERATION PRINCIPLES OF  MAGNETIC CIRCULATORS  \n The S -matrix of an ideal circ ulator can in general be written \nin the following form  \n \n \n0 0 1\n1 0 0\n0 1 0S\n\n , (1) \n \n Equation  (1) shows that an input signal at, say, p ort 1 is \nexclusively routed to port 2 without any reflections or leakage to \nport 3. Similarly, an input signal at port 2 or 3 is routed to port 3 \nand 1, respectively. In other words, an ideal circulator is \ncyclic -symmetric and exhibits zero insertion loss,  perfect \nmatching, and infinite isolation. In reality, however, not only the \nS-matrix is limited by the inevitable dissipation in the \nconstituent materials of the circulator, but it is also frequency \ndispersive and maintains non -reciprocity over a finite \nbandwidth (BW). Despite such practical imperfections, the \nharmonic re sponse of a linear non -dispersive cyclic -symmetric \ncirculator can still give us an invaluable insight into the physical \nprinciples of breaking reciprocity. For example, an excitation at \none port of an ideal circulator, say port 1, can be decomposed \nusing superposition into three sub -circuits, as shown in Fig. \n1(a)-(c).  In Fig. 1(a), the applied voltage sources are identical in \nboth magnitude and phase while in Fig. 1(b) and Fig. 1(c), the \nsources have different phases, which increase by 120 deg either \nin the clockwise or counter -clockwise direction. Because of \nsuch distinctive phase pattern, we refer to such excitations as the \nin-phase (0), clockwise (+), and counter clockwise ( –) modes, \nrespectively. In general, each of these modal excitations \ngenerates a current component at the output ports and their \nlinear combination gives the solution to the original excitation at port 1. If the circuit is symmetric and has only three terminals \n(no int ernal connections to ground), then a non -zero in -phase \ncurrent violates Kirchhoff’s  current law, hence it cannot be \nexcited. Furthermore, if a preferred sense of precession is \nprovided to the counter -rotating currents, then they will be \nexcited with an equ al amplitude, say \ngI , and opposite phase, \nsay \n . Consequently, the total current at the n -th port \nbecomes  \n \n \n22 cos 13ngI I n   , (2) \n \nwhere  n is the port index. For \n30 deg, Equation  (2) results \nin \n30I  and \n12 3g I I I  . In other words, the impinging \nsignal at port 1 is exclusively routed to port 2 and isolated from \nport 3. This implies that it is possible to implement an ideal \ncirculator by imparting a preferred sense of precession to a pair \nof counter -roting modes in  a symmetric three -port network.  \n This principle indeed governs the way in which magnetic \ncirculators operate: consider a ferrite cavity connected \nsymmetrically to three ports at 120 deg intervals  as depicted in \nFig. 2. If a strong magnetic bias is applied orthogonally to the \ncavity in order to align the spinning electron dipole moments \nalong a particular direction, the spin imparts a form of \nangular -momentum bias at the microscopic  level, which in turn \ntranslates  into a preferred sense of precession for the rotating \nmodes of the cavity.  As a result, if these modes are excited by an \nimpinging signal at one port,  say port 1,  they accumulate \ndifferent phases as they appear the other two ports. For a \nparticular strength of th e bias, these modes can sum up in phase \nat one of the output ports , say port 2,  and destructively interfere \nat the other,  i.e., port 3. If power dissipation inside the cavity is \n \n \nFig. 1.  Eigenmodes  of a generic  three -port circulator  [35]. (a) In -phase mode. (b) Clockwise mode. (c) Counterclockwise mode.  \n \n \nFig. 2. Stripline Y -junction ma gnetic circulator  [10]. Submitted to the IEEE Antennas and Propagation Magazine  \n 3 \nnegligible, then this means that the incident power is exclusively \nrouted from  port 1 to 2. Due to the geometrical three -fold \nsymmetry of the circulator, excitation at ports 2 or 3, will \nsimilarly be routed to ports 3 and 1, respectively, which yields \nthe S -matrix of (1).  \nIII. LIMITATIONS OF  ACTIVE CIRCULATORS  \n Despite the fact that ma gnetic -circulators can achieve \nexcellent S -parameters, power handling and noise performance \nwith negligible power consumption, they suffer from large size \nand high cost due to  the fact that they require an external bulky \nmagnet and rare -earth ferrite mater ials. These critical problems \nprohibited an ubiquitous use of such devices in wireless  \ncommunication  systems  and played a n important  role in \nconsolida ting the false assumption that full -duplex radios are \nimpossible to realize in practice. For this reason, an interest in \ninvestigating magnet -free approaches to design circulators has \nemerged  since the advent of  solid -state transistors . When  a BJT \ntransistor, for instance,  is biased with a dc current, it  operate s as \na trans -conductance amplifier which converts  ac voltage s \napplied  at its base terminal into large current s flowing from  its \nemitter  to collector  terminals . A reverse voltage applied at \neither of the output terminals, however, is isolated from the \nbase. In other words, a current -biased transistor oper ates as an \nisolator with forward gain. By combining three of such active \nisolators in a loop as shown in Fig. 3(a), an active magnet -free \ncirculator can be constructed as explained in [20]. Variants of \nthis circu its at different frequency ranges were also investigated over the years. For example, a robust low -frequency \nimplementation using operational amplifiers (op -amps) was \nalso presented in  [21]. More recently, a synt hetic active \nmetamaterial consisting of an array of transistor -loaded \nmicrostrip ring resonators  was also proposed in [22] as an \nartificial medium to replace magnetic -biased ferrites. In such \nmaterials, the tran sistors simply kill one of the rotating modes \nsupported by the ring, thus allowing to realize active circulators \nas depicted in Fig. 3(b). \n It is also worth mentioning that nonlinear  materials have also \nbeen explored as an alternative medium t o realize mag net-free \ncirculators , especially  at optical frequencies where transistors \ndo not exist  [27]-[29]. However , the resulting devices  were only \nable to work when only one po rt is excited at a time . When two \nor more ports are concurrently excited, the S -parameters suffer \nfrom hysteresis and strongly depend on the relative phase \nbetween the input signals. T herefore, this approach is  less useful \nthan its acti ve transistor -based counterpart  which can support \nsimultaneous excitation from different ports and at the same \ntime offers distinct advantages  in terms of weight, size, cost, and \nscalability compared to  magnetic -biased ferrite devices . \nUnfortunately, however, active circulato rs still suffer from low \npower handling and poor noise figure . A comprehensive \nanalysis of these fundamental limitations was presented in [26] \nleading to the conclusion that active circulators are not suitable \nfor use as frontend modules in  the vast majority of microwave  \napplications where Tx power is either high or  Rx sensitivity is \ncritical.  \nIV. PARAMETRIC STM -AM  CIRCULATORS  \n In recent years, time-varying  circuits were presented as a \npromising solution to realize c irculators that could handle much \nhigher power, exhibit better linearity, and achieve lower noise \nfigure than active implementations , while at the same time \neliminating the necessity for bulky magnet s and expensive \nmaterials needed  in ferrite devices.  Table I summarizes these \nadvantages in comparison to traditional magnetic and active \ntechnologies presented in previous sections.  Among all recent TABLE I  \nCOMPARISON BETWEEN DIFFERENT CIRCULATOR DESIGN  APPROACHES.  \n \nApproach  Magnetic  Active  Parametric  \nMagnet  Required  Not required  Not required  \nSize Large  Small  Medium  \nCost High  Low Low \nInsertion loss Low Low \n(or gain)  Medium  \nIsolation  Medium  Medium  High  \nBandwidth  Wide  Average  Average  \nPower handling  High  Low Medium  \nNoise figure  Low High  Medium  \nPower dissipation  Low High  Medium  \n \n \n \nFig. 3. (a) Active circulator consisting of three common -emitter amplifiers in \na loop, each acting as an isolator [20]. (b) Active circulator consisting of three \ncoupled transistor -loaded ring resonators forming an artificial metamaterial \nnonreciprocal core [22]. Submitted to the IEEE Antennas and Propagation Magazine  \n 4 \nworks on parametric circulators, STM -AM circuits, which is the \nmain focus of this article, have shown significant  promise in \nsatisfying the challenging requirements of practical systems at \ndifferent portions of the electromagnetic spectrum. I n \nparticular, our group presented in [56] a circulator for airborne \nsound by synth esizing an effective  angular momentum bias, in \nthe form of mechanical rotation of air, inside a circularly \nsymmetric resonant acoustic cavity. Such bias  imparted a \nmacroscopic sense of precession to the degenerate modes of \nopposite handedness supported by the cavity, which translates \ninto non -reciprocity and isolation  as explained in Sec. II . Using \nsimilar principles, one could imagine spinning a circuit board \nsufficiently fast to break reciprocity for the electromagnetic \nsignals traveling across a resonant  device, but arguably this \nmechanical spin would be even less practical than magnetic \napproaches to non -reciprocity. However, spatiotemporal \nmodulation can impart a form of synthetic angular momentum \nbias, enabling the implementation of compact STM -AM \ncirculators as described in the following.  \nA. Single -Ended Topologies  \n Consider  a symmetric resonant three -port network  consisting \nof three series or parallel LC tanks  connected  in a loop or to a \ncentral node, result ing in four possible combinations as shown \nin Fig. 4. Let us further assume that t he instantaneous oscillation \nfrequencies of the tanks are spatiotemporally modulated as \nfollows  \n \n02cos 2 13nmf f f f t n     , (3) \nwhere \n  is the natural oscillation frequency of the \ntanks, \n  is the modulation depth, and \n  is the \nmodulation frequency . In practice, (3) is implemented using \nvaractors, switched capacitors or Jose phson junctions, \ndepending on the target application. In the case of varactors, for \ninstance, \n  can be written as \n  where k is the slope of \nthe C -V cur ve. The modulation scheme of  (3) synthesizes an \neffective angular momentum bias i n the clockwise direction \nsince the phases of the modulation signals increase by 120 deg \nin that direction thus lifting the degeneracy of the rotating \nmodes and allowing them to oscillate at different frequencies, \nsay \n . Between these two frequ encies, the skirts of the \nresonances do allow transmission, whether the original \nresonance was bandpass or bandstop, and they exhibit opposite phases. Therefore, if an input signal is incident at one port in the \nmiddle of these two frequencies, i.e., at \n , it will excite the two \nresonances with equal magnitude and opposite phases \n , \nwhich is consistent with the generic description of the previous \nsection based on providing a preferred sense of precession. It is \nalso worth menioting t hat the amount of splitting \n  \ndepends on the modulation parameters \n  and \n , thus \nallowing to control the value of \n . As explained earlier, for \n deg, ideal circulation can be achieved at  \n. In light of \nthis discussion, this condition will correspond to a certain \ncombination of \n  and \n. \n As an example, the bandstop/delta topology  of Fig. 4(c) was \nbuilt and tested in [33]. The measured insertion loss (IL), return \nloss (RL), and isolation (IX) at the center frequency of 1 GHz \nwere 3.3 dB, 10.8  dB, 55 dB, respectively, and the measured \nBW was 2.4% (24 MHz). This circuit can also be reconfigured \nfor operation at different channels over a frequency range of 60 \nMHz by simply changing the DC bias of the varactors. The \nmeasured 1 - and 20 -dB compression points of IL and IX, \nrespectively, were both +29 dBm. Also, the measured IIP3 was \n+33.8, which are reasonable nonlinearity metr ics meeting the \nspecifications for a range of applications. It is important to \nstress that these metrics stem from the properties of the \nemployed circuit components, not from a fundamental limit in \nthe approach to non -reciprocity. It can also be proven tha t \nSTM -AM circulators are inherently passive linear circuits [42], \nhence we expect that the overall noise figure (NF) is \napproximately equal to the IL but slightly larger due to noise \nfolding from the intermodula tion products (IMPs) and phase \nnoise in the modulation signals. Indeed, the measured NF was \n4.5 dB at 1 GHz and less than 4.7 dB over the BW . These results \n \n \nFig. 4. Single-ended topologies of STM -AM circulators [35]. (a) Bandpass/wye. ( b) Bandstop/wye. ( c) Bandstop/ delta. (d) Bandpass/ delta . \n  \n   \n   \n    \n \n         \n   \n        \n \n \nFig. 5. Differential  configurations  of STM -AM circulators [34]. (a) \nCurrent -mode. (b) Voltage -mode.  Submitted to the IEEE Antennas and Propagation Magazine  \n 5 \nare summarized in Table I I and the interested reader is refer red \nto [33] for further technical details.  \nB. Differential Configurations  \n As illustrated in previous section , single -ended STM -AM \ncirculators can achieve good performance in terms of several \nmetrics, particularly in terms of power handling and noise \nfigure. Nevertheless, these circuits suffer from large \nintermodulation products (IMPs) in close proximity to the \ndesired BW, resulting from parametric mixing induced by the \ntemporal modulation and from the inevitable non -linearities \nassociated with its circu it components. These IMPs draw power \nfrom the input signal, which limits IL to about 2 dB in practice. \nThey are also considerably large, i.e., only 10 dB below the \nfundamental harmonic, and they equally appear at all ports for \nexcitation at a single port. Therefore, the IMPs resulting at the \nRx port from the Tx excitation could saturate the Rx front -end \nregardless of how much isolation is achieved at the fundamental \nfrequency. Similarly, the IMPs resulting from the Rx signal at \nthe Tx port can cause instabi lity issues to the power amplifier \n(PA) because of load -pull effects. More importantly, the IMPs \nat the antenna (ANT) port, either due to the Tx or the Rx signal, \npose a serious interference problem to adjacent channels and \nwould prohibit compliance with t he spectral mask regulations of \nmost commercial systems. Therefore, it is crucial to reject these \nproducts.  This problem was overcome in [34] by combining two \nsingle -ended circuits similar to those presented in the previous \nsection in a differential voltage - or current -mode architecture, \nas shown in Fig. 5 and maintaining a constant 180 deg phase \ndifference between their modulation signals, i.e.,   \n,02cos 2 13nmf f f f t n      (4) \nwhere \n,nf  and \n,nf  are the modulation signals of the two \nconstituent single -ended circuits. As an example,  in [34], the \nvoltage -mode configuration of Fig. 5(b) was built and tested \nusing two of the single -ended bandstop/delta circuits depicted in \nFig. 4(c). The measured IL, RL, and IX after de -embeding the \nbaluns were 0.78 dB, 23 dB, and 2 4 dB, respectively, and the \nfractional BW was 2.3% (23 MHz). Table I I also summarizes \nthese results  along with other metrics  and the interested reader is \nreferred  to [34] for further details.  \nC. N-Way Architectures  \n While the differential STM -AM circulators presented in \nprevious section  dramatically improved the performance over \nmany metrics compared to the single -ended implementations, \ntheir spurious emission can remain strong when non -idealities in \nthe circuit are  considered. In order to address this problem, we \ncan extend the differential configuration to an N-way \narchitecture, consisting of N unit cells connected in parallel or in \nseries as shown in Fig. 6 [35]. The modu lation scheme in this \ncase becomes  \n \n022cos 2 1 13ni mf f f f t n iN        (5) \nwhere  \n  is the unit index and \n  is the tank index \nin the i-th unit. Equation  (5) reduces to (4) for \n , and this \ngeneralization is more effective at eliminating the IMPs that are \nnot multiples of \n  in the presence of nonidealities, such as \nnon-linearities, phase synchronization errors and random \nmismatches between the constituent elements. As  a by-product,  \nN-way circulators provide better power handling, enhanced by a \nfactor of N since the input power is split amongst N units. \nNonetheless, the benefits of the N -way architectures come at the \nexpense of an overall increased size and power consum ption. As \nbefore, Table I I summarizes the results and the interested reader \nis refe rred to [35] for further technical details.  \nD. Bandwidth Extension  \n The instantaneous BW of all circuits presented thus far is \nlimited by the modulation frequency \n , the loaded quality \nfactor \n  of the resonant junction, and the order of the \n \n \nFig. 6. N-way architectures of STM -AM circulator s [35]. (a) Parallel \ninterconnection  based on current combination . (b) Series interconnection  based \non voltage summation .  \n \n \nFig. 7. Bandwidth extension of STM -AM circulators based on terminating \nthe ports of a narrowband differential or N -way circuit with three properly \ndesigned matching filters . Submitted to the IEEE Antennas and Propagation Magazine  \n 6 \nTABLE I I \nSUMMARY OF THE MEASURED RESULTS IN COMPARISON TO PREVIOUS WORKS.  \n \nReference  [33]  [34]  [35] ¥ [36] [37] [41] \nArchitecture  Single -ended  \nBandstop/Delta  Differential  \nBandstop/Delta  Parallel N -way \nBandstop/Delta  Broadband  \nCurrent -mode  \nBandpass/Wye  Current -mode  \nBandpass/Wye  Current -mode  \nBandpass/Wye  \nTechnology  PCB  PCB  N/A PCB  CMOS  \n180-nm MEMS/PCB  \nCent. frequency (GHz)  1 1 1 1 0.91 2500  \nMod. frequency (%)  19 10 10 11 11.6 1.6 \n20 dB IX BW (%)  2.4 2.3 3 14 2.4 0.72 \nMax IX (dB)  55 24 25 31 65 30 \nMin IL (dB)  3.3 0.8 † 2.2 4.2 4.8 4 \nMax RL (dB)  11 23 24 16.7 11 15 \nP1dB (dBm)  +29 +29 +39 +23 N/A +28 \nIX20dB (dBm)  +29 +28 +41 N/A N/A N/A \nIIP3 (dBm)  +33 +32 +43 N/A +6 +40 \nMin NF (dB)  4.5 2.5 2.2 N/A 5.2 N/A \nMax. IMP (dBc)  –11 \n@ \n  –30 \n@ \n  –62 \n@ \n  –22 \n@ \n  –20 \n@ \n  –30 \n@ \n  \nPower diss. (mWatt)  N/A N/A N/A N/A 64 0.15 \nSize (mm2) 143 286 N/A 480 36 225 * \n  N/A: Not available.   ¥ Simulation results.   † Baluns are de -embeded.   * MEMS die area = 0.25 m m2. \n constituent resonators. For example, first -order resonators, i.e., \nseries or parallel LC tanks, with 10~20% modulation fre quency \nand 50 Ohm termination led to a BW of about 3~4% at best. \nOne solution to improve this further is to increase \n , but this \napproach increases power consumption, prohibits the use of \nthick -oxide low -speed technologies that could handle hig h \npower, and complicates scaling the center frequency to the \nmm-wave bands. Alternatively, one may think of replacing the \nLC tanks with high -order LC filters but this would increase the \ncircuit complexity and size dramatically. In [36], an optimal \nsolution based on terminating  a narrowband circulator with \nthree identical bandpass filters , as depicted in Fig. 7, was \nproposed.  The added filters are essentially matching networks \nthat shape the frequency of the combi ned network and maintain \na certain level of IX over a wider BW. A theoretical bound on \nthe maximum possible bandwidth for a given modulation \nfrequency was also derived in [36]. As one may expect, in order to app roach such a bound, t he order of the matching filters  must  \nbe very large. In practice, however, the maximum order is \nlimited by  the additional losses exhibited by the filters \nthemselves which increase  the circulator’s overall  IL. In order \nto experimentally  validate this technique , [36] relied on a \n2nd-order filter and a current -mode bandpass/wye STM -AM \ncircuit resulting in a three -fold increase in the  BW from 4% (40 \nMHz) to 14% (140 MHz). The measured results  of all other \nmetrics  are again summarized in Table I I and the interested \nreader is referred to  [36] for further details.  \nE. Integrated Implementations  \n The results presented in the previous section s were all based \non printed  circuit board (PCB) prototypes  which , despite \noutperforming previous works in many metrics, were still \nlimited in size. To overcome this problem and to achieve \nsignificant miniaturization, an IC implementation is highly \ndesirable. Towards this g oal, [37] modified the bandpass/wye \n \n \nFig. 9. Photographs  of the first MEMS STM -AM circulator  [41]. (a) C hip. \n(b) board.  \n \n \nFig. 8. Photographs  of the  first CMOS STM -AM circulator  [37]. (a) Chip. \n(b) Board.  Submitted to the IEEE Antennas and Propagation Magazine  \n 7 \ntopology of Fig. 4(a) to be compatible with CMOS technologies \nand experimentally validated the resulting circuit in a \ncurrent -mode architecture using a 180nm process. In partic ular, \nthe modulated capacitors were implemented using s witched \ncapacitors which result  in a larger modulation index than \nvaractors, alleviate the tradeoff between the index and the \nquality factor, and permit the use of digital clocks  which can be \neasily ge nerated on chip.  Fig. 8 shows photographs of  the \nfabricated chip and the testing board presented in [37]. It is also \nworth mentioning that this was the  first CMOS implementation \nof STM -AM circulators  ever-present ed. The measured  RL, IL, \nand IX at the center frequency of 910 MHz were about 12 dB, \n4.9 dB, and 60 dB, respectively. Also, the measured IIP3 was +6 \ndBm which was limited by the ESD protection circuitry and the \nNF was 5.2 dB. Similarly, these results are a ll summarized in \nTable I I and the interested reader is refe rred to [37] for further \ndetails.  \n While a CMOS implementation of STM -AM circulators does \nindeed  reduce the size significantly compared to a PCB design, \nextreme miniaturization  is still limited by the fact that on -chip \ninductors consume large silicon area and exhibit low quality \nfactors. This problem can be overcome by using \nmicro -machined integrated devices such as MEMS resonators. \nSuch devices can exhibi t a series resonance with a very high Q \nwhich mimics the response of a series LC tank while being much \nsmaller in size. However, MEMS resonators also suffer from a \nparasitic parallel resonance in close proximity to its series \nresonance. For devices operati ng in the low GHz range, the \nparasitic resonance can be shifted to higher frequencies with a \nsmall inductor of 0.5~1 nH which can be realized using a \nbondwire thus maintaining the benefit of reducing the \ncirculator’s area.  As an example, [41] presented  the first MEMS implementation of STM -AM circulators  as depicted in \nFig. 9. The measured  RL, IL, and IX at the center frequency of \n2.5 GHz were about 15 dB, 4 dB, and 30 dB, respectively. A \ndistinct advantage of MEM S implementations is also \nmanifested by the fact that they allow a significant reduction of \nthe modulation frequency. In this design, the modulation \nfrequency was as small as 40 MHz (1.6%) which resulted in a \nvery low power consumption of only 150 µW. Howe ver, this \nadvantage comes at the expense of a narrowband bandwidth \nwhich was about of 18 MHz (0.72%). These results are \nsummarized in Table I I and the interested reader is refer red to \n[41] for further details.  \nV. METAMATERIAL APPLICATIONS  \n Applications of the STM -AM concepts presented thus far \nhave been heavily investigated over the last few years. While \nfull-duplex communication is an obvious example, as explained \nearlier in this article, other applications have als o been \nexplored . Specifically, in the field of metamaterials STM -AM \nbiasing has been recently exploited to synthesize Faraday \nrotation for free -space propagating waves hitting a metasurface \nformed by arrays of such elements, and topological phenomena \nfor s urface wave propagation along such arrays. A \nmagneto -optical material biased by a DC magnetic field \nsupports circular bi -refringence and Farday rotation, which may \nbe used as a way to realize non -reciprocal wave propagation for \nfree-space radiation. Simila rly, an array of STM -AM elements \nlike those described in the previous sections can impart circular \nbirefringence over a thin surface, causing Faraday rotation to a \nlinearly polarized impinging plane wave that may be directly \ntranslated in the implementatio n of free -space isolators [30]. \nThe non -reciprocal polarization rotation imparted by such \nmetasurfaces can reach 1,000s of degrees per wavelength, a \nvery impressive metric considering that these surfaces can be \nimplemented in standard CMOS technology without need of an \napplied magnetic bias. Beyond electro -optical modulation of \nthese arrays, similar phenomena can be achieved using \nall-optical modulation via nonlinearities. By parametrically \npumping a nonlinear eta lon with two circularly polarized waves \nat detuned frequencies, it is possible to impart an effective \nangular momentum bias that replaces the modulation schemes \ndiscussed in the previous sections [65].  \n Opportun ities arise also to impart non -reciprocal responses to \nsurface waves propagating along an array of STM -AM biased \ndevices. Arguably a striking example of this type is the \nopportunity to mimic the response of a topological insulator \n(TI) for electromagnetic waves. A TI is a material that acts as an \ninsulator in the bulk and as a conductor on its edge or boundary, \nindependent of the way in which the boundary is deformed or \nchanged [66]-[69]. What is unique about such materials is that \nits edge conduction states are symmetry -protected by the \ntopology of their conduction bands delimiting the bandgap, \nhence they allow unusually robust conduction properties at the \nfoundations of  the quantum Hall effect, among many other \nunusual phenomena. Translating these concepts from \nelectronics to electromagnetics, a photonic TI enables \nunidirectional reflection -free propagation on the edges of the \n(meta) -material witnin a photonic bandgap, i mmune against a \n \n \nFig. 10. (a) Hexagonal lattice  consisting  of acoustic STM -AM circulators  \nforming a Floquet  topological insulator  for sound waves [81]. (b) Unit cell \nconsisting of two acoustic STM -AM circulators. (c) Comparison between the \nband structure of an infinite lattice with and without modulation showing a \nband gap opening in the vicinity of 22 MHz when modulation is turned on.  Submitted to the IEEE Antennas and Propagation Magazine  \n 8 \nwide range of structural imperfections. The existence of such \nmodes can be directly predicted from the topology of the bands \nof an infinite lattice, by calculating the so -called topological \ninvariant, a quantity that describes the nature of  the bandgap \nboundaries in momentum -frequency space. If the topological \ninvariant of the bands delimiting the bandgap are different, then \nthe bandgap necessarily supports a boundary mode, which is \ninherently topologically protected from disorder and back \nreflections.  \n While TIs were originally envisioned and discovered in \ncondensed -matter physics, their extension to classical photonic \n[69]-[78], acoustics [79]-[83], and circuits [83]-[85]  has been \ndriving a lot of excitement. In such a case, symmet ry breaking at \na degeneracy point in momentum space is typically sufficient to \ninduce topological order and enable edge states. Breaking \ntime-reversal symmetry is a particularly effective way to realize \nelectromagnetic TIs, since it supports robust, broadb and \nnon-reciprocal propagation and guarantees the absence of \nreflected modes regardless of the nature of the possible disorder \nin the system.  \n Here, we consider an array of STM -AM elements that realize \nan electromagnetic TI with non -reciprocal edge propaga tion. \nSince the edge states in this case emerge from periodic temporal \nvariations, the resulting system is refer red to as a Floquet TI. An \nexample is shown in Fig. 10(a) [81]: the unit cell of the \ncorresponding h exagonal lattice is depicted in Fig. 10(b) and it \nconsists of two waveguide STM -AM circulators in which the \ntime modulation synthesizes a rotation, similar to what \ndescribed in the previous sections, to impart an effective angular \nmomentum bias. The STM -AM waveguide circu lator is \nconsistent with the ba ndpass/delta topology of Fig. 4(d), and the \nmodulation can be achieved by modulating the volume of each \ncavity using piezoelectric actuators, or in other similar schemes \nas described in the previous sections.  Fig. 10(c) shows a \ncomparison between the band diagrams of such a photonic \ncrystal with and without modulation. Without modulation (blue \ncurves), four propagating bands are found around 22 kHz, two \nof which appear to be slow in nature corresponding to deaf \nmodes, while the other two correspond to fast Dirac bands with \ndegeneracy at both G and K points. On the other hand, when the \nmodulation is turned on (orange curves), the band diagram folds \nwith periodicity equal to the modulation frequency, as would be expected from the Floquet -Bloch theorem. More importantly, \nsince the modulation breaks the temporal symmetry, the \ndegeneracy at both G and K points is lifted and a bandgap \nopens. Ref. [81] theoretically predicted  the existence of a \nunidirectional edge state in such a gap by calculating the \ntopological invariant of the first Chern class for the four \nconduction bands. Such a prediction was then validated through \nfull-wave simulations of a truncated lattice confirmin g the \ntheoretical analysis.   \n In [85], an electronic STM -AM TI was also presented. \nSimilar to the TI of Fig. 10, the lattice was also hexagonal but \nthe meta -atom was based on a bandpass/wye STM -AM \ncirculator, as in Fig. 4(a). Without modulation, the lattice \nexhibits a continuous energy spectrum with six Dirac points. \nWhen the STM -AM biasing is applied to each meta -atom, the \ndegeneracy of the Dirac points is lifted  and a bandgap opens in \nthe bulk. Furthermore, in o rder to test the presence of an edge \nmode, the lattice can be divided it into two regions, one with a \nclockwise STM -AM bias and the other with a counter clockwise \nbias. Such a division results in a domain wall  between the two \nregions that can support an ed ge state and is easier to calculate \nnumerically. Fig. 11(a) shows the spatial distribution of the \nenergy density of such a mode when excited at the middle of the \nbandgap. As depicted in the left inset, the energy is localized \naround the artificial boundary  and decays sharply in the bulk. \nFig. 11(b) shows the band diagram for such a scenario, which \nindeed mainfests an edge mode that can carry energy only in the \n+x direction. An experimental demonstration  for elastic  waves \nhas also been recently reported usin g arrays of piezo -electric \nelements [82], also showing dynamic reconfigurability and \nopportunities for multiplexing . \nSTM -AM Floquet TIs do not require fast modulation speeds nor \nphase uniformity between different  unit cells, which eases their \nimplementation and makes them relevant for pratical \napplications, for instance for multiplexing. T he channels can be \ndynamically reconfigured thanks to the possibility of creating \nartificial domain walls by simply controlling  the direction of the \nSTM -AM bias, which is of significant importance in today’s \nprogrammable systems. This, in turn, opens the door t o new \nvenues and possibilities in future research directions.  \nVI. CONCLUSION S \nIn this paper, we reviewed the concept and recen t progress on \nmagnet -free STM -AM devices. We outlined the underlying \nphysical principles and presented its basic topologies and \nadvanced architectures, with a focus on the single device \nelement to realize circulators. For completeness, the results of \nall fabricated prototypes are summarized in Table I I. We also \nprovided an outlook on the potential impact STM -AM \ncirculators may have on future technologies such as wireless \nfull-duplex communications and in the field of metamaterials \nsuch as Floquet topologica l insulators. The antennas and \npropagation community can play a pivotal role in pushing \nforward the application  research on these systems, in which \nmany outstanding challenges remain, both from the modeling \nand theoretical perspective, for instance in the efficient \ncalculation of time -modulated systems, and from the practical \n \n \nFig. 11. (a) Hexagonal lattice  consisting of electronic  STM -AM circulators  \nbased on the bandpass wy e topology of Fig. 2(a) [85]. An artificial boundary is \ncreated at the middle of two domains inside the lattice, each with an opposite \ndirection of STM -AM bias. Schematic and energy density of a single \nmeta -atom near the artific ial interface are shown in the insets. (b) B and \ndiagram  with four conduction bands (blue) and a topologically protected edge \nstate (purple) at the artificial interface.  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Fleury, S. H. Mousavi, and A. Alù, “Topologically \nrobust sound propagation in an angular -momentum -biased graphene -like \nresonator lattice,” Nat. Commun., vol. 6, no. 1, Dec. 2015.  \n[81] R. Fleury, A. B. Khanikaev, and A. Alù, “Floquet topological insulators \nfor sound,” Nat. Commun., vol. 7, p. 11744, Jun. 2016.  \n[82] A. Darabi, X. Ni, M. Leamy, and A. Alù, “Reconfigurable Floquet \nElastodynamic Topological Insulators Ba sed on Synthetic Angular \nMomentum Bias,” Science Advances, 6, eaba8656, July 2020.  \n[83] C. H. Lee et al., “Topolectrical Circuits,” Commun. Phys., vol. 1, no. 1, \nDec. 2018.  \n[84] S. Imhof et al., “Topolectrical -circuit realization of topological corner \nmodes,” Nat. P hys., vol. 14, no. 9, pp. 925 –929, Sep. 2018.  \n[85] M. Tymchenko and A. Alù, “Circuit -based magnetless floquet topological \ninsulator,”  in Proc. 10th International Congress on Advanced \nElectromagnetic Materials in Microwaves and Optics \n(METAMA TERIALS), 2016.  \n \n \nAhmed Kord  (GS’15) receiv ed the \nB.S. and M.S. degrees from Cairo \nUniversity, Egypt, in 2011 and 2014, \nrespectively, and the Ph.D. degree \nfrom the University of Texas at \nAustin, TX, USA, in 2019, all in \nelectrical engineering.  In 2015 and \n2018, he was with  Qualcomm Inc. \nand Intel Labs, respectively, as a \nsummer intern. From Dec. 2016 to \nAug. 2017, he was with Eureka Aerospace Inc. as a part -time \nconsultant . From May 2019 to June 2020,  he was with \nColumbia University as a postdoctoral research scientist. He is  \ncurrently with  Qualcomm Inc. as a senior RFIC design engineer. \nIn Dec. 2019, he was named to the Forbes 30 Under 30 in \nScience.  \nSubmitted to the IEEE Antennas and Propagation Magazine  \n 11 \nDr. Kord  is a co-author  of more than 30 scientific papers and a \nco-invent or of  two patents, one of which is licensed to the US \nstartup company Silicon Audio/RF Circulator.  He is also a \nrecipient of several prestigious awards including the Qualcomm \nInnovation Fellowship, the IEEE MTT -S graduate fellowship, \nthe IEEE AP -S doctoral research award, the graduate student \nexcellence award , the graduate dean's fellowship, the Douglas \nWilson fellowship, the undergraduate student excellence award, \nand several travel fellowships and first place awards in different \nstudent paper, poster and design competitions.  \n \nAndrea Alù  (Fellow 2014, IEEE) is \nthe Founding Director of the Photonics \nInitiative at the CUNY Advanced \nScience Research Center. He is also \nthe Einstein Professor of Physics at the \nCUNY Graduate Center, Professor of \nElectrical Engineering at the City \nCollege of New York, and Adjunct \nProfessor and Senior Research \nScientist at the University of Texas at \nAustin. He received the Laurea , MS \nand PhD degrees from the University of Roma Tre, Rome, Italy, \nrespectively in 2001, 2003 and 2007. After spending one year as \na postdoctoral research fel low at UPenn, in 2009 he joined the \nfaculty of the University of Texas at Austin, where he was the \nTemple Foundation Endowed Professor until 2018. His current \nresearch interests span over a broad range of areas, including \nmetamaterials and plasmonics, elec tromangetics, optics and \nnanophotonics, acoustics, scattering, nanocircuits and \nnanostructures, miniaturized antennas and nanoantennas, RF \nantennas and circuits. He is a Fellow of the National Academy \nof Inventors, AAAS, IEEE, OSA, APS and SPIE, a Highly \nCited Researcher since 2017, a five times Blavatnik awardee \nand a Simons Investigator in Physics. He has received several \nawards for his scientific work, including the IEEE Kiyo \nTomiyasu award (2020), the NSF Alan T. Waterman award \n(2015), the Vannevar Bush  Faculty Fellowship from DoD \n(2019), the ICO Prize in Optics (2016), the OSA Adolph Lomb \nMedal (2013), and the URSI Issac Koga Gold Medal (2011).  \n \n \n \n"    },    {        "title": "1408.0125v1.Effect_of_the_techniqie_of_drawing_with_shear_on_the_structure_and_the_properties_of_low_carbon_wires.pdf",        "content": "E.G. Pashinskaya, V.N. Varyukhin, A.A. Maksakova, A.I. Maksakov1, A.A. Tolpa1, A.V. Zavdoveev \n \nEFFECT OF THE TECHNIQIE OF DRAWING WITH SHEAR ON THE STRUCTURE AND \nTHE PROPERTIES OF LOW-CARBON WIRES \nDonetsk Institute for Physics and Engineering named after А.А. Galkin NAS of Ukraine \n72 R.Luxemburg Str., 83114, Donetsk, Ukraine \n1SPA «Donix», Donetsk, Ukraine \n \n \n \nTo obtain the materials of ultrafine grain (UFG) structure, different methods of severe plastic \ndeformation (SPD) with shear in both cold and hot state are used: equal-channel angular pressing, \ntwist extrusion, and combination of these methods with succeeding rolling, upset, drawing etc. The \napplication of these methods allows substantial increase in the strength of the material at certain \nconserved reserve of plasticity. \nHowever the mentioned combination of the methods can not be realized at wiredrawing \nproduction plants, whereas the last are very interested in new technological and operation \ncharacteristics of long-length wire products [1,2]. One of constraints  imposed upon the production of \nUFG wire is that the total output of the materilas  produced by the listed SPD methods is measured \nby tens of kilograms and tons but the required capacity  of wiredrawing production is hundreds of \nthousand tons. \nA possible solution of the problem of the long wire products can be application of drawing with \nshear. As may be supposed, an increase in the plasticity reserve of rods and wires allows cheapening \nand simplification of the production technique due to the abandonment of the intermediate anneal. \nIn [1–9], different methods of severe plastic deformation of long metal billets of varied \nconfiguration are described. The works [5,6,9] are of special interest here. \nThe authors of [5] analyze the application of sign-alternation twisting of cold-drawn furniture \nwithout an additional heating. The main advantages of the method are continuous operation and \npossible application to the production of long products with enhanced mechanical properties. In [6], a \nmethod of plastic structure formation in the material of long billets is tested, and a facility of the \nrealization is presented that is based on sign-alternating deformation within intersecting channels. \nThe deformation zone is formed within the billet due to a shift of the symmetry axes of the channels \ncombined with uniaxial tension. The presented method is intermittent, and the finite billets of several \nmeters in length can be produced. The advantage of the method is formation of a fine grain structure. \nBut the deforming block is even more complex than that in the case of [5]. Both the methods [5,6] do \nnot permit production of the wire of small diameter. \nA drawing-based method of obtaining of UFG structure in long products is described in [9]. The \nmain advantage is continuous operation and possible application to the large-scale drawing \nproduction. A drawback is labor intensity of drawing because a complex technical facility is used that \nrequires disassembling and new installation in the course of replacement of wire drawing dies. \nThe present paper reports the developed technique of drawing with shear aimed at enhancement of \ntechnological plasticity of low-carbon steels without thermal treatment. The technology should \nprovide definite physical and mechanical properties of the wire. It also should be cheap, simple and  \nreliable in the course of operation. Experimental technique \nThe experiment was carried out with using the billet of steel ER70S-6 (Table 1). The drawing was \nrealized at the plant АZТМ 7000/1 by the developed (experimental) technique (with shear dies) and \nby the classical one (with the standard round dies). The drawing routes of the both techniques are \nlisted in Table 2. \nТable 1 \nChemical composition of ER70S-6  steel, % \nC Mn Si S P Cr Ni Cu N \n0.071  1.98 0.84 0.015  0.018  0.015  0.009  0.016  0.0055  \nTi As B Al V Mo W Co  \n< 0.005 < 0.005  < 0.0005  0.005  0.006  < 0.01  0.024  0.01  \n \nТable 2 \nDrawing routes for the wires produced by the experimental and the standard techniques  \nTechnique  Die diameter, mm \nExperimental  6.15 5.4 5.2* 5.0 4.30  3.90 3.5 3.06  2.70 2.39 \nConventional    5.30        \nNote . * – die with shear  \n \nThe obtained samples from 6.15 tо 3.90 mm in diameter were subjected to mechanical tests, in \nparticular, the ultimate tension strength and the contraction ratio were measured (Fig. 1). \nThe microstructures of the annealed sample and the deformed ones were studied at the 100–1000 \npower device «Neophot-32» after repetitive polishing and  etching of the grain boundaries (the \ncomposition of the etching agent: 4% nitric acid, 97% alcohol). The photos were made by the optical \nmicroscope Axiovert 40 MAT .                  \n \n   \n  \nа b  \nFig. 1.  Mechanical properties of the ER70S-6 steel wire obtained by the conventional ( ■) and  the \nexperimental ( ) technique: а – the ultimate tensile strength σ uts, b – the contraction ratio  \nТable 3 \nElongation ratio k of the  ER70S-6 wire \nTechnique  k \n 6.5 5.0 3.9 \nExperimental  1 0.35 0.19 \nConventional   0.21 0.10 The estimation of the grain size and the fragment size was performed in the transversal and \nlongitudinal directions of the samples. 100 measurements were made on every photo. The elongation \nratio was calculated as  \n k  = D1/D2, (1) \nwhere D1, D2 were the lengths of the grain along the grain elongation and in the cross-section of the \nsample, respectively, mm. \nBesides, Vickers hardness HV (loading of 200 g) and microhardness Hμ (loading of 100 g) were \nmeasured. The measurement error was 5%. The density of the samples was measured by hydrostatic \nweighing. Strength tests were carried out with using UММ-50 test machine at the temperature of 293 K \nand the rate of loading of 10 mm/min according to GOST 25.601–80. \nExperiment results \nThe tests have demonstrated that when the diameter of the sample is reduced, the ultimate tensile \nstrength remains at the same level irrespective of the conventional or experimental technique   (Fig. \n1,а), and the contraction ratio of the sample processed by the experimental technique is higher as \ncompared to the conventional drawing  (Fig. 1, b). \nOn the basis of the mean values, it is seen that the reduction of the diameter after the conventional \ntechnique is associated with the increase in the ultimate tensile strength by 380 N/mm2,  a n d  t h e  \nexperimental technique yields 240 N/mm2. \nAfter the conventional technique, the value of the contraction ratio is substantially decreased from  \n69 to 28%, and the drop after the experimental technique is from 69 to 55%. \nOne of the distinguishing features of the experimental technique is a decrease in the structure \nanisotropy in the longitudinal samples. An evidence of the fact is an increase in the elongation ratio  \n(Table 3). The photos of microstructures confirm the calculated data (Fig. 2). \n   \n  \nа b  \nFig. 2.  Microstructure of the samples of the wire of ER70S-6  5.0 obtained by the experimental technique ( а) \nand the conventional technique ( b), 100. The scale  represents 10 μm \nThe analysis of the structures revealed the following principal structure features. \n1. The etchability of perlite colonies is higher at the conventional technique that is related to non-\nequilibrium of perlite after drawing. In the longitudinal section, along the section at the conventional \ntechnology is larger that of the experimental technique. \n2. It has been established that the experimental technique results in decreased anisotropy of the \ngrains, and the grain size is reduced as a whole, when the degree of deformation increases. But at \ncertain stages, successive decrease and increase in the grain size is observed that is supposed to be \nrelated to the progress in competitive processes of fragmentation and dynamical polygonization. For instance, in the tested samples of 5.2 and 5.0 in cross-section, the structure is enlarged: the \nferrite grains and the sizes of perlite colonies grow as compared to the sample of 6.15. In the \ncross-section of the experimental sample,  the structure was larger in comparison with the sample \nobtained by the conventional technique. The structure of the last sample contained more perlite \nphase. \n3. In the cross-section of the samples of 3.9 obtained by the conventional technique, there is a \nsurface zone of about  200–300 m  that differs in etchability. This effect is not found after the \nexperimental technique, the metal is homogeneous, the ferrite grains are lager, the size and the \namount of perlite is much less than that after the conventional technique. \n4. The samples of 2.39 by the conventional technique are etched better, so, the degree of non-\nequilibrium is higher. In the longitudinal section of the conventional samples, the anisotropy is more \nintensive, the ferrite grains are smaller, the perlite colonies are elongated as stripes (Fig. 3). The same \neffect is observed in the cross-section. Besides, healing of  pores and microcracks is registered in the \ncourse of the experimental drawing   (compare Fig. 3, c and d). \nConclusions \n1. The application of the experimental dies allows improvement of the mechanical properties of \nthe samples when the diameter is decreased after drawing: the contraction ratio of the wire is \ninsignificantly reduced and stays at a relatively high level as compared to the conventional technique \nwhere the contraction ratio is halved. \n2. The use of the experimental technique permits variation of the ferrite grain size (successive \nincrease and decrease) in comparison with the conventional technique where the increase in the \ndegree of deformation results in unambiguous grain reduction. \n3. The application of the experimental technique generates pore healing in the wire of a small \ndiameter. \n   \n  \nа b \n   \n  \nc d  Fig. 3.  Microstructure of the central zone of the ER70S-6  wire samples of 2.39 obtained by th4e \nconventional technique ( a, c) and by the experimental technique ( b, d) : а, b – the cross-section, c, d –  th e \nlongitudinal section, 100. The scale is as in Fig. 2 \n4. To prevent the heating of the wire and the die, it is necessary to pass to the experimental drawing \nthrough two experimental dies with shear separated by a standard die. This configuration will result in \nenhanced fabricability and reduction of the drawing force. \n \n \n1. E. Astafurova, G. Zakharova., E. Naydenkin., G. Raab., P. Odesskiy., S. Dobatkin , Materials letters 1. #4, 198 \n(2011). \n2. E. Astafurova, G. Zakharova, E. Naydenkin, S. Dobatkin,  G. Raab,  FMM 110, 275 (2010).  \n3. N. Kolbasnikov, O. Zotov., V.Duranichev, Metall treatment , # 4, 25 (2009).  \n4. S. Yakovleva, S. Makharova, News of Samarian scientific centre of RAS,  12, # 1(2), 589 (2010).  \n5. E. Kireev, M. Shulyak, A. Stolyarov,  Steel # 3, 56 (2009).  \n6. Patent RF #  2440865, The method of structure of plastic material of long workpieces and device for its \nimplementation, A. Matveev, R. Kazakov, Yu. Shumkina, V. Kurganskiy, application # 2010121631/02, \n27.05.2010.  \n7. E. Astafurova, G. Zakharova., E. Naydenkin., G. Raab., S. Dobatkin ,Е.Г. Астафурова, Physical \nmesomechanic 13, № 4, 91 (2010).  \n8. A. Zakirova, R. Zaripova, V. Semenov,  News UGATU, Engineering, materials science and thermal \nprocessing of metals 11, № 2 (29), 123 (2008).  \n9. Patent RF #  2347633, A method for producing ultrafine semis drawing shift, G. Raab, A. Raab, \napplication # 2007141899/02, 12.11.2007. \n "    },    {        "title": "2204.08293v1.Experimental_Visualization_of_Dispersion_Characteristics_of_Backward_Volume_Spin_Wave_Modes.pdf",        "content": " 1 \nExperimental Visualization of Dispersion Characteristics of \nBackward Volume Spin Wave Modes  \n \nSergey  V. Gerus , Alexander Yu. Annenkov  and Edwin H. Lock   \n \n(Kotel’nikov Instutute  of Radio Engineering and Electronics (Fryazino branch), \nRussian Academy of Sciences,  Fryazino, Moscow region , Russia ) \n \nBasing on the measurement of spatial spectra (spectra of wavenumbers), the dispersion \ncharacteristics of t he first three modes of backward volume spin wave, propagating along the direction \nof a constant uniform magnetic field in a tangentially magnetized ferrite film, were  visualized  firstly . \nThe study was carried out by microwave probing of spin waves with su bsequent use of spatial Fourier \nanalysis of the complex wave amplitude for a series of frequencies. It was found that every m-th mode \nof the backward volume spins wave can be split into n satellite modes due to the existence of layers \nwith similar magnetic  parameters in ferrite film.  It was found that satellites of the first mode of this \nwave are excited most effectively, while satellites of the third mode – least effectively, and the \neffectiveness of satellites excitation decreases as the number n increase s. It is found that the \ntheoretical dispersion dependencies of the first three modes of the wave coincide well with the \nexperimental dispersion dependencies of the satellite mode that are excited most effectively.  \n \n1. Introduction  \nIt is well known that the only dipole spin wave (SW) that can propagate in a ferrite \nplate along the direction of a tangent homogeneous magnetic field is backward volume \nspin wave (BVSW) , also called the backward volume magnetostatic wave since it was \nfirst described in magnetostatic approximation [1].  In the following years, the main \ncharacteristics of BVSW  and various devices using th is wave have been described in \nseveral monographs and review work s (see [2 -9] and references given in these work s), \nas well  as in the number of recent papers [10 -17], in which new results concerning \nBVSW  were obtained.  \nFrom the publication of work [1] to the twenty -first century, researchers were \nmainly interested by the BVSW dispersion dependence, the amplitude -frequency \ncharacteristic  of BVSW  transfer coefficient and the delay time of BVSW between two \ntransducers perpendicular to the external magnetic field H0 [1]1. That is, the \ncharacteristics of the BVSW s with collinear orientation of wave vector k and group \nvelocity v ector V were investigated  to construct various filters, delay lines and other \ndevices for analogue signal processing at microwave frequencies.   \n \n1 Including cases when transducers of various shapes were used or various inhomogeneities were placed between them – \na lattice of conductive strips, etched grooves, etc.   2 \nTheoretical and experimental stud ies of BVSW characteristics with noncollinear \norientation of the vectors k and V really started with the works [10, 11]. In particular, \nthe microwave field distribution of the BVSW  propagating at an arbitrary angle to \nvector H0 has been investigated theoretically and the instantaneous vector lines patterns \nof magnetic field for the first mode  of BVSW  have been calculated [10].  In addition, it \nhas been shown theoretically and experimentally that excitation of BVSW  by a linear \ntransducer which is not perpendicular to the H0 vector gives rise to appearance of two \nwaves characteri zed by oppositely directed wave vectors and different magnetic \npotential distribution s2 across the thickne ss of ferrite plate , that course to the significan t \ndiffer ence between amplitudes of these two excited  waves3 [11]. The condition under \nwhich two refle cted beams appear and the condition under which a negative reflection4 \noccurs have also been described for the geometry where BVSW  incidents on the \nstraight edge of ferrite film [11,  12]. As a result of further theoretical studies of the \nproperties of BVSW , it was proved that propagation of this wave, like the propagation  \nof the surface SW, is characteri zed by the presence of wave vector cut -off angles [15, \n17]. In addition, it was found th at an extremum point5 appears on the distribution of \nmagnetic potential amplitude for the first BVSW mode at a certain orientation of its \nwave vector, and the coordinate of this point corresponds to the one of ferrite plate \nsurfaces  [17]. It has also been  found that the degree of wave non -reciprocity, defined \nas the ratio of normalized  amplitudes of the magnetic potential on a ferrite plate surface \nfor two waves with oppositely directed wave vectors, depends significantly on the \norientation of these vector s with respect to the vector H0 [17].  \nRather recently, the diffraction properties of the first mode of the BVSW  have been \ninvestigated theoretically and experimentally [14, 16]. In particular, based on \ntheoretical results obtained in [18], for the first BVSW  mode it was calculated the wave \nvector orientations [14], at which the angular width of the wave beam is zero and the \nwave is characterized by super -directional propagation wh ich was then discovered \nexperimentally in [16].  \nThe description of BVSW  without magnetostatic approximation has shown that the \nelectromagnetic wave, named in [1] as BVSW , has all six microwave electromagnetic \nfield components (three magnetic and three ele ctric) both in the ferrite layer and in the \n \n2 This means that BVSW  is characteri zed by non -reciprocal properties which were previously described only for the \nsurface SW [1] .  \n3 An exception is the case where both waves propagate parallel to the external magnetic field vector in opposite directions. \nOnly in this case both waves have the  same magnetic potential distribution across the thickne ss of ferrite plate and, \ntherefore, are excited with the same amplitude s.  \n4 That is , when the incident and reflected beam s are on the same side of the boundary normal .  \n5 This point is the only extre mum point on the distribution of magnetic potential  amplitude  for the first mode  of BVSW . \nIt should also be noted that the coordinate of this point gradually shifts towards the middle of ferrite plate as the angle \nbetween the BVSW  wave vector and vector H0 increases.    3 \nadjacent half -spaces: that is, both E -wave and H -wave, which are coupled to each other \ndue to the presence of ferrite plate, arise in adjacent half -spaces as a result of the \nsatisfaction of electrodynamic boundar y conditions [13].  As a result of these studies \nthe exact dispersion equation for this wave was obtained (for the case of wave \npropagation along the vector H0) and it was shown that the distribution of all its \nmicrowave components across the thickness of t he ferrite plate is not purely \ntrigonometric (which follows from [1]) but is a sum of exponential and trigonometric \nfunctions, and the cross components of the wave number included in trigonometric and \nexponential functions have different values [13]. Obvio usly, this fact means that \nBVSW  is not a purely volume wave and gives the reason to rename th is wave . \nHowever, over the past time , the term BVSW  has become widespread  and we think \nthat, to avoid confusion  of terms , it would not be appropriate to use anothe r term with \nrespect to  this wave just because new its properties have been discovered or other \nequations have been used for its description : nevertheless, this wave is an objective \nreality and its name should not depend on the way of its theoretical description .  \nIt should be noted that all the studies listed above are mainly devoted to the first \nmode of BVSW . The characteristics and properties of the higher mod es of BVSW  are \nstill almost not investigated : only the dispersion dependences and magnetic potential \ndistribution of higher modes of BVSW  have been calculated for waves propagating \nalong vector H0 [1] and for waves propagating in arbitrary direction s [10, 17], and it \nwas also shown that all modes of BVSW  are characterized by the same cut -off angles \nof the wave vector [15].  However, the properties and characteristics of the higher \nmodes of BVSW  have not yet been studied experimentally.   \nFilling this gap, we present below an experimental study of the spatial spectrum of \nBVSW modes which are excited in a tangentially magnetized ferrite plate by a linear \ntransducer perpendicular to the vector H0. \n2. Experimental setup and method for the probing of spin waves  \nAt present, measurement of SW characteristics is generally  carried out using the \nmethod of Brillouin light scattering on SW (see, for example, [19, 20]) or the method \nof SW probing [16, 21, 22], w hich allow to obtain visuali zed patterns describing the \namplitude and the phase distribution of SW in the plane of ferrite film or ferrite \nstructure.  To carry out the planned experiments, the re was used SW probing method,  \nwhich, as will be shown below, allowed to decide all experimental problems .  \nThe method of SW  probing  emerged from the step -by-step development of the \nmoving antenna method6 in which a receiver transducer was moved over a ferrite \n \n6 It is the first method used to measure the dispersion characteristics of SWs in the 1970s - 1990s    4 \nstructure surface along the SW path to measure the phase shift of SW at certain distance \nand then to find the wave number at fixed frequency (see, for example, [23, 24]).  \nSubsequently, both the measurement method and the experimental setup were \nimproved significantly (see Fig.  1).  \n \nFig. 1. Simplified schem e of experimental setup .  \nThe microwave probe, having form of a loop of thin gold -plated tungsten wire with \nan aperture of ~0.5 mm , began to be used in place of the receiving transducer (which \nwas previously identical to the exciti ng transducer ) in the experimental setup.  Both the \nprobe and the exciting transducer were equipped with position sensors, could rotate \naround the axis normal to the ferrite structure  plane , and could move freely over the \nstructure  surface along y and z axes by means of movement mechanism. SW \nchara cteristics were measured in the following way: for a number of fixed coordinates \ny, a continuous probe movement over the structure surface along z axis was carried out \nwith simultaneous digitization of both the instantaneous z coordinate values and the \ncomplex amplitude of the microwave transfer coefficient between transducers [21].  \nSince it is SW that carries the microwave signal in ferrite structure, then the probe , in \nfact, measures the complex amplitude of SW during its movement. As a result of \nprocessi ng the experimental data coming from the vector analy zer into the computer, \nit was possible to obtain visuali zed patterns of the amplitude and phase distribution for \nthe studied S W in its propagation area at a fixed frequency f = fconst (see, for example, \nthe patterns presented in [16, 22]).   \n 5 \nThen this method was improved again to obtain visualized wave distribution \npatterns over a wide frequency range during a single probe pass along the surface of \nthe structure.  For this purpose, during slow movement of the probe over the surface of \nthe structure along z-axis a fast sawtooth change of the vector analyzer frequency was \ncarried out simultaneously in a certain frequency range  fstart < f < fend, and the probe \nshift du ring the period of change of sawtooth voltage was very small.   \nAs a result of this improve ment, it was possible to measure the distribution of the \ncomplex transfer coefficient K(y, z, f) as a function of the frequency f and the y, z \ncoordinates along the s tructure surface.  \n3. Method for determining the spatial spectrum of spin waves  \nBy processing the K(y, z, f) distribution in different ways, a number of SW \ncharacteristics can be obtained.  For example, using a discrete Fourier transform of the \nK(y, z, f) distribution along the wave vector direction normal to the phase fronts7, one \ncan obtain the dependence of the Fourier amplitude A on the wave number k for each \nfrequency f from the frequency range  fstart < f < fend. Obviously, the position s of the \nmaxima on the A(k) dependence correspond to the spin wave numbers exciting at a \ncertain frequency f in experiment . Thus, in general, the dependence A(k) is a spatial \nspectrum or wave numbers spectrum, the maxima of which correspond to the excited \nwave numbers f or SW, or various SW  modes, or any other waves that can excite and \nreceive the used transducer and probe. It is clear that the sum of the maxima of all \nspatial spectra at the studied frequencies A(k, f) represent the dispersion relations of \nobserved waves.  It should be noted, however, that in order to successfully use the \ndescribed method to find the spatial waves spectrum A(k, f) in ferrite structures, the \nexperiments must be carried out with taking into account the well-known feature of \nFourier analysis.  In particular, it is known from the theory of spectral analysis of signals \nbased on the Fourier transform that the longer the du ration of a pulse, the narrower its \nspectrum (see, for example, §2.2. in [25]).   \nSince we use spatial Fourier analysis of the K(y, z, f) distribution, the following \nstatement is valid: the greater is the distance at which the K(y, z, f) distribution is \nmeasured, the narrower are the maxima of the investigated spatial wave spectrum \nA(k, f) and, therefore the greater are the accuracy and resolution one can obtain in the \nexperiment to determine  the spatial wave spectrum.  This statement has been recently \nconfirmed in experiment al studies of  SW characteristics [26].   \n \n7 It is assumed that in experiment  one studies a sufficiently wide wave beam, in the middle of which the phase fronts are \nstraight lines, i.e., the wave is practically like a homogeneous plane wave. It is clear that in the experiment, more \ncomplex distributions of K(y, z, f) may occur, when the phase fronts are not straight lines( such fronts often appear in \ndiffraction patterns ). However, a discussion of such complex distributions is beyond the framework of this paper.    6 \n4. Spatial spectrum of spin waves propagating along the vector H 0 \nThe spatial spectrum of the spin wave was studied on a YIG film grown by liquid \nphase epitaxy on a 0.5 mm thick galliu m gadolinium garnet (GGG) substrate. The YIG \nfilm, having a diameter of 76 mm, a thickness of d = 39 µm and a saturation \nmagnetization of 4πM0 = 1750 G, was magnetized to saturation by a tangential \nhomogeneous magnetic field of H0 = 483  ±5 Oe. A 10 mm long linear transducer was \nused to excite SW. The transducer was oriented perpendicular to vector H0 and \npositioned near the edge of the film by such means that the normal passing through the \ncenter of the transducer did not pass through the center  of the film circle  (Fig. 1) . This \norientation of the transducer provided excitation of any types of SW (including BVSW  \nmodes)  with wave vectors directed along z-axis, and asymmetric location of transducer \nrespect to the film center prevented reflection of  the waves from the near film edge \nback towards the transducer and interference of these waves with the waves running \ntowards the far film edge, where the prob ing area  was located.   \nTo describe the obtained results, a Cartesian coordinate system is used in which the \nx-axis is perpendicular to the film plane and the z-axis is directed along the vector H0, \nwith coordinate  z = 0 corresponding to the projection of excitin g transducer midd le \nonto the film plane.   \nThe spatial distribution of the complex BVSW  amplitude measured during probe \nmovement parallel to the magnetic field H0 is shown in Fig . 2 for arbitrarily frequency \nvalue. As can be seen from Fig.  2, the oscillation amplitude decreases quite  rapidly \nwith increasing distance between the probe and the excitation transducer, that is related \nmainly with BVSW  beam diffraction divergence (rather than losses), which is quite \nlarge for this geometry of wave excitation [17, 18].  The subtle distortions of the \noscillation curves show that besides the most intensely excited first mode BVSW, there \nare modes with higher values of the wave number.  This is also confirmed by Fig.  3, \nwhich shows the wave vectors spectrum A(k) obtained from a Fourier analysis of the \nmeasured complex amplitude of BVSW  for an arbitrary fixed frequency . The relative \nmagnitude of the maxima on the dependence A(k) differs greatly and characterizes the \nintensity of BVSW  modes excitation: the greatest amplitude has a maximum located \nnear the theoretical value of the wave number \nt\n11()kf for the first BVSW  mode (Fig. \n3a), and the smallest – a maximum located near the theoretical value of the wave \nnumber \nt\n31()kf for the third BVSW  mode (Fig. 3c).  \nNote that the complex amplitude of BVSW  was measured when the probe was \nmoving in the negative z-axis direction, i.e., in the wave energy transfer direction \ndescribed by the group ve locity vector. However, from the Fourier analysis we \nobtained positive values of wave numbers k1, k2 and k3 for BVSW  mode s, that  7 \ncorresponds to the positive orientation of wave vectors k1, k2 and k3 with respect to the  \nz-axis. Thus, opposite orientations of wave vectors and corresponding group velocity \nvectors confirm that we are dealing with the backward  waves.   \n \nFig. 2. Real R (1) and imaginary I (2) parts of the complex amplitude as a  \nfunction of the z-coordinate when microwave energy is transferred between the \nexcitat ion transducer and the probe by means of BVSW at the frequency f1 = 2808 \nMHz .  \nSince the maxima of each spectrum A(k) correspond to the wave numbers of the \nBVSW  modes at a fixed frequency f (e.g., in Fig. 3, f = f1), so obviously, by uniting all \nthese spectra for different frequencies,  we obtain the surface A(k, f), on which the \nmaxima of the spectra correspond to the dispersion dependences f(k) for BVSW  modes \nexcited in the experiment. Such a surface is shown in Fig. 4.   \nThe darkest regions of the spectrum A(k, f) correspond to amplitudes A that are \nclose to zero, wh ile the lightest regions correspond to the wave numbers maxima \nobtained by Fourier analysis. The set of these maxima in Fig.  4 represents the \nvisualized dispersion dependencies f(k) of all waves observed in this experiment. In \naddition, the white points in Fig. 4 correspond to the several frequency and wave \nnumber values describing theoretical dispersion dependences of the first, second, and \nthird modes of BVSW  in magnetostatic approximation for the experimental \nparameters. A s can be seen from Fig. 4, the theoretical points generally agree well with \nthe experimental curves.   \n 8 \n \nFig. 3. Fourier amplitude versus wave number A(k) at the frequency f1 = 2808 \nMHz near the theoretically calculated wavenumber values \nt\n11()kf  \nt\n21()kf  \ncorresponding to the vertical lines for the first, second and third BVSW modes (a, b \nand c, respectively).  \n 9 \n \nFig. 4. Graphical illustration of the experimental surface A(k, f), visualizing the \ndispersion dependences f(k) for the first three BVSW  modes . The white circles \ncorrespond to the theoretical dispersion dependences for these three BVSW  modes \n(mode numbers are shown near the curves).   \nHowever, it should be noted that the maxima of the experimental spectrum A(k, f) \nvisualize in Fig. 4 much more dispersion dependences then follows from the theory \n[1]. Th ese additional dispersion dependences are especially well seen near the \ntheoretical di spersion dependence for the first mode of BVSW  (Fig. 4). This fact is also \nclearly seen in Fig. 3a: near the theoretical value of the wave number \nt\n11()kf for the first \nBVSW  mode, there is not one experimental maximum, but n closely located maxima \nat \nexp\n11()nkf  values.   \nIt may be assumed that series of closely located  dispersion dependences observed \nexperiment ally are explained by splitting of the first BVSW  mode into satellite modes8, \n \n8 Apparently, the distribution of the magnetic potential of all modes -satellites over the thickness of the ferrite film is like \na similar distribution for the first mode of the BVSW  according to the theory [1]: that is, the distribution of modes -\nsatellites i s described by an odd, sinusoidal, centrally symmetric relative to the middle of the film wave function (see, \nfor example, [17, Figure 2a, curve 1]).   \n 10 \nthe excitation of which is caused by the stratification of YIG film in the process of its \ngrowth with formation of several layers having similar parameters of saturation \nmagnetization or growth anisotropy .  \nFragments of the dispe rsion dependences in which satellite modes  of the first \nBVSW  mode are clearly distinguishable are shown in Fig. 5 in a more detail ed scale . \nThese dispersion dependences correspond to the maxima of the A(k) spectra (such as \nare seen in Figure 3a) for the certain frequency interval . Bold lines in Fig. 5 identify \nparts of the dispersion curves with maxima A(k) having highest values. For example, \nin Fig.  3a this is the left maximum, which is significantly larger than the others. In \nFig. 4 such curves, to which the highest maxima of A(k) correspond, are shown by the \nlightest lines.   \n \nFig. 5. Experimental dispersion dependences f(k) for the first BVSW  mode in a \nmore detailed scale (dependences f(k) correspond to the dependences A(k, f), in Fig.  4). \nThe circles correspond to the theoretical dispersion dependences of this mode in the \nYIG1 –GGG –YIG2 structure.   \nThe presence of layers in garnet ferrite films was observed earlier in the study of \nfilms with cylindrical domains [27]: when the layers were chemically etched off \n 11 \nsuccessiv ely, the domain structure was changed by  jumps.  However, there is still no \ncomplete understanding of the cause of the formation of such layers. There are several \nhypotheses. It is supposed that during epitaxial growth of the ferrite film on a non -\nmagnetic gadolinium -gallium substrate elastic stresses arise and build up d ue to some  \nlattice mismatch, and these stresses at some moment  decrease by jump due to the \nrestructuring of the ferrite film lattice. In addition, during the growth process a subtle, \nbut significant for the growth, change in the melt composition may occur.  A \ncombination of these mechanisms is possible  too.  \nIn addition, by analyzing Fig. 5, one can notice that in the range of small wave \nnumbers k < ~70 cm-1 (Fig. 5a) there are observed dispersion curves, which are absent \nin the region of higher values of k. The appearance of these curves is evidently related \nto the interaction between BVSW  modes, which propagate in different YIG films \ngrown on opposite subst rate surfaces of h = 0.5 mm thickness. Indeed, since along the \nx-axis, normal to YIG plane, the microwave  fields decrease according to the exp( -kx) \nlaw, so at k = 20 cm-1 and x = h the exponent index becomes equal to unity, which \nshows the possibility of i nteraction between these BVSW  modes  and the possibility of \nexcitation of both modes by the transducer located on one of the film surfaces (a similar \neffect occurring for surface SW was described earlier in [21]).   \nTo verify the mentioned interaction the dispersion dependences of the first BVSW \nmodes, which propagate in two ferrite layers separated by substrate of thickness h, were \ncalculated. In calculations it was assumed that the layers had the same thickness of \n39 μm and saturation magnetizations of 1852.1 and 1874.3 G. The calculated \ndispersion dependences are plotted  by dots in Fig. 5, which shows that the curves \ncorresponding to the two almost  identical ferrite layers differ most at f > ~ 2750 MHz \nand k < ~ 70 cm-1, while at f < ~ 2750 MHz and k > ~70 cm-1 these curves gradually \napproach each other by a distance of ~ 7 cm-1.  \nSince it was assumed that several ferrite layers are formed on each side of the \nsubstrate, so a series of dispersion dependences for satellite mode s of the first BVSW  \nmode in the form of a stretched loop is observed in the frequency range ~ 2750 MHz \n< f < ~2900 MHz. With increasing wave number at k > ~70 cm-1 the interaction between \nBVSW  mode s localized in different layers decreases, which leads to weakening of the \nsecond and subsequent experimental maxima  observed in Fig. 3.  \nCalculations have shown that closely located dispersion dependences of BVSW  \nmode s do not arise for the geometry of contacting or closely locate d ferrite layers : a \ncertain gap (or non -magnetic substrate) between layers with a thickness considerably \nless than the gap thickness is necessary for such dependencies to occur.  However, a \ndetailed study of this phenomenon is beyond the framework of this p aper.   12 \n5. Summary  \nThe spatial spectrum of waves propagating in a tangentially magnetized ferrite film \nalong the direction of a constant homogeneous magnetic field is investigated \nexperimentally. The study was carried out by means of microwave probing with the \nfollowing use of Fourier analysis to visualize the spatial spectrum of the waves. As a \nresult, it was found that the experimental spatial wave spectrum for this excitation \ngeometry is significantly different from the theoretical BVSW  spectrum calculate d \naccording to the theory [1]. In particular, it was found that each m-th mode of the \nBVSW  predicted by the theory [1] can additionally split into n satellite mod es. It was \nfound that the satellites of the first BVSW  mode are excited most efficiently, whil e the \nsatellites of the third BVSW  mode are excited least efficiently, and the excitation \nefficiency of satellites decreases with the increasing of number n.  The greatest number \nof mode satellites, about seven, was observed for the first BVSW  mode, whose \ntheoretical dispersion dependence matched well with the experimental dependence of \nits first satellite, which was most efficiently excited. The splitting of the second  BVSW  \nmode into satellite modes was inefficient, and the splitting of the th ird BVSW  mode \nwas practically not observed due to inefficient excitation of these modes, so the \ntheoretical dispersion dependences of the second and third BVSW  modes coincided \nwell with the experimental dependences, which as a rule corresponded to a single  \nmaximum obtained by Fourier analysis (one may suppose that only one satellite mode \nwas effectively excited for both the second and the third BVSW  modes ).  \nAlso,  the interaction between BVSW  modes, which propagate in YIG films grown \non opposite substrate s urfaces, was observed experiment ally for the wave number s \nk < ~70 cm-1. It was found that t his interaction lead s to a notable distortion of dispersion \ndependences for satellite modes of the first BVSW  mode.   \nIt can be assumed that BVSW modes split into satellite modes due to stratification \nof the ferrite film into several layers with the same magnetic parameters during \nepitaxial growth. The observed effect of splitting of the first BVSW mode into satellites \ncan be used in practice to study the parameters of grown  ferrite films.   \nFunding  \nThis work was performed as part of State Task .  \n   13 \nReferences  \n1. Damon R.  W. and Eshbach J.  R., J. Phys. Chem. Solids , 19 (1961) 308.  \n2. Pizzarello F.  A., Collins J.  H., Coerver L.  E., J. Appl. Phys. , 41 (1970 ) 1016.   \n3. Adam. J. D., Daniel M. R. , IEEE Trans. on Magnetics, MAG -17 (1981 ) 2951 .  \n4. Castera J. -P., J. Appl. Phys., 55 (1984 ), 2506.   \n5. Danilov V. V., Zavislyak I. V., and Balinskii M. G., Spin Wave Electrodynamics \n(Libid ’, Kiev) 1991).  \n6. 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Avaeva I.G., Kravchenko V.B. , Lisovsky F.V. et al., Microelectronics 7 (1978 ) \n444.  \n "    },    {        "title": "2201.08923v1.A_Detailed_Investigation_of_the_Onion_Structure_of_Exchanged_Coupled_Magnetic_Fe_3_delta_O4_CoFe2O4_Fe_3_delta_O4_Nanoparticles.pdf",        "content": " 1 This is the Preprint version of our article accepted for publication in ACS \nApplied Materials & Interfaces  [ACS Applied Materials & Interfaces 13, \n16784- 16800 (2021)]. The Version of Record is available online: \nhttps://doi.org/10.1021/acsami.0c18310  . \n \nA Detailed Investigation of the Onion Structure of Exchanged Coupled \nMagnetic Fe 3-δO4@CoFe 2O4@Fe 3-δO4 Nanoparticles  \n \nKevin Sartori,1,2,3 Anamaria Musat,1 Fadi Choueikani,2 Jean Marc Grenèche,4 Simon Hettler,5,6 Peter \nBencok,7 Sylvie Begin- Colin,1 Paul Steadman,7 Raul Arenal,5,6,8 Benoit P. Pichon*,1,9  \n \n \n1 Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, F -\n67000 Strasbourg, France  \n2 Synchrotron SOLEIL, L’Orme des Merisiers, Saint Aubin –  BP48, 91192 Gif -sur-Yvette, France \n3 Laboratoire Léon Brillouin, UMR12 CEA -CNRS, F -91191 Gif -sur-Yvette, France  \n4 Institut des Molécules et Matériaux du Mans, IMMM, UMR CNRS -6283 Université du Maine,  avenue Olivier \nMessiaen, 72085 Le Mans Cedex 9, France  \n5 Instituto de Nanociencia y Materiales de Aragon (INMA), CSIC -Universidad de Zaragoza, Calle Pedro Cerbuna, \n50009 Zaragoza, Spain  \n6 Laboratorio de Microscopias Avanzadas (LMA), Universidad de Zarago za, Calle Mariano Esquillor, 50018 \nZaragoza, Spain   \n7 Diamond Light Source, Didcot OX11 0DE, UK  \n8 Fundacion ARAID, 50018 Zaragoza, Spain  \n9 Institut Universitaire de France, 1 rue Descartes, 75231 Paris Cedex 05, France  \n \n \nKeywords  : exchanged -coupling, onion -type, nanoparticles, magnetism, XMCD, Mössbauer, EELS \nCorresponding Author  \n*E-mail: Benoit.Pichon@unistra.fr  \n   2 Abstract  \n \nNanoparticles which combine several magnetic  phases offer wide perspectives for cutting edge \napplications because of the high modularity of their magnetic properties. Besides the addition of the \nmagnetic characteristics intrinsic to each phase, the interface that results from core -shell and, further , \nfrom onion structures leads to synergistic properties such as magnetic exchange coupling. Such a \nphenomenon is of high interest to overcome the superparamagnetic limit of iron oxide nanoparticles which hampers potential applications such as data storage or sensors. In this manuscript, we report \non the design of nanoparticles with an onion -like structure which have been scarcely reported yet. \nThese nanoparticles consist in a Fe\n3-O4 core covered by a first shell of CoFe 2O4 and a second shell of \nFe3-O4, e.g. a Fe 3-O4@CoFe 2O4@Fe3-O4 onion -like structure. They were synthesized by a multi -step \nseed mediated growth approach which consists to perform three successive thermal decomposition \nof a metal complexes in a high boiling point solvent (about 300 °C). Alt hough TEM micrographs clearly \nshow the growth of each shell from the iron oxide core, core sizes and shell thicknesses markedly differ \nfrom what is suggested by the size increase. We investigated very precisely the structure of \nnanoparticles in performing high resolution (scanning) TEM imaging and geometrical phase analysis \n(GPA). The chemical composition and spatial distribution of atoms were studied by electron energy loss spectroscopy (EELS) mapping and spectroscopy. The chemical environment and oxidatio n state of \ncations were investigated by Mössbauer spectrometry, soft X -ray absorption spectroscopy (XAS) and \nX-ray magnetic circular dichroism (XMCD). The combination of these techniques allowed us to estimate \nthe increase of Fe\n2+ content in the iron oxide  core of the core@shell structure and the increase of the \ncobalt ferrite shell thickness in the core@shell@shell one, while the iron oxide shell appears to be much thinner than expected. Thus, the modification of the chemical composition as well as the siz e of \nthe Fe\n3-O4 core and the thickness of the cobalt ferrite shell have a high impact on the magnetic \nproperties. Furthermore, the growth of the iron oxide shell also markedly modifies the magnetic properties of the core -shell nanoparticles, thus demonstr ating the high potential of onion -like \nnanoparticles for tuning accurately the magnetic properties of nanoparticles according to the desired applications.     3 Introduction \n Bimagnetic nanoparticles open huge perspectives toward potential applications in fields such as biomedicine, sensors, or data storage because of the high modulation of their magnetic properties.\n1 It \nis very well established that, at the nanoscale, the surface contribution predominates on the volume contribution.\n2 Therefore, slight modifications of the size and the shape significant ly influence the \nmagnetic properties. Besides controlling the size and the shape of nanoparticles, the design of multicomponent nanoparticles allows the intrinsic magnetic properties of different phases to be \ncombined. In addition, core@shell structures wh ich result in large interfaces at the nanoscale usually \nfavor synergistically enhanced magnetic properties such as effective magnetic anisotropy energy. This approach represents a high potential to reduce the amount of rare earths used to produce permanent  \nmagnets and classified by the European Union as critical raw materials.\n3,4,5 Metal oxide nanoparticles \nsuch as iron oxide (Fe 3-O4, magnetite / maghemite)6 - which is cheap, nontoxic and abundant –  \nrepresent an interesting alternative . Although the magnetic anisotropy energy is not high enough to \nproduce permanent magnets at room temperature – iron oxide nanoparticles are superparamagnetic -\n, it can be markedly enhanced by coating nanoparticles with a harder magnetic metal oxide.7–12,13 The \ndesign of core -shell nanoparticles gives rise to an interesting interfacial magnetic properties which \nconsists in the pinning of soft spins of the iron oxide core by the harder spin of the harder shell, e.g. the so -called exchange bias coupling.\n14 This phenomenon is of great interest to push the \nsuperparamagnetic limit over room temperature.15 Fe3-O4 is us ually combined to antiferromagnetic \nphases such as CoO that have a magnetic anisotropy constant which is two order of magnitude higher than the one of Fe\n3-O4.16,16 –20,21 Nevertheless, exchange bias coupling only happens below the Néel \ntemperature (T N = 290 K for  CoO). Indeed, above the Néel temperature, the antiferromagnetic order \nvanishes and loses its ability to pin the spins of the soft phase.  \nIn contrast, ferrite (MFe 2O4) phases open interesting perspectives because of their common spinel \nstructure and close  lattice parameters, while their magnetic hardness and softness markedly depend \non the transition metal M.22 Such ferrimagnetic (FiM) materials display Curie temperatures that are \nusually much higher than room temperature (T C = 790 K for CoFe 2O4), thus avoiding the Néel \ntemperature limitation of antifferomagnetic phases. Exchange coupled nanoparticles which  combine \nseveral ferrite phases into a core -shell structure showed remarkably tunable magnetic properties such \nas enhanced magnetic anisotropy and magnetization saturation.23,24–30 The exceptional properties of \nFerrite@Ferrite nanoparticles result from much more complex structure than the ideal picture of a well- define inte rface in core -shell structure. Although each crystal phase is selected because of the \nlow lattice mismatch, defects may occur at the interface because of the shape of the nanoparticles.\n31 \nIndeed, an isotropic shape which is close to sphere induces a high curvature radius and facets with different surface energies which result in the complex growth of the shell component at the core surface. Considering defects at the surface of the nanoparticle which result from the break of symmetry, diffusion of cations leading to an interfacial composition gradient between the core and the shell has been regularly reported.\n8,32 –34 The growth of the shell usually occurring at high \ntemperatures (200 –  300 °C) results from a synthesis m echanism which does not systematically consist \nin a simple seed mediated growth process. Indeed, it usually consists in the partial solubilization of the seeds which is usually followed by a recrystallization of the monomers issued from the Ostwald ripenin g process and with those that remain from the decomposition of the reactant.\n34,35 All these \nfeatures significantly alter the expected ideal chemical composition of the core -shell. Therefore, the \nmagnetic properties deviate from those initially expected and are difficult to anticipate.  \nNon -hydrol ytic synthesis techniques have been demonstrated to be particularly effective to design \ncore -shell nanoparticles with a very high control of their structure. Hence, the effect of the core size \nand of the shell thickness on their magnetic properties can be systematically studied.8,19,20,24,29 The new \nchallenge is to design nanop articles with more complex structure in order to precisely tune their \nmagnetic properties. In this aim, onion -like structures, which consist of a core covered by several \nshells, have recently driven a tremendous interest, although they have been scarcely r eported so far.36–\n38 Although the synthesis of such complex nanoparticles is not trivial, the fine understanding of the  4 onion structure is necessary to rationalize the study of their magnetic properties. Re cently, we \nreported on the synthesis of nanoparticles which consist in a Fe 3-O4@CoO@ Fe3-O4 onion -like \nstructure.39 Both soft/hard and hard/soft interfaces resulted in blocked magnetization at room \ntemperature, although these nanoparticles are mostly composed of iron ox ide with a size below 16 \nnm. Such enhanced magnetic properties account from a more complex structure that resulted from \nthe partial replacement of CoO by cobalt ferrite at both interfaces during the synthesis steps.40 \n \nIn this context, we have designed new multi -component nanoparticles in order to rationally investigate \ntheir structure by combining advanced characterization techniques. Hence, we report here on onion -\nlike magnetic nanoparticles which consist of an iron oxide core combined w ith a first shell of cobalt \nferrite and a second shell of iron oxide, i.e. a Fe 3-O4@CoFe 2O4@Fe3-O4 structure. We have deeply \nstudied the structure in order to bring a better understanding on its relationship with the  magnetic \nproperties. The onion -like structure of nanoparticles was systematically compared to those of seeds – \niron oxide and core@shell nanoparticles –  by means of highly complementary and advanced \ncharacterization techniques. Spatially -resolved energy electron loss spectroscopy (EELS) mapping and \nspectrum analysis, Mössbauer spectrometry, X -ray absorption spectroscopy (XAS) and X -ray magnetic \ncircular dichroism (XMCD) were used in order to accurately characterize the chemical composition and the cationic  distribution. The combination of such advanced techniques allowed us to show that the \ncore size and the shell thicknesses markedly differ from what is suggested by size variations observed on TEM micrographs. Finally, the magnetic properties were correlat ed to the structure of the \nnanoparticle, in order to evaluate the effect of the modification of the iron oxide core size, the cobalt ferrite shell thickness and the growth of the second iron oxide shell.  \n   5 Experimental section  \n \nMetal (Fe or Co) stearate sy nthesis.  Iron stearate was synthesized according to our previous work41 \nwhile the synthesis of cobalt stearate was adapted. In a 1 L two -necked round bottom flask, 9.8 g (32 \nmmol) of sodium stearate (98.8 %, TCI) were poured and 320 mL of distilled water were added. The mixture was heated to reflux under magnetic stirring until all the stearate was dissolved. Afterwards, 3.80 g (16 mmol) of iron (II) chloride tetrahydrated (or 3.16 g (16 mmol) of cobalt (II) chloride \nhexahydrated) dissolved in 160 mL of distilled water were poured in the round bottom flask. The  \nmixture was heated to reflux and kept to this temperature for 15 minutes under magnetic stirring \nbefore cooling down to room temperature. The colored precipitate was collected by centrifugation \n(15 000 rpm, 5 min) and washed by filtration with a Buchner f unnel. Finally, the powder was dried in \nan oven at 65 °C for 15 hours.  \n Nanoparticle synthesis.  Fe\n3-O4@CoFe 2O4@Fe 3-O4 nanoparticles were synthesized using a three steps \nthermal decomposition method in a similar way that we reported recently.39 First, iron oxide \nnanopartic les were synthesized according to our previous work.42 A two -necked round bottom flask \nwas filled with 1.38 g (2.22 mmol) of iron (II) stearate, 1.254 g (4.44 mmol) of oleic acid (99% Alfa \nAesar) and 20 mL of ether dioctyl (B P = 290 °C, 97 % Fluka). The brown ish mixture was heated at 100 \n°C under a magnetic stir for 30 min in order to remove water residues and to homogenize the solution. \nThe magnetic stirrer was then removed and the flask was connected to a reflux condenser before heating the solution to reflu x for 2 h with a heating ramp of 5°C/min. At the end, the mixture was \nallowed to cool down to 100 °C and 4 mL of the solution were removed and washed to serve as a reference (sample C).  \nSecondly, 0.29 g (0.46 mmol) of cobalt (II) stearate, 0.791 g (2.8 mm ol) of oleic acid and 32 mL of 1 -\noctadecene were added to the reaction medium. The mixture was heated to 100 °C for 30 min under magnetic stirring to remove water residues and to homogenize the solution. After removal of the magnetic stirrer, 0.585 g (0.94  mmol) of iron (II) stearate was added. The flask was then connected to \na reflux condenser in order to heat the solution at reflux for another 2 h with a heating ramp of 1 °C/min. After cooling down to room temperature, the nanoparticles were precipitated by the addition of acetone in order to wash them by centrifugation with a mixture of chloroform: acetone (1 : 5). The final nanoparticles (sample CS) were stored in chloroform.  \nThirdly, half of the volume of the washed CS suspension was poured in a two -necked round bottom \nflask with 0.548 g (0.88 mmol) of iron (II) stearate, 0.497 g (1.76 mmol) of oleic acid and 20 mL of ether dioctyl. The mixture was then heated to 100 °C under magnetic stirring for 30 min. As mentioned above, after removing the magnetic s tirrer, the mixture was heated to reflux for 2 h with a heating \nramp of 1 °C/min. After cooling down, the nanoparticles were collected and washed in the same way \nas for CS nanoparticles. The final nanoparticles (sample CSS) were stored as a colloidal suspe nsion in \nchloroform.  \n \nTransmission electron microscopy (TEM)  was performed by using a JEOL 2100 LaB6 with a 0.2nm \npoint -to-point resolution and a 200 KV acceleration voltage. EDX was performed with a JEOL Si(Li) \ndetector. The average size of the nanopartic les was calculated in measuring at least 300 nanoparticles \nfrom TEM micrographs by using the Image J software. The average shell thickness was calculated as \nthe half of the difference between the size of the nanoparticles before and after the thermal \ndecom position step. The size distribution was fitted by a log -normal function.  \n \nScanning transmission electron microscopy (STEM)  experiments were carried out using a probe \naberration corrected Titan (Thermo Fisher Scientific) equipped with a high -brightness fi eld emission \ngun. While the electron gun was operated at 300 keV for acquisition of high -angle annular dark field \n(HAADF, acceptance angle 47.9 mrad) STEM images to obtain maximum spatial resolution \n(convergence angle 25 mrad), the high energy was lowered to 80 keV for electron energy -loss  6 spectroscopy (EELS) to minimize beam damage, to increase the EELS signal and to improve energy \nresolution (~1 eV). The Gatan imaging filter (GIF, Gatan Inc) was operated at a dispersion of 0.2 eV /px in order to simultane ously analyze O K, FeL and Co L edges with a collection angle of 119 mrad. EELS \nspectra and spectrum images (SI) were treated using a custom Matlab software including principal \ncomponent analysis (PCA) for noise reduction. Quantification was done using power- law background \nsubtraction and an integration width of 40 eV for the C and 30 eV for the CS/CSS nanoparticles. The \nsample preparation was done by drop casting 2 μL of the nanoparticle suspension on Holey -C grids \nfollowed by 14 s of plasma cleaning.   X-ray diffraction (XRD)  was performed using a Bruker D8 Advance equipped with a monochromatic \ncopper radiation (Kα = 0.154056 nm) and a Sol- X detector in the 20− 80°  2θ range with a scan step of \n0.02°. High purity silicon powder (a  = 0.543082 nm) was systemati cally used as an internal standard. \nCrystal sizes were calculated by the Scherrer’s equation and cell parameters by the Debye’s law.  \n \nFourier transform infra -red (FT-IR) spectroscopy was performed using a Perkin Elmer Spectrum \nspectrometer in the energy ra nge 4000−400 cm\n−1 on samples diluted in KBr pellets.  \n \nGranulometry measurements were performed using a nano -sizer Malvern (nano ZS) zetasizer at a \nscattering angle of 173°. A measure corresponds to the average of 7 runs of 30 seconds.  \n Themogravimetry  analyses (TGA) were performed using a SDTQ600 from TA instrument. \nMeasurements were performed on dried powders under air in the temperature range of 20 to 600 °C at a heating rate of 5 °C/min.  \n X-ray absorption. XAS and XMCD spectra were recorded at the L\n2,3 edges of Fe and Co, on the DEIMOS \nbeamline at SOLEIL (Saclay, France)43 and on I10 (BLADE) beamline at Diamond Light Source (Oxford, \nUnited Kingdom). All spectra were re corded at 4.2 K under UHV conditions (10-10 mbar) and using total \nelectron yield (TEY) recording mode. The measurement protocol was previously detailed by Daffé et al.\n44 An external parallel magnetic field H+ (respectively antiparallel H-) was applied on the sample while \na σ + polarized (σ - polarized respectively) perpendicular beam was directed on the sample. Isotropic \nXAS signals were obtained by taking the me an of the σ ++σ- sum where σ + = [σ L(H+)+ σ R(H-)]/2 and σ - = \n[σL(H-)+ σ R(H+)]/2 with σ L and σ R the absorption cross section measured respectively with left and right \ncircularly polarized X -rays. XMCD spectra were obtained by taking the σ +-σ- dichroic signal with a ± 6.5 \nT applied magnetic field.  \nAt DEIMOS beamline, the circularly polarized X -rays are provided by an Apple -II HU -52 undulator for \nboth XAS and XMCD measurements while EMPHU65 with a polarization switching rate of 10 Hz was used to record hysteresis cycle at fixed energy.\n43.  Measurements were performed between 700 and \n740 eV at the iron edge and between 770 and 800 eV at the cobalt edge with a resolution of 100 meV and a beam size of 800*800 µm. Both XMCD and isotropic XAS signals presented here are normalized by dividing the raw signal by the edge jump of the isotropic XAS.  \nAt BLADE beamline, the circularly polarized X -rays were provided by a helical undulator with 48 cm \nperiodicity. Monochromatic X -rays in the soft X -ray range (400 -1600 eV) were provided with a plane \ngrating monochromator\n45 beamline giving an energy resolution o f 100meV and a beam size of \n100*100 μm2 (root mean square) at the sample position. Sample cooling and applied field were \nsupplied with an Oxford Instruments cryomagnet.  \n \nThe samples consist of drop casted suspension of nanoparticles in chloroform onto a sil icon substrate. \nThe substrates were then affixed on a sample holder.  \n \nMössbauer spectrometry. 57Fe Mössbauer spectra were performed at 77 K using a conventional \nconstant acceleration transmission spectrometer with a 57Co(Rh) source and a bath cryostat. Th e \nsamples consist of 5 mg Fe/cm2 powder concentrated in a small surface due to the rather low  7 quantities. The spectra were fitted by means of the MOSFIT program46 involving asymmetrical lines \nand lines with Lorentzian profiles, and an α -Fe foil was used as the calibration sample. The values of \nisomer shift are quoted relative to that of α -Fe at 300 K.  \n \nSQUID magnetometry. M agnetic measurements were performed in using a Superconducting \nQuantum Interference Device (SQUID) magne tometer (Quantum Design MPMS -XL 5). Temperature \ndependent zero -field cooled (ZFC) and field cooled (FC) magnetization curves were recorded as \nfollows: the sample was introduced in the SQUID at room temperature and cooled down to 5 K with \nno applied magneti c field and after applying a careful degaussing procedure. Then, a magnetic field of \n7.5 mT was applied, and the ZFC magnetization curve was recorded upon heating from 5 to 400 K. The sample was then cooled down to 5 K under the same applied field, and the  FC magnetization curve \nwas recorded upon heating from 5 to 400 K. Magnetization curves as a function of a magnetic field (M(H) curves) applied in the plane of the substrate were measured at 5 and 400 K. The sample was also introduced in the SQUID at high temperature and cooled down to 5 K with no applied magnetic \nfield (ZFC curve) and after applying a subsequent degaussing procedure. The magnetization was then \nmeasured at constant temperature by sweeping the magnetic field from +7 T to −7 T, and then from \n−7 T to +7 T. To evidence exchange bias effect, FC M(H) curves were further recorded after heating up \nat 400 K and cooling down to 5 K under a magnetic field of 7 T. The FC hysteresis loop was then \nmeasured by applying the same field sweep as for the ZFC c urve. The coercive field (H\nC) and the M R/M S \nratio were measured from ZFC M(H) curves. The exchange bias field (H E) was measured from FC M(H) \ncurves. Magnetization saturation (M S) was measured from hysteresis recorded at 5 K.  \n \n   8 Results and discussion  \n \n  \n \nFigure 1.  Schematic illustration of the synthesis pathway of onion -like nanoparticles (CSS) using a three -\nstep thermal decomposition method.  \n Core@shell@shell nanoparticles were synthesized by performing successively the thermal decomposition of a metal stearate three times (Figure 1). First, the thermal decomposition of iron (II) \nstearate (FeSt\n2) in ether dioctyl (B P = 290 °C) was performed in presence of oleic acid in order to \nsynthesize Fe 3-O4 nanoparticles (C). Second, cobalt (II) stearate (CoSt 2) and FeSt 2 (molar ratio 1:2) were \ndecomposed together in octadecene (B P = 320 °C), in order to grow a cobalt ferrite (CoFe 2O4) shell at \nthe surface of pristine Fe 3-O4 nanoparticles. The aim was to synthesize core -shell Fe3-O4@CoFe 2O4 \nnanoparticles (CS). Finally, FeSt 2 was again decomposed in ether dioctyl in the presence of CS \nnanoparticles with the aim to grow a second Fe 3-O4 shell, i.e. to synthesize Fe 3-O4@CoFe 2O4@Fe3-O4 \n(CSS) nanoparticles.  \n TEM micrographs (Figure 2) show that C nanoparticles display a homogeneous shape close to sphere \nand a narrow size distribution centered at 8.0 ± 0.9 nm. The nanoparticle size increases from CS (10.0 ± 1.5 nm) to CSS (13.1 ± 2.2 nm) corresponding to average shell thicknesses of 1.0 nm and 1. 6 nm, \nrespectively. The broadening of size distribution and the deviation of shape from sphere to CS and CSS \nis ascribed to the inhomogeneous growth of both shells. Indeed, the nanoparticle surface consists in \nfacets which feature different surface energies according to the corresponding hkl planes.  \n \n 9  \nFigure 2. Conventional TEM micrographs of a), b) C nanoparticles; d), e) CS nanoparticles and g), h) CSS \nnanoparticles with (c), f), i)) their corresponding size distributions.  \n \nTable 1. Structural characteristics of nanoparticles. Mean core sizes and shell thicknesses were \ncalculated from TEM micrographs. Cell parameters and crystal sizes were calculated from XRD patterns.  \n  C CS CSS \nSize (nm)  8.0 ± 0.9  10.0 ± 1.5  13.1  ± 2.2  \nSize variation (nm)  - 2.0 3.1 \nFe : Co at. Ratio by EDX  - 86 : 14  94 : 6  \nHydrodynamic diameter (nm)  8.7 13.5  18.2  \nCell parameter ( Å) 8.37 ± 0.01  8.41 ± 0.01  8.41  ± 0.01  \nCrystal size (nm)  7.4 ± 0.5  10.1 ± 0.5  12.0 ± 0.5  \n \n   \n \n \n 10  \nFigure 3. STEM -HAADF micrographs of a) C, b) CS and c) CSS nanoparticles showing their \nmicrostructures. Inter- reticular distances are highlighted by double red arrows. A stacking defect may \nbe observed by the change of the atomic c olumn contrast along the highlighted area in yellow, also \nvisible in neighboring lines. (d) -f)) FFT from the C, CS and CSS nanoparticles obtained from the core \nregion of the nanoparticle using a circular smoothed mask. FFT from the shell region are overlai d in \ngreen color in the right half for CS and CSS nanoparticles. Colored circles evidence the related hkl  \nreflections attributed to magnetite (JCPDS card n° 19 -062) and to cobalt ferrite (JCPDS card n° 00 -022-\n1086). Colors refer to hkl plan families.  \n \nSTEM -HAADF micrographs of C, CS and CSS nanoparticles display straight and continuous lattice fringes, \nevidencing the single crystal -like structure of the nanoparticles resulting from the successive epitaxial \ngrowth of the different shells ( Figures 3 a-c). Minor crystal defects were observed in a few nanoparticles, \ne.g. in the CS nanoparticle (Figure 3b). These defects are recognizable by the changing contr ast of the \natomic columns from dots to a line along the highlighted area. As the pattern remains undisturbed, the \ndefects are attributed to stacking defects. Figure S1 shows additional STEM micrographs of the \nnanoparticles suggesting that minor defects wer e already present in few of the C nanoparticles, \nalthough an induction by electron beam damage cannot be excluded. The inter -reticular distances \nbetween two fringes were attributed to the spinel ferrite phase (including magnetite, maghemite and cobalt ferr ite) for each sample. These results are supported by FFT calculated from STEM -HAADF \nmicrographs that show spots corresponding to ( hkl) directions of the spinel phase ( Figures 3 d-f). A \ncomparison of FFT calculated from core and shell regions revealed perfect overlap of the spots, agreeing with good epitaxial relationships.  \n \nEnergy dispersive X -ray spectrometry (EDX) showed that CS nanoparticles consist of F e (86 at. %) and \nCo (14 at. %) which agree with values calculated for a Fe\n3-O4 core of 8.0 nm (Fe: 84 at. %) and a CoFe 2O4 \nshell thickness of 1.0 nm (Co: 16 at. %), as measured from TEM micrographs. CSS nanoparticles display \nthe increase of Fe (94 at. %) vs. Co (6 at. %) ratio, in agreement with the size variation measured from TEM micrographs, which corresponds to the growth of iron oxide at the surface of CS (Fe : 93 at. % and Co : 7 at. %). Nevertheless, EDX does not give any information on the spatial arrangement of Fe and Co \n 11 within the nanoparticle volume. Therefore, we performed complementary measurements using \nadvanced characterization techniques.  \n \n  \nFigure 4. Elemental mapping performed by EELS -SI on isolated nanoparticles (a)-c)) C, (d) -g)) CS, (h) -k)) \nCSS with a), d), h) the sum of the composite (Fe in green + Oxygen in red + Cobalt in blue), b), e), i) Fe -\nedge, c), f), j) O -edge and g), k) Co -edge, which is displayed in magenta to improve visibility compared \nto blue.  \n The spatial distribution of Fe, O and Co atoms was investigated by performing elemental mapping (~ 0.2 \nnm resolution) with electron energy loss spectroscopy spectrum- imaging (EELS -SI) of the Fe L -edge \n(green), Co L -edge (magenta, blue in composite) and O K -edge (red) (Figure 4). EELS -SI micrographs of \nC nanoparticles evidence the homogeneous atomic distribution of Fe and O atoms all across the nanoparticle which agrees with an iron oxide structure. In the case of CS nanoparticles, Fe, O and Co \nspatial distrib utions also overlap. A slight increase of the Co content at the edges of the nanoparticle \n(blue border of NP in Figure 4d) was observed, which agrees with the expected Fe\n3-O4@CoFe 2O4 \ncore@shell structure. Finally, CSS nanoparticles display a homogeneous s patial distribution of Fe, Co \nand O atoms all across the nanoparticle. The Fe/Co atomic ratio tends to increase in comparison to CS \nnanoparticles, in agreement with the expected Fe 3-O4@CoFe 2O4@Fe 3-O4 structure and in line with EDX \nresults. Line profiles of the composition across CS and CSS confirmed that the Co content increases at \nthe edges of both nanoparticles (Figures S2 and S3). Therefore, the second shell of iron oxide is much thinner than the size variation between CS and CSS (3.1 nm).  \n  \n 12   \nFigure 5. Dark field images of a) C, b) CS and c) CSS nanoparticles. d) Exemplary EELS spectra obtained \nfrom positions marked in (a) -c)) showing the O -K edge (532 eV, red vertical line), Fe- L (708 eV, green \nvertical line) edges as well as  the Co -L edge (779 eV, blue vertical line) in case of CS and CSS \nnanoparticles. The spectra are vertically displaced to improve visibility. e) -g) Comparison of the \nbackground -subtracted EELS spectra of the e) O -K, f) Fe- L and g) Co -L edges reveal fine structure \nchanges induced by the presence of Co in comparison to the C spectra. The Fe -L3 peak shifts to lower \nenergies in case of high Co percentages (CS shell). All spectra are normalized to the Fe -L3 peak.  \n \nTo further investigate the chemical compositio n of C, CS and CSS nanoparticles, EELS spectra were \nrecorded at precise positions within the EELS -SI data by averaging over the corresponding area, \nexcluding the shell (Figure 5). Spectra recorded for C (magenta) do not vary within the nanoparticle and corresponds to a homogeneous chemical composition of Fe\n3-O4 (Figure S5).47 In contrast, spectra of CS  \nclearly show a higher Co (779 eV) / Fe (708 eV) intensity ratio at the edge than in the center of t he \nnanoparticle (red and blue line in Figure 5 d), which agrees with a Fe 3-dO4@CoFe 2O4 core@shell \nstructure. In contrast, spectra recorded for CSS (yellow ) nanoparticles show that O K and Fe L signals are \nclosely related to that of the C spectrum and displays a weaker signal at the Co -L edge than CS. The \n 13 Fe/Co ratio is thus increased compared to CS nanoparticles which is coherent with EDX results. A \ncomparison of spectra obtained from different CS and CSS nanoparticles of the same batch reveals the reproducibility and homogeneity within the batches (Figures S6 and S7).  \n \nThe background- subtracted oxygen signal of the same spectra is shown in Figure 5e and reveals a clear \ndependence on the different cation environments. While the appearance of the broad peak around 540 \neV does not differ much between the spectra, the intensity of the sharp peak at 532 eV and the intensity of the following valley at 534  eV strongly vary with the Co content. The spectrum from the C \nnanoparticle displays a similar shape to the spectrum of magnetite Fe\n3O4 (Figure S5),47–49 exhibiting a \nsharp and intense peak at 532 eV followed by a deep valley at 534 eV. With increasing the presence of Co in the material, the peak intensity at 532 eV decreases while the intensity in the valley at 534 eV \nincreases. In the spectrum obtained from the shell of the CS nanoparticle, i.e. the area with the highest Co content, the intensities at 530 eV and 533 eV almost level out. This behavior is consistent with the one observed in CoO spectra.\n50 The comparison of the local intensity ratio between the pre -peak and \nthe following valley allows to map the chemical composition (Figure S8). While the O -K edge is highly \nsensitive to the Co con tent in the crystal, an effect on the Fe -L edge is only visible at considerable Co \ncontents leading to a shift of the peak position to lower energies as it is observed in the shell of the CS nanoparticle (red and blue lines in Figure 5f). A fit of the Fe -L peak allows to determine the exact peak \nposition and to map the chemical composition within the CS nanoparticles (Figure S8). Unfortunately, a fine edge analysis of the Co peak is not possible because of the low intensity in the Co peak (Figure 5g).  \n \n \n \nFigure 6. XRD patterns of C, CS and CSS nanoparticles. Black and blue bars correspond to the Fe 3O4 \n(JCPDS card n° 19 -062) and CoFe 2O4 (JCPDS card n°00 -022-1086) phases, respectively.  \n XRD patterns recorded for each nanoparticle show  peaks that were attributed to the spinel structure \n(Fd-3m space group) ( Figure 6 ). Unfortunately, Fe\n3O4 and CoFe 2O4 cannot be discriminated because of \nsimilar cell parameters (a(Fe 3O4) = 8.396 Å, JCPDS card n°19 -062 and  a(CoFe 2O4) = 8.392 Å, JCPDS card \n 14 n°00 -022-1086) . Nevertheless, peaks become narrower from C, CS to CSS, which correspond to larger \ncrystal sizes of 7.4, 10.1 and 12.0 nm ( Table 1 ), respectively. These values are consistent with the \nnanoparticles’ size measured from TEM micrographs. Hence confirming the good epitaxial relationship \nbetween  each core and shells as observed in high resolution (HR) TEM micrographs. The slight increase \nof the XRD signal at low angles arises from the presence of oleic acid which is used as a ligand.  \nThe cell parameter of C nanoparticles (8.37 Å) is intermediate to that of magnetite (a = 8.396 Å, JCPDS card n° 19 -062) and maghemite (a =  8.338 Å, JCPDS card n° 39 -1346) which confirms the partial \noxidation of C at their surface.\n51 The cell parameter of CS (8.41 Å) and CSS (8.41 Å) are larger than those \nof magnetite and cobalt ferrite which can be attributed to cr ystal strains resulting from the small size of \nthe nanoparticles (high curvature radius). Indeed, crystal strains up to 10 % were observed by performing GPA on isolated CS and CSS nanoparticles using HR -STEM images (Figure S9). Furthermore, \nit may also par tially account from the high content of Fe\n2+.39 These results confirm those of Lopez -Ortega \net al.52 who reported cell parameters of 8.40 -8.42 Å for Co 0.6-0.7Fe2.4-2.3O4 nanoparticles of different sizes. \nThey attributed such observation to the stabilization of a pure cobalt- doped magneti te phase with Fe2+ \nthat were not oxidized and to the presence of strains for such a small size.  \n \n \nFigure 7. FT-IR spectra recorded at low wavelength for C, CS and CSS. Stars correspond to bands that were \nascribed to some residue of iron and cobalt stearates (see Figure S10 for more information).  \n Fourier transform infrared (FT -IR) spectra exhibit large bands in  the region from 800 to 450 cm\n-1, which \ngive additional indications on the chemical composition of the nanoparticles ( Figure 7 ).53 C nanoparticles \ndisplay a broad band centered at 602 cm-1 which agree with the partial oxidation of Fe 3O4 in -Fe2O3, \ndenoted as Fe 3-O4.51  Indeed, Fe 3O4 displays a single band at 574 cm-1 with a shoulder at 700 cm-1 while \n-Fe2O3 maghemite shows a maximum centered at 639 cm-1 with several oscillations from 800 to 600 \ncm-1.53 This band shifts down to 600 cm-1 and becomes narrower for CS, which agree with a higher \ncontent in Fe 3O4. The CoFe 2O4 shell partially avoids the oxidation of the core when exposed upon to air \nafter washing. Furthermore, as cobalt fer rite displays a band at 590 cm-1,54 it also contributes to shift \ndown the band. This band shifts down even lower to 581 cm-1 for CSS, getting closer to that of magnetite \n(574 cm-1). Hence, the second shell in CSS would mainly  consists of magnetite, although it was expected \nto be fully oxidized according to our previous work on single Fe 3-O4 nanoparticles.51 The narrower band \nof CS and CSS than C, and the concomitant disappearance of oscillations attributed to maghemite, confirm these observations. Thus, FT- IR shows the inc rease of Fe\n2+ content from C, CS to CSS and the \npresence of CoFe 2O4 in CS and CSS. Small bands around 725 cm-1 were attributed to the H -C-H scissoring \n 15 bond and correspond to negligible amounts of remaining stearates that could not be removed after \nwashing without avoiding nanoparticle aggregation.  \n The chemical composition was investigated deeper by means of cationic distribution in O\nh and T d sites \nand oxidation state by performing Mössbauer and XAS/XMCD spectroscopies. 57Fe Mössbauer \nspectrometry brings information on the valence state of each Fe species, the local electronic structure and the magnetic environment which are described by the isomer shift δ’, the quadrupolar shift ε and the hyperfine field B\nhf, respectively (Table 2).  \n  \nTable 2. Refined values of hyperfine parameters calculated from the fit of 57Fe Mössbauer spectra \nrecorded at 77 K.  \nSample  isomer shift \nrelative to α -\nFe \n  \n(mm/s)  \n±0.01  quadrupole \nshift or  \nquadrupole \nsplitting  \n(mm/s)  \n±0.01  Hyperfine \nfield  \n  \n(T) \n±0.5  Relative sub -\nspectral area  \n  \n(%) \n±2 Fe species  Site \noccupancy  \nC <0.44>  <0.04>  <42.8>  100 Fe3+ - \nCS 0.53  -0.04  53.4  36 Fe3+ Oh \n0.41  0.02  50.9  51 Fe3+ Td \n0.65  0.02  47.4  9 Fe2-3+ Oh \n1.11  2.13  34.5  3 Fe2-3+ Oh \n1.27  2.38   - 1 Fe2+ \n<0.50>  <0.07>  <50.5>   99 - - \nCSS 0.50  0.08  52.5  46 Fe3+ Oh \n0.33  -0.07  51.8  45 Fe3+ Td \n0.53  -0.09  47.1  6 Fe3+ Oh \n1.04  1.74  32.4  3 Fe2+ Oh \n<0.49>  <0.05>  <51.2>  100 - - \n Mössbauer spectra were recorded at 77 K for each sample ( Figure 8\n). They all display a resolved sextet, \nconsistent with rather magnetic blocked state. Mössbauer spectrum of C sample displays the broadest sextet lines which are ascribed to the fas ter relaxation time of a fraction of spins than the measurement \ntime of the experiment (τ\nm = 10-10 – 10-7 s). Such a superparamagnetic contribution can be attributed to \na fraction of nanoparticle with small size (about 6.0 nm) as shown by the size distribu tion (Figure 2c). \nThese results are consistent with the literature: Iron oxide nanoparticles of 11 nm measured at 77 K display no superparamagnetic contributions in Mössbauer spectra,\n55 while smaller iron oxide \nnanoparticles of 4.6 nm measured at 77 K show significant superparamagnetic contributions.56 The \nsuperparamagnetic contribution decre ases significantly for CS (1 %) and is not observed for CSS.  \n  16 0.991.00-12 -6 0 6 12\n0.960.970.980.991.00\n-12 -6 0 6 120.960.981.00 \n \nC\nCS\n  V (mm/s)\nV (mm/s)Relative transmission\nCSS\n  \n \nFigure 8. Mössbauer spectra of C, CS and CSS recorded at 77 K recorded at 77 K: black, blue and red \nlines correspond to the total theoretical  spectrum, the total contribution of Fe3+ and Fe2-3+ components \nand the Fe2+ components, respectively.  \n \nThe fitting procedure requires great care to describe the Mössbauer spectra, which consist of magnetic \nsextets composed of wide, asymmetrical lines.hey have been well described by means of a discrete number of magnetic and/or quadrupolar components with independent values of isomer shift, quadrupolar shift and hyperfine field. As the solution is not unique, we report in Table 2 the \ncorresponding refined v alues obtained from one example of it, but it is important to note that the mean \nvalues of hyperfine parameters are independent of the fitting procedure. Although the broad sextet \nlines of C were not accurate for spectrum refinement, the mean isomer shift (0.44 mm/s) is much closer \nt o  t h a t  o f  m a g h e m i t e  ( δ  =  0 . 4 0  m m / s )  t h a n  m a g n e t i t e  ( δ  =  0 . 6 1  m m / s ) .\n57,58 According to a linear \nextrapolation, C consists of approximately 19 % of magnetite and 81 % of maghemite.  The mean isomer \nshift for CS increases to 0.50 mm/s which is correlated to a larger amount of Fe2+ than C. More precisely, \nthe spectrum refinement evidences two contributions with the typical Fe2+ isomer shift (1.11 and 1.27 \nmm/s), the first one as a sextet corresponding t o Fe2+ in magnetically blocked magnetite, the second \none as quadrupolar doublet to Fe2+ in superparamagnetic nanoparticles containing a substantial core of \nmagnetite. The main contribution centered at 0.53 mm/s was attributed to Fe3+ in O h sites that account \nfor 36 %. A second contribution centered at 0.41 mm/s was attributed to Fe3+ in T d sites and accounts \nfor 51 %. In addition, a third component centered at 0.65 mm/s was assigned to some intermediate \nFe2+,3+ species, which occurs below the Verwey transition. The 𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇3+𝐹𝐹𝐹𝐹𝑂𝑂ℎ2+,3+�  ratio calculated for CS is \napproximately 1, which is much higher than the theoretical value of 0.5 for pure magnetite. Considering \nthe core- shell structure where Co2+ partially replaces Fe2+ in Oh sites, an intermediate value was \nexpected. It may be attributed to a super stoichiometry in oxygen or to the presence of vacancies.59  17 Furthermore, high hyperfine fields of 53.4 T and 50.9 T for Fe3+ in O h and T d sites, respectively, agree \nwith the presence of Co species in the vicinity of Fe species, consistent with CoFe 2O4.60,61 In contrast, \nhyperfine fields of 47.4 and 34.5 T measured for Fe3+ and Fe2+ in O h sites correspond to the Fe 3-O4 core. \nAccording to isomer shifts reported previously for Fe 2.95O4 (0.61 mm/s)62 and CoFe 2O4 (0.45 mm/s)63 \nnanoparticles, the mean isomer shift of CS (0.51 mm/s), would correspond to a composition of 63 % of \nCoFe 2O4 and 37 % of Fe 3-dO4, i.e. a core size of 6.8 nm and a shell thickness of 1.6 nm, assuming a simple \ncore@shell model with a radial structure. Accor ding to the size variation measured from TEM \nmicrographs, we expected a cobalt ferrite shell thickness of 1 nm. Therefore, such a thicker cobalt ferrite shell may result from the partial solubilisation of the iron oxide core followed by the recrystallizati on\n35 \nof Fe monomers with Co monomers in a similar way we reported earlier.23,39,40 Considering that a \nstoichiometric ratio of Fe and Co stearate was used, we expect the cobalt ferrite shell to be under stoichiometric.  \n In CSS, the mean isomer shift decreased slightly to 0.49 mm/s which evidences a slightly lower content \nof Fe\n2+ than in CS. The contribution centered at 1.04 mm/s felt down to 3 %. Additional contributions \ncentered at 0.53 mm/s and 0.50 mm/s were attributed to Fe3+ in O h sites (6 % and 46 %, respectively). \nA third contribution centered at 0.33 mm/s was attributed to Fe3+ in T d sites (45 %). The lower \n𝐹𝐹𝐹𝐹𝑇𝑇𝑇𝑇3+𝐹𝐹𝐹𝐹𝑂𝑂ℎ2+,3+�  ratio (0.82) for CSS than for CS (1) agrees with higher amount of Fe2+ that may be \nlocalized in the second shell of iron oxide that was grown at the surface of the CoFe 2O4 shell. High \nhyperfine fields (52.5 and 51.8 T) calculated for Fe3+ in O h and T d sites also confirm the presence of \ncobalt ferrite within CSS. The slight increase of their sub spectral areas in comparison with CS is indicative of a larger fraction of cobalt ferrite. It is clearly confirmed by the higher mean isomer shift which corresponds to 75 % of cobalt ferrite and 25 % of Fe\n3-O4. Thus, the composition of CSS \nnanoparticl es would consist in a core of 6.8 nm, a 2.9 nm thick cobalt ferrite shell and a 0.3 nm thick \nFe3-O4 shell, which is in agreement with STEM -EELS measurements. Therefore, the cobalt ferrite shell \nin CSS would be thicker than in CS (1.6 nm).  \n \nMössbauer spec trometry has shown that the Fe2+ content increases from C to CS but slightly decreases \nfrom CS to CSS. It has also shown that the Fe 3-O4 core size decreases concomitantly to the cobalt ferrite \nshell which increases further from CS to CSS thus resulting in  a much thinner Fe 3-O4 shell than expected. \nAccording to the results obtained with the above -mentioned techniques (the increase of the mean cell \nparameter (XRD) and the shift of the M -O band to lower frequencies (FT -IR) from C, CS to CSS as well as \nthe br oad Co distribution in CSS nanoparticles revealed by STEM -EELS (mapping and spectra), it seems \nthat the evolution of the cobalt ferrite phase predominates over the variation of the Fe2+ content \nbetween CS and CSS.  \n  18  \nFigure 9. a), c) Isotropic XAS and b, d) XMCD spectra at the a), b) Fe L2,3 edges and at the c), d) Co L2,3 \nedges of C, CS and CSS nanoparticles.  \n \nThe isotropic XAS and XMCD spectra recorded at the Fe L 2,3 edges ( Figure 9a and b ) are all typical of a \nspinel ferrite structure.64–66 XAS spectra evidenced two main contributions in the  L3 region that were \nascribed to Fe2+ in O h sites (peak I 1) and to Fe3+ in O h and T d sites (peak I 2). Hence, the intensity ratio I 1/I2 \nreported as 0.7167 for Fe 3O4 and 0.35 for  -Fe2O367 brings further information on the Fe2+ content within \nthe nanoparticles. The value calculated for C (0.53) agrees with an intermediate composition of between magnetite and maghemite. Then, it incr eases for CS (0.64) corresponding to a higher content of Fe\n2+. \nThese results agree with those of XRD, FT- IR spectroscopy and Mössbauer spectrometry and those of \nour previous work on similar core -shell nanoparticles.23 The I 1/I2 ratio slightly decreases for CSS (0.62) \nwhich is ascribed to a slightly lower content in Fe2+ (as obs erved from Mössbauer spectrometry).  \n XMCD spectra recorded at the Fe L\n2,3 edges display three main peaks in the L 3 region where the S1 and \nS3 peaks were respectively attributed to Fe2+ and Fe3+ in O h sites, while the S2 peak corresponds to Fe3+ \nin T d sites , which spins are coupled antiparallel to Fe cations in O h sites. Such consideration is typical of \nthe ferrimagnetic coupling of Fe spins in the reverse spinel structure of iron oxide and cobalt ferrite. The intensity ratio S=(S1+S2)/(S2+S3) brings further  information on the oxidation state of iron cations. \nHence, magnetite displays a higher ratio (1.14) than maghemite (0.69).\n67 C sample displays the closest \nratio (0.77) to maghemite, while it increases for CS (0.85) and get even higher for CSS (0.90). Such an increase of Fe\n2+ content from CS to CSS is contradictory with the I 1/I2 ratio measured from XAS spectra \nand to Mössbauer spectrometry. XMCD being a polarized mode, it may favor Fe2+ uncompensated spins \nat the nanoparticle surface resulting from the break of symmetry vs. Fe3+ spins which are coupled \nantiparallel. In addition, we observed an excess of Fe3+ in T d sites in the XMCD spectra of CS and CSS \n 19 (Figure S13) which may result from a preferential occupancy of the O h sites by the Co2+ cations in the \ncobalt ferrite structure.  \n \nIsotropic XAS spectra recorded at the Co  L2,3 edges confirmed the presence of Co2+ in O h sites of a spinel \nstructure.66,68 The I 4 peak is slightly higher than the I 3 peak for CS which  is more obvious for CSS.  Such \nobservation qualitatively shows the increase of the cobalt ferrite content23,69 from CS to CSS \nnanoparticles. The XMCD spectra recorded at the Co L 2,3 edges are all typical of Co2+ cations in O h \nsites.68,70, 23 All spectra being normalized at the edge of the energy jump, the intensity of the S4 peak (95 \n% for CS and 108 % for CSS with respect to XAS  signal) agree with an increase of uncompensated Co2+ \nspins from a CoFe 2O4 phase between CS and CSS samples.71,44 It confirms the increase of Fe2+ from C, CS \nto CSS and XAS and XMCD spect ra unambiguously show the presence of a CoFe 2O4 phase, which \nincreases from CS to CSS.  \n Element -specific magnetization curves were recorded at the Fe S2, S3 and Co S4 peak energies for CS, \nand at the Fe S2 and Co S4 peak energies for CSS (\nFigure 10 ). The selective hysteresis curves recorded \nfor CS at different energies showed similar coercive fields (H C) of about 6.5 kOe (Table 3). It shows that \nFe spins i n T d and O h sites and Co spins in O h sites are magnetically coupled which confirms the presence \nof CoFe 2O4.23,44 CSS behaves similarly although H C is larger (about 9.5 kOe), agreeing with a thicker cobalt \nferrite layer than for CS, as shown by Mössbauer spectrometry. These values may differ from those \nobtained by magnetometry (see below) because the sample’s preparation is different, which induces some variations of dipolar interactions between nanoparticles.  \n \nFigure 10. Element -specific magnetization curves recorded at 4 K by XMCD at the Fe and Co L 2,3 edges in a) \nCS, b) CSS.  \nTable 3. Magnetic characteristics of element specific magnetization curves.  \nSample  HC \nFe S2  \n(kOe)  HC \nFe S3  \n(kOe)  HC \nCo S4  \n(kOe)  <H C> \n(kOe)  \nCS 6,3 6,7 6,6 6,5 ±  \nCSS 9,7 - 9,4 9,5 ±  \n \n \nThe magnetic properties of C, CS and CSS were investigated by SQUID magnetometry ( Figure 11 ). \nMagnetization curves recorded against temperature (M(T)) after zero field cooling (ZFC) show a \nmaximum at T max which is usually assimilated to the transition temperature between blocked magn etic \nmoments and the superparamagnetic behavior. T max measured for C (86 K) agrees with values reported \nfor iron oxide nanoparticles of similar sizes.51 It increases for CS (290 K), and further for CSS (300 -  350 \n 20 K), agreeing with higher magnetic anisotropy energy (E a) which results from the modification of the \nnanoparticle structure. Although M(T) curves recorded for C are very typical of magnetic iron oxide \nnanoparticles, the ones recorded for CS and CSS correspond to different magnetic properties. For CS, a kink at temperatures slightly bel ow T\nmax refers to a minor fraction of nanoparticles which exhibit a lower \neffective magnetic anisotropy energy than the rest of the sample. As long as the size distribution of CS is rather narrow, it may results from inhomogeneous spatial distribution and weaker dipolar interactions which may be partially due to the oleic acid.\n72 For CSS, the increase of magnetization at temperature \nhigher than T max can be ascribed to super magnetic doma ins resulting from strong dipolar interactions \nbetween nanoparticles, e.g. superferromagnetism.2,73 Indeed, the M(T) field cooled  (FC) curve show \nalmost constant magnetization at low temperatures which agrees with dipolar interactions between nanoparticles.\n74,75 Therefore, T max corresponds to a broad distribution of temperatures which is difficult \nto assess precisely.  \n \nThe transition between blocked and flipped magnetic moments is more accurately described by the \nblocking temperature (T B) which corresponds to a  distribution of energy barriers. T B can be easily \nextracted from the ZFC -FC M(T) curves using the following equation:76 \n \n  𝑓𝑓(𝑇𝑇𝐵𝐵)= −[𝑇𝑇𝑀𝑀𝑍𝑍𝑍𝑍𝑍𝑍−𝑀𝑀𝑍𝑍𝑍𝑍]\n𝑇𝑇[𝑇𝑇𝑇𝑇]  (1) \n \nTB corresponds to temperature distributions centered at 48, 239 and 280 K for C, CS and CSS, \nrespectively ( Figure 11 b). The dramatic enhancement of T B from  C to CS is attributed to arise from \ninterfacial exchange coupling between the soft Fe 3-O4 and the hard CoFe 2O4 phases. Temperature \ndistributions of CS and CSS are more complex than that of C. For CS, an additional contribution centered at 278 K correspon ds to the kink observed in M(T) curves. CSS displays a minor contribution centered \nat 390 K which may be attributed to the presence of super ferromagnetic domains.\n2,73 A second \ncontribution centered at 220 K can be attributed to CSS nanoparticles which are partially or not covered by a second shell.  \n  21  \nFigure 11. Magnetic characterizations of C, CS a nd CSS. a) Magnetization curves recorded against \ntemperature after zero field cooling (ZFC) and field cooling (FC). b) Distribution of blocking temperatures \n(TB). Magnetization measurements recorded against a magnetic field at c) 300 K, d) 5 K after ZFC, a nd \ne) 10 K after FC under 7 T.  \n Considering the anisotropy constants (K(CoFe\n2O4)8,77 ≈ 1–6.105 J/m3  and K(Fe 3O4)73 ≈ 1–5.104 J/m3 ) and \nthe volume V of each phase,  \n \nK(CoFe 2O4).V(CoFe 2O4) >> K(Fe 3O4).V(Fe 3O4) (2) \n \nTherefore, i n exchange coupled nanoparticles, the effective magnetic anisotropy (E eff) of CS and CSS can \nbe assimilated to:78  \n \nEeff = K effV = K(CoFe 2O4).V(CoFe 2O4) = 25K BTB (3) \n 22  \nwith V the total volume of the nanoparticle, and k B the Boltzmann constant. K eff calculated for CS (1.58 \n105 J/m3) agrees with the anisotropy constants reported for CoFe 2O4 nanoparticles. Then, K eff \nsignificantly decreases for CSS (8.2 104 J/m3) and gets very close to C (6 .2 104 J/m3) although the soft -\nshell volume is low (150 nm3) and the volume of the CoFe 2O4 shell is increased by four times. According \nto the anisotropy constant calculated for CS, a much higher T B (400 K) was expected for CSS. In contrast, \nthe increase of  TB (40 K) from CS to CSS is higher than that (20 K) corresponding to the volume of the \nFe3-O4 shell. Therefore, the increase of T B is not ascribed to the volume increase but to exchange \ncoupling at the CoFe 2O4/Fe 3-O4 that contributes to the enhancement of the effective magnetic \nanisotropy energy of nanoparticles.  \nHC temperature dependent curves of CS and CSS (Figure S12) also bring information on the magnetic \nbehavior of CS and CSS. H C decreases with increasing the temperature until the onset at similar \ntemperatures (about 265 K) for both CS and CSS (Figure S12). Nevertheless, H C is higher for CS than for \nCSS and decreases faster when the temperature rises up. According to the Stoner -Wohlfarth model, 79 \nit is ascribed to a higher effective magnetic anisotropy energy for CS than for CSS.  \n \n𝐻𝐻𝐶𝐶=0.48𝐻𝐻𝐾𝐾�1−�𝑇𝑇\n𝑇𝑇𝐵𝐵�0.5\n� (4) \n \nwith the anisotropic field 𝐻𝐻𝐾𝐾=2𝐾𝐾𝑒𝑒𝑒𝑒𝑒𝑒\n𝑀𝑀𝑆𝑆 \n \nMagnetization curves recorded against an applied magnetic field at 300 K perfectly overlap for each sample which agree with superparamagnetic behavior. In contrast, M(H) curves recorded at 5 K show \nopened hysteresis corresponding to blocked magnetic moments. The coercive field (H\nC) measured for \nC (300 Oe) is typical of  iron oxide nanoparticles.51 It increases dramatically for CS (19.2 kOe) because of \nthe strong interfacial exchange -coupling between the hard CoFe 2O4 shell and the soft Fe 3-O4 core .1 \nThen, it decreases for CSS  (13.1 kOe) because of the presence of the soft Fe 3-O4 shell. It is consistent \nwith the decrease of the effective magnetic anisotropy constant in comparison with CS, as observed for \nCoFe 2O4@MnFe 2O4 core -shell nanoparticles.24 Same trends were observed for the remanent \nmagnetization (M R) and the M R/M S ratio (Tabl e 4). The volume of the Fe 3-O4 shell (160 nm3) is much \nsmaller than that corresponding to the increase of the CoFe 2O4 shell (500 nm3). Therefore, H C being \ndependent on the fractions of hard and soft phases,80 it was expected to increase. Indeed, the Fe 3-O4 \nphase favors the  magnetic reversal of interfacial spins of CoFe 2O4 which results in lower H C. \nFurthermore, no kinks were observed in M(H) curve, agreeing with an effective exchange coupling between both Fe\n3-O4 and CoFe 2O4 phases, that propagate through the entire volume of nanoparticles, \nwhatever the interface in both CS or CSS structures.  \nM(H) curves recorded for CS and CSS after field cooling (7 T) from 300 K to 10 K show larger H C than ZFC \nM(H) curves. It is ascribed to the alignment of soft spins with the applied mag netic field that favors their \ncoupling with hard spin, thus resulting in magnetic reversal at higher magnetic fields. Moreover, the hysteresis curves were not shifted to low magnetic fields. Such a behavior being typical of exchange bias coupling between s oft FiM and hard antiferromagnetic phases, it agrees with the absence of CoO.\n14,39,81 \n The increase of saturation magnetization (M\nS) from C (58 emu/g) to CS (78 emu/g) and CSS (77 emu/g) \nis also indicative of their chemical composition which agree with a thicker CoFe 2O4 shell.50,60 Indeed, for \nsimilar sizes, CoFe 2O4 nanoparticles84,85 display a higher M S than Fe 3-O4 nanoparticles.42 Although  these \nvalues are lower than bulk values (Fe 3O4 : 98 emu/g, CoFe 2O4 : 94 emu/g) because of surface effects,42 \nthe higher M S of CS and CSS than C agrees with the coupling of the Fe 3-O4 interfacial spins by that of \nthe CoFe 2O4. The slight decrease of M S from CS to CSS agrees with the formation of a very thin shell of \nFe3-O4. \n   23 Table 4. Magnetic characteristics of C, CS and CSS.  \n C CS CSS \nSize (nm)  8.0 10.0  13.1  \nShell thickness (nm)  - 1.0 1.5 \nHC 5 K (ZFC) kOe  0.3 19.2  13.1  \nHC 10 K (FC) kOe  0.3 24.1  15 \nHE 10 K (FC) Oe  0 0 0 \nTmax (K) 86 290 301-400 \nTB (K) 48 239 280 / 327  \nKeff (104 J.m-3) 6.2 15.8  8.2 \nMS 5K (ZFC) emu/g  58 78 77 \nMR 5K (ZFC) emu/g  15 69 63 \nMR/M S 0.26  0.89  0.82  \n \n \nDiscussion  \n TEM micrographs showed that the size of the nanoparticles increased from 8.0 ± 0.9 nm (C) to 10.0 ± 1.5 nm (CS) and, further to 13.1 ± 2.2 nm (CSS), in agreement with the successive growth of shells. Nevertheless, the complexity of crystal growth processes resulted in broader size distributions and in \nshape deviation from spheres.  It can be ascribed to various param eters such as kinetics (reagent \nconcentration, temperature, mass transport, capping agent) and thermodynamics (energy barrier, \nsurface energy) parameters.\n86  Iron oxide nanoparticles exhibit a faceted shape that consists in the {100} \nand {110} planes as usually observed for cubic crystall ographic structures.31,42 hkl reflections  featuring \ndifferent surface energies, they may favor heterogeneous seed -mediated growth. Other processes such \nas the selective binding of oleic acid acting as capping agent on specific { hkl} planes and the competition \nof adsorption vs. diffusion of atoms on crystal surface markedly alter crystal growth.86 \nHAADF -HRSTEM micrographs revealed a single crystal -like structure for each nanoparticle, thanks to the \nnegligible lattice mismatch between both spinel structures. FFT calculated from HR -TEM micrographs \nrecorded on the edge and at the center of the nanoparticles evidenced the good epitaxial relationship \nof the shells with the core. These results were confirmed by the increase of the crystal size from C to CS \nand further to CSS, as it has been measured from XRD patterns. The cell parameter calculated for the \niron oxide core corresponds to partially oxidized Fe 3-O4 nanoparticles which agrees with the literature. \nValues calculated for CS and CSS are higher than that of C and correspond to higher Fe2+ content and to \nthe growth of a cobalt ferrite shell. However, they are larger than the theoretical value of cobalt f errite \nwhich is related to the presence of crystal strains, as shown by GPA.84 Slight stacking defaults in the \ncrystal structure were also observed from HR -TEM micrographs at the edge of the nanoparticles.  \n Depending on the chemical composition of nanoparticles, since XRD cannot accurately discriminate Fe\n3-\nO4 from CoFe 2O4 due to the very similar cell parameters of the two crystalline structures, a wide set of \ncomplementary analysis techniques was used. The EELS -SI mapping performed on individual \nnanoparticles clearly showed the presence of Co atoms at the edges of the CS na noparticles. In CSS, \nEELS data suggest that the Co atoms are not confined to the first shell, but are distributed in the core and in the second shell. Although EELS cross -sections performed at different position –  i.e. at the center \nand on the edges -   of CSS did not show a decrease in the Co content at the very edge of CSS, EDX \nmeasurements performed on groups of nanoparticles showed that the Fe:Co atom ratio increased from CS to CSS which is consistent with the growth of an iron oxide shell at the surface  of CS. The FT- IR spectra \ngave more details on the evolution of the chemical composition of the nanoparticles. The spectra showed that the M -O vibration band became narrower and was shifted to lower frequencies from C to  24 CS and, further to CSS, which is co nsistent with the formation of cobalt ferrite as well as the increase of \nFe2+ content after the growth of each shell.  \n \nThe chemical structure of the nanoparticles was investigated more deeply by performing Mössbauer \nspectrometry and XAS/XMCD spectroscopy.  Mössbauer spectrometry gave information on the \noxidation state and the site occupancy of Fe atoms. The spectra showed that the content of Fe2+ \nincreased from C nanoparticles to CS which is consistent with the preservation of the core against \noxidation upo n exposure to air through the formation of the cobalt ferrite shell. The increase of the \naverage hyperfine field with respect to that of Fe 3-O4 nanoparticles is indicative of a significant fraction \nof Fe3+ cations in a cobalt ferrite structure. Considerin g the respective fractions of the different cations \n(oxidation state and site occupancy), CS consists of a core of 6.8 nm, surrounded by a 1.6 nm thick CoFe\n2O4 shell. Such a reduction of the iron oxide core (8.0 nm) while the size of the nanoparticles \nincreases from C to CS, was ascribed to the partial solubilisation -recrystallization process upon heating \nat high temperature.35 \nFe monomers generated by the partial solubilisation of  the Fe 3-O4 core contribute to the formation of \nthe CoFe 2O4 shell. Although the CoFe 2O4 shell becomes thicker, Fe and Co stearates being added as \nstoichiometric, the CoFe 2O4 shell is certainly sub stoichiometric and corresponds to about Co 0.79Fe2.21O4. \nIn CSS, the CoFe 2O4 shell became thicker (up to 2.9 nm) than CS because some Co stearate that remains \nin the nanoparticle suspension (see FTIR, Figure 8) reacted with Fe stearate that was added in the reaction medium. Therefore, most of Fe monomers contribut e to the extension of the cobalt ferrite \nshell, while a small fraction led to the formation of the second Fe\n3-O4 shell which is very thin (0.3 nm). \nThis value being smaller than the cell parameter, such a shell did not grew homogeneously at the surface of CSS. These results were confirmed by STEM -EELS line profiles performed across CS and CSS \nnanoparticles (Figures S2 and S3). These STEM -EELS analyses showed that the Co content exists in larger \nvolume than what was expected from the size variation measured  from TEM micrographs. Furthermore, \nthe Co content increases at the edge of CS as expected (Figure S4). It is much lower for CSS, which agree with the formation of a very thin shell of Fe\n3-O4 although it was expected to disappear (Figure S4).  \n \nThe XAS an d XMCD spectra recorded at the Fe and Co edges provide additional information on the \nchemical structure of the nanoparticles. The evolution of several peaks in XAS spectra at Fe L3 edge \nagrees with an intermediate phase between magnetite and maghemite. It confirmed that the Fe2+ \ncontent increased from C to CS, and slightly decreased in CSS, in accordance with XRD, FT -IR \nspectroscopy  and Mössbauer spectrometry. XMCD spectra are signatures of a ferrimagnetic coupling of \nFe spins in the reverse spinel structur e of iron oxide and cobalt ferrite. However, XMCD shown a weak \nincrease of Fe2+ content from CS to CSS which may be due Fe2+ uncompensated spins at the nanoparticle \nsurface resulting from the break of symmetry vs. Fe3+ spins in O h and T d sites which are coupled \nantiparallel. Furthermore, XAS and XMCD spectra at the Co L 3 edge unambiguously demonstrate the \npresence of CoFe 2O4 in CS and CSS with a Co2+ cations in O h sites.  The magnitude of XMCD confirms the \nincrease of the cobalt ferrite phase form CS to C SS. This result was confirmed by element- specific \nmagnetization curves recorded at Fe and Co edges. Similar H C agree with strong exchange interactions \nbetween Fe and Co spins in a spinel structure.  \n \nThese results show that the chemical composition and the  structure of CS and CSS ( Figure 12) differ \nsignificantly from what was expected from the size variation of the nanoparticles measured from TEM micrographs and the Fe/Co molar ratio calculated from EDX analysis.  \n   25  \n \nFigure 12. Schematic illustration of the nanoparticle structure a) expected from size variation \nmeasured from TEM micrographs and Fe/Co molar ratio measured from EDX, and b) from data \ndeduced from Mössbauer and XAS/XMCD spectra.  \n \n The magnetic properties of CS and C SS are directly correlated to their chemical composition and crystal \nstructure. The significant increase of T\nB from C to CS, and further to CSS agrees with strong hard- soft \nmagnetic exchange coupling at Fe 3-O4/CoFe 2O4 and CoFe 2O4/Fe 3-O4 interfaces. The h ighest T B value of \n280 K is very close to that (300 K) measured for Co 0.68Fe2.32O4 nanoparticles with similar size.85 It shows \nthat interfacial exchange coupling mostly compensates the lack of magnetic anisotropy energy corresponding to the volume of the Fe\n3-O4 core. The contribution of the exchange coupling \nphenomenon at the soft/hard interface to T B is more significant than that of the hard/soft shell. It can \nbe ascribed to the  inhomogeneous coating of the Fe 3-O4 shell which limits the hard/soft interface and \nthe coupling efficiency. Nevertheless, the combination of soft/hard and hard/soft interfaces contributes \nto the enhancement of the effective magnetic anisotropy energy. Fu rthermore, the smooth variation of \nmagnetization in each M(H) curves (no kinks) confirms strong hard- soft exchange coupling at core/shell \nand shell/shell interfaces which propagates efficiently  through the entire core -shell structure. The \ndecrease of H C from CS to CSS is also ascribed to the growth of the second Fe 3-O4 shell. The increase of \nthe hard CoFe 2O4 shell volume from CS to CSS has no significant effect on these parameters. In contrast, \nMS increased significantly from C to CS. Interestingly, CSS displays much stronger dipolar interactions \nthan CS although they display similar M S. It means that the super ferromagnetic behavior is significantly \ndependent on the amount of oleic acid grafted at the nanoparticle surface which control interparticle \ndistances. \n    \nConclusion  \n \nFe3-O4@CoFe 2O4@Fe 3-O4 (CSS) nanoparticles  were synthesized  by the thermal decomposition method \nthrough two successive seed -mediated growth steps. The chemical composition and the crystal \nstructure of the nanoparticles were investig ated by a wide set of analysis techniques. First of all, the \nthickness of the shells in CS and CSS (Figure 12) vary significantly from values expected from TEM \n 26 micrographs and EDX analysis. Indeed, the reaction mechanism consists in the partial solubilisation of \nthe nanoparticles followed by the recrystallisation of monomers which results in significantly different spatial distributions of Fe and Co cations in the nanoparticle volume. While such a mechanism leads to a thicker cobalt ferrite shell, it is und er stoichiometric with respect to Co content. In contrast, the second \niron oxide shell is so thin that it does not homogeneously cover the cobalt ferrite shell. Nevertheless, each nanoparticle exhibits a single crystal -like structure because of the very lo w lattice mismatch \nbetween both spinel phases. Some crystal defects which arise from the nanoparticle geometry and the different surface energy of facets could be observed. Although cobalt ferrite could not be discriminated from iron oxide by XRD, its pres ence was unambiguously showed by XAS/XMCD measurements and \nconfirmed by Mössbauer spectrometry and FT -IR spectroscopy. The CoFe\n2O4 shell avoids the oxidation \nof Fe2+ at the surface of the iron oxide core which results in a higher Fe2+content in CS that slightly \ndecreases in CSS.  \nCS and CSS nanoparticles display enhanced magnetic anisotropy energies in comparison to C as a result \nof strong exchange coupling at the soft/hard and hard/soft interfaces. Although the Fe 3-O4 shell is very \nthin and discontinuou s, it has a significant influence on the magnetic properties of CS nanoparticles. In \ncomparison, pure CoFe 2O4 nanoparticles with similar size to that of CSS display close value of T B to ours, \nalthough a much lower amount of Co atoms was incorporated. Furth ermore, the softness of the Fe 3-O4 \nshell resulted in the decrease of H C and M R while M S was preserved. Such a high control of the structure \nof the nanoparticles is particularly interesting to modulate their magnetic properties, thus extending \ntheir potent ial to a wide range of applications.  \n \nASSOCIATED CONTENT  \nSupporting Information. Additional data related to STEM -HAADF micrographs, EELS spectra, EELS -SI mapping, GPA, FTIR, \ngranulometry, magnetism, XMCD spectra. The Supporting Information is available fr ee of charge on the ACS Publications \nwebsite.  \n \nAUTHOR INFORMATION  \nCorresponding Author  \nbenoit.pichon@ipcms.unistra.fr  \n \nNotes  \nThe authors declare no competing financial interests.  \n \nACKNOWLEDGMENT  \nK.S. was supported by a PhD grant from the French Agence Nationale de la Recherche (ANR) under the reference ANR- 11-\nLABX -0058- NIE within the Investissement d'Avenir program ANR- 10-IDEX -0002- 02 and SOLEIL synchrotron / Laboratoire Léon \nBrillouin fellowship. The authors are grateful to SOLEIL syn chrotron light source for providing the access to DEIMOS beamline \nand to DIAMOND synchrotron light source for providing access to BLADE beamline. HR- STEM and STEM -EELS studies were \nconducted at the Laboratorio de Microscopias Avanzadas, Universidad de Zara goza, Spain. S.H. is grateful to DFG (HE 7675/1 -\n1). 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Exchange Bias Phenomenology and Models of \nCore/Shell Nanoparticles. Journal of Nanoscience and Nanotechnology  2008 , 8 (6), 2761.  \n(82)  Salazar -Alvarez, G.; Sort, J.; Uheida, A.; Muhammed, M.; Suriñach, S.; Baró, M. D.; Nogués, J. \nReversible Post -Synth esis Tuning of the Superparamagnetic Blocking Temperature of γ -Fe 2 O 3 \nNanoparticles by Adsorption and Desorption of Co( II ) Ions. J. Mater. Chem.  2007 , 17 (4), 322 –328. \n(83)  Polishchuk, D.; Nedelko, N.; Solopan, S.; Ślawska -Waniewska, A.; Zamorskyi, V.; Tovstolytkin, A.; \nBelous, A. Profound Interfacial Effects in CoFe2O4/Fe3O4 and Fe3O4/CoFe2O4 Core/Shell Nanoparticles. Nanoscale Research Letters  \n2018 , 13 (1). \n(84)  López -Ortega, A.; Lottini, E.; Fernández, C. de J.; Sangregorio, C. Exploring the Magnetic \nProperties of Cobalt- Ferrite Nanoparticles for the Development of a Rare- Earth -Free Permanent \nMagnet. Chem. Mater.  2015 , 27 (11), 4048 –4056.  \n(85)  Torres, T. E.; Lim a, E.; Mayoral, A.; Ibarra, A.; Marquina, C.; Ibarra, M. R.; Goya, G. F. Validity of \nthe Néel -Arrhenius Model for Highly Anisotropic CoxFe3−xO4 Nanoparticles. Journal of Applied Physics  \n2015 , 118 (18), 183902.  \n(86)  Xia, Y.; Xia, X.; Peng, H. -C. Shape -Cont rolled Synthesis of Colloidal Metal Nanocrystals: \nThermodynamic versus Kinetic Products. J. Am. Chem. Soc. 2015 , 137 (25), 7947 –7966.  \n    32 For Table of Contents Only  \n \n \n"    },    {        "title": "1202.6194v2.An_original_unified_approach_for_the_description_of_phase_tranformations_in_steel_during_cooling___first_application_to_binary_Fe_C.pdf",        "content": "For Peer Review Only \n \n \n \n \n \nAn original unified approach for the description  \nof phase transformations in steel during cooling  \n \n \nJournal:  Philosophical Magazine & Philosophical Magazine Let ters  \nManuscript ID:  Draft \nJournal Selection:  Philosophical Magazine Letters \nDate Submitted by the Author:  n/a \nComplete List of Authors:  Bouaziz, Olivier; Arcelor Research, ;   \nKeywords:  phase transformations, phase stability, phase trans itions, solid-state \ntransformation, steel \nKeywords (user supplied):  phase transformation, steel, order parameter \n  \n \n \nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine LettersFor Peer Review OnlyAn original unified approach for the description \nof phase transformations in steel during cooling  \n \n \nO. Bouaziz *1,2  \n1ArcelorMittal Research, Voie Romaine-BP30320, 57283  Maizières-lès-Metz Cedex, \nFrance \n2Centre des Matériaux, Ecole des Mines de Paris, CNR S UMR 7633, B.P. 87, 91003 \nEvry Cedex, France \nEmail : olivier.bouaziz@arcelormittal.com \n \n \nKeywords : phase transformation, steel, austenite, ferrite, bainite, martensite, Landau, \norder parameter. \n \nAbstract \n \nExploiting Landau’s theory of phase transformations , defining an original order \nparameter and using the phenomenological transforma tion temperatures, it is reported \nthat it is possible to describe in a global approac h the conditions for the formation of \neach phase (ferrite, bainite, martensite) from aust enite during cooling in steel. It allowed  \nto propose a new rigorous classification of the dif ferent thermodynamic conditions \ncontrolling  each phase transformation. In a second  step, the approach predicts naturally \nthe effect of cooling rate on the bainite start tem perature. Finally perspectives are \nassessed to extend the approach in order to take in to account the effect of an external \nfield such as applied stress. \n \n \n1. Introduction \nIn the field of solid-state phase transformation in  metallic alloys [1], the transformation \nof austenite in steel on cooling can occur by a var iety of mechanisms including the \nformation of ferrite, bainite and martensite. The b ainitic transformation occurs in a \nrange between purely diffusional transformation to ferrite or pearlite and low \ntemperature transformation to martensite by a displ acive mechanism. Thus the bainite \ntransformation exhibits features of both diffusiona l and displacive transformations and \nhas given rise to a large amount of research activi ty (see [2-3] for reviews). A major \npart of the research has concerned modelling of the  kinetics of the transformations by \ndetailed descriptions of the thermodynamic conditio ns operating at the interface [4-7]. \nHowever the crucial understanding of the physically  based conditions of the start of \neach phase transformation is less understood, espec ially for bainite.  \nUsually the 3Ar  temperature is defined as the minimum temperature  for any phase \ntransformation of austenite to ferrite during cooli ng [8] : \n() Mo32Si45Ni15Mn21C230910CAro\n3 ++−−−=  (1) Page 1 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review OnlyFor temperature lower than 3Ar, the basical tools of physical metallurgy are the \ndefinition of the martensite start temperature sMand the bainite start temperature sB \nexpressed phenomenologically as functions of chemic al composition as :  \n() Si11Cr. 1 .12Ni7 .17Mn4 .30C423539CMo\ns −−−−−=  (2 ) \nwhere alloying element are expressed in weigth% [9] , and [10] : \n() Mo83Cr70Ni37Mn90C270870CBo\ns −−−−−=  (3) \n \nIn addition another characteristic temperature is d efined as the temperature where \naustenite and ferrite has the same thermo-chemical free energy determined as [2,11]: \n() Mo87Si15Cr73Ni36Mn91C .198835CTo\no −−−−−−=  (4) \n \nIn this publication, it is showed that it is possib le to describe in a global approach the \nconditions for each phase transformation exploiting  completely Landau’s theory [12-13] \nof phase transformations including an original para meter of order and to propose a new \nclassification of the different phase transformatio n in steel during cooling and to predict \nnaturally the effect of cooling rate on bainite sta rt temperature. \n \n2. The proposed approach \n \nIn the framework of Landau’s theory [12-13], for a second order phase transition, the \nfree energy is expressed as  : \n()()()4 2\no . C .TATFT ,F χ+χ+=χ  (5) \nwhere T is the temperature,  χ the order parameter, ()TFothe thermo-chemical free \nenergy, A(T) a function of temperature and C a posi tive constant. \n \nThe simplest expression for A(T) is : \n()()c oTTATA −=  (6) \nwith oA a positive constant and cT a critical temperature where A(T) changes of sign.  \n \nUsually in phase transformation of steels, ()TFo is known [2,3,7]. So the identification \nof the total free energy ()T ,Fχ requires the determination of three parameters :  \nC,T ,Aco . \n \nFor cTT>, ()T ,Fχexhibits one minimum as a function of χ : \n()0T ,F=χ∂χ∂ (7 ) \nfor 0=χ \n \nFor cTT≤, ()T ,Fχexhibits two minima as a function of χ : Page 2 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only()0T ,F=χ∂χ∂ (8) \nfor  \n()\nC . 2TTAco−±=χ  (9) \n \nIn order to exploit this approach to a classificati on of phase transformations in steel, it is \nnow assumed that : \ns cBT= (10) \nwhere sB  is the bainite start temperature. \n \nBy convention, it is chosen to have an order of par ameter for martensite : \n()1C . 2MBAs s o=−=χ  (11) \nGiving a first relationship : \ns so\nMB1\nC . 2A\n−=  (12 ) \n \nTherefore the order parameter is : \ns ss\nMBTB\n−−=χ  (13) \n \nIn order to illustrate quantitatively this law, the  evolution of the order parameter from \nBs to Ms temperatures for two different Fe-C compos itions has been plotted in Fig1. \n \nIt is now proposed to define clearly what could be the parameter of order for phase \ntransformation in steels. If C is the carbon conten t of the phase appearing, it is \nreasonable to propose that :  \neq,eq,\nCCCC\nαγα\n−−=χ  (14) \nWhere eq,Cα is the solubility of carbon in ferrite at equilibr ium and γC  is the carbon in \naustenite. So the order parameter can be a sursatur ation in carbon in the binary Fe-C \nsystem. \n \n \n \nFigure 1. : Evolution of the order parameter from B s to Ms temperatures for two \ndifferent Fe-C compositions.  \n \n \nIn addition chemical free energy at sM has been determined as [12]:  \n()()()s o s o s o MT.S MFT ,MF −+=  (15) Page 3 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Onlywith () mol/ J 1250 MFs o= and 1 1\no K . mol. J 8 . 6 S−−−= , wich are independent of the \ncomposition. \nThe energy value ()s oMF should correspond to the maximum at sM  for 0=χ. As for \nany sBT< this maximum exist for the same value of 0=χ, it is written for  \ns s BTM<≤  : \n()()()s o s o MF MTFT , 0F +−=  (16) \nwith \n()()s o s o MT.S MTF −=−  (17) \nwhen T is near sM but it can be completely different for higher temp erature. \nIn order to determine oAand C, as the total free energy at sM for 1=χ has to be zero, \nit comes : \n()() 0CBMA MFs s o s o =+−+  (18) \nor \n()s oMFC2C−=  (19) \nand \n()s oMFC=  (20 ) \n \nConsequently : \n().MBMF . 2A\ns ss o\no−=  (21 ) \n \nFinally it is possible to express completely the fr ee energy : \n( ) ( ) ( )()( ) ( )4\ns o2\ns\ns ss o\ns o s o .MF.BTMBMF . 2MF MTFT ,F χ+χ−−++−=χ\n (22) \nor \n( ) ( ) ( ) ( )()\n\n\n\nχ+−−χ+−−=χ2\ns ss 2\ns o s o s oMBBT. 2.MF MF MTFT ,F  (23) \n \nIn order to highlight the key role of the order par ameter, it has been drawn in Fig3. the \nevolution of the right-end side term of the express ion :  \n( ) ( ) ( ) ( )()\n\n\n\nχ+−−χ+=−−χ2\ns ss 2\ns o s o s oMBBT. 2.MF MF MTFT ,F  (24) \n \n \n \n \n \nFigure 2. : Evolution of the ()()s oMTFT ,F −−χ  as a function of the order parameter at \nsB  and at sM  temperatures. Page 4 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only \nFinally, the quantitative developed approach can be  used in order to summarized the \nthermodynamic conditions for the formation of each phase, as it is summarized in Table \n1, providing more rigorous occurrence criterion esp ecially to distinguish ferrite, bainite \nor martensite formation. \n \n \nTab.1 : Classification of phase transformation cond itions from austenite during cooling \n \n \n \n2. Extended approach including cooling rate \n \nIt is well known that the formation of the bainitic  phase depends on the cooling rate \napplied to austenite [2]. It means that the critica l temperature cT  defined in Eq.6 can be \nchosen equal to sB only for the minimum cooling rate min, rCfor bainite formation. A \nmaximum cooling rate max, rCis also commonly used indicating the end of bainite  \nformation and the start of martensite occurrence [2 ,16]. Therefore, in the case of a \ncontinuous cooling (i.e. constant cooling rate), it  is also possible to define the order \nparameter as a function of cooling rate as : \np\nmin, r max, rmin, r r\nC CCC\n\n\n\n\n−−=χ  (25)  \nwhere rC is the constant applied cooling rate and p is a pa rameter. \n \nIn order to have a complete consistency with the de finition of χdetermined in Eq13,  \np\nmin, r max, rmin, r r\ns ss\nC CCC\nMBTB\n\n\n\n\n−−=−− (26 ) \n \nThe temperature T respecting Eq.26 is the “generali zed” Bainite Start temperature \nincluding the effect of cooling rate expressed as :  \n( )p . 2\nmin, r max, rmin, r r\ns s s g , sC CCCMBB B\n\n\n\n−−−−=  (27) \n \n \n \nA value of the parameter p has been identied by com parison with experimental data of \nthe evolution of g , sB as a function of cooling rate which can be extract ed from \nContinuous Cooling Transformation diagram [17]. For  the first assessment of the \napproach, a chemical composition as simple as possi ble has been selected (Fe-0.4%C-\n0.7%Mn-0.2%Si). The experimental evolution of g , sB is reported in Fig.3 and \ncompared with the model (Eq.27). A very good agreem ent is found for p equal to 41 . It \nmeans that a law for g , sB can be expressed as : Page 5 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only( )21\nmin, r max, rmin, r r\ns s s g , sC CCCMBB B\n\n\n\n−−−−=  (28) \n \n \n \nFigure 3. : Comparison between experimental and pre dicted evolution of the Bainite \nStart temperature as a function of cooling rate for  Fe-0.4%C-0.7%Mn-0.2%Si \ncomposition.  \n \n \n \n4. Conclusions \n \nExploiting Landau’s theory of phase transformations , defining an original order \nparameter and using the phenomenological transforma tion temperatures, it is reported \nthat it is possible to describe in a global approac h the conditions for the formation of \neach phase (ferrite, bainite, martensite) from aust enite during cooling in steel. It allowed  \nto propose a new more rigorous classification of th e different thermodynamic conditions \ncontrolling  each phase transformation. In a second  step, the approach predicts naturally \nthe effect of cooling rate on the bainite start tem perature.  \nIn perspectives, the approach can be extended to ta ke into account external fields by \nadding a linear term in free energy linearly propor tional to the order of parameter and \nproportional to the potential energy of the externa l field [12-13]. For instance, in the \ncase of an uniaxial applied stress σ, the contribution to the free energy is expressed as : \n( ) χσ±=σχ .E.2),F2\n (2 9) \nwhere E is the elastic modulus and the sign depends  on the compressive or tensile \nstress. \nThis term breaks the symmetry of the total free ene rgy as a function of the parameter of \norder and it can change the occurrence conditions o f each phase. A lot of experimental \ndata are available is the literature in order to va lidate this last point which will be \ninvestigated in a next future. \n \n \nAcknowmedgement : The author thanks Dr. F. Levy for  stimulating discussions. \n \n \nReferences \n \n[1] M. Hillert, Phase Equilibria, Phase diagrams an d Phase tranformations, second ed., \nCambridge, University Press, Cambridge, UK, 2008.  \n[2] H.K.D.H. Bhadeshia, Bainite in Steels, second e d., IOM Communication Ltd., \nLondon, 2001. Page 6 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only[3] M. Hillert, Paraequilibrium and other restricte d equilibria, L.H. Bennet T.B. \nMassalski, B.C. Giessen editors, Alloy phase diagra ms, Materials Research Society \nSymposium Proceedings vol.19 (1983) p.295. \n[4] J.S. Kirkaldy, Can. J. Phys. 36 (1958) p.907. \n[5] A. Hultgren, Trans. ASM 39 (1947) p.915. \n[6] M. Takahashi, Solid State Mat. Sci. 8 (2004) p. 213. \n[7] H.S. Zurob, C.R. Hutchinson, , Y. Bréchet, H. S eyedrezai, G. Purdy, Acta Mat. 57 \n(2009) p.2781. \n[8] F.B. Pickering, Steels: Metallurgical Principle s, in Encyclopedia of Materials \nScience and Engineering, vol. 6, The MIT Press, Cam bridge, 1986. \n[9]  K.W. Andrews, J. of Iron and Steel Institute 2 03 (1965) p.721. \n[10] W. Steven, A.G. Haynes,  JISI 183 (1956) p.349 . \n[11] S.M.C. Van Bohemen, Metall. Trans. A 41 (2010)  p.285. \n[12] L.D. Landau, E.M. Lipschitz, Statistical Physi cs, Pergamon, London, 1959. \n[13] R.A. Cowley, Advanced in Phys. 29, (1980) p.1.  \n[14] J. Wang, S. Van der Zwaag, Metall. Trans. A 32  (2001) p.1527. \n[16] N. Yurioka, Met. Const. 19 (1987) p.217 \n[17] Atlas of Time-Temperature Diagrams for Irons a nd Steels, G. Vander Voort Ed., \nASM International Publishers, 1991. \n Page 7 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review OnlyTables with captions \n \nPhase Free energy Order parameter \nFerrite ()0T ,F=χ∂χ∂ 0=χ \nBainite   \n()0dB,Fs=χχ∂ \n()0B,F\n2s2\n=χ∂χ∂ \n sBT<,\ns ss\nMBTB\n−−=χ  \nMartensite ()0 M,Fs=χ  \n()0M,Fs=χ∂χ∂ 1=χ \nTab.1 : Classification of phase transformation cond itions from austenite during cooling \n \n Page 8 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review OnlyFigures captions \n \nFigure 1. : Evolution of the order parameter from B s to Ms temperatures for two \ndifferent Fe-C compositions.  \n \nFigure 2. : Evolution of the ()()s oMTFT ,F −−χ  as a function of the order parameter at \nsB  and at sM  temperatures. \n \nFigure 3. : Comparison between experimental and pre dicted evolution of the Bainite \nStart temperature as a function of cooling rate for  Fe-0.4%C-0.7%Mn-0.2%Si \ncomposition.  \n Page 9 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only  \n \n \n \n256x159mm (150 x 150 DPI)  \n \n \nPage 10 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only  \n \n \n \n256x159mm (150 x 150 DPI)  \n \n \nPage 11 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60For Peer Review Only  \n \n \n \n256x159mm (150 x 150 DPI)  \n \n \nPage 12 of 12\nhttp://mc.manuscriptcentral.com/pm-pmlPhilosophical Magazine & Philosophical Magazine Letters \n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n36\n37\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n54\n55\n56\n57\n58\n59\n60"    },    {        "title": "2102.08082v2.Layer_and_spontaneous_polarizations_in_perovskite_oxides_and_their_interplay_in_multiferroic_bismuth_ferrite.pdf",        "content": "Layer and spontaneous polarizations in perovskite oxides and their interplay in\nmultiferroic bismuth ferrite\nNicola A. Spaldin,1Ipek Efe,1Marta D. Rossell,2and Chiara Gattinoni1\n1Department of Materials, ETH Zurich, CH-8093 Z urich, Switzerland\n2Electron Microscopy Center, Swiss Federal Laboratories for Materials\nScience and Technology, Empa, 8600, D ubendorf, Switzerland\n(Dated: February 26, 2021)\nWe review the concept of surface charge, \frst in the context of the polarization in ferroelectric\nmaterials, and second in the context of layers of charged ions in ionic insulators. While the former\nis traditionally discussed in the ferroelectrics community, and the latter in the surface science\ncommunity, we remind the reader that the two descriptions are conveniently uni\fed within the\nmodern theory of polarization. In both cases, the surface charge leads to electrostatic instability\n| the so-called \\polar catastrophe\" | if it is not compensated, and we review the range of\nphenomena that arise as a result of di\u000berent compensation mechanisms. We illustrate these\nconcepts using the example of the prototypical multiferroic bismuth ferrite, BiFeO 3, which is\nunusual in that its spontaneous ferroelectric polarization and its layer charges can be of the same\nmagnitude. As a result, for certain combinations of polarization orientation and surface termination\nits surface charge is self-compensating. We use density functional calculations of BiFeO 3slabs and\nsuperlattices, analysis of high-resolution transmission electron micrographs as well as examples\nfrom the literature to explore the consequences of this peculiarity.\nI. INTRODUCTION\nBound charge at the surface of an insulator, or at\nan interface between two insulating materials, must be\nscreened in order to avoid a so-called polar catastrophe\ncaused by a divergence of the electrostatic energy [1].\nSuch bound surface charge, \u001bsurf, exists whenever there\nis an uncompensated component of the bulk polarization,\n~Pbulk, perpendicular to a surface, and is given by\n\u001bsurf=~Pbulk\u0001~ n (1)\nwhere~ nis the unit vector along the surface normal. A\nbulk polarization can occur of course in ferroelectric ma-\nterials due to their spontaneous polarization . It can also\noccur in centrosymmetric crystals in which the ions form\ncharged layers; we refer to this latter contribution as the\nlayer polarization .\nMany mechanisms are known for screening bound sur-\nface charge, ranging from metal electrodes [2, 3], to for-\nmation of charged defects [4{8] or adsorption of charged\nspecies [9{12], and even loss or reorientation of ferro-\nelectric polarization [13{15]. Note that it is not possible\nto screen the bound surface charge by surface relaxation\nalone, without the addition or removal of charged species.\nIn this work we explore the special case of multifer-\nroic perovskite-structure bismuth ferrite, BiFeO 3, which\nhas both a spontaneous polarization from its ferroelec-\ntric distortion and a layer polarization from its ionic\ncharges. BiFeO 3is particularly unusual because the size\nof its spontaneous ferroelectric polarization in the com-\nmon [001] and [111] growth directions is close to the size\nof the layer polarization from the charged ionic layers\nin \rat (100) and (111) planes. As a result, for cer-\ntain choices of polarization orientation and surface ter-mination, the spontaneous and layer polarizations self-\ncompensate, leading to uncharged surfaces that are sta-\nble without external screening mechanisms.\nWe begin by reviewing in the next section (Sec. II) two\nkey results of the modern theory of polarization | the\nmultivaluedness of the polarization lattice and the con-\ncept of the polarization quantum | that are key to this\nwork. We then brie\ry review compensating mechanisms\nat the surfaces of ferroelectric materials with charge-\nneutral layers (Sec. III). Next, we discuss centrosymmet-\nric materials with charged layers (Sec. IV), and show how\nthe layer polarization associated with charged ionic lay-\ners is conveniently described within the modern theory of\npolarization. We then combine the concepts of layer po-\nlarization and spontaneous polarization in the example\nof bismuth ferrite (Sec. V), discussing in turn its inter-\nface with metallic electrodes, and with insulators with\ndi\u000berent layer polarizations.\nII. REMINDER OF KEY RESULTS FROM THE\nMODERN THEORY OF POLARIZATION\nWe begin with a reminder of a fundamental result of\nthe modern theory of polarization [16], that the polariza-\ntion,~P, of a bulk periodic solid, is not a single number\nbut rather a lattice of values, separated by the polar-\nization quantum, ~Pq=e~R\nV. The polarization quantum\ncorresponds to the change in polarization on moving an\nelectronic charge eby a lattice vector ~R(Vis the unit cell\nvolume), which changes the polarization by an amount\n~Pq, but does not change the physical system. (For a more\nextensive introductory discussion, see Ref. 17.)\nA consequence of this property is that the polarizationarXiv:2102.08082v2  [cond-mat.mtrl-sci]  25 Feb 20212\nlattice of a centrosymmetric crystal, which must also be\ncentrosymmetric by symmetry, can take one of two sets\nof values,\n~P= 0 +n~Pq;or (2)\n~P=~Pq\n2+n~Pq; (3)\nwherenis any integer. All insulating, centrosymmet-\nric, periodic solids can be classi\fed as belonging to one\nof these classes, which we will refer to as having \\zero-\ncontaining\" (Eq. 2) or \\half-quantum-containing\" (Eq. 3)\npolarization lattices. (In principle, it is possible for a\ncentrosymmetric material to contain zero for one com-\nponent of its polarization and a half quantum along an-\nother component, although we do not know of an exam-\nple). Examples of half-quantum-containing centrosym-\nmetric crystals are the high-symmetry paraelectric phase\nof BiFeO 3, and the wide-band-gap insulating perovskite,\nLaAlO 3. Note also that these materials have charged\nionic layers perpendicular to their usual [001] growth di-\nrection; in section IV we will show that these proper-\nties are formally connected. SrTiO 3is an example of the\nzero-containing polarization lattice type; correspondingly\ncharge-neutral layers, such as SrO and TiO 2(001) planes,\ncan be readily identi\fed.\nWhile the fact that a centrosymmetric crystal can have\na polarization lattice that does not contain zero is some-\nwhat unintuitive, it is reconciled by a second impor-\ntant result of the modern theory of polarization: Dif-\nferences in polarization, de\fned as the change in polar-\nization along a given branch of the polarization lattice\nas the system is modi\fed along an insulating pathway,\nare single-valued. As a result, the spontaneous polar-\nization, which is the di\u000berence in polarization between\nthe ferroelectric structure and its high-symmetry cen-\ntrosymmetric counterpart, is single-valued. Since only\npolarization di\u000berences (for example when a ferroelectric\nis switched between domains, or heated above its Curie\ntemperature) are experimentally accessible, the theory is\nconsistent with experimental reality. Indeed, when dis-\ncussing the properties of in\fnite, bulk periodic crystals,\nthe question of whether the polarization lattice contains\nzero or a half quantum is not generally relevant.\nAt surfaces and interfaces, however, the question of\nthe origin of the polarization lattice should not be disre-\ngarded. The bulk polarization, ~Pbulk, which gives rise to\nthe bound charge at the surface of a crystal (Eqn. 1) con-\ntains the contributions from both the half-polarization\nquantum (layer charges) andthe spontaneous polariza-\ntion [18]. Considering only the spontaneous polarization\nin evaluating the bound charge on the surface of a ma-\nterial will yield an incorrect result for centrosymmetric\ncrystals with half-quantum-containing polarization lat-\ntices, as well as for ferroelectrics whose paraelectric ref-\nerence structures' polarization lattices contain the half\nquantum.\nTo illustrate these concepts in this work, we use ABO 3\nperovskite-structure oxides in the usual pseudo-cubic[001] growth orientation. (For the generalization to other\nsurface planes see Ref. 1). In this orientation, the layers\nhave alternating AO / BO 2chemistry, and the planar\nsurfaces are formed from either entirely AO or entirely\nBO2layers. Since oxygen has formal charge -2, the (001)\nlayers in so-called II-IV perovksite oxides (in which the\nA-site cation is divalent and the B-site cation has for-\nmal charge +4) are charge neutral. In III-III perovskite\noxides (in which both A- and B-site cations have formal\ncharge +3), the AO layer has charge +1 and the BO 2\nlayer has charge -1.\nAs example II-IV ferroelectric materials with charge-\nneutral layers we choose PbTiO 3and BaTiO 3; the polar\ndiscontinuity at their surfaces (that is their interface with\nthe vacuum), and hence their bound surface charge, de-\nrives entirely from their spontaneous polarization. To\nillustrate the two possible behaviors of centrosymmetric\nionic insulators, we choose SrTiO 3, with its neutral lay-\ners and zero-containing polarization lattice, and LaAlO 3,\nwhose layers are charged, and whose polarization lattice\ncontains the half quantum. The polar discontinuity be-\ntween these two materials at their (001) interface has\nbeen of particular recent interest [19{21]. The main part\nof this paper combines the concepts developed for both\nof these pairs of example materials to treat the case of\nBiFeO 3, with its combined spontaneous and layer polar-\nizations.\nIII. SURFACE EFFECTS IN FERROELECTRIC\nMATERIALS WITH ZERO LAYER\nPOLARIZATION\nWe begin with a brief review of relevant results for the\nsurface properties of the prototypical II-IV perovskite fer-\nroelectrics, PbTiO 3and BaTiO 3. In both cases, when the\nions have the centrosymmetric arrangement of the high-\nsymmetry reference phase, then the polarization lattice\ncontains zero, and is given by Eqn. 2. Correspondingly,\nthe (001) layers have zero formal charges (Fig. 1). As\na result, the only contribution to the ~Pbulkof Eqn. 1,\nand hence to the surface charge, is from the spontaneous\npolarization. In PbTiO 3the spontaneous polarization\nin the ferroelectric phase is oriented along a Cartesian\naxis ([001] in Fig. 1) and has the value \u001880\u0016C cm\u00002;\nin BaTiO 3there are a series of phase transitions with\nthe polarization (of magnitude \u001825\u0016C cm\u00002) reorient-\ning from the cubic [111] to [011] to [001] directions as\nthe temperature is lowered; we show the low-temperature\n[001] case in Fig. 1b. The bound surface charge on a (001)\nsurface (Fig. 2) is then given trivially by the component\nof the spontaneous polarization perpendicular to the sur-\nface (80 or 25 \u0016C cm\u00002for PbTiO 3and BaTiO 3respec-\ntively). It is negative (positive) on the upper surface for\ndownward (upward) pointing polarization, and does not\ndepend on the choice of layer (AO or BO 2) termination.\nIn all cases, the uncompensated (001) surface is electro-\nstatically unstable when the ferroelectric polarization is3\na) Paraelectricb) Ferroelectric\nPbTiO3\nBaTiO3\nPbTiO3\nBaTiO3\nPb+2O-2Pb+2O-2Pb+2O-2Ti+4O-22Ti+4O-22\nBa+2O-2Ba+2O-2Ba+2O-2Ti+4O-22Ti+4O-220000000000PPTO=80!C/cm2\nPBTO=25!C/cm2\nFIG. 1. Crystal structure and (001) layer charges of PbTiO 3\n(top) and BaTiO 3(bottom). a) Arrangement of ions in the\nhigh-symmetry centrosymmetric reference structures. On the\nleft of the crystal structure is the total charge for each layer,\non the right the composition of each layer. b) Arrangement\nof ions in the ferroelectric structure. The arrows indicate the\ndirection of the ferroelectric polarization. Pb is in black, Ti\nin blue, Ba in green and O in red.\nalong the [001] axis.\nPspontµC/cm2+  +  +  +  ++  +  +  +  ++  +  +  +  ++  +  +  +  +_  _  _  _  __  _  _  _  __  _  _  _  __  _  _  _  _-PspontµC/cm2[001]+PspontµC/cm2\nFIG. 2. In ferroelectric materials with uncharged layers, such\nas BaTiO 3and PbTiO 3, the sign and magnitude of the sur-\nface charge are determined by the spontaneous polarization,\n~Pspont.\nThe electrostatic instability associated with the spon-\ntaneous polarization in ferroelectrics is often discussed in\nterms of a depolarizing \feld, which acts in the opposite\ndirection to the polarization to suppress the ferroelec-\ntricity in thin \flms. For a \flm with no compensation\nfrom external species, four main responses are known,\nas sketched in Fig. 3. Polarization reorientation into\nthe plane of the \flm (Fig. 3a) completely eliminates thebound surface charge, and so is energetically favorable if\nit is not prohibited by, for example, strain e\u000bects [22]. If\nin-plane rotation of the polarization is unfavorable, the\nformation of domains can occur. The example of small\ndomains of opposite orientation [23{25] shown in Fig. 3b\nreduces the overall charge on each surface; other more\nexotic textures such as polar skyrmions have also been\nreported [26{29]. Screening surface charges can in prin-\nciple be generated by electron-hole excitation across the\nband gap [1] (Fig. 3c), although, since band gaps are typ-\nically of the order of an eV in ferroelectrics this is ener-\ngetically expensive. Finally, complete suppression of the\npolarization (Fig. 3d) can occur, usually manifesting as\na critical thickness of the paraelectric reference structure\nbefore the ferroelectric phase emerges [23, 30, 31].\nScreening can also occur from extrinsic factors. When\nmetallic electrodes are present, then carriers from the\nmetal can provide compensating external surface charge\nto screen the polarization discontinuity. Unless the\nscreening is completely e\u000bective, however, the magni-\ntude of the polarization and the ferroelectric Curie tem-\nperature,Tc, tend to be reduced from their bulk val-\nues [3, 13, 32{34] and there still tends to be a criti-\ncal thickness below which the paraelectric phase is sta-\nble [13, 35]. In the absence of electrodes (or in combi-\nnation with a bottom electrode), compensating charge\ncan be provided by ions from the environment [8]; this\nis the physics behind the well-known pyroelectric e\u000bect,\nin which the reduction in polarization on heating re-\nleases charged species from the surface [36]. Recently,\nthe reciprocal e\u000bect has been demonstrated, in which\nadsorption of adsorbates carrying a speci\fc charge was\nshown to switch the ferroelectric polarization to achieve\nan electrostatically stable surface con\fguration [4, 8, 11].\nFinally, we mention that the presence of charged ions\nin the growth chamber atmosphere has been exploited\nto enable growth of single-domain ultra-thin ferroelec-\ntric \flms of PbTiO 3on SrRuO 3through metalorganic\nchemical vapor deposition of PbTiO 3on SrRuO 3[2]. A\ncharged atmosphere has even been shown to be more ef-\nfective in screening the polarization than a top electrode\nfor BaTiO 3\flms grown on SrRuO 3using pulsed laser\ndeposition [10].\nIV. SURFACE EFFECTS ARISING FROM THE\nLAYER-CHARGE POLARIZATION IN\nNON-FERROELECTRIC MATERIALS\nA pair of well-known centrosymmetric materials\nthat illustrate the two cases of \\zero-containing\" or\n\\half-quantum-containing\" polarization lattices are the\nperovskite-structure oxides strontium titanate, SrTiO 3,\nand lanthanum aluminate, LaAlO 3, shown in Fig. 4.\n(Note that rotations of the oxygen octahedra, which\nlower the symmetry from the ideal cubic perovskite struc-\nture, occur in both materials; since these rotations pre-\nserve the center of inversion they do not change the polar-4\nPa)b)c)d)Ph+e-P=0\nFIG. 3. E\u000bects of the depolarizing \feld on a ferroelectric thin\n\flm. a) In-plane polarization; b) Formation of domains; c)\nAccumulation of interfacial charge, for example by electron-\nhole excitation across the gap; d) Phase transition to a non-\npolar phase.\nization behavior and we do not consider them here.) We\ndiscuss next how the di\u000berent polarization lattice types\ncorrespond to their di\u000berent layer charges and result in\ndi\u000berent surface charges, focussing on the (001) surface\nfor conciseness. For a more comprehensive discussion we\ndirect the reader to Ref. 1.\n(a) SrTiO3(b) LaAlO3Sr+2O-2Sr+2O-2Sr+2O-2Ti+4O-22Ti+4O-22La+3O-2La+3O-2La+3O-2Al+3O-22Al+3O-2200000+1+1+1-1-1[001]\nFIG. 4. Crystal structure and (100) layer charges of a) SrTiO 3\nand b) LaAlO 3. On the left of the crystal structure is the\ntotal charge for each layer, on the right the composition of\neach layer.\nSince SrTiO 3is a II-IV perovskite, and LaAlO 3is a\nIII-III perovskite, it is trivial to show, by calculating the\npolarization as ~P=1\nV\u0006iZi~ ri, that SrTiO 3has the zero-\ncontaining polarization lattice of Eqn. 2, and LaAlO 3the\nhalf-quantum-containing form of Eqn. 3. Here, Ziare the\nformal ionic charges and ~ ritheir positions within any\nchoice of unit cell. (Note that rigorous calculation using\nthe Berry phase formalism gives an identical result, and\nthe use of the formal charges when calculating the lat-\ntice polarization of the paraelectric structure is formally\ncorrect [1].) As a result, the (001) surface of SrTiO 3has\na charge of zero or n~Pq, whereas that of LaAlO 3has a\n(b) AlO2surface+1-1+1-1+1-1\n-1EV½ h+EV-1+1-1+1-1+1-1(a)[001]!\"#$%=−()*=−+*,-./\nLa+3Al+3\nAl+3La+3\n+1+1-1+1-1+1-1V½ e-EV+1+1-1+1-1+1-1(c) LaO surface\nE!\"#$%=+()*=++*,-./FIG. 5. a) Unit cell of LaAlO 3with AlO 2(left) and LaO\n(right) termination. The dotted-line box indicates the basis\nused to calculate ~Pbulk, and the surface termination is the\ntopmost layer in the box. ~Pbulkin the [001] direction is of\nopposite sign for the two systems. b-c) Electric \feld E and\npotential V along the [001] direction for LaAlO 3with b) AlO 2\nand c) LaO termination. The system on the left in each panel\nhas no compensation and so V diverges with thickness. The\nsystem on the right has a surface compensation of 1 =2 of the\nlayer charge and therefore is stable.\ncharge ofn~Pq\n2.\nA convenient recipe was provided in Ref. 1 to deter-\nmine which branch of the polarization lattice (that is,\nwhich value of n) is the relevant one for a particular\nchoice of surface plane and chemistry: Take the unit\ncell that tiles the semi-in\fnite slab containing the sur-\nface of interest, and calculate the dipole per unit volume\nfor that unit cell. The answer is ~Pbulkwith the appropri-\nate choice of polarization lattice branch. For the (001)\nsurface of SrTiO 3, both smooth surfaces (containing SrO\nor TiO 2) yield~Pbulk= 0 with this recipe. Therefore they\nhave no bound charge and do not require any external\ncharge compensation. For the (001) surface of LaAlO 3\n(Fig. 5a), the LaO surface has [001] polarization value\n~Pbulk= +1\n2jej\na2\n0= +~Pq\n2, requiring compensation by a neg-\native of charge of this size and the AlO 2surface has\n~Pbulk=\u00001\n2jej\na2\n0=\u0000~Pq\n2, requiring compensation by the\ncorresponding positive charge. ( a0is the length of the\npseudo-cubic unit cell, and the polarization quantum is\n~Pq=jej\na2\n0.) The required compensation of half an elec-\ntronic charge per simple cubic unit cell corresponds to\nthe convenient value of 50 \u0016C cm\u00002in the conventional\nunits used in the ferroelectrics literature, taking a cubic\nlattice constant of 4 \u0017A which is slightly larger than the\nvalues for SrTiO 3(\u00183.9\u0017A ) and LaAlO 3(\u00183.8\u0017A ).5\nAn alternative picture that is intuitively appealing, al-\nthough not as rigorously well-founded, is to decompose\nthe materials into planes of ions and consider the net\ncharges of these planes, as shown in Fig. 4. For the case\nof II-IV perovskites such as SrTiO 3, the (001) planes are\nalternately SrO and TiO 2, both of which are charge neu-\ntral. Therefore any planar (001) surface in a II-IV per-\novskite carries no net surface charge and so is stable.\nIn III-III perovskites such as LaAlO 3, the (001) planes\nare alternately LaO and AlO 2with charges +1 eand\n\u00001eper unit cell respectively. And so, depending on the\nchoice of termination, the surface has the corresponding\npositive or negative charge per surface unit cell, that is\n\u0006100\u0016C cm\u00002. As illustrated in Fig. 5b, the positive\nlayer charge of the LaO surface requires a compensating\nnegative charge of halfthe layer charge, that is \u00000:5e\nper unit cell or\u000050\u0016C cm\u00002, to prevent a divergence\nof the electrostatic potential and stabilize the surface.\n(A compensating charge equal to the surface charge just\ndisplaces the problem to a new terminating layer [37].)\nLikewise, the formally negatively charged AlO 2surface\n(Fig. 5c) requires a compensating positive charge of the\nsame amount. Thus we reach the same conclusion as that\nderived from consideration of the bulk polarization.\nThe implication of the di\u000berent bulk polarization lat-\ntices of LaAlO 3and SrTiO 3for the interface between the\ntwo materials is profound: The polarization discontinuity\nbetween the two materials means that it is not possible\nto make a stoichiometric interface that is electrostatically\nstable [38]. Speci\fcally, an SrO / AlO 2interface requires\na compensating positive charge of magnitude half an elec-\ntronic charge per unit cell, and the LaO / TiO 2interface\nrequires half an electronic charge per unit cell of negative\ncharge. In the latter case, the extra electrons occupy the\nbroad Ti 3d-derived energy bands at the bottom of the\nvalence band. The compensating electrons are therefore\nmobile and form an interfacial two-dimensional electron\ngas [19], a remarkable behavior for the interface of two\nrobust band insulators. The electron gas has even been\nshown to be superconducting at low temperature [21].\nNote that these considerations are not limited to III-III\nperovskites, but are relevant for the surfaces and inter-\nfaces of all centrosymmetric insulators that have a \\half-\nquantum containing\" polarization lattice. Another ex-\nample is provided by the I-V perovskites, such as KTaO 3,\nwhere surface reconstructions [9] and surface and inter-\nface 2D electron gases [39, 40] have been observed.\nV. SURFACES OF FERROELECTRIC\nMATERIALS WITH CHARGED LAYERS { THE\nINTERPLAY OF LAYER CHARGE AND\nSPONTANEOUS POLARIZATION IN BISMUTH\nFERRITE\nNext we turn to the case of ferroelectric materi-\nals whose polarization lattice in their high-symmetry\ncentrosymmetric prototype structure contains the half-polarization quantum. We choose the example of the\nIII-III ferroelectric perovskite BiFeO 3, which, as men-\ntioned in the introduction, combines a half-quantum-\ncontaining polarization lattice in its centrosymmetric ref-\nerence structure, with a spontaneous polarization of al-\nmost exactly 50 \u0016C cm\u00002in the [001] direction. In par-\nticular, we will explore the consequences of the accidental\nlayer- and spontaneous polarization-surface charge com-\npensation on the stability of thin \flms and heterostruc-\ntures of BiFeO 3.\nThe ground state of bulk BiFeO 3has theR3cstruc-\nture, which is reached from the prototypical cubic per-\novskite structure by alternating rotations of the oxygen\noctahedra around the [111] axis, combined with oppo-\nsite displacements of anions and cations along the [111]\ndirection. The latter results in a large spontaneous po-\nlarization of magnitude \u001890\u0016C cm\u00002oriented along\n[111]. In Fig. 6 we show the evolution of the polarization\n(calculated using the Berry phase approach in Ref. 41) as\na function of the amplitude of the ferroelectric distortion\nfrom the high-symmetry reference structure (0% distor-\ntion) to the ground-state ferroelectric structure (100%\ndistortion), for several branches of the polarization lat-\ntice. The spontaneous ferroelectric polarization along\n[111] is highlighted in red. Interestingly, and completely\ncoincidentally, the value of the spontaneous polarization\nis very close to half of the polarization quantum of \u0018180\n\u0016C cm\u00002along the [111] direction (highlighted in blue\nin Fig. 6) for BiFeO 3. Since the polarization lattice for\nthe centrosymmetric reference structure is of the half-\nquantum type, we see that there are two combinations\nof the centrosymmetric layer polarization and the spon-\ntaneous polarization (+~Pq\n2withPspont =\u000090\u0016C cm\u00002,\nand\u0000~Pq\n2withPspont = +90\u0016C cm\u00002) that combine to\ngive a bulk polarization value, ~Pbulk, in the ferroelectric\nstructure that is very close to zero (in fact \u00062:3\u0016C cm\u00002\nin Fig. 6).\nThis in turn leads to a cancellation of the bound sur-\nface charge, \u001bsurf\u00190. A consequence of this cancellation,\ntherefore, is that free-standing thin \flms of BiFeO 3are\nelectrostatically stable for one choice of polarization for\neach surface; this has been referred to as the \\happy\"\ncon\fguration in the literature [42].\nIn Fig. 7 we illustrate this with a cartoon of a free-\nstanding BiFeO 3slab in the commonly grown [001] orien-\ntation. The projection of the [111]-oriented ferroelectric\npolarization into the [001] direction results in a sponta-\nneous [001] polarization of \u0018\u000650\u0016C cm\u00002. As we saw\nin the case of LaAlO 3in Section IV, the unit cell corre-\nsponding to the BO 2(FeO 2in this case) surface selects\nfor the branch on the centrosymmetric polarization lat-\ntice with value\u00001\n2e\na2\n0=\u000050\u0016C cm\u00002. Therefore an\nFeO 2surface with a positive (i.e. pointing towards it,\nor upwards in Fig. 7a) value of spontaneous polarization\nhas zero surface charge and is stable; conversely the AO\n(BiO in this case) surface selects for the +1\n2e\na2\n0= +50\u0016C\ncm\u00002half quantum, and requires a negative (i.e. point-6\nPolarization quantumFerroelectric polarization\nFIG. 6. Polarization in the [111] direction for BiFeO 3, cal-\nculated using the LSDA+ Uand Berry phase methods within\ndensity functional theory in Ref. 41. \u0006100% distortion cor-\nresponds to the ground-state R3cstructures of opposite po-\nlarization; 0% distortion corresponds to the ideal cubic per-\novskite structure. Three full branches of the polarization lat-\ntice are shown. The central branch illustrates that, starting\nfrom a centrosymmetric polarization of 92.8 \u0016C cm\u00002(=~Pq\n2)\nand introducing a negative spontaneous polarization yields a\nbulk polarization close to zero (in fact -2.3 \u0016C cm\u00002) and\na correspondingly small surface charge. Introducing a posi-\ntive spontaneous polarization in this branch results in a very\nlarge bulk polarization (187.8 \u0016C cm\u00002) and an unfavorably\nlarge surface charge. The lower branch illustrates the oppo-\nsite scenario. Reproduced from Ref. 41. Copyright 2005 by\nthe American Physical Society.\ning away from it) polarization to ensure stability. Note\nthat the opposite combinations are twice as unfavorable\n(\\unhappy\") as they would be in a II-IV perovskite with\nthe same magnitude of spontaneous polarization but un-\ncharged layers (Fig. 7b), since they would have a surface\ncharge of\u0006100\u0016C cm\u00002. In the alternative charged-\nlayers picture, the centrosymmetric BiFeO 3is composed\nof alternating (001) layers of positively charged BiO (+1 e\nper unit cell or\u0018+100\u0016C cm\u00002) and negatively charged\nFeO 2(-1eper unit cell or\u0018\u0000100\u0016C cm\u00002). The appro-\npriately oriented spontaneous polarization of magnitude\n50\u0016C cm\u00002then provides the required compensating\nsurface charge of half that amount.\nNext, we discuss the consequences of this layer and\nspontaneous polarization cancellation, examining exam-\nples from the literature as well as presenting new results\nof behaviors that are caused by the happiness or unhap-\npiness of BiFeO 3surfaces and interfaces. We consider\nthree scenarios: \frst, BiFeO 3on a metallic substrate,\nfollowed by interfaces with centrosymmetric II-IV then\nIII-III insulators.\n!= 100 µC/cm2!= -100 µC/cm2\n!= 0 µC/cm2FeO2-BiO+-+PBiO+FeO2-+-Pa)!= 0 µC/cm2b)\nFIG. 7. Combinations of ferroelectric polarization direction\nand surface termination leading to a) stable (happy) and b)\nunstable (unhappy) BiFeO 3(001) surfaces. In a), the spon-\ntaneous polarization, Pspont = 50\u0016C cm\u00002is compensated\nby the layer polarization. In b) the layer polarization adds to\nthe spontaneous polarization Pspont =\u000050\u0016C cm\u00002to give\na surface charge of \u0006100\u0016C cm\u00002.\nA. Interaction of BiFeO 3thin \flms with a metal\nsubstrate\nWe begin with the case of BiFeO 3\flms grown on\nsubstrates that are metallic, and therefore provide good\nscreening of any interfacial charge at the bottom inter-\nface. We expect, therefore, that the orientation of the\nspontaneous polarization will be determined by the na-\nture of the top surface with the vacuum.\nWe take the examples of BiFeO 3on two metallic\noxides, La 0:7Sr0:3MnO 3(LSMO) and SrRuO 3. Het-\nerostructures of these combinations were grown and char-\nacterized in Ref. 43, and we begin by analyzing the results\nof that work in the context of the surface electrostatics\nintroduced above.\nIn Ref. 43, the SrRuO 3substrate was terminated with\nan SrO layer, and so the bismuth ferrite \flm, which grows\nin complete BiFeO 3unit cells, began with an FeO 2layer\nand ended with a BiO surface. As expected, the polar-\nization spontaneously adopted the down orientation, cor-\nresponding to zero surface charge. An FeO 2surface was\nachieved for BiFeO 3on SrRuO 3by inserting a monolayer\nof TiO 2at the interface so that the BiFeO 3\flm began\nwith a BiO layer. This caused a spontaneous upwards\npolarization in the BiFeO 3, again corresponding to the\nzero surface-charge con\fguration as expected.\nGrowth of BiFeO 3on LSMO shows a similar behavior.\nIn Fig. 8 we show two high-angle annular dark-\feld scan-\nning transmission electron microscopy (HAADF-STEM)\nimages of the BiFeO 3on LSMO heterostructures grown\nin Ref. 43. (For additional details about the thin \flm\ngrowth see section VII and Ref. 43.) In panel a the\nLSMO is terminated with MnO 2, so the BiFeO 3layer has\nan FeO 2surface, while in panel b the LSMO is (La,Sr)O7\nterminated, so the bismuth ferrite starts with a FeO 2\nlayer and has a BiO surface. We have overlaid arrows,\nwhich are vector maps indicating the local polarization\nextracted from the measured atomic positions, in the\nBiFeO 3layers. Again, as expected from the surface elec-\ntrostatics, we see that case (a) develops a spontaneous\nup-pointing polarization ( ~Ppointing towards the FeO 2\nsurface) and case (b) a spontaneous down-pointing ( ~P\npointing away from the BiO surface). In all four scenar-\nios switching of the polarization was achieved using a tip\nin a piezoforce geometry, but with considerable exchange\nbias favoring the spontaneous orientation.\nIn our discussion so far, we have assumed that the\nLSMO and SrRuO 3substrates behave like ideal metals,\nand have disregarded the fact that their constituent ions\nhave di\u000berent formal layer charges ( \u0006\u0018 0:7efor LSMO\nat 0.3 Sr concentration, and neutral for SrRuO 3). The\ndi\u000berent formal layer charge discontinuities could play\na role if the metallic screening is incomplete. Indeed,\nit is known that ionic relaxation is an important con-\ntributor to surface-charge screening in oxide electrodes\n[44], and the use of polar metals as electrodes has been\nproposed as a route to overcoming the critical thickness\nin ferroelectric capacitors [35]. To investigate the role\nof the formal layer charges in metallic oxide electrodes,\nwe next perform density functional calculations of [001]-\noriented BiFeO 3/SrRuO 3superlattices in both the happy\n(Fig. 7a) and sad (Fig. 7b) interfacial orientations (for\ndetails see the methods in Section VII).\nFIG. 8. Cross-sectional HAADF-STEM images of two dis-\ntinct BiFeO 3/La0:7Sr0:3MnO 3heterointerfaces with overlaid\nvector maps showing the polarization in the BiFeO 3layers. a)\nThe sample with the MnO 2-terminated (La 0:7Sr0:3O-MnO 2-\nBiO-FeO 2) interface develops a spontaneous up-pointing po-\nlarization ( ~Ppointing towards the FeO 2surface). b) The\nsample with the La 0:7Sr0:3O-terminated (MnO 2-La0:7Sr0:3O-\nFeO 2-BiO) interface develops a spontaneous down-pointing\npolarization ( ~Ppointing away the BiO surface). The scale\nbar is 1 nm.We constructed two superlattices each containing six\nlayers of BiFeO 3and four layers of SrRuO 3, with one\nSrO/FeO 2and one BiO/RuO 2interface, see Fig. 9. The\ntwo supercells had opposite orientation of the BiFeO 3\nferroelectric polarization, such that one system had self-\ncompensating (Fig. 9a) and the other charged interfaces\n(Fig. 9b). In both cases, the entire heterostructure\nadopts the a\u0000a\u0000c\u0000tilt pattern of BiFeO 3(Fig. 9, left\npanels), and both materials maintain their bulk magnetic\norderings (G-type antiferromagnetic for BiFeO 3and fer-\nromagnetic for SrRuO 3). As expected, both from our\nelectrostatic arguments and from the experimentally ob-\nserved exchange bias [43], the happy system is energeti-\ncally the most stable, \u00182 eV per supercell lower in energy\nthan the unhappy system.\nThe calculated structures, layer-by-layer polarizations\nand layer-resolved densities of states are shown in Fig. 9.\nNote that, in both the \\happy\" and \\unhappy\" systems,\nthe SrRuO 3layers are metallic (in green in the density\nof states graph on the right-hand-side on Fig. 9), with\na \fnite density of states at the Fermi energy in all lay-\ners, and the BiFeO 3(in purple) is insulating, with the\nFermi energy (shown as a vertical red line) lying in the\ngap. The layer-resolved densities of states for the happy\nsystem (right-most panel of Fig. 9a) indicate that there\nis no band bending and hence no internal electric \feld in\nthe happy system, consistent with the absence of surface\ncharge in the happy BiFeO 3slabs. The polarization (mid-\ndle panel) has its full bulk value throughout the slab, and\ndrops abruptly to zero in the \frst layer of the SrRuO 3.\nFor the six-unit-cell BiFeO 3heterostructures that\nwe present here, we \fnd that the unhappy system is\nmetastable in our DFT calculations. (For thinner \flms\nthe polarization orientation reverses and the structure\nrelaxes to the happy system.) We \fnd, however, a sup-\npressed layer polarization compared with the happy sys-\ntem, as visible in the middle graph of Fig. 9b, as well as\na pronounced shift in the BiFeO 3band edges from layer\nto layer indicating a strong internal electric \feld result-\ning from the large uncompensated surface charges in the\nBiFeO 3slab. In addition, the interfacial SrRuO 3layers\nundergo a polar ionic distortion to further reduce the po-\nlar discontinuity, similar to that observed in Ref. 35 for\nthe SrRuO 3/BaTiO 3interface.\nIn summary, our calculations indicate that, even with\nmetallic screening, the direction of polarization preferred\nby the interplay between the lattice and spontaneous po-\nlarization of the ferroelectric layer is strongly preferred.\nWhile screening by the metal is able to stabilize the un-\nhappy polarization orientation it is still energetically un-\nfavorable. This behavior explains the strong electric-\feld\nexchange bias e\u000bects [45], as well as the highly asymmet-\nric resistive switching [46] found in BiFeO 3capacitors.\nAs a \fnal example from the literature, we choose the\ncase of heterostructures between BiFeO 3and the metallic\nlightly-doped II-IV magnetic insulator, CaMnO 3. Het-\nerostructures of Ca 1{xCexMnO 3/BiFeO 3were grown on\nYAlO 3, whose small lattice constant causes a strongly8\nFIG. 9. Calculated structures (left), layer by layer polar-\nizations (middle) and layer density of states (right) for the\ntwo BiFeO 3/SrRuO 3heterostructures studied in this work.\nPanel a) shows the happy BiFeO 3slab con\fguration; panel b)\nthe unhappy. Green symbols and shading indicate SrRuO 3;\nBiFeO 3is shown in purple.\ncompressive biaxial in-plane strain and correspondingly\nlarge out-of-plane lattice constant and polarization (see\ndetailed discussion in Subsection C) [47]. The ferro-\nelectric polarization was determined to have an out-of-\nplane value of\u0018100\u0016C/cm\u00002, and to point towards\nthe BiFeO 3/CaMnO 3interface, which was of the FeO\u0000\n2{\nCaO type. While this is the least unhappy arrangement,\nin this case, because of the unusually large out-of-plane\npolarization, the FeO\u0000\n2layer provides only partial com-\npensation of the bound surface charge. An additional\nelectronic charge accumulation of \u00180:65 electrons per\nunit cell area was found using electron energy loss spec-\ntroscopy (EELS) to accumulate in the near-interfacial\nCa1{xCexMnO 3layers.\nB. Interfaces of bismuth ferrite with\ncentrosymmetric II/IV insulating perovksites\nSince the (100) surface of a centrosymmetric II/IV\ninsulating perovskite such as SrTiO 3has zero bound\ncharge, we expect it to behave electrostatically similarly\nto the vacuum in its interface with BiFeO 3. That is,\nwe expect that BiFeO 3surfaces that are happy in free-\nstanding slabs to form stable interfaces with SrTiO 3,\nwith bulk-like ferroelectricity in the BiFeO 3layer down\na)\nP\nP\nb)\n-80-4004080SrTiO3\nSrTiO3BiFeO3BiO(1+)FeO2(1-)\n----3-2-10123-3-2-10123E [eV]\nE [eV]P [!C/cm2]-90-50-103070SrTiO3\nSrTiO3BiFeO3BiO(1+)FeO2(1-)\nSrTiO3BiFeO3-80-4004080P [!C/cm2]FIG. 10. Polarization along the [001] direction of a\n(SrTiO 3)4/(BiFeO 3)6heterostructure with one BiO/TiO 2in-\nterface and one FeO 2/SrO interface. The polarization direc-\ntion leading to a happy system is in panel a, and to an un-\nhappy system in panel b. The systems are shown on the left,\nin the middle their corresponding unit cell-by-unit cell polar-\nization and on the right the unit cell-by-unit cell local density\nof states. Orange data points correspond to SrTiO 3and pur-\nple ones to BiFeO 3. In the atomic structure Sr is green, Ti\nblue, Bi purple, Fe brown and O red.\nto small thicknesses, whereas electrostatically unsta-\nble surfaces will have similarly unhappy interfaces with\nSrTiO 3and their ferroelectric polarization will tend to\nreverse [42]. All the examples that we have been able to\n\fnd of experimental BiFeO 3/SrTiO 3superlattices and\nheterostructures have their as-grown polarization orien-\ntation in the direction which compensates the layer po-\nlarization, consistent with this assumption [48{51]. Like-\nwise, literature calculations of PbTiO 3/BiFeO 3superlat-\ntices, with the polarization of the PbTiO 3entirely in the\nplane of the superlattice [52], found a stable solution with\nBiFeO 3in its happy con\fguration.\nTo explore the details of the behavior, we perform DFT\ncalculations for two [001] superlattices, each containing\nfour layers of SrTiO 3and six layers of BiFeO 3with one\nSrO / FeO 2and one BiO / TiO 2interface, but with the\nBiFeO 3polarization initialized to opposite orientations.\nAs expected, we \fnd that the happy case is stable, with\nthe layer polarization close to the bulk value throughout\nthe \flm (Fig. 10a, middle panel). As in the case of the\ninterface with SrRuO 3, the unit cell-by-unit cell density\nof states indicates zero internal electric \feld in ferroelec-\ntric BiFeO 3since the charge compensation between the\nspontaneous and layer polarization leads to zero surface\ncharge. In contrast, we are unable to stabilize the un-9\nhappy state in our DFT calculations unless we constrain\nthe polarization orientation in the middle layers of the\nBiFeO 3slab (Fig. 10b). The layer-by-layer density of\nstates indicates a strong band bending due to the large\ninternal electric \feld, and the formation of metallic lay-\ners by electron-hole excitation across the band gap at\nthe interfaces. (Similar metallicity at the unhappy BiO\n/ TiO 2interface was seen in an earlier DFT calculation\nfor a non-stoichiometric BiFeO 3slab with both of its in-\nterfaces set to BiO sandwiched between two SrTiO 3lay-\ners [53].) Additionally, the SrTiO 3, which is an incipient\nferroelectric and therefore readily polarizable, develops a\npolarization parallel to the spontaneous polarization of\nBiFeO 3to reduce the polar discontinuity.\nC. Interfaces of bismuth ferrite with insulating\ncentrosymmetric III-III perovskites\nFinally, we consider the case of the interface between\nBiFeO 3and a centrosymmetric insulating III-III per-\novskite, which contains~Pq\n2in its polarization lattice.\nHere the centrosymmetric contributions to the polariza-\ntion lattice are similar in both materials (they will dif-\nfer slightly if the lattice vectors and unit cell volumes\nare di\u000berent) and so the interfacial polar discontinuity\nis given by the di\u000berence in spontaneous polarizations.\nIf the second material is centrosymmetric, then the po-\nlar discontinuity is equal to the spontaneous polarization\nof BiFeO 3, similar to the case of the interfaces between\nthe II-IV ferroelectrics PbTiO 3or BaTiO 3and vacuum\ndiscussed above.\nA number of di\u000berent routes to avoiding the polar-\nization discontinuity at the interface, of the types we\nsummarized in Fig. 3, have been observed. An in-plane\npolarization (Fig. 3a) associated with an orthorhombic\nphase has been reported for BiFeO 3on a NdScO 3sub-\nstrate, which also imparts a small biaxial tensile strain\n[54]. Since many low-energy metastable non-polar, anti-\npolar and even antiferroelectric phases of BiFeO 3are\nknown [55, 56], the BiFeO 3\flm can also lose its polariza-\ntion entirely (Fig. 3d). For example, in superlattices and\nheterostructures of BiFeO 3with centrosymmetric Pnma\nLaFeO 3, BiFeO 3has been reported to adopt the antifer-\nroelectric PbZrO 3structure [57, 58], or even observed in\nan entirely new antiferroelectric structure, which has not\nbeen reported in the bulk [15]. Density functional cal-\nculations indicated that, for the strain conditions of the\nsample, this antiferroelectric phase is only slightly higher\nin energy than the ground-state polar phase, and it is fa-\nvored because of its lower electrostatic energy cost [15].\n(Note that a DFT calculation for a BiFeO 3/LaFeO 3slab\nin vacuum suggested the formation of a metallic layer at\nthe interface, although that study did not explore the\nformation of non-polar BiFeO 3phases and it is unclear\nhow the polar discontinuties at the surface were treated\nin the calculation [59].)\nAnother route to the compensation of the polar dis-continuity is the creation of extended defects; we dis-\ncuss the example of LaAlO 3/BiFeO 3, where this behav-\nior has been observed, next. Under strong biaxial com-\npressive strain, imposed by a small-lattice-constant sub-\nstrate such as LaAlO 3, BiFeO 3is known to undergo a\nphase transition to a tetragonal or tetragonal-like phase\n(T-BiFeO 3) with a large c=aratio of\u00181:3 and a giant,\nalmost entirely out-of-plane spontaneous polarization of\n\u0018150\u0016C cm\u00002[60{62]. This spontaneous polarization\nis roughly three times the [001] spontaneous polariza-\ntion of the usual rhombohedral phase of BiFeO 3, and\ncorrespondingly roughly three times the half-polarization\nquantum. (Note that the half-polarization quantum for\nT-BiFeO 3is slightly larger, at 59 \u0016C/cm2, than that of\nthe usual rhombohedral phase, because of its di\u000berent\nlattice parameters. We obtain values of a=b= 3:67\n\u0017A andc= 4:64\u0017A in our calculations for the lowest en-\nergy tetragonal structure.) This giant spontaneous po-\nlarization has two implications: First, the spontaneous\npolarization can be at best only partially compensated\nby the layer polarization at a \rat and stoichiometric BiO\nor FeO 2(001) surface. Second, the giant polarization\nmeans that the electrostatic potential diverges strongly\nat an interface, and only a few layers can form before\na compensation mechanism is required. In Fig. 11 we\nshow a HAADF-STEM image of a 100nm-thick \flm of T-\nBiFeO 3on LaAlO 3, in which, we observe such a compen-\nsation mechanism in the formation of an extended planar\ndefect just a few unit cells above the T-BiFeO 3/LaAlO 3\ninterface. We indicate the local ferroelectric polarization\n(plotted opposite to the atomic displacements of the Fe\ncations) by the yellow arrows in Fig. 11; this vector map\nreveals that the \frst \fve to seven T-BiFeO 3unit cells\nabove the T-BiFeO 3/LaAlO 3interface develop a sponta-\nneous up-pointing polarization. Then, perhaps unexpect-\nedly, above the extended planar defect the polarization\nin the T-BiFeO 3lattice is reversed and a down-pointing\npolarization forms. This results in a head-to-head po-\nlarization con\fguration, with a giant discontinuity of the\nspontaneous polarization of\u0018300\u0016C cm\u00002. For both\nthe top and bottom layers, however, the stoichiometric\nBiFeO 3terminates with an FeO 2layer, so the absolute\npolarization of each layer is reduced from the sponta-\nneous polarization by half a quantum, to \u0018100\u0016C cm\u00002.\nNote that this is the \\happiest\" con\fguration possible\nfor T-BiFeO 3, which, we emphasize again, does not have\nthe accidental cancellation between its spontaneous po-\nlarization and the half-polarization quantum seen in the\nrhombohedral ground state. Correspondingly, the abso-\nlutepolar discontinuity between the two T-BiFeO 3layers\nis reduced to the (still very large!) value of \u0018200\u0016C\ncm\u00002.\nThe planar defect (highlighted with red rectangles in\nFig. 11) consists of a characteristic Bi 2O2+\n2layer, which\nshifts the perovskite layers above the defect half a per-\novskite block along the orthogonal horizontal directions\nin the manner of an Aurivillius phase, surrounded by two\nO2\u0000layers (see cartoon on the right side of Fig. 11; the10\nhorizontal red lines indicate the boundary of the defect).\nThe total stoichiometry of the defect consists of one for-\nmally Bi 2O2+\n2block plus two O2\u0000ions per surface BiFeO 3\nunit cell, leading to a net defect charge of two electrons,\nor\u0018\u0000100\u0016C cm\u00002, per primitive unit cell cross-sectional\narea. This is exactly half of the polar discontinuity, and\nso, as sketched in the lower panel of Fig. 5, is exactly the\nlayer charge needed for compensation. Note that oxygen\nnon-stoichiometry in the Aurivillius-like layer, yielding\nBi2O2\u0006\u000erather than precisely Bi 2O2, is likely, and will\nchange the exact amount of compensating charge that it\nprovides. Indeed, a similar planar defect, with a double\nBi2O2+\n2layer, was observed in a BiFeO 3\flm grown on\nan LaAlO 3substrate in spite of an intermediate metallic\nelectrode between the \flm and the substrate [63]. In this\ncase the BiFeO 3layer adjacent to the electrode was non-\npolar, and that above the defect developed a downward-\npointing polarization as in our example.\nInterestingly, a similar extended defect has been re-\nported as a surface \\skin\" in BiFeO 3, in all cases when\nthe polarization points in the upwards (towards the sur-\nface) direction [64{66]. In Ref. 65 a \flm of rhombohe-\ndral BiFeO 3was grown in [001] orientation on an insulat-\ning DyScO 3substrate, and two opposite domains, sepa-\nrated by a 180\u000edomain wall were imaged using HAADF-\nSTEM. The down-polarization domain had a pristine\nBiO surface and so was in the `happy' con\fguration. The\nup-polarization domain, which would have been in the\n`unhappy' con\fguration with an excess positive charge in\nits pristine form, had a capping layer of the negatively-\ncharged Bi 2O2Aurivillius-type extended defect to com-\npensate. This \fnding has clearly unfavourable implica-\ntions for the switching of BiFeO 3domains. In Ref. 66,\nthe BiFeO 3\flm on TbScO 3had domains of strongly\nsuppressed polarization alternating with domains of en-\nhanced upward-pointing polarization; the latter had the\nsurface skin overlayer. Finally, we mention that similar\nAurivillius structures have also previously been observed\nas intergrowths in rhombohedral BiFeO 3thin \flms [67].\nIt would be interesting to analyze the interfaces between\nthe intergrowths and the surrounding BiFeO 3regions to\ndetermine the nature of the interface chemistry and the\npolarization orientation in the context of the compensa-\ntion principles discussed here.\nVI. SUMMARY AND OUTLOOK\nIn summary, we have reviewed how the spontaneous\npolarization associated with ferroelectricity combines\nwith the layer polarization associated with the ionic\ncharges of the lattice to determine the electrostatic sta-\nbility of the surfaces and interfaces of insulators. We\nreminded the reader that the two contributions are con-\nveniently treated on the same footing by the modern the-\nory of polarization, allowing straightforward determina-\ntion of the amount of bound charge at a general surface\nor interface. The bound charge is important for the de-\n  BiBiBi\nBiBiBiBiBiBiBiBiBiBiBiBiBiBiBiPP(Bi2O2)+2O-2O-2FIG. 11. Cross-sectional HAADF-STEM image showing the\npresence of planar defects just above the BiFeO 3/LaAlO 3in-\nterface. The stepped Bi 2O2structural units are highlighted\nby the red rectangles and the overlaid vector map of the fer-\nroelectric polarization plotted opposite to the displacement\nof the Fe cations (yellow arrows) reveals that the polarization\nin the pseudo-tetragonal BiFeO 3lattice changes from up to\ndown direction across the defect. An atomic model of the\ndefect is shown on the right. The scale bar is 1 nm.\nsign of thin-\flm heterostructures and for the stability of\nsurfaces, since any non-zero surface or interfacial charge\nmust be compensated to avoid divergence of the electro-\nstatic potential.\nAfter brie\ry discussing examples of materials with a\nspontaneous polarization but no layer polarization ([001]-\noriented PbTiO 3and BaTiO 3) and a material with no\nspontaneous polarization but a non-zero layer polariza-\ntion ([001]-oriented LaAlO 3) we focused on the case of\nmultiferroic BiFeO 3, which combines both spontaneous\nand layer contributions. BiFeO 3is of particular interest,\nbecause the spontaneous and layer contributions to the\npolarization in the ground-state R3cstructure have the\nsame size along the usual [001] growth direction, leading\nto combinations of polarization and surface termination\nthat are uncharged and therefore electrostatically sta-\nble. The opposite combinations have double the surface\ncharge of a ferroelectric with the same spontaneous po-\nlarization but with uncharged layers. These have been\nreferred to as the happy and unhappy combinations in\nearlier work [42].\nWe considered three scenarios: BiFeO 3on a metal-\nlic substrate, BiFeO 3in a superlattice or heterostruc-\nture with a zero-layer-charge insulator, and \fnally\na BiFeO 3/insulator superlattice or heterostructure in\nwhich the insulator has the same layer charges as the\nBiFeO 3. In each case we illustrated the di\u000berent possible\nbehaviors with examples from the literature, as well as\nwith density functional calculations and HAADF-STEM\nanalyses performed for this work.\nWe can summarize the di\u000berences in behavior in the\nthree cases as follows: i) Both the happy and the unhappy\npolarization orientations can be stabilized by a metallic\nelectrode, although the unhappy case is higher in energy11\nand has a lower spontaneous polarization. This is con-\nsistent with the known large electric-\feld exchange bias\nof BiFeO 3\flms, and implies that symmetric switching of\na BiFeO 3capacitor will be di\u000ecult to achieve. ii) The\ninterface of BiFeO 3with a zero-charge-layer insulator be-\nhaves like a free BiFeO 3surface, with the happy combi-\nnation of polarization and surface termination strongly\nfavored. The unhappy combination can be stabilized by\nstrong band bending to generate metallic layers at the in-\nterface and/or polarization of the adjacent insulator. iii)\nThe interface of BiFeO 3with another III-III perovskite\nhas a polar discontinuity equal to the spontaneous po-\nlarization, and so is equally energetically unfavorable for\nboth orientations of the polarization. As a result of the\nrich Bi-Fe-O low energy phase space, many responses are\npossible, including stabilization of phases with zero out-\nof-plane polarization, and the formation of extended pla-\nnar defects.\nWhile the electrostatic concepts discussed in this pa-\nper are not new, we hope that their collection in this ar-\nticle will be helpful in guiding the design of BiFeO 3and\nrelated thin \flms or heterostructures with targeted elec-\ntrical properties, as well as in interpreting experimental\nobservations.\nVII. METHODS\nDensity functional Theory. Density functional cal-\nculations were performed within the periodic supercell\napproach using the VASP code [68{71]. We chose the\nPBEsol functional [72] for all calculations because i) it\ngives a good band alignment between metallic SrRuO 3\nand BiFeO 3and no pathological situation arises [73], and\nii) it yields a paraelectric ground state for SrTiO 3. In\norder to obtain a band gap for BiFeO 3close to the ex-\nperimental value the Hubbard U, in the Dudarev [74] ap-\nproach, was used with U\u0000J= 4 eV on the Fe 3 dstates,\nandU\u0000J= 2:0 eV on the Ti and Ru dstates. Core elec-\ntrons were replaced by projector augmented wave (PAW)\npotentials [75], while the valence states (5e\u0000for Bi, 8e\u0000\nfor Fe, 6e\u0000for O, 10e\u0000for Sr, 4e\u0000for Ti, 8e\u0000for Ru)\nwere expanded in plane waves with a cut-o\u000b energy of\n500 eV. In all calculations the in-plane lattice parameters\nwere set to that of SrTiO 3,aPBE sol= 3:90\u0017A, as it is the\nsubstrate commonly used in epitaxial growth of BiFeO 3\nthin \flms. The in-plane surface area isp\n2a\u0002p\n2a. A\nMonkhorst-Pack k-point grid of (5\u00025\u00021) was used for\nall ionic relaxations, which had an optimization thresh-\nold on the forces of 0.01 eV/ \u0017A. For the density of states\ncalculations, Monkhorst-Pack k-point grid of 11 \u000211\u0002\n1 was used. An antiferromagnetic G-type ordering was\nimposed in BiFeO 3, which gave a magnetic moment of\n4.15\u0016Bper Fe ion in the bulk. SrRuO 3instead is ferro-\nmagnetic and the magnetic moment is 1.4 \u0016Bper Ru ion\nin the bulk.\nThe unit cell-by-unit cell polarization along the [001]\ndirection shown in Fig. 9 and 10 was calculated by com-puting the displacement of each ion from the high sym-\nmetry position and multiplying it by the Born e\u000bective\ncharges from Ref. [41].\nThin Film Growth. The BiFeO 3/La0:7Sr0:3MnO 3\n(BFO/LSMO) heterostructures shown in Fig. 8 were\ngrown by pulsed laser deposition on SrTiO 3(001) (STO)\nsingle crystal substrates. Before the growth, a bu\u000bered\nHF acid-etch and thermal treatment process was used to\nobtain fully TiO 2-terminated surfaces. The sample with\nthe MnO 2-terminated (La 0:7Sr0:3O-MnO 2-BiO-FeO 2) in-\nterface was designed by growing whole LSMO unit cells\ndirectly on the STO substrate, followed by the growth\nof the BFO layer. For the sample with the La 0:7Sr0:3O-\nterminated (MnO 2-La0:7Sr0:3O-FeO 2-BiO) interface, 1.5\nunit cells of SrRuO 3(SRO) were deposited on STO to\nswitch the termination of the STO from TiO 2to SrO.\nThe SRO layer was grown at 650\u000eC in 100 mTorr of\noxygen pressure. Both the LSMO and BFO layers were\ngrown at 690\u000eC in 150 mTorr of oxygen pressure. A\npostannealing process was carried out at 400\u000eC under\nan oxygen ambient for 1 h to ensure the samples were\nfully oxidized. For additional information, see Ref. [43].\nThe BFO thin \flm shown in Fig. 11 was grown by molec-\nular beam epitaxy (MBE) on single-crystal substrates\nof (001) LaAlO 3(LAO). The studied 100-nm-thick \flm\nshowed the coexistence of two interspersed BFO phases:\na rhombohedral-like (R) phase and a tetragonal-like (T)\nphase. Further growth and characterization details can\nbe found in Ref. [60].\nTransmission Electron Microscopy. Cross-sectional\nspecimens for transmission electron microscopy analy-\nsis were prepared by mechanical polishing using a tri-\npod polisher followed by argon ion milling until elec-\ntron transparency. High-angle annular dark-\feld scan-\nning transmission electron microscopy (HAADF-STEM)\nwas carried out using the TEAM 0.5 microscope lo-\ncated at the National Center for Electron Microscopy\n(NCEM). The TEAM 0.5 is a FEI Titan 80-300 mi-\ncroscope equipped with a high-brightness Schottky-\feld\nemission X-FEG electron source, a source monochroma-\ntor, a CEOS DCOR spherical-aberration probe correc-\ntor, and a CEOS CETCOR spherical-aberration image\ncorrector. The microscope was operated at 300 kV, the\nprobe semi-convergence angle set to 16.5 mrad (which\nyields a calculated probe size of 0.63 \u0017A), and the annular\nsemi-detection range of the HAADF detector calibrated\nat 45{290 mrad. This setting was chosen to allow for a\nsu\u000eciently large depth of \feld in order to enhance the\ncontrast of the atomic columns. The positions of the\natomic columns were \frst \ftted by means of a center of\nmass peak-\fnding algorithm, and subsequently re\fned by\nsolving a least-squares minimization problem (using the\nLevenberg{Marquardt algorithm). This iterative re\fne-\nment was carried out using a custom-developed script\nthat makes use of 7-parameter two-dimensional Gaus-\nsians and allows estimation of the atomic column peak\npositions with picometer precision [76, 77]. Then, polar-\nization maps were calculated from the relative displace-12\nments of the two cation sublattices present in the ferro-\nelectric perovskite-type structures with general formula\nABO 3. Thus, the local ferroelectric polarization was cal-\nculated by measuring the polar displacement in the im-\nage plane of the B position from the center of mass of\nits four nearest A neighbors. 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The implementation of the self-consistent mean ﬁeld met hod in the KineCluE code allows for the\ncalculationoftransportcoeﬃcients extended to arbitraryintera ctionranges, crystalstructures, and diﬀusion\nmechanisms. In addition, the code gives access to the diﬀusion and d issociation rates of small solute-defect\nclusters – in this case, vacancy- and dumbbell-solute pairs. The res ults show that the diﬀusivity of P, Mn,\nand Cr solute atoms is dominated by the dumbbell mechanism, that of Cu by vacancies, while the two\nmechanisms might be in competition for Ni and Si, despite the fact tha t the corresponding mixed dumbbells\nare not stable. Systematic positive radiation-induced segregation (RIS) at defect sinks is expected for P\nand Mn solutes due to dumbbell diﬀusion, and for Si due mainly to vaca ncy drag. Vacancy drag is also\nresponsible for Cu and Ni enrichment at sinks below 1085 K. The RIS b ehavior of Cr is the outcome of a\nﬁne balance between enrichment due to the dumbbell diﬀusion mecha nism and depletion due to the vacancy\none. Therefore, for dilute Cr concentrations global enrichment o ccurs below 540 K, and depletion above.\nThis threshold temperature grows with solute concentration. The ﬁndings are in qualitative agreement with\nexperimental observations of RIS and clustering phenomena, and conﬁrm that solute-defect kinetic coupling\nplays an important role in the formation of solute clusters in reactor pressure vessel steels and other alloys.\nKeywords: Iron alloys, Atomic diﬀusion, Radiation-induced segregation, Trans port coeﬃcients, Density\nFunctional Theory\n1. Introduction\nAtomic transport plays a key role in driving the evolution of structur al properties of metals and alloys\nduring fabrication, processing, and operation. For instance, solu te diﬀusion determines the ﬁnal microstruc-\nture during phase transformations in heat treatments [1, 2, 3], an d the knowledge of the atomic diﬀusion\nmechanismsisnecessarytoadjusttheprocessparametersando btainthedesiredproperties. Atomicdiﬀusion\ncontrols also the microstructure evolution of irradiated materials d riven by the permanent excess of point\ndefects (PD). Due to kinetic coupling between PD and atomic ﬂuxes, solutes can migrate to PD sinks under\nthe eﬀect of thermodynamic driving forces acting only on PD. This ma y cause radiation-induced segregation\n(RIS) on grain boundaries, dislocations, PD clusters, surfaces, o r precipitate-matrix interfaces [4, 5, 6]. In\nturn, RIS can catalyze solute-defect clustering and be the precu rsor of phase precipitation in otherwise\nundersaturated alloys, if saturation conditions are reached locally . Several studies have shown for instance\nthe formation of solute-enriched clusters (containing Cr, P, Mn, N i, and Si) in undersaturated FeCr alloys\n[7, 8] as well as in reactor pressure vessel (RPV) model alloys and s teels [9, 10, 11, 12, 13, 14, 15], and even\nthe precipitation of a secondary phase in an ion-irradiated Fe-3%Ni alloy [16] well below the Ni solubility\nlimit. The thermal stability of Mn-Ni-Si precipitates in RPV steels, pos sibly enhanced by a locally low Fe\ncontent and by the presence of irradiation-induced defects [17, 18], has been predicted by thermodynamics-\ninformed models and simulations [19, 20, 21] and conﬁrmed experime ntally [14, 22, 23, 24]. In addition to\nthe thermodynamic driving forces at play, solute-defect kinetic co upling can contribute to the formation of\n∗Corresponding author\nEmail addresses: luca.messina@cea.fr (Luca Messina), maylise.nastar@cea.fr (Maylise Nastar)\nPreprint submitted to Acta Materialia April 16, 2020solute-rich precipitates by increasing the solute concentration at sinks. Moreover, in under-saturated solid\nsolutions with no driving forces for precipitation, it can provide a loca l driving force whenever RIS leads\nto local solute concentrations above the solubility limit. Due to this int erplay between thermodynamics\nand kinetics, a radiation-enhanced mechanism for nucleation and gr owth of solute precipitates may arise.\nFor instance, defect clusters immobilized by small amounts of solute s can turn into nucleation sites where\nother solutes are progressively accumulated by PD-assisted tran sport [17]. This is conﬁrmed by several\nobservations of toroidal-shaped and planar solute clusters in RPV s teels [25, 12, 26, 27], and has been re-\ncently rationalized in an advanced model able to reproduce the micro structure evolution of various RPV\nsteel types [28]. The ”pinning” eﬀect on clusters of self-interstitia l atoms (SIA) has been inferred from the\ninterpretation of experimental results for Mn, P, and Ni [12, 7, 16 , 8], and by modeling for Cr, P, and Ni\n[29, 30, 31]. Recent ab initio calculations of solute-loop interactions suggest a similar pinning tend ency for\nMn, Cu, and Si [32].\nHowever, the capability of PDs (vacancies and self-interstitials) to carry solute atoms to the nucleation\nsites is yet not fully characterized. Nowadays, precise analytical m odels based on the Self-Consistent Mean\nField (SCMF) theory [33] or the Green-function approach [34], in com bination with ab initio calculations\nof defect jump rates, allow for a highly accurate analysis of the intr insic atomic-transport properties by\ncalculation of the transport (Onsager) matrix [35]. By the latter me thods, it has been proven that solute\ndragby vacancies is a widespread phenomenon arisingbelow a given te mperature threshold in body-centered\ncubic (bcc) [36, 37], face-centered cubic (fcc) [38, 39, 40], and he xagonal close-packed (hcp) metals [41],\nprovided that the vacancy-solute interaction is suﬃciently strong . In dilute ferritic alloys, the threshold\ntemperature has been systematically determined for all transition -metal impurities [42, 37]. This threshold\nis near or above 1000 K for Cu, Mn, Ni, P, and Si, whereas it lies near 30 0 K for Cr. Therefore, vacancies are\ncapable of transporting all solutes to sinks, with the exception of C r for which depletion at sinks is expected.\nOn the other hand, the transport eﬃciency of self-interstitial at oms (SIA) has been only superﬁcially\ninvestigated. Speculations based on ab initio evaluations of the stability of mixed dumbbells (MD) [43, 44,\n45], and the interpretation of resistivity-recovery (RR) experime nts [46, 47, 48] lead to the assumption that\ntransport of Cr, Mn, and P should occur, since the corresponding MDs are stable, whereas for the opposite\nreason it should not occur for Ni, Cu, and Si. As a matter of fact, fo r Ni and Si conﬂicting assumptions\nand interpretations are found in the literature [47, 48, 7]. Sometim es, even more simpliﬁed assumptions are\nmade based only on solute size, namely that enrichment is expected f or undersized solutes, and depletion\nfor oversized ones [49]. Only Cr and Ni solutes have been addressed in more detail with full calculations\nof transport coeﬃcients. Choudhury et al.[50] found that no transport of Ni by SIAs should occur.\nHowever, the drawn conclusions on the RIS behavior disregarded t he eﬀects of ﬂux coupling and predicted\nNi depletion, in disagreement with the experimental evidence [16]. Th e FeCr alloy has been the object of\na series of studies based on rate-theory [51], atomistic kinetic Mont e Carlo (AKMC) [52], and phase-ﬁeld\n[53, 6] models. They showed that SIA transport plays a crucial role and yields Cr enrichment at sinks in\nopposition to the vacancy-induced depletion. The balance of the tw o eﬀects provides an explanation for\nthe experimentally-observed [54] existence of a switchover temp erature between enrichment and depletion,\nand has been shown to be extremely sensitive to microstructure fe atures (e.g., composition, strain ﬁelds, or\nsink density) [52, 55]. On the contrary, no studies have yet fully unc overed the reasons for the systematic\nenrichment tendencies observed for the other solutes (Cu, Mn, N i, P, Si) on grain boundaries, dislocations\nlines, and loops [54, 12, 13, 7]. It is unclear whether this is due to vaca ncy drag, SIA transport, or a\ncombination of both.\nThe objective of this study is to perform an accurate analysis of SI A-driven solute transport in a set of\ndiluteFe- Xalloys(X=Cr,Cu, Mn, Ni, P,Si), andprovideacomprehensiveoverviewofthe expectedintrinsic\nRIS behavior of said chemical species. In addition, this work establis hes a general modeling framework\naimed at accurate RIS predictions by vacancies and interstitials in dilu te alloys, providing in this way a\nsound knowledge of the kinetic properties necessary to the interp retation of diﬀusion-driven microstructure\nphenomena. This is achieved by means of the recently published KineC luE code [56] that implements the\nSCMF method in its cluster-expansion form [57]. For SIA diﬀusion, Kine CluE allows for a considerable\nadvancement with respect to the available analytical models [58, 59] developed in the SCMF framework\nand featuring limited solute-SIA interaction ranges. KineCluE gener alizes this framework to arbitrarily\nlong ranges, solute concentration eﬀects, and complex SIA migrat ion mechanisms in any periodic crystal.\nIn the next sections, after a brief overview over the cluster-exp ansion framework, KineCluE is applied to\nSIA-solute migration in the aforementioned bcc ferritic alloys, base d on accurate ab initio calculations of\nbinding energies and migration rates. The Onsager matrices are the n combined with the vacancy-related\ntransport coeﬃcients [42] and used to analyze ﬂux coupling and RIS tendencies, in comparison with the\n2current experimental and modeling knowledge.\n2. Methodology for the calculation of transport coeﬃcients\n2.1. Cluster expansion of transport coeﬃcients\nIn the framework of linear thermodynamics of irreversible process es, transport coeﬃcients relate atomic\nﬂuxestothe underlyingthermodynamicforces,e.g., chemicalpote ntialgradients(CPG)[35]. Thesymmetric\nOnsager matrix allows for the analysis of ﬂux coupling between diﬀere nt species, for instance PD-induced\nsolute transport. The components Lijare deﬁned as:\nJi=−/summationdisplay\njLij∇µj\nkBT, (1)\nwhereJiis the ﬂux of species iand∇µjthe CPG of species j. In binary alloys, the independent coeﬃcients\nneeded for each PD type are Lδδ,LδB, andLBB, whereδ= (V,I) marks vacancies or interstitials, and B the\nsolute species. The solvent (A) coeﬃcients LAA,LAB, andLδAcan be inferred from the former (cf. Table\nB.6).\nThe transport coeﬃcients are computed here within the SCMF theo ry [33]. The latter has been applied\nmultiple times to vacancy diﬀusion [36, 40, 42, 37, 57, 60], but the only model available for SIA-assisted\nsolute transport is that by Barbe et al.[58, 59, 61, 62]. The latter is here substantially improved with\nthe KineCluE code [56], which implements a recent development of SCMF where the Onsager matrix is\nexpanded in terms of cluster contributions L(c)\nij[57]:\nLij=C/summationdisplay\ncfcL(c)\nij. (2)\nCis the total concentration, while fcis the concentration fraction of cluster cproportional to the cluster\npartition function (cf. Appendix B). In the dilute limit, the matrix is sp lit into a ”monomer” (an isolated\ndefect) and a(solute-defect) ”pair”contribution, while immobile sp eciessuch assubstitutional solutes donot\ncontribute to the total transport matrix. Cluster transport co eﬃcients and partition functions are computed\nwith KineCluE and then combined with a concentration model to obtain the cluster fractions appearing in\nEq. 2. In the case of solute transport by vacancies and interstitia ls in a dilute binary alloy, the total Onsager\nmatrix reads [63]:\n\nLVVLVILVB\nLVILIILIB\nLVBLIBLBB\n=C\nfV\nL(V)\nVV0 0\n0 0 0\n0 0 0\n+fI\n0 0 0\n0L(I)\nII0\n0 0 0\n+fVB\nL(VB)\nVV0L(VB)\nVB\n0 0 0\nL(VB)\nVB0L(VB)\nBB\n+fIB\n0 0 0\n0L(IB)\nIIL(IB)\nIB\n0L(IB)\nIBL(IB)\nBB\n\n.\n(3)\nThe vacancy and interstitial contributions to the LBBcoeﬃcient, respectively LBB(V) =CfVBL(VB)\nBBand\nLBB(I) =CfIBL(IB)\nBB, arecomputedseparately. The LVIcoeﬃcientissettozero,whichmeansthatanykinetic\ncoupling between vacancies and interstitials is neglected. The conce ntration model and the procedure to\ncompute the total Onsager coeﬃcients are explained in detail in App endix B. Note that in case detailed\nbalance is not satisﬁed, e.g., as an eﬀect of ballistic atomic relocation u nder irradiation at low temperature\n[64], the cluster transport matrices are not symmetric.\nKineCluE extends the reach of SCMF to an arbitrarily long kinetic radiu srkin, which deﬁnes the maxi-\nmum extent of the kinetic trajectories included in the calculation. Th e transport coeﬃcients converge with\nincreasing rkinto an asymptotic value [56]. After a speciﬁc convergence study on S IA-solute migration, the\ncoeﬃcients were found to be well converged at rkin= 4a0(a0is the lattice parameter), and to reach an\nacceptable accuracy already at rkin= 2a0, whereas they are not converged with a kinetic radius as short\nas the ﬁrst nearest-neighbor (nn) distance (corresponding to B arbe’s model). The kinetic radius is thus set\nhere to 4 a0, unless otherwise speciﬁed.\nFor each cluster, the KineCluE calculation proceeds in three steps:\n1. A ”symbolic run” that depends only on the crystal geometry and the defect migration mechanisms.\nThis yields the Lijsemi-analytical expressions, as well as the list of symmetry-unique conﬁgurations\nand jumps required to perform the numerical calculations, within th e interaction radius of choice (cf.\nSec. 2.2).\n32. The calculation of the binding and migration energy for each item in t he lists, here obtained with ab\ninitiocalculations (cf. Sec. 2.3).\n3. A ”numeric run” to compute the cluster transport coeﬃcients a nd the partition function in a given\ntemperature range (cf. Sec. 4).\n2.2. Equilibrium conﬁgurations and jump frequencies\nIn bcc iron, the most stable SIA conﬁguration is the /angb∇acketleft110/angb∇acket∇ightdumbbell [65, 43], with three distinct jump\nmechanisms [66, 58]: 60 °rotation-translation jumps (RT) to four target sites out of the e ight 1nn, pure-\ntranslation jumps (T) to the same target sites, and 60 °onsite rotations (R). The target sites, labelled\n’1b’ in Fig. 1, are those in a so-called ”compressed” position. In addit ion, it is possible to have a 2nn\n90°jump [65]. For a solute-dumbbell pair, it is necessary to deﬁne two ad ditional jump mechanisms,\nnamely the MD migration and dissociation. The former Barbe’s model lim ited to 1nn interactions is here\ngeneralized by extending the kinetic radius to 4 a0, which yields a list of 94 symmetry-unique conﬁgurations\nand 260 symmetry-unique RT jump frequencies, compared to the 3 and 8 featured in Barbe’s model. This\nallows for a much better precision in the transport coeﬃcient calcula tion, thanks to the inclusion of wider\nsolute-dumbbell correlated trajectories. The computational loa d can be then reduced by setting a smaller\nthermodynamic radius, which depends on the extent of the thermo dynamic interaction between dumbbell\nand solute. Consistently with the ab initio binding energies (cf. Table 2), it is chosen to set rthto the 5nn\ndistance. This yields a subset of 13 conﬁgurations and 13 RT jump fr equencies that are computed by precise\nab initio calculations, and are shown in Figs. 1 and 2.\nxzy\n<110>3b\n2bFe atom\nXαSolute atom\nposition\n5a\n3a\nM2a\n2b\n2b3c3b5b 3b\n1a 1b 4c4a 4b\nFigure 1: Nomenclature of solute-dumbbell equilibrium con ﬁgurations. The red circles mark the solute position relati ve to\nthe dumbbell defect: xis the solute nearest-neighbor shell with respect to the dum bbell, and αthe symmetry class within\nthe same shell (see Table A.4). The colored atoms are located outside the interaction shells. ’M’ marks the mixed-dumbbe ll\nconﬁguration.\nIn Fig. 1, sites in the same nn shell are categorized depending on the distance from each atom of\nthe dumbbell. Each site is thus labeled ’ Xα’, where Xstands for the nn shell, and αfor the speciﬁc\nsymmetry class within that shell (with the exception of the MD conﬁg uration, marked as ’M’). For each\nsymmetry class, sample atomic coordinates are provided in Appendix A. In Fig. 2, the jumps connecting\nthese conﬁgurations are marked with ω,τ, andωR, for RT, T, and R jumps, respectively. The MD jumps\nare marked with ’1’, while the jumps of the Fe-Fe dumbbell are either la beled with ’0’, when far away from\nsolute atoms, or according to the initial ( Xα) and ﬁnal ( Yβ) conﬁguration when nearby solutes. Refer to\nTable A.5 for a summary.\nThejumpsshowninFig. 2aremodeledaccordingtotransition-state theory(TST)asthermally-activated\nprocesses with frequency [35]:\nωij=νijexp/parenleftigg\n−Emig\nij\nkBT/parenrightigg\n, (4)\n4ω1b2b\nω1a3cω1a2a\nω1b3b\nω1a3b\nω1b5bω1a2b\nω1ωM 1bω1b M\nτ1b2a\nτ1a3bτ1a2b\nτ1b3c\nτ1a3b\nτ1b5bτ1a2bτ1\nτM 1bτ1b M\nωR1b1a\nωR1a1bωR1a1b\nωR1b1a\nωR1a1a\nωR1b1bωR1a1aωR1ωR1b1b\nRotation−translation ( ω) Translation ( τ) Onsite rotation ( ωR)\nω2a4cω2b4b\nω2b4c\nτ2a4bτ2b4c\nτ2b4c\nωR2a2bωR2b2a\nωR2b2bBefore jump\nAfter jump\nSolute position\nFigure 2: Representation of all dumbbell jumps (rotation-t ranslations ω, translations τ, and onsite rotations ωR) connecting\nthe solute-dumbbell conﬁgurations in Fig. 1, i.e., for an in teraction range rthextending to the 5nn shell.\nwhereνis the attempt frequency and Emigthe migration energy. Beyond rth, the binding energies are set to\nzero, and the jump frequencies are obtained with a classic kinetically -resolved activation (KRA) approach\n[67], which was proven suﬃciently accurate for far-away jumps [42 ]:\nEmig\nij=Emig\n0+Eb\ni−Eb\nj\n2. (5)\nEb\niandEb\njare the binding energy (positive when attractive) of the initial and ﬁ nal state, and Emig\n0is the\nmigration barrier of the ω0RT jump. This approach with a reduced rthallows for a better accuracy with\nthe same amount of ab initio calculations. Furthermore, detailed-balance requirements are au tomatically\nfulﬁlled.\n2.3. Ab initio methodology\nThe binding and migration energiesare calculated in the frameworkof Density Functional Theory (DFT)\nwith the Vienna ab initio Simulation Package ( vasp) [68, 69, 70]. The vaspfull-core pseudopotentials\ndeveloped within the projector augmented wave (PAW) method [71, 72] are employed for all elements. The\nPerdew-Burke-Ernzerhof (PBE) parameterization [73] of the ge neralized gradient approximation (GGA) is\nused to describe the exchange-correlation function. Calculations are spin-polarized, and make use of the\nVosko-Wilk-Nusair (VWN) interpolation scheme of the correlation po tential. Particular attention has been\npaid to achieve the correct antiferromagnetic state of Mn and avo id local minima [74], by providing a precise\ninitial guess of the magnetic moment on Fe (2 .23µB) and Mn ( −2.24µB) atoms [32], and by applying linear\nmixing in the starting guess of the charge dielectric function [37].\nIonic relaxations are performed in a 251-atom supercell, whose volu me and shape are maintained to that\nof a bcc iron cell. Following previous convergence studies [75], the eq uilibrium lattice parameter is found\nto bea0= 2.831˚A. The Brillouin zone is sampled with a Monkhorst-Pack 3 ×3×3k-point grid, and a\nplane-wave cut-oﬀ of 350 eV is used. More accurate parameters ( 4×4×4 k-points and 400 eV cut-oﬀ) are\nused to reﬁne the calculations with Cr atoms, because the energy d iﬀerences among distinct conﬁgurations\nare small and the results are thus more sensitive to the input param eters.\nThe dumbbell formation energy is:\nEf=E[N+1]−N+1\nNE[N], (6)\n5whereE[N+ 1] and E[N] are the energies of the supercell with and without the dumbbell, re spectively.\nThe binding energies of the solute-dumbbell conﬁgurations shown in Fig. 1 are evaluated as:\nEb\nXα=−E[I,B]+E[I]+E[B]−Eref, (7)\nwhere the terms on the right-hand side are the energy of the supe rcell containing one dumbbell and one\nsolute, one dumbbell, one solute, and no defect, respectively. In t his convention, positive signs stand for\nattractive interactions.\nThe migrationbarriers(Eq. 4) areobtained by means ofnudged-ela sticband (NEB) calculations[76, 77],\nimplemented with the climbing-image algorithm [78] and three intermedia te images. In order to reduce the\ncomputational load, calculations are mainly limited to the RT mechanism , as the latter is characterized by\nthe lowest migration barriers. The accuracy on each migration barr ier is estimated in less than 5 meV.\nThe attempt frequencies of the Fe-Fe dumbbell ( ω0) and the MDs RT jumps ( ω1) are computed by\nmeans of DFT frozen-phonon calculations [79, 80, 37] in 128-atom c ells, with a cut-oﬀ energy of 400 eV. All\nother jumps shown in Fig. 2 are assigned the attempt frequency of ω0. The vibrational frequencies of each\ndumbbell in the equilibrium andsaddle-pointpositions νE\nkandνS\nkareobtainedin twosteps: ﬁrst, by relaxing\nthe conﬁguration with a force convergencecriterion of 0.01 eV/ ˚A on each atom, and restraining the supercell\nvolume to the value stemming from the equilibrium lattice parameter a0mentioned above. Secondly, by\napplying four displacements of ±0.010 and ±0.020˚A on each atom and diagonalizing the Hessian matrix.\nThese values ensure an accurate numerical estimation of the Hess ian and its phonon spectrum [81]. The\nattempt frequency is given by [79]:\nν=/producttext3NE−3\nk=1νE\nk/producttext3NE−4\nk=1νS\nk(8)\nwhereNEis the total number of atoms in the supercell. In addition, the dumbb ell formation entropy in\npure Fe can be obtained as [82]:\nSf\nkB=−/bracketleftigg\nln/parenleftigg3NE−3/productdisplay\nk=1νE\nk/parenrightigg\n−NE\nNUln/parenleftigg3NU−3/productdisplay\nk=1νU\nk/parenrightigg/bracketrightigg\n, (9)\nwhereνU\nkandNUare the eigenfrequenciesand the number of atoms in the undefect ed supercell, respectively.\n2.4. Vacancy transport coeﬃcients\nThe vacancy coeﬃcients were already calculated within the SCMF the ory in a previous work [42], but\nthey are recomputed here with a longer kinetic radius ( rth= 4a0) and the cluster-expansion approach (Eq.\n3), for the sake of consistency with the SIA calculations. The ther modynamic radius is set to the 5nn\ndistance (√\n3a0), yielding 12 jump frequencies that correspond exactly to the one s depicted in Fig. 1 of\nreference [42]. The DFT calculations are repeated with the same k-point grid (3 ×3×3) but a higher cut-oﬀ\nenergy (400 eV in place of 300 eV) and a larger simulation cell (249 ato ms instead of 127). No substantial\ndiﬀerences are found, with the exception of larger ω15barriers in FeP and FeSi (0.85 and 0.82 eV instead\nof 0.72 and 0.71 eV, respectively). Finally, jumps for non-compact c onﬁgurations (i.e., from√\n3 to 4a 0) are\nobtained with the KRA approach (Eq. 5). To this purpose, the solut e-vacancy binding energies are set to\ntheirab initio value up to the 10nn (2 .6a0), and to zero beyond this distance.\n3. DFT results\n3.1. Dumbbell properties in pure iron\nThe dumbbell properties in pure Fe are summarized in Table 1. At 0 K, t he/angb∇acketleft110/angb∇acket∇ightorientation, with a\nformationenergyof4.08eV,ismorefavorablethanthe /angb∇acketleft111/angb∇acket∇ightorientationbyamarginof0.75eV,inagreement\nwith previous studies performed with other DFT codes or functiona ls [65, 43]. The /angb∇acketleft110/angb∇acket∇ightformation entropy\n(0.05 k B) is much lower than that of the Ackland–Mendelev potential ([83, 84 ]), but closer to Lucas and\nSch¨ aublin’s DFT calculations based on the same functionals and a lowe r cut-oﬀ energy [80]. According to\nthe latter study, the large formation-entropy diﬀerence betwee n/angb∇acketleft110/angb∇acket∇ight(0.05 k B) and/angb∇acketleft111/angb∇acket∇ightdumbbells (4.17\nkB) sensibly reduces the formation free-energy gap at ﬁnite temper atures.\nThe migration barriers of the diﬀerent jump types are in good agree ment with previous calculations,\nconﬁrming that rotation-translation is the most probable mechanis m. For a /angb∇acketleft110/angb∇acket∇ight–/angb∇acketleft011/angb∇acket∇ightonsite rotation,\n6the saddle-point orientation is /angb∇acketleft1h1/angb∇acket∇ight(h≈2). In the RT jump, the hopping atom forms a symmetric\ndouble dumbbell with the initial and ﬁnal atom with orientations /angb∇acketleftkj1/angb∇acket∇ightand/angb∇acketleft1jk/angb∇acket∇ight(j= 3.3 andk= 5,\napproximately). The 2nn jump has a slightly lower barrier (0.46 eV) th an previous computations [65] and\nis the second most probable jump.\nFinally, the RT attempt frequency of 4.44 THz is close to the Debye fr equency in Fe (6 THz) [85]\nand lower than that of a vacancy jump (10.8 THz) [37]. The resulting dumbbell diﬀusion prefactor D0=\na2\n0exp(Sf/kB)ν0= 3.74·10−7m2/s is about an order of magnitude lower than the results of molecular -\ndynamics simulations based on the Ackland–Mendelev potential [86]. T he reason is twofold. Firstly, since\nthe latter simulations are performed at much higher temperatures , they account for anharmonic eﬀects.\nSecondly, even within the harmonic approximation the force ﬁeld bas ed on the Ackland–Mendelev potential\nprovides for higher vacancy-diﬀusion prefactors than DFT [87]. Bo th topics are beyond the scopes of this\npaper and will be the object of future studies.\nTable 1: Summary of the DFT dumbbell properties in pure Fe, re ferring to the /angbracketleft110/angbracketrightorientation unless otherwise speciﬁed.\nQuantity This work Previous works\nLattice parameter a0 2.831˚A\nDumbbell formation entropy Sf\n/angb∇acketleft110/angb∇acket∇ight0.050 k B 0.24 k B[80], 1.41[84], 2.84[84]\nDumbbell formation enthalpy Ef\n/angb∇acketleft110/angb∇acket∇ight4.082 eV 3.64[65], 3.94[88]\n/angb∇acketleft111/angb∇acket∇ight-dumbbell formation enthalpy Ef\n/angb∇acketleft111/angb∇acket∇ight4.832 eV 4.34[65], 4.66[88]\nRotation-translation energy barrier ω0 0.335 eV 0.32[89], 0.37[44], 0.34[30], 0.34[75], 0.35[50]\n2nn rotation-translation energy barrier ω0(2nn) 0.459 eV 0.50[65]\nTranslation energy barrier τ0 0.785 eV 0.78[75], 0.84[50], 0.80[44], 0.78[30]\nOnsite rotation energy barrier ωR0 0.611 eV 0.63[44]\nRotation-translation attempt frequency ν0 4.445 THz\nDumbbell diﬀusion prefactor D0 3.745·10−7m2/s 2 .268·10−6m2/s[86]\n3.2. Solute-dumbbell interactions and jump frequencies\nTable 2 reports the relevant binding energies and migration barriers of dumbbell-solute pairs, following\nthe nomenclature of Figs. 1 and 2. In addition to RT jumps, the table lists the results obtained by previous\nstudies [44, 30, 45] for 2nn jumps leading to a MD displacement ( ω1,2nn) or dissociation ( ωM2b). The\ncomplete set of DFT results, including those related to vacancy diﬀu sion, is available in the associated\nsupplementary database [90]. In a few cases, no barrier between t he initial and ﬁnal state is found, namely\nfor athermal transitions where the end-point energy diﬀerence is large. In FeP, since conﬁguration ’2b’ is\nunstable and relaxes spontaneously to a MD, the related barriers h ave not been computed.\nThe binding energies are in good agreement with previous DFT calculat ions that were limited to 1nn\nsites [43, 45, 44, 30]. They fade already at the 2nn shell, with a few ex ceptions. The most important one is\nthe ’5b’ conﬁguration in FeP, with a non negligible binding (+0.21 eV) due to the fact that the in-between\natom (along the /angb∇acketleft111/angb∇acket∇ightdirection) is pushed towards P and approaches a MD conformation. As the MD\ncan migrate in a fully 3D path without dissociating, solute diﬀusion can t ake place even in the absence of a\nstrong 1nn binding, as opposed to vacancy drag where 1nn binding is a necessary condition, and 2nn binding\na strong additional enhancement factor [37].\nClassic strain-reliefarguments suggest that oversized impurities ( Cu, Ni, Mn, Cr) should hold a repulsive\ninteraction in conﬁgurations ’M’ and ’1b’, and an attractive one in ’1a’ – the opposite for undersized\nimpurities (P, Si) [45]. This is indeed true for P, and to a certain extent for Cu, Ni, and Si, but does\nnot apply to Cr and Mn, since the corresponding ’M’ and ’1b’ conﬁgur ations are stable. Many studies have\nshown that solute-defect interaction energies cannot be explaine d based on size-related arguments only and\nare mainly determined by electronic interactions [91, 92, 45, 37]. In particular, the out-of-trend interactions\nof Cr and Mn with vacancies and dumbbells are related to electronic an d magnetic arguments. Both species\nhold antiferromagnetic (AFM) moments in a perfect Fe lattice, but t he presence of a dumbbell nearby\nweakens their AFM character and induces a ”rearrangement” of t he local moments (cf. Fig. 3), as it was\nalready mentioned in a previous FeCr study [75]. In FeMn, this is consis tent with the decrease of magnetic\nmoments occurring next to a foreign interstitial impurity (C, N, O), which was ascribed to an increase of\nthe local charge [74]. However, the magnetic moments in Fig. 3 for Cr and Mn follow a similar trend, so\nthe diﬀerences in solute-dumbbell binding-energies cannot be expla ined based on magnetism only.\n7Migration barriers and trapping conﬁgurations are further discus sed in Section 6.1 in relation to RR\nexperiments. In summary, the MD is stable for P, Mn, and Cr to a less er extent, and not favorable for\nSi, Ni, and Cu. The stability of the ﬁrst three MDs was already known f rom resistivity recovery (RR)\nexperiments in bcc iron [46, 47, 48] and previous theoretical stud ies [43, 30, 75, 45], and suggests that these\nsolutes are likely to diﬀuse via a dumbbell mechanism. However, the MD migration and dissociation rates\nmay play an important role. Furthermore, while Cu and Ni diﬀusion see ms unlikely, the Fe-Si dumbbell has\na nearly zero interaction, so it is not possible to tell a priori if solute transport takes place.\nsubstitutional-2.24\n-1.80\npure dumbbell-0.18-0.18\nmixed dumbbell+0.35-0.89+0.30-1.01\nconfiguration 1b-0.14+0.18+0.11-0.07\nconfiguration 1a-0.18-0.18-0.25-0.25Fe atom Mn/Cratom\n-1.32-1.30-1.94\n-2.79\nFigure 3: DFT local magnetic moments (in µB) in diﬀerent dumbbell-solute conﬁgurations, in the presen ce of Mn (red values)\nor Cr (blue values) solutes. The reference local magnetic mo ment in bulk Fe is +2 .23µB.\n3.3. Dumbbell attempt frequencies\nThe RT attempt frequencies of the mixed dumbbells are listed in Table 2 . The values vary in a small\nrange around the attempt frequency in pure Fe, with the exceptio n of Si. According to the Meyer-Neldel\ncompensation rule (MNR) [94], the prefactors νshould be correlatedto the correspondingmigration barriers\nEmigas:\nln/parenleftigν\nν∗/parenrightig\n=/parenleftbiggEmig\nε∗/parenrightbiggα\n(10)\nwhere the parameters ν∗,ε∗, andαdepend on the diﬀusion process, and provide information about the type\nof vibrational modes involved in the transition. The prefactors sho uld thus tend to be higher for high-barrier\njumps, and viceversa. This rule was successfully applied to vacancy -solute exchange in Fe [37] in spite of\nthe missing multi-phonon excitations in TST harmonic calculations. In F ig. 4, Eq. 10 is applied to the\nMD jumps, and a perfect correlation ( R2= 1) is found for P, Cr, Mn, and Si with an exponent α= 1.625\nand ﬁtting parameters ν∗= 2.830 THz and ε∗= 0.38 eV. Fe is also not far from the MNR ﬁt. Such a high\nαexponent might suggest that the migration is guided by optical mode s, as was the case for 5d elements\nexchanging with a vacancy in Fe. On the other hand, Ni and Cu do not fulﬁll this condition, so diﬀerent\nvibrational phenomena might be at play. Further investigations bey ond the harmonic approximation [95]\nare needed to clarify this anomaly.\nFigure 5 shows the phonon spectra for each dumbbell in the equilibriu m conﬁguration and at the saddle\npoint. The equilibrium spectrum of the /angb∇acketleft110/angb∇acket∇ightpure dumbbell is in good agreement with Lucas et al.[80].\nWith respect to the bulk spectrum, it contains a soft mode (s1) and some hard modes (h1, h2, h3, and\nH). According to Lucas, the hard modes are related to the stretc hing of the dumbbell bond (H) or the\nsurrounding ones (h1, h2, h3) (their higher frequency is due to bo nd compression). The soft mode is instead\nassociated to a translation of the dumbbell along the /angb∇acketleft110/angb∇acket∇ightdirection, possibly favoring the RT mechanism.\nExcept Fe-Cr and Fe-Mn, the mixed dumbbells have similar high phonon modes, reaching up to 500\nTHz (Fe-Ni). Likewise in pure Fe-Fe, these high modes are due to the stretching of the bonds. The Fe-\nCu and Fe-Ni dumbbells have very low frequency modes that compen sate the strongly negative binding\nenergies, whereas for all the other solutes, with positive binding en ergies, these instabilities are missing.\nThe pronounced soft modes in Fe-Cu and Fe-Ni induce a ﬂat energe tic landscape and can stabilize these\nmixed dumbbells at higher temperatures. At the saddle point, the Fe -Cu and Fe-Ni dumbbells havethe same\nlow-frequency modes and ﬂat energetic landscape. The migration p athway for Cu and Ni is thus probably\nrelated to low-frequency modes, which might explain the mismatch wit h the MNR rule for these elements.\n8Table 2: DFT solute-dumbbell binding energies (cf. Fig. 1) a nd migration barriers (cf. Fig. 2) in eV, compared with previ ous\ncalculations [44, 30, 93, 50] and resistivity-recovery exp eriments [89]. Energy barriers for the mixed-dumbbell tran slation,\nonsite rotation, and 2nn jump are also reported. The second c olumn refers to Barbe’s nomenclature [59].\nSolute-dumbbell binding energies (positive = attraction)\n(Fig. 1) P Mn Cr Si Ni Cu\nM +1.025 +0.555 +0.045 -0.002 -0.191 -0.380\n1a, 1b -0.331, +0.855 +0.107, +0.305 -0.065, +0.038 -0.173, +0.274 +0.016, +0.065 +0.188, +0.065\n2a, 2b -0.012, –a-0.006, +0.083 -0.082, -0.081 -0.064, +0.045 -0.047, +0.027 +0.099, +0.064\n3a, 3b, 3c -0.05, +0.04, -0.12 -0.06, +0.01, +0.01 -0.06, -0.03, -0.03 -0.01, -0.02, -0.07 -0.03, -0.00, -0.04 +0.02, +0.06, +0.10\n4a, 4b, 4c -0.01, +0.04, -0.02 -0.03, +0.02, -0.00 -0.04, -0.05, -0.03 -0.01, +0.08, +0.02 -0.02, +0.03, +0.01 +0.01, +0.07, +0.05\n5a, 5b -0.031, +0.212 -0.004, +0.013 -0.023, -0.033 -0.000, +0.038 +0.011, -0.024 +0.052, +0.017\nJump frequencies\n(Fig.2) (Barbe)[59] P Mn Cr Si Ni Cu\nMixed dumbbell (MD) jumps\nω1 ω1 0.217 0.316 0.241 0.568 0.464 0.364\nνω1 – 4.229THz 5.930THz 4.555THz 19.268 THz 2.828THz 2.685THz\nτ1 τ1ω1 0.493 0.648 0.407 0.575 0.634 0.368\nωR1 ωR1 0.327 0.314 – – – –\nMD association/dissociation\nωM1b,ω1bM ω2,ω3 0.230 0.060 0.448 0.198 0.365 0.358 0.069 0.345 0.083 0.339 0.000b0.445\nJumps from conﬁg. 1b\nω1b2b,ω2b1b ω4,ω5 – –c0.378 0.156 0.373 0.254 0.439 0.210 0.312 0.275 0.305 0.305\nω1b3b,ω3b1b ω4,ω5 0.810 0.000b0.521 0.227 0.386 0.318 0.475 0.184 0.396 0.327 0.304 0.302\nω1b5b,ω5b1b ω4,ω5 0.642 0.000b0.511 0.219 0.401 0.330 0.476 0.240 0.405 0.316 0.357 0.309\nJumps from conﬁg. 1a\nω1a2a,ω2a1a ω6,ω7 0.185 0.504 0.351 0.239 0.326 0.309 0.294 0.403 0.353 0.289 0.357 0.269\nω1a2b,ω2b1a ω6,ω7 – –c0.374 0.350 0.411 0.395 0.229 0.447 0.311 0.323 0.321 0.198\nω1a3b,ω3b1a ω6,ω7 0.124 0.499 0.367 0.272 0.321 0.356 0.278 0.434 0.361 0.341 0.388 0.263\nω1a3c,ω3c1a ω6,ω7 0.283 0.490 0.368 0.270 0.352 0.383 0.312 0.411 0.366 0.315 0.359 0.269\nJumps from 2nn\nω2a4c,ω4c2a – 0.438 0.432 0.377 0.382 0.309 0.362 0.388 0.470 0.303 0.358 0.352 0.302\nω2b4b,ω4b2b – – –c0.334 0.274 0.314 0.349 0.289 0.325 0.289 0.288 0.256 0.265\nω2b4c,ω4c2b – – –c0.376 0.293 0.314 0.366 0.362 0.335 0.336 0.316 0.302 0.286\nPrevious calculations\nω1 ω1 0.18[30]0.34[44]0.23[93]0.25[50]0.52[44]0.41[50]0.41[44]0.32[44]\nτ1 τ1ω1 0.24[30]0.66[44]0.42[93]0.48[50]0.37[44]0.69[50]0.46[44]0.26[44]\nωR1 ωR1 0.24[30]0.45[44]0.36[93]0.48[44]0.36[44]0.32[44]\nω1(2nn) – 0.18[30]0.53[44]0.43[93]0.67[44]0.84[44]0.59[44]\nωM1b,ω1bM ω2,ω3 1.26, 0.34[30]0.49, 0.22[44]0.33, 0.30[93]0.06, 0.35[44]0.09, 0.33[50]0.00, 0.50[44]\nωM2b,ω2bM – – – 0.80, 0.04[44]0.52, 0.36[93]0.26, 0.14[44]0.23, 0.45[44]0.05, 0.33[44]\nω1b2b,ω2b1b ω4,ω5 – – – – 0.35, 0.22[93]– – 0.31, 0.27[50]– –\nω1a2b,ω2b1a ω6,ω7 – – – – 0.36, 0.37[50]– – 0.36, 0.34[50]– –\naConﬁguration 2b in Fe(P) is unstable, as it relaxes into a mix ed dumbbell.\nbAthermal jump with no energy barrier.\ncNot computed because of the instability of the initial or ﬁna l conﬁguration.\n0.0 0.1 0.2 0.3 0.4\n(Emig)1.6251.01.52.02.53.0log(ν)\n  Fe\n PM   \nCr   Si\n  Ni  CuMNR fitti g:\nα = 1.625\nν* = 2.83 THz\nε* = 0.38 eV\nFigure 4: Fitting of the pure- and mixed-dumbbell migration energies and attempt frequencies according to the Meyer-Ne ldel\nrule (Eq. 10), achieved by excluding the Fe, Cu, and Ni data po ints from the ﬁtting.\n90 50 100 150 200 250 300 350 400 450 500Frequency [cm−1]\nbulk\nequilibrium saddle_pointx10 x10Fe-Fe\ns1 h1h2h3 H1\nx10 x10Fe-Mn\nx10 x10Fe-Cr\nx10 x10Fe-Si\nx10 x10Fe-P\nx10 x10Fe-Cu\n0 50 100 150 200 250 300 350 400 450 500x10 x10Fe-Ni\nFigure 5: DFT-computed phonon spectra for the Fe-Fe and Fe- Xdumbbells in the equilibrium conﬁguration (continuous lin es)\nand at the saddle point (dashed lines) of the rotation-trans lation jump. Areas marked with ’x10’ indicate that the spect rum\nmagnitude is ampliﬁed by 10 times.\n4. Solute transport\nThe DFT data presented in the previous section is used as input in the numerical part of KineCluE to\nobtain the vacancy- and dumbbell transport coeﬃcients for each Fe-Xbinary alloy, applying the kinetic\ncluster-expansion approach of Eq. 3. The complete set of results is available in the associated database [90].\nSolute-defect ﬂux coupling and transport is then analyzed based o n the following quantities:\n- Theﬂux-couplingratiosinvolvingtheoﬀ-diagonalcoeﬃcients: L(VB)\nVB/L(VB)\nBBandL(δB)\nδB/L(δB)\nδδ(δ= V,I),\nwhich determine the mutual directions of defect and solute ﬂuxes.\n- The solute tracer diﬀusion coeﬃcients Dδ\nB∗obtained from the solute diagonal coeﬃcients L(δB)\nBB; the\ncomparison between DV\nB∗andDI\nB∗reveals the preferential diﬀusion mechanism for each solute.\n- The ratios of partial-diﬀusion coeﬃcients Dδ\npd, describing the diﬀusion speed (and direction) of solute\natoms relative to matrix atoms, thus the RIS tendencies induced by each mechanism.\n10The mathematical framework used to derive these quantities from the KineCluE output is presented in\nAppendix B and summarized in Table B.6. Even though the total conce ntrations of defects CV,CIand\nsolutesCBare required as additional parameters, GδandDδ\nB∗are independent of CB, and so is Dδ\npdfor\nsuﬃciently low concentrations. The total defect concentrations appear in Dδ\nB∗but not in GδnorDδ\npd.\nTable 3: Summary of solute (B) transport by vacancies (V) and dumbbells (I), solute-defect cluster properties, and radi ation-\ninduced segregation (RIS) results. The diﬀusion coeﬃcient s, cluster mobility and lifetimes, and the solute-vacancy c orrelation\nfactors are shown in terms of activation energy and prefacto r, after Arrhenius interpolation between 300 and 1000 K. The solute\ndiﬀusion coeﬃcients are proportional to the corresponding defect concentration, which is here set to 1. In this dilute- limit\nmodel, the RIS switchover temperatures are nearly independ ent of solute concentration CB, except that of Cr that can vary\nby±30 K with respect to the shown value obtained at CB= 0.1 at.%.\nCr Cu Mn Ni P Si\nVacancy-assisted diﬀusion\nB diﬀusion coeﬀ.EV\nact[eV] 0.632 0.462 0.508 0.527 0.321 0.453\nDV\nB0[m2/s] 1 .31·10−61.14·10−61.34·10−65.15·10−77.66·10−78.70·10−7\nB correlation factor(300-1000 K) 0.012 – 0.37 0.0012 – 0.29 0.00034 – 0.19 0.89 – 0. 73 0.00010 – 0.17 0.00055 – 0.23\nEV\nFB[eV] 0.124 0.205 0.235 -0.008 0.276 0.223\nFV\nB01.68 3.30 3.00 0.665 4.35 3.14\nVB pair mobilityE(VB)\nmig[eV] 0.650 0.693 0.623 0.700 0.668 0.720\nM(VB)\n0[m2/s] 2 .12·10−86.58·10−84.32·10−82.89·10−86.77·10−87.64·10−8\nVB pair lifetimeE(VB)\ndiss[eV] 0.706 0.908 0.805 0.853 1.042 0.965\nτ(VB)\n0[s] 3 .24·10−141.01·10−141.65·10−141.08·10−146.83·10−156.45·10−15\nDumbbell-assisted diﬀusion\nB diﬀusion coeﬀ.EI\nact[eV] 0.219 0.744 -0.231 0.654 -0.803 0.569\nDI\nB0[m2/s] 5 .21·10−71.29·10−64.12·10−71.34·10−62.39·10−73.90·10−6\nB correlation factor (300-1000 K) 0.11 – 0.20 0.99 – 1.00 0.11 – 0.14 0.99 – 1.00 0.19 – 0.23 0.96 – 1.00\nIB pair mobilityE(IB)\nmig[eV] 0.217 0.881 0.319 0.674 0.213 0.824\nM(IB)\n0[m2/s] 1 .85·10−99.12·10−92.97·10−84.61·10−93.07·10−82.20·10−7\nIB pair lifetimeE(IB)\ndiss[eV] 0.348 0.448 0.884 0.355 1.290 0.579\nτ(IB)\n0[s] 6 .72·10−143.64·10−143.45·10−157.30·10−147.20·10−151.08·10−14\nDominant diﬀusion mechanism dumbbell vacancy dumbbell vacancy/ both dumbbell both\nVacancy drag max temperature 243 K 1085 K 997 K 1074 K 2060 K 1308 K\nRIS switchover temperature 543±30 K 1086 K enrichment 1089 K enrichment enrichment\nRIS tendency by dumbbells enrichment (negligible) enrichment depletion enrichment depl./enrich.\nRIS tendency by vacancies depletion enrichment enrichment enrichment enrichment en richment\nforT <1086 K for T <1084 K for T <2218 K for T <1606 K\n4.1. Properties of solute-defect pairs\nIn addition to transport coeﬃcients, KineCluE yields cluster proper ties that can be of direct use in\ncoarse-grained methods such as object Kinetic Monte Carlo (OKMC ) or cluster dynamics: the average\nmobility, lifetime, and mean free path (MFP) before dissociation. To t his purpose, the kinetic radius is set\nto a smaller value that corresponds to the cluster cut-oﬀ radius, i.e ., the mutual distance of atoms beyond\nwhich the cluster is considered as dissociated. The chosen cut-oﬀ r adius is√\n3a0(5nn distance). Following\nthe theoretical framework explained in a previous work [57], each clu ster transport coeﬃcient is expressed\nas a sum of two contributions related respectively to mobility and ass ociation-dissociation jumps. Mobility\nis obtained by zeroing all dissociative jump frequencies. The dissocia tion rate is equal to the sum of the\ndissociationfrequencies, multiplied by the total probabilityofbeing in a conﬁgurationfrom which the cluster\ncan dissociate. The lifetime is then the inverse of the dissociation rat e. Both quantities can be ﬁtted with\nan Arrhenius curve as:\nM(δB)=M(δB)\n0exp/parenleftigg\n−E(δB)\nmig\nkBT/parenrightigg\n, (11)\nand\nτ(δB)=τ(δB)\n0exp/parenleftigg\nE(δB)\ndiss\nkBT/parenrightigg\n, (12)\n11whereE(δB)\nmigandE(δB)\ndissare respectively the pair migration and dissociation energy, and M(δB)\n0,τ(δB)\n0the\nassociated prefactors. The MFP for 3-dimensional migration is the n given by ∆ R=√\n6Mτ.\n400400\n700700\n10001000\nTemperature  [K]10−310−210−1100101102103104105Vacancy-solute MFP [nm]Cr\nCu\nMnNi\nP\nSi\n400400\n700700\n10001000\nTemperature [K]10−310−210−1100101102103104105\nDumbbell -solute MFP [nm]\n(lattice parameter)\nFigure 6: Mean free paths (MFP) of vacancy- (left) and dumbbe ll-(right) solute pairs, computed in KineCluE with a cluste r\ncut-oﬀ radius of√\n3a0(5nn distance).\nThe results ofthe Arrheniusﬁtting in the range300-1000Karerep orted in Table3, and the MFPs in Fig.\n6 as functions of temperature. The results compare qualitatively w ell with previous AKMC calculations of\nvacancy-soluteproperties,performedwiththesameDFTmigratio nbarriersbutdiﬀerentattemptfrequencies\nand under slightly diﬀerent assumptions [96]. In this sense, KineCluE represents a computationally eﬃcient\nalternative method to AKMC to parameterize coarse-grained simula tions, which can be especially useful at\nlow temperatures and in the presence of trapping conﬁgurations.\nThe mobility term represents the kinetic properties of the cluster in dependently of its stability. As such,\nit stems from the combination of the solute-defect jump rate ( ω2for vacancies, ω1for dumbbells) with the\nother jump frequencies comprised in the kinetic radius. For vacanc y-solute pairs, many jump frequencies\ncontribute to the cluster mobility because the ω2jump alone is not suﬃcient to produce a net cluster\ndisplacement. For dumbbells, this is not true because the RT jump alo ne can yield an actual cluster jump.\nFor this reason, the dumbbell-solute mobilities for Cr, Mn, and P are e qual to the migration barriers of the\nω1jumps.\nThe dissociation energy is the sum of the cluster average binding ene rgy and the PD migration energy in\nthe periphery of the cluster (very close thus to ω0). Table 3 shows that Ediss> Emigfor all stable pairs, i.e.,\nall solute-vacancypairs and the mixed Fe-Cr, Fe-Mn, and Fe-P dum bbells. On the contrary, Emig> Edissfor\nthe remaining mixed dumbbells, which are thus more likely to dissociate t han migrate. This reﬂects directly\non their MFPs that are much shorter than the 1nn distance. The Fe -Cu, Fe-Ni, and Fe-Si dumbbells can\nbe therefore regarded as unstable and sessile. Instead, the MFP s of the Fe-P and Fe-Mn dumbbells are\nremarkably longer than the corresponding vacancy pairs thanks t o the combination of high dissociation and\nlow migration barriers, and can reach distances of the order of µm and higher. The vacancy-solute MFPs\nare as long as 1 nm or shorter between 500 and 700 K, except the va cancy-P pair whose MFP is roughly\none order of magnitude higher. Finally, the MFPs of pairs containing C r atoms are rather limited for both\nmechanisms.\n4.2. Solute diﬀusion coeﬃcients\nThe solute tracer diﬀusion coeﬃcients, normalized to the correspo nding defect concentration, are shown\non the left-hand side of Fig. 7. The ratio of the coeﬃcients stemming from each mechanism is reported on\nthe right-hand side of the same ﬁgure. In addition, Table 3 compiles t he solute diﬀusion activation energies\n12and prefactors, ﬁtted in the range 300-1000 K according to a clas sic Arrhenius ﬁt:\nDδ\nB=Dδ\nB0exp/parenleftbigg\n−Eδ\nact\nkBT/parenrightbigg\n. (13)\nThe same table shows the solute correlation factor Fδ\nB, which accounts for the solute slowdown (with respect\nto a random walk) due to exchanges with the defects yielding no net d isplacement. FV\nBfollows an Arrhenius\nbehavior analogous to Eq. 13, and is therefore reported with a ﬁtt ed activation energy and prefactor, while\nFI\nBis found to be rather independent of temperature.\nThe diﬀusion activation energy is roughly given by Eact=Emig−Ebind+EFB, whereEmigis the solute\nmigration energy ( ω2for vacancies [42] and ω1for dumbbells). An attractive (positive) solute-defect binding\n(Ebind) decreases the energy barrier, while correlations( EFB) increase it. In thermal-equilibrium conditions,\nthe defect formation energy should be added to the activation ene rgies shown in Table 3. This does not\napply in irradiation conditions where the defect population is ﬁxed.\nFigure 7 shows that P is the fastest diﬀuser for both mechanisms. F or vacancy diﬀusion, Cu, Mn, and Si\nhave similar diﬀusivities, while Ni and Cr are the slowest species. On the other hand, dumbbell coeﬃcients\nare very high for Mn, very low for Cu and Ni, and in an intermediate ran ge for Cr and Si. It is noteworthy\nthat the ﬁtted energy for DI\nPandDI\nMnis negative, which leads to the counter intuitive conclusion that, if th e\ndumbbell formation energy is neglected, P and Mn diﬀuse faster at lo w temperature. This stems from the\ncomparison between the MD migration and binding energies: since the dumbbell correlation factor is quite\nweak(about 0.1-0.2forP,Mn, Cr, andaroundunityforCu, Ni, Si), Eactisroughlyequalto Emig−Ebind, and\nfor Fe-Mn and Fe-P dumbbells Ebind> Emig. In other words, with decreasing temperatures, the mobility\nof single P and Mn atoms decreases, but the total diﬀusivity rises be cause of the increasing population of\nmixed dumbbells.\nOwing to weak correlations and lower migration barriers than vacanc ies, dumbbell diﬀusion is faster\nthan vacancies for those solutes whose MD is stable. On the right-h and side of Fig. 7, it can be seen\nthat P, Mn, and Cr diﬀuse preferentially by dumbbells, while Cu diﬀuses exclusively by vacancies. On the\nother hand, the two mechanisms are surprisingly in competition for N i and Si atoms, although the balance\ndepends as well on the ratio of PD concentrations CV/CI, which can span a large range of values depending\non irradiation and microstructure conditions. Despite the lack of MD stability, and the high ω1migration\nbarrier, Si can diﬀuse by both mechanisms. The compensation of th e highω1attempt frequency has thus\nan important impact on diﬀusion. For Ni, the MD has a strong repulsive interaction, and yet the vacancy\ndiﬀusion coeﬃcient is just 3 to 10 times larger than the dumbbell one b etween 450 and 700 K. Dumbbell\ndiﬀusion can therefore be as important as vacancy even if the MDs a re unstable, not mobile and have short\nMFPs. Even though the dissociation rate of a single MD is high, the pro bability of associating with another\nincoming dumbbell and performing some jumps is still comparable to th e energy barriersinvolvedin vacancy\ndiﬀusion. Ni and Si interstitial diﬀusion should thus not be neglected in the interpretation of experiments\nand in microstructure evolution models.\n4.3. Flux-coupling ratios\nThe ﬂux-coupling ratios are shown in Fig. 8. Above, the ”convention al” vacancy drag ratio GV=\nL(VB)\nVB/L(VB)\nBBindicates if solute and vacancies diﬀuse in the same ( GV>0) or opposite ( GV<0) direction.\nBelow, the oﬀ-diagonal coeﬃcient ratio with respect to the PD tran sport coeﬃcient gδ=L(δB)\nδB/L(δB)\nδδcan\nbe interpreted as the probability for a given diﬀusing PD to be coupled to a solute atom, or in other words,\nas the fraction of defect jumps causing a correlated solute jump. Note that the L(VB)\nVBcoeﬃcient can switch\nsign, but when negative, the gVratio remains close to zero. On the other hand, the L(IB)\nIBcoeﬃcient is\nalways positive because a ﬂux of dumbbells cannot induce a solute ﬂux in the opposite direction.\nAs discussed in a previous work [42], vacancy drag occurs systemat ically below a solute-dependent\ntemperature threshold directly related to the extent of solute-v acancy binding. The coupling progressively\nfades with increasing temperature. At the temperatures of inter est for ferritic steels, including RPV steels,\nall solutes here investigated except Cr are expected to diﬀuse by v acancy drag. The slightly diﬀerent set of\nDFT data and SCMF model here used with respect to the previous st udy did not lead to any substantial\ndiﬀerence. Concerning dumbbell diﬀusion, the gIratio shows that the defect is strongly correlated to P and\nMn atoms, which means that a (mixed) dumbbell jump is likely to produc e a solute displacement. P and\nMn transport by dumbbells is therefore expected to play an importa nt role.\n13400400\n700700\n10001000\nTemperature [K]10−1410−1210−1010−810−610−410−2100102DδB / Cδ [ 2/s]\ndumbbells\nvacancies\n400400\n700700\n10001000\nTemperature  [K]10−810−610−410−21001021041061081010\n( DIB / DVB ) * ( C V / C I )\nCr\nCu\nMnNi\nP\nSi\nFigure 7: (left) Solute tracer diﬀusion coeﬃcients for the d umbbell mechanism (continuous lines) and the vacancy mecha nism\n(dashed lines), divided by the corresponding defect concen tration. (right) Ratio between the diﬀusion coeﬃcients of t he two\nmechanisms, showing that P, Mn, Cr are transported preferen tially by dumbbells, and Cu by vacancies.\n4.4. Partial diﬀusion coeﬃcient ratios\nFigure 9 shows the partial diﬀusion coeﬃcient (PDC) ratios Dδ\npd:\nDδ\npd=(1−CB)\nCB·LδB\nLδA(14)\n(cf. Table B.6) for vacancy ( δ= V) and dumbbell ( δ= I) diﬀusion. With respect to the solute diﬀusion\ncoeﬃcients, the PDC ratios evaluate solute diﬀusion relative to host atoms, which determines the solute\nsegregation or depletion tendency at sinks. As opposed to the solu te-to-solvent diﬀusion coeﬃcient ratio\n(DB/DA) commonly used to discuss RIS, the PDC ratio includes ﬂux coupling an d is therefore more suitable\nin systems with correlated ﬂuxes (e.g., in the presence of vacancy d rag).\nThe sign of the diﬀerence ( Dvac\npd−Dsia\npd) determines the global RIS tendency (cf. Eq. 16). Therefore,\nthe diﬀusion mechanism predominantly driving the RIS behavior of eac h solute can be discussed based on\nthe magnitudes of the two ratios. The contribution of each mechan ism can be singled out by setting the\nPDC ratio of the other mechanism to 1. Consequently, enrichment b y vacancies takes place when Dvac\npd<1,\nand by dumbbells when Dsia\npd>1.Dsia\npdis always positive due to the fact that the dumbbell-solute coupling\ncannot be negative ( LIB>0), whereas Dvac\npdis negative in case of vacancy drag. In the latter case, the\ndiﬀerence ( Dvac\npd−Dsia\npd) is always negative, which means that an enrichment tendency due t o vacancy drag\ncannot be overturned by a dumbbell-induced depletion.\nThe vacancy PDC ratios are unchanged with respect to the previou s study [42]. Enrichment by vacancies\nis expected for all solutes but Cr due to vacancy drag up to high tem peratures. The enrichment regime\nextendsfurtherthandragbecause, when0 < Dvac\npd<1, solutesdiﬀusingbytheinverseKirkendallmechanism\nare slower than host atoms, especially for Ni where Dvac\npd<1 up to much higher temperatures than other\nsolutes (Mn, Cu, Si) with similar or stronger drag tendencies. On the other hand, Cr depletion occurs\nbecause vacancy drag is absent above 260 K and Cr diﬀusion is faste r than self diﬀusion.\nThe dumbbell PDC ratios show a clear demarcation between solutes u ndergoing enrichment (P, Mn, Cr)\nand depletion (Cu, Ni, and Si up to 1310 K), which only incidentally corr esponds to the categorization in\nstable and unstable mixed dumbbells. For P and Mn, dumbbell-induced e nrichment is expected due to the\nMD stability and mobility discussed in Section 4.1. The diﬀerence betwee n Cr and Si, on the other hand, is\nin apparent contradiction with the small diﬀerence between the res pective MD binding energies, but can be\nexplained by the fact that the ω1migration barrier is lower than ω0for Cr (0.24 vs 0.33 eV), while higher\nthanω0for Si (0.57 vs 0.33 eV).\n14500 1000 1500Temperature  [K]\n−1.5−1.0−0.50.00.51.0GV (drag ratio)\nCrCuMnNiPSidrag\nno drag\n5009001300\nTem erature [K]0.00.20.40.60.81.0gV\n5009001300\nTem erature [K]0.00.20.40.60.81.0\ngI\nFigure 8: Above, vacancy drag ratio GV=L(VB)\nVB/L(VB)\nBBas a function of temperature. Vacancy drag occurs for positi ve values\nofGV. Below, ratio of the oﬀ-diagonal coeﬃcients gV=L(VB)\nVB/L(VB)\nVVandgI=L(IB)\nIB/L(IB)\nIIwith respect to the vacancy (left)\nand dumbbell (right) transport coeﬃcient, respectively. T he latter can be interpreted as the probability for a defect j ump to\nproduce a solute displacement.\n5. Radiation-induced segregation\nThetransportcoeﬃcientsareherecombinedtoanalyzetheintrins icsoluteRIStendenciesstemmingfrom\nthe coupling with PDs. To this extent, the steady-state solution of Wiedersich’s model [97] is used. This\nwas later adapted by Nastar, Soisson, and Martinez [4, 53, 55] to deﬁne the partial and intrinsic diﬀusion\ncoeﬃcients in terms of transport coeﬃcients and alloy/PD driving fo rces. In this model, the relationship\nbetween vacancy and solute concentration gradients can be writt en as:\n∇CB=−α∇CV\nCV, (15)\nwhere factor αis:\nα=ℓAIℓAV\nℓAI/parenleftbig\nDvac\nB+KDsia\nB/parenrightbig\n+ℓBI/parenleftbig\nDvac\nA+KDsia\nA/parenrightbig\nCB/parenleftig\nDvac\npd−Dsia\npd/parenrightig\n1−CB\n, (16)\nwithℓiδ=Liδ/Cδ(δ= V,I). The deﬁnition of the normalized intrinsic diﬀusion coeﬃcients ( Dvac\nA,Dvac\nB,\nDsia\nA,Dsia\nB), more details about the underlying assumptions, and the analytica l derivations can be found in\nAppendix D, in the formula summary of Table B.6, and in the original wor ks [97, 4, 55]. Since the vacancy\ngradient is negative near the sink interface, the sign of α, controlled by the diﬀerence of PDC ratios (Eq.\n14), determines whether solute enrichment ( α >0) or depletion ( α <0) occurs.\nIn Eq. 16, αis in ﬁrst approximation proportional to CB. Factor K=CI/CVgroups all dependencies\nonCVandCI, and can be adapted to speciﬁc irradiation and microstructure con ditions. In order to analyze\nRIS regardless of any external parameter other than the intrins ic ﬂux-coupling tendencies, it is chosen to\n15300 600 900 1200 1500\n−10.0−7.5−5.0−2.50.02.55.0Vacancy PDC ra iosolute enrichment\nby vacancy dragsolute enrichmentsolute depletion\n300 600 900 1200 1500\nTempera ure [K]10−510−310−1101103105Dumbbell PDC ra ioCr Cu Mn Ni P Si\nsolute depletionsolute enrichment\nFigure 9: Ratios of partial diﬀusion coeﬃcients Dδ\npd(δ= V,I) (cf. Table B.6) for vacancy (top) and dumbbell (bottom)\ndiﬀusion. Solute enrichment by vacancies occurs via vacanc y drag ( DV\npd<0), or by the inverse Kirkendall mechanism when\nvacancies exchange preferentially with host atoms (0 < DV\npd<1). Enrichment by dumbbells occurs when DI\npd>1.\nsetCI/CV=DV(Fe)/DI(Fe), where DV(Fe) and DI(Fe) are the diﬀusion coeﬃcients of PDs in pure Fe\n(i.e., solute enhancement is neglected). This choice of CI/CVcorresponds to the steady-state solution of\nthe rate-theory equations describing the evolution of the PD popu lation, when PD recombination, as well\nas sink and source bias, are neglected [98].\nFigure 10 shows the α/CBratio obtained for CB= 0.1%; this ratio is roughly independent of CBexcept\na small eﬀect on the normalized intrinsic diﬀusion coeﬃcients. It is wor th reminding that non-linear solute\nconcentration eﬀects are not included in this dilute model, due to the lack of multiple-solute and multiple-\ndefect interactions. Relevant temperatures for vacancy drag a nd RIS, alongside qualitative indications of\nthe dominant mechanisms, are reported in Table 3.\nThe model predicts systematic enrichment of P, Mn, and Si, and enr ichment below ≈1085 K for Cu and\nNi, with mutually similar behaviors. In FeCr alloys, a switchoverbetwee n Cr enrichment and depletion takes\nplace at temperatures relevant for nuclear structural materials (543±30 K, depending on the chosen, yet\ndilute, Cr concentration). Based on the vacancy and dumbbells PDC ratios shown in Fig. 9, it is possible\nto identify the dominant diﬀusion mechanism determining the RIS beha vior of each solute.\n- InFeCu, thedepletioncontributionofdumbbellsisnegligible. Thetot alRISbehavioristhuscontrolled\nby vacancy drag: the maximum drag temperature coincides with the switchover between enrichment\nand depletion (1086 K).\n- The behavior of Ni is apparently identical to Cu, with a switchover a t the same temperature. Indeed,\nin the vacancy-drag regime Ni enrichment takes place. However, o utside the drag regime the vacancy\nand dumbbell PDC ratios are close to one another, and both compris ed between 0 and 1. So, even\nthough vacancy diﬀusion would still yield an enrichment tendency, de pletion eventually occurs because\nof the dumbbell contribution. The RIS tendency is therefore cont rolled by vacancies below 1089 K,\nand by dumbbells above.\n16300 600 900 1200 1500\nTemperature  [K]−0.75−0.50−0.250.000.250.500.751.00RIS facto  α/cB\nC \nCu\nMnNi\nP\nSisolute enrichment\nsolute depletion\nFigure 10: Radiation-induced segregation (RIS) tendencie s (factor αin Eq. 16 divided by the solute concentration CB) obtained\nwith the KineCluE transport coeﬃcients at CB= 0.1%. Solute enrichment occurs for positive αvalues.\n- Si follows a similar reasoning as Ni, but with a diﬀerent conclusion. Vac ancy drag determines Si\nenrichment up to the temperature at which it disappears (1308 K). Above that, dumbbell diﬀusion\nprovides an enrichment tendency that overturns vacancy-drive n depletion, keeping Si in an enrichment\nregime in spite of the absence of vacancy drag.\n- In FeP and FeMn, both mechanisms yield enrichment, but the dumbb ell contribution is dominant by\nseveral orders of magnitude. Hence, P and Mn enrichment occurs almost exclusively by dumbbell\ntransport.\n- For Cr, the two mechanisms give rise to opposite tendencies with sim ilar magnitudes, as pointed out in\nprevious studies [51, 52]. The bottom panel of Fig. 11 (thick red cur ve) shows that the total PDC ratio\ndetermining the sign of αremains very close zero in a wide range above the switchover temper ature\n(TRIS). Any minor inﬂuence from various parameters or external condit ions can thus lead to large\nchanges in TRIS. This is discussed in more detail in Section 6.3.\nQuite surprisingly, the RIS magnitude is smaller in a dumbbell-driven reg ime than in a vacancy-driven\none. This is the case for P and Mn, as well as for Si in the temperatur e range where dumbbells are dominant.\nThis might be explained in terms of back diﬀusion, i.e., defect ﬂuxes att empting to restore a homogeneous\nconcentration gradient near sinks. The eﬃciency of solute-dumbb ell transport might be favoring RIS and\ncontrasting it at the same time by facilitating back diﬀusion, thus red ucing the total eﬀect.\n6. Discussion\n6.1. Solute-dumbbell transport in view of previous ab initi o and experimental studies\nThe obtained dumbbell transportpropertiesarediscussed herein termsofprevious ab initio calculations,\nevidence from RR experiments, and common empirical assumptions.\nIn the FeP system, P migrates in a continuous oscillation between the MD and the ’1b’ conﬁguration,\nwith a very small dissociation probability. The combination of strong b inding and high mobility produces\na very long-ranged migration that goes far beyond common grain siz es (cf. Fig. 6). These results conﬁrm\nprevious RR experiments according to which the Fe-P dumbbell has a higher mobility than self-interstitials,\nandslightly higherthan Fe-Crdumbbells [48]. Thisseems to excludethe presenceoftrappingconﬁgurations.\nHowever, previous ab initio works [43, 30] have shown that the addition of a second P atom has a strong\ntrapping eﬀect on the Fe-P dumbbell. Therefore, P transport migh t become less eﬀective with growing\nsolute concentration. On the other hand, this may be counterbala nced by the fact that P migration is likely\nto involve foreign interstitial sites. According to previous siestacalculations [30], the P formation energy\n17in octahedral sites is lower than in the MD conﬁguration, and a MD jum p passing through the octahedral\nconﬁguration has actually a lower barrier (0.17 eV) than the RT mech anism. In addition, the barrier for a\n2nn MD jump was found to be as low (0.18 eV). In this work, the diﬀere nce between MD and octahedral\nstability (+0.25 eV) is actually larger than previous calculations with siesta(-0.08 eV [30]) and vasp\nultrasoft pseudopotentials (+0.05 eV [43]), so the octahedral diﬀu sion pathway might be less important.\nNevertheless, the matter should be further investigated.\nThe Fe-Mn dumbbell is very stable and has a mobility close to that of se lf-interstitials, as was suggested\nby RR experiments [47] and later proven by AKMC simulations of isochr onal annealing [99]. The resulting\nMFPs are not as long as for Fe-P, but still comparable to common gra in sizes. In a regime of defect\nabsorption at sinks, hence, P and Mn can be expected to reach gra in boundaries rather easily. For this\nreason, the frequent observation of P/Mn grain-boundary segr egation is unsurprising. In addition, previous\nab initio calculations have found a very low MD association barrier via a 2nn jum p (0.04 eV) [44]. If\nconﬁrmed, this would entail that the solute-dumbbell pair dissociat ion is practically impossible, and may\nincrease correlations reducing the diﬀusivity of Mn solute atoms.\nThe Fe-Cr mixed-dumbbell interaction is attractive but weak, similar ly to the vacancy-Cr interaction\n[42]. Moreover, the ’M’ and ’1b’ binding energies are very close to one another. According to earlier DFT\ncalculations based on the Perdew and Wang (PW91) functional, the M D should be slightly more stable [45].\nPBE is in agreement with PW91 only when the accuracy is increased to 4 ×4×4 k-points and 400 eV.\nThe obtained migration barriers are in good agreement with a previou s work [50], except a mismatch in the\nMD association/dissociation rate. The high MD mobility is in accordance with RR experiments [46] and\nprevious molecular-dynamics simulations [100]. However, the MFP of t he Fe-Cr dumbbell is rather short\nwith respect to Fe-P and Fe-Mn. This means that Cr migration occur s by exchange with several Fe-Fe\ndumbbells that in turn form a mixed dumbbell, displace the Cr solute by a few˚ angstr¨ om, and then quickly\ndissociate. It should also be mentioned that a 2nn association jump is possible with a barrier of 0.36 eV, as\npointed out by Olsson [93].\nSi and Ni present the most surprising behavior. Whereas RR exper iments were interpreted by assuming\nthe formation and migration of the Fe-Si and Fe-Ni mixed dumbbells [ 101, 48], there is no unanimous\nconsensus on the eﬀective possibility of Si/Ni migration by a dumbbell mechanism. A previous ab initio\nstudy excluded this hypothesis for both solutes based on the unfa vorable MD repulsive energy and the high\ndissociation probability [44]. Indeed, it is conﬁrmed here that the Fe- Si and Fe-Ni are not stable, have\na strong tendency to dissociate (low ωM1bbarrier), and their MFP before dissociation is essentially null.\nHowever, since for both Si and Ni the vacancy- and dumbbell-relat ed diﬀusion coeﬃcients (divided by defect\nconcentration)havecomparablemagnitudes(cf. Fig. 7), the dom inantmechanismis determinedbythe ratio\nCV/CI. When the latter is larger than 1, vacancy diﬀusion dominates: for in stance, if CV/CI≈DI/DV,\nthis ratio is roughly equal to 103at the RPV operation temperature of roughly 300 °C, which indicates that\nunder these assumptions Si and Ni diﬀuse predominantly by vacanc ies. However, this can change depending\non the speciﬁc microstructure and irradiation conditions. It is also w orth mentioning that the same solutes\nwereobservedto trap self-interstitial atoms in RR experiments [1 01, 47]. In FeSi, this can be clearlyascribed\nto the ’1b’ conﬁguration (strong binding and high migration barriers ). In FeNi, it could be due to the ’1b’\nconﬁguration as well, although with a weaker binding strength.\nFinally, the strongly repulsive interaction of the FeCu mixed dumbbell, combined with its spontaneous\ndissociation (zero ωM1bbarrier), conﬁrms the common assumption that Cu diﬀusivity by dum bbells is neg-\nligible. However, Cu can aﬀect the migration of self interstitials by tra pping them in the ’1a’ conﬁguration:\nthe migration barriers towards ’1a’ are sensibly lower than the reve rse ones, and also lower than ω0. This\ntrapping conﬁguration was shown in isochronal annealing simulations [99] to be the cause for the observed\ndisappearance of the single-SIA migration peak in RR experiments [4 7]. The same experiment mentions as\nwell further trapping caused by a second Cu atom, so the slowdown eﬀect on dumbbells might grow with\nCu concentration.\n6.2. Prediction of dumbbell transport and RIS for other Fe-b ased alloys\nEarlier systematic investigations allow for a qualitative prediction of s olute transport and RIS for the\nothertransition-metalsolutesin Fe. Thevacancy-transportpr opertieshavebeen accuratelydetermined with\nan analogous ab initio-SCMF model [37]. After removing the vacancy formation energy, t he solute diﬀusion\nactivation energies range between 0.2 and 0.9 eV, with a bell-shaped p roﬁle across the 4d and 5d elements\nreaching a maximum in the mid-row elements (Ru and Os). Vacancy dra g and solute enrichment extends\nto high temperatures for the early and late elements. For the mid-r ow elements (and Ti), the maximum\ndrag temperature is in the range 500-700 K, and depletion occurs a bove this range. A few exceptions were\n18mentioned: depletion due to the lack of vacancy drag takes place fo r V and Cr, while for a set of ”slow\ndiﬀusers” (Co, Re, Os, Ir) enrichment always occurs because of t he lower diﬀusivity than host atoms.\nThere has been no systematic analysis of the same kind for dumbbell diﬀusion, but a systematic cal-\nculation of MD binding energies is available [45]. All MDs but Fe-Mn and Fe- Cr have a strong repulsive\ninteraction ( <−0.8 eV), except Co and V that have a milder repulsion ( −0.25 and−0.55 eV, respectively).\nBased on this evidence, it would be tempting to assume that none of t hese solute diﬀuse by dumbbells.\nHowever, this must be carefully assessed by comparing the activat ion energies. For dumbbell diﬀusion, it\nholdsEI\nact≈EI\nmig−EI\nbind. Hence, neglecting the prefactors, solute diﬀusion by dumbbells is q uicker than\nby vacancies if EI\nact< EV\nact, i.e., ifEI\nmig< EI\nbind+EV\nact. Since the MD migration energies EI\nmigcannot be\nnegative, combining the vacancy activation energies with the MD bind ing energies leaves only two candi-\ndates: vanadium, if the MD migration energy is lower than 0.19 eV, and cobalt, if lower than 0.58 eV. The\nlatter case is deﬁnitely possible, given that for instance the Fe-Cu M D has a migration barrier of 0.36 eV in\nspite of a strong repulsive interaction (-0.38 eV).\nBy a similar logic, it is possible to make predictions on the dumbbell PDC ra tios. In this case, if ﬂux\ncoupling is negligible (which is not necessarily the case), the PDC ratio c an be evaluated by comparing the\nMD activation energy with that of the Fe-Fe dumbbell: namely, DI\npd>1 ifEI\nmig< EI\nbind+E0\nmig, where\nE0\nmigis the Fe-Fe migration energy (0.33 eV). The only solute that might fu lﬁll this condition is cobalt,\nif the Fe-Co migration energy were smaller than 0.08 eV, which seems u nlikely. Thus, it is possible to\nconclude that depletion by dumbbells is expected for all solutes ( DI\npd<1). It is therefore expected to have\nvacancy-induced enrichment in the vacancy-drag regime, and vac ancy-induced depletion in the absence of\ndragwhen DV\npd>1. In the interval0 < DV\npd<1, theremight be acompetition between the twomechanisms,\ndepending on the magnitude of DI\npd. For slow diﬀusers, and especially Co, this prediction must be suppor ted\nby further ab initio calculations, including at least the MD dumbbell migration energy.\n6.3. Comparison with previous transport coeﬃcient calcula tions\nFull sets of Onsager coeﬃcients have been previously computed on ly for the FeCr and FeNi systems.\nChodhury et al.[50] used the Le Claire/Serruys model [102, 103] for vacancies an d Barbe’s SCMF model\n[59] for dumbbells to compute the full matrix, but then based the RI S discussion on the ratio of diﬀusion\ncoeﬃcients DB/DA, without taking ﬂux coupling into account. The PDC ratios in Fig. 9 com pare qualita-\ntively well with Choudhury’s results concerning Cr-dumbbell, Ni-dumb bell, and Cr-vacancy diﬀusion, since\nﬂux coupling in these systems is negligible. On the contrary, for Ni-va cancy diﬀusion the ratio is smaller\nthan 1, while according to Choudhury’s model it is consistently larger than 1 (i.e., depletion) due to the\nlack of ﬂux coupling.\nA series of combined AKMC-phase ﬁeld studies, based on an ab initio -ﬁtted pair interaction model,\nhas achieved a more accurate analysis of Cr segregation in FeCr, inc luding the eﬀects of ﬂux coupling and\nnon-dilute concentrations [52, 53, 6]. The AKMC-computed PDC ra tios are compared to the results of\nthis work in Fig. 11, where the bottom panel reports the diﬀerence between vacancy and dumbbell PDC\nratios. Our dilute model does not provide the variation with Cr conce ntration, but in the most dilute case\n(0.25% Cr) it compares very well with Senninger’s results [52] both in t erms of switchover temperature TRIS\nand magnitude. This clearly proves that the mismatch in TRISbetween KineCluE and the experimental\nobservation is at least partially due to Cr concentration eﬀects. It is also possible to observe that, according\nto Senninger’s work, TRIShas a maximum somewhere between 5% and 10% Cr, before decreasin g again,\nin striking agreement with Wharry’s experimental observations [54]. This occurs because, while vacancy\ndepletion increases monotonically, dumbbell enrichment reaches a m aximum at 10% Cr and then decreases.\nThemobilityoftheFe-Crdumbbellseemsthustobereducedbythep resenceofmoreCratoms,inaccordance\nwith Terentyev’s molecular-dynamics simulations [100].\nVacancy depletion and dumbbell enrichment in FeCr have very similar m agnitudes. This makes the RIS\nbehavior very sensitive to any parameter that can perturb this ba lance, and is most likely the reason why\nconﬂicting conclusions (enrichment vs depletion) can be found in the literature for ferritic alloys [104, 105,\n106]. Besides Cr concentrations, other parameterssuch as sink d ensity, sink and source bias, and strains may\naﬀect RIS. In particular, strain ﬁelds induced by defect sinks are s urely playing a major role. Thuinet et al.\n[6] performed a ﬁrst strain-dependent investigation of RIS in FeCr alloys with phase-ﬁeld simulations, and\nconcluded that vacancy-attracting compressionzones should be more prone to Cr depletion (the opposite for\ntraction zones). This is suﬃcient to overturn the results of the st rain-free model. However, these conclusions\nwere drawn neglecting the eﬀect of strains on ﬂux coupling. Since st rain-dependent transport coeﬃcients\ncan be computed in KineCluE, this topic will be investigated in future st udies.\n19400 600 800 1000 1200 1400\nTemperature [K]1357Vacancy\nPDC ratio0.25% Cr0.25% Cr1% Cr5% Cr10% Cr15% Cr\ndeplet %n\nTemperature [K]1357Dumbbell\nPDC rat %Th s w%rk\nSenn nger\n0.25% Cr0.25% Cr\n1% Cr5% Cr\n10% Cr\n15% Cr\nenr chment\n400 600 800 1000 1200 1400\nTemperature [K]−2\n0\n2T%tal\nPDC rat %0.25% Cr\n0.25% Cr1% Cr5% Cr\n10% Cr\n15% Crenr chment\ndeplet %ne−per mental\nsw tch%ver\nFe-8.9\\%Cr\nFigure 11: Comparison between the partial diﬀusion coeﬃcie nt (PDC) ratios obtained in this work in the dilute limit, and the\nCr concentration-dependent ones computed by Senninger et al.[52] with an AKMC model. From top to bottom: PDC ratios\nfor vacancy diﬀusion ( DV\npd), for dumbbell diﬀusion ( DI\npd), and diﬀerence ( DV\npd−DI\npd) determining the sign of the global RIS\ntendency (enrichment when negative). The experimental swi tchover range refers to the RIS measurements by Wharry et al.in\na T91 alloy [54].\n6.4. Experimental diﬀusion coeﬃcients and ﬁnite-temperat ure eﬀects\nExperimental data for dumbbell-assisted diﬀusion are very diﬃcult t o obtain and are therefore not\navailable for comparison. For vacancy migration, the solute diﬀusion coeﬃcients computed in the SCMF\nframework were already successfully compared to experimental m easurements in previous publications [42,\n37]. A very close match in activation energies and only slight diﬀerence s in the prefactors (one order of\nmagnitude or less) were found. In these earlier works, a magnetic m odel [107] accounted for the eﬀect of the\nmagnetic transition, in the aim of a meaningful comparison with exper iments in the range around the Curie\ntemperature. In that model, it was assumed that the variation of t he magnetic enthalpy with temperature\nhad an absolute eﬀect on the self-diﬀusion activation energy, but n o relative eﬀects on solute behavior. This\nassumption proved valid for all solutes except Mn, which is known to h old odd magnetic properties in Fe\nthat are currently under investigation [74]. For these reasons, no magnetic model is implemented in this\nwork. Conclusionsonsolute-transportpropertiesandRISarest illvalid,especiallyatlowtemperatureswhere\nﬁnite-temperature eﬀects due to magnetism are negligible [107]. Vibr ational-entropyharmonic contributions\nto the defect formation and solute migration energies are included. However, this is not the case for the\nsolute-defect binding entropy, which can lead to variations of the r elative stability of solute-defect pairs, as\nwas shown for instance by Murali et al.[108].\n6.5. Comparison with experimental RIS observations\nThe RIS tendencies presented in Fig. 10 are consistent with many ex perimental observations. Firstly,\nas pointed out by Rehn and Okamoto, no case of depletion of unders ized solute species in dilute alloys has\never been reported [109]. According to Ardell [5], this holds true st ill today. The consistent enrichment of\nthe undersized elements (P and Si) shown in Fig. 10 conﬁrm these obs ervations.\nFurthermore, our results match well those of Wharry’s RIS study in several FeCr alloys between 300 and\n700°C [54] . Wharry found a systematic enrichment of Si, Cu, and Ni at gr ain boundaries. In addition, Cr\nwas found to switch from (weak) enrichment to depletion across 60 0-700 °C. In view of this diﬀerence, they\nsuggested that the mechanisms driving the RIS behavior of these s olutes might be diﬀerent. Indeed, the\nresults presented here conﬁrm all of these ﬁndings, and prove th at Si/Cu/Ni enrichment is due to vacancy\n20drag, while Cr enrichment to dumbbell transport. The mismatch in sw itchover temperature (543 K vs ≈\n900 K) can be ascribed to its dependence on solute concentration, as shown in Fig. 11 and discussed in\nSection 6.3.\nIn support to our ﬁndings, P and Mn have been repeatedly observe d to segregate on grain boundaries,\ndislocation lines, and loops [13, 12, 7]. Furthermore, all solutes con sidered here have been observed to form\nvacancy-solute clusters in model alloys and RPV steels [110, 111, 11 2], which is consistent with a positive\ncoupling between solutes and point defects. In RPV steels, the the rmal stability of Mn-Ni-Si clusters\n[22, 20, 23, 24] points towards the existence of a thermodynamic d riving force for precipitation, although\nthe actual solubility limit in these complex multi-component alloys is yet t o be precisely determined. In\nthis context, positive solute RIS tendencies should enhance the he terogeneous nucleation of Mn-Ni-Si-rich\nsecondary phases at PD sinks, such as pinned SIA clusters [17, 28 ]. Our results show that Mn, Ni, and Si\ncan indeed be dragged to sinks by point defects, and thus conﬁrm t he important role of kinetic coupling in\nthe precipitation process.\nSimilar phenomena have been observed in other alloys. For instance, Ni enrichment at dislocation loops\nhas been observed to occur in FeNi alloys, as Ni atoms bind with loops and then act as sinks for mobile\ndefect-Ni complexes. This mechanism could be the precursor of th e observed secondary-phase precipitation\n[16]. Our results suggest that such mobile defect-Ni complexes are likely to be vacancy-Ni pairs. In Cr-rich\nferritic alloys, Cr-Si-Ni-P clusters are formed due to P atoms diﬀusin g to dislocations and catalyzing the\nformation and growth of clusters around them [7, 8]. Fig. 7 shows th at P is indeed the fastest diﬀuser\nthanks to the Fe-P dumbbell, which might thus be responsible for the onset of cluster precipitation in these\nalloys.\nRinghals Wharry Pareige Gomez\nFerrer10−510−410−310−2RIS factor α\nCr\nCu\nMnNi\nP\nSi0.07%8.9% 9.16%14.8% 0.07%0.15%1.48%\n0.45%1.50%\n0.20%\n0.057%0.098%\n0.016% 0.016% 0.013%0.065%0.42%0.55% 0.55%0.42%\nFigure 12: KineCluE radiation-induced segregation (RIS) f actorsα(cf. Eq. 16) marking solute enrichment in several multi-\ncomponent alloys, namely an RPV steel (Ringhals) [113], and three FeCr alloys: T91 (Wharry) [54], Fe-9%Cr (Pareige) [7] , and\nFe-15%Cr (Gomez-Ferrer) [8]. Each data point reports also t he nominal concentration (in at.%) of the corresponding ele ment.\nFigure12reportsamorequantitativeevaluationofsolutesegrega tioninthefollowingalloys: theRinghals\nhigh-Ni, low-Cu RPV steel [113] and three FeCr alloys, including Wharr y’s T91 [54] and the two model\nalloys where P-catalyzed precipitation took place [7, 8]. The solute no minal concentrations are reported\nnext to each data point. The calculation of the transport coeﬃcien ts has been modiﬁed in order to add\na ”multicomponent eﬀect” to the total partition function (cf. App endix C for the mathematical details).\nThis is to take into account the fact that, in a multicomponent syste m with ﬁxed PD concentration, the\nmobility of a given solute might be reduced if the majority of PDs binds p referentially to other solutes. For\ninstance, in the Ringhals RPV steel the vast majority of SIAs is boun d to P atoms, and just a small fraction\n(≈0.6% of the total dumbbell population at 300 °C according to our calculations) is bound to Mn atoms,\neven though the nominal P concentration is much lower (0.016% vs 1.4 8%) and the formation of Fe-Mn\ndumbbells is favorable. Consequently, Mn diﬀusivity in the multicompon ent alloy is lower than in a binary\nFeMn alloy, which might explain why Ni segregation (predominantly driv en by vacancies) is higher than\nMn segregation even though the starting Mn and Ni concentration s are similar. However, the predicted\nsegregation magnitudes are to some extent in disagreement with th e experimental compositions of Mn-Ni-Si\nclusters, showing a majority of Mn and Ni atoms, with Ni >Mn, a lower Si content, and traces of Cu and P\n21[13]. Such mismatch in cluster composition likely supports the argumen t that RIS is the precursor and an\nimportant enhancement factor of the formation of some stable ph ases, but not the only cause. Concerning\nthe FeCr alloys, the segregation magnitudes compare well with Whar ry’s study [54], in which Si, Ni and Cu\nsegregation (in decreasing order of magnitude) was observed, bu t less well in the other cases: our model\nyields a stronger Si segregation than Ni, in place of the experimenta lly observed similar magnitudes.\nLet us summarize the limitations of our RIS model. The latter provides the kinetic contribution to segre-\ngation due to solute-PD coupling in a dilute limit approach, i.e., it neglects non-linear solute-concentration\neﬀects and the possible coupling with small PD clusters, which might be relevant when solutes and PDs\naccumulate locally next to sinks. Under such conditions, the solute d riving forces, not considered in this\nmodel, might reduce solute back ﬂuxes and promote segregation du e to solute stabilization. Furthermore,\nany eﬀect of sink absorption bias, local sink-induced strains, and b inding free-energy variations due to ﬁnite\ntemperatures is neglected, and might aﬀect the predicted temper ature thresholds. More importantly, the\nthermodynamic interaction between solutes and sinks is currently n ot included in the model, and might add\na localized equilibrium segregation proﬁle in addition or in contrast to RI S. Typical examples of such inter-\nplay are the so-called ”W-shaped” proﬁles arising when RIS promote s solute depletion at sinks characterized\nby attractive thermodynamic interactions with solutes [4].\nIntroducing chemical potential gradients, in combination with spec iﬁc irradiation and microstructure\nconditions, would allow for a more signiﬁcant comparison to experimen ts, and for the prediction of the\nbell-shaped segregation proﬁles found, for instance, by Wharry et al.[54]. This shape arises because,\nat low temperature, segregation is limited by slow diﬀusivities, wherea s at high temperature, the total\nPD driving force gets progressively weaker due to the increasing eq uilibrium concentrations. It must be\nmentioned, however, that according to Mart` ınez et al.[55], for low equilibrium PD concentrations the total\namount of segregated solutes depends only on the RIS factor αand the sink density, even when the PD\ndriving forces are taken into account. For this reason, the compa rison shown in Fig. 10 is still relevant,\nand the fundamental solute transport behaviors here uncovere d can be used as a guide for experimental\ninterpretations and microstructure models.\n7. Conclusions\nThis work has provided an in-depth investigation of solute diﬀusion by interstitial-type defects in binary\ndilute ferritic alloys including Cr, Cu, Mn, Ni, P, and Si solute atoms. In combination with previous\nknowledge on vacancy-assisted diﬀusion, a general overview of th e intrinsic radiation-induced segregation\n(RIS) behavior of these solutes has been achieved. The most relev ant ﬁndings can be summarized as follows.\n1. P, Mn, and Cr form stable mixed dumbbells, while the Fe-Si dumbbell is neither binding nor repulsive.\nThe Fe-Cu and Fe-Ni dumbbells are repulsive at 0 K, but due to soft v ibrational modes, they might be\nmore stable at higher temperatures. Si, Cu, and Ni solutes can tra p Fe-Fe dumbbells in a ﬁrst-nearest\nneighbor position.\n2. Vacancy-solute pairs are stable and mobile. At 300 °C, their mean free paths can reach up to 1 nm,\nand up to 10 nm for P. The Fe-Cu, Fe-Ni, and Fe-Si dumbbells have a n egligible mobility and a high\ndissociation rate, while the Fe-Cr, Fe-P, and Fe-Mn dumbbells are mo bile. The mean free path of a\nsingle Fe-P and Fe-Mn dumbbell can exceed 1 µm.\n3. Even if the mixed dumbbell is unstable, dumbbell diﬀusion of solutes can be in competition with\nvacancy diﬀusion. Ni diﬀusivity by vacancies (normalized to the vaca ncy concentration) is ﬁve times\nfaster than dumbbells at 300 °C and only twice as fast at 630 °C. For Si atoms, vacancy diﬀusion is\nslightly predominant below 300 °C, but perfectly equivalent to dumbbell diﬀusion at RPV tempera-\ntures. Dumbbells provide the predominant contribution to P, Mn, an d Cr diﬀusivity, while vacancies\ndominate that of Cu. P is the fastest diﬀusing species by both mecha nisms. The predominance of\na mechanism over the other depends on the ratio of point defect co ncentrations CV/CI. Among all\nother transition-metal (TM) solutes, Co is the only one for which th e dumbbell mechanism might be\ncomparable with the vacancy one.\n4. Systematic enrichment is predicted for P and Mn due to dumbbell d iﬀusion, and for Si due to vacancy\ndrag. Cu and Ni enrichment by vacancy drag takes place below 1085 K. Right above this limit,\ndepletion should occur. Enrichment by vacancy drag is dominant for all other TM solutes when\npresent [37]; in the opposite case, dumbbell depletion can be in compe tition with inverse-Kirkendall\ndiﬀusion.\n225. The RIS behavior of Cr is the outcome of a ﬁne balance between du mbbell enrichment and vacancy\ndepletion, and this leads to a switch from enrichment to depletion at TRIS= 540 K. The match\nis so close that small variations of any parameter (e.g., Cr concentr ation, strains, sink biases) can\nlead to large changes of TRIS. This is in agreement with experimental RIS observation and previou s\ncalculations, which pointed out that TRISgrows with Cr concentration up to about 10% Cr, where\nTRISis comprised between 600 and 700 °C.\n6. These ﬁndings are consistent with the interpretation of the kno wn RIS and solute clustering phenom-\nena. They conﬁrm that a drag mechanism by point defects is active f or all solutes, and can provide an\nimportant kinetic mechanism that enhances the formation of solute clusters and precipitates in many\ntypes of Fe alloys, including RPV alloys.\nThis study was the ﬁrst application of a novel, multi-scale framework that combines ab initio calcula-\ntions (or any other energy model) with the self-consistent mean-ﬁ eld model and a computationally eﬃcient\nnumerical tool (KineCluE) to produce the transport coeﬃcients o f an alloy to a high degree of accuracy.\nThis can be used to analyze solute transport and diﬀusion propertie s, ﬂux coupling with point defects, and\nRIS behaviors for a wide range of periodic crystal structures and diﬀusion mechanisms. Even though the\ncurrent study is limited to kinetic segregation in the dilute-limit approa ch, it can be extended by adding\nthermodynamic interactions with sinks and the energetics of cluste rs larger than single defect-solute pairs,\nwith the aid of an appropriate energy model. It also allows for strain- dependent calculations of transport\ncoeﬃcients, which will be the object of future studies. The presen ted framework is a valuable asset for gain-\ning fundamental knowledge that can be useful to interpret exper imental observation and devise physically\naccurate microstructure evolution models.\nAcknowledgments\nThis work was ﬁnancially supported by Vattenfall AB, G¨ oran Gusta fsson Stiftelse, and the Euratom\nresearch and training programme 2014-2018 under the grant agr eement No 661913 (SOTERIA). The paper\nreﬂects only the authors’ view and the European Commission is not r esponsible for any use that may be\nmade of the information it contains. 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Bonny, Thermal stability and t he structure of vacancysolute clusters in iron alloys, Acta Mate-\nrialia 85 (2015) 107–111. doi:10.1016/j.actamat.2014.11.026 .\nURLhttp://www.sciencedirect.com/science/article/pii/S1 359645414008799\n[113] P.Efsing,J.Rouden, M.Lundgren, Long Term Irradiati on Eﬀects on the Mechanical Properties of Reactor Pressure V essel Steels from Two Commercial PWR Plants,\nin: T. Yamamoto (Ed.), Eﬀects of Radiation on Nuclear Materi als: 25th Volume, ASTM International, 100 Barr Harbor\nDrive, PO Box C700, West Conshohocken, PA 19428-2959, 2013, pp. 52–68. doi:10.1520/STP104004 .\nURLhttp://www.astm.org/doiLink.cgi?STP104004\nAppendix A. Conﬁgurations and jump frequencies for dumbbel l-solute pairs in bcc\nTable A.4 reports the atomic coordinates corresponding to the dum bbell-solute conﬁgurations shown\nin Fig. 1. Furthermore, Table A.5 summarizes all dumbbell migration ev ents in bcc crystals where the\ndumbbell moves by a 1nn distance, depicted also in Fig. 2. Pure trans lations (τ) and onsite rotations (R)\nare shown in addition to the rotation-translation jumps.\nAppendix B. Computation of total transport coeﬃcients from cluster contributions\nIn the cluster development framework [57, 56] of SCMF [33], each to tal transport coeﬃcient Lijis\nobtained as a weighed sum of cluster contributions:\nLij=C/bracketleftigg/summationdisplay\nmfmL(m)\nij+/summationdisplay\ncfcL(c)\nij/bracketrightigg\n, (B.1)\nwhereCis the total concentration and fm(fc) the fraction of each monomer m(clusterc). The monomer\nand cluster transport coeﬃcients Lm\nijandLc\nijare outputs of KineCluE. Cincludes the concentrations of all\nmonomers and clusters:\nC=Cmono+Cclusters. (B.2)\nThe total monomer concentration is given by Cmono=/summationtext\nm[m]Zm, where [ m] is the concentration of\nmonomer mandZmits partition function. The total cluster concentration Cclustersdepends on the speciﬁc\n28Table A.4: List of symmetry-unique conﬁgurations of a dumbb ell-solute pair up to the 5nn (cf. Fig. 1). A and B mark the\ntwo atoms of the dumbbell, located respectively in [ −δ,−δ,0] and [δ,δ,0]; 2δis the distance between them. The dumbbell is in\nthe center of the reference system and oriented along the /angbracketleft110/angbracketrightdirection. Lengths are given in units of a0/2.\nSolute site Label Distance to A Distance to B\n[δ,δ,0] M 2 δ 0\n[−1,1,1] 1a√\n2δ2+ 3√\n2δ2+ 3\n[1,1,1] 1b√\n2δ2+ 4δ+ 3√\n2δ2−4δ+ 3\n[0,0,2] 2a√\n2δ2+ 4√\n2δ2+ 4\n[2,0,0] 2b√\n2δ2+ 4δ+ 4√\n2δ2−4δ+ 4\n[−2,2,0] 3a√\n2δ2+ 8√\n2δ2+ 8\n[2,0,2] 3b√\n2δ2+ 4δ+ 8√\n2δ2−4δ+ 8\n[2,2,0] 3c√\n2δ2+ 8δ+ 8√\n2δ2−8δ+ 8\n[−1,1,3] 4a√\n2δ2+ 11√\n2δ2+ 11\n[1,1,3] 4b√\n2δ2+ 4δ+ 11√\n2δ2−4δ+ 11\n[3,1,1] 4c√\n2δ2+ 8δ+ 11√\n2δ2−8δ+ 11\n[−2,2,2] 5a√\n2δ2+ 12√\n2δ2+ 12\n[2,2,2] 5b√\n2δ2+ 8δ+ 12√\n2δ2−8δ+ 12\nTable A.5: Dumbbell migration events in a bcc crystal ( τfor pure translation, ωfor rotation-translation, and R for onsite\nrotation). Mixed-dumbbell transitions involving solute m igration are marked with ’1’. 2nn jumps (e.g., 2b →M), as well as 1nn\njumps involving sites further than the 2nn, are not shown.\nM 1a 1b 2a 2b 3b 3c 4b 4c 5b\nM ω,τ, R ω,τ\n1a R R ωω,τω,τω\n1b ω,τ R R τ ω ω τ ω,τ\n2a ω τ R τ ω\n2b ω,τω R R ωω,τ\n3b ω,τω\n3c ω τ\n4b τ ω\n4c ωω,τ\n5b ω,τ\nconditions of the system (e.g., thermal equilibrium or irradiation), an d can be obtained for instance with\ncluster-dynamics models. Under equilibrium conditions and within a dilut e approach, the cluster concen-\ntrations can be expressed in a low-temperature expansion framew ork as the product of the concentrations\nof each cluster component. In this case, Cclustersbecomes:\nCclusters=/summationdisplay\nc[c]Zc=/summationdisplay\nc/parenleftigg/productdisplay\nk[k]Mk/parenrightigg\nZc, (B.3)\nwhere [k] is the concentration of each component kof cluster c, andMkthe component multiplicity. For\ninstance,thecontributionfromaclusterwithtwovacancies(V)an donesolute(B)wouldbe: [V]2[B]Z(VVB).\nThe partition functions of each monomer and cluster ( ZmandZc) are directly output by KineCluE and\ncorrespond to the sum of Boltzmann terms exp( Ex/kBT) over all conﬁgurations xof the cluster. For\nmonomers, this is equivalent to counting the symmetry-equivalent c onﬁgurations: for instance, ZV= 1 for\nvacancies, whereas ZI= 6 for/angb∇acketleft110/angb∇acket∇ightdumbbells due to their 6 possible orientations in a cubic crystal. The\nmonomer and cluster fractions are then expressed as:\nfm=[m]Zm\nC; (B.4)\nfc=[c]Zc\nC. (B.5)\nIn this study where only one mobile defect monomer δ(V or I for vacancy and dumbbell, respectively)\n29and one defect-solute pair δB are concerned, the total concentration (Eq. B.2) is given by:\nC=Cmono+Cpair−Ccorr, (B.6)\nwhere the monomer and pair concentrations are respectively Cmono= [δ]ZδandCpair= [δ][B]ZδB.Ccorris\na correction term accounting for the sites that monomers are pre vented to occupy due to the geometrical\ndeﬁnition of the pair, for kinetic radii well beyond the range of solut e-defect thermodynamic interactions.\nThis term amounts to [ δ][B]Z0\nδB, whereZ0\nδB(the non-interacting cluster partition function) corresponds\nto the count of possible pair geometric conﬁgurations within the deﬁ ned kinetic radius rkin. With this\ncorrection, the total transport coeﬃcients are nearly independ ent ofrkin. Equation (B.6) thus becomes:\nC= [δ]/bracketleftbig\nZδ+[B]/parenleftbig\nZδB−Z0\nδB/parenrightbig/bracketrightbig\n, (B.7)\nwhereZδ,ZδB, andZ0\nδBare the monomer, pair, and non-interacting pair partition function s computed by\nKineCluE. The fractions of monomers and pairs are written respect ively as:\nfδ=Cmono−Ccorr\nC=Zδ−[B]Z0\nδB\nZδ+[B](ZδB−Z0\nδB); (B.8)\nfδB=Cpair\nC=[B]ZδB\nZδ+[B](ZδB−Z0\nδB). (B.9)\nThus, the total transport coeﬃcients (Eq. (B.1)) are:\nLδδ=C/bracketleftig\nfδL(δ)\nδδ+fδBL(δB)\nδδ/bracketrightig\n; (B.10)\nLδB=CfδBL(δB)\nδB; (B.11)\nLBB=CfδBL(δB)\nBB, (B.12)\nwhereL(δ)\nijandL(δB)\nijare the monomer and pair cluster transport coeﬃcients yielded by K ineCluE.\nThe correction term Ccorrcan be applied as long as fm>0. This implies that Ccorr< Cmono, hence\nyielding a constraint on the maximum solute concentration that can b e treated in this dilute model:\n[B] =cB<Zδ\nZ0\nδB. (B.13)\nThis means that the maximum solute concentration is inversely propo rtional to Z0\nδBand thus to the chosen\nkinetic radius.\nFinally, the solute and defect concentrations depend on the speciﬁ c target-alloy composition and con-\nditions. They are assumed here to be ﬁxed by the nominal solute con centration ([B] = CB), and by the\nsteady-state defect concentration established under irradiatio n (C=Cﬁx\nδ). The latter is then split into\nmonomers and pairs according to the fractions in Eqs. (B.8) and (B.9 ). This way, the monomer concentra-\ntion [δ] can be derived from Cﬁx\nδvia Eq. (B.7), and play no role in solute-transport and RIS behaviors , as\nshown in the formula summary in Table B.6.\nAppendix C. Multicomponent partition function\nThe presence of other solutes than species B in a multicomponent allo y can reduce the amount of solute\ncarriers that actively contribute to the transport of B. This can b e taken into account in ﬁrst approximation\nwith a small modiﬁcation of the framework in Appendix B, if the multicom ponent eﬀect on the global\nequilibrium between the vacancy and interstitial population is neglect ed. To this purpose, the concentration\nof additional defect-solute clusters ’ e’ is added to the total cluster concentration in Eq. B.2 as:\nC=Cmono+Cclusters+Cext, (C.1)\nwithCext=/summationtext\ne[e]Ze. Consequently, since Cis larger, the fraction of monomers fmand clusters fccon-\ntributing to the total transport coeﬃcients (Eqs. B.4 and B.5) is sm aller. In a monomer/pair framework,\nthis writes:\nC=Cmono+Cpair−Ccorr+(Cext−Ccorr\next), (C.2)\n30whereCext=/summationtext\nα[δ][α]ZδαandCcorr\next=/summationtext\nα[δ][α]Z0\nδα(αmarks all chemical species diﬀerent from B).\nEquation B.7 is now:\nC= [δ]\nZδ+[B]/parenleftbig\nZδB−Z0\nδB/parenrightbig\n+/summationdisplay\nα/negationslash=B[α]/parenleftbig\nZδα−Z0\nδα/parenrightbig\n. (C.3)\nThe external correction Ccorr\nextmust be then subtracted from the monomer fraction in Eq. B.8:\nfδ=Cmono−Ccorr−Ccorr\next\nC, (C.4)\nwhile the pair fraction in Eq. B.9 remains unchanged.\nAs a practical example, the multicomponent eﬀect on dumbbell-Mn pa irs in an Fe-MnNiP alloy can be\naccounted for by writing Cas:\nC= [I]/bracketleftig\nZI+[Mn]/parenleftbig\nZIMn−Z0\nIMn/parenrightbig\n+\n+[Ni]/parenleftbig\nZINi−Z0\nINi/parenrightbig\n+[P]/parenleftbig\nZIP−Z0\nIP/parenrightbig/bracketrightig\n.(C.5)\nThe monomer and dumbbell-Mn pair fractions then become:\nfI=ZI−[Mn]Z0\nIMn−[Ni]Z0\nINi−[P]Z0\nIP\nC; (C.6)\nfIMn=[Mn]ZIMn\nC, (C.7)\nfrom which the transport coeﬃcients LII,LIMn, andLMnMnare computed according to Eqs. B.10, B.11,\nand B.12. When one of the external species (e.g., P) has a very stro ng binding with dumbbells, the\ncorresponding partition function is large ( ZIP>> ZIMn). As a consequence, the fraction of dumbbell-Mn\npairsfIMnbecomes small due to the increased total concentration C.\nAppendix D. Radiation-induced segregation model\nThe RIS tendencies are inferred from the transport coeﬃcients w ith the mean-ﬁeld model derived by\nNastar, Mart` ınez et al.[4, 55] from Wiedersich’s theory [97] and used in several following st udies [42, 52, 53,\n6]. The solute concentration gradient next to defect sinks is relate d to the vacancy concentration gradient\nas [52]:\n∇CB=−α∇CV\nCV, (D.1)\nwhere the α(RIS)factor is given by:\nα=LAILAV\nLAIDB+LBIDA/parenleftbiggLBV\nLAV−LBI\nLAI/parenrightbigg\n. (D.2)\nThe expressions of the intrinsic diﬀusion coeﬃcients DAandDBas functions of the transport coeﬃcients\ncan be found in one of the aforementioned studies [4]. This model rep resents the steady-state solution of\nEq. 1 where JV=JI,JB= 0, and the chemical potential gradients due to point defects are developed\nin a low-temperature expansion framework [4]. In addition, the abs ence of signiﬁcant sink bias and a low\nsink density are assumed. Multiple-solute and multiple-defect eﬀect s are also neglected. In this dilute-limit\napproximation, the thermodynamic factor φis equal to 1, and the ξfactors can be assumed to be negligible\n(ξ≈0). The sign of αdetermines if solute enrichment (positive) or depletion (negative) o ccurs, and is fully\ncontrolled by the diﬀerence of partial diﬀusion coeﬃcient ratios (las t term in parenthesis in Eq. D.2). In\nthe extremely dilute approach ( CB→0),αis roughly proportional to CBdue to the fact that LBVandLBI\nare themselves proportional to CB; hence, in this case the ratio α/CBis nearly constant.\n31Since each transport coeﬃcient is proportional to the total defe ct concentrations C=Cﬁx\nδ(Eqs. B.10,\nB.11, B.12), it is convenient to re-arrange Eq. D.2 to isolate Cﬁx\nVandCﬁx\nIand analyze αin terms of the\npoint-defect concentration ratio. The factor becomes:\nα=ℓAIℓAV\nℓAI/parenleftbig\nDvac\nB+KDsia\nB/parenrightbig\n+ℓBI/parenleftbig\nDvac\nA+KDsia\nA/parenrightbig/parenleftbiggℓBV\nℓAV−ℓBI\nℓAI/parenrightbigg\n, (D.3)\nwhere each intrinsic diﬀusion coeﬃcient has been split into a vacancy a nd interstitial contribution, and\nnormalized by the corresponding defect concentration (cf. Table B.6). All transport coeﬃcients ℓij=\nLij/Cﬁx\nδare now independent of Cﬁx\nV,Cﬁx\nI, and the latter come into play only in the ratio K=Cﬁx\nI/Cﬁx\nV.\nThis allows for a custom choice of the defect concentration model, e .g., the outcome of a time-dependent\nrate-theory model adapted to the target conditions. In the latt er case, the steady-state solution depends on\ntemperature and sink density, and is in most cases equal to DV/DI[98]. Since the defect diﬀusion coeﬃcient\nis given by Dδ=Lδδ/C, andLδδis proportional to C(Eq. (B.10)), the Kfactor can be written as:\nK=DV\nDI=ℓVV\nℓII=fVL(V)\nVV+fVBL(VB)\nVV\nfIL(I)\nII+fIBL(IB)\nII, (D.4)\nwhich reduces, for very low solute concentrations ( fδB→0,fδ→1), to a simple ratio of monomer transport\ncoeﬃcients:\nK=L(V)\nVV\nL(I)\nII. (D.5)\n32Table B.6: Summary of formulas used to infer transport prope rties and RIS tendencies from the KineCluE output [4, 57, 55, 56].\nSubscripts ’(V)’ and ’(I)’ refer to monomers (isolated defe cts), ’(VB)’ and ’(IB)’ to solute-defect pairs. ZδBis the pair partition\nfunction, while Z0\nδBmarks the number of possible geometric conﬁgurations assoc iated to the δB pair.ZV= 1 and ZI= 6 are\nthe possible geometric conﬁgurations of single vacancies a nd single dumbbells. The thermodynamic factors φandξare set to 1\nand 0, respectively. Total defect concentrations Cﬁx\nVandCﬁx\nIare kept as variable parameters. Labels ’V’ and ’I’ are perfe ctly\ninterchangeable, except for the host-related coeﬃcients a nd the partial diﬀusion coeﬃcient ratio.\nDumbbell diﬀusion Vacancy diﬀusion\nKineCluE output\nmonomer: ZI= 6,L(I)\nIImonomer: ZV= 1,L(V)\nVV\npair:L(IB)\nII,L(IB)\nIB,L(IB)\nBB,ZIB,Z0\nIB pair:L(VB)\nVV,L(VB)\nVB,L(VB)\nBB,ZVB,Z0\nVB\nTotal transport coeﬃcients\nC= [I]/bracketleftbigZI+ [B]/parenleftbigZIB−Z0\nIB/parenrightbig/bracketrightbig=Cfix\nI C= [V]/bracketleftbig\nZV+ [B]/parenleftbig\nZVB−Z0\nVB/parenrightbig/bracketrightbig\n=Cfix\nV\nfI=ZI−[B]Z0\nIB\nZI+[B]/parenleftBig\nZIB−Z0\nIB/parenrightBig fV=ZV−[B]Z0\nVB\nZV+[B]/parenleftBig\nZVB−Z0\nVB/parenrightBig\nfIB=[B]ZIB\nZI+[B]/parenleftBig\nZIB−Z0\nIB/parenrightBig fVB=[B]ZVB\nZV+[B]/parenleftBig\nZVB−Z0\nVB/parenrightBig\nLsia\nBB=Cfix\nIfIBL(IB)\nBBLvac\nBB=Cfix\nVfVBL(VB)\nBB\nLIB=Cfix\nIfIBL(IB)\nIBLVB=Cfix\nVfVBL(VB)\nVB\nLII=Cfix\nI/parenleftBig\nfIL(I)\nII+fIBL(IB)\nII/parenrightBig\nLVV=Cfix\nV/parenleftBig\nfVL(V)\nVV+fVBL(VB)\nVV/parenrightBig\nHost (’A’)-related transport coeﬃcients\nLsia\nAB=LIB−Lsia\nBBLvac\nAB=−LVB−Lvac\nBB\nLsia\nAA=LII−2Lsia\nAB−Lsia\nBB Lvac\nAA=LVV−2Lvac\nAB−Lvac\nBB\nLAI=Lsia\nAA+Lsia\nABLAV=−Lvac\nAA−Lvac\nAB\nNormalized transport coeﬃcients\nℓsia\nBB=Lsia\nBB\nCfix\nI;ℓIB=LIB\nCfix\nI;ℓII=LII\nCfix\nIℓvac\nBB=Lvac\nBB\nCfix\nV;ℓVB=LVB\nCfix\nV;ℓVV=LVV\nCfix\nV\nℓsia\nAB=Lsia\nAB\nCfix\nI;ℓAI=LAI\nCfix\nI;ℓsia\nAA=Lsia\nAA\nCfix\nIℓvac\nAB=Lvac\nAB\nCfix\nV;ℓAV=LAV\nCfix\nV;ℓvac\nAA=Lvac\nAA\nCfix\nV\nFlux-coupling ratios\nGI=LIB\nLsia\nBB=L(IB)\nIB\nL(IB)\nBB;gI=L(IB)\nIB\nL(IB)\nIIGV=LVB\nLvac\nBB=L(VB)\nVB\nL(VB)\nBB;gV=L(VB)\nVB\nL(VB)\nVV\nSolute tracer diﬀusion coeﬃcient\nDsia\nB∗=Lsia\nBB\nCB=Cfix\nI/bracketleftBigg\nZIBL(IB)\nBB\nZI+[B]/parenleftBig\nZIB−Z0\nIB/parenrightBig/bracketrightBigg\nDvac\nB∗=Lvac\nBB\nCB=Cfix\nV/bracketleftBigg\nZVBL(VB)\nBB\nZV+[B]/parenleftBig\nZVB−Z0\nVB/parenrightBig/bracketrightBigg\nRatio of partial diﬀusion coeﬃcients\nDsia\npd=(1−CB)LBI\nCBLAI=(1−[B])fIBL(IB)\nIB\n[B]/bracketleftbigg\nfIL(I)\nII+fIB/parenleftbigg\nL(IB)\nII−L(IB)\nIB/parenrightbigg/bracketrightbigg Dvac\npd=(1−CB)LBV\nCBLAV=−(1−[B])fVBL(VB)\nVB\n[B]/bracketleftbigg\nfVL(V)\nVV+fVB/parenleftbigg\nL(VB)\nVV−L(VB)\nVB/parenrightbigg/bracketrightbigg\nNormalized intrinsic diﬀusion coeﬃcients\nDsia\nA=φ/parenleftbigg\nℓsia\nAA\n1−CB−ℓsia\nAB\nCB−ℓAIξAI\nφ(1−CB)/parenrightbigg\nDvac\nA=φ/parenleftbigg\nℓvac\nAA\n1−CB−ℓvac\nAB\nCB−ℓAVξAV\nφ(1−CB)/parenrightbigg\nDsia\nB=φ/parenleftbigg\nℓsia\nBB\nCB−ℓsia\nAB\n1−CB−ℓBIξBI\nφCB/parenrightbigg\nDvac\nB=φ/parenleftbigg\nℓvac\nBB\nCB−ℓvac\nAB\n1−CB−ℓBVξBV\nφCB/parenrightbigg\nRadiation induced segregation\n∇CB=−α∇CV\nCV, α=ℓAVℓAI\nℓAI/parenleftBig\nDvac\nB+KDsia\nB/parenrightBig\n+ℓBI/parenleftBig\nDvac\nA+KDsia\nA/parenrightBig/parenleftBigℓBV\nℓAV−ℓBI\nℓAI/parenrightBig\nK=CI\nCV≈DV\nDI=ℓVV\nℓII=fVL(V)\nVV+fVBL(VB)\nVV\nfIL(I)\nII+fIBL(IB)\nII\n33"    },    {        "title": "1210.0765v1.Magnetic_Relaxation_in_Bismuth_Ferrite_Micro_Cubes.pdf",        "content": " 1 Magnetic Relaxation in Bismuth Ferrite Micro-Cubes \nB. Andrzejewski, K. Chybczy ńska, K. Pogorzelec-Glaser, B. Hilczer, T. Toli ński \nInstitute of Molecular Physics \nPolish Academy of Sciences \nSmoluchowskiego 17, PL-60179 Pozna ń, Poland \nbartlomiej.andrzejewski@ifmpan.poznan.pl \nB. Ł ęska, R. Pankiewicz \nFaculty of Chemistry \nAdam Mickiewicz University \nUmułtowska 89b, PL-61614 Pozna ń, Poland \nP. Cieluch \nResearch Centre of Quarantine, Invasive and Genetic ally Modified Organisms \nInstitute of Plant Protection – National Research I nstitute \nWęgorka 20, PL-60318 Pozna ń, Poland \n \n \nAbstract — The process of magnetic relaxation was studied in  \nbismuth ferrite BiFeO 3 multiferroic micro-cubes obtained by \nmeans of microwave assisted Pechini process. Two di fferent \nmechanisms of relaxation were found. The first one is a rapid \nmagnetic relaxation driven by the domain reorientat ions and/or \npinning and motion of domain walls. This mechanism is also \nresponsible for the irreversible properties at low temperatures. \nThe power-law decay of the magnetic moment confirms  that this \nrelaxation takes place in the system of weakly inte racting \nferromagnetic or superferromagnetic domains. The se cond \nmechanism is a longterm weak magnetic relaxation du e to spin \nglass-phase. \nKeywords-component; magnetic relaxation, domain wal ls, spin-\nglass, bismuth ferrite  \nI.  INTRODUCTION \nBismuth ferrite BiFeO 3 (BFO) belongs to a group of materials \ncalled magnetoelectric (ME) multiferroics, that exh ibit charge \nand magnetic ordering with some mutual coupling bet ween \nthem [1-4]. ME multiferroics recently have attracte d the \nattention of numerous groups of researchers because  of their \nvery interesting and rich physical properties and p rospective \ntechnological potential [5]. BFO compound is  \na rhombohedrally distorted perovskite with space gr oup R3c at \nroom temperature. The ferroelectric properties appe ar in BFO \nbelow Curie ferroelectric temperature TC=1100 K due to charge \nordering caused by the ordering of lone electron pa irs of Bi 3+  \nions. The magnetic properties together with weak \nferromagnetic (FM) moment results from the complex ordering \nof Fe 3+  spins and they appear below the Néel temperature \nTN=643 K. The magnetic ordering in BFO is an \nantiferromagnetic state (AFM) exhibiting G-type str ucture and \nsuperimposed long range incommensurate cycloidal \nmodulation with the period λ=62 nm [3, 6]. The spin cycloid \npropagates along three equivalent crystallographic directions [1,-1,0], [1,0,-1] and [0,-1,1] (pseudocubic notati on). The spins \nin the cycloid rotate in the plane determined by th e direction of \ncycloid propagation and the [1,1,1] direction of sp ontaneous \nelectric polarization. ME type of Dzyaloshinskii-Mo riya \ninteraction induces small canting of the spins out of the rotation \nplane, which leads to a local ferromagnetic orderin g. This local \nFM ordering in the form of weak ferromagnetic domai ns was \nindeed recently found experimentally in BFO by mean s of \nneutron diffraction [7]. The mean size of the domai ns is 30 nm \ni.e.  a half of modulation period of the spin cycloid. \nA system containing magnetic domains or clusters li ke \nBFO can exhibit variety of magnetic orderings depen ding on \nthe strength of interdomain interactions. For negli gible energy \nof interaction, the system attains a superparamagne tic (SP) \nstate. With increasing energy of interdomain coupli ng the \nsystem start to attain collective magnetic states; first  \na superspin glass (SSG) state for moderate interact ions and \nnext superferromagnetic (SFM) state if the coupling  becomes \nstrong enough [8]. Moreover, the properties of SFM or FM \nsystems are modified by growth and/or reorientation s of \nmagnetic domains and also by pinning or motion of d omain \nwalls which makes their behavior very complex. Magn etic \ndomain walls can be pinned at pinning centers like;  local lattice \nstrains, structural defects, grain boundaries, and in \nmultiferroics even at ferroelectric domains due to \nflexomagnetic interactions [9]. This last phenomeno n is \nanother interesting example of the coupling between  electric \nand magnetic properties in multiferroics. Domain wa lls are \nbowed between the pinning centers but they can move  or jump \nto the another center due to magnetic interaction o r thermal \nexcitations. The motion of magnetic domain walls ca uses \ndomain growth, domain wall reconformations and also  \nrelaxation of a magnetic moment.   2 \nBesides weak ferromagnetism and ferromagnetic domai ns \nfound in BFO [7], a spin-glass (SG) transition at l ow \ntemperatures has been also recently postulated by S ingh et al. \n[10, 11]. However, as noted in [10, 11], it is very  difficult to \ndistinguish SG phase from SP one and from ferroics with \ndomain wall pinning and motion or relaxors, because  all these \nsystems exhibit existence of Almeida de Thouless li ne (AT-\nline) [11], aging, rejuvenation, magnetic relaxatio n and \nsometimes also the memory effect. \nThe aim of this paper is to shed some light on the question \nwhich phenomenon brings a dominant effect on the ma gnetic \nproperties of BFO: SG phase or relaxation due to FM  domain \ngrowth, domain reorientations and wall pinning or m otion. This \nproblem is very essential because it is expected th at the domain \nwalls in multiferroics will be active elements in f uture device \napplications due to extremely short switching time and low \nconsumption of energy instead of devices using ferr oelectric or \nferromagnetic domains [12]. \nII.  EXPERIMENTAL \nA.  Sample Synthesis \nBFO powder-like samples were obtained by means of \nmicrowave assisted hydrothermal Pechini process. An  aqueous \nsolution necessary for the synthesis was prepared b y dissolving \nappropriate amounts of nitrates; Bi(NO 3)3·5H 2O, \nFe(NO 3)3·9H 2O, sodium carbonate Na 2CO 3 and potassium \nhydroxide KOH in distilled water. This solution was  \ntransferred and sealed in PTFE reactors. Next, it w as processed \nin CEM Mars-5 microwave oven at 200 0C for about 30 \nminutes. During the reaction, the water vapour pres sure inside \nthe reactors was about 2·10 6 Pa. After the reaction was \ncompleted, the oven was cooled gradually to about 5 0 0C. Than \nthe as obtained, brown in coloration suspension was  filtered off \nto collect the fine BFO powder. This powder was rin sed with \nwater, dried in air and some samples were also air calcined at \n500 0C for 1 hour. \nB.  Sample Characterization \nThe crystallographic structure of the samples and m inority \nphase content was studied by means of x-ray diffrac tion (XRD) \nusing an ISO DEBYEYE FLEX 3000 diffractometer equip ped \nwith a Co lamp ( λ=0.17928 nm). Their morphology was \nexamined by S3000N Hitachi Scanning Electron Micros cope \n(SEM), whereas magnetometric measurements were perf ormed \nusing the Quantum Design Physical Property Measurem ent \nSystem (PPMS) fitted with a superconducting 9T magn et and \nwith a Vibrating Sample Magnetometer (VSM probe). B efore \nmeasurements of magnetic relaxation, the supercondu cting \nmagnet was switched to driven mode for about 1 h to  remove \nthe remnant magnetic flux. \nIII.  RESULTS  AND  DISCUSSION \nFig. 1 presents the XRD pattern of as-obtained micr owave-\nsynthesized product. The solid line corresponds to the best \nRietveld profile fit to the experimental data calcu lated by \nmeans of FULLPROF software. This analysis reveals a  well \ncrystallized BFO rhombohedral phase with R3c  space group and a small content of Bi 25 FeO 40  parasitic phase labeled by \nasterisk. The line below the XRD data shows the dif ference \nbetween the experimental data and the fit. The vert ical sections \nindicate the positions of Bragg peaks. The amount o f Bi 25 FeO 40  \nparasitic phase evaluated with respect to the most intensive \nBFO peaks (110, 104) is about 2%. The low content o f the \nparasitic phase is a result of a short time of micr owave reaction \nwhich promotes the synthesis of major BFO phase. Al so it was \nobserved that the Bi 25 FeO 40  undesired phase is usually formed \nmainly during an early stage of the reaction at a l ow \ntemperature. For microwave synthesis, duration of t his stage of \nreaction is very short because of even heating of t he solution \nvolume and high rate of temperature increase in the  microwave \noven.  \nFigure 1.  XRD-pattern of BFO sample. The solid line is the be st fit to the \nexperimental data represented by open points. The s olid line below is the \ndifference between the data and the fit. The vertic al sections represent \npositions of the Bragg peaks. The asterisk indicate s the trace of Bi 25 FeO 40  \nphase. \nFig. 2 presents a SEM micrograph illustrating the \nmorphology of BFO sample. It turns out that the BFO  grains \nare actually the agglomerates composed of almost re gular \nmicro-cubes of the mean size of about 1 µm, however the size \ndistribution of the grains is rather wide. The magn etization \nloops of BFO sample M(H) at three selected temperatures i.e.  \n10 K, 100 K and 300 K are shown in Fig. 3. Complete  \nsaturation of magnetization is never achieved even at the \nmagnetic field 9 T, much stronger than that applied  to record \nthe data plotted in Fig. 3 (not presented here). At  high \ntemperatures 100 K and 300 K, the magnetization is reversible, \nhowever it exhibits pronounced hysteresis at 10 K. The shape \nof the hysteresis loops, its evolution with tempera ture and \nappearance of irreversibility in low temperatures r esembles the \nbehavior of SP, SSG phases reported by Shen et al. [13] or  \na weak ferromagnetic order caused by the presence o f SFM \nphase. The presence of FM order was verified by mea ns of \nArrot construction [14] where the magnetization M(H) data are \nplotted in the form of the square of magnetization M2 versus \ndimensionless variable H/M. According to Arrot approach \nbased on the Weiss molecular-field theory [15], the  relation  3 \n between magnetization M(H) and internal magnetic field is as \nfollows: \n \n (1) \n \nwhere µ, M0, k B=1.38·10 -23  J/K and Tc are the magnetic \nmoment per atom, the spontaneous magnetization at z ero \ntemperature, Boltzmann’s constant and the Curie tem perature \nrespectively.  \nFigure 2.  SEM micrograph of BFO sample.  \nFigure 3.  BFO magnetization M(H) loops for the temperatures 10 K, 100 K \nand 300K.  \nIf the data are plotted in the form of a square of the \nmagnetization M2 vs. H/M, they ordinate in the Arrot \nconstruction along the straight isotherms for each measuring \ntemperature and can be fitted by a linear function.  The slope of \na linear fit is positive and increases with decreas ing \ntemperature. The intercept of the linear fits with the ordinate axis evolves from negative to positive when the mag netic \nordering changes from paramagnetic to ferromagnetic . The \nintercept with the origin of the Arrot plot occurs for the \nisotherm corresponding to the Curie temperature. Th e Arrot \nconstruction for the BFO sample measured at a few \ntemperatures selected from the range 2 K-300 K is s hown in \nFig. 4. The parts of isotherms recorded for high ma gnetic field \nare well fitted by linear functions and length of t hese linear \nsections of isotherms increases as the temperature decreases. \nHowever, for low magnetic fields the isotherms exhi bit an \napparent positive curvature, which can indicate a s econd-order-\ntransition [16]. The ordinate intercept is always n egative which \nmeans that there is no net ferromagnetic order. For  the BFO \nsample studied, the intercept approaches the origin  of the Arrot \nplot when temperature decreases, which indicates th e \nincreasing role of interdomain ferromagnetic intera ctions. \nHowever, these interactions are not strong enough t o induce net \nFM ordering. The lack of evidence for FM ordering i n the \nArrot plot in Fig. 4 may be also explained in anoth er way, in \nterms of a weak BFO ferromagnetism, which results f rom \ncomplex spin ordering. Without any distortion of th e structure, \nBFO compound should be an ideal antiferromagnet wit h no net \nmagnetic moment because all magnetic moments of Fe 3+  spins \nare compensated. Also the local magnetic moments du e to spin \ncanting in the spin cycloid should be balanced out by the \nmoments of the neighboring cycloids. Then, the non- zero \nmagnetic moment can originate only from the diminut ive tilt of \nthe moments in the adjacent (111) planes which indu ces weak \ncanted ferromagnetism [17]. \nFigure 4.  The Arrot plots for BFO sample at various selected temperatures.  \nThe inverses of zero field cooling (ZFC) and field cooling \n(FC) susceptibilities 1/ χ(T) areshown in Fig. 5. Above about  \n70 K, the ZFC and FC inverses of susceptibility are  identical \nand they deviate from the linear relation derived f rom the \nCurie-Weiss law: 1/ χ(T)=( T-Θ)/C (where C is the Curie \nconstant and Θ is the Weiss temperature). The nonlinear \ndependence of the inverse of susceptibility 1/ χ(T) usually is \nattributed to the formation of SP domains and to th e increase in \n 2\n03\n0 21 3k3MTT\nMH\nTMMc\nB\n\n− +\n\n=µ 4 \n domain volume with decreasing temperature [13]. A d ifferent \nbehavior is observed below 70 K, where the inverses  of ZFC \nand FC susceptibilities start to bifurcate. In this  temperature \nrange the inverse of ZFC susceptibility increases w ith \ndecreasing temperature, whereas the inverse of FC \nsusceptibility is almost constant. This means that the value of \nFC susceptibility χ(T) also does not change at low \ntemperatures, a feature that usually indicates the onset of \ninterdomain interactions [18-23]. The nonzero inter actions \nbetween domains lead to the appearance of a SSG or a SFM \nphase, if the interactions are strong enough.  \nFigure 5.  The inverse of ZFC and FC susceptibilities 1/ χ(T) (main panel) \nand the dependence of TB(H) temperature vs. field (inset). The solid line is the \nlinear fit to the FC data at low temperature range.  \nThe bifurcation between ZFC and FC susceptibilities , \nwhich appears at TB is usually explained in terms of SP domain \nblocking. The blocking temperature depends on the d omain \nvolume V and the anisotropy constant K and can be evaluated \nfrom the relation: K V≈25k BTB. However, for BFO the mean \nsize of weak ferromagnetic domains is d≈30 nm [7] and the \nanisotropy constant equals K=6·10 4 J/m 3 [3] so this formula \ngives an unrealistic blocking temperature of TB≈5000 K. \nMoreover, in the case of BFO sample investigated, t he field \ndependence of TB(H) temperature cannot be fitted with usual \nrelations describing the AT-line in the form of TB~H2/3 , \npredicted for the SP or SG models of noninteracting  spin \nsystem [24] and TB~H1/2  in the Néel model with ferromagnetic \ninteractions [25]. Instead, the relation of TB temperature vs. H \nis linear when presented in a semilogarithmic plot ln TB-H (see \ninset to Fig. 5). The linear dependence of ln TB on H means that \nthe relation between the temperature of bifurcation  TB and the \napplied magnetic field is: \n \n (2) \n Here H0 is the magnetic field value that suppresses the \nbifurcation temperature TB to absolute zero and b is  \na phenomenological parameter. It turns out that the  relation (2) \nis similar to the formula for the energy density of  \nferromagnetic domain wall pinning: E=E0exp(-b T) [26, 27]. \nTherefore, we can deduce that TB is the temperature of pinning \nor blocking of the motion of FM or SFM domain walls  rather \nthan the SP, SG or the SSG freezing temperature Tf. \nIn the model of weak ferromagnetic domains, above TB the \nthermal energy exceeds the energy of domain wall pi nning, \nwhich implicates free motion of domain walls and/or  the \ndomain growth. In this regime ZFC and FC 1/ χ curves are \nidentical and magnetization is reversible. Below TB the thermal \nenergy is not enough to overcome the pinning energy  barrier. \nThe domain walls are pinned and their conformation depends \non the applied magnetic field and on history of the  sample. \nNamely, when the system is field cooled it assumes the domain \nconfiguration close to the equilibrium single domai n state, \nwhereas when the system is zero field cooled it exh ibits  \na spontaneous polydomain state. Upon ZFC, the magne tization \nis lower than upon FC but it starts to increase as temperature \nincreases because the thermally activated domain wa ll motion \nbecomes more rapid. The volume of the energetically  favorable \ndomains increases at the expense of other domains a nd the \nsystem tends to a monodomain state as for FC case. The \nminimum in the ZFC inverse of susceptibility (maxim um in \nZFC susceptibility) appears at the mean blocking te mperature \nTB,mean , whereas the temperature TB of bifurcation between ZFC \nand FC curves corresponds to the maximum blocking \ntemperature above which the thermal energy exceeds the \nenergy of pinning and the motion of domain walls is  no longer \nhindered. A large difference between TB and TB,mean  usually \nhappens in the ferromagnetic systems with a wide di stribution \nof domain sizes and pinning energies. For our BFO s ample \nstudied TB-TB,mean ≈6 K which confirms that the distribution of \ndomain sizes is quite narrow [13]. \nThe blocking of domain wall motion influences also the \ncoercivity Hc and remanence Mr of the hysteresis loops, which \nis presented in Fig. 6. In high temperature range w here the \ndomain wall pinning is ineffective, both the coerci vity and \nremanence are negligible. However, these quantities  start to \nincrease rapidly below 100 K, a value comparable to  the \ntemperature of blocking TB determined using the criterion of \nZFC and FC magnetization bifurcation (see Fig. 5). The \nmaximum value of remanence Mr for BFO sample reported \nhere, measured at lowest temperature available i.e.  2 K was 6.1 \nAm 2/kg. The coercivity determined at the same temperat ure \nwas about 0.09 T. \nBesides the irreversible properties, the systems li ke \ndisordered FM phase, SG, SP and SFM phases exhibit also \nstrong time evolution of magnetic properties [28, 2 9]. The \nresults of measurements of magnetic moment relaxati on on \ntime in our BFO sample are shown in Fig. 7. To bett er \nvisualize the magnetic moment changes in time, all data in Fig. \n7 are normalized so that the initial value of magne tic moment \nm(τ) for τ=1 is always equal to unity. The variable τ=t/t0 is the \ntime normalized with respect to the dead time t0≈10 s needed to \nstabilize magnetic field and to complete the first measurement. () [ ]HT H HBb exp 0 − = 5 \n \n The time evolution of the magnetic moment m(τ) was \nmeasured during three following experiments, which allows the \nstudy of different mechanisms of magnetic relaxatio n:  \na)  ZFC to selected low temperature followed by rap id \nswitching on of the magnetic field \nb)  FC to a given low temperature followed by switc h off of \nthe magnetic field \nc)  FC in the magnetic field to the lowest temperat ure \navailable (2 K) followed by temperature increase to   \na selected temperature at which the measurement was  next \nperformed \nThe magnetic field in all experiments was 0.1 T, wh ich \ncorresponds to the blocking temperature TB of about 70 K. \nFigure 6.  The dependence of coercivity Hc and remanence Mr of BFO \nsample on temperature. The solid lines are guides f or eyes, only. \nIn the case “ a”, a time increase in the magnetic moment \nm(τ) is observed because of the progress in formation of \ndomain aligned state due to domain reorientations, growth \nand/or domain wall motion in response to the applie d magnetic \nfield (Fig. 7a). When this process is completed the  magnetic \nmoment attains maximum value and stabilizes. Howeve r, after \na long time ( τ≈10 3 corresponding to t≈10 4 s) a small decrease \nin the magnetic moment appears. The time evolution of the \nmagnetic moment due to domain reorientation, until it reaches \na maximum, can be described in terms of a modified power law \nrelaxation: \n (3) \n \nwhere the parameters m0, mR define the initial magnetization \nand the relaxing part of the magnetic moment and n is the \nfitting co-efficient. The best fit of the model (3)  to the \nexperimental data was obtained for m0=0.57, mR=0.43 and \nn=1.033. Equally good fit can be also obtained by me ans of the \nsaturating stretched exponential law used by Chen e t al. [30] to \ndescribe the SFM domain systems:  \n \n(4)/uniF029 where τ0 and β are fitting parameters. For the best fit (not \nshown for clarity in Fig. 7a) for 10 K they are as follows: \nm0=0.959, mR=0.142, τ0=13 and β=0.42. However, model (4) \nbesides SFM phase also correctly describes relaxati on in the \nSG phase [20, 31] and in spin cluster glass [32] an d therefore \ngives no indication as to which phase actually occu rs in the \nsample investigated.  \nFigure 7.  Time evolution of the normalized magnetic moment ob served in \nthe experiments “ a”, “ b” and “ c” (panels a, b and c, respectively). \nMagnetic measurements performed at 80 K indicate no  \nrelaxation except a small decay of the magnetic mom ent for \nvery long time, a case which will be discussed latt er. At 80 K \nthe magnetic moment immediately attains a constant value \nbecause the domain walls are not pinned above the b ifurcation \ntemperature TB and thus the process of domain ordering and \ndomain wall motion is instantaneous. After this pro cess of very \nrapid saturation of magnetic moment, the moment doe s not \nchange during three decades of time (corresponding to two \ndecades of normalized time τ). Next, for a very long time \nanother mechanism of relaxation appears, which lead s to a \ndecrease in the moment. Similar, nonmonotonic relax ations \nhave also been observed for SFM systems [33] and at tributed \nto intradomain thermally activated relaxation.  \nThe panel “ b“ in Fig. 7 shows a decrease in the magnetic \nmoment with time after FC and switching off of the magnetic \nfield according to procedure “ b”. When the data are \nrepresented in log m(τ)-log τ plot, they ordinate linearly, except \nthe measurements performed at 80 K. Therefore the r elaxation \nproceeds according to the power-law relation:  \n \n(5) \n () ( ) [ ]βττ τ0 0 exp 1 − − + =Rm m m()n\nRm m m−+ =1\n0 τ τ()1\n0−+ =n\nRm m m τ τ 6 \n where m(τ) denotes the normalized moment, m0 is the remnant \nmagnetic moment and mR is the initial value of the relaxing \npart of the magnetic moment. The best fit is obtain ed for m0≈0, \nmR=1 and n=1.033. The last co-efficient is identical to the n \nparameter obtained from the fit of model (3) to the  data \nrecorded in experiment “ a”. The power-law dependence of \nmagnetic moment on time equation (5), similar to th at observed \nin the BFO sample studied, has been predicted theor etically for \nan assembly of single ferromagnetic nanodomains wit h dipolar \ninteractions [34] and also found experimentally in SFM \nsystems [30]. The power-low relaxation it is also a  strong \nargument for SFM behaviour [8]. The value of the co -efficient \nn>1, indicates weak SFM state with ferromagnetic dom ains, \nwhich can be separated by ferromagnetic walls. \nThe values of n co-efficient determined in experiments “ a” \nand “ b”, are almost identical and this also supports the \ninterpretation of the results obtained in terms of relaxation \ndriven by domain reorientations and domain wall mot ion. The \nnegligible value of m0≈0 determined from the best fits to the \nexperimental data by means of model (5), correspond s to the \nsystems exhibiting low concentration of ferromagnet ic domains \nand no long range FM ordering. This is consistent w ith the \nconclusions educed from the Arrot construction, Fig . 4, which \nshows no net ferromagnetic order and the absence of  \ninterdomain ferromagnetic interactions even at lowe st \ntemperatures available in our experiment.  \nThe mechanism of magnetic relaxation in experiment “ b” \ncan be understood in terms of temporal decompositio n of initial \nalmost single domain state, obtained after FC, to a  polydomain \nstate which appears immediately after the field is suppressed. \nThis sample exhibits no remnant moment m0 after long time \nmeasurements, which confirms that BFO is a weak \nferromagnet with negligible inter domain interactio ns. At the \n80 K, i.e . above the bifurcation temperature TB, the pinning of \nmagnetic domains vanishes and the relaxation cannot  be \ndescribed by means of the power-law behaviour (4) b ecause it \nproceeds according to different mechanism like rela xation in \nSG phase.  \nThe results of measurements after FC in a magnetic field of \n0.1 T obtained in experiment “ c” are shown in Fig. 7c. Both at \n10 K and 80 K, the magnetic moment only slightly in creases \nwith time. The total change in the moment is below 1% of its \ninitial value and becomes noticeable for long time \nmeasurements only. Therefore this slow relaxation c annot be \nexplained by a relatively fast domain wall motion a nd/or \ndomain reorientations and has probably the same ori gin as the \nlong-time processes observed in experiment “ a”. It turns out \nthat the data in Fig. 7c can be well fitted using a  modification \nof the stretched exponential law equation (6):  \n \n(6) \n \nwhere the parameters of the best fit at 10 K (or 80  K) are: \nm0=0.9989 (0.9998), mR=6.7 ⋅10 -4 (1.5 ⋅10 -4), τ0=27 (identical at \n10 K and 80 K), β=0.25 (0.27). The stretched exponential \nmodel usually well describes intra domain relaxatio n due to SG \nphase or spin cluster phase. This allows us to supp ose that in BFO compound actually there are two processes of \nmagnetization relaxation: one stronger that occurs at low \ntemperatures due to domain reorientation, growth an d/or \npinning and motion of domain walls and another more  subtle \nwhich dominates long term intradomain relaxation ca used by \nSG phase. This last conclusion is consistent with t he point of \nview of Singh et. al [10, 11] about a possible pres ence of SG \nphase in BFO compound.  \nThe effects of aging and rejuvenation are shown in Fig. 8. \nTo study these effects, the BFO sample was first FC  in the \napplied field of 0.1 T to 50 K. Next the cooling wa s interrupted \nand the relaxation of magnetic moment was measured during \nthe period tw=10 4 s (data represented by open squares in Fig. \n8). After that the cooling was resumed until low te mperature \nwas reached and in next step the magnetization was measured \nwhen heating the sample (filled circles in Fig. 8).  The decay of \nmagnetic moment, during the process of aging is ver y small, \nbelow 1% of the total signal (see the inset to Fig.  8). Thus the \nmagnitude of this decay is comparable to the long-t ime \nvariations of the magnetic moment observed in exper iment “ c” \nand probably driven by the same mechanisms of intra domain \nrelaxation caused by SG phase or spin cluster glass . However, \ncontrary to the experiment “ c”) where an increase in the \nmagnetic moment has been observed, this time the mo ment \ndecreases. It seems that there is a simple correlat ion between \nthe magnetic moment evolution and the history of th e sample. \nNamely, if the initial state of the sample exhibits  magnetization \n(as in experiment “ c” in which temperature was increased) \nhigher than that in the state in which heating was interrupted \nand aging is studied, than the magnetic moment incr eases with \ntime. Conversely, if the initial state exhibits a m agnetic \nmoment lower than that in the state in which relaxa tion is \nmeasured (as in the latter case in which temperatur e was \ndecreased) than temporary decay of the magnetic mom ent is \nobserved.  \nFigure 8.  Aging and rejuvenation of BFO sample (main panel). Inset shows \ndecay of the magnetic moment.  () ()βττ τ0 0 exp Rm m m + = 7 IV.  CONCLUSIONS \nThe results of present studies of ZFC and FC magnet ization \ndependence on temperature, magnetization dependence  of field \nM(H) and relaxation processes after various procedures  applied \nseem to be well explained in terms of two different  \nmechanisms of magnetic relaxation in bismuth ferrit e. The first \nmechanism of rapid magnetic relaxation occurs at lo w \ntemperatures and is related to domain growth and/or  domain \nwall pinning and motion in the system of weak FM or  SFM \ndomains. This point of view is supported by a field  dependence \nof the temperature TB(H) of bifurcation between ZFC and FC \nmagnetization curves which can be understood in ter ms of the \nmodel of effective energy of domain wall pinning. T he power-\nlaw dependence of the magnetic moment on time is an other \nstrong argument for the relaxation to take place du e to FM \ndomains or SFM phase, whereas a negligible value of  the \nremnant magnetic moment m0 indicates a low concentration of \ndomains. The second mechanism is long-term weak rel axation \nof the magnetic moment which appears in SG phase as  \nsuggested earlier by Singh et al. [10] or spin clus ter phase. In \nthis case the time dependence of magnetic moment fo llows the \nusual stretched-exponential dependence on time obse rved for \nthe SG phase. 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Phys \nRev B. 2003;67:024416-1-4 "    },    {        "title": "1911.02704v1.Magnetic_monopoles_and_toroidal_moments_in_LuFeO__3__and_related_compounds.pdf",        "content": "Magnetic monopoles and toroidal moments in LuFeO 3and related compounds\nFrancesco Foggetti,1, 2Sang-Wook Cheong,3and Sergey Artyukhin1\n1Quantum Materials Theory, Italian Institute of Technology, Via Morego 30, 16163 Genova, Italy\n2Department of Physics, University of Genova, Via Dodecaneso, 33, 16146 Genova GE\n3Rutgers Center for Emergent Materials and Department of Physics and Astronomy,\nRutgers University, Piscataway, New Jersey 08854, USA\nMagnetic monopoles and toroidal order are compelling features that have long been theorized\nbut remain elusive in real materials. Multiferroic hexagonal ferrites are an interesting realization of\nfrustrated triangular lattice, where magnetic order is coupled to ferroelectricity and trimerization.\nHere we propose a mechanism, through which magnetic monopolar and toroidal orders emerge from\nthe combination of 120\u000eantiferromagnetism and trimerization, present in hexagonal manganites\nand ferrites. The experimentally observable signatures of magnetic monopolar and toroidal orders\nare identi\fed in the inelastic neutron scattering cross section, simulated from a microscopic model\nof LuFeO 3. The non-reciprocal magnon propagation is demonstrated.\nIntroduction { Multiferroics are a class of materials\nthat are attracting attention due to their promise in sens-\ning, IT and spintronic applications [1]. The contempo-\nrary presence of properties such as ferroelectricity, elec-\ntrostriction, magnetostriction, piezoelectricity, and mag-\nnetoelectricity [2] allows to build devices with interesting\nproperties related to e.g electric transport, heat trans-\nport, information storage and manipulation. Compet-\ning magnetic interactions often lead to peculiar magnetic\nstates and associated symmetry breaking, resulting in\nthe interaction between magnetism and structural dis-\ntortions. These states support a rich landscape of topo-\nlogical defects and elementary excitations that may en-\nable novel devices [2, 3]. Particular attention was con-\ncentrated on magnetic toroidal order [4, 5] and magnetic\nmonopoles [6{8] that give rise to peculiar magnetoelec-\ntricity, non-reciprocal e\u000bects [9] and also lead to peculiar\nexcitations and dynamics in spin ice [10, 11]. One of the\nreasons for such a variety of properties in these systems\nis the coupling between magnetic and electric degrees of\nfreedom, i.e. magneto-electric e\u000bect [12], which enables\nthe manipulation of currents and charges through mag-\nnetic \felds or engineering particular spin con\fgurations.\nAntiferromagnetic triangular lattices are a compelling\ncase of geometrically frustrated magnets because they\nhost peculiar orders and excitations that may be uti-\nlized in the next-generation electronic devices to manipu-\nlate information without electric currents, thus reducing\nheat dissipation [3]. Hexagonal manganites and ferrites\nRTMO 3(TM=Mn,Fe) are multiferroics with triangular\nlayers of magnetic ions, where unit cell-tripling buck-\nling of bipyramids (trimerization) induces electric polar-\nization [13, 14]. The energy landscape has six minima\nalong the rim of a distorted mexican hat, correspond-\ning to alternating directions of electric polarization[15].\nThat leads to trimerization vortices, at which six trimer-\nization domain meet, and polarization changes sign six\ntimes around a vortex core [15, 16]. In this work\nwe study hexagonal LuFeO 3(Fig. 1), a rare room-\ntemperature multiferroic possessing weak ferromagneticmoment [17, 18]. Iron atoms constitute a triangular lat-\ntice and the spin con\fguration is determined by the com-\npeting antiferromagnetic superexchange interactions be-\ntween Fe spins.\nFIG. 1. The structure of LuFeO 3showing triangular layers of\nFeO 5bipyramids interspaced with Lu layers.\nIn order to identify the INS signatures of various\nstates we compute magnon dispersion in LuFeO 3, sim-\nulate inelastic neutron scattering (INS) experiment and\ncompare the results with existing data[19]. We show\nsignatures of magnetic monopolar and toroidal orders\nthat are present in the lattice due to the combination\nof 120\u000eantiferromagnetism and trimerization that lifts\nthe cancellation of contributions from neighboring trian-\ngles. The presence of the toroidal moment gives rise to\nmagnon non-reciprocity along the caxis while the pres-\nence of monopoles is directly related to the emergence\nof magneto-electric (ME) e\u000bect with diagonal ME tensor\n[6, 8, 20]. Ab-initio calculations suggest the magnetoelec-\ntric tensor in LuFeO 3\u000bxx= 0:26 ps/m,\u000bzz=\u00003 ps/m\n[21].\nThe Model { In LuFeO3Fe has spin 5 =2 while Lu3+\nis non-magnetic. The system is composed of 2D trian-arXiv:1911.02704v1  [cond-mat.str-el]  7 Nov 20192\nFIG. 2. 120\u000espin con\fguration in 2D triangular lattice. The trimers with stronger in-plane interactions are marked with\ncolored triangles. An alternating pattern of toroidal (a) and monopolar con\fgurations (b) emerges in this ordered phase.\nSymmetry considerations for non-reciprocity: c) the presence of a toroidal moment allows non-reciprocal magnon propagation;\nd) Monopolar moment does not support k-linear invariant, hence the magnon is reciprocal; e) if wavevector and the magnetic\n\feld are in the [ ab] plane and perpendicular to each other, and the polarization along caxis is present, magnon non-reciprocity\nis possible.\ngular layers of Fe spins. The frustration due to antifer-\nromagnetic exchange on the triangular lattice results in\na 120\u000espin structure (Fig. 2). The single triangles that\nform the structure can have spins ordered in two ways,\nde\fning the toroidal and the monopolar con\fgurations,\nas shown in Fig. 2(a,b) respectively. In this framework\nthe Hamiltonian takes the following form:\nH=X\nij\u0010\nJij~Si\u0001~Sj+~Dij\u0001~Si\u0002~Sj\u0011\n+\nX\ni\u0010\n\u0000K(~Si\u0001~ ni)2+K0(Sz\ni)2\u0000g\u0016B~H\u0001~Si\u0011\n;(1)\nwhere the \frst term describes the nearest-neighbor AFM\nHeisenberg exchange Jij=Jand FM interlayer exchange\nJij=J0; the second { Dzyaloshinkii-Moriya (DM) in-\nteraction [22{24] between the nearest neighbors in the\nsame layer [25]. DM vectors were computed from exper-\nimental structural data as ~Dij=\u000bDM~ rij\u0002~\u000e, where~ rij\nare the vectors connecting Fe ions, and ~\u000eare the vec-\ntors connecting the middle of Fe-Fe line to the closest\noxygen, with \u000bDM = 0:05 meV/ \u0017A2. The term with\nKstands for the easy-plane anisotropy governed by the\nshifts~ niin theab-plane of apical oxygens in the trimer-\nized state. The term with K0accounts for the hard-axis\nanisotropy perpendicular to the layers, forcing spins into\nthe plane. The last term represents the interaction be-\ntween the spins and an external magnetic \feld ~Hwith\nthe gyromagnetic ratio g=\u00002. The parameter values\nJ= 2:8 meV,jJ0j= 0:3 meV,Hx= 2 T,K0= 0:3 meV,\njKj= 0:68 meV/ \u0017A2are chosen to reproduce the experi-\nmental INS spectra [19]. We started from the published\nparameter values [19] for JandK. In this case the spec-\ntra did not capture the experimentally observed gap at\n\u0000 point and the energy of the plateaux between Aand\nBpoints (Fig. 3). We then chose hard axis anisotropy\nand DM parameters for the simulated spectra to capture\nthese features, and adjusted Jto position the plateauxat the correct energy.\nOf a particular importance is the DM term that results\nfrom the displacements of oxygen ions away from the Fe-\nO-Fe bond center induced by trimerization and polariza-\ntion modes. This polarization drives the magneto-electric\ne\u000bect with a diagonal or o\u000b-diagonal magneto-electric\ntensor for monopolar and toroidal states, respectively.\nMagnon non-reciprocity | In the presence of inversion\nor time-reversal the magnons are reciprocal, in the pres-\nence of trimerization, the DM interactions break inver-\nsion and allow for non-reciprocity, giving rise to peculiar\ntransport properties. We can infer information about\nmagnons from symmetry considerations. Table I shows\nhow di\u000berent observables and orders transform under the\nsymmetry operations, Fig. 2 represents the possible cases\nof reciprocal or non-reciprocal magnons described by our\nsymmetry analysis. We can build the invariants entering\n!k;\u001bof the system by multiplying the signatures of the\ndi\u000berent quantities in the table.\n2001j(001\n2)2110I3zT\n(~ ri\u0002~Si)z +\u0000\u0000+\u0000\n~ ri\u0001~Si + +\u0000+\u0000\nkz +\u0000\u0000+\u0000\nPz +\u0000\u0000++\nHz +\u0000++\u0000\n(~k\u0002~H)z +\u0000\u0000++\nPhases\nA1 +\u0000\u0000+\u0000\nA2 + +\u0000+\u0000\nB1\u0000 +++\u0000\nB2\u0000\u0000++\u0000\nTABLE I. (Left) Transformation properties under the gen-\nerators of the symmetry group P6 3/mmc (#194 in the Inter-\nnational Tables). (Right) Di\u000berent possible spin orders, red\nand blue arrows represent spins from to two di\u000berent layers.3\nFIG. 3. (a) Constant energy cut of INS cross section in kx; kyplane at != 20 meV. The black line de\fnes the q-space path\nused to plot the cross section. (b) Magnon spectrum along the path de\fned in a) for a single layer ( J0= 0) of LuFeO 3with\nmonopolar spin con\fguration. Colorscale encodes the INS cross-section.\nThe following situations are possible:\n\u000fIf the toroidal moment along caxis is present then\nnon-reciprocal spin wave propagation along caxis\nis possible, and appears in the simulation as seen in\nFig. 4 (e). The product kz(~ ri\u0002~Si)zis in fact an in-\nvariant for the system, thus the magnon energy and\nthe INS scattering cross-section will have contribu-\ntions, proportional to kzand hence non-reciprocal:\n\u000e!/kz.\n\u000fThe presence of monopoles alone (with Pz= 0)\ndoes not induce non-reciprocity of spin wave prop-\nagation along caxis, since kz(~ r\u0001~S) is not an in-\nvariant. The absence of the linear k-term in!\nmeans reciprocal propagation, as seen in the simu-\nlated spectra in Fig. 4 (f).\n\u000fWhenkand~Hare both in the [ ab] plane and the\npolarization along caxis is present, non-reciprocity\nof the spin wave is possible. As before, the term\nkz\u0001[~H\u0002~P] is an invariant, meaning that a linear\nk-term in!is allowed . This can generate non-\nreciprocity even when the system is in a monopolar\ncon\fguration.\n\u000fMagneto-electric e\u000bect is possible in the presence\nof monopoles since the term ( ~ r\u0001~S)PzHzis allowed\nby symmetry.\nWe compute the magnetic susceptibility \u001fij(!;k) us-\ning linear spin wave theory as described in the Supple-\nmentary, and use it to evaluate the INS intensity. For a\nnon-polarized neutron beam the INS cross section due to\ndipole-dipole interactions with neutrons is given by [26]\nd2\u001b\ndEd\n\u0018X\nij\u0012\n\u000eij\u0000kikj\nk2\u0013\n\u001fij (2)\nThe imaginary part of susceptibility gives the magnon\nspectral function and represents all possible magnon ex-\ncitations in the material. The resulting simulated INScross section along the path, connecting high-symmetry\nk-points marked in Fig. 3, is shown in Fig. 3 for a single\nlayer of LuFeO 3.\nRepeating these calculations for toroidal and monopo-\nlar orders we obtain the spectra shown in Fig. 4, and iden-\ntify di\u000berences in the INS plots between monopolar and\ntoroidal con\fguration. The di\u000berences are subtle in the\nin-plane dispersion plotted in Fig. 4(a-b). However, they\nare evident in the kzdispersion, Fig. 4(e-f), as predicted\nby the symmetry analysis. These di\u000berences can there-\nfore be used to identify monopolar and toroidal orders\nin the future experiments. Supplementary Fig. S1, S2\nshows how these features depend on the strength of DM\ninteractions in all phases.\nThe toroidal and monopolar orders are stabilized by\nmanipulating the easy axis anisotropy term of Hamilto-\nnian (1) and the interlayer coupling J0. The reversal of\nsign ofKin the model turns the easy direction into a hard\none, thus rotating the easy axis by 90\u000ein the [ab] plane,\nwhile reversing the J0term allows for ferromagnetic or\nantiferromagnetic interlayer coupling hence allowing to\nselect the phase of interest. Panels (c-d) of Fig. 4 show\na close-up view near Dpoint, where the di\u000berences be-\ntween the two cases are evident.\nQuadrupole contributions | In addition to magnetic\nmonopoles, dipoles and toroidal moments, an extra term\ndue to quadropolar moment appears in the multipole ex-\npansion of the vector potential ~Aat the same order as\nthe toroidal moment [4]. While the monopole is the zero\norder term, the toroidal moment and the quadrupolar\nmoment both enter the expansion in the second order\nhA(2)\nquadii=\u0000\u000fijkqkl@j@l1\nR\nh~A(2)\ntori=r(~t\u0001r)1\nR+ 4\u0019~t\u000e(~R)(3)\nwhereqand~tare the quadrupolar and toroidal moments,4\nFIG. 4. Simulated INS spectra for (a) monopolar ( K > 0) and (b) toroidal ( K < 0) states, stabilized by in-plane easy axis\nanisotropy. Colorscale encodes the INS cross-section. Di\u000berences in the intensity of the signal can be seen between the two\n\fgures. Arrows highlight the most evident ones close to BandDpoints and between \u0000 and B; (c,d) Close-up view of the area\nin red from panels (a) and (b). Arrows point to the di\u000berences between the two con\fgurations. (e,f) The INS cross-section on\nthe peak between Cand \u0000 points along kzin the BZ. Magnon non-reciprocity is evident in the scattering cross section for the\ntoroidal order (e), while the magnon propagation is reciprocal in the monopolar state (f).\nwith~t=\u00001\n2g\u0016BP\n\u000b~ r\u000b\u0002~S\u000b, and\nqij=\u0000g\u0016B\n2X\n\u000b(S\u000bir\u000bj+S\u000bjr\u000bi): (4)\nHeregis the gyromagnetic factor and \u0016B{ the Bohr\nmagneton. The three-fold symmetry only allows for qzz\nto be non-zero. As the symmetry analysis shows, the\ntoroidal moment contributes to magnon non-reciprocity.\nIt's interesting to study if the quadrupolar moment con-\ntributes to the non reciprocity too, i.e.if there exists an\ninvariant, linear in ~k, that contains q. In the absence of\nexternal \felds the tensor qijcan only be contracted with\nthe vectors ~kand~P, and since ~Pis along (001) the only\npossible combination is kiqi3P3. Using Eq. (4), we verify\nthat the components qi3are present in the B1phase but\nare small compared to the contributions of the toroidal\nmoment in A1phase.\nConclusions | We presented the mechanism, through\nwhich magnetic monopolar and toroidal orders emerge\nfrom the combination of 120\u000eantiferromagnetism and\ntrimerization, present in hexagonal manganites and fer-\nrites. Symmetry considerations regarding the non-\nreciprocal propagation of magnons are presented and cor-\nroborated by the simulations, based on a realistic mi-\ncroscopic model and the spin wave approximation. The\nsimulated INS spectra allow to discriminate betweenmonopolar and toroidal orders. We hope the results\ncould pave the way to manipulating magnetic monopoles\nin hexagonal manganites and ferrites. The e\u000bects could\nbe useful in magnon-based devices and magnonic circuits,\nutilizing monodirectional magnon propagation.\nSWC was supported by the DOE under Grant No.\nDOE: DE-FG02-07ER46382.\n[1] D. Khomskii, Physics 2, 20 (2009).\n[2] S.-W. Cheong and M. Mostovoy, Nature Materials 6, 13\n(2007).\n[3] R. Ramesh and N. A. Spaldin, Nature Materials 6, 21\n(2007).\n[4] N. A. Spaldin, M. Fiebig, and M. Mostovoy, Journal of\nPhysics: Condensed Matter 20, 434203 (2008).\n[5] J. Lehmann, C. Donnelly, P. M. Derlet, L. J. Heyderman,\nand M. Fiebig, Nature Nanotechnology 14, 141 (2019).\n[6] N. A. Spaldin, M. Fechner, E. Bousquet, A. Balatsky,\nand L. Nordstr om, Phys. Rev. B 88, 094429 (2013).\n[7] F. Th ole and N. A. Spaldin, Philosophical Transactions\nof the Royal Society A 376(2018).\n[8] Q. N. Meier, M. Fechner, T. Nozaki, M. Sahashi,\nZ. Salman, T. Prokscha, A. Suter, P. Schoen-\nherr, M. Lilienblum, P. Borisov, I. E. Dzyaloshinskii,\nM. Fiebig, H. Luetkens, and N. A. Spaldin, Phys. Rev.\nX9, 011011 (2019).5\n[9] S.-W. Cheong, D. Talbayev, V. Kiryukhin, and A. Sax-\nena, NPJ Quantum Materials 3, 19 (2018).\n[10] D. I. Khomskii, Nature Communications 3, 904 (2012).\n[11] D. I. Khomskii, Nature Communications 5, 4793 (2014).\n[12] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,\nand Y. Tokura, Nature 426, 55 (2003).\n[13] B. B. Van Aken, T. T. M. Palstra, A. Filippetti, and\nN. A. Spaldin, Nature Materials 3, 164 (2004).\n[14] C. J. Fennie and K. M. Rabe, Phys. Rev. B 72, 100103\n(2005).\n[15] S. Artyukhin, K. T. Delaney, N. A. Spaldin, and\nM. Mostovoy, Nature Materials 13, 42 (2014).\n[16] T. Choi, Y. Horibe, H. T. Yi, Y. J. Choi, W. Wu, and\nS.-W. Cheong, Nature Materials 9, 253 (2010).\n[17] W. Wang, J. Zhao, W. Wang, Z. Gai, N. Balke, M. Chi,\nH. N. Lee, W. Tian, L. Zhu, X. Cheng, D. J. Keavney,\nJ. Yi, T. Z. Ward, P. C. Snijders, H. M. Christen, W. Wu,\nJ. Shen, and X. Xu, Phys. Rev. Lett. 110, 237601 (2013).[18] S. M. Disseler, X. Luo, B. Gao, Y. S. Oh, R. Hu, Y. Wang,\nD. Quintana, A. Zhang, Q. Huang, J. Lau, R. Paul, J. W.\nLynn, S.-W. Cheong, and W. Ratcli\u000b, Phys. Rev. B 92,\n054435 (2015).\n[19] J. C. Leiner, T. Kim, K. Park, J. Oh, T. G. Perring, H. C.\nWalker, X. Xu, Y. Wang, S.-W. Cheong, and J.-G. Park,\nPhysical Review B 98, 134412 (2018).\n[20] F. Th ole, M. Fechner, and N. A. Spaldin, Phys. Rev. B\n93, 195167 (2016).\n[21] M. Ye and D. Vanderbilt, Phys. Rev. B 92, 035107\n(2015).\n[22] I. E. Dzyaloshinskii, Sov. Phys. JETP 10, 628 (1960).\n[23] I. E. Dzyaloshinskii, Sov. Phys. JETP 19, 960 (1964).\n[24] T. Moriya, Phys. Rev. 120, 91 (1960).\n[25] H. Das, A. L. Wysocki, Y. Geng, W. Wu, and C. J.\nFennie, Nature Communications 5, 2998 (2014).\n[26] J. Jensen and A. R. Mackintosh, \\Rare earth magnetism:\nStructures and excitations,\" (Clarendon Press, Oxford,\n1991) p. 174.6\nSUPPLEMENTARY INFORMATION\nCALCULATION OF THE INELASTIC NEUTRON SCATTERING CROSS SECTION\nIn order to compute INS intensity, Hamiltonian (1) is expanded to the second order in the deviations of spherical\nangles of classical spins ~Si(\u0012i;\u001ei) from their ground state values, \u0012i=\u0012i0+\u000bi; \u001ei=\u001en\ni+\fi.\u001en\niis governed by the\neasy axis direction ^ ni. The dynamics of ( \u000bi;\fi) near the energy minimum are governed by Hamilton equations,\nsin\u0012i0_\u000bi=\u0000@H\n@\fisin\u0012i0_\fi=@H\n@\u000bi(5)\nIn the Fourier space the Eq. (5) take the form of an eigenvalue problem,\n(A\u0000i!1) \n\u000bk\n\fk!\n= 0; A= \n\u0000@\fi\u000bj\u0000@\fi\fj\n@\u000bi\u000bj@\u000bi\fi!\nH; (6)\n\fj=eikr j\u0000i!t\fk; \u000bj=eikr j\u0000i!t\u000bk (7)\nwhereAis the 2n\u00022nmatrix of the second derivatives of the Hamiltonian with respect to \u000bk,\fkandnis the number\nof spins in the unit cell. The neutron beam used in INS is modelled with an external time-dependent magnetic \feld\n~h. The equations of motion describing the steady-state dynamics driven by the neutron beam now take the form:\n(A\u0000i!1) \n\u000bk\n\fk!\ne\u0000i!t= \nh\u000b\nh\f!\ne\u0000i!t; (8)\nwithh\u000bandh\fbeing the terms of ~S\u0001~hlinear in\u000biand\fi. The steady-state response appears at the frequency !of\nthe oscillating magnetic \feld associated with neutrons INS experiments. After solving Eq. 8 for all !we express \u000bk\nand\fkas!dependent and \fnd the magnetic susceptibility tensor as\n\u001fij(!;k) =g\u0016B\nV@S(i)\nk\n@h(j)(\u000bk;\fk\u0006Q); (9)\nwhereQis the wave vector of the spin texture. For a non-polarized neutron beam the INS cross section due to\ndipole-dipole interactions with neutrons is given by [26]\nd2\u001b\ndEd\n\u0018X\nij\u0012\n\u000eij\u0000kikj\nk2\u0013\n\u001fij (10)\nINELASTIC NEUTRON SCATTERING SPECTRA FOR DIFFERENT PHASES AND THEIR\nDEPENDENCE ON THE DZYALOSHINSKII-MORIYA INTERACTION STRENGTH\nFig. S1, S2 illustrate the e\u000bect of Dzyaloshinskii-Moriya interactions on the INS spectra in di\u000berent magnetic phases.\nThey show how DM increases the spin wave bandwidth and enhances the non-reciprocal INS signatures in A1and\nB2phases, although the non-reciprocity is already evident at zero DM strength in A1phase. With increasing DM,\nthe spins deviate slightly from the 120\u000econ\fguration. The two sets of bands, separated at zero DM strength, merge\ntowards\u000bDM= 0:5 meV/ \u0017A2, as seen in Fig. S1. The bandwidth of the overall dispersion in the hexagonal plane\nincreases with \u000bDM, as seen in Fig. S2. The dispersion along caxis di\u000bers signi\fcantly for all phases, and therefore\nbe used in order to distinguish them.7\nFIG. S1. INS cross-section on the peak between Cand \u0000 points along kzin the BZ for di\u000berent phases and for di\u000berent\nvalues of Dzyaloshinskii-Moriya interaction. The values of the model parameters are J= 2:8 meV,jJ0j= 0:3 meV, hx= 2 T,\nK0= 0:3 meV,jKj= 0:68 meV/ \u0017A2.\nFIG. S2. Simulated INS spectra for di\u000berent phases and di\u000berent values of Dzyaloshinskii-Moriya interaction. The values of\nthe model parameters are J= 2:8 meV,jJ0j= 0:3 meV, hx= 2 T, K0= 0:3 meV,jKj= 0:68 meV/ \u0017A2."    },    {        "title": "1401.2099v1.Mechanical_behavior_of_steels__from_fundamental_mechanisms_to_macroscopic_deformation.pdf",        "content": "UNIVERSITE DE LORRAINE - INSTITUT NATIONAL POLYTECH NIQUE DE LORRAI NE \nECOLE DOCTORALE EMMA \n \n \n \n \n \n \nMémoire d’Habilitation à Diriger des Recherches  \n \nprésen té par \n \nSébastien ALLAIN \n \n \n \n \n \nComportement mécani Comportement mécani Comportement mécani Comportement mécanique des aciers que des aciers que des aciers que des aciers    : des mécanismes : des mécanismes : des mécanismes : des mécanismes \nfondamentaux à la déformation macroscopique fondamentaux à la déformation macroscopique fondamentaux à la déformation macroscopique fondamentaux à la déformation macroscopique     \n \n \n \n \n \n \nSoutenu publiquement le 6 Décembre 2012 à l’Institu t Jean Lamour, devan t le jury composé de : \n \nMr Marc FIV EL  Directeur de Recherche CNRS, SIM A P G renoble Rapporteur \nMr Xavier FEAUGAS  Professeur, LEMMA La Rochelle   Rapporteur \nMr Armand COUJOU  Professeur, CEMES Toulouse    Rap porteur \nMr Javier GIL SEVILLANO Professeur, CEIT-TECNUM  San  Sebastian  Examinateur \nMr Michel VERGNAT  Professeur, IJL Nancy     Examin ateur \nMr Mikhail LEB EDKIN Directeur de Recherche CNRS, LE M 3 Metz  Examinateur \nMr Olivier BOUAZIZ  Docteur HDR, Arcelormittal Maiz ières les Metz Examinateur \nMr Alain JACQUES  Directeur de Recherche CNRS, IJL Nancy  Parrain Scientifique \nMr Thierry  IUNG  Docteur, Arcelormittal Maizières l es Metz  Invité \n \nDisciplines: Milieux Den ses et Matériaux, Chimie de s Matériaux, Mécanique \n \nArcelorm ittal Maizières Research SA, Voie Romaine, BP 30320 F-57283 Maizières les Metz \n \n  1   \n \n \n \n \n \n \n \n \n \n \n \n \n \nA ma Femme, Nathalie, \nmes Filles, Fanny, En ora \net Chloé, mon  petit ange \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  2   \n \n  3  Remerciements Remerciements Remerciements Remerciements    \n \nLa pluspart des travaux présentés dans ce mé m oire o n t été financés e n totalité ou en partie \npar le groupe Arcelormittal.  \n \nJe suis tout d’abord très reconnaissant en vers les rapporteurs de ce m é moire, Arman d Coujou, \nXavier Feaugas et Marc Fivel d’avoir accepté très s pontan ément cette charge et m’avoir \napporté par leurs grandes e xpériences de n ouveaux q uestionnem ents sur mes travaux. \nJ’aim erais aussi remercier chaleureusem e nt Javier G il Sevillano pour l’in térêt qu’il a pu \nman ifester pour m es travaux, son  accueil à de n ombr euses reprises au CEIT et notre fructeuse \ncollaboration. Merci aussi à Mikhail Lebedkin pour sa participitation à mon jury, sa confiance \nau cours de ces dernières années et sa patie nce de pédagogue avec moi sur les problématiques \nde viellissem e nt dynamique. Je voudrais égalem e nt t ém oigner de toute ma gratitude à Michel \nVergnat pour avoir accepté de pr ésider le jury et d irigé les discusssions lors de la soutenance. \nMerci aussi à Alain  Jacques pour sa confiance dans mon  poten tiel de chercheur, ses conseils et \npour son en gagement dans ma démarche d’habiliation.  \n \nJe souhaite exprim er toute ma gratitude et m on am it ié à Jean-Philippe Chateau-Cornu et \nOlivier Bouaziz, que je considère mes m entors en sc iences des matériaux, qui m’ont dirigé en \nDEA et en thèse, puis accompagn é et en fin e ncouragé  dans m a démarche de recherche depuis \nmain te nant 12 ans. Merci aussi à Colin Scott et Moh am ed Gouné leurs amitiés, leurs soutiens \ndans les moments les plus durs et nos grandes discu ssions en métallurgie au cours de ces \ndern ières an nées.  \n \nJe voudrais aussi remercier Thierry Iun g pour m’avo ir offert un cadre dès mon arrivée ch ez \nArcelor pour poursuivre mes travaux scie ntifiques d ans un  contex te industriel et soutenu \ndans ma démarche d’habilitation d’un point de vue a cadémique. Sa relecture attentive et la \npertinence des questions soulevées m’on t permis d’a méliorer significativement le m émoire. Je \ntenais aussi à rem ercier Jean-Hubert Schmitt et Dav id Em bury, qui m’ont apporté une \ncertaine motivation exterieure et auprès de qui j’a i toujours pu toujours trouver une écoute \natten tive et de nouvelles idées ces dernières ann ée s. \n \nComme l’a soulign é Olivier lors de la soutenan ce, n ous avon s eu la chance de vivre avec le \ndéveloppem ent des aciers TWIP à Arcelor une véritab le « ave nture », en toute in souciance, \nsans anticiper et compren dre les implications de no s travaux  au début des années 2000. Ces \ntravaux m’ont apporté depuis une certain e reconnais sance de la communauté et m’ont permis \nde rencontrer et collaborer avec des chercheurs for midables, que je souhaiterais rem ercier ici. \nJe pense à N icolas Guelton, Philippe Cugy et Michel  Faral et à l’équipe FeMn. J’ai aussi une \npensée particulière pour Maurita Roscini et son com bat.  4   \nJe voudrais remercier et f éliciter pour leurs trava ux les doctorants que j’ai co-encadré, Alexis \nDum ay, Steven Dillie n et en particulier David Barbi er et Jean-Christophe Hell, maintenant \ncollègues à Arcelormittal et ceux  avec qui j’ai sim plem ent collaboré, Najeeb Shiekhelshouk, \nBenoit Krebs, Ir ina Pushkareva, Krystel Renard, mai s aussi Anis Aouafi, Jean -Philippe Masse, \nGuillaume Badinier, mainte ntant tous les trois auss i collègues.  \n \nLa recherch e scien tifique est devenue de nos jours le fruit d’une activité collaborative. Il est \nen effet humaine m ent, c’est en tout cas vrai pour m oi, difficile de pouvoir embrasser seul \nl’ensemble des connaissances, des techniques et des  résultats concern ant un sujet. J’ai donc au \ncours de ces années, eu le plaisir et même souven t le privilège de collaborer avec de \nnom breux cherch eurs issus non seuleme nt du mon de in dustriel mais aussi du monde \nacadé mique. J’aimerais rem ercier à ce titre Chad Si nclair, Hateem Zurob, Francisca Caballero, \nCarlos Garcia-Mateo, Frédéric et Raphaele Danoix , X avier Sauvage, Alain Iost, Gildas \nGuille mot, Paul Van Houtte, Marc Seefeldt, Alexis R usinek, Moh amed Cherkaoui, Alain \nHazotte, Lionel Germain, Nathalie Gey, Tatiana Lebe dkina, Amandin e Roth, Michel \nHumbert, Diego Gonzales, Juan-Manuel Esnaola, Man ab u Takahash i, M ixing Huang, Pascal \nJacques, Marc Legros, Sylvie Migot, Moukrane Dehmas , les m e mbres des équipes 304 et 303 \nde l’Institut Jean Lamour, et bien entendu m es coll ègues d’Arcelormittal, surtout Xavier \nLem oine, Jean-Luc Christen, Dominique Kaplan, Miche l Soler, Antoine Bui-Van, Astrid \nPerlade, Artem Arlazarov, Kevin Tihay, Philippe Mau gis, Catherine Vinci, Juan Puerta \nVelasquez, Antoin e Moulin, Jan  Mahieu, François Mud ry, Michel Babbit ainsi que tous les \ningénieurs, techniciens et secrétaires des équipes MPM, AH SS et AUP2 et du campus de \nMaizières avec qui j’ai pu collaborer. Après mainte nan t plus de 12 ans d’activités \nprofessionn elles et de collaborations, il est toujo urs délicat de se livrer à cet ex ercice des \nrem erciements et j’espère très sincèrem ent n’avoir om is personne, sans quoi, je m ’en e xcuse \npar avance. \n \nMerci aussi à Sylviane Tranchan t pour sa bonn e hum e ur permanen te et l’organisation de la \nsoutenance. \n \nPlus personnellem e nt, j’aimerais remercier Mohamed,  Olivier, Thierry et Tibault pour avoir \nété là dans les très mauvais mom ents de 2009. La pe rte d’un  enfant rend inconsolable mais \nleur soutie n  m’a permis de sur m onter cette épreuve.  Je voudrais aussi rem ercier m a femm e, \nNath alie, qui m’a aidé non seulement à la relecture  du mém oire m ais m’a aussi permis d’y \nconsacrer le temps nécessaire. Merci de m’avoir sup porté, dans les différentes acceptations du \nterm e. Merci enfin à mes filles, Fanny et Enora pou r avoir égayé durant cette période m es \njours, et surtout mes nuits … \n \n  5  Table des Matières Table des Matières Table des Matières Table des Matières    \n \nRemerciements...................................... ................................................... ........................................... 3  \nTable des Matières................................. ................................................... ........................................... 5  \nSigles et localisation des entités de recherche cit ées ................................................ .................................. 7  \nRésumé............................................. ................................................... ............................................... 9  \n1.  Introduction – démarche scientifique............... ................................................... ..........................11  \n2.  Microstructure et comportement des aciers austéniti ques FeMnC à effet TWIP............................ .....15  \n2.1.  Introduction....................................... ................................................... ..............................15  \n2.2.  Morphogénèse de la microstructure de maclage....... ................................................... ...............21  \n2.3.  L’effet TWIP : La relation entre maclage mécanique et comportement.................................... ........44  \n2.4.  Effet de la composition chimique : le rôle particul ier du carbone..................................... ..............67  \n2.5.  Conclusions et Perspectives........................ ................................................... .........................92  \n2.6.  Pour le plaisir des yeux........................... ................................................... ...........................93  \n3.  Comportement des aciers Dual-Phase ; des composites  modèle ?.......................................... ............94  \n3.1.  Introduction....................................... ................................................... ..............................94  \n3.2.  Extension d’un modèle monophasé en plasticité polyc ristalline pour des applications en rhéologie \nappliquée.......................................... ................................................... .........................................101  \n3.3.  Plateforme de modélisation « générique » des aciers  Ferrite-Martensite................................ .......107  \n3.4.  Modélisation par EF de VER : Effet de la morphologi e de la martensite................................. ......137  \n3.5.  Conclusions et perspectives........................ ................................................... .......................150  \n4.  Conclusion personnelle............................. ................................................... ..............................152  \n5.  Références bibliographiques........................ ................................................... ............................153  \n6.  Annexe............................................. ................................................... ....................................163  \nCurriculum vitae................................... ................................................... ........................................165  \nRapport de soutenance.............................. ................................................... .....................................167  \n \n \n \n \n \n  6    7  Sigles Sigles Sigles Sigles et localisation des entités de recherche et localisation des entités de recherche et localisation des entités de recherche et localisation des entités de recherche cité  cité  cité  citéee eess ss    \n \nAM AM AM AM : Arcelormittal, Arcelormittal M aizières Researc h SA, Maizières les Metz, FRANCE \n Ancienne m ent IRSID, Institut de Recherche SIDérurg ique \n \nNSC NSC NSC NSC : Nippon Steel Corporation, Steel Research  Labo ratories, Futtsu, JAPON  \n \nIJL IJL IJL IJL  : Institut Jean Lamour, Université de Lorrain e,  Nancy, FRANCE \n \nLPM LPM LPM LPM : Laboratoire de Physique des Matériaux, Ecole des Mines de Nancy, Nancy, FRANCE \n Main te nant intégré dans l’IJL \n \nLSGS LSGS LSGS LSGS : Laboratoire de Science et Génie des Surfaces , Ecole des Mines de Nancy, Nan cy, \nFRANCE \n Main te nant intégré dans l’IJL \n \nLEM3 LEM3 LEM3 LEM3 : Laboratoire d’Etude des Microstructures et M écanique des Matériaux, Université Paul \nVerlaine, Metz, FRANCE \n \nLETA LETA LETA LETAM M MM : Laboratoire d’Etude des Te xtures et Application  aux Matériaux , Université Paul \nVerlaine, Metz, FRANCE \n Main te nant intégré dans le LEM3 \n \nLPMM LPMM LPMM LPMM : Laboratoire de Physique et Mécanique des Mat ériaux, Université Paul Verlaine, \nMetz, FRAN CE \n Main te nant intégré dans le LEM3 \n \nCEIT CEIT CEIT CEIT- - --TECN UN TECN UN TECN UN TECN UN : Centro de Estudios e In vestigaciones Técni cas de Guipúzcoa, Technological \nCam pus of the University of Navarra, San Sebastian,  ESPAGNE \n \nGPM GPM GPM GPM : Groupe de Physique des Matériaux, Université de Rouen, Saint Etienne du Rouvray, \nFRANCE \n \nHK University HK University HK University HK University : Th e University of Hong Kong, Depart ment of Mechanical Engineering, Hong \nKon g, CHIN E \n \nICMCB ICMCB ICMCB ICMCB : In stitut de la Matière Condensée de Bordeau x, Université de Bordeaux, Pessac, \nFRANCE \n  8  KUL KUL KUL KUL : Kath olieke Universiteit Leuven, Departm ent of  Metallurgy an d Materials En gineering, \nHeverlee, BELGIQUE \n \nLPMTM LPMTM LPMTM LPMTM : Laboratoire des Propriétés Mécaniques et The rmodynamiques des Matériaux, \nUniversité Paris XIII, Villetaneuse, FRANCE \n \nMcMaster University McMaster University McMaster University McMaster University : McMaster Un iversity, Departme n t of Materials Science and \nEngineering, Hamilton, CANADA \n \nMon ash University Mon ash University Mon ash University Mon ash University : Monash University, Departmen t o f Materials Engineering, Clayton, \nAUSTRALI E \n \nMPI MPI MPI MPI : Max Planck Institute, Düsseldorf, ALLEMAGNE \n \nSIMAP SIMAP SIMAP SIMAP : Laboratoire de Science et Ingénierie des Ma tériaux  et Procédés, Université Joseph \nFourier, St Martin d’Hères, FRAN CE \n \nUBC UBC UBC UBC : The University of British Columbia, Departmen t of Materials Engineering, Vancouver, \nCAN ADA \n \nUCL UCL UCL UCL : Université Catholique de Louvain, Département  des Sciences des Matériaux et des \nProcédés, Louvain la Neuve, BELGIQUE \n \n  9  Résumé Résumé Résumé Résumé    \nMon  activité de recherche scientifique concerne pri ncipale ment la compréhen sion et la \nmodélisation du comportement mécaniques des aciers ; des mécanismes fondamentaux à la \ndéformation  macroscopique. Ce mém oire est con sacré en particulier à l’effet TWIP \n(TWinning Induced Plasticity) des aciers austénitiq ues FeMn C à haute teneur en m angan èse \net l’effet Dual-Phase des aciers Ferrite-Martensite . \nL’effet TWIP est un mécanisme d’écrouissage spécifi que des aciers austénitiques FeMnC lié à \nun processus de maclage m écanique, mécanism e de déf ormation com pétitif au glisseme nt des \ndislocations. L’accumulation de ces macles, défauts  plans d’épaisseur nanom étrique, crée au \ncours de la déform ation une microstructure enchevêt rée et difficilem ent franchissable par les \ndislocations mobiles à l’intérieur des grains austé nitiques. Au cours de nos travaux, ces \nmicrostructures on t été expliquées et quantifiées à  différentes échelles. Nous avons ainsi pu \nmodéliser la double contribution  du m aclage à l’écr ouissage grâce à une augmentation  de \ndensité de dislocations statistiquement stockées et  à une contribution  de nature cin ématique, \nassociée à l’incom patibilité de déform ation entre m acles et matrice. L’influence de ce \nmécanisme a en conséquence été mieux comprise lors de trajets de mise en forme complex es. \nAfin  de pouvoir optimiser le comportem ent mécanique  de ces aciers TWIP, notre second axe \nde recherche a porté sur l’effet de leurs compositi ons chimiques sur ces m écanism es \nd’écrouissage, en particulier au travers de la rela tion entre m aclage mécanique et énergie de \ndéfaut d’em pilement (EDE). Ces travaux ont débouch é s sur l’identification  d’un « paradoxe \ncarbone » que nous sommes en passe de résoudre. \nMes travaux  ultérieurs de modélisation du com portem ent des aciers Dual-Phase (DP) Ferrite-\nMartensite se sont aussi attachés à décrire systéma tique ment les effets de fraction et de tailles \ndes m icrostructures. Ils on t eu différentes finalit és : \n• l’extension en plasticité polycristalline d’un m odè le m onoph asé analytique pour des \napplications en rhéologie appliquée (prévision des surfaces de charges sous \nsollicitation s comple xes).  \n• le développement d’un m odèle biphasé générique pour  des utilisations en « alloy-\ndesign » métallurgique. Le modèle intègre en outre nos travaux les plus récents sur les \naciers martensitiques (Approche Composite Continu) et a été ajusté sur  une large base \nde données issues de la littérature. \n• l’approfondissemen t de nos conn aissances sur les ef fets de morphologie et de topologie \nde la microstructure DP sur le comporteme nt et la r upture de ces aciers composites. Il \npasse par le développemen t d’une chaîn e de sim ulati on  à champs locaux par Eléments \nFinis (EF), allant de la num érisation aux calculs s ur Volume Elémentaire Représentatif \n(VER) de la microstructure, sensibles aux gradients  de déformation , et intégrant les \nmécanismes d’endommageme nts pertin ents. L’approche est encore incomplète mais \nperm et de traiter des question s au premier ordre co mme l’aspect néfaste d’une \nstructure en  bandes sur l’endomm agement.  10    11  1. 1. 1. 1.  Introduction Introduction Introduction Introduction    –– –– démarche scientifique  démarche scientifique  démarche scientifique  démarche scientifique    \n \n« Les hommes construisent trop de murs et pas assez  de ponts » \nIsaac Newton \n \nMon  activité de recherch e scientifique concerne pri ncipale ment la compréhen sion et la \nmodélisation du comportement mécaniques des aciers ; des mécanismes fondamentaux à la \ndéformation  macroscopique. Cette connaissance du li en entre microstructure et propriété est \nstratégique pour un sidérurgiste. En effet, celui-c i produit dans ses usines des microstructures \net des revêtements particuliers, mais il ve nd et ga ran tit des propriétés d’emploi et d’usage \ndurables dans le temps. \n \nMa formation initiale en métallurgie physique, de «  plasticien », m’a donc mis à l’interface \nentre « métallurgistes » et « mécanicien s » et le s e ns de mon action a finale ment été de \nconstruire des pon ts entre ces différents métiers. Mes travaux , à la fois expérimen taux et de \nmodélisation, ont toujours visé à mettre en  évidenc e quels son t le ou les paramètres \nmicrostructuraux pertin ents pouvan t expliquer les c omportements m écaniques \nmacroscopiques, et ce, avec le souci constant de va lider la démarche à plusieurs échelles \n(structuration de la plasticité) ou sous plusieurs angles (contraintes internes, effets de la \nvitesse de déformation ou de la température). \n \nMes principales productions « pratiques » ont ainsi  été : \n• soit des outils pratiques à destination des « métal lurgistes », et utilisables par des non \ninitiés en mécanique, pour des travaux d’optimisati on (par exem ple, un modèle \nd’én ergie de défaut d’em pileme n t en fonction  de la composition chimique pour les \naciers austénitiques FeMn C ou un modèle générique d e comporteme n t des aciers DP à \ndes fins « d’alloy design »), \n• soit des modèles de comportement à base microstruct urale pour les « mécan iciens » et \nles rhéologues (par exemple, des modèles de comport ement à composantes isotrope et \ncinématique pour les aciers ferrito-perlitiques ou un m odèle de sensibilité à la vitesse \npour les aciers ferritiques). \nCes travaux  s’appuient sur une bonne connaissance d es m écanism es fon damentaux de la \nplasticité. En fonction des travaux antérieurs de l a littérature, j’oriente et dirige des activités \nde recherche de nature plus expérim entale pour éluc ider la nature des m écanism es \nd’écrouissage (par exemple, mes travaux sur la micr ostructure de maclage à différen tes \néchelles de la microscopie électronique à transm iss ion (MET) à la microscopie optique(MO)).  \n \nMes centres d’intérêt actuels con cernen t la compréh ension fine de phé nomènes h autem ent \ncouplés comme le paradox e carbone dans les aciers T WIP ou dans lesquels les hétérogénéités  12  spatio-temporelles de contraintes et de déformation  jouent un rôle particulier (les \nmécanismes de localisation et d’endommagement des a ciers DP, les m écanismes d’écrouissage \ndes aciers martensitiques ou les m écanism es de viei llisse ment dynamique). \n \nCette position particulière à l’interface de nombre uses spécialités e xplique la diversité de m es \nactivités de recherche. Le premier support de ces é tudes a été pour moi les aciers \nausténitiques à hautes teneurs en man ganèse présen t ant un effet TWIP (pour Twinning \nInduced Plasticity) au cours de m a thèse. Ces acier s «redécouverts» depuis les années 1990 par \nles grands sidérurgistes présenten t de formidables potentiels pour la construction automobile \nen particulier grâce à des mécanismes de déformatio n et d’écrouissage multiples et en \ninteractions complexes (glisse m ent des dislocations , maclage mécanique, transfor mation \nmartensitique induite, vieillissem e nt dynamique). C ette gran de rich esse en fait un système \nd’étude intéressant à différentes échelles, des int eractions entre dislocations / solutés aux \nproblématiques de mise en forme et de rupture en pa ssant par la nature de la microstructure \nde m aclage mécanique.  \nDepuis mon  arrivée en 2004 à l’I RSID, le plus grand  centre de Rech erche et Développem ent \nd’Usinor (m ainte nant AM), j’ai aussi eu l’opportuni té de diversifier m on champ d’applications \net de connaissances en travaillant sur le comportem ent mécanique des nombreuses phases \nferritiques « basses températures » qu’offre le sys tème FeMnC (comm e les aciers bainitiques, \nmartensitiques, perlitiques ou tout simpleme nt ferr itiques).  \n \nA titre d’illustration, la Figure 1 m ontre le posit ionne m ent des sujets de mes publications dans \ndes revues à comité de lecture par rapport aux diff érentes familles d’aciers utilisés dans la \nconstruction autom obile. Elles sont représentées tr aditionnellement dans un plan  résistance \nmécanique / allongeme nt à rupture. \n \n \nFig ure Fig ure Fig ure Fig ure 1 1 11    : Po : Po : Po : Positionn ement relatif des sitionn ement relatif des sitionn ement relatif des sitionn ement relatif des sujets de  sujets de  sujets de  sujets de  mes  mes  mes  mes publications internationales depuis 2002 par r a pport aux  publications internationales depuis 2002 par ra ppo rt aux  publications internationales depuis 2002 par ra ppo rt aux  publications internationales depuis 2002 par ra ppo rt aux \ngrandes familles d’aciers utilisés dans le domaine  automobile. grandes familles d’aciers utilisés dans le domaine  automobile. grandes familles d’aciers utilisés dans le domaine  automobile. grandes familles d’aciers utilisés dans le domaine  automobile.     13   \nMa position  de ch ercheur-ingén ieur de recherche dan s le secteur privé m ’a amené aussi à \ndiriger des projets de rech erche de développement p roduit court et m oyen terme. En tant que \nchef de projet, ma mission est d’affecter les resso urces, assurer le suivi techn ique et le \nman ageme n t de la qualité des projets. Cette positio n particulière m’a aussi permis de conduire \ndes discussions avec des experts de différents doma ines de la sidérurgie (de l’acier liquide à la \nmise en forme chez les clie nts constructeurs automo biles). A ce titre, je suis actuelleme nt très \nimpliqué dans l’industrialisation de produits relev ant des aciers THR (Très H aute Résistan ce) \ndits « de troisième génération » (3rd Gen sur Figur e 1) et dans la coordination technique et \nscien tifique des activités associées à ces métallur gies spécifiques (Quenching and Partionning \nsteels, Carbide-Free bainitic steels, TRIP steels w ith annealed marten sitic matrix par exemple). \nCette activité s’est concrétisée par la rédaction  d e deux brevets produits et procédés sur ces \naciers. Cependant, mes travaux « publics » relevant  de cette thématique sont encore peu \nnom breux et concernent principaleme n t des études m é tallurgiques sur les aciers bainitiques \nsans carbure. J’ai fait le ch oix de ne pas détaille r ces travaux dans ce mé moire, mais le lecteur \npourra se reporter à quelques références bibliograp hiques ( [ALLAIN  2008_3][HELL \n2011_1][HELL 2011_2] ). \n \nPour ce mémoire, j’ai ch oisi un e grille de lecture de mes travaux  associée non  pas à des \nmécanismes ou des échelles métallurgiques, mais à d es microstructures particulières.  \nCe m émoire est donc divisé en deux grands chapitres  : \n• le premier est d édié à la mise en perspectives de m es travaux sur les aciers \nausténitiques FeMn C TWIP dans un con te xte in ternati onal très actif. \n• le second est consacré aux  aciers ferritiques Dual- Phase. C’est aussi une occasion pour \nprésenter mes travaux récents et en cours sur le co mportement des aciers \nmartensitiques.  \nCe choix permet de maintenir un e unité dans les con cepts micromécaniques utilisés, mê m e si \nde n ombreuses sim ilitudes existent entre ces deux s ystèmes, comme je le démontrerais. Cet \nordre de présentation résulte d’un e mise en perspec tive chron ologique de m es travaux et non \nde la comple xité des systèmes étudiés, bien au cont raire. Il permettra par contre de montrer \ncomment certains concepts développés pour les acier s TWIP ont ain si été adaptés aux aciers \nDP. \n \nCes travaux qui on t pour la plupart fait l’objet de  publications scie ntifiques sont bien entendu \ndes œuvres collectives, réalisées en partie en asso ciation avec le monde acadé mique et \nuniversitaire et en  partie avec certains chercheurs  du secteur privé, collègues d’Arcelormittal \nou concurrents, que je tiens à rem ercier ici encore . Les références aux travaux que j’ai dirigés \nou auxquels j’ai été associé seront indiquées en bl eu dan s la suite. \n \nLa rédaction  de ce mémoire a aussi été pour m oi l’o ccasion de finaliser et discuter certain es \nétudes non publiées à ce jour. Je pen se en particul ier aux comparaison s entre modèles  14  cinématiques et composites de l’effet TWIP (cf. cha pitre 2.3.5 page 52), au m odèle biphasé de \nl’effet DP (cf. chapitre 3.3 page 107) ou d’autres disponibles uniquem ent dans des travaux  de \nthèse comm e l’étude des change m ents de trajets sur les aciers TWIP et la modélisation des DP \npar EF. Ces résultats importants d’un point de vue scientifique et révélateurs de ma dé marche \nscien tifique ont donc été détaillés dans ce manuscr it. Je suis conscie nt que ce choix peut \nmalh eureuse ment nuire dans certaines parties à l’es prit de synthèse qui doit anim er un tel \nmém oire. \n  15  2. 2. 2. 2.  Microstructure et comportement des aciers austéniti ques FeM nC à Microstructure et comportement des aciers austéniti ques FeM nC à Microstructure et comportement des aciers austéniti ques FeM nC à Microstructure et comportement des aciers austéniti ques FeM nC à \neffet TWIP effet TWIP effet TWIP effet TWIP    \n \n« C’est l’incon nu qui m’attire. Quan d je vois un éc heveau bien enchevêtré  \nje me dis qu’il serait bien de trouver un fil condu cteur » \nPierre Gilles de Gen nes \n2.1. 2.1. 2.1. 2.1.  Introduction Introduction Introduction Introduction    \n2.1.1. 2.1.1. 2.1.1. 2.1.1.  Con texte Con texte Con texte Con texte    technique et technique et technique et technique et industriel industriel industriel industriel    \n \nLes aciers austénitiques TWIP (TWinning Induced Pla sticity) à haute teneur en m angan èse \nfont actuelle ment l’objet d’un en goue m ent important  dans la communauté des matériaux  et \nde la métallurgie. Cet intérêt est non seule ment dû  à des perspectives d’application structurale \nimportantes dans le dom aine de construction autom ob ile mais aussi à la rich esse et la \ncomplexité de questions scie ntifiques posées par ce tte m étallurgie. \n \nCon crètement, il se traduit depuis ces 5 dernières ann ées par une augme n tation importante \ndes publications scie ntifiques et de brevets dédiés  à ces aciers comm e le m ontre la Figure 2. \nUne première con férence dédiée spécifiquem e nt à ces  aciers a même récem ment été \norganisée en  Corée, souten ue par un sidérurgiste d’ envergure internationale (1st International \nCon ference on High Man ganese Steels - HMn S 2011 - S éoul) et des num éros spéciaux de \ngran des revues scientifiques ont été consacrés à ce tte m étallurgie ([VIEWPOINT 2012]). \n \n \nFigure Figure Figure Figure 2 2 22    : Evolution du nombre de publications et de brevet s conc ernant les aciers FeM nC  TWIP : Evolution du nombre de publications et de brevets  conc ernant les aciers FeM nC  TWIP : Evolution du nombre de publications et de brevets  conc ernant les aciers FeM nC  TWIP : Evolution du nombre de publications et de brevets  conc ernant les aciers FeM nC  TWIP de  1985 à 2010 de  1985 à 2010 de  1985 à 2010 de  1985 à 2010 \n[BOUAZIZ 2011_1] [BOUAZIZ 2011_1] [BOUAZIZ 2011_1] [BOUAZIZ 2011_1]     \n \nLes perform ances de ces aciers « austénitiques » so nt uniques et bien  supérieures à la fois en \nterm es de résistance et de ductilité aux  aciers à m atrice « ferritiques » généralement utilisés \ndans le dom aine de la construction automobile. Depu is peu, on parle d’ailleurs de ces aciers  16  comme étan t la seconde génération des aciers THR (T rès Haute Résistance) pour l’automobile \n(cf. Figure 3).  \n \n(a) (b) \nFigure Figure Figure Figure 3 3 33    : (a) C ourbes de traction conve ntionnel les : (a) C ourbes de traction conve ntionnel les : (a) C ourbes de traction conve ntionnel les : (a) C ourbes de traction conve ntionnel les typique typique typique typiques s s s d’aciers ferritiques (aciers actuellement utilisé s dans d’aciers ferritiques (aciers actuellement utilisés dans d’aciers ferritiques (aciers actuellement utilisés dans d’aciers ferritiques (aciers actuellement utilisés dans \nla construction automobile) (b) Positionnement dans  un  plan résistance mécanique / allong ement à  ruptu re des la construction automobile) (b) Positionnement dans  un  plan résistance mécanique / allong ement à  ruptu re des la construction automobile) (b) Positionnement dans  un  plan résistance mécanique / allong ement à  ruptu re des la construction automobile) (b) Positionnement dans  un  plan résistance mécanique / allong ement à  ruptu re des \n3 grandes générations d’aciers dits à THR 3 grandes générations d’aciers dits à THR 3 grandes générations d’aciers dits à THR 3 grandes générations d’aciers dits à THR (Très Ha u te Résistance)  (Très Ha ute Résistance)  (Très Ha ute Résistance)  (Très Ha ute Résistance). . ..    \n \nCes perform ances en traction sont comparables à cel les des aciers austénitiques in oxydables \ncomme le m ontre la Figure 4. Leur intérêt techno-éc on omique provient don c principalem ent \nde leur coût sur des m archés de grande diffusion, c om me l’automobile, et non de \nperformances spécifiques. En effet, le m anganèse co mme matière première présen te des prix \nbien  inférieurs à ceux du nickel (de l’ordre de gra ndeur de ceux du chrome) tout en ayant un \npouvoir de stabilisation des phases austénitiques ( effet dit gammagène) supérieur.  \n \n \nFig ure Fig ure Fig ure Fig ure 4 4 44    : C ourbes d e : C ourbes d e : C ourbes d e : C ourbes d e traction traction traction traction conventionnelles typiques d’aciers FeM nC  T WIP, de deux  aciers  conventionnelles typiques d’aciers FeM nC  TWIP, de deux  aciers  conventionnelles typiques d’aciers FeM nC  TWIP, de deux  aciers  conventionnelles typiques d’aciers FeM nC  TWIP, de deux  aciers austénitiques austénitiques austénitiques austénitiques \nFeNiCr FeNiCr FeNiCr FeNiCr ino x ydables  de type 301 (à effet TRIP  ino x ydables  de type 301 (à effet TRIP  ino x ydables  de type 301 (à effet TRIP  ino x ydables  de type 301 (à effet TRIP    –– –– TRans formation Induced Plastic ity  TRans formation Induced Plastic ity  TRans formation Induced Plastic ity  TRans formation Induced Plastic ity) ou 304L (stable ) d ) ou 304L (stable) d ) ou 304L (stable) d ) ou 304L (stable) d’après ’après ’après ’après \n[TAL YAN 1998] [TAL YAN 1998] [TAL YAN 1998] [TAL YAN 1998]. . ..    \n \nGrâce au développeme nt de cette nouvelle famille d’ aciers, un e mutation rapide des pratiques \ndans le design et la construction automobile était atten due avant 2020 [CORNETTE \n 17  2005] [SCOTT 2006] , mais ce ch angement n’a pas eu lieu. L’introductio n des aciers TWIP \nreste à l’h eure actuelle confidentielle. La Figure 5 mon tre par exem ple la première \nimpléme ntation d’un acier TWIP1000 produit par POSC O sur véhicule FIAT [MAGGI 2012]. \nDe façon surprenante, il s’agit d’une pièce anti-in trusion (poutre de pare-choc) n écessitant \nplutôt des produits à fortes limites d’élasticité ( comme les aciers martensitiques). \n \nLes principales raisons qui ont conduit au retard d ans le développe m ent et l’impléme ntation \nindustrielle de ces produits sont de plusieurs ordr es : \n• problématique process : élaboration nécessitant un e  filière com pliquée et des \ncapabilités proches de la production des aciers ino xydables (élaboration d’acier liquide, \ncoulée continue, laminage par ex emple) \n• problématique d’usage : soudage hétérogène difficil e entre les aciers ferritiques et ces \naciers TWIP [BEAL 2011] sur véhicules automobiles l ié aux compatibilités de \npropriétés thermiques (liquidus, conductivité), \n• problématique de durée de vie : casses différées et  de fragilisation  par l’h ydrogène, \nproblématique bien  connue des aciers austénitiques inoxydables, \n• problématique de valorisation : faible potentiel d’ allègement de ces nuances pour des \napplications en substitution pure, car les limites d’élasticité des aciers austénitiques \nsont faibles. Des grandes déformations sont nécessa ires sur pièces pour pouvoir \nbénéficier du poten tiel de durcisseme nt de ces alli ages. Les pièces et fonctions doivent \ndonc être entièrem ent repensées. Dans l’état actuel  du marché automobile, peu de \nconstructeurs sont prêts à investir sur ces nouvell es possibilités. Il existe toutefois des \npossibilités métallurgiques mais onéreuses pour amé liorer ce paramètre (précipitation, \npré-déform ation) [BOUAZIZ 2011_1].  \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 5 5 55    : : : : Première implémentation des aciers  Première implémentation des aciers  Première implémentation des aciers  Première implémentation des aciers  TWIP TWIP TWIP TWIP dans le domaine automobile dans le domaine automobile dans le domaine automobile dans le domaine automobile - - -- Poutre pare  Poutre pare  Poutre pare  Poutre pare- - --choc mono choc mono choc mono choc mono- - --\ncoquille et « crash box » en TWIP 1000 fourni par Pcoquille et « crash box » en TWIP 1000 fourni par Pcoquille et « crash box » en TWIP 1000 fourni par Pcoquille et « crash box » en TWIP 1000 fourni par P OSCO OSCO OSCO OSCO sur véhicule  sur véhicule  sur véhicule  sur véhicule de série de série de série de série FIAT FIAT FIAT FIAT (véh icule  (véh icule  (véh icule  (véh icule actuellement  actuellement  actuellement  actuellement    \ncomme rcialisé) [M comme rcialisé) [M comme rcialisé) [M comme rcialisé) [M AGGI  AGGI  AGGI  AGGI  2012] 2012] 2012] 2012]    \n \n  18  2.1.2. 2.1.2. 2.1.2. 2.1.2.  Mise en perspect Mise en perspect Mise en perspect Mise en perspective des travaux personn els ive des travaux personn els ive des travaux personn els ive des travaux personn els et collectifs  et collectifs  et collectifs  et collectifs    \n \nCes aciers ont été découverts par Sir Robert Hadfie ld en 1888. Beaucoup de travaux depuis \ncette époque ont été consacrés à l’identification  p uis à la com préhe n sion des mécanismes de \ndurcissemen t de ces alliages. L’identification du m aclage mécanique comm e mécanisme de \nplasticité dans ces structures date des années 1960  et la première interprétation  de l’effet \nTWIP est proposée par Rémy dans les années 1970 [RE MY  1975]. Suite à ces travaux, ces \nmétallurgies ont principalemen t été considérées pou r applications cryogéniques ou de \nrésistance à la corrosion grâce à de forts ajouts d ’aluminium . Les premiers alliages destinés \nspécifiquem ent au march é autom obile sont dus au sid érurgiste POSCO à la fin des ann ées \n1990. Le lecteur pourra se reporter à l’historique détaillé dans l’article de revue récent de \nnotre équipe  [BOUAZIZ 2011_1] . \n \nJ’ai eu la ch ance de participer au début de l’aven t ure du développement des aciers TWIP pour \nAM dès 1999, au cours de mon DEA puis de ma thèse a u LPM, intitulée :  \n• « Caractérisation et modélisation thermomécaniques multi-éch elles des mécanismes de \ndéformation  et d’écrouissage d’aciers austé nitiques  à haute teneur en manganèse – \napplication à l’effet TWIP » [ALLAIN 2004_1]  \nCes travaux  ont ensuite donné lieu à deux thèses fi nancées par AM que j’ai directement co-\nencadrées : \n• Thèse de A. Dum ay, au LPM, intitulée : « Am éliorati on des propriétés ph ysiques et \nmécaniques d’aciers TWIP FeMn XC : in fluence de la s olution  solide, durcisse ment par \nprécipitation et effet composite » [DUMAY 2008_1].  \n• Thèse de D. Barbier, au LETAM/LPMM, intitulée : « E tude du com portem ent \nmécanique et évolutions microstructurales de l’acie r austénitique Fe22Mn0.6C à effet \nTWIP sous sollicitation s complexes. Approche ex péri mentale et modélisation » \n[BARBIER 2009_1] . \nJ’ai aussi eu l’occasion de collaborer à la thèse d e N. Shiekhelsouk au LPMM (directeurs de \nthèse : M. Cherkaoui, V. Favier) et à la thèse de K . Renard à l’UCL (directeur de thèse : \nP .Jacques). En parallèle de ces thèses, cette thém atique de recherche m’a aussi permis de \ncoopérer directem ent ou indirectement avec de nombr eux chercheurs, je pense en particulier \nà M. Lebedkin, T. Lebedkina, A. Roth (LEM3), A. Des champs (SIMAP), D. Embury, A. Zurob \n(Mc Master university), C. Sinclair (UBC), Y . Estri n (Monash university), et M. Huang \n(Hon g-Kong university). \n \n2.1.3. 2.1.3. 2.1.3. 2.1.3.  Problématique Problématique Problématique Problématique     \n \nLa composition de référence à laquelle la plupart d e mes travaux ont été consacrés est une \nnuance austé nitique à haute ten eur en  manganèse et carbone Fe22Mn0.6C. Elle perm et \nd’atteindre des résistances mécan iques élevées supé rieures à 1 GPa associées à un e ductilité  19  supérieure à 50%. Les meilleures performan ces ducti lité / résistance mécaniques sont \nobtenues à température ambiante, grâce à la com bina ison d’un  glissement faiblement \nthermiquem e nt activé et à l’activation d’un  mécanis me de déformation com pétitif au \nglisse ment, le maclage mécanique. A plus basse temp érature, la transformation martensitique \nε se substitue au maclage, permettant d’atteindre d es résistan ces mécaniques plus élevées au \ndétriment de la ductilité alors qu’à haute tem pérat ure, le m aclage mécanique est inhibé à \ncause d’une énergie de d éfaut d’empilement (EDE) pl us élevée. Les résistances m écaniques \nsont alors réduites. D’autres hypothèses sur l’orig ine du fort taux d’écrouissage de ces nuan ces, \ncomme le vieillissement dynamique, sont proposées d ans la littérature et seron t discutées \ndans ce mé m oire. \n \nLa Figure 6 présen te ainsi les évolution s des param ètres du comportemen t en traction en \nfonction de la tem pérature d’essai pour cette nuanc e avec une taille de grain  d’environ 3 µm. \nL’allongement total (en rouge) passe par un optimum  à température am biante quand le \nmaclage mécanique est activé et la lim ite d’élastic ité conventionnelle (en vert) atteint un \npalier athermique. A haute température, les résista nces mécan iques (en bleu) sont faibles et à \nbasse température, les capacités d’écrouissage sont  réduites (faible différen ce entre la limite \nd’élasticité et la résistance mécanique). Les procé dures suivies pour la réalisation de ces essais \nsont détaillés dans [ALLAIN 2004_1] . La Figure 7 montre des micrographies en m icroscop ie \nélectronique à transmission (MET) après 10% de défo rmation de cet alliage à différen tes \ntempératures caractéristiques, pour illustrer les p ossibles m écanism es de déform ation. De \ngauche à droite, on reconnaîtra une microstructure après transformation  martensitique ε \ninduite par la déformation  (obte n ue à 77K), une mic rostructure de m aclage mécanique (à 298 \nK) et enfin  une distribution hom ogène de dislocatio n s (à 673K). N ous reviendrons dans le \nchapitre suivant sur l’origine et la caractérisatio n de cette forte dépendance des m écanism es \nde plasticité à la température. \n \n \nFigure Figure Figure Figure 6 6 66    : Evolution de : Evolution de : Evolution de : Evolution des paramètres de traction d’un a cie s paramètres de traction d’un a cie s paramètres de traction d’un a cie s paramètres de traction d’un a cier Fe22Mn0.6C  (tai lle d e grain r Fe22Mn0.6C  (taille d e grain r Fe22Mn0.6C  (taille d e grain r Fe22Mn0.6C  (taille d e grain = = = = 33 33    µm), résistance µm), résistance µm), résistance µm), résistance \nmécanique (bleu), limite d’élasticité (vert) et all ongement à  rupture (ro uge), en fonction de la tempé rature  mécanique (bleu), limite d’élasticité (vert) et all ongement à  rupture (ro uge), en fonction de la tempé rature  mécanique (bleu), limite d’élasticité (vert) et all ongement à  rupture (ro uge), en fonction de la tempé rature  mécanique (bleu), limite d’élasticité (vert) et all ongement à  rupture (ro uge), en fonction de la tempé rature  \nd’essai. Sont indiqués en o utre les m d’essai. Sont indiqués en o utre les m d’essai. Sont indiqués en o utre les m d’essai. Sont indiqués en o utre les mécanismes d’éc rouissage ac écanismes d’écrouissage ac écanismes d’écrouissage ac écanismes d’écrouissage actifs tifs tifs tifs [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n 20   \n(a) (b) (c) \nFigure Figure Figure Figure 7 7 77    : M icrographies en  M ET : M icrographies en  M ET : M icrographies en  M ET : M icrographies en  M ET en cha mp clair de la nuance de référence en cha mp clair de la nuance de référence en cha mp clair de la nuance de référence en cha mp clair de la nuance de référence après après après après 10% de 10% de 10% de 10% de déformation en déformation en déformation en déformation en \ntr action à différentes températures, met tr action à différentes températures, met tr action à différentes températures, met tr action à différentes températures, mettant en lum ière les mécanismes de déformation actifs (a) tant en lumière les mécanismes de déformation actif s (a) tant en lumière les mécanismes de déformation actif s (a) tant en lumière les mécanismes de déformation actif s (a)     martensite  ε martensite  ε martensite  ε martensite  ε à à à à \n77 K 77 K 77 K 77 K    (b) maclage méca nique (b) maclage méca nique (b) maclage méca nique (b) maclage méca nique à 298 K à 298 K à 298 K à 298 K (c) glissement (c) glissement (c) glissement (c) glissement    à 673 K à 673 K à 673 K à 673 K [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n \nL’effet TWIP est un mécanisme d’écrouissage spécifi que de ces aciers lié à l’apparition  de \nnanomacles au cours de leur déformation. Ces défaut s plans d’épaisseur nan ométrique créent \nune microstructure enchevêtrée et difficilemen t fra nchissable par les dislocations mobiles à \nl’intérieur des grains austénitiques. Il en  résulte  une double contribution au durcisse ment, la \nprem ière de nature isotrope (augme n tation de densit é de dislocation s statistiquem ent \nstockées) et la seconde de nature cin ématique (méca nism e de type composite associé à \nl’incompatibilité de déformation entre m acle et mat rice). Dan s ce mémoire, nous aborderons \naussi largem ent le cas de la transformation martens itique ε, transformation présentant de très \nfortes analogies avec le m aclage mécanique, et dont  les contributions à l’écrouissage sont \nsimilaires. \n \nLe processus de m aclage m écanique est contrôlé par l’énergie de défaut d’empile m ent (EDE) \nde l’alliage considéré (composition). Con trairem ent  aux éléments d’alliage substitutionnels, la \nteneur en carbone augmen te cette EDE et donc serait  défavorable pour le maclage m écanique, \nmais on observe au contraire qu’elle con tribue à ac croître le taux d’écrouissage de ces aciers. \nDan s la littérature, un grand rôle sur le comportem ent de ces aciers était attribué au carbone \npar certains auteurs à travers un mécanisme de viei llissement dynam ique. Cette contribution \nindépendan te du maclage mécanique s’est avérée êt re  négligeable. Deux  hypothèses sont \nactuelle men t investiguées pour résoudre ce paradoxe  en considérant la relation entre carbone \net m aclage mécanique : \n• Le carbone inhiberait les processus de relaxation p lastique dans les m acles conduisant \nà un e augm e ntation de l’effet com posite. \n• Le carbone modifie très sensibleme nt la mobilité de s dislocations parfaites via un \nprocessus thermiquement activé et la planéité du gl issement (glisseme nt dévié). Ces \nconditions favoriseraient le maclage mécanique en a baissant la contrainte critique de \nmaclage, en  compétition avec le glissement. \n \n 21  Dan s ce mémoire, on se propose donc de discuter du comportement de ces alliages selon trois \naxes : \n• Le premier concerne la m orphogénèse de la m icrostru cture de maclage m écanique, \nc’est-à-dire de la structuration spatiale et la dis tribution de la microstructure de \nmaclage ainsi que ses conditions d’apparition (trav aux principalement de nature \nexpérimentale). \n• Le second axe concerne la modélisation  proprement d ire de l’effet TWIP, c'est-à-dire \nde l’impact de la microstructure de maclage sur les  propriétés mécaniques. On \ns’intéressera en particulier à la n ature de l’écrou issage dû à l’effet TWIP et son rôle \nlors des ch argements thermomécaniques présentant de s com ple xités croissantes, de \nplus e n plus représentatives de ch argement réels (e mboutissage). \n• Le troisièm e concerne les effets de la composition chimique sur le comportement et le \nrôle paradoxal du carbone dans ces aciers. \nLe cas des aciers austénitiques sans carbone FeMnSi Al TWIP ou les alliages à m émoire de \nform e FeMnSi sera aussi bien e n tendu évoqué pour m i eux compren dre les effets complexes \ndu carbone dans les systèm es tern aires FeMnC. \n \n2.2. 2.2. 2.2. 2.2.  Morphogénèse de la microstructure de maclage Morphogénèse de la microstructure de maclage Morphogénèse de la microstructure de maclage Morphogénèse de la microstructure de maclage    \n \nLes travaux  détaillés dans ce chapitre concernent l ’étude de la microstructure de maclage et \nsont tirés de mon travail de thèse et de celles de D. Barbier (EBSD, MET, RX) et d’A. Dum ay \n(MET). Après un court rappel bibliographique, nous discuterons des caractéristiques clefs de \ncette microstructure ayant une conséque nce pour l’e ffet TWIP : la fraction maclée, les \népaisseurs de macles et la topologie. \n \n2.2.1. 2.2.1. 2.2.1. 2.2.1.  Introduction Introduction Introduction Introduction    :: :: maclage mécanique, transfor mation martensitique  maclage mécanique, transfor mation martensitique  maclage mécanique, transfor mation martensitique  maclage mécanique, transfor mation martensitique ε et EDE ε et EDE ε et EDE ε et EDE    \n \nLe m aclage mécanique apparaît dans la plupart des m atériaux  cristallins (structures cubiques \ncentrées CC, cubiques à faces cen trées CFC, hexagon aux com pactes HC ou de structures plus \ncomplexes) sous certaines conditions de déform ation s. Ce mécanism e ne joue souve nt qu’un \nrôle mineur dans les matériaux ductiles, mais se ré vèle être indispen sable dans les matériaux \ndans lesquels le nombre de système de glissement es t limité [FISCHER 2003]. Il compense le \nman que de systèm es de glisse ment indépendan ts pour accommoder la déformation dans les \nagrégats polycristallins de structures hex agonales [CAHN 1953] (cf. Figure 8(a)) ou les alliages \nordonnés par exem ple [JIN 1995][FARENC 1993]. Le ma clage mécanique joue cependant un \nrôle important dan s certains matériaux  ayant de plu s nombreuses symétries comme les CC \nCu-Ag [SUZUKI 1958], Fe-Si [YANEZ 2003] (appelées a ussi bandes de Neum an, sur la Figure \n8(b)), structures m artensitiques lenticulaires FeNi C (dites martensite vierges [MAGEE 1971]) \nou les CFC comm e les alliages Cu-Si [COUJOU 1983], Cu-Al [MORI 1980], Co-Ni [REMY  22  1978] et mêmes les aciers inoxydables austénitiques  [LECROI SEY 1972] ou les aciers FeMnC \nHadfield [KARAMAN 2001]. Un e revue bibliographique très complète a été proposée par \nChristian et Mah ajan [CHRISTIA N 1995] dans laquelle  le lecteur pourra trouver de \nnom breux détails et commentaires sur le maclage m éc anique dans différentes structures \ncristallographiques et en particulier dans les alli ages CFC. \n \n (a) \n  (b) \nFigure Figure Figure Figure 8 8 88    : Exemples de microstructures : Exemples de microstructures : Exemples de microstructures : Exemples de microstructures de maclage de maclage de maclage de maclage dans des allia ges mét  dans des allia ges mét  dans des allia ges mét  dans des allia ges méta lliques. Micrographies optiq ues issu es a lliques. Micrographies optiques issu es a lliques. Micrographies optiques issu es a lliques. Micrographies optiques issu es \nde la littérature (a) dans l’u ranium de la littérature (a) dans l’u ranium de la littérature (a) dans l’u ranium de la littérature (a) dans l’u ranium α d’apr α d’apr α d’apr α d’après [CAHN 1953] (b) dans un  ès [CAHN 1953] (b) dans un  ès [CAHN 1953] (b) dans un  ès [CAHN 1953] (b) dans un  acier dit électrique, acier dit électrique, acier dit électrique, acier dit électrique, Fe3Si  Fe3Si  Fe3Si  Fe3Si, , ,, d’après  d’après  d’après  d’après \n[YANEZ 2003] ( [YANEZ 2003] ( [YANEZ 2003] ( [YANEZ 2003] (microstructures connues  sous le nom d e microstructures connues  sous le nom de microstructures connues  sous le nom de microstructures connues  sous le nom de « « ««     Bandes de Neuman Bandes de Neuman Bandes de Neuman Bandes de Neuman    »). »). »). »).    \n \nBien  que ce mémoire soit principalemen t dédié aux a ciers austénitiques, nous revien drons à la \nfin de ce chapitre sur l’analogie qu’il peut existe r avec les structures à faibles nombres de \nsystèmes de glissem e nt. N ous donnerons aussi quelqu es pistes de recherche pour compren dre \npourquoi ces aciers présentent ce mécanism e com péti tif au glisse ment et comment la \ncomposition  chimique semble con trôler l’activation de ce mécanisme. \n \n2.2.1.1.  Maclage mécanique \n \nLe maclage mécanique et la transformation  martensit ique ε sont des mécanismes de \ndéformation  compétitifs au glissement des dislocati ons. Ils son t très similaires du point de vue \nde leur mécanisme de germination et de croissance ( en  termes de réactions entre dislocations), \nmais aussi de celui de la m orphologie résultante. C ette similitude est illustrée par ex emple sur \nla Figure 7, qui m ontre les micrographies en  MET de s deux types de m icrostructures de \ndéformation . \n \nDan s les alliages CFC à faible EDE, il est maintena nt bie n établi que le maclage mécanique est \nle résultat d’un glissement collaboratif de disloca tions partielles de Sh ockley a/6<112> tous les \nplan s parallèles {111} successifs, définissant la d irection de maclage et le plan de maclage \nrespectivem ent. La nature du défaut traîné par cett e dislocation partielle de tête est alors \nsupposée de nature intrinsèque. Des m odèles de germ ination suppose nt que la nature peut \nêtre extrinsèque n écessitant alors de faire passer les dislocations partielles de tête tous les  23  deux  plans atomiques. Les macles mécaniques apparai sse nt donc au cours de la déformation \nde l’alliage. Elles n e doivent pas être confondues avec les m acles de recuit qui apparaissent \nlors de la recristallisation des alliages CFC. \n \nLa structure recon struite entre les plans fautés, l es joints de macle, est toujours de structure \ncristallographique CFC m ais e n orie ntation de macle  Σ3 par rapport à la m atrice comme le \nmon tre la Figure 9. \n \n(a) (a) (a) (a) \n (b) (b) (b) (b)    \nFigure Figure Figure Figure 9 9 99    : : : : (a) représentation schém (a) représentation schém (a) représentation schém (a) représentation schém atique d’une macle dans un arrangement périodique. Les joints de  ma atique d’une macle dans un arrangement périodique. Les joints de  ma atique d’une macle dans un arrangement périodique. Les joints de  ma atique d’une macle dans un arrangement périodique. Les joints de  macles sont cles sont cles sont cles sont \ndes plans de symétrie des plans de symétrie des plans de symétrie des plans de symétrie (b) Représentation sché matiqu e  (b) Représentation sché matique  (b) Représentation sché matique  (b) Représentation sché matique typique typique typique typique du réarra ngement des plans compa cts autour du réarra ngement des plans compa cts autour du réarra ngement des plans compa cts autour du réarra ngement des plans compa cts autour \nd’une macle mécaniqu e créée par le déplacement de d islocations partielles i d’une macle mécaniqu e créée par le déplacement de d islocations partielles i d’une macle mécaniqu e créée par le déplacement de d islocations partielles i d’une macle mécaniqu e créée par le déplacement de d islocations partielles intrinsèques tous les plans atomiq ues. ntrinsèques tous les plans atomiq ues. ntrinsèques tous les plans atomiq ues. ntrinsèques tous les plans atomiq ues. \nLa flèche indique la po sition du joint de  macle pla n miroir pour le  réseau C FC. La flèche indique la po sition du joint de  macle pla n miroir pour le  réseau C FC. La flèche indique la po sition du joint de  macle pla n miroir pour le  réseau C FC. La flèche indique la po sition du joint de  macle pla n miroir pour le  réseau C FC.    \n \nLes élémen ts cristallographiques du maclage dans un  réseau CFC sont, selon les conve ntions \nhabituelles : \n• K1 le plan  de m aclage de type {111} correspondant a u plan de glissement des \ndislocations partielles de Shockley (correspondant aussi aux joints de macle), \n• η1 la direction de cisaillem e nt de type <112> corre spon dant à la direction des vecteurs \nde Burgers des dislocation s partielles, \n• K2 le plan n on distordu au cours du cisailleme n t, d e type {111}, \n• η2 l’intersection des plans K2 et K1 de type <112>.  \n \nLe glissement produit par une macle parfaite d’orig in e intrinsèque dans la direction η1 est \négale à γ T = b 112  / d 111  avec d 111  la distance réticulaire entre deux plans denses {1 11} et b 112  le \nvecteur de Burgers des dislocations de type a/6<112 >. Si l’on considère par contre un \nmécanisme mettan t e n jeu des défauts d’origine extr insèques, le glissement résultan t est divisé \npar 2 et est égal à γ T / 2 [COUPEAU 1999].  \n \nCertains auteurs ont souligné que ce processus ne p ermet pas de reproduire tout à fait à \nl’identique la maille de la matrice dans le cas des  alliages ternaires. En effet, Adler et al.  ont \nmon tré que les atomes de carbone se situaie nt préfé rentiellement dans les sites interstitiels \noctaédriques de la structure CFC et que ces sites é taient e n fait tran sfor més en sites \ntétraédriques lors du processus de maclage [ADLER 1 986]. D’après les auteurs, ce processus de \n« pseudo-maclage » induit un durcissement important , responsable de l’excellent taux  24  d’écrouissage dans ces alliages. Toutefois, cette a pproche ne permet en aucun cas d’expliquer \nles taux d’écrouissage des alliages sans carbone co mme les aciers FeMnSiAl étudiés par Grässel \net al. [GRASSEL 2000] par exemple. \n \nCon cernant le processus de maclage mécanique dans l es structures CFC, il ressort aussi de la \nlittérature que : \n• la propagation des macles est e xtrêmement rapide, d e l’ordre de grandeur de la vitesse \ndu son dan s le matériau [LUBEN ETS 1985][REMY 1975].  L’étape de germ ination est \ndonc le processus limitant, \n• la germination des macles est hétérogèn e à partir d ’une configuration particulière de \ndislocations [REMY 75][VENABLES 1964][LUBENETS 1985 ][CH RISTIAN 1995] \n[CH RISTIAN 1969]1, \n• Il n écessite en outre l’activation préalable de plu sieurs systèmes de glisse ments \n[CH RISTIAN 1969]. \n \n2.2.1.2.  Tran sfor mation martensitique ε et α’  \n \nLa transformation marten sitique ε est un mécanisme très similaire au maclage et se produit \nquan d des dislocations partielles de Shockley de na ture intrinsèque se propagent tous les deux \nplan s atomiques {111} ou tous les plans atomiques d ans le cas de défaut de n ature extrinsèque \n[IDRISSI 2009]. Contrairement au m aclage, ce mécani sm e collectif de dislocations ne \nreconstruit pas une structure CFC en volume m ais un e structure hexagonale compacte (H C). \nLes plans {111} d’h abitat de la m artensite dans l’a ustén ite correspondent aux plans de nses de \nla structure hexagonale. Maclage et transformation  martensitique sont si proches qu’ils \npeuvent co-exister au sein  d’une  même bande de déf ormation comm e l’ont observé certains \nauteurs [BRACKE 2007_2]. \n \nMaclage mécanique et transform ation marten sitique ε  sont fortem ent reliés à l’Energie de \nDéfaut d’Em pile m ent (EDE) de l’alliage qui contrôle  le coût énergétique de la création des \ndéfauts d’em pile m ent et donc la distance de dissoci ation entre les dislocations partielles \nbordant ce défaut [FRIEDEL 1964][FERREIRA 1998][BUY N 2003][KARAMAN 2000_1]. Une \nrelation e xiste d’ailleurs entre cette énergie et l ’ enthalpie libre de tran sformation γ \u0001 ε ∆G γ\u0001ε. \nCette formule due à Hirth  [HIRTH 1970] a été repris e et ren due populaire par les travaux  de \nOlson et Cohen. Cette approche con siste à établir u ne équivale nce entre un défaut \nd’em pilement de n ature intrinsèque et une plaquette  de martensite ε ayant une épaisseur de \ndeux  plans atomiques com pacts et créant deux n ouvel les interfaces γ/ε. Il e n résulte : \n \n                                                 \n1 un processus de mac lage homogène tel que discuté p ar Lubenets et a l.  [LUBENETS 1985] nécessiterait des \ncontraintes de l’ordre de 20% du module d’Young.  25   γ/ε ε γ\n111 int 2σ ∆Gρ 2 EDE + =→ (1) \n \nAvec ρ 111  la densité surfacique m olaire sur les plans d’habi tat {111} et σγ/ε l’énergie par un ité \nde surface des interfaces γ/ε. \n \nUn second type de transform ation marten sitique est aussi observé dans cet alliage \ncorrespondant à un e transformation γ \u0001 α’ de structure tétragonale centrée. A l’instar de  la \ntran sformation martensitique ε, elle peut avoir lie u au cours d’une trempe (tran sformation \nthermique) ou de la déformation de ces alliages (tr ansfor m ation induite mécaniquement). \nCette transformation martensitique particulière tou che les alliages les moins riches en \ncarbone et/ou Mn ou lors de déformation  à basse tem pérature [SCHUMANN  1972]. \n \nCes transformations son t très largement documentées  dans la littérature car elles \ncorrespondent à celles ren contrées traditionnelleme nt dans les aciers ferritiques (on  pourra se \nreporter par exem ple à la revue de Krauss [KRAUSS 1 999]). N ous reviendrons sur les \npropriétés de cette phase à la section 3.3.2 (page 111). Dans le cas des aciers austénitiques, \ncette transformation au cours de la déformation ind uit un fort effet TRIP (Transfor mation \nInduced Plasticity) dans ces alliages qui peut cond uire à des résistances m écaniques élevées \n(comme par exem ple l’acier inoxydable de type 301 s ur la Figure 4). Par contre, ces aciers \nmon trent alors des ductilités généralement moins él evées que les aciers TWIP et sont sujets à \nune forte sensibilité à la casse différée [DAGBERT 1996]. C’est pourquoi, ces alliages \nausténitiques instables sont moins regardés actuell em e nt dans la littérature et seront peu \nsouvent abordés dans le cadre de ce mém oire.  \n \nUne confusion est souvent faite dan s la littérature  entre EDE et occurrence de la \ntran sformation martensitique α’. Il n’est en effet pas rar e de r etrouver des germes de \nmartensite α’ aux intersections de plaquettes de ma rtensite ε de différents variants  ou de \nbandes de cisaillem ent [BRACKE 2007_1][OLSON 1975][ STRINGFELLOW 1992][TOM I TA \n1995]. Cependant, la nature de cette transform ation  n e fait pas intervenir n écessaireme nt de \nréaction γ \u0001 ε. La coïncidence e n tre occurrence de la marten sit e α’ et les très faibles EDE est \ndonc purem e nt « fortuite » et ne peut être systém at isée. \n \n2.2.1.3.  EDE et mécanismes de déformation  \n \nPour une composition donnée, Rémy a démontré que l’ EDE est nécessairem ent un e fonction \ncroissante de la te m pérature au travers de la relat ion suivante : \n \n ε γ\n2\nCFCint∆S\na1\n38\ndTEDEd →\nℵ−=  (2)  26   \navec a CFC  le param ètre de maille de l’austénite, א le nombre d’Avagadro et ∆Sγ\u0001ε la variation \nd’en tropie au cours de la transformation γ \u0001 ε. Si la te mpérature est supérieure à Es, la \ntempérature de transform ation marten sitique ε spont anée au refroidisse ment, ∆Sγ\u0001ε est \nnécessairem ent négative et conduit donc à une sensi bilité positive de l’EDE à la température.  \n \nCette relation est intéressante car elle permet d’e xpliquer en grande partie l’évolution des \nmicrostructures de déform ation en fonction de la te m pérature de déformation présentée sur \nla Figure 7. Cette séquence dans l’apparition des m écanismes de déformation com pétitifs au \nglisse ment est valable pour une composition donnée.  \n \nA haute tem pérature par rapport à Es, l’EDE est gra nde. Le coût é nergétique de la dissociation \nde dislocations parfaites est grand et donc défavor able. Le seul mécan isme possible est don c le \nglisse ment de dislocations (com me observé par de no mbreux auteurs sur la nuance de \nréférence à 400°C). Les hautes températures et les EDE élevées sont favorables au glissement \ndévié et à l’activation de n ombreux systèmes de gli ssem ent [RAUCH 2004]. \n \nQuand la température diminue, le glisseme nt des dis locations devie n t de plus en plus planaire, \ncar le glissement dévié devient défavorable (ce pro cessus sera discuté en détail au chapitre \n2.4.3.4 page 89). En dessous d’une certaine valeur d’EDE, de larges dissociations de \ndislocations parfaites sont possibles à un  coût éne rgétique faible [BY UN 2003] et l’apparition \ndu maclage mécanique est favorisée. Actuellement, l e mécan isme réactionnel en tre \ndislocations conduisant à for mer un germe de m acle n e fait pas l’objet d’un consen sus dan s la \ncommunauté. Les mécanismes de croissance et d’épais sissem e nt ne se mble nt pas non plus \névidents. Des expériences récen tes in situ  au MET [IDRISSI 2010_1] suggèrent que le \nmécanisme de germination puisse être une réaction p olaire avec déviation  comm e proposé \npar Cohen and Weertman [COH EN 1963_1] [COHEN  1963_2 ] ou par Miura, Takamura et \nNarita [M IURA 1968] alors que la croissance de la m acle (épaississe men t) soit due à un \nmécanisme polaire simple comme proposé initialeme nt  par Venables [VENABLES 1974]. \n \nA plus basse température, la tran sformation martens itique ε remplace le maclage m écanique. \nLa transition e ntre ces deux mécanism es de plastici té est expliquée par une évolution du \ncaractère des défauts d’empilement responsables de leur germ ination  (intrinsèque et \nextrinsèque pour une macle et la martensite respect iveme nt) comm e l’a montré récemment \nIdrissi et al.  [IDRI SSI 2009]. Le calcul proposé par Lecroisey et  Pin eau [LECROI SEY 1972] \nsoutie nt cette hypothèse, dans la mesure où l’EDE e xtrinsèque est e n viron 1,5 fois plus élevée \nque l’EDE intrinsèque. Des températures très basses  sont don c nécessaires pour permettre la \ntran sformation martensitique ε par rapport au macla ge. Comme dans le cas du maclage \nmécanique, la tran sfor mation martensitique ε est un  m écanisme compétitif au glissement. Ce  27  dern ier assure toujours la plus grande contribution  de la déformation macroscopique, mais \npeut être significativemen t inhibé par l’ajout d’él éme nts d’alliage 2. \n \nAinsi, pour un acier de composition donné, différen ts mécanismes de déformation peuvent \nêtre activés en fonction de la te m pérature de solli citation mécanique. De même, il est possible \nen modifiant la composition chimique de l’alliage d ’activer certains mécanismes de \ndéformation  à une température donnée. Ce con stat pe rmet d’envisager des stratégies « d’alloy \ndesign » tout à fait e xcitan tes. \n \nLa première étude majeure du système ternaire FeMn C  date de 1972 perm ettant de relier la \ncomposition  chimique et la stabilité de l’austé nite  à te mpérature ambiante vis-à-vis des \ntran sformations m artensitiques ε et α’ et a été r éa lisée par Schumann  [SCHUMANN  1972]. Il \na établi la première carte du plan Mn/C définissant  les compositions stables au cours de la \ndéformation  (incluant le maclage mécanique, diffici le à identifier à l’époque) et instables \n(subissant une transfor m ation m artensitique ε). La limite entre ces deux domaines peut \ns’écrire : \n \n 32 %C20 %Mn +−=  (3) \n \nAvec %Mn  et %C les teneurs massiques en élém ents d’ alliage en pourcents. \n \nCette form ule montre par exemple que les alliages F e35Mn ou Fe22Mn0.6C (acier de \nréférence de cette étude) ou Fe12Mn 1.1C (acier Hadf ield) sont 100% austénitiques à \ntempérature ambiante et ne présenteront pas de tran sfor mation martensitique ε au cours de \nleurs déform ations. \n \nDepuis Schumann, nous avons contribué à l’instar de  nom breuses équipes à élaborer des \nrelations m oins em piriques e ntre compositions, EDE et mécanismes de déformation , sans réel \nconsensus. Saeed-Akbari et al.  [SAEED-AKBARI 2009] ont récemment réalisé une revu e \nbibliograph ique de ce sujet et m ontré que la plupar t des auteurs concluait que cette lim ite \ndéfin ie par Schum ann correspondait à une iso-valeur  d’EDE d’environ 20 mJ.m -2. Nous \nreviendrons dans le chapitre 2.4.1 page 68 sur les stratégies de mesure et de modélisation  de \ncette EDE, et sur le rôle particulier joué par le c arbone sur cette valeur. \n \n                                                 \n2 Le glissement est largement inhibé pa r l’ajout mas sif d’éléments d’alliage comme le silic ium. C’est l e cas par \nex emple des Alliages à Mémoire de Forme (A M F) Fe30M n6Si,  austénitiques à température ambiante, qui se \ndéfor ment principalement par  transfor mation martens itique ε [ M YAZAKI 1989]. Dans c e cas, c ette dernièr e \ntransformation marten sitique est massive et contrib ue d e façon majoritaire à la déformation macroscopi que de \nl’allia ge (cf. Figure 58 page 93).  28  2.2.2. 2.2.2. 2.2.2. 2.2.2.  Microstructure de maclage Microstructure de maclage Microstructure de maclage Microstructure de maclage    : : : : nn nnanomacle anomacle anomacle anomacles s ss et faisceaux  et faisceaux  et faisceaux  et faisceaux    \n \nLe m aclage mécan ique dans ces structures austénitiq ues adopte une structuration multi-\néchelle particulière dont les caractéristiques de t aille, de morphologie et de distribution \nspatiale expliquent en grande partie l’effet TWIP. Une partie importante de nos travaux a \ndonc été consacrée à la description et la compréhen sion de cette structure, dans la nuance de \nréférence ou des compositions proches. \n \nNous avons caractérisé ces microstructures de défor mation à plusieurs échelles, de celle de la \nmicroscopie électronique à transmission (MET) pour la caractérisation des macles \nindividuelles à la microscopie optique (MO) pour la  structuration des systè m es de m aclage en \npassant par la microscopie électronique à balayage (MEB) couplée à des analyses en mode \nElectron Back Scattering D iffraction (EBSD) pour l’ an alyse systématique des relations en tre \nmaclage mécanique et cristallographie. Le couplage de ces différentes tech niques permet de \ndonn er un aperçu statistique ment admissible de la m icrostructure de m aclage et de son \névolution au cours de la déformation. \n \nLes macles mécaniques observées dans ces alliages s ont en réalité des nanom acles, c’est-à-dire \ndes objets très minces dont l’épaisseur est de l’or dre de quelques dizaines de nanomètres \ncomme le montre la Figure 10. Le nombre de dislocat ion s partielles impliquées dans le \nprocessus reste donc limité. A titre d’ordre de gra ndeur, environ 27 dislocations sont \nnécessaires pour former une macle de 100 nm d’épais seur.  29   \n(a) \n (b) \n(c) \n (d) \nFigure Figure Figure Figure 10 10 10 10    : : : : (a ), (b), (c) et (d) (a ), (b), (c) et (d) (a ), (b), (c) et (d) (a ), (b), (c) et (d) Nanomacles organi sées en faisc eaux  (paquets) observé e par MET en cha mp sombre Nanomacles organi sées en faisceaux  (paquets) observ é e par MET en cha mp sombre Nanomacles organi sées en faisceaux  (paquets) observ é e par MET en cha mp sombre Nanomacles organi sées en faisceaux  (paquets) observ é e par MET en cha mp sombre \ndans la nuance de référence après  déforma tion dans la nuance de référence après  déforma tion dans la nuance de référence après  déforma tion dans la nuance de référence après  déforma tion [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n \nPeu d’études ont été consacrées à la mesure de l’ép aisseur de macles de façon systématique et \nstatistique. Cette lacune s’explique par la nécessi té de réaliser ce travail e n MET après \nindexation systématique des zones (norm ale au plan K1 perpendiculaire au faisceau incident) \nsans aucun e garan tie d’obtenir une netteté suffisan te à cause des densités importantes de \ndislocations sur les interfaces [IDRISSI 2010_2]. C ette carence dans la littérature n’en reste \npas moins paradoxale car l’épaisseur des macles est  un  paramètre clef dans la compréhe nsion \ndes mécanismes d’écrouissage (cf. chapitre 2.3.5.1 page 52). Ces fin es épaisseurs expliquent \nque les zon es maclées supporten t des contrain tes él evées de l’ordre de plusieurs GPa sans \npossibilité de relaxation. [SEVILLANO 2012]. \n  30  Afin  de pallier ce manque, nous avons développé une  approche pour prédire l’épaisseur des \nmacles en fonction  de paramètres cristallographique s et therm ochimiques [ALLAI N 2004_1] \n[ALLAIN 2004_4] . Ce modèle est basé sur une exten sion des travaux de Friedel [FRIEDEL \n1964]. Celui-ci assimile une macle à un assemblage de boucles de dislocations partielles \ncirculaires et concentriques et calcule son facteur  de form e S en  fonction de la cission \nappliquée τ a pp . \n \n app\n112111τ\nμb2d\nDeS==  (4) \n \nAvec e l’épaisseur de la m acle considérée, d 111  l’espacement réticulaire entre les plans {111}, \nb112  le vecteur de Burgers des dislocations partielles de Shockley, µ le module d’élasticité en \ncisaillement. D est le rayon de la macle. Nous avon s étendu ce concept en  introduisant une \ncontrainte critique d’émission qui dépend de l’EDE in trinsèque. Cet ajout permet de définir \nl’épaisseur minim ale des macles e min  issue d’un  processus de germination (avant \népaississe m ent) : \n \n D\nbEDEJD\nbEDE\nμb2de\n112int analytique\nisolé\n112int\n112111\nmin \n\n\n=\n\n\n=  (5) \n \nCette formule mon tre que l’épaisseur minimum  des m a cles doit augm enter avec l’EDE et son \ndiam ètre (son libre parcours moyen). Numérique ment,  avec µ = 62 GPa, EDE = 20 m J.m -2, b112  \n= 0,147 nm , le facteur de forme minimal d’une macle  isolée est de 3,1x10 -3 selon ce modèle. \nPour une taille de grain de 15 µm , ceci correspond à une épaisseur de 47 nm , en accord avec \nles ordres de gran deurs observés [ALLAIN 2004_1]  et ceux rapportés dans la littérature \n[REN ARD 2012]. \n \n(a) \n (b) \nFigure Figure Figure Figure 11 11 11 11    : Schéma  de princ ipe du modèle 2D  de : Schéma  de princ ipe du modèle 2D  de : Schéma  de princ ipe du modèle 2D  de : Schéma  de princ ipe du modèle 2D  de dyna mique disc rète de s disloca tions (DDD) dyna mique discrète de s disloca tions (DDD) dyna mique discrète de s disloca tions (DDD) dyna mique discrète de s disloca tions (DDD) simplifié  pour  simplifié pour  simplifié pour  simplifié pour \nsimuler le fonctionnement d’u n germe  de micromacle et la  formation d’u n « simuler le fonctionnement d’u n germe  de micromacle et la  formation d’u n « simuler le fonctionnement d’u n germe  de micromacle et la  formation d’u n « simuler le fonctionnement d’u n germe  de micromacle et la  formation d’u n «    ps ps ps pseudo eudo eudo eudo- - --empilement empilement empilement empilement    » en front de » en front de » en front de » en front de \nmacle sur  un obstacle (a) c onfigura tion 3D (b) macle sur  un obstacle (a) c onfigura tion 3D (b) macle sur  un obstacle (a) c onfigura tion 3D (b) macle sur  un obstacle (a) c onfigura tion 3D (b) g g ggrandeurs cara ctéristiq ues du modèle randeurs cara ctéristiq ues du modèle randeurs cara ctéristiq ues du modèle randeurs cara ctéristiq ues du modèle    [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n  31  Les résultats de ce modèle analytique ont été confi rmés par la suite à l’aide d’un modèle \nsimplifié de dynam ique discrète des dislocations (D DD), schém atisé sur la Figure 11. Il perm et \nde simuler le fonctionneme nt d’un germe de macle et  l’établissem ent d’une con figuration \nd’équilibre sous contrain te de ce « pseudo-empilem e nt ». Ce m odèle est détaillé dans \n[ALLAIN 2004_1][ALLAI N 2004_4]  et permet de retrouver la même sensibilité à l’EDE  et à la \ntaille des m acles, m ais la constante de proportionn alité J est sensible m ent plus élevée (environ \n7,57 10 -5 MPa -1 au lieu de 2,28 10 -5 MPa -1). L’épaisseur prévue pour les micromacles par cett e \nsimulation DDD est de 158 nm avec les valeurs usité es ci-dessus. \n \nLa forme du front de macle calculée par ce m odèle d e DDD simplifié est très proche d’un \nempile ment de dislocations. En effet, seule un e fai ble proportion des dislocations partielles \nform ant la macle arrive au contact de l’obstacle. L a plupart reste distribuée sur l’interface à \nl’instar d’un  empilement. Cette constatation est im portante à double titre : \n• l’approximation proposée par Mullner et al [MULLNER  2002] assimilant un front de \nmacle à un  dipôle de disclinations est acceptable à  grande échelle m ais ne reproduit \npas fidèle m ent le champ de contraintes autour d’un  front de macle. Elle ne peut \nrendre com pte de la disposition particulière des di slocations le long du fron t observée \npar de nom breux auteurs [FAREN C 1993][CHRISTIA N  199 5]. \n• les dislocations d’interface sont donc n ombreuses e t contribuent à rendre ce défaut \ninfranchissable en  pratique par des dislocation s in cidentes, mê me si de n ombreuses \ncombinaisons d’interactions macle/glisse ment sont t h éoriquement possibles [REMY \n1975][IDRISSI 2010_2]. \nUne autre spécificité est que ces nanom acles sont g roupées généralement e n paquets. Cette \nstructuration que nous avons qualifiée de « faiscea ux  » est particulièrement visible sur les \nprem iers pourcents de déformation et devie nt diffic ile à distin guer à plus grande déformation. \nLa Figure 10(a) montre un bel exemple de cette conf iguration. Ces structures déjà mises en \névidence par Rém y [REMY 1975], son t assimilées à de s zones entièrem ent maclées lors \nd’études à plus faibles grossissements (MEB ou MO).  Cette assim ilation sans précaution \nconduit alors à un e surestimation de la fraction ma clée. La Figure 12 mon tre les grandeurs \ncaractéristiques des nanomacles et des faisceaux ob servées dans la nuance de référence après \ndéformation . \n \nFig ure Fig ure Fig ure Fig ure 12 12 12 12    : Grandeurs caractéristiques des nano : Grandeurs caractéristiques des nano : Grandeurs caractéristiques des nano : Grandeurs caractéristiques des nanomacles et des fa isceaux  observé macles et des fa isceaux  observé macles et des fa isceaux  observé macles et des fa isceaux  observés dans la nuance de  référenc e s dans la nuance de  référenc e s dans la nuance de  référenc e s dans la nuance de  référenc e \naprès déformat après déformat après déformat après déformation. ion. ion. ion.     32  2.2.3. 2.2.3. 2.2.3. 2.2.3.  SS SStructuration tructuration tructuration tructuration multi multi multi multi- - --échelle échelle échelle échelle du maclage au cours de la déformation du maclage au cours de la déformation du maclage au cours de la déformation du maclage au cours de la déformation (MET,  (MET,  (MET,  (MET, \nEBSD, MO) EBSD, MO) EBSD, MO) EBSD, MO) en relation avec la texture  en relation avec la texture  en relation avec la texture  en relation avec la texture    \n \nAu delà de la structuration en faisceaux, la micros tructure de maclage présente un second \nniveau d’organisation spatiale qui apparaît au cour s de la déformation à l’échelle du grain \nausténitique. Les m acles se développent en eff et su r plusieurs plans d’habitats parallèles ou \nsécants, correspondant à une ou deux familles de pl ans {111}. La première famille de plans sur \nlesquels apparaissent les macles est souve nt qualif iée de prim aire, et les autres de secondaires \n(par abus de langage, on parlera de systèmes de mac lage prim aires et secon daires). La Figure \n10 (a) mon tre par exemple un grain d’austénite con t enant un seul et unique système de \nmaclage. Quand un système de maclage secondaire est  activé, les macles de ce nouveau \nsystème ne peuven t s’éten dre sur l’ense m ble du grai n m ais restent générale m ent bloquées par \nles macles du système primaire. Cela conduit à la f ormation de structures caractéristiques \ndites en éch elle comme le montre la Figure 13. \n \n \nFigure Figure Figure Figure 13 13 13 13    : Schéma  de principe de la  mise en place de  la mic rostructure de maclage. Con figurations après : Schéma  de principe de la  mise en place de  la micr ostructure de maclage. Con figurations après : Schéma  de principe de la  mise en place de  la micr ostructure de maclage. Con figurations après : Schéma  de principe de la  mise en place de  la micr ostructure de maclage. Con figurations après \nactivation séquen activation séquen activation séquen activation séquentielle o u simultanée des systèmes de maclage tielle o u simultanée des systèmes de maclage tielle o u simultanée des systèmes de maclage tielle o u simultanée des systèmes de maclage    [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n \nSi les deux  systèm es de maclages apparaissen t non p as de manière séquentielle, mais de \nman ière concomitante, alors les deux systèm es de m a clage se retrouven t ench evêtrés de \nman ière complexe, comme l’illustre aussi la Figure 13 3. \n \nLa m ise en  place de ces structures particulières à l’échelle des grains dans la nuance de \nréférence a été observée en microscopique optique p our les grains de relativement grande \ntaille (20 µm ), e n EBSD ou e n microscopie MET. Les observations en MET de la Figure 7(b) et \nde la Figure 14, confirm e nt toutes que les micromac les sont des obstacles forts pour \n                                                 \n3 Dans la littérature, la plupart des auteurs rappor te l’activation de deux systèmes d e macla ge pour de s \ndéfor mations en traction. De ra res cas de trois sys tèmes ont toutefois été rapportés [RENARD 2012].  33  l’extension de micromacles incidentes sécantes 4. Ce constat semble être aussi valable dan s le \ncas de la transform ation martensitique ε comme le m on tre la Figure 7(a). \n \n \n \n(d) \n (e) \nFigure Figure Figure Figure 14 14 14 14    : Micrographies M ET de la microstructure ma clage d e la nuance de référence (a) : Micrographies M ET de la microstructure ma clage de  la nuance de référence (a) : Micrographies M ET de la microstructure ma clage de  la nuance de référence (a) : Micrographies M ET de la microstructure ma clage de  la nuance de référence (a), (b) et (c ) , (b) et (c ) , (b) et (c ) , (b) et (c )    en en en en champ champ champ champ \nc l air c l air c l air c l air après 5% de déforma tion en traction après 5% de déforma tion en traction après 5% de déforma tion en traction après 5% de déforma tion en traction – – –– grains maclés par un ou de ux  systè mes  grains maclés par un ou de ux  systè mes  grains maclés par un ou de ux  systè mes  grains maclés par un ou de ux  systè mes – – –– agrandissement de  agrandissement de  agrandissement de  agrandissement de \nl’interaction entre deux  systè mes de maclage l’interaction entre deux  systè mes de maclage l’interaction entre deux  systè mes de maclage l’interaction entre deux  systè mes de maclage [BARBIER 2009 [BARBIER 2009 [BARBIER 2009 [BARBIER 2009_1 _1 _1 _1]] ]]    (d) et (e) (d) et (e) (d) et (e) (d) et (e) en en en en cha mp cha mp cha mp cha mps s ss sombres  sombres  sombres  sombres après 33% de après 33% de après 33% de après 33% de \ndéformation en traction déformation en traction déformation en traction déformation en traction– – –– mise  en évidence de  mise  en évidence de  mise  en évidence de  mise  en évidence de l’enchevêtr  l’enchevêtr  l’enchevêtr  l’enchevêtrement des ement des ement des ement des deux systèmes de mac lage  deux systèmes de mac lage  deux systèmes de mac lage  deux systèmes de mac lage par la par la par la par la sélection sélection sélection sélection \nsimultanée des conditions de diffraction simultanée des conditions de diffraction simultanée des conditions de diffraction simultanée des conditions de diffraction des deu x s ystèmes  des deu x systèmes  des deu x systèmes  des deu x systèmes    [ALLA IN 2004_1] [ALLA IN 2004_1] [ALLA IN 2004_1] [ALLA IN 2004_1]. . ..    \n \nL’observation de l’évolution des microstructures de  maclage à de moindres grossisse ments en \nMO permet de quantifier l’activation séquentielle d es systèmes de maclage, comme le mon tre \n                                                 \n4 Rémy a prouvé que c ertaines configurations de tran smi ssions sont toujours théoriquement possibles à d es c oûts \nénergétiques élevés da ns des structures maclées mod èles [COUJO U 1992]. Dans le cas de ces aciers, les interfaces \ndes micromac les saturées de dislocations incidentes  ou de réac tions comme l’a montré récemme nt [IDR ISS I \n2010_2], ce q ui rend encore plus difficile la  possi bilit é d’interactions entre deu x  micro macles sécant es.  Par \ncontre, il est tout à  fait possible d e considérer q u’une intersection de micromacles agisse  comme  un \nconcentrateur de contraintes suffisant pour pa rtici per à la  germination d’une nouvelle microma cle, don nant \nl’illusion à une échell e trop grande d’u n franchiss ement direct. La nouvelle mic romacle sera donc cons idérée \ncomme un mécanisme potentiel de rela x ation. C ’est p robablement le cas de la  Figur e 14(c).  34  par exemple les Figure 15 et Figure 16. Elles mette nt en lumière que tous les grains ne \nprésentent pas un e microstructure de maclage identi que pour un  niveau de déformation \ndonn ée. Certains grains ne sont pas ou peu maclés a près 30% de déformation alors que \nd’autres présenten t depuis 10% de déformation plusi eurs systèmes de maclage. Cette \ndifférence de comportement entre les g rains s’expli que par un e forte sensibilité du maclage à \nl’orientation  cristallograph ique du grain matrice p ar rapport à la direction de sollicitation. \n \n   \n  \n(a)     (b) \n    \n  \n(c)     (d) \n   \n  \n(e)     (f) \nFigure Figure Figure Figure 15 15 15 15    : : : : M icrographies de la  nuance M icrographies de la  nuance M icrographies de la  nuance M icrographies de la  nuance de référe n de référe n de référe n de référe nce (taille de grain environ ce (taille de grain environ ce (taille de grain environ ce (taille de grain environ 2 2 220 µm) en m 0 µm) en m 0 µm) en m 0 µm) en m icroscopie optique a près icroscopie optique a près icroscopie optique a près icroscopie optique a près \nattaque électrolytique. Les échantillons ont été dé formés de (a ) 4.8 %, (b) 9.5 %, (c) 13.9  %, (d) 18. 2 %, (e) attaque électrolytique. Les échantillons ont été dé formés de (a ) 4.8 %, (b) 9.5 %, (c) 13.9  %, (d) 18. 2 %, (e) attaque électrolytique. Les échantillons ont été dé formés de (a ) 4.8 %, (b) 9.5 %, (c) 13.9  %, (d) 18. 2 %, (e) attaque électrolytique. Les échantillons ont été dé formés de (a ) 4.8 %, (b) 9.5 %, (c) 13.9  %, (d) 18. 2 %, (e) \n26.3 % et (f) 33.6 % respectivement. La  direction d e traction est horizon 26.3 % et (f) 33.6 % respectivement. La  direction d e traction est horizon 26.3 % et (f) 33.6 % respectivement. La  direction d e traction est horizon 26.3 % et (f) 33.6 % respectivement. La  direction d e traction est horizonta le ta le ta le ta le    [AL [AL [AL [ALLAIN 2004_1] LAIN 2004_1] LAIN 2004_1] LAIN 2004_1]. . ..    \n  35   \nFigure Figure Figure Figure 16 16 16 16    : Evolution de la  proportion de grains mac lés par u n ou deu x systèmes de ma clage mesurée en : Evolution de la  proportion de grains mac lés par u n ou deu x systèmes de ma clage mesurée en : Evolution de la  proportion de grains mac lés par u n ou deu x systèmes de ma clage mesurée en : Evolution de la  proportion de grains mac lés par u n ou deu x systèmes de ma clage mesurée en \nmicroscopie optique microscopie optique microscopie optique microscopie optique  sur la nuance de référence (ta ille de grain environ 20 µm)  sur la nuance de référence (taille de grain enviro n 20 µm)  sur la nuance de référence (taille de grain enviro n 20 µm)  sur la nuance de référence (taille de grain enviro n 20 µm)  [ALLA  [ALLA  [ALLA  [ALLAIN 2004_1] IN 2004_1] IN 2004_1] IN 2004_1] .. ..    \n \nDès 2004, grâce à une étude en EBSD [ALLAIN  2004_1] , nous avons vérifié statistiquem ent \npour un grain donn é que les macles du systè me prima ire apparaissaien t sur le plan (111) ayant \nle système de dislocation s partielles de Shockley a vec le m eilleur facteur de Schmid par \nrapport à la direction de sollicitation (par abus d e langage, on  parlera dans la suite du facteur \nde Schmid du système de maclage). Ce résultat a été  approfondi dans le cadre de la thèse de \nDavid Barbier [BARBIER 2009_1][BARBIER 2009_3] . Cette étude confirme que la sélection \ndes systèm es de maclage dans un grain dépend bien d ’une loi de Schmid et que par \nconséquent la texture de déform ation joue un rôle i mportant sur l’évolution  de la fraction  de \nmacle en fonction  de la déform ation. Par abus de la ngage, nous parlerons de cin étique de \nmaclage dan s la suite de ce mé moire. \n \nLa traction suivant un axe DT (Direction Tran sverse ) de la nuance de référence produit une \ntexture très prononcée caractérisée par quatre com p osantes principales de déformation ; \nlaiton {110}<112>, cuivre tournée {112}<110> appart en ant à la fibre <111>//DT, Goss tournée \n{110}<110> et cube {001}<100> appartenant à la fibr e <100>//DT. Ces deux dern ière fibres \nsont particulièrem ent visibles (grains en bleu et e n rouge) sur la Figure 17 après 30 % de \ndéformation . Ces différen tes composantes se m ettent  en place dès les premiers pourcents de \ndéformation  et se renforcent au cours de la déforma tion progressiveme nt, aux dépens des \norientations Goss et Cuivre présentes initialement (absence de texture sur la n uance de \nréférence après traitemen t de recristallisation). C ette texture de déformation est typique des \naciers austénitiques à faible EDE. \n \n 36  \n \nFig ure Fig ure Fig ure Fig ure 17 17 17 17    : Cartographie EBS : Cartographie EBS : Cartographie EBS : Cartographie EBSD de la D de la D de la D de la nuance nuance nuance nuance de référence après 30% de déformation vraie  (a) en contraste de  de référence après 30% de déformation vraie  (a) en  contraste de  de référence après 30% de déformation vraie  (a) en  contraste de  de référence après 30% de déformation vraie  (a) en  contraste de \nbandes (b) en orientation selon les couleurs du  tri angle  standard. M ise en évidence des fibres <111>// DT (e n bandes (b) en orientation selon les couleurs du  tri angle  standard. M ise en évidence des fibres <111>// DT (e n bandes (b) en orientation selon les couleurs du  tri angle  standard. M ise en évidence des fibres <111>// DT (e n bandes (b) en orientation selon les couleurs du  tri angle  standard. M ise en évidence des fibres <111>// DT (e n \nbleu) et <100>//DT (en rouge) bleu) et <100>//DT (en rouge) bleu) et <100>//DT (en rouge) bleu) et <100>//DT (en rouge)    [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] .. ..    \n \n \nFig ure Fig ure Fig ure Fig ure 18 18 18 18    :: ::    Cartogra phie EBSD de la Cartogra phie EBSD de la Cartogra phie EBSD de la Cartogra phie EBSD de la nuance nuance nuance nuance de référence après 30% de déformation vraie  (a) en contraste de  de référence après 30% de déformation vraie  (a) en  contraste de  de référence après 30% de déformation vraie  (a) en  contraste de  de référence après 30% de déformation vraie  (a) en  contraste de \nbandes (b) en  orientation selon les couleurs du tri angle standard. La tex ture bandes (b) en  orientation selon les couleurs du tri angle standard. La tex ture bandes (b) en  orientation selon les couleurs du tri angle standard. La tex ture bandes (b) en  orientation selon les couleurs du tri angle standard. La tex ture est alors peu prononcée est alors peu prononcée est alors peu prononcée est alors peu prononcée et 50% des  et 50% des  et 50% des  et 50% des \ngrains sont déjà ma clés comm grains sont déjà ma clés comm grains sont déjà ma clés comm grains sont déjà ma clés comme le montre (a) en part iculier ceu e le montre (a) en particulier ceu e le montre (a) en particulier ceu e le montre (a) en particulier ceux de la fibre <10 1>//DT x de la fibre <101>//DT x de la fibre <101>//DT x de la fibre <101>//DT [BARBIER  2009_1] [BARBIER  2009_1] [BARBIER  2009_1] [BARBIER  2009_1] .. ..    \n \nDan s la nuance de référence, les premières macles a pparaisse nt très tôt au cours de la \ndéformation . L’activation  des macles dépend alors d e deux  facteurs, l’orientation \ncristallographique et la taille de grain. Les grain s ayan t une orientation proche des \ncomposantes Goss et Cuivre (axe <101>//DT, en vert sur la Figure 18) et un e taille suffisante \n(>5µm) présentent des m acles mécaniques. Cette fibr e pr ésente des facteurs de Schm id \nfavorable au maclage mécanique comm e le m ontre la F igure 19 [BARBIER 2009_1] [SATO \n2011]. \n  37  \n \nFig ure Fig ure Fig ure Fig ure 19 19 19 19    : Figures de pôles inverses des facteurs de Sc hmid  en traction pour le glissem : Figures de pôles inverses des facteurs de Sc hmid en traction pour le glissem : Figures de pôles inverses des facteurs de Sc hmid en traction pour le glissem : Figures de pôles inverses des facteurs de Sc hmid en traction pour le glissement et le ma clage. Les ent et le ma clage. Les ent et le ma clage. Les ent et le ma clage. Les \nlignes d’iso lignes d’iso lignes d’iso lignes d’iso- - --intensité < 0.44 varient au pas de 0.02 (d’après [ BARBI ER 2009_1]). intensité < 0.44 varient au pas de 0.02 (d’après [B ARBI ER 2009_1]). intensité < 0.44 varient au pas de 0.02 (d’après [B ARBI ER 2009_1]). intensité < 0.44 varient au pas de 0.02 (d’après [B ARBI ER 2009_1]).     \n \nCes deux dernières composantes disparaisse nt au cou rs de la déformation au delà de 30 % de \ndéformation . Par contre, les grains apparten ant aux  orien tations principales de la fibre \n<111>//DT, possèdent également des facteurs d’orien tations plus favorables au maclage qu’au \nglisse ment cristallographique. Ainsi, l’intensifica tion  de la fibre <111>//DT sera plutôt \nfavorable au maclage contrairement à l’in tensificat ion de la fibre <100>//DT. \n \nLes orientations cristallographiques locales influe nce nt donc la formation de la \nmicrostructure de maclage. Réciproquement, le macla ge génère de nouvelles orientations \ncristallographiques susceptibles de modifier la tex ture de déformation. Toutefois, les macles \nform ées dans les grains de la fibre <111>//DT ou <1 10>//DT contribuent à renforcer la fibre \n<100>//DT et inversemen t, celles formées dans les g rains de la fibre <100>//DT renforcen t la \nfibre <111>//DT. C e tran sfert en tre fibres préexist antes n’ajoute pas de nouvelles fibres au \ncours de la déformation. Les aciers TWIP en tractio n ne développent donc pas de texture \nparticulière par rapport aux aciers austé n itiques s tables ! \n  38  \n(a) \n (b) \nFigure Figure Figure Figure 20 20 20 20    :: :: (a) Figures de pôle inverse pour l’a x e de tractio n de la nuance de ré férence  (a) Figures de pôle inverse pour l’a x e de traction  de la nuance de ré férence  (a) Figures de pôle inverse pour l’a x e de traction  de la nuance de ré férence  (a) Figures de pôle inverse pour l’a x e de traction  de la nuance de ré férence a près 5 et  30% de  a près 5 et  30% de  a près 5 et  30% de  a près 5 et  30% de \ndéformation vraie montrant les grains maclés (point s noirs) et non maclés (en rouge) déformation vraie montrant les grains maclés (point s noirs) et non maclés (en rouge) déformation vraie montrant les grains maclés (point s noirs) et non maclés (en rouge) déformation vraie montrant les grains maclés (point s noirs) et non maclés (en rouge) [GUTIERREZ 2010] [GUTIERREZ 2010] [GUTIERREZ 2010] [GUTIERREZ 2010]. (b) . (b) . (b) . (b) \nFigure s équiva lentes po ur 2 aciers TWIP après 20  % dé Figure s équiva lentes po ur 2 aciers TWIP après 20  % dé Figure s équiva lentes po ur 2 aciers TWIP après 20  % dé Figure s équiva lentes po ur 2 aciers TWIP après 20  % déformation formation formation formation [SATO 2010] [SATO 2010] [SATO 2010] [SATO 2010]. . ..    \n \nCes différen tes con statations permettent d’explique r parfaitem ent les observations rapportées \nsystématiquement dans la littérature m ontrant qu’ap rès déformation, les grains n on maclés \ngénéralement présentent une orientation <100>//DT ( en rouge) et les grains m aclés une \norientation <111>//DT (en bleu) sur les cartographi es EBSD précédentes. Cette répartition \nparticulière intervient très tôt au cours de la déf ormation (dès 5% de déformation  comme le \nmon tre la Figure 20(a)) [GUTIERREZ 2010][SATO 2010] . Ce résultat suggère que le maclage \nest bien un mécan isme supplétif au glissement car d ans l’orientation  <111>//DT le facteur de \nSchm id du glisse m ent est très faible (0,3) du m ê me ordre de grandeur que celui du maclage. \nC’est dans cette fibre en particulier que les deux mécanismes sont susceptibles de rentrer en  \ncompétition . Si les grains sont bien orie ntés vis-à -vis du glissemen t, l’acier ne semble pas \navoir besoin  de macler pour assurer la vitesse de d éformation  imposée. Nous reviendrons sur \ncette compétition probable entre glissem ent et macl age mécanique au chapitre 2.4.3.3 page 88. \n \nCette discussion détaillée dans le cas de la tracti on uniaxiale s’applique aussi à d’autres modes \nde chargem ent [BARBIER 2009_1] . La structuration du maclage, les fractions et tex tures \nrésultantes vont donc beaucoup dépen dre du mode de chargemen t (cisailleme nt, traction, \nexpansion…). Nous discuterons de ces implications d an s la suite de ce mé moire. \n  39  2.2.4. 2.2.4. 2.2.4. 2.2.4.  Discussion Discussion Discussion Discussion    \n2.2.4.1.  Notion de contrain te critique de maclage  \n \nLe fait que l’apparition des systèmes de maclage pr im aires puis secondaires suit une loi de \nSchm id con forte n écessairement l’existence d’une ci ssion résolue critique pour le maclage \nmécanique, i.e. un e cission seuil pour la germ inati on  des macles, aussi appelée contrainte \ncritique de maclage [GUTIERREZ 2010] [BRACKE 2009] [CHRISTIA N 1995] [KARAMAN  \n2000_1] [MEYERS 2001]. \n \nCette contrainte critique pour l’apparition du macl age a été observée et m esurée aussi bien \ndans des mono que des polycristaux de différen ts él ém ents purs ou alliages métalliques. Cette \nvaleur dépend au premier ordre de l’EDE pour les au sténitiques mais peut dépendre aussi du \ntype de charge (traction/compression), de la tempér ature, de la pré-déform ation, de la taille \nde grain ou de la vitesse de déformation. Une récen te revue de ce sujet a été publiée par \nMeyers et al.  [MEY ERS 2001]. \n \nSi l’on considère un mécanisme générique de formati on d’une micromacle reposant sur la \ncission nécessaire pour émettre une première boucle  de dislocation partielle de Shockley \ntraîn ant un  défaut d’empile men t, la cission critiqu e de maclage s’exprim e sous la forme \nsuivante : \n \n \nC112\n112int intC\nRμb\nbEDEτ + =− (6) \n \navec R c un rayon critique d’émission de boucle de dislocat ion [LUBEN ETS 1985][C H RISTI AN \n1969]. Les dislocations suivantes constitutives de la micromacle ne ressentent pas la création \ndu défaut et sont donc émises rapidemen t à la suite . C’est ce processus qui a été utilisé dan s le \nmodèle de DDD décrit ci-dessus. En con sidéran t une EDE de 20 mJ.m -2 , un rayon  Rc de 50 \nnm, et un facteur de Taylor de 3, la con trainte cri tique de m aclage calculée est de l’ordre de \n1200 MPa, bie n supérieure à la contrainte d’écoulem e nt de ces alliages. Ces valeurs de \ncontraintes ne son t attein tes qu’après plus de 30 %  de déform ation alors qu’un maclage déjà \nintense est observé dans la nuance de réf érence à g ros grain. \n \nLes valeurs déterminées par cette équation sont don c bien supérieures aux valeurs de \ncontraintes critiques de m aclage mesurées à l’échel le m acroscopique. Il existe un accord dans \nla littérature pour suggérer que la germin ation des  macles est bie n un  processus assisté par des \nconcentrations de contraintes comme des e mpilements  de dislocation s perm ettant localem ent \nde dépasser la con trainte critique locale définie c i-dessus. Le prem ier modèle in cluant ces  40  différents m écanism es a été proposé en 1964 par Ven ables sur la base d’un m écanisme polaire \n[VEN ABLES 1964] \n \n ()\n112int twinC twinC\n110pile\nbEDEτ τ\nμb 1,84L υ 1\n31=\n\n\n−+− − (7) \n \navec L pile  la longueur caractéristique des empileme nt de disl ocations parfaites, ν le coefficient \nde Poisson, b 110  le vecteur de Burgers des dislocations parfaites. D epuis ce premier m odèle, des \nversions plus soph istiquées ont été proposées se di stin guant principaleme nt par le modèle de \nréactions en tre dislocation s pour la germ ination de s m acles, comme Karaman et al.  repren ant \nun m odèle par  glissement dévié ou Buyn  basé sur la simple e x tension  de fautes d’e m pilem ent \n[KARAMAN 2000_1] [MEYERS 2001][GUTIERREZ 2010][BUYN  2003]. \n \nAu-delà de son aspect fonctionn el, cette équation s uggère la relation profonde qu’il existe \nentre maclage mécanique et la nature du glissement dans ces structures. Nous reviendrons au \nchapitre 2.4.3 page 85 de manière détaillée sur cet te relation complexe. La question de la \ngerm ination  possible des macles sur d’autres sites privilégiés comme des précipités est \ndiscutée en  Anne x e 6 page 163. \n \n2.2.4.2.  Mesure de la fraction maclée  \n \nIl ressort aussi de cette présentation un e difficul té centrale et inhérente à l’étude de l’effet \nTWIP, celle de la m esure de la fraction de phase ma clée. Les deux raisons principales en sont : \n• la zone reconstruite dans la macle est de structure  CFC donc non différentiable de la \nmatrice d’un point de vue cristallographique. La fr action de macle n e peut dont être \nmesurée par une analyse volumique classique par dif fraction des rayons X, comme on \npourrait l’en visager dans le cas d’une transformati on martensitique ε ou α’ 5.  \n• les nanomacles ne peuven t être r ésolues qu’à l’éche lle de la microscopie électronique à \ntran smission (MET). Com pte tenu du volume des zones  analysées par cette technique, \non peut mettre en  doute rapidement sa représentativ ité statistique pour estimer de \nfaçon fiable des fractions. Aucun es des techniques à plus grandes échelles comme le \nMEB, l’EBSD ou pire encore la MO qui pourraient per mettre une meilleure statistique \nne peuvent garantir une résolution suffisante perme ttant de distinguer les nanomacles \n                                                 \n5 Cette problé matique a été abordée da ns le cadre de  la thèse de J.L. Collet à l’ES RF de Grenoble finan cée par \nAM  [COLLET 2009]. Ces travaux  ont confirmé  l’intérê t des rayonneme nts X à haute én ergie pour l’étude d u \nmaclage dans ces ac iers. Toutefois, les résultats s ont soumis à interprétation par le biais d’un modèl e \nd’interaction entre rayonnement et défauts d’empile ment. En conséquence, les cinétiques de mac lage \n« mesurées » sont donc très dépendantes de l’ajuste ment de ces multiples paramètres et n e peuvent donc  être \nconsidérées c omme absolues.  41  de leur faisceaux ou de révéler toutes les micromac les isolées (plus minces que la \nrésolution théorique de la microscopie optique). \n \nPour pallier cette difficulté d’ordre expérimentale , plusieurs m éthodes alternatives indirectes \nont été développées au cours de nos travaux . La pre mière consiste à utiliser la relation \nstéréologique de Fullman  [FULLMAN 1953], relation e ssentielle pour la compréh ension  de \nces aciers, qui relie la distance moye n ne entre mac les t, leur épaisseur moyenne e et la \nfraction maclée F. \n \n ( )F1F\n2e1\nt1\n−=  (8) \n \nCette distance moyenne entre m acles t peut être déd uite de la mesure ex périmentale L du \nlibre parcours moyen dan s la microstructure, m esure  intégrant joints de m acles et de grains \n(D la taille de grain  efficace). \n \n \nt1\nD1\nL1+=  (9) \n \nCes deux relation s perm ettent d’estim er la fraction  maclée F en  fonction de la mesure \nexpérimentale de L (par la technique des intercepte s par exem ple) et de e (en  MET). \n \n \n−+=\nL1\nD1\n2e111F  (10) \n \nCette relation peut être affinée en tenan t compte d e la structuration  en faisceaux du maclage \n[ALLAIN 2004_1] .  \n \n \n( )( )1DL λe 2NLD1\nλeeF\nfaisceau faisceaufaisceau+\n\n\n\n− +\n\n\n\n+=  (11) \n \nAvec N faiscea u  le nombre m oye n de micromacles par faisceaux et λ f aisceau  l’espacem ent inter-\nmacles moyen dan s les faisceaux . Appliquée au cas d e la nuance de référen ce, cette relation \nperm et de confirm er que la fraction de phase maclée  reste faible mêm e après 50 % de \ndéformation  (environ 12%) (cf. Figure 21). \n  42  \n(a) \n (b) \nFigure Figure Figure Figure 21 21 21 21    : (a) Evolution du libre parcours moye n L par la m éthode des interceptes au cours de la déformation : (a) Evolution du libre parcours moye n L par la mé thode des interceptes au cours de la déformation : (a) Evolution du libre parcours moye n L par la mé thode des interceptes au cours de la déformation : (a) Evolution du libre parcours moye n L par la mé thode des interceptes au cours de la déformation en  en en en \ntr action tr action tr action tr action de la  nuance de référence (mesures en de la  nuance de référence (mesures en de la  nuance de référence (mesures en de la  nuance de référence (mesures en MO MO MO MO après  attaque) (b) Evolution de  la fraction  après  attaque) (b) Evolution de  la fraction  après  attaque) (b) Evolution de  la fraction  après  attaque) (b) Evolution de  la fraction de ma c les déduite  de ma cles déduite  de ma cles déduite  de ma cles déduite \nde l’équation de l’équation de l’équation de l’équation (11) (11) (11) (11) [ALLAIN 2004_1]. [ALLAIN 2004_1]. [ALLAIN 2004_1]. [ALLAIN 2004_1].     \n \nDepuis mainte nan t une dizaine d’année, la technique  EBSD e n MEB-FEG a perm is de faire \ndes progrès dans la mesure des fractions maclées gr âce à des temps d’acquisition raisonnables. \nCette techn ique permet d’analyser de g randes plages  en con servant des distances entre pas \nd’an alyse inférieures à 100 nm , progrès auquel n ous  avons pu contribuer [BARBIER \n2009_1][BARBIER 2009_2] . Toutes les macles ne peuvent être indexées compte  tenu de la \nrésolution ou tout simplement révélées, rendant ina déquate l’utilisation directe des \ninformation s concernant les orien tations cristallog raph iques. Pour contourner cette difficulté, \nRen ard a utilisé les informations supplé mentaires c ontenue dans les cartographies de \ncontrastes de bandes. Les difficultés locales d’in d exation sont utilisées alors comme marqueurs \nde la présen ce de macles [RENARD 2012]. Cette méth o de reste bien  sûr plus précise que des \nmesures en  microscopie optique et élimine tous les biais liés à l’attaque m étallographique, \nmais est toujours im parfaite. \n \nEn jouant sur cette difficulté d’inde xation des nan omacles par EBSD, Barbier a proposé une \nméth ode de mesure tout à fait originale et automati sable. Cette t echnique repose sur les \nconstats suivants : \n• l’EBSD permet de déterm iner l’orientation des grain s m ais ne permet pas de résoudre \nles macles. Pour un e orien tation donnée, seules les  matrices n on maclées participeront \nà la mesure. \n• L’an alyse des den sités de distribution d’orie ntatio ns obtenues par DRX intègre les \ncontributions des m atrices et des macles de façon i ndifférenciées.  \n• Pour une orientation donnée, la fraction  de ph ase m aclée est donc la différence en tre \nla m esure par DRX (matrice+macle) F DRX  et par EBSD (matrice uniqueme nt) F EB SD . \n• L’augmentation de la fibre <100>//DT est principale m ent due au maclage au cours de \nla déformation \n \nOn peut donc alors écrire \n \n //DT 100\nEBSD//DT 100\nDRX F FF>< ><− =  (12)  43   \nLa validité de l’approche pourrait être améliorée e n in tégrant ces constats sur de n ombreuses \norien tations. En l’état, les mesures pour l’acier d e référ ence et la seule orientation <100> //DT \nsont représentées pour différents niveaux de déform ation sur la Figure 22. Ces valeurs restent \nraisonnablement proche des résultats de l’autre mes ure indirecte présentés sur la Figure 21(b). \n \n \nFig ure Fig ure Fig ure Fig ure 22 22 22 22    : : : : Evolution de la fraction volumique  de macle en fo nc tion de la  déformation vraie estimée grâce à Evolution de la fraction volumique  de macle en fonc tion de la  déformation vraie estimée grâce à Evolution de la fraction volumique  de macle en fonc tion de la  déformation vraie estimée grâce à Evolution de la fraction volumique  de macle en fonc tion de la  déformation vraie estimée grâce à \nl’équation l’équation l’équation l’équation (12)  (12)  (12)  (12) sur la nuance de ré f  sur la nuance de ré f  sur la nuance de ré f  sur la nuance de ré férence déformée  en traction érence déformée  en traction érence déformée  en traction érence déformée  en traction    [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] .. ..    \n \nMalgré un constat d’échec partagé dans la littératu re, voire une certaine résignation, peu de \ntravaux ont été dédiés spécifiquement à cette quest ion  et l’on  peut s’en étonner compte tenu \ndes enjeux. Il serait probablem ent intéressan t de r edécouvrir des techn iques comme la \nmicroscopie à force atom ique pour y répondre. Cette  tech nique utilisée pour l’étude du \nmaclage par Coupeau et al.  [COUPEAU 1999] possède une résolution spatiale suf fisante pour \nrésoudre la présence de micromacles incidente à une  surface polie. Toutefois, il est \nindispensable de connaître l’orien tation cristallog raphique du grain considéré pour définir de \nman ière un ivoque l’épaisseur de la macle et son deg ré de perfection (fautes résiduelles). Il \nserait donc intéressant de coupler cette technique avec une mesure préalable par EBSD des \norientations des grains analysés. \n \n2.2.5. 2.2.5. 2.2.5. 2.2.5.  Con clu Con clu Con clu Con clusion s interm édiaire sion s interm édiaire sion s interm édiaire sion s interm édiaires s ss    \n \nLa m orphogénèse de la microstructure de m aclage dan s les aciers TWIP fait maintenant \nl’objet d’un  large consensus dans la littérature sc ientifique, auquel n ous avons con tribué, en \nparticulier sur les aspects suivants : \n• l’identification des nanom acles, leur taille et org anisation en faisceaux. \n• L’activation  séque n tielle de différents systèmes de  maclage suivant une loi de Schm id \net ses conséque nces sur la texture de déformation. \nOn notera paradoxalement que les progrès accomplis dans le domaine de la mesure de la \nfraction de phase maclée resten t marginaux. En con s éque n ce les m odèles de cin étique de \nmaclage depuis la thèse de Rémy ont peu progressé e n l’absence de données expérime ntales  44  fiables. Cette évolution de fraction de phase maclé e reste souvent le dernier paramètre \nd’ajuste men t des m odèles structure-propriétés dont nous allons maintenant discuter. \n \n2.3. 2.3. 2.3. 2.3.  L’ L’ L’ L’ee eeffet TWIP ffet TWIP ffet TWIP ffet TWIP    : La relation entre maclage mécanique et comportem ent : La relation entre maclage mécanique et comporteme nt : La relation entre maclage mécanique et comporteme nt : La relation entre maclage mécanique et comporteme nt    \n \nCom pte ten u de son fort potentiel pour le marché au tomobile, il existe une forte attente des \n« mécaniciens » pour comprendre les spécificités du  comportement m écanique de ces alliages, \nen particulier leurs très forts taux  d’écrouissage,  de nature principaleme nt cinématique ou la \nform e particulière de leurs surfaces de charge. Pou r répondre à ce besoin, n ous avons \ndéveloppé des lois de com portem ent à base physique pour la mise en forme et l’usage de ces \naciers (com portem ent à h aute vitesse de déformation  par  ex emple pour des applications en \nconditions de crash), en prenan t e n compte les prin cipaux mécanismes éléme ntaires de la \nplasticité, glissement et m aclage. \n \nCes travaux  nous ont naturellem ent am e nés à considé rer des chargements de plus en plus \ncomplexes pour se rapprocher des conditions d’utili sation réalistes (des essais de traction aux \nessais d’em boutissage en passant par les changement s de trajets th ermom écaniques). Nous \nmettrons en  évidence dans ce mémoire l’origin e et l es conséquences des contraintes internes \ndans ces résultats.  \n \nMes travaux de thèse et au-delà ont principalement permis de mieux  appréhender et \nquan tifier l’impact du m aclage mécanique et son int eraction avec le glissement sur le \ncomportem ent mécanique macroscopique, appelé mainte nant communément effet TWIP. Je \ntiens d’ailleurs ici à rendre homm age aux travaux p récurseurs du Professeur L. Rém y dans les \nannées 1970 dans ce domaine, qui m’ont fourni une b ase précieuse et dont les interprétations \nrestent plus que jam ais d’actualité. \n \nAu cours de mes travaux de thèse, j’ai mis en évide nce que l’effet TWIP pouvait être \ninterprété comme un mécanisme d’effet  « Hall et Pet ch » dyn amique. Nous avons en effet pu \nrelier les propriétés d’écrouissage de ces aciers à  la mise en  place de la m icrostructure de \nmaclage, en particulier leur sous-structures nanom é triques et l’activation de multiples \nsystèmes n on coplanaires au cours de la déform ation , créan t des défaut planaires \ninfranchissables pour les dislocations et induisant  une rapide dim inution de leurs libres \nparcours moyens. \n \nPlus récemm e nt, avec mes collègues, O. Bouaziz et C .Scott, n ous avons développé un modèle \nmicromécanique basé sur les travaux de Sinclair et al.  (extension du modèle de Mecking-\nKocks-Estrin) [SIN CLAIR 2006] intégrant une composa nte cinématique se n sible aux effets de \ntaille et un e description phénom énologique de la mi crostructure de maclage. Ce nouveau \nmodèle a permis de décrire de façon complète non se ulement les effets de tailles de grain sur  45  le comporteme nt en traction mais aussi la com posan t e cinématique de l’écrouissage de ces \naciers, associée à la microstructure de maclage. Ce s travaux on t ensuite été in tégrés comme un \nprolongement naturel dan s le cadre de la thèse de D . Barbier.  \n \nDan s cette partie, n ous considérerons im plicitem ent  que seul le maclage mécanique contribue \nau durcissement des aciers TWIP FeMn C. Les autres m écanismes potentiels de durcissem ent \ncomme le vieillisseme nt dynamique seront discutés a u chapitre suivant. Nous utiliserons les \nnotions classiques d’essai Bauschinger, d’écrouissa ge ciné matique ou isotrope et de surface de \ncharge. Toutefois, nous ne revien drons pas sur ces con cepts. Le lecteur pourra se référer par \nexem ple au mé moire d’Habilitation à Diriger des Rec herches d’O. Bouaziz pour plus de \ndétails [BOUAZIZ 2005]. \n \n2.3.1. 2.3.1. 2.3.1. 2.3.1.  Prem ier modèle de l’effet TWIP Prem ier modèle de l’effet TWIP Prem ier modèle de l’effet TWIP Prem ier modèle de l’effet TWIP    : : : : Rôle Rôle Rôle Rôle précurseur de  précurseur de  précurseur de  précurseur de L.  L.  L.  L. Remy  Remy  Remy  Remy    \n \nLa première modélisation du rôle du maclage m écaniq ue sur l’écrouissage de ces aciers a été \nproposée par Rém y en 1978 [REM Y 1978]. Afin de rend re com pte de l’effet des larges \nempile ments de dislocations sur les joints de macle s, il propose par analogie avec une \napproche de type « Hall et Petch » : \n \n ( ) ( )Tr\nT matrice matricet1K ε σ EΣ \n\n+ =  (13) \n \navec Σ et E les con traintes et déformations macrosc opiques, σ matrice  et ε matrice  les con traintes et \ndéformation s de la matrice sans macle mécanique, t la distance moyenne entre macles sur \nlesquelles se stockent les dislocations et K T et r T deux constantes. r T pouvan t pren dre \nclassiquement des valeurs entre 1 et ½, en fonction  du mécanisme d’écrouissage con sidéré. La \ndistance moyenne entre m acle t est alors évaluée gr âce à la r elation de Fullman (cf. équation \n(10)).  \n \nAfin  d’estim er le terme K T, Rém y propose une autre m odélisation de la con trai nte basée sur \nl’équilibre à grandes distances en tre con traintes e n retour d’un empileme nt de n T dislocations \net la contrainte appliqué τ app  de façon analogique à l’approche de Friedel [FRIED EL 1964] \nprésentée au chapitre 2.2.2 page 28 : \n \n ( )\nt1\n2μbnMσΣ\nt1\n2μbn τ110\nTmatrice 110\nT app =−⇒ =  (14) \n \nAvec M un facteur de Taylor. Cette relation conduit  alors à la relation  suivan te : \n  46   ( ) ( )Tr\n110\nT matrice matricet1\n2bMμn ε σEΣ \n\n+ =  (15) \n \nDan s les travaux de Rémy sur les alliages Ni-Co, n T prend des valeurs relativeme nt grandes de \nl’ordre de 30-50 dislocations. Stricto sensu, cette  équation est un modèle de durcissem ent \nintracristallin et ne permet pas de prendre en comp te de façon paradoxale l’effet de taille de \ngrain  initial. Par contre, en considérant le champ de con traintes à lon gue distance des \nempile ments de dislocations, ce modèle d’écrouissag e est le prem ier d’essence purem ent \ncinématique. \n \n2.3.2. 2.3.2. 2.3.2. 2.3.2.  Prem ière Prem ière Prem ière Prem ières s ss extension  extension  extension  extensions s ss du  du  du  du m odèle m odèle m odèle m odèle M  M  M  Meckin g eckin g eckin g eckin g- - --Kocks Kocks Kocks Kocks- - --Estrin Estrin Estrin Estrin    \n \nA la suite de ces travaux  précurseurs, Karam an et al.  [KARAMAN  2000_2] et Bouaziz et \nGuelton [BOUAZI Z 2001_1] ont proposé au début des a nnées 2000, des approches de l’effet \nTWIP considéran t une exten sion du modèle de Mecking -Kocks-Estrin [MECKING \n1981][ESTRIN 1984]. La nature de ces approches est puremen t isotrope. Le maclage \nmécanique est supposé réduire le libre parcours moy en des dislocations et donc favoriser \nl’augmentation de la de nsité de dislocations statis tiquem ent stockées (DSS). \n \n \n\n\n− + =DSS DSS\n110 110 MDSSf.ρ ρ\nbk\nLb1M.\ndεdρ (16) \n \nAvec L don né par \n \n ( )F1F\n2e1\nD1\nL1\n−+=  (17) \n \nAvec ρ DSS la densité de DSS, ε M la déformation  plastique dan s la matrice non maclé e, k et f \nsont deux paramètres caractérisant respective m ent u n  mécan isme d’écrouissage latent et un \nmécanisme de restauration  dynam ique. \n \nLa contrain te d’écoulemen t est alors déduite d’un m odèle de Taylor (durcissement de la forêt \nde DSS) et s’écrit : \n \n DSS 110 0 ρ b μMα σΣ+=  (18) \n \navec α une constante (durcisse ment de la forêt), M un factor de Taylor moyen et σ 0 une \ncontrainte de friction dépe ndan t unique ment de la s olution solide. On peut prendre en \ncompte dans ces approches la con tribution du maclag e à la déformation macroscopique :  47   \n () .dFγ .dγF1 dT M+−=Γ  (19) \n \navec Γ le cisaillement macroscopique, γ M le cisailleme n t dû au glissement des dislocations,  γ T \nle cisaillem ent produit par une macle et F la fract ion de m acles. γ T est une constante qui \ndépend uniqueme n t de la nature des macles (cf. chap itre 2.2.1.1 page 22).  \n \nCette dernière équation permet de déterminer la déf ormation totale en introduisant le facteur \nde Taylor. La con tribution totale des macles à la d éformation en traction ε T peut donc \ns’écrire : \n \n MFγεT\nT=  (20) \n \nSi l’on considère une fraction de phase maclée de l ’ordre de 10% (soit environ la fraction  de \nphase maclée de la nuan ce de référen ce vers 50% de déformation  – cf. Figure 21), cette \ncontribution vaut alors à peine 2%. Ce calcul d’ord re de gran deur m ontre pourquoi ce terme \nest souvent négligé dans les modèles micromécanique s les moins élaborés. \n \n2.3.3. 2.3.3. 2.3.3. 2.3.3.  Extension polycristalline Extension polycristalline Extension polycristalline Extension polycristalline    \n \nA l’instar de Karaman et al.  [KARAMAN 2001_2], nous avons étendu ce modèle isot rope dans \nle cadre de la plasticité polycristalline en consid érant que les équations précédentes étaient \napplicables pour tous les systèmes de glissement d’ un grain [ALLAIN 2002][ALLAIN \n2004_1][ALLAIN 2004_2] . Nous avons in troduits deux  originalités : \n• la m atrice décrivan t les in teractions latentes en tr e les systèmes de glissemen t (matrice \nde type Franciosi) s’est révélée in suffisan te pour décrire l’écrouissage dû à l’effet TWIP. \nNous avons donc rajouté un term e suppléme ntaire cor respon dant à un term e d’auto-\nécrouissage des systè mes de glisse ment. Nous avons qualifié ce term e de contribution \n« d’H all et Petch » (HP) car il était aussi nécessa ire pour décrire les effets de taille de \ngrain  et correspon dait dan s l’esprit à un e contrain te en retour due à des em pile ments \nde dislocations (back-stress). Nous n’avons pas fai t à l’époque explicite ment le lien \navec une contribution de nature ciné matique. \n• une matrice d’intersection s entre les systèmes de m aclage et de glissement, qui perm et \nde calculer la réduction du libre parcours moye n d’ un système de g lissem ent donné \nseulement par les m icromacles sécantes à ce système . \n \nCes différentes lois ont été intégrées dans un sché ma polycristallin  viscoplastique uniax iale \nselon  trois modèles de transition d’échelle simples , Taylor relaxé (iso-déformation en tre \ngrain s), Sach s (iso-contrainte) ou isoW (iso-travai l dissipé entre les grains) [BOUAZIZ 2002].  48  Un m odèle de cinétique de maclage em pirique mais ba sé sur la notion de contrain te critique \nde m aclage a été in troduit pour chaque système de m aclage. \n \nLa Figure 23 montre les courbes de traction prédite s par le m odèle avec 100 grains \n(orientation s aléatoires) selon les trois modèles d e transition d’échelle retenus. Les deux \nhypothèses polycristallines les plus réalistes donn ent des résultats satisfaisants et permettent \nde décrire assez fin eme nt l’évolution du taux d’écr ouissage. \n \n \nFigure Figure Figure Figure 23 23 23 23    : : : : Courbes de traction ex pér imentales de la n uance Courbes de traction ex pér imentales de la n uance Courbes de traction ex pér imentales de la n uance Courbes de traction ex pér imentales de la n uance de référenc e de référenc e de référenc e de référenc e à 298 K et simulées selo n les troi s à 298 K et simulées selo n les troi s à 298 K et simulées selo n les troi s à 298 K et simulées selo n les troi s \nmodèles modèles modèles modèles de transition d’éc helles  de transition d’éc helles  de transition d’éc helles  de transition d’éc helles    [A LLAIN 2004_1] [A LLAIN 2004_1] [A LLAIN 2004_1] [A LLAIN 2004_1]. . ..    \n \nCette approche est d’autan t plus intéressante qu’el le permet de décrire avec un bon  accord les \nobservations expérimentale de l’évolution des m icro structures, comm e le montre la Figure 24. \nLa Figure 24(a) montre l’évolution du libre parcour s moyen dans la microstructure pour des \ndislocations entre obstacles forts (joints de grain s et de macles) et la Figure 24(b) la proportion \nde grains m aclés par 1 ou 2 systè m es. \n \n(a) \n (b) \nFigure Figure Figure Figure 24 24 24 24    :: :: (a) Evolution du libre parcours moyen dans la mic rostructure pour des dislocations entr e obstacles  (a) Evolution du libre parcours moyen dans la micr ostructure pour des dislocations entr e obstacles  (a) Evolution du libre parcours moyen dans la micr ostructure pour des dislocations entr e obstacles  (a) Evolution du libre parcours moyen dans la micr ostructure pour des dislocations entr e obstacles \nforts (joints de grains et de mac les) et (b) propor tion de grains ma clés par 1 ou 2 systèmes de forts (joints de grains et de mac les) et (b) propor tion de grains ma clés par 1 ou 2 systèmes de forts (joints de grains et de mac les) et (b) propor tion de grains ma clés par 1 ou 2 systèmes de forts (joints de grains et de mac les) et (b) propor tion de grains ma clés par 1 ou 2 systèmes de maclag e en maclage en maclage en maclage en fonc tion  fonc tion  fonc tion  fonc tion \nde la déformation de la déformation de la déformation de la déformation vraie prévues par le  modèle  selon  de ux modèles vraie prévues par le  modèle  selon de ux modèles vraie prévues par le  modèle  selon de ux modèles vraie prévues par le  modèle  selon de ux modèles de de de de tra nsition d’échelle tra nsition d’échelle tra nsition d’échelle tra nsition d’échelle    [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n \nCette description de la microstructure de maclage e t de la réduction du libre parcours moyen \ndes dislocations par  les macles e n plasticité crist alline a e nsuite été utilisée dans les \nmodélisations micromécaniques viscoplastiques des t h èses de N . Shiehkelshouk \n[SHI EKHELSOUK 2006] et de D. Barbier [BARBIER 2009_1] .  49   \n2.3.4. 2.3.4. 2.3.4. 2.3.4.  Distinction  et couplage des composantes ciném atiqueDistinction  et couplage des composantes ciném atiqueDistinction  et couplage des composantes ciném atiqueDistinction  et couplage des composantes ciném atique ss ss et  et  et  et isotropes isotropes isotropes isotropes    \n \nLa principale limitation des modèles décrits précéd emme nt et développés avant 2008 par \nnotre équipe est de prévoir un écrouissage de natur e principale m ent isotrope (excepté la \ndispersion de con traintes d’écoulement entre les gr ains dans le modèle de plasticité \npolycristalline). Même si le term e de type HP corre spond à un mécanisme de durcissem ent \ncinématique, il n’a jamais été exploité comm e tel. Ces premiers modèles, y compris les \nextensions en plasticité cristallin e [BARBIER 2009_1] , ont donc été mis en défaut par les \nprem ières mesures d’effet Bauschinger [BOUAZIZ 2008_1] . Depuis, ces mesures ont été \nconfir mées plusieurs fois en  particulier par l’équi pe du CEIT [SEV ILLANO \n2009][SEVI LLANO 2012] montrant que l’écrouissage de s aciers TWIP est principalem ent \nd’origine ciné matique, comme en visagé initiale m ent par Rémy. \n \nLe premier modèle incluant les deux composantes d’é crouissage a été proposé par notre \néquipe en 2008 afin d’expliquer le fort effet Bausc hinger de ces aciers, m ais aussi de bien \nreproduire simultanéme nt le sim ple effet « Hall et Petch » (évolution  de la limite d’élasticité \navec la taille de grain austé nitique) de ces nuance s, généraleme nt mal capté par tous les \nmodèles précédents (y com pris celui de Rémy). \n \nLe principal apport de ce modèle par rapport au mod èle de Bouaziz et Guelton est \nd’introduire explicite men t la notion de dislocation s stockées sur les obstacles forts de la \nstructure (dislocations géométrique men t nécessaires  DGN) et de définir la con trainte en \nretour résultante. \n \nCette contrainte en retour σ b s’écrit de façon  très analogue à la proposition de  Rémy (cf. \néquation (14)) : \n \n nLμ.bM σ110\nb=  (21) \n \navec n le n ombre moyen de dislocations mobiles stop pées sur des joints de grains ou de \nmacles (L tient compte des deux  types d’obstacles f orts) 6. Cette approche est justifiée par les \nobservations par EBSD des très forts gradients loca ux de déformation  observés par D. Barbier \nsur les structures m aclées. La Figure 25 montre par  ex emple les gradie nts cumulés et locaux \nobservés dans un grain maclé de la nuance de référe nce après seulem ent 20% de déformation. \n                                                 \n6 Ce modèle repose au ssi sur l’hypothèse que les jo ints de macles sont instantanément saturés par des DGN. Si ce \nn’éta it pas le cas, les nouvelles macles créées ne contribueraient pas autant au durci ssement cinémati que .   50  D’un  bout à l’autre du grain, la désorientation var ie de 12° et localem ent près des m acles de 3° \nsur des distances très faibles. \n \n \nFigure Figure Figure Figure 25 25 25 25    : C arto : C arto : C arto : C artographie EBSD  des désorientations intragranul air es graphie EBSD  des désorientations intragranulair es graphie EBSD  des désorientations intragranulair es graphie EBSD  des désorientations intragranulair es d e de de de la nuance de référence défor mé  la nuance de référence défor mé  la nuance de référence défor mé  la nuance de référence défor mée e ee de 20%  de 20%  de 20%  de 20% \nen traction (a ) en contraste de bandes (b) en gra di ent d’orientations à longue dista nce par rapport à la moyenne en traction (a ) en contraste de bandes (b) en gra di ent d’orientations à longue dista nce par rapport à la moyenne en traction (a ) en contraste de bandes (b) en gra di ent d’orientations à longue dista nce par rapport à la moyenne en traction (a ) en contraste de bandes (b) en gra di ent d’orientations à longue dista nce par rapport à la moyenne \ndu grain  (c) en gradient d’orientations à courte didu grain  (c) en gradient d’orientations à courte didu grain  (c) en gradient d’orientations à courte didu grain  (c) en gradient d’orientations à courte di sta nce (désorientation angulaire entr e pixels voisi ns) sta nce (désorientation angulaire entr e pixels voisi ns) sta nce (désorientation angulaire entr e pixels voisi ns) sta nce (désorientation angulaire entr e pixels voisi ns)    \n[BARBIER  2009_1]. [BARBIER  2009_1]. [BARBIER  2009_1]. [BARBIER  2009_1].     \n \nLe flux de dislocations arrivant sur les joints par  bandes de glissement s’écrit : \n \n \n\n\n−=\n0 110 matrice nn1\nbλ\ndεdn (22) \n \navec λ l’espace moyen en tre ban des de glisse m ent, n 0 le nombre maximum de boucles de \ndislocations arrêtées sur les joints de grain et de  macle. Le rapport λ/b 110  représente le nom bre \nde dislocations par bandes de glissemen t nécessaire  pour fournir la déform ation. Au-delà de \nn0 les joints de grains ne stocke nt plus de DGN car d es mécan ismes de relaxation sont activés, \ncomme l’ém ission de dislocation s derrière l’obstacl e ou l’activation de glissement sur des \nplan s différents. L’équation de MKE utilisée par Bo uaziz et Guelton [BOUAZIZ 2001_1] est \nalors modifiée pour tenir compte de la saturation e t de la relaxation possible des joints. La \ndensité peut augm e nter locale ment mais le vecteur d e Burgers total résultant reste constant. \nCette nouvelle équation est directemen t inspirée de s travaux de Sinclair et al.  [SINCLAIR \n2006] [BOUAZIZ 2008_2]  : \n \n \n\n\n\n\n− +−\n=DSS DSS\n110 1100\nmatriceDSSf.ρ ρbk\n.Lbnn1\nM.dεdρ (23)  51   \nPar extension de l’équation (18), la contrainte d’é coulement m acroscopique s’écrit : \n \n nLμ.bM ρ b μMα σ σ σ σΣ110\nDSS 110 0 b F 0 + +=++=  (24) \n \nLa cinétique de m aclage dans ce modèle est encore c hoisie de manière empirique, en \nl’absence de modèle à base physique dans la littéra ture : \n \n pour ε > ε init , ( )( )Tinit Tm ε ε β\n0 e1FF−−−=  (25) \n \nOu ε init , F 0, β T et m T sont 4 paramètres à identifier. F 0 correspond à la fraction totale de ph ase \nmaclées et ε init  une déformation minimum avant l’apparition des pre mières macles. \n \nCette approche après ajusteme nt des cin étiques de m aclages permet de décrire \nsimultanément : \n• les courbes de comportement en  traction pour différ entes tailles de grain s [SCOTT \n2006] , avec une bonne description en particulier des lim ites d’élasticité, \n• le comportement estimé de l’alliage san s maclage et  pour un e gran de taille de grain \n[ALLAIN 2004_1]  (cf. chapitre 2.3.6.3 page 65), \n• l’écrouissage cinématique en fon ction de la déforma tion qui a la particularité d’être \nnon saturan t. \n \n(a) \n (b) \nFigure Figure Figure Figure 26 26 26 26    : (a ) Cour : (a ) Cour : (a ) Cour : (a ) Courbes  de traction ex périmentales bes  de traction ex périmentales bes  de traction ex périmentales bes  de traction ex périmentales et simulé et simulé et simulé et simulée e ees pour différentes tailles de gra ins s pour différentes tailles de gra ins s pour différentes tailles de gra ins s pour différentes tailles de gra ins [SCOTT 2006] [SCOTT 2006] [SCOTT 2006] [SCOTT 2006]  et  et  et  et \nsans effet TWIP sans effet TWIP sans effet TWIP sans effet TWIP [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1]  et (b) prédic tion de l’écrouissage cinématique en fonc  et (b) prédic tion de l’écrouissage cinématique en fonc  et (b) prédic tion de l’écrouissage cinématique en fonc  et (b) prédic tion de l’écrouissage cinématique en fonction de la déformation tion de la déformation tion de la déformation tion de la déformation \n–– ––    «« ««    back back back back-- --stress stress stress stress    » me surée » me surée » me surée » me surée par des essa is Bauschinger en fonction de  la pré  par des essa is Bauschinger en fonction de la pré  par des essa is Bauschinger en fonction de la pré  par des essa is Bauschinger en fonction de la pré- - --déformation déformation déformation déformation [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] .. ..    \n \nSeul le paramètre ε init  a été ajusté pour prédire individuelle ment les cou rbes de la Figure 26. \nLes 3 autres paramètres du maclage son t pris consta n ts pour tous les aciers. Ce décalage de \ndéformation  pour l’apparition d’un maclage effectif  est reporté sur la Fig ure 27 ainsi que la \ncontrainte d’écouleme nt macroscopique correspondan t e pour chaque acier étudié. Afin  de \nreproduire le comportement expérimental, le décalag e en déformation aug mente avec la taille  52  de grain mais la contrain te d’apparition  semble con stante, de l’ordre de 550 MPa. Ceci est \ncohérent avec la notion de contrainte critique de m aclage.  \n \nToutefois, ce résultat suggère aussi que les macles  apparaissent à de plus faibles niveaux  de \ndéformation  dans les aciers à petits grains. Ce rés ultat est contradictoire avec les observations \nde n ombreux auteurs, y compris D. Barbier, qui mont rent que le maclage est plus susceptible \nd’apparaitre dans les grain s de grande taille, c'es t-à-dire ceux  dont la contrainte est la moins \nélevée. Pour cette question, l’élément clef reste l a  mesure expérimentale de la cin étique de \nmaclage. Ces différentes modèles n’am élioreront leu rs capacités de prédiction que si des \nprogrès son t faits pour la m odélisation à base ph ys ique des cinétiques de maclage. \n \n \nFig ure Fig ure Fig ure Fig ure 27 27 27 27    :: ::    Paramètres Paramètres Paramètres Paramètres ε ε εεinit init init init  aju stés en fonction de la taille de gra ins austéni tiques et contra intes d’écoulement  aju stés en fonction de la taille de gra ins austéni tiques et contra intes d’écoulement  aju stés en fonction de la taille de gra ins austéni tiques et contra intes d’écoulement  aju stés en fonction de la taille de gra ins austéni tiques et contra intes d’écoulement \nrespectives respectives respectives respectives    [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] [BOUAZIZ 2008_1] .. ..    \n \nNous revien drons principaleme n t au ch apitre 2.4.3 p age 85 sur l’utilisation et le calage de ce \nmodèle sur une base de données d’aciers FeMnC. \n \n2.3.5. 2.3.5. 2.3.5. 2.3.5.  Approches composites et contraintes dan s les m aclesApproches composites et contraintes dan s les m aclesApproches composites et contraintes dan s les m aclesApproches composites et contraintes dan s les m acles     \n2.3.5.1.  Analogie en tre les approch es  \n \nSevillano et al.  [SEV ILLANO 2009][SEVILLANO 2012] ont proposé récem m ent un e nouvelle \ninterprétation de l’effet TWIP basée sur une approc he « composite » 7. Cette dern ière relève \nd’un e vision  « mécanicie n ne » mais nous allons mont rer que cette n ouvelle approche et celle \ndéveloppée par notre équipe, découlant d’une vision  « plasticitenne », sont en  fait très proches \net duales. \n \n                                                 \n7 Une  autre a pproche introdu isant un aspect « compos ite » a été propos ée par Kim et a l.  [KIM  2009]. C ette \napproche ne vise pas à considérer les macles co mme des objets d urs dans  une ma trice molle, mais  à cons idérer \ndeux populations de g rains, les grains non mac lés e t les  autres s ur-durcis  par le maclage mécaniqu e (c f. Fi gure \n20).  53  Sevillano et al.  distingue nt deux con tributions à l’écrouissage de la microstructure de \nmaclage : \n• la diminution de la taille de grain effective par l es joints de macles conduisant à un \nsur-durcissement isotrope de la m atrice du composit e, \n• la présence de macles con sidérées comm e les seconde s phases « dures » du composite, \nqui apparaissent dynamique men t au cours de la défor mation. Ces secondes phases \ncontribuent à l’écrouissage en supportant des contr aintes intern es élevées avant \nrelax ation. \nDan s le cadre de cette approche, la contrainte d’éc oulemen t macroscopique du composite \ns’écrit : \n \n ()T MFσ σF1Σ +−=  (26) \n \navec σ M et σ T les contraintes dans la matrice et les macles resp ectiveme nt. Ces deux valeurs \npermettent de définir les contraintes internes de l a manière suivante : \n \n ()\n( )M M intT T int\nσΣ σσΣ σ\n−=−= (27) \n \nLa somme des contraintes internes pondérée par les fractions respectives de phase maclée F et \nde matrice (1-F) est bien entendu nulle. \n \n ()()() 0 σF1 σFM int T int = −+  (28) \n \nCette configuration classique est sché matisée sur l a Figure 28 en reprenant les notations de \nSevillano et al. .  \n  54   \nFigure Figure Figure Figure 28 28 28 28    : Représentation schématique du mécanisme d’écroui ssage par les macles selon une approche : Représentation schématique du mécanisme d’écrouis sage par les macles selon une approche : Représentation schématique du mécanisme d’écrouis sage par les macles selon une approche : Représentation schématique du mécanisme d’écrouis sage par les macles selon une approche \ncomposite (comportement sans  macle, c composite (comportement sans  macle, c composite (comportement sans  macle, c composite (comportement sans  macle, comportement de  la matric e durcie omportement de  la matric e durcie omportement de  la matric e durcie omportement de  la matric e durcie  par les  par les  par les  par les macles macles macles macles    σσ σσMM MM, contraintes dans les , contraintes dans les , contraintes dans les , contraintes dans les \nma cles ma cles ma cles ma cles    σσ σσTT TT, c ontrainte macroscopique Σ). , c ontrainte macroscopique Σ). , c ontrainte macroscopique Σ). , c ontrainte macroscopique Σ).    \n \nPar rapport à l’équation (24), le terme σ M est tout à fait analogue au terme σ 0 + σ F. La seule \napproximation réalisée à ce stade est (1-F) ≈ 1. No us allons maintenant mon trer que le terme \nσb est en fait équivalent au terme Fσ T. \n \nD’après notre modèle, la contrain te de back-stress σb est défin ie comm e : \n \n ( )\n\n\n\n\n\n\n−−\n\n\n\n−+ = =\n1100M\n0 110110\nbbnλεexp1n\nF1F\n2e1\nD1bMμn\nLμbM σ  (29) \n \nComme discuté précédem ment, les macles son t très ra pidem e nt saturées d’empilements de \ndislocations, c'est-à-dire que le terme expon e ntiel  devient vite n égligeable en  quelques \npourcents de déformation . De même, dès la formation  des premières macles, le libre parcours \nmoyen L devie nt in dépendant de D. Ces deux approxim ations successives donnent alors : \n \n ( )0n Fσ n\n2eMµbF\nF1n\n2eMµbF n\nF1F\n2e1bMμn\nLμbM σLB\nT 0110 0 110\n0 110110\nb = ≈\n−=\n\n\n\n−= =  (30) \n \nCe t erme est très similaire à celui proposé par Sev illan o et al.  du point de vue de la forme et \nde la signification. Ce terme est proportionnel à F  et dépend d’un m écanisme de relaxation \ndes empilem e nts au niveau de la macle, c'est-à-dire  faisant interven ir soit le com portem ent \nplastique des macles soit des mécanismes de relaxat ion. Le seul term e additionnel est n 0 qui \ncaractérise la longueur maximum  des e m pilements de dislocations sur les macles et représente \ndonc un facteur de concentration de contraintes dan s les macles elles- m êmes. Ce terme σM Σ σT  \n(σ int )T \n(σ in t )M \nσM sans macle  \nDéform ation  Contrainte \nεM  55  perm et d’ex pliquer pourquoi Sevillano et al.  on t estim é des contrain tes dan s les macles bien \nsupérieures à la borne inférieure σ TLB  [SEVILLANO 2012]. \n \nEn retenan t la valeur des param ètres pour la nuance  de référence, cette borne inférieure \nserait de 800 MPa seule m ent (e = 30 nm , µ = 65 GPa,  b = 2.5x 10 -10  m, M = 3), très rapidem ent \ninférieure à la con trainte d’écoulement moye nn e de l’acier. Cela sign ifie nécessairement que \nles m écanismes de relaxation son t plus difficiles à  activer que ne le laisse penser cette borne \ninférieure et que des mécanismes concentrateurs de contrain tes son t requis pour activer ce \nprocessus. On pourra penser aux très nombreuses dis locations sessiles observées par Idrissi et \nal.  [I DRISSI  2009] pour expliquer cette différence. On  retrouve en  tout cas dans ces deux \nmodèles l’im portan ce de la finesse de macles comme un probable frein pour le mécanisme de \nrelax ation plastique au niveau des macles. \n \n2.3.5.2.   Con traintes intern es et essais Bauschinger  \n \nDan s notre approche développée au chapitre 2.3.4, n ous avons comparé directement les \nvaleurs du back-stress σ b à une composante ciné matique d’écrouissage X mesur ée par essai \nBauschinger (cf. Figure 26(b)). Cette comparaison é tait clairement m otivée par la volonté de \nse placer dans un  cadre mécanique classique de type  Lem aitre et  Chaboche [LEMAITRE \n2004], défin issant l’écrouissage macroscopique comm e la somme de termes isotrope et \ncinématique. Par contre, d’un point de vue microméc anique, cette comparaison est abusive. \n \nIl est souvent délicat de relier certaines grandeur s m icromécaniques, comme la contrainte \ndans les macles, à une valeur de contrainte in terne  unique m esurées par des essais de type \nBauschinger (écrouissage cinématique m acroscopique X) [FEAUGAS 1999]. En effet, ce type \nde changem e nt trajet est un processus complexe, qui  fait intervenir m ê me dans les cas les plus \nsimples plusieurs mécan ismes successivemen t (relaxa tion  des boucles de dislocations \n[ALLAIN 2010_3] , émission de n ouvelles dislocations, …) et surtout  un processus transitoire \nvers un nouvel état stationnaire [ALLAIN 2012]  (mis à part peut-être l’effet Bauschin ger \nperm ane nt).  \n \nLa m esure de limite d’élasticité lors de ce changem ent de trajet, donc la mesure de X, dépend \naussi très fortement du choix fait de la valeur de décalage en déformation plastique (le \n« plastic strain onset »). La valeur déterminée d’e ffet Bausch inger peut varier du simple au \ndouble com me le montre expérimentaleme nt Sevillan o et al.  [SEVILLANO 2012]. La Figure \n29 illustre cette difficulté [ALLAIN 2010_3] . \n \nMesurer rigoureusement une valeur d’écrouissage X a u travers d’un essai Bauschin ger \nprésente donc une vraie difficulté expérimentale et  th éorique. Par contre, cela reste un des  56  seuls essais mécan iques possibles pour investiguer les contraintes internes à l’échelle des \nmicrostructures. \n \n \nFigure Figure Figure Figure 29 29 29 29    : Détermina tion de l’effet Ba uschinger, illustra nt  l’ : Détermina tion de l’effet Ba uschinger, illustra nt l’ : Détermina tion de l’effet Ba uschinger, illustra nt l’ : Détermina tion de l’effet Ba uschinger, illustra nt l’ambigüité ambigüité ambigüité ambigüité de cette mesure sur  la base d’un critère  de cette mesure sur  la base d’un critère  de cette mesure sur  la base d’un critère  de cette mesure sur  la base d’un critère \nde « de « de « de «     plastic strain onset plastic strain onset plastic strain onset plastic strain onset    » » » » [ALLAIN 2010 [ALLAIN 2010 [ALLAIN 2010 [ALLAIN 2010_3 _3 _3 _3]] ]].. ..    \n \nDan s leur première publication, Sevillano et al.  relient cette composante X à la valeur de \n(σ int )M, hypothèse rigoureusemen t dérivée d’un  modèle comp osite de type Masing. Ce ch oix \nsignifie que lors du changement de trajet on choisi t de défin ir la nouvelle limite d’élasticité \ndu composite comme étant la limite d’élasticité de la ph ase la plus « molle » (celle en \ncompression lors du chargement initiale). Cette hyp oth èse nécessite alors de définir X pour de \nfaibles valeurs du décalage de déform ation plastiqu e. Dan s leurs travaux postérieurs, ils \nreviennent sur cette hypothèse stricte en spécifian t que (σ in t )M représente en fait une large \nfraction de X. \n \nNous avons déjà été confrontés à cette question au cours d’une étude sur des aciers ferrite-\nperlite que l’on pourrait qualifier de m icrostructu re composite « modèle ». Afin de pouvoir \nutiliser les valeurs conventionn ellement mesurées p ar des essais Bauschinger (avec un \n« onset » conventionnel de 0,2%), nous avons propos é une relation reliant cette composante \ncinématique X mesurée à des valeurs microm écaniques  dans chacune des phases 1 et 2 \n[ALLAIN 2008_1]  : \n \n 2 1 2 1 2 2 1 1 σ σF F XF XFX − ++=  (31) \n \nAvec F i, σ i et X i, les fractions, con traintes et composantes cinémat iques propres de la phase i (i \nvalant 1 ou 2). Cette relation est phén oménologique  mais permet de reproduire toutes les  57  tendances connues d’évolution de la com posante ciné matique d’écrouissage des com posites et \na été validée expérimentaleme nt dans le cadre des a ciers ferrito-perlitiques (pour des valeurs \n« d’onset » conventionnelles).  \n \nDan s le cas de cette étude, cette expression de X p eut se réduire au terme non linéaire \nd’interactions entre les phases, c'est-à-dire entre  la matrice et les macles. En  effet, la \ncomposante ciném atique de durcisse men t dans les mac les est probable ment nulle (X 2 = 0) car \nelles restent prin cipale m ent élastiques au cours de  la déform ation. La composante \ncinématique de la matrice austénitique dépend dans le cas général de l’effet taille de grain \n[FEAUGAS 1999].  Dans le cas de valeurs « d’on set » conventionnelles et de grandes tailles de \ngrain  austénitique, ce term e X 1 peut être négligé par rapport au terme composite. \n \nIl vient alors : \n \n ()()M Tσ σF1 F X −−=  (32) \navec \n \n ()T MFσ σF1Σ +−=  (33) \n \nOn peut donc en déduire les composantes de contrain tes dans chacune des phases. \n \n F1XΣ σetFXΣ σM T−−= +=  (34) \n \nOn retrouve dans cette expression selon les n otatio ns de Sevillano et al.  que la contrainte \ninterne (σ int )M est bie n une valeur négative, proche de X dans la m esure où (1-F) reste limitée : \n \n ( ) 0F1XΣ σ σM M int <−−=−=  (35) \n \nPar contre, cette valeur (σ int )M est n écessairement supérieure en valeur absolue à la \ncomposante d’écrouissage cinématique X mesurée expé rimentalemen t. On n otera aussi que la \ncomposante cinématique s’écrit alors en fonction de  la contrainte dans les m acles : \n \n () FΣ FσΣ σFXT T +=−=  (36) \n \nSi σ T >> Σ, on retrouve alors l’approxim ation que nous a vons utilisée précédemment, c'est-à-\ndire que le back-stress peut être comparé à X ≈ σ b = Fσ T .  \n  58  Un cas d’application concret de ces relations s’est  présenté dans les travaux de th èse de K. \nRen ard [RENARD 2012] [BOUAZIZ 2013]8. Au cours de ce travail, des essais Bauschinger on t \nété réalisés par cisaillement réversible et les fra ction s de m acles correspondantes mesurées \npour deux niveaux  de prédéform ation. Le Tableau 1 r épertorie pour ces deux niveaux de pré-\ndéformation  les valeurs de contraintes, de fraction  maclées et d’effet Bauschinger, ainsi que \nles contrain tes locales déduites des équations préc éden tes. \n \n                                                 \n8 M a lheureusement, c e type d’analyse n’a pu être mené  après les travaux de D. Barbier pour deux raisons \nprinc ipales :  \n• les fractions maclées n’ont pa s été me surées lors d es essais Bau schinger  et il semble que  les taux de \nmaclage soient fonction du mode de chargement (prin cipalement  pour des  raisons de tex ture). \n• la ta ille de grain initiale est faible (3 µm) c e qu i pourrait c ontribuer significativement à l’e ffet \nBauschinger en début de déformation. Le terme X 1 serait alors non négligeable. Cette contribution \nex pliquerait les mesures de X d e 310 MPa pour des c ontraintes d’écoulement de 720 M Pa a près 10% de \ndéfor mation a lors que seulement 30% des grains sont  faiblement maclés. \nDans ce cas, il conviendra de c onserver dans l’équa tion précédente la contribution au durc issement cin ématique \nde la  matrice X M, sacha nt que c e terme va saturer très rapidement a près quelques pourcents  de défor mation : \n \n()()()M M T XF1 σ σF1 F X −+−−=   59   \nParamètres Paramètres Paramètres Paramètres    Signification Signification Signification Signification    Valeurs Valeurs Valeurs Valeurs    \nE Déform ation \nm acroscopique \néquivale nte 0 0.1 0.2 \nF Fraction de macle \nestim ée 0 0.09 0.11 \nΣ Contrainte \nm acroscopique \néquivale nte 332 600 800 \nX0.2%  Effet Bauschinger \n(onset de 0.2%) 0 170 275 \nX0.5%  Effet Bauschinger \n(onset 0.5%) 0 115 200 \nF1XΣ σ0.2% 0.2%\nM−−=  Contrain te dans la \nm atrice (onset de \n0.2%) 332 413 489 \nF1XΣ σ0.5% 0.5%\nM−−=  Contrain te dans la \nmatrice (onset \nde0.5%) 332 474 574 \nεM Déform ation \nplastique dans la \nmatrice 0 0.06 0.16 \nFXΣ σ0.2% 0.2%\nT+=  Contrainte dans les \nm acles (onset de \n0.2%) - 2489 1878 \nFXΣ σ0.5% 0.5%\nT+=  Contrainte dans les \nmacles (on set 0.5%) - 3191 2539 \nTableau Tableau Tableau Tableau 1 1 11    : C ontraintes dans l es macles et la ma trice après déformation de la nuance de réf : C ontraintes dans l es macles et la ma trice après d éformation de la nuance de réf : C ontraintes dans l es macles et la ma trice après d éformation de la nuance de réf : C ontraintes dans l es macles et la ma trice après d éformation de la nuance de référence, déduites des érence, déduites des érence, déduites des érence, déduites des \nmesures d’effet Bausc hinger (deux mesures d’effet Bausc hinger (deux mesures d’effet Bausc hinger (deux mesures d’effet Bausc hinger (deux niveaux  de plasti ques  strain on niveaux  de plastiques  strain on niveaux  de plastiques  strain on niveaux  de plastiques  strain onset set set set –– –– 0.2 et 0.5%)  0.2 et 0.5%)  0.2 et 0.5%)  0.2 et 0.5%)    –– –– données initia les issues  de  données initia les issues  de  données initia les issues  de  données initia les issues  de \n[RENARD 12]. [RENARD 12]. [RENARD 12]. [RENARD 12].    \n \nLe Tableau 1 montre que les contraintes dans les ma cles prenn e nt des valeurs comprises en tre \n2 et 3 GPa, en coh érence avec les valeurs proposées  par Sevillano et donc avec les épaisseurs \nde m acles rapportées dans la littérature. \n \nLa Figure 30 montre la comparaison de ces valeurs d e contraintes dans la matrice avec les \ncourbes de com portem e nt d’un acier austé nitique bin aire Fe30Mn [ BOUAZIZ \n2011_2][HUANG 2011]  et Fe36Mn0.6C stables sans effet TWIP. A la limite  d’élasticité près \n(effet de la taille de grains), le comportement de l’austénite sans maclage déduite de cette \nanalyse est comparable aux autres aciers de réf éren ce. Ce nouveau résultat (Fe22Mn0.6C(II))  60  est aussi comparé avec la courbe de comportem ent de  l’acier de réf érence Fe22Mn0.6C sans \nmaclage déduite par une autre méthode indirecte (dé taillé au chapitre 2.3.6.3 page 65) \n(Fe22Mn0.6(I)). \n \nDe façon plus surprenante, il apparaît sur la Figur e 30 que le carbon e n’influe pas ou peu sur \nl’écrouissage de ces structures austénitiques. Ce r ésultat confirme notre choix   dans la \nmodélisation micromécan ique des aciers TWIP de ne p as faire varier le paramètre de \nrestauration  dynam ique avec la teneur en carbone [BOUAZIZ 2011_2] . \n \n \nFigure Figure Figure Figure 30 30 30 30    : : : : Courbes de compor Courbes de compor Courbes de compor Courbes de comportement en trac ti tement en trac ti tement en trac ti tement en trac tion d’ac iers Fe30 M n et Fe36Mn0.6C on d’ac iers Fe30 M n et Fe36Mn0.6C on d’ac iers Fe30 M n et Fe36Mn0.6C on d’ac iers Fe30 M n et Fe36Mn0.6C stables sans effet  TWIP . stables sans effet TWIP . stables sans effet TWIP . stables sans effet TWIP . \nComparaison avec le c omporte ment estimé d’un  acier Fe22M n0.6C sans maclage par des changements de Comparaison avec le c omporte ment estimé d’un  acier Fe22M n0.6C sans maclage par des changements de Comparaison avec le c omporte ment estimé d’un  acier Fe22M n0.6C sans maclage par des changements de Comparaison avec le c omporte ment estimé d’un  acier Fe22M n0.6C sans maclage par des changements de tra j ets tra jets tra jets tra jets    \nthermoméc anique (voir ci thermoméc anique (voir ci thermoméc anique (voir ci thermoméc anique (voir ci- - --dessous) dessous) dessous) dessous) (I) (I) (I) (I) ou l’ana lyse des essais ou l’ana lyse des essais ou l’ana lyse des essais ou l’ana lyse des essais Ba uschinger Ba uschinger Ba uschinger Ba uschinger    de de de de K. Renard  K. Renard  K. Renard  K. Renard (II) (II) (II) (II) en fonction de la  en fonction de la  en fonction de la  en fonction de la \ndéformation vraie  dans la matrice déformation vraie  dans la matrice déformation vraie  dans la matrice déformation vraie  dans la matrice. . ..    \n2.3.6. 2.3.6. 2.3.6. 2.3.6.  Influence du mode de chargemen t Influence du mode de chargemen t Influence du mode de chargemen t Influence du mode de chargemen t    \n2.3.6.1.  Influence du mode de chargemen t (trajets monotones)  \n \nUne des difficultés majeures dans l’analyse des con train tes internes au cours d’essais \nBauschinger dans ces alliages, est que la microstru cture de maclage (fraction et \nmorphogén èse) sem ble très sensible aux conditions d e chargem ent, y compris le lon g de trajet \ndirect. Par conséquent, on peut s’attendre à des di fférences notables de comportement \nmécanique équivalent entre traction uni-axiale et c isailleme nt dans ces alliages. Il convien dra \npar exemple de pratiquer avec prudence des comparai sons entre des données d’essais \nBauschinger en cisaillement réversible et des donné es en traction. 9 \n                                                 \n9 Cette précaution n’a  par exemple pas été prise  en compte dans le cadre de notre modèle  [BOUAZIZ 2008_1]  \nprése nté ci-dessus. Toutefois, l es cinétiques de ma clage ont été adaptées pour la modélisation les dif férents essais \nde traction et la  mesure des contraintes internes X  en c isaillement uniquement a pprochée en termes de forme \npar le modèle .  61   \nCon trairem ent aux  aciers TRIP montrant une transfor mation  γ \u0001α’ [JACQUES 2007], cette \ndifférence de comportement ne réside pas dans une c inétique de tran sfor mation (de maclage) \ncontrôlée par la valeur de triaxialité du ch argemen t car le processus de m aclage ne \ns’accompagne pas de variation de volume mesurable. Par contre, au cours de sa thèse, D. \nBarbier a m ontré que les microstructures de m aclage  en traction ou en cisaille m ent sim ple \nn’évoluaien t pas de manière sim ilaire [BARBI ER 2009_2] . Au cours d’un essai de traction, \ndeux  systèm es de maclages apparaissent séquentielle m ent au cours de la déformation, alors \nqu’un seul systèm e est généralement activé par grai n en  cisaillement. Cette différence \ns’explique notamm ent par des différences de texture  de déform ation, qui contrôle les \nsystèmes de glisse m ent et de maclage activés. De ma n ière quantitative, les fraction s de ph ase \nmaclées sem ble nt être équivale n tes après 30 % de dé formation équivale nte, voire supérieure \nen traction. Ce résultat est contradictoire avec le s résultats récents de K. Renard [RENARD \n2012] dans un alliage Fe20Mn1.2C qui au contraire m ontren t une cinétique de maclage très \nélevée dans le cas du cisaillemen t et un e saturatio n précoce au cours de la déformation. Ils \nconfir ment la prévalence d’un seul système de macla ge par grain. On pourra peut-être \nexpliquer cette différence par des tailles de grain s austénitiques sensiblement différentes en tre \nles deux études. \n \nCes microstructures de maclage expliquent probable m ent les différen ces observées de \ncomportem ent équivalent en cisaille men t et en tract ion dans le cas de l’utilisation de surfaces \nde charge simple (Von Mises ou Hill 48, figures tir ées de [ALLAIN  2004_1]  et [ BARBI ER \n2009_1]  reprises sur la Figure 31). On notera que dans ces  deux cas (petits et gros grains de la \nnuance de référen ce) les courbes de comportement di vergen t seuleme nt après 20%-30% de \ndéformation . Cette observation viendrait plutôt con firmer les conclusions de D. Barbier sur la \nsensibilité de la microstructure de maclage au m ode  de sollicitation, discutées ci-dessus. \n \n(a) \n (b) \nFigure Figure Figure Figure 31 31 31 31    : C omparaison entr e le comportement équiva lent en traction et en cisaillement (a) dans le cas d’une : C omparaison entr e le comportement équiva lent en t raction et en cisaillement (a) dans le cas d’une : C omparaison entr e le comportement équiva lent en t raction et en cisaillement (a) dans le cas d’une : C omparaison entr e le comportement équiva lent en t raction et en cisaillement (a) dans le cas d’une \nsurface de charge de type Mises (nua nce de ré férenc e à gros grains) surface de charge de type Mises (nua nce de ré férenc e à gros grains) surface de charge de type Mises (nua nce de ré férenc e à gros grains) surface de charge de type Mises (nua nce de ré férenc e à gros grains)    [AL LAI [AL LAI [AL LAI [AL LAIN 2004_1] N 2004_1] N 2004_1] N 2004_1]    (b) dans le  cas d’une (b) dans le  cas d’une (b) dans le  cas d’une (b) dans le  cas d’une \nsurface de charge de type H ill48 (nuance de r éféren ce à petits grains) surface de charge de type H ill48 (nuance de r éféren ce à petits grains) surface de charge de type H ill48 (nuance de r éféren ce à petits grains) surface de charge de type H ill48 (nuance de r éféren ce à petits grains)    [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1]. . ..    \n \nSelon le trajet de chargem ent direct, les microstru ctures de m aclage produites sont différen tes \navec un fort impact sur les taux d’écrouissage. Les  modèles de plasticité polycristalline les plus  62  récents de l’effet  TWIP basés sur des considération s de texture de déform ation sont \nsusceptibles de reproduire ce processus (avec des c inétiques de maclage ajustées). Toutefois, \naucun des modèles de la littérature ainsi ajusté n’ est capable de prédire le comportement lors \nde trajets alternés type Bauschin ger, comme le mont re la synthèse récente de Favier et al.  \n[FAVIER 2012]. La forme de la surface de charge des  aciers TWIP est donc aussi curieuse et \ninattendue comme le mon tre l’étude récente de [CHUN G 2011], reproduite sur la Figure 32. \n \n \nFigure  Figure  Figure  Figure  32 32 32 32    : Sur face de c harge (écrouissage purement isotr : Sur face de c harge (écrouissage purement isotr : Sur face de c harge (écrouissage purement isotr : Sur face de c harge (écrouissage purement isotropes ) d’un  acier TWIP [ opes) d’un  acier TWIP [ opes) d’un  acier TWIP [ opes) d’un  acier TWIP [CHUNG 2011 CHUNG 2011 CHUNG 2011 CHUNG 2011] ] ]].. ..    \n \nLa formulation em pir ique retenue pour l’écrouissage  est purement isotrope et nécessiterait \nl’introduction d’un e composante cinématique. Ce rés ultat con fir me aussi de manière indirecte \nle rôle prépondérant du maclage mécanique dans l’in terprétation du com portem e nt de ces \naciers par rapport à un mécanism e supposé de vieill issement dynamique. \n \n2.3.6.2.  Chan gemen t de trajets de chargem ent m écanique  \n \nL’apparition  de la m icrostructure de maclage condui t à une évolution profon de de contrain tes \ninternes révélées macroscopiquement lors de changem e nts de trajets dits altern és (essais \nBauschinger). La morphogénèse de cette structure es t orientée aussi suivant une loi de \nSchm id. Cela suggère que les aciers TWIP présenten t  une sensibilité très particulière aux \nautres chan geme nts de trajets m écaniques (par exe mp le, essai de traction d’une éprouvette \npré-déform ée en cisaillem e nt). Différen ts cas de fi gures ont donc été étudiés dans le cadre de \nla th èse de D. Barbier. \n \nLa Figure 33 montre les courbes de con traintes et d éformations équivalentes obtenues selon \nles différen ts scenarios suivants : \n• Traction monotone  63  • Cisailleme n t monotone \n• traction plane DT puis cisaillement DL (θ = 0 – cha ngement de trajet dur) \n• traction large DT (4% ou 8% de déformation équivale n te) puis traction DL (θ = -0.5) \n• expansion equi-biaxiée (9% et 17 % de déformation é quivalen te) puis traction DL (θ = \n+0.5) \n \nL’an gle (θ) correspond à l’indicateur de la dureté d’un ch angement de trajet au sens de \nSchm itt : \n \n \n2 2 1 12 1\nε : ε ε : εε : εθ=  (37) \n \navec ε 1 et ε 2 définissant classiqueme n t les tenseurs de déformat ion du trajets de pré-\ndéformation  (1) et de caractérisation (2) respectiv eme n t [SCHMITT 1994]. \n \n(a) \n (b) \nFigure Figure Figure Figure 33 33 33 33    : (a) Comportement mécanique de la nuance de : (a) Comportement mécanique de la nuance de : (a) Comportement mécanique de la nuance de : (a) Comportement mécanique de la nuance de référe nce après les différents types de pré  référence après les différents types de pré  référence après les différents types de pré  référence après les différents types de pré- - --déforma tion déforma tion déforma tion déforma tion \n–– –– représentation en contrainte et défo rmation équiv ale nte déca lée du niveau de  représentation en contrainte et défo rmation équiva le nte déca lée du niveau de  représentation en contrainte et défo rmation équiva le nte déca lée du niveau de  représentation en contrainte et défo rmation équiva le nte déca lée du niveau de pré pré pré pré-- --déformation déformation déformation déformation    [BARBIER [BARBIER [BARBIER [BARBIER \n2009_1] 2009_1] 2009_1] 2009_1]  (b) Essais  (b) Essais  (b) Essais  (b) Essais Ba uschinger Ba uschinger Ba uschinger Ba uschinger en cisaillement  en cisaillement  en cisaillement  en cisaillement r éversible r éversible r éversible r éversible de la nuance de référenc e de la nuance de référenc e de la nuance de référenc e de la nuance de référenc e représentés représentés représentés représentés    en déformation en déformation en déformation en déformation \ncumulée cumulée cumulée cumulée    [B [B [B [BOU OU OU OU AA AAZIZ 2008_1] ZIZ 2008_1] ZIZ 2008_1] ZIZ 2008_1] .. ..    \n \nLes courbes obten ues après pré-déformation sont rep résen tées en tenant compte de ce \ndécalage en déformation équivalente générée lors de  la sollicitation initiale. Les \ncomportem ents équivalen ts ont été déduits d’un  crit ère de Hill 48, e n  l’absen ce de critère plus \nadapté. Ce choix n ’affecte toutefois pas les conclu sions ci-dessous. Dans cette an alyse, ont \naussi été in tégrés bien entendu les essais de cisai lleme nt altern és discutés ci-dessus et \ncorrespondant à un angle θ = -1 caractéristique des  change m ents de trajet Bauschinger. Ces \nessais ont été repris sur la Figure 33(b) permettan t de mettre en valeur n on seulement les \ncontraintes internes (écrouissage cinématique) m ais  aussi le fort effet Bausch inger permanent. \n \nLa Figure 33 met en évidence que selon la nature du  ch angem ent de trajets,  \n• les n iveaux de con traintes à la recharge peuven t êt re très supérieurs ou inférieurs à la \ncourbe de comportement de référence pour un niveau de pré-déform ation donnée.  64  • par contre, après une faible déformation , les courb es de com portem e nt retrouven t le \ntaux  d’écrouissage de la nuance de référence. Les e ssais Bauschinger en cisaillement de \nla Figure 33(b) présentent en conséque nce un fort a doucisse men t perm anent. Ce \nphénomè ne est aussi claireme nt visible lors des ess ais d’expansion-traction. \n \nLes rapports R θ en tre con trainte équivale nte à la recharge et cont rainte de référen ce ont été \ndéfin is pour chaque essai selon la procédure représ entée sur la Figure 34(a). Ces rapports sont \nensuite reportés sur la Figure 34(b) en fonction  de  l’an gle θ des différents essais. Ces valeurs \nrelatives on t été comparées à celles d’un acier dou x [SCHMITT 1994] et au com portem ent \natten du d’un acier Dual-Phase [DILL IEN  2010_2] .  \nL’acier doux « classique » présente un faible effet  Bauschinger (uniqueme nt liés à la taille de \ngrain ) mais un durcissement important lors des chan geme nts de trajets durs (cf. Figure 35(a)). \nCe phénom è ne est lié à la structure orientée et pol arisée de cellules de dislocation s générées \nlors de la pré-défo rmation . Les cellules ainsi créé es in duisent un sur-durcissement important \nlors d’un chargement orth ogonal ultérieur. Ce sur-d urcissem ent n’est que transitoire car les \ncellules de dislocations subisse nt rapide m ent un  re modelage. \nLes aciers DP a contrario présen tent un  fort effet Bauschinger lié à la présence d’une ph ase \ndure (martensite) dans un e matrice molle (ferrite) com me nous le verrons au chapitre 3. Les \nfortes contraintes internes conduise nt à un retard dans l’apparition des structures de cellules \npolarisées [GARDEY 2005]. En conséquence, le durcis se ment lors d’un changeme n t de trajet \ndur est ine x istant dans les aciers DP (cf. Figure 3 5(b)) \n \nLe cas des aciers TWIP est donc particulièrement in téressant. Les essais de cisaillem ent \nréversible m ontren t bien entendu un fort effet Baus ch inger, par con tre, les mesures avec des \nangles θ proches de 0 m ontren t que ces aciers prése ntent certain ement un durcissem ent \nimportant lors des changem e nts de trajet durs10 . Ce sur-durcisseme nt est probablem ent dû à la \nstructure de maclage qui se trouve être orientée pa r rapport à la sollicitation initiale et donc \nnon propice pour le second chargement 11 . Con trairem ent aux  aciers DP, contraintes internes  \net genèse d’une structure fortement polarisée en  re lation avec la cristallographie ne sont donc \npas incompatibles dans les aciers TWIP. \nLe deuxièm e résultat fondamental de cette étude est  que ces effets dus au pré-ch argem ent \nsont permane nts dans le cas des aciers TWIP (adouci sseme nt permanent) contrairement aux \naciers DP ou IF. Cette permane n ce s’explique simple m e nt par l’impossibilité de remodeler la \nmicrostructure de maclage au cours du second charge ment, contrairement aux structures de \ncellules de dislocations. \n                                                 \n10  Un essai de changement de trajet dur a aussi été r éalisé dans le cadre de la thèse de D. Barbier (tra ction plane \nsuivi e d’un cisailleme nt). Toutefois, les conclusio ns de cet essai sont assez incoh érentes. Ce résulta t unique et  \nisolé n’a donc  pas été présenté dans ce mémoire. \n11  Ce comportement spécifique  ne peut être expliqué u niquement par les chang ements de facteur de Ta ylor \ncomme l’a démontré D. Barbier dans sa  thèse, mais b ien à la microstructure de mac lage.  65  Ce résultat est tout à fait origin al par rapport à la littérature et fera bientôt l’objet d’une \npublication  spécifique. \n \n(a) \n (b) \nFigure Figure Figure Figure 34 34 34 34    : (a ) : (a ) : (a ) : (a ) M éthode de me sure de la contrainte équivalent e a près pré M éthode de me sure de la contrainte équivalente a prè s pré M éthode de me sure de la contrainte équivalente a prè s pré M éthode de me sure de la contrainte équivalente a prè s pré- - --déformation pour  le calcul du rapport déformation pour  le calcul du rapport déformation pour  le calcul du rapport déformation pour  le calcul du rapport \nde contrainte R de contrainte R de contrainte R de contrainte R θθ θθ pour chaque  essai (b) Représentation du rapport R  pour chaque  essai (b) Représentation du rapport R  pour chaque  essai (b) Représentation du rapport R  pour chaque  essai (b) Représentation du rapport R θθ θθ en fonction de l’angl e  en fonction de l’angl e  en fonction de l’angl e  en fonction de l’angl e θ θ θθ carac térisant la  carac térisant la  carac térisant la  carac térisant la \ndureté du cha ngement de trajet pour l’a cier de r éfé rence et les comportement dureté du cha ngement de trajet pour l’a cier de r éfé rence et les comportement dureté du cha ngement de trajet pour l’a cier de r éfé rence et les comportement dureté du cha ngement de trajet pour l’a cier de r éfé rence et les comportements s ss attendus typiques d’aciers dou x et  attendus typiques d’aciers dou x et  attendus typiques d’aciers dou x et  attendus typiques d’aciers dou x et \nDual Dual Dual Dual- - --Phase Phase Phase Phase    [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1] [BARBIER 2009_1]. . ..    \n \n \nFigure Figure Figure Figure 35 35 35 35    : : : : C omporte ment C omporte ment C omporte ment C omporte ments s ss typique  typique  typique  typiques s ss attendues  attendues  attendues  attendues lors de c hangeme nts de trajets de déform ations (a) d’acier do lors de c hangeme nts de trajets de déformations (a) d’acier do lors de c hangeme nts de trajets de déformations (a) d’acier do lors de c hangeme nts de trajets de déformations (a) d’acier doux ux ux ux \n(IF) (b) d’ac ier DP (IF) (b) d’ac ier DP (IF) (b) d’ac ier DP (IF) (b) d’ac ier DP [DILLIEN [DILLIEN [DILLIEN [DILLIEN 2010_2 2010_2 2010_2 2010_2] ] ]].. ..    \n2.3.6.3.  Chan gemen t de trajets thermomécaniques  \n \nDès mes travaux de thèse, nous avons envisagé un au tre type de changement de trajets, que \nnous avons qualifié de thermom écanique, con sistant à chan ger la température entre pré-\ndéformation  et essai de traction proprem ent dit. La  pré-déformation  permet d’introduire une \nmicrostructure initiale différente d’un trajet dire ct. Des essais de pré-déform ation ont été par \nexem ple réalisés à 400°C puis les éprouvettes ont é té déformées à température ambiante afin \nd’évaluer l’impact d’une densité importantes de dis location s parfaites (seul mécanisme de \ndéformation  actif à 400°C) sur le maclage mécan ique  (à température ambian te). Ces différents \nessais sont représen tés sur la Figure 36.  \n  66  \n \nFig ure Fig ure Fig ure Fig ure 36 36 36 36    : Essais de traction à température : Essais de traction à température : Essais de traction à température : Essais de traction à température ambiante ambiante ambiante ambiante a près différents ta ux de pré  a près différents ta ux de pré  a près différents ta ux de pré  a près différents ta ux de pré- - --déformat déformat déformat déformation en traction à ion en traction à ion en traction à ion en traction à \n400°C . Les courbes de 400°C . Les courbes de 400°C . Les courbes de 400°C . Les courbes de comportement attendues comportement attendues comportement attendues comportement attendues à ces deu x températures sont  aussi représentées  à ces deu x températures sont aussi représentées  à ces deu x températures sont aussi représentées  à ces deu x températures sont aussi représentées (e n bleu et ro uge  (en bleu et ro uge  (en bleu et ro uge  (en bleu et ro uge \nrespective ment) respective ment) respective ment) respective ment)    [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] .. ..    \n \nCette expérience originale a permis de mettre en lu mière deux résultats fondam entaux  au \nsujet de l’effet TWIP : \n• Les lieux des limites d’élasticité à la recharge à température ambian te correspondent \naux contraintes d’écoulem e nt qu’aurait l’austénite après déformation mais en  l’absence \nde m acles m écaniques (cette contrainte tenant compt e aussi de la con trainte effective). \nLa courbe d’évolution de ces con traintes en fon ctio n des pré-déform ations représente \ndonc la courbe de comportemen t de l’alliage sans ma clage. Cette an alogie repose sur \nl’hypothèse que les densités de dislocations généré es à 400°C pour une déformation \ndonn ée son t identiques à celles pouvan t être généré es à tem pérature ambiante sans \nmaclage. La comparaison entre ce comportement estim é sans maclage et la courbe de \ncomportem ent de l’alliage de référen ce est représen tée sur la Figure 37(a). Ces \nrésultats présenten t une parfaite concordance avec les courbes de comportement de \nl’alliage bin aire Fe30Mn ou le ternaire Fe35Mn 0.6C stables san s effet TWIP présentées \nsur la  Figure 30 au chapitre 2.3.5.2. Ce résultat vient naturelle ment confirm er le rôle \nimportant du maclage mécanique sur l’écrouissage de  ces aciers. \n• Le second résultat majeur est révélé en comparant l es taux d’écrouissage de la nuance \nselon  un trajet direct ou après déform ation pour un e contrainte donnée. Pour les \nfaibles déformations (in férieures à 15%), les courb es de com portem ent sont \nconfondues. Par contre, au-delà, comme le montre le  Figure 37(b), les courbes \ndivergent. L’écrouissage des aciers TWIP est donc c on trôlé par deux  variables d’état 12 , \ntrès probableme nt la densité de dislocations et la microstructure de maclage, comme \nnous l’avon s implicitemen t supposé dans notre modèl e ciné matique. \n \nLes résultats d’un essai m écanique sur ces aciers v on t donc être très sen sibles à l’histoire \nthermomécanique de l’échantillon, con trairem ent aux  aciers doux  par exemple [RAUCH \n1997] . \n                                                 \n12  Ces conclus ions sont sensibl ement différentes de c elle présentées dans une de nos publications [BOUAZIZ \n2010] .  67  \n(a) \n (b) \nFigure Figure Figure Figure 37 37 37 37    : (a ) Lieu : (a ) Lieu : (a ) Lieu : (a ) Lieux x xx des limite s d’élasticité à températur e a mbiante  des limite s d’élasticité à températur e a mbiante  des limite s d’élasticité à températur e a mbiante  des limite s d’élasticité à températur e a mbiante en  fonction des nivea ux de pré  en fonction des nivea ux de pré  en fonction des nivea ux de pré  en fonction des nivea ux de pré- - --déformation déformation déformation déformation \nà 400°C permettant d’estimer le comporteme nt de la n uance de référence sans e ffet TWIP. Les courbes de  à 400°C permettant d’estimer le comporteme nt de la n uance de référence sans e ffet TWIP. Les courbes de  à 400°C permettant d’estimer le comporteme nt de la n uance de référence sans e ffet TWIP. Les courbes de  à 400°C permettant d’estimer le comporteme nt de la n uance de référence sans e ffet TWIP. Les courbes de  \ncomportement a ttendues à 25°C et à 400°C  sont aussi  représentées comportement a ttendues à 25°C et à 400°C  sont aussi  représentées comportement a ttendues à 25°C et à 400°C  sont aussi  représentées comportement a ttendues à 25°C et à 400°C  sont aussi  représentées    [BOUAZIZ 2010] [BOUAZIZ 2010] [BOUAZIZ 2010] [BOUAZIZ 2010] . (b) Courbes de . (b) Courbes de . (b) Courbes de . (b) Courbes de \ncomportement comportement comportement comportement après pré après pré après pré après pré- - --dé formation à 400°C  recalées en déformation pour f aire correspondre le ur limite  dé formation à 400°C  recalées en déformation pour fa ire correspondre le ur limite  dé formation à 400°C  recalées en déformation pour fa ire correspondre le ur limite  dé formation à 400°C  recalées en déformation pour fa ire correspondre le ur limite  \nd’élasticité avec la  courbe de comportement à tempé rature ambiante d’élasticité avec la  courbe de comportement à tempé rature ambiante d’élasticité avec la  courbe de comportement à tempé rature ambiante d’élasticité avec la  courbe de comportement à tempé rature ambiante    [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1] [ALLAIN 2004_1]. . ..    \n \n2.3.7. 2.3.7. 2.3.7. 2.3.7.  Con clusion  interm édiaire Con clusion  interm édiaire Con clusion  interm édiaire Con clusion  interm édiaire    \n \nDepuis les travaux  précurseurs de L. Rémy, notre pr incipale contribution scientifique a été \nfinalement de découpler et quan tifier les contribut ion s d’origine cin ématique et isotrope de \nl’effet TWIP. La première contribution s’apparente à un durcisseme n t dû aux joints de grains \n(empile men ts de dislocations). Ce processus est dyn amique car la densité de ces joints \naugm e nte au cours de la déform ation. Ce term e peut avoir une interprétation duale si l’on \nconsidère les macles et la matrice comm e un composi te à inclusions dures. Le second terme \nisotrope est lié à la morph ogénèse de la m icrostruc ture de maclage (texture).  \n \nCes contributions relatives sont révélées en partic ulier lors des ch angements de trajets de \nsollicitation s, com me nous l’avons montré expérim e n talement. Ces résultats origin aux \ns’expliquent par la morphogénèse de m icrostructure de maclage et sa persistan ce lors de \nchan geme n ts de trajets. Ils ont des conséquences im portantes pour l’utilisation de ces aciers \n(sollicitations mécaniques de pièces après mise en form e par exemple). \n \nLes meilleurs modèles micromécaniques polycristalli ns de l’effet TWIP à l’h eure actuelle sont \nsusceptibles de décrire assez correctement des effe ts de textures mais s’avèrent encore \ninsuffisants pour décrire correctement les changeme n ts de trajets de ces aciers et reposent \ntoujours sur des cin étiques de maclage empiriques. \n \n2.4. 2.4. 2.4. 2.4.  Effet de l a composition chimique Effet de l a composition chimique Effet de l a composition chimique Effet de l a composition chimique    : : : : ll lle rôle e rôle e rôle e rôle particulier particulier particulier particulier du du du du carbone  carbone  carbone  carbone    \n \nCom prendre et mettre en  éviden ce les mécanismes d’i nflue n ce de la composition chimique, \net en  particulier du carbon e en solution solide, su r le comportement m écanique de ces aciers a  68  toujours été un objectif important pour moi. Cet ax e de travail permet de fournir aux \n« métallurgistes » des outils et des règles métier pour optimiser les performances m écaniques \nde ces alliages [SCOTT 2006] . \n \nJe m e suis intéressé dans un premier temps à l’infl ue nce de la composition ch imique de \nl’alliage sur son EDE, et j’ai élaboré dan s le cadr e de m a thèse un m odèle thermodynamique \ncomplet pour la prédiction de cette énergie intégra nt pour la première fois les effets  du \ncarbone dans le système FeMnC. Ce modèle a ensuite été éten du dans le cadre de la thèse d’A. \nDum ay à d’autres éléments d’alliages comm e l’alumin ium ou le cuivre. Ce paramètre \nthermochimique  permet de prédire les mécanismes de  déformation activables dans ces \nalliages mais il est insuffisant pour com prendre et  prédire l’écrouissage de ces alliages. Nous \nmon trerons en particulier le rôle paradoxal du carb one. \n \nDan s un second temps, m on intérêt s’est donc porté sur la compréhension du rôle spécifique \ndu carbone sur le comportemen t mécanique des aciers  TWIP tern aires. En effet, il ne se \nréduit pas à augm enter l’EDE, mais contribue aussi à un  mécanisme de vieillissem ent \ndynamique, modifie la m obilité des dislocations et leur structuration indépendamment des \nmacles. On reviendra aussi sur son rôle présumé dan s le processus de maclage. \n \nCes différen ts travaux ont principaleme n t été m enés  en  collaboration  avec l’IJL de N ancy et le \nLEM3 de Metz. \n \n2.4.1. 2.4.1. 2.4.1. 2.4.1.  Com position Com position Com position Com position, EDE et écrouissage , EDE et écrouissage , EDE et écrouissage , EDE et écrouissage    \n2.4.1.1.  Modélisation therm ochim ique de l’EDE  \n \nLa m esure directe de l’EDE d’un acier est un exerci ce de MET particulièrement délicat, \nreposant sur l’analyse de la configuration de jon ct ions triples de dislocations parfaites [REMY \n1975], la méthode de référence, ou des distances de  dissociations entre dislocations partielles \n[BRACKE 2007_2]. Ces différentes méthodes sont suje ttes à de nom breux biais (l’épinglage \npar exemple des dislocations partielles par des ato mes de carbone au cours de traitements \nthermiques [REM Y  1975]). Elles sont donc rares dan s  la littérature et souvent sources de \ncontroverses, comme le jeu de m esures de Volosevitc h et al.  [VOLOSEVI TCH 1976]. Pour \nune composition d’acier donnée comme les aciers Had field (typiqueme nt Fe12Mn 1.2C), les \nEDE mesurées à température am biante peuven t varier du sim ple au double selon les auteurs \n(23 mJ.m -2 pour Karaman  et al.  [KARAMAN 2000_1] ou 50 m J.m -2 pour Dastur et Leslie par \nexem ple [BAYRAKTAR 2004][DASTUR 1981]) \n \nC’est donc un exercice auquel n ous n’avons pas dire ctemen t participé ! N otre équipe s’est \nplutôt consacrée à la modélisation de cette EDE sur  des bases thermodynamiques par une  69  méth ode indirecte, les résultats et paramètres étan t en suite ajustés en fon ction des \ntempératures de transform ation martensitique de ces  alliages [COTES 1998]. \n \nCes calculs reposent sur l’existence d’une relation  entre EDE et fo rces m otrice de \ntran sformation  martensitique ε, reprise ci-dessous  : \n \n γ/ε ε γ\n111 int 2σ ∆Gρ 2 EDE + =→ (38) \n \nAvec ρ 111  la densité molaire surfacique des plans denses {111} et σγ/ε l’énergie de surface γ/ε. \n \nDan s ce type de calcul, la détermination de l’énergie de Gibbs pour la transfor mation \nvolumique γ \u0001ε reste le point clef. De nombreux auteurs on t part icipé à la description du \nsystème bin aire FeMn [MIODOWNIK 1998][HUANG 1989][R EMY  1975][LIN 1997] ou du \nsystème ternaire FeMnSi [COTES 1998], mais nous avo ns été parmi les premiers à proposer \nune approche complète in tégran t de manière semi-empi rique les effets du carbone [ALLAIN \n2004_3][ALLAIN 2004_1] . Ces travaux ont ensuite été repris et étendu à des systèmes plus \ncomplexes comme aux FeMn AlC par Saeed-Akbari et al.  [SAEED-AKBARI 2009], \nFeMnCuSiAlC par Dumay et al.  [DUMAY 2008_2]  ou aux FeMnCN par Curtze et al.  \n[CURTZE 2010]. La révision la plus récente et la pl us complète a été proposée par Nakano et \nal.  [NAKANO 2010] avec, pour la première fois, une app roche globale, compatible avec le \ndiagramme des phases stables du système FeMn binaire complet et cohéren t avec l’approche \ndu SGTE (Scientific Group Thermodata Europe). \n \n \nFig Fig Fig Figure ure ure ure 38 38 38 38    : Cartographie d’EDE prédite par notre modèle en f onction de la teneur massique en ca rbone et : Cartographie d’EDE prédite par notre modèle en fo nction de la teneur massique en ca rbone et : Cartographie d’EDE prédite par notre modèle en fo nction de la teneur massique en ca rbone et : Cartographie d’EDE prédite par notre modèle en fo nction de la teneur massique en ca rbone et \nma nganèse. Sont identifiés les domain es d’apparitio n de la martensite ma nganèse. Sont identifiés les domain es d’apparitio n de la martensite ma nganèse. Sont identifiés les domain es d’apparitio n de la martensite ma nganèse. Sont identifiés les domain es d’apparitio n de la martensite ε au refroidissement (thermique)  et au ε au refroidissement (thermique) et au ε au refroidissement (thermique) et au ε au refroidissement (thermique) et au \ncours de la déformation (athermique) défini s par Sc h cours de la déformation (athermique) défini s par Sc h cours de la déformation (athermique) défini s par Sc h cours de la déformation (athermique) défini s par Sc h umann [SCHUMANN 1972] et les domaines d’états umann [SCHUMANN 1972] et les domaines d’états umann [SCHUMANN 1972] et les domaines d’états umann [SCHUMANN 1972] et les domaines d’états \nmagnéti magnéti magnéti magnétiq ues calculés (en rouge) q ues calculés (en rouge) q ues calculés (en rouge) q ues calculés (en rouge)    [AL LAIN 2004_1] [AL LAIN 2004_1] [AL LAIN 2004_1] [AL LAIN 2004_1][ALLAIN 2004_3] [ALLAIN 2004_3] [ALLAIN 2004_3] [ALLAIN 2004_3]. . ..    \n  70  Notre modèle reste très souvent utilisé en pratique  par d’autres équipes car il ne nécessite pas \nl’utilisation  de logiciels spécifiques (type THERMO CALC) et fonction ne sur un simple tableur. \nLa Figure 38 montre le calcul des EDE en fonction d es teneurs en carbone et manganèse des \nalliages FeMnC à te mpérature am biante. Le modèle a été ajusté pour que les lignes d’iso-EDE \nsuivent les limites des domaines de transfor mation martensitique ε. Sont reportés en outre les \ndomaines d’états magnétiques (séparés par la ligne rouge). Les deux éléments Mn et C \ncontribuent donc à augm e nter l’EDE, mais dans des p roportions différen tes à température \nambiante. La Figure 39 montre l’influen ce de l’ajou t d’autres éléments d’alliage \nsubstitution nels comme le Cr, Al, Si ou le Cu d’apr ès le modèle de A. D umay et al.  [DUMAY \n2008_2] . \n \n \nFig ure Fig ure Fig ure Fig ure 39 39 39 39    : : : : II IInfluenc e des éléments d’a lliage nfluenc e des éléments d’a lliage nfluenc e des éléments d’a lliage nfluenc e des éléments d’a lliage substitutionnels su r  l’EDE relativement à la  nuance de référenc e  substitutionnels sur  l’EDE relativement à la  nuanc e de référenc e  substitutionnels sur  l’EDE relativement à la  nuanc e de référenc e  substitutionnels sur  l’EDE relativement à la  nuanc e de référenc e \nFe22M n0.6C Fe22M n0.6C Fe22M n0.6C Fe22M n0.6C [DUM AY 2008_2] [DUM AY 2008_2] [DUM AY 2008_2] [DUM AY 2008_2]     \n \nLe dernier terme de l’équation précédente, correspo ndant à l’énergie d’interface γ/ε est \nsouvent considéré comm e une variable d’ajusteme nt d e ces modèles. Saeed-Akbari et al.  \n[SAEED-AKBARI 2009] a proposé une revue pertine nte des différen tes valeurs trouvées dans \nla littérature pour ce paramètre (parfois variable en fonction de la chimie). On peut atten dre \nbeaucoup dans l’avenir des calculs ab initio  pour définir cette én ergie sur des bases plus \nphysiques ([KIBEY  2006] par exem ple). \n \n2.4.1.2.  Etat magnétique et module d’élasticité  \n \nCes études récentes dédiées à l’estimation therm ody namique de l’EDE reposent en  partie sur \nla connaissance de l’état magnétique (para- ou anti ferro-magnétique) de l’austénite, \ninformation  clef sur la stabilité des phases. Dan s certains cas, la con tribution magn étique en \nexcès peut contrebalancer le terme chimique, et con duit à une stabilisation à elle seule de  71  l’austé nite. Contrairemen t à la plupart des aciers austé nitiques inoxydables, la température \ncritique de transition en tre ces deux états m agnéti ques (appelé température de Néel) est \nsupérieure à 250K dans la plupart des alliages FeMn C. Ce changem ent d’état magnétique a \ndonc des conséquences directes et significatives su r leur comportemen t à température \nambiante. \n \nCette transition en tre état paramagnétique de haute  température et antiferromagn étique de \nbasse température est con nue de longue date dans ce s alliages par les « physiciens » (comm e le \nmon tre la revue de L. Rémy) [REM Y 1975]. Elle a pou rtan t souvent été négligée par les \n« métallurgistes » malgré de très forts ef fets atte ndus sur les transformation s de ph ases et les \npropriétés mécaniques. Le module d’élasticité en ci saillemen t µ de la nuance de référence \nFe22Mn0.6C mesuré par Dynamical Mechanical Thermal Analysis (DTMA) en fonction de la \ntempérature est r eprésen té sur la Figure 40(a). Cet te évolution présen te une anomalie \névidente autour de la tem pérature de N éel. A basse température, les modules d’élasticité sont \nfaibles et quasime n t insen sibles à la température c omm e dans les alliages Invar. Au dessus de \nla température de Néel, dans l’état paramagnétique,  il diminue quasi linéairement avec la \ntempérature comm e atten du par le modèle em pirique d e Ghosh et Olson [GHOSH  2002], et \nce après une très forte augmentation vers 290 K (+1 5 GPa e n 50°C en viron). On notera que la \ntempérature de Néel défin ie d’un  point de vue magné tique correspond toujours au milieu de \nla transition comm e l’a remarqué expérimen talement Rém y. Cette anom alie de module \nd’élasticité est d’im portance car toutes les compos antes de la contrain te d’écoule ment sont des \nfonctions de ce m odule, y compris les termes de con trainte effective ou de frottement de \nréseau. \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 40 40 40 40    : (a) : (a) : (a) : (a) Evolution Evolution Evolution Evolution du module d’ du module d’ du module d’ du module d’élasticité élasticité élasticité élasticité en  en  en  en cisa ill cisa ill cisa ill cisa illement ement ement ement µ de la nuance de référenc e en fonction de la   µ de la nuance de référenc e en fonction de la  µ de la nuance de référenc e en fonction de la  µ de la nuance de référenc e en fonction de la \ntempérature température température température  dans les domaines antife rro  dans les domaines antife rro  dans les domaines antife rro  dans les domaines antife rro- - -- et  et  et  et pa ramagnétiques pa ramagnétiques pa ramagnétiques pa ramagnétiques    [ALLAIN 2010_2 [ALLAIN 2010_2 [ALLAIN 2010_2 [ALLAIN 2010_2] ] ] ] (b) Effet de l’ajout de cuivre (b) Effet de l’ajout de cuivre (b) Effet de l’ajout de cuivre (b) Effet de l’ajout de cuivre \ndans la n uance de  dans la n uance de  dans la n uance de  dans la n uance de  référence référence référence référence sur la  sur la  sur la  sur la température température température température de Néel de l’a lliage  de Néel de l’a lliage  de Néel de l’a lliage  de Néel de l’a lliage [DUM AY 2008_1] [DUM AY 2008_1] [DUM AY 2008_1] [DUM AY 2008_1]     \n \nLa contribution m agnétique en  excès à l’EDE est gén éralement calculée en utilisant la \nméth ode de Hillert et Jarl [IN DEN 1981][HILLERT 197 8]. Cette équation requiert les \ntempératures de N éel et les moments magnétiques de chacune des phases (austénite γ et \nmartensite ε) en fonction de la composition des all iages. Dans le cadre de la th èse de A. \nDum ay, les températures de Néel de nom breux  alliage s ont été mesurées pour déterminer les \n 72  effets relatifs du cuivre, m anganèse et carbone en solution solide. La Figure 40(b) m ontre par \nexem ple l’évolution de la température de Néel de la  nuance de référence en fonction de sa \nteneur en cuivre. Ces différents résultats nous ont  permis de définir une équation statistique \nde cette tem pérature T γNé el  critique pour l’austénite en  fonction de sa compos ition  chimique \n[HUANG 1989][MIODOWNIK 1998][ZHANG 2002] [DUMA Y  2008_1]  : \n \n []() 720 13x 6.2x 2.6x 222x x 4750x x 250ln K TSi Al Cr Cu MnC Mnγ\nNéel +−−− − − =  (39) \n \navec x i la fraction atomique de l’élément chimique i considé ré et T γNée l  en K. \n \nPar contre, peu de progrès ont été faits dans la me sure de moments mag nétiques de ces \nalliages. Ces termes resten t la plupart du temps des paramètres d’ajusteme nt du modèle d’EDE \n[ALLAIN 2004_2] [COTES 1998]. Là encore, des progrès significatifs s ont attendus des calculs \nab in itio  dans le futur. \n \n2.4.1.3.  EDE et écrouissage : le paradoxe carbone  \n \nComme nous l’avons discuté dans la section précédente , la susceptibilité d’un e nuan ce FeMnC \nà macler est fortement corrélée à son EDE. Cette én ergie con trôle en effet les possibilités de \ndissociation  des dislocations parfaites et la contrai nte critique de maclage, comme le mon tre \nl’équation (7). \n \nLa conséquence directe de cette observation est que l’ajout d’éléments d’alliage aug mentant \nl’EDE devrait avoir pour effet de ralentir la cinétiq ue de maclage et donc dimin uer \nglobale men t le taux d’écrouissage d’un acier donné.  Ce résultat a été confirmé \nexpérimentaleme n t par A. Dumay en étudiant l’ajout de Cu dans la nuance de réf érence. Le \ncuivre aug mente l’EDE de 3.5 mJ.m-2.%Cu -1 selon la Figure 39. Les courbes de traction de 3 \naciers Fe22Mn0.6CxCu sont représentées sur la Figur e 41(a), montran t clairement la chute du \ntaux  d’écrouissage pour les nuances les plus riches en cuivre à partir de 15% de déformation \n[DUMAY 2008_1] . \n \n \n  73  \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 41 41 41 41    : (a) Effet de l’ajout de cuivr e sur le c omporte me nt en  traction de la nuance de r éférence : (a) Effet de l’ajout de cuivr e sur le c omporte men t en  traction de la nuance de r éférence : (a) Effet de l’ajout de cuivr e sur le c omporte men t en  traction de la nuance de r éférence : (a) Effet de l’ajout de cuivr e sur le c omporte men t en  traction de la nuance de r éférence [DUM AY [DUM AY [DUM AY [DUM AY \n2008_1] 2008_1] 2008_1] 2008_1]  (b) Courbes de traction rationnelles de différe  (b) Courbes de traction rationnelles de différe  (b) Courbes de traction rationnelles de différe  (b) Courbes de traction rationnelles de différents  a lliages ternaires FeMnC TWIP nts a lliages ternaires FeMnC TWIP nts a lliages ternaires FeMnC TWIP nts a lliages ternaires FeMnC TWIP [HUANG [HUANG [HUANG [HUANG \n2011][BOUAZIZ 2011] 2011][BOUAZIZ 2011] 2011][BOUAZIZ 2011] 2011][BOUAZIZ 2011]. . ..    \n \nDe façon très surprenante, c’est l’effet inverse qu i est observé avec l’ajout de carbone, nous \nparlerons donc dans la suite de « paradoxe carbone ». Nous avons compilé dans [HUANG \n2011][BOUAZIZ 2011]  les courbes de traction de différents aciers FeMn C  reprises sur la \nFigure 41. Toutes les courbes de traction  présen ten t globalem e nt le même taux d’écrouissage \ninitial, très similaire à celui d’une austén ite à f aible EDE sans maclage, comme l’acier binaire \nFe30Mn de la Figure 30. Les différences apparaissen t après 5% à 10% de déform ation. Les \nnuances les plus riches en carbone voie nt leur taux  d’écrouissage augmenter, avec un \nchan geme n t de courbure dans certains cas (comme les  aciers Fe17Mn 0.95C ou Fe30Mn1.0C).  \n \nComme nous l’avons montré précédem ment, la contrain te d’écoulement des aciers TWIP \npeut s’écrire : \n \n () ()()()T T M M 0 σEF εσ σEΣ ε+ +≈  (40) \n \nAvec σ 0 la contrain te de friction dû aux  élémen ts en solut ion  solide, σ M la contrain te dan s la \nmatrice austénitique et Fσ T la contrainte supportée par la fraction maclée (la  con trainte de \nback-stress σ b) et ε T la déformation plastique dans les macles (très fai ble). Dans la mesure où \nσM n e se mble pas dépendre de la composition en  carbon e, seule la contribution à l’écrouissage \nde n ature cinématique due au maclage peut expliquer  la différence en tre ces alliages.  \n \nCe second terme apparaît empiriquem ent comme un e fo nction polynomiale simple de la \ndéformation  plastique et de la com position e n m anga nèse et carbone [BOUAZIZ 2011]  : \n \n ( ) ( ) ( )\n5 MnCz avecE 261874z 60661z εσEF2\nT T−= − ≈  (41) \n \nDe façon très surprenan te et paradoxale, il apparaî t alors que les iso-valeurs de z ainsi \ndéterminées suivent dans le plan M n/C des lig nes qu asimen t orthogonales aux iso-valeurs \nEDE (cf. Figure 42). Autrement dit, l’écrouissage d es nuances TWIP n e dépend pas de l’EDE !  74   \n \nFigure Figure Figure Figure 42 42 42 42    : C artographie d’EDE en fonction de la te : C artographie d’EDE en fonction de la te : C artographie d’EDE en fonction de la te : C artographie d’EDE en fonction de la teneur en ca rbone et en  neur en carbone et en  neur en carbone et en  neur en carbone et en  manganèse manganèse manganèse manganèse. Surimposition des lign es . Surimposition des lign es . Surimposition des lign es . Surimposition des lign es \niso iso iso iso-- --z , valeur représentative du taux d’écrouissage des  alliages à grande déformation. Qualita tivement, le s lignes z , valeur représentative du taux d’écrouissage des alliages à grande déformation. Qualita tivement, les  lignes z , valeur représentative du taux d’écrouissage des alliages à grande déformation. Qualita tivement, les  lignes z , valeur représentative du taux d’écrouissage des alliages à grande déformation. Qualita tivement, les  lignes \niso iso iso iso-- --z sont normales a ux  lignes iso z sont normales a ux  lignes iso z sont normales a ux  lignes iso z sont normales a ux  lignes iso- - --EDE. EDE. EDE. EDE.    \n \nDan s la littérature, on attribue aussi au carbon e u n grand rôle sur le com portem ent de ces \nalliages au travers d’un m écanisme de vieillisseme n t dynamique. Comme nous le verrons, \ncette contribution  directe est probablement néglige able et ne permet pas d’ex pliquer le \nparadoxe ci-dessus. Par contre, la teneur en carbon e va modifier sensiblement la m obilité des \ndislocations et leur structuration  au cours de la d éform ation, donc avoir un  effet indirect en \nretour sur la microstructure de maclage et sur les propriétés d’écrouissage. Dans la section \nsuivante, nous allons don c présenter nos travaux su r l’activation thermique du glisse men t et \nsur le mécanisme de vieillisseme nt dynamique dans c es aciers. Nous présen terons et \ndiscuterons alors un certain nombre d’hypothèses pe rmettant d’expliquer cette relation \ncomplexe entre carbone, glisse ment et maclage mécan ique et donc effet TWIP.  \n \n2.4.2. 2.4.2. 2.4.2. 2.4.2.  Activation thermique du glisse ment, vieillissemen t dyn amique et leurs Activation thermique du glisse ment, vieillissemen t dyn amique et leurs Activation thermique du glisse ment, vieillissemen t dyn amique et leurs Activation thermique du glisse ment, vieillissemen t dyn amique et leurs \nconséquences conséquences conséquences conséquences    \n \nDan s un premier temps, on s’intéressera en particul ier à l’influence des éléments d’alliage et \ndu carbone en particulier sur la dynamique du gliss e m ent sur une large gamme de \ntempérature et vitesse de déform ation. Ces travaux sont basés sur une analyse thermique de \ntype interaction « dislocation / obstacle ponctuel fix e ». Cette étude fon damen tale de la \nmobilité des dislocations a souve nt été négligée da ns la communauté au profit des \nmicrostructures de déformation  observables comme le  m aclage ou les transformations \nmartensitiques qui fournissent un e explication dire cte de l’écrouissage de ces nuances.  \n  75  Com pte ten u de la vitesse de diffusion du carbone d ans la structure CFC de ces aciers, il existe \ndes domain es de températures et de vitesses de défo rm ation plus restreints où des in teractions \ndynamiques entre ces atomes mobiles et dislocations  sont possibles, c’est-à-dire, un \nmécanisme de vieillissement dyn amique. C’est en par ticulier le cas de la nuance de référence \nFe22Mn0.6C à tem pérature ambiante lors d’essais qua si-statiques. De nombreuses études ont \npar contre été dédiées à ce processus. Nous revien d rons dans un second tem ps sur ce \nmécanisme, ses conséquen ces supposées et avérées.  \n \n2.4.2.1.  Elém ents d’alliage et mobilité des dislocations  \n \nLe point de départ de cette recherche a été une lar ge étude bibliographique sur l’évolution  de \nla limite d’élasticité en traction de ces n uances e n fon ction de la tem pérature d’essai et de la \nvitesse de déformation [ALLAIN  2010_1] . Ce paramètre est supposé être représen tatif de la  \nmobilité des dislocations (contrainte d’écoulem ent en  l’absence de macles mécaniques dans \ndes structures recristallisées à grandes tailles de  grains). Le Tableau 2 reprend les \ncomposition s des alliages considérés, les tailles d e grains, températures de Néel estim ées et les \nconditions d’essais tirées des publications respect ives [ALLAIN 2004_1] [REMY \n1975][TOMOTA 1986][CHOI 99][KIM 1986][KUNTZ 2007][A DLER 1986]. La Figure 43(a) \nmon tre l’évolution  des limites d’élasticité en trac tion (contraintes d’écoulement pour des \ndécalages de déform ation plastique faibles de 0.2% à 1%) en fonction de la température d’essai. \nLa Figure 43(b) présente les mê m es données mais nor malisées par la contrainte d’écoulem ent \nà température ambiante de chacun des alliages. Cett e norm alisation perm et de se départir \nd’un e contribution  structurale liée à la taille de grain. \n \n \nTableau Tableau Tableau Tableau 2 2 22    :: ::    Aciers ter naires Fe M nC étudiées d’a près Aciers ter naires Fe M nC étudiées d’a près Aciers ter naires Fe M nC étudiées d’a près Aciers ter naires Fe M nC étudiées d’a près [Allain 2004 [Allain 2004 [Allain 2004 [Allain 2004_1 _1 _1 _1]] ]][Rémy 1975] [Tomota 1986][Choi 99][Kim [Rémy 1975] [Tomota 1986][Choi 99][Kim [Rémy 1975] [Tomota 1986][Choi 99][Kim [Rémy 1975] [Tomota 1986][Choi 99][Kim \n1986] 1986] 1986] 1986][Kuntz 2007][Adler  1986] [Kuntz 2007][Adler  1986] [Kuntz 2007][Adler  1986] [Kuntz 2007][Adler  1986]. . ..    \n \nLes aciers austénitiques à haute teneur en man ganès e présentent le comportement typique \ndes structures CFC durcies par des interstitiels [K OCKS 1995] lors de sollicitations quasi-\nstatiques, c’est-à-dire :  76  • Une rapide dimin ution de la contrainte d’écoulemen t  avec la tem pérature sous la \ntempérature ambiante, définissant le domaine du gli sse men t therm iquement activé. \nAutrement dit, le glissem e nt des dislocations néces site le franchissement d’obstacles \nlocaux dont la barrière enthalpique est trop grande  pour la température considérée. \nLeur franchisse ment nécessite l’application d’une c on trainte supplémentaire, appelée \ncontrainte effective. \n• Un plateau au-dessus d’un e température critique, pr oche de la te mpérature ambiante \n(en réalité, la limite d’élasticité continue à dimi nuer faible ment avec la température et \ncette dimin ution s’explique comm e nous le verrons p ar une baisse linéaire du module \nde cisaillem e nt). Ce plateau correspond à un régime  athermique et est souvent associé \nà un  processus de vieillisse men t dynamique. Dans ce  dom aine de température, la \ncontrainte effective est quasiment nulle. \n \nLa Figure 44 mon tre par exemple le comportement d’a utres alliages austénitiques avec des \nteneurs en carbone variables, des alliages NiC et F eNiC, qui présenten t un comportement tout \nà fait similaire, m ais des températures critiques d e transition en tre régimes différen tes \n(environ 150K pour les alliages de nickel). \n \n(a) (b) \nFig ure Fig ure Fig ure Fig ure 43 43 43 43    : (a) E : (a) E : (a) E : (a) Evolution des  limites d’élasticité en trac tio n (contraintes d’écoulement pour des décalages de volution des  limites d’élasticité en trac tion (cont raintes d’écoulement pour des décalages de volution des  limites d’élasticité en trac tion (cont raintes d’écoulement pour des décalages de volution des  limites d’élasticité en trac tion (cont raintes d’écoulement pour des décalages de \ndéformation plastique faibles de 0.2% à  1%) en fonc tion de la températur e d’essai déformation plastique faibles de 0.2% à  1%) en fonc tion de la températur e d’essai déformation plastique faibles de 0.2% à  1%) en fonc tion de la températur e d’essai déformation plastique faibles de 0.2% à  1%) en fonc tion de la températur e d’essais s ss pour les  aciers détaillés da ns  pour les  aciers détaillés da ns  pour les  aciers détaillés da ns  pour les  aciers détaillés da ns \nll lle e e e Tablea u Tablea u Tablea u Tablea u 2 2 22 (b) m  (b) m  (b) m  (b) mêmes données êmes données êmes données êmes données qu e (a) qu e (a) qu e (a) qu e (a) ma is norma lisées par la contrainte d’écoule ment à température ma is norma lisées par la contrainte d’écoulement à t empérature ma is norma lisées par la contrainte d’écoulement à t empérature ma is norma lisées par la contrainte d’écoulement à t empérature \nambiante ambiante ambiante ambiante    [ALLAIN 2010_1] [ALLAIN 2010_1] [ALLAIN 2010_1] [ALLAIN 2010_1] .. ..    \n \n(a) (b) \nFig ure Fig ure Fig ure Fig ure 44 44 44 44    : E : E : E : Evolution des limites d’éla sticité en  traction en  fon volution des limites d’éla sticité en  traction en fo n volution des limites d’éla sticité en  traction en fo n volution des limites d’éla sticité en  traction en fo nc tion de la  température d’essai pour  différents c tion de la  température d’essai pour  différents c tion de la  température d’essai pour  différents c tion de la  température d’essai pour  différents \nalliages (a) NiC [NAKADA 1971] et (b)FeNiC [KOCKS 1 995]. alliages (a) NiC [NAKADA 1971] et (b)FeNiC [KOCKS 1 995]. alliages (a) NiC [NAKADA 1971] et (b)FeNiC [KOCKS 1 995]. alliages (a) NiC [NAKADA 1971] et (b)FeNiC [KOCKS 1 995].    \n 77   \nLa contrain te effective σ e ffective  peut se m odéliser grâce à un potentiel viscoplasti que tiré de la \nthéorie des interactions entre dislocations et défa uts discrets fixes [ALLAI N 2009][ALLAIN \n2010_1]  : \n \n ( )\n* VTk\nTk∆Gexp\nνb 2MρEsinh\nM1ET, σB\nB0\nDebye2\n110 m1\neffective\n\n\n\n\n\n\n=−&&  (42) \n \nAvec M un  facteur de Taylor moyen, E la déformation , ρ m la densité de dislocations mobiles, \nνDebye  la fréquence de Debye, k B la constante de Boltzm ann, V* et ∆G 0 les volumes et énergies \nd’activation  apparents du processus.  \n \nCe m odèle a été appliqué pour décrire les donn ées d e la Figure 43. La procédure d’ajustem ent \nest décrite dans [ALLAIN  2010_1] . Tous les alliages présentent les mêmes températur es de \ntran sition entre régime athermique et therm iquem ent  activé et donc la mêm e énergie \nd’activation  apparente. Par contre, le volume d’act ivation varie très fortement avec la ten eur \nen carbone de l’alliage comme le montre la Figure 4 5. \n \n \nFig ure Fig ure Fig ure Fig ure 45 45 45 45    : Evolution du volu me d’ac tivat : Evolution du volu me d’ac tivat : Evolution du volu me d’ac tivat : Evolution du volu me d’ac tivation apparent de la  c ontrainte e ffective en fonction de la teneur e n ion apparent de la  contrainte e ffective en fonction  de la teneur e n ion apparent de la  contrainte e ffective en fonction  de la teneur e n ion apparent de la  contrainte e ffective en fonction  de la teneur e n \ncarbone des alliages considérés carbone des alliages considérés carbone des alliages considérés carbone des alliages considérés    [AL LAIN 2010_ [AL LAIN 2010_ [AL LAIN 2010_ [AL LAIN 2010_1 1 11]] ]].. ..    \n \nCette figur e démontre l’effet n égligeable de l’état  magnétique (antiferromagn étique ou \nparamagnétique), de la teneur en Mn, en Al ou micro alliage sur ce processus d’activation \nthermique du glissement. Dans le cas de l’acier de référence il est possible de déterminer la \ntran sition entre les deux régimes car le volume de données disponibles est important (th èse \npersonnelle et thèse M. Kuntz [KUNTZ 2008]) mais au ssi car les limites d’élasticité peuvent \nêtre décorrélées de l’évolution des modules d’élast icité connus par ailleurs en fonction de la \ntempérature (cf. Figure 40) \n 78   \nLa Figure 46 mon tre l’évolution du rapport entre la  limite d’élasticité et le m odule de \ncisaillement de la n uance de référence en fonction de la te mpérature. Cette figure m ontre que \ndans le domaine défini comme atherm ique précédem me n t, la con trainte d’écoulement est \nconstante jusqu’à des températures élevées (700K). Par contre, la température de transition  est \nsignificativeme nt supérieure à la température ambia n te (environ 350K). Le glissement dans \nl’acier de référence est donc « thermiqueme nt activ é » à température ambian te. La contrainte \neffective à température am biante est de l’ordre de 70 MPa pour des essais à 7x10 -4 et 4x10 -3 s -1. \nCette valeur est très cohérente par rapport aux ess ais de relaxation, que nous avons réalisés \npar ailleurs, qui donnent une valeur d’en viron 60 M Pa [ALLAIN 2011] (cf. Figure 46(b)). \n \n(a) (b) \nFigure Figure Figure Figure 46 46 46 46    : : : : (a) évolution du rapport entre la l imite d’élasti cité et le module de ci saillement de la n (a) évolution du rapport entre la l imite d’élastici té et le module de ci saillement de la n (a) évolution du rapport entre la l imite d’élastici té et le module de ci saillement de la n (a) évolution du rapport entre la l imite d’élastici té et le module de ci saillement de la nuance de uance de uance de uance de \nréférence en f référence en f référence en f référence en fonction  de la température. Donnée onction  de la température. Donnée onction  de la température. Donnée onction  de la température. Données de s de s de s de [ALLAIN [ALLAIN [ALLAIN [ALLAIN 2010_2 2010_2 2010_2 2010_2] ] ]] e n bleu, données de [KUNTZ 2008] en  e n bleu, données de [KUNTZ 2008] en  e n bleu, données de [KUNTZ 2008] en  e n bleu, données de [KUNTZ 2008] en \nrouge (b) rouge (b) rouge (b) rouge (b) Mesure de la contrainte effec tive à tempé rature ambiante Mesure de la contrainte effec tive à température amb iante Mesure de la contrainte effec tive à température amb iante Mesure de la contrainte effec tive à température amb iante    par u n essai d e relax ation sur la  nu par u n essai d e relax ation sur la  nu par u n essai d e relax ation sur la  nu par u n essai d e relax ation sur la  nuance de ance de ance de ance de \nréfére nce référe nce référe nce référe nce [AL LAIN 2011 [AL LAIN 2011 [AL LAIN 2011 [AL LAIN 2011]. ]. ]. ].     \n \nLe m ême comportement en température a été observé d ans des nuances stables sans carbone \ncomme les aciers Fe25Mn3Si3Al. Ceux-ci présentent u n e tem pérature critique bien \nsupérieure à celles des aciers ternaires FeMn C (400  K environ, révélateur d’un e énergie \nd’activation  apparente plus élevée) mais un volume d’activation plus faible que la nuance de \nréférence Fe22Mn 0.6C [GRASSEL 2000]. Com pte tenu de s résultats précédents, on peut \npenser que ce son t les atomes de silicium qui contr ôle nt le processus et joue n t un rôle \nsimilaire au carbon e (d’où les énergies et volum e d ’activation différents). Ce résultat explique \nen grande partie le com portem ent des Alliages à Mém oire de Forme (AMF) FeMnSi. Le \nglisse ment des dislocations est presque complèteme n t inhibé dans les nuances Fe30Mn6Si, au \nprofit de la seule transformation martensitique ε [ MYAZAKI 1989]. \n \n 79  (a) (b) \nFig ure Fig ure Fig ure Fig ure 47 47 47 47    : (a) Evolution ex périmenta le et modélisée de  la c ontrainte effective en fonction de : (a) Evolution ex périmenta le et modélisée de  la co ntrainte effective en fonction de : (a) Evolution ex périmenta le et modélisée de  la co ntrainte effective en fonction de : (a) Evolution ex périmenta le et modélisée de  la co ntrainte effective en fonction de la te mpérature  la te mpérature  la te mpérature  la te mpérature \npour la n uance de référenc e et une nuance sans carb one Fe25 Mn3SiSAl [GRASSEL 2000] (b) Evolution pour la n uance de référenc e et une nuance sans carb one Fe25 Mn3SiSAl [GRASSEL 2000] (b) Evolution pour la n uance de référenc e et une nuance sans carb one Fe25 Mn3SiSAl [GRASSEL 2000] (b) Evolution pour la n uance de référenc e et une nuance sans carb one Fe25 Mn3SiSAl [GRASSEL 2000] (b) Evolution \nex périmenta le et modélisée de la contrainte effecti ve en fonction de la vitesse de déformation pour  le s mêmes ex périmenta le et modélisée de la contrainte effecti ve en fonction de la vitesse de déformation pour  le s mêmes ex périmenta le et modélisée de la contrainte effecti ve en fonction de la vitesse de déformation pour  le s mêmes ex périmenta le et modélisée de la contrainte effecti ve en fonction de la vitesse de déformation pour  le s mêmes \nnuances que (a) nuances que (a) nuances que (a) nuances que (a)    [ALLAIN 2010 [ALLAIN 2010 [ALLAIN 2010 [ALLAIN 2010_1 _1 _1 _1]] ]].. ..    \n \nDan s le cas des aciers Fe25Mn3Si3Al sans carbon e, l ’équation(42) permet de d’écrire \nsimultaném ent les effets de la vitesse de déformati on et de température sur la contrainte \neffective comme le mon tre la Figure 47.  Par con tre , elle se révèle malheureusem ent \ninsuffisante dans le cas des aciers FeMnC. En effet , ces aciers à température ambiante son t le \nsiège d’un m écanisme de vieillissement dynamique qu i modifie profondément la m obilité des \ndislocations (vitesse de glisseme n t interdites) et rend négative la sensibilité des alliages à la \nvitesse de déformation.  \n \nAux  faibles vitesses de déformations, le paramètre classique m vitesse  de sensibilité à la vitesse de \ndéformation  vaut pour la nuance de référence : \n \n ()\n( )()\n( )170\nΣ∆lnE∆ln\nΣlnElnm\n298KT 298KTvitesse − = ≈\n∂∂=\n= =& &\n (43) \n \nDan s le cas de la n uance de référence, ce mécanisme  spécifique d’interactions n’agit qu’aux \nfaibles vitesses de déform ation (< 1 s -1). La  Figure 47(b) m ontre d’ailleurs qu’au-delà, l a \nvariation de la con trainte avec la vitesse de défor mation est bien décrite par le modèle (pen te). \nLes données à hautes vitesses peuven t donc être exp loitées pour estim er le profil en \ncontrainte des obstacles au glissem ent des dislocat ions, dont le franch issemen t est le processus \nthermiquem e nt activé. Suivant la th éorie de Kocks [ KOCKS 1995], l’aire apparente \nd’activation  des obstacles s’exprim e en fonction  de  la contrain te : \n \n ( )()\nΣbTkΣ mΣ∆a\n110B vitesse=  (44) \n \nLa Figure 48 représente les contraintes d’écoulemen t en fonction des aires apparen tes \nd’activation  pour les essais réalisés à des vitesse s de déform ation supérieures à 1 s -1 de la \n 80  Figure 47. Quelle que soit la valeur de contrainte,  le glisse m ent est therm ique m ent activé \navec des aires d’activation  apparentes très faibles  (éch elle du processus de franchisseme nt de \nl’obstacle) de l’ordre de 30 b² cohéren te avec un d urcissem ent par atomes interstitiels. La \ncohérence de cette analyse (mesure et interprétatio n ) est renforcée par l’extrapolation aux \nfaibles aires. Cette extrapolation montre une valeu r de « mech anical threshold » de 1000 MPa \nen parfait accord avec les valeurs estimées de limi te d’élasticité à 0K de 950 MPa (cf. Figure 48 \nqui reprend les données de la Figure 46(a) prolongé es linéairem e nt à 0K). \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 48 48 48 48    : (a) Profil apparent en contrainte des obstacles au glissement des disloc ations en fonction de : (a) Profil apparent en contrainte des obstacles a u glissement des disloc ations en fonction de : (a) Profil apparent en contrainte des obstacles a u glissement des disloc ations en fonction de : (a) Profil apparent en contrainte des obstacles a u glissement des disloc ations en fonction de l’aire   l’aire  l’aire  l’aire \nd’activa tion déterminé grâce a ux  essais à ha ute vit e sse de dé formation à température ambiante d’activa tion déterminé grâce a ux  essais à ha ute vit e sse de dé formation à température ambiante d’activa tion déterminé grâce a ux  essais à ha ute vit e sse de dé formation à température ambiante d’activa tion déterminé grâce a ux  essais à ha ute vit e sse de dé formation à température ambiante  (hors  (hors  (hors  (hors \nvieillissement dynamique) établi da ns le cas d’une interaction vieillissement dynamique) établi da ns le cas d’une interaction vieillissement dynamique) établi da ns le cas d’une interaction vieillissement dynamique) établi da ns le cas d’une interaction « « ««     dislocations / obstacles ponctuels fix es dislocations / obstacles ponctuels fix es dislocations / obstacles ponctuels fix es dislocations / obstacles ponctuels fix es    »» »». . . . \nDéfinition d’un seuil mécanique (« Définition d’un seuil mécanique (« Définition d’un seuil mécanique (« Définition d’un seuil mécanique («    mech mech mech mechanical threshold anical threshold anical threshold anical threshold    ») par ex trapolation à 0K. (b) Extrapolation à 0K de ») par ex trapolation à 0K. (b) Extrapolation à 0K d e ») par ex trapolation à 0K. (b) Extrapolation à 0K d e ») par ex trapolation à 0K. (b) Extrapolation à 0K d e \nl’évolution des limites d’élasticité de la  nuance d e référence  avec la température l’évolution des limites d’élasticité de la  nuance d e référence  avec la température l’évolution des limites d’élasticité de la  nuance d e référence  avec la température l’évolution des limites d’élasticité de la  nuance d e référence  avec la température    [ALLAIN 2010_2 [ALLAIN 2010_2 [ALLAIN 2010_2 [ALLAIN 2010_2] ] ]].. ..    \n \nOn regrettera toutefois l’absence de données ex péri mentales qui perm ettrait un ajustement du \nmodèle san s avoir à considérer le dom aine de vieill issement dyn amique, c'est-à-dire des \nrésultats d’essais en vitesses à basse te m pérature principalem e nt. Ces donn ées permettraient \nde dresser un profil d’obstacle complet en l’absenc e d’interaction s dynamiques avec les \natom es de carbone et ce avec un e approche conve ntio nnelle, comm e dans le cas des alliages \nFeMnSiAl. Dans un  second temps, ce modèle pourrait être com plété par la prise en compte de \nla contribution du vieillisseme nt dynam ique, comme le propose Hong par exemple [HONG \n1986] ou par une approch e de type Kocks pour les in teractions dyn amiques avec des solutés \nmobiles [KOCKS 1995]. \n2.4.2.2.  Caractéristiques du processus de vieillissement dyn amique  \n \nComme le m ontre la Figure 49, les courbes de tracti on de la nuance de référ ence Fe22Mn0.6C \nprésentent des instabilités e n contrainte typiques à des te mpératures proch es de l’ambian te. \nLa forme de ces perturbations de contraintes est si milaire à celle reportée dans le cas de l’effet \nPortevin-Le Chatelier (PLC) dan s des alliages d’alu minium (morphologie de type A, B ou C \npar exemple). Ces instabilités, tout comm e la sensi bilité négative de la contrainte à la vitesse \nde déformation, sont les symptômes d’un mécanisme d e vieillissem ent dynamique lié au à \nl’interaction  dynam ique entre atomes de carbon e mob ile et glisseme n t des dislocations.  \n  81  En effet, les aciers austénitiques sans carbone ne présenten t jamais de courbes de traction \nperturbées, y com pris les nuances à très faible EDE  [REMY 1975]. Le phénomène est \nd’ailleurs réduit dans les alliages contenant de l’ aluminium, qui est supposé réduire l’activité \ndu carbone [ZUIDEMA 1987][SHUN 1992] ou dans les al liages avec substitution partielle du \ncarbone par de l’azote [BRACKE 2007_1]. Ce m écanism e et ses conséque nces macroscopiques \nont fait l’objet de nombreuses études ces dern ières  années, sans toutefois faire émerger un \nconsensus sur la nature du mécanisme d’interaction dislocation  / carbone. \n \n(a) \n (b) \nFigure Figure Figure Figure 49 49 49 49    : (a : (a : (a : (a) Courbes de traction rationnelles de la n uance  de référence à différentes te mpératures ) Courbes de traction rationnelles de la n uance de référence à différentes te mpératures ) Courbes de traction rationnelles de la n uance de référence à différentes te mpératures ) Courbes de traction rationnelles de la n uance de référence à différentes te mpératures – – –– les  les  les  les \ndomaines d’a pparition d’insta bilités sur les courbe s sont indiqués (b) Evolution des contraintes d’éc o ulement à domaines d’a pparition d’insta bilités sur les courbe s sont indiqués (b) Evolution des contraintes d’éc o ulement à domaines d’a pparition d’insta bilités sur les courbe s sont indiqués (b) Evolution des contraintes d’éc o ulement à domaines d’a pparition d’insta bilités sur les courbe s sont indiqués (b) Evolution des contraintes d’éc o ulement à \ndifférents niveaux de déformation en fonction de l différents niveaux de déformation en fonction de l différents niveaux de déformation en fonction de l différents niveaux de déformation en fonction de la  vitesse de traction a vitesse de traction a vitesse de traction a vitesse de traction – – ––    sensibilité sensibilité sensibilité sensibilité néga tive à la vitesse de   néga tive à la vitesse de   néga tive à la vitesse de   néga tive à la vitesse de  \ndéformation dans le domaine  des basses vitesses (CTdéformation dans le domaine  des basses vitesses (CTdéformation dans le domaine  des basses vitesses (CTdéformation dans le domaine  des basses vitesses (CT     : Machine de traction conve ntionnelle, SD : Machine de traction conve ntionnelle, SD : Machine de traction conve ntionnelle, SD : Machine de traction conve ntionnelle, SD    : Machine de : Machine de : Machine de : Machine de \ntraction à traction à traction à traction à grande grande grande grande vitesse, SHB  vitesse, SHB  vitesse, SHB  vitesse, SHB    : Barre s d’Hopkinson) : Barre s d’Hopkinson) : Barre s d’Hopkinson) : Barre s d’Hopkinson) [ALLAIN 2008 [ALLAIN 2008 [ALLAIN 2008 [ALLAIN 2008_2 _2 _2 _2]] ]].. ..    \n \nLa Figure 49 présente l’évolution  des contrain tes d ’écoule ment de la nuan ce de référence à \ntempérature ambiante après des taux de déformation données en traction  sous différen tes \nvitesses de déform ations (CT = machine de traction conve ntionn elle, SD = m achine de \ntraction grande vitesse, SHB = Barre d’H opkin son). Elle confir me la sensibilité négative à la \nvitesse de déformation de cette nuance à tem pératur e ambiante et aux faibles vitesses de \ndéformation . C’est dans ce domaine que l’on observe  la présence d’in stabilités sur les courbes \nde traction. Ces domaines de tem pératures et vitess es de déformation  (domaine de sensibilité \nnégative à la vitesse de déformation, in cluant le d omaine d’apparition des instabilités) sont \nrelativemen t bien identifiés dans la littérature, n on seulement pour la nuance de référence \nmais aussi pour d’autres nuances comme les aciers H adfield [DASTUR 1981] [ALLAIN \n2004_1] [BRACKE 2007_1][SHUN  1992][RENARD 2010].  \n \nLa Figure 49(a) montre que selon ces con ditions de tem pérature et de vitesse de déformation, \nles instabilités n e semblent apparaître qu’après un e certaine déformation  critique, \ncaractéristique aussi comm une avec l’effet PLC, et don c essentielle, car il s’agit d’un  processus \nthermiquem e nt activé [SHUN 1992]. Toutefois, la dét ermination précise de cette déformation \ncritique est difficile, comm e le m ontre nos travaux  sur l’analyse multi-fractal des courbes de \ncomportem ent [CH ATEAU 2009]  ou les mesures récentes de champs de déformation l ocaux \nde Roth et al.  [ROTH 2012] [LEBYODKIN 2012] . Ces deux  travaux mon trent que malgré  82  l’absence d’instabilités formées sur la courbe de t raction, les éprouvettes de traction son t le \nsiège de fluctuations locales de vitesses de déform ation  corrélées de façon spatio-temporelle. \nCette difficulté de mesure explique en  grande parti e la dispersion  dans la littérature des \nvaleurs d’én ergie d’activation apparente de ce proc essus. Kuntz, Allain and Bracke rapportent \ndes valeurs proches de 15 kJ.mol -1 pour des nuances Fe-22Mn-CN alors que Dastur et al.  et \nShun  et al.  propose nt des valeurs de 146 et 60 kJ.mol -1 respectivement pour des aciers \nHadfield. La découverte de ces fluctuations de défo rmation s préalables aux instabilités en \ncontrainte ouvre un cham p d’in vestigation nouveau e t pertinent pour mieux quantifier ce \nprocessus de vieillisse ment dyn amique dans les acie rs TWIP et tenter de rétablir un \nconsensus sur la nature du  mécanisme. \n \nLes instabilités sur les courbes de traction sont d ues com me dans le cas du PLC à des \nhétérogénéités très locales de vitesse de déform ati on dans les éprouvettes, sous la for me de \nbandes de déform ation. Nous avons été parm i les pre miers à identifier et caractériser la \ndynamique de ces bandes dans les aciers TWIP ([CHEN  2007] [ALLAIN 2008_2][LEBEDKINA \n2009] ) et ce en utilisant différentes techniques. La Fig ure 50 m ontre par ex emple des \nobservations de bandes de déformation et leurs dépl acements sur le fût d’une éprouvette au \ncours d’un essai de traction par des techniques de thermographie infrarouge (I R) ou des \nmesures de champs locaux par corrélation d’images d igitales [ALLAIN  2008_2] . Elles \nillustrent la propagation des bandes de déformation  dans le cas d’instabilités de type A. \n \n \nFigure Figure Figure Figure 50 50 50 50    : : : : Observation de la propagation de ba ndes de déform ation corrélées aux instabilités sur les courbes de  Observation de la propagation de ba ndes de déformat ion corrélées aux instabilités sur les courbes de Observation de la propagation de ba ndes de déformat ion corrélées aux instabilités sur les courbes de Observation de la propagation de ba ndes de déformat ion corrélées aux instabilités sur les courbes de \ntr action (a) en caméra  infrarouge (mesure de l’éc ha uffement) et ( tr action (a) en caméra  infrarouge (mesure de l’éc ha uffement) et ( tr action (a) en caméra  infrarouge (mesure de l’éc ha uffement) et ( tr action (a) en caméra  infrarouge (mesure de l’éc ha uffement) et (b b bb) en corrélation d’images digitales (mesure de ) en corrélation d’images digitales (mesure de ) en corrélation d’images digitales (mesure de ) en corrélation d’images digitales (mesure de \nc h amps locaux ) c h amps locaux ) c h amps locaux ) c h amps locaux ) [[ [[ALLAIN 2008 ALLAIN 2008 ALLAIN 2008 ALLAIN 2008_2 _2 _2 _2]] ]].. ..    \n \nLes caractéristiques spatio-temporelles de ces ban d es on t été docum entées dans de \nnom breuses nuances par ces mêmes techniques [LEBEDKIN A 2009] ,[ZAVATIERRI 2010] \n[DECOOMAN 2009] [ALLAIN 2008_2] [RENARD 2010] [CAN ADINC 2008][CHEN 2007]. Les \nconclusions de ces différentes études son t de façon  surprenante très cohérentes entre elles :  83  • La vitesse des ban des de déform ation augme nte avec la vitesse de déform ation mais \ndiminue avec la déformation \n• La déformation dans les bandes de déform ation augme n te avec la déformation \nmacroscopique \n• La vitesse de déformation dans les bandes est très supérieure de 10 à 20 fois à la vitesse \nde déformation macroscopique appliquée. \nLa Figure 51 tirée de [LEBEDKIN A2009]  montre par  exemple ces évolutions dans le cas de l a \nnuance de référence. \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 51 51 51 51    : (a) Evolution de la déformation loc ale dans les bandes et vite sse de propagation de ces bandes su r : (a) Evolution de la déformation loc ale dans les b andes et vite sse de propagation de ces bandes su r : (a) Evolution de la déformation loc ale dans les b andes et vite sse de propagation de ces bandes su r : (a) Evolution de la déformation loc ale dans les b andes et vite sse de propagation de ces bandes su r \nl’éprouvette de traction en fonction de  la défo l’éprouvette de traction en fonction de  la défo l’éprouvette de traction en fonction de  la défo l’éprouvette de traction en fonction de  la déformat ion (b) Vitesse de déformation en rmation (b) Vitesse de déformation en rmation (b) Vitesse de déformation en rmation (b) Vitesse de déformation en- - --dehors et dans les bandes dehors et dans les bandes dehors et dans les bandes dehors et dans les bandes \nde déformati de déformati de déformati de déformation en fonction du temps d’essai on en fonction du temps d’essai on en fonction du temps d’essai on en fonction du temps d’essai    [LEBEDKINA2009] [LEBEDKINA2009] [LEBEDKINA2009] [LEBEDKINA2009]    lors d’essai lors d’essai lors d’essai lors d’essais s ss de traction à 2.2x10  de traction à 2.2x10  de traction à 2.2x10  de traction à 2.2x10 -- --33 33 s  s  s  s -- --11 11.. ..    \n \nLa plupart des études sur les aciers TWI P FeMnC rap porte des instabilités de type A selon la \nclassification conve ntionn elle du PLC. Elles se car actérisent par la formation de bandes de \ndéformation  relativeme nt minces mais mobiles. Ces b andes se forment lors des sauts de \ncontraintes et se déplacement lors des « plateaux » , à contrainte quasimen t constante. Des \ninstabilités de type B ou C sont moins souven t décr ites, sauf à plus hautes te mpératures ou \naux très faibles vitesses de déformation [REN ARD 20 12]. Contrairement à un effet PLC \nconve ntionnel, le domain e de vitesse de déform ation  dans lequel les instabilités sont de type \nA est beaucoup plus étendu que dans le cas des alli ages d’alum inium [LEBEDKINA 2009] . On \npourrait ainsi soupçonner que vieillisseme nt dynami que et fort écrouissage dû au maclage \nmécanique interagissent pour stabiliser ce processu s. Le processus de vieillissem ent \ndynamique dans ces aciers reste encore peu com pris et présen te donc les caractéristiques d’un \nPLC atypique. \n \nLes conditions d’apparition des instabilités resten t ainsi un ph énomène surprenant comme le \nmon tre l’analyse du comportement après des essais d e relaxation sur la Figure 52. De façon \ninattendue, les essais de relaxation réalisés après  différents taux de pré-déform ation ont \ndéclenché après recharge des in stabilités alors que  la déform ation critique était loin d’être \natteinte pour ces conditions de température et vite sse de défo rmation. Ce résultat reste pour  84  l’instant in expliqué, mais pourrait être lié à une intensification  « catastrophique » des \nfluctuations observés par Roth et al. . \n \n \nFigure Figure Figure Figure 52 52 52 52    : Courbes de traction : Courbes de traction : Courbes de traction : Courbes de traction conventionnelles conventionnelles conventionnelles conventionnelles suite à des essais de rela xation ( 1000  suite à des essais de rela xation (1000  suite à des essais de rela xation (1000  suite à des essais de rela xation (1000    s) après différents niveaux s) après différents niveaux s) après différents niveaux s) après différents niveaux \nde pré de pré de pré de pré- - --déformation. Les courbes ont été décalées en défor mation déformation. Les courbes ont été décalées en déform ation déformation. Les courbes ont été décalées en déform ation déformation. Les courbes ont été décalées en déform ation pour des raisons de lisibilité  pour des raisons de lisibilité  pour des raisons de lisibilité  pour des raisons de lisibilité    [ALL [ALL [ALL [ALLAIN 2011_2] AIN 2011_2] AIN 2011_2] AIN 2011_2] .. ..    \n \nLa contribution relative du vieillissem ent dynamiqu e à l’écrouissage et au com portem ent \nmécanique de cette famille d’aciers est certain eme n t ce qui fait le plus débat actuellem ent \ndans la littérature. Cette position  proposée initia le men t par Dastur et Leslie et suivie par de \nnom breux autres auteurs [DASTUR 1981][ZUIDEMA 1987] [LAI 1989_1][LAI 1989_2] \nsuppose que le seul mécanisme de vieillissem ent dyn amique peut expliquer le fort taux \nd’écrouissage de ces aciers, de manière indépendant e du maclage mécanique et de la \ntran sformation martensitique ε. Sur la base de calc uls ab initio , Owen et Grujicic ont suggéré \nde plus le rôle particulier de dipôles MnC dans ce processus [OWEN 1999] 13 .  \nCette explication a le mérite en  effet d’expliquer sim plement le résultat de la Figure 41, \nmon trant le rôle particulier du carbone sur le comp ortement. Elle ex plique en outre pourquoi \ncertains alliages très chargés en  carbone (avec une  forte EDE) présentent un fort taux \nd’écrouissage en l’absence de maclage. \n \nA contrario, notre équipe s’est plutôt toujours tou rnée vers des ex plications relevant de la \nmicrostructure de déform ation (maclage mécanique ou  tran sformation martensitique) pour \nexpliquer ce comportement, scénario largement détai llé dan s le ch apitre précédent. Cette \nposition rend par contre plus difficile l’explicati on  des effets du carbone. Les quatre \nprin cipales raisons motivant ce ch oix son t les suiv antes : \n• Dès mes travaux de thèse, nous avions estimé la dif férence de comporteme nt de l’acier \nde référence avec et sans maclage dans des conditio ns de vieillissement dynamique \niden tiques qui prouve l’ex istence d’un durcissem ent  d’origine structural important (cf. \nFigure 37). \n• Les mécanismes associés à des interactions locales avec le carbone son t nécessairem ent \ndes contributions isotropes à l’écrouissage. Il ser ait impossible d’expliquer les forts \neffets Bauschinger dans ces structures (cf. Figure 26). \n                                                 \n13  Cette ex plication très spécifique rend difficile l ’interprétation du mécanisme de vieillissement dyna mique \naussi  observé dans les alliages FeNiC, comme le Fe2 2Ni0.6C [ALLAIN 2008_2] .  85  • La forme particulière de la surface de ch arge lors de trajets monoton es directs tels que \nrepr ésentés sur la figure n e peut s’expliquer que p ar des effets de la texture au travers \ndu m aclage mécanique (cf. Figure 32). \n• L’évaluation  de la contribution du vieillissemen t d ynamique par l’équipe de Y. Estrin, \nmon tre que la contribution au durcissem e nt isotrope  est additive de l’ordre de 20 MPa \nau m ieux dans une nuance Fe18Mn0.6C1.5Al [KIM 2009] .  \nCe scénario n’exclut cependant pas que le processus  de vieillisse men t dynam ique ait un effet \nsur le maclage m écanique et donc un  effet sur le co mportemen t macroscopique. Cette \ndiscussion sur les effets indirects du carbone sur l’effet TWIP au travers du glisseme nt est \nl’objet de la section  suivan te. \n \n2.4.3. 2.4.3. 2.4.3. 2.4.3.  Les effets indirects du carbone, de la Les effets indirects du carbone, de la Les effets indirects du carbone, de la Les effets indirects du carbone, de la structuratio n structuration structuration structuration des dislocations des dislocations des dislocations des dislocations au au au au maclage maclage maclage maclage \nmé ca mé ca mé ca mé can ique n ique n ique n ique. . ..    \n \nContrairem ent aux éléments substitutionnels qui aug m enten t l’EDE et con tribue donc à un \nadoucissem ent de ces aciers, le carbone augmente le  taux d’écrouissage de ces n uances, en \njouant très probablement sur l’in tensité de sa cont ribution cin ématique [BOUAZIZ 2011] . La \nplupart des nuances ternaires FeM nC est sujette à t empérature ambiante à un processus de \nvieillisse ment dyn amique. Ce processus ne contribue  que très peu à l’écrouissage de \nl’austé nite, par con tre cette interaction entre car bone et dislocations contribue à la contrainte \nd’écoulement sous la forme d’une contrainte effecti ve. Cette dernière contribution est \nsignificative à température ambiante et dépend de l a teneur en  carbon e. \n \nA ce jour, il n’existe dans la littérature aucun co nsensus sur le lien en tre carbone, m écanism es \nde déformation et écrouissage. Les scén arios d’expl ications présentés ci-dessous ne sont donc \nque des h ypothèses de travail docume ntées qu’il ser ait intéressant de détailler et \nd’approfondir dans de futures études. \n \nSi l’on considère l’équation (40), reprise ci-desso us \n \n () ()()()T T M M 0 σEF εσ σEΣ ε + +≈  (45) \n \nLes différents résultats montrent que σ M(ε M) ne dépend que peu de la teneur carbone (cf. \nFigure 30). Selon cette équation, il ne reste alors  que deux façons possibles d’expliquer une \naugme ntation du taux d’écrouissage des aciers TWIP avec leur teneur en carbone : \n• l’augmentation de la contrainte supportée par les m acles σ T \n• l’augmentation de la cinétique de maclage (propensi on au maclage) F \n  86  2.4.3.1.  Carbone et contrain tes dans les m acles  \n \nLe premier scénario a été suivi par Huan g et al.  [HUAN G 2011]  pour expliquer les courbes de \ncomportem ents présentées sur la Figure 41. A l’aide  de notre m odèle de durcissem ent \ncinématique [BOUAZIZ 2008_1] , les courbes de traction ont été simulées en ajust ant \nunique men t le terme n 0 en fonction de la ten eur en carbon e, sans changer la cinétique de \nmaclage d’un acier à l’autre. Les valeurs de n 0 ainsi ajustées évoluent linéairement avec la \nteneur en carbone, comm e le m ontre la Figure 53(a).  La figure (b) reprend les courbes de \ntraction sim ulées et expérimen tales de deux  aciers Fe22Mn0.6C et Fe22Mn1.2C à titre \nd’ex emple. \n(a) \n (b) \nFigure Figure Figure Figure 53 53 53 53    : (a ) Evolution du pa ramètre n : (a ) Evolution du pa ramètre n : (a ) Evolution du pa ramètre n : (a ) Evolution du pa ramètre n 00 00 de notre modèle de durcissement cinématique pour r eproduire l’effet  de notre modèle de durcissement cinématique pour r eproduire l’effet  de notre modèle de durcissement cinématique pour r eproduire l’effet  de notre modèle de durcissement cinématique pour r eproduire l’effet \ncarbone carbone carbone carbone sur le comportement des aciers ternaire s Fe MnC  (b) ex emple d’ajustement du modèle pour  reprodu ire  sur le comportement des aciers ternaire s FeMnC  (b)  ex emple d’ajustement du modèle pour  reprodu ire  sur le comportement des aciers ternaire s FeMnC  (b)  ex emple d’ajustement du modèle pour  reprodu ire  sur le comportement des aciers ternaire s FeMnC  (b)  ex emple d’ajustement du modèle pour  reprodu ire \nla c ourbe de traction de la nuance de ré la c ourbe de traction de la nuance de ré la c ourbe de traction de la nuance de ré la c ourbe de traction de la nuance de référenc e et d’un  férenc e et d’un  férenc e et d’un  férenc e et d’un  acier acier acier acier Fe22M n1.2C Fe22M n1.2C Fe22M n1.2C Fe22M n1.2C [HUANG 2011] [HUANG 2011] [HUANG 2011] [HUANG 2011] .. ..    \n \nCon sidérer que le terme n 0 augmente avec la te neur en carbone revient à pense r que les \nprocessus de relaxation des e mpileme nts sur les mac les s’active nt plus difficilement (ou que la \ncontrainte supportée par une macle est plus grande,  selon une vision composite). L’explication \nmise en avant par ce mécanisme est basée sur les ob servations expérimen tales récentes en \nMET de Idrissi et al.  [IDRISSI 2009] de l’interface des macles sur des a ciers TWIP. Les joints \nde m acle des aciers chargés en carbone contiendraie n t de nombreuses dislocations sessiles a \ncontrario des aciers sans carbon e. En conséquence, les empile ments de dislocations sur ces \ninterfaces seraient facilités. \n \n2.4.3.2.  Carbone et cinétique de m aclage  \n \nLe second scenario repose sur une plus grande facil ité des nuances au carbone à macler, \nautrement dit, la teneur en carbone serait suscepti ble de jouer sur la contrainte critique et sur \nla cinétique de maclage des aciers.  \n  87  Il n’existe m alheureusement pas dans la littérature  d’étude comparée complète des cinétiques \nde m aclages (y com pris avec toutes les réserves dis cutées dans la section précédente) pour des \nteneurs en carbone variables. La Figure 54 mon tre p ar contre les propriétés de traction et les \nmécanismes de déformation d’aciers ternaires issus de différentes études mais présentant une \nEDE faible mais identique, c’est-à-dire une même pr ope nsion théorique à macler. La figure \n(a) représen te les limites d’élasticité con ve ntionn elles et con traintes d’écouleme nt à rupture \nde ces aciers FeMn C à gros grains (> 20 µm) et la f igure (b) leurs positionnem e nts relatifs dans \nune carte Mn/C de Schumann par rapport à la droite délim itant les domaines d’austénite \nstables et in stables. Ce positionnement leur con fèr e des EDE estimées très proches. \n \nLa figure (a) confirme le « paradoxe carbone ». La limite d’élasticité de ces aciers augmente \npeu avec la te neur en carbone, avec une sensibilité  de 250 MPa/% environ . Cette valeur est \nrelativemen t comparable à celle retenue par Bouaziz  et al.  de 187 MPa.% [BOUAZIZ 2011] . \nCette évolution s’explique quasime nt pour moitié pa r l’augmen tation de la contrainte \neffective à température ambian te (cf. Figure 43) qu i dépend fortement de la teneur en \ncarbone. L’autre contribution est probable men t d’or dre structural. Par  con tre, la contrainte \nd’écoulement augm ente elle très sensibleme nt avec l a teneur en carbone (taux d’écrouissage). \n \nLe résultat le plus surpren ant est que ces 5 aciers  ne présenten t pas les mê m es mécanismes de \ndéformation . L’acier binaire Fe30Mn ne présen te ni m aclage ni transformation martensitique \nε induite, alors que les 4 autres aciers ternaires FeMn C se déforment e n partie par maclage \nmécanique et leurs courbes de traction présentent d es instabilités en contrain tes \ncaractéristiques, manifestation d’un mécanisme de v ieillissement dyn amique. \n \n(a) \n (b) \nFigure Figure Figure Figure 54 54 54 54    : (a) : (a) : (a) : (a) Limites d’élasticité et c ontrainte vraie à  ru pture en fonction de la teneur en carbone Limites d’élasticité et c ontrainte vraie à  rupture en fonction de la teneur en carbone Limites d’élasticité et c ontrainte vraie à  rupture en fonction de la teneur en carbone Limites d’élasticité et c ontrainte vraie à  rupture en fonction de la teneur en carbone    d’ac d’ac d’ac d’aciers iers iers iers \nFeM nC pr ésentant la même  EDE estimée (b) Position  d es aciers de (a) dans une carte de  Schuma nn M n/C  FeM nC pr ésentant la même  EDE estimée (b) Position  d es aciers de (a) dans une carte de  Schuma nn M n/C  FeM nC pr ésentant la même  EDE estimée (b) Position  d es aciers de (a) dans une carte de  Schuma nn M n/C  FeM nC pr ésentant la même  EDE estimée (b) Position  d es aciers de (a) dans une carte de  Schuma nn M n/C  \n[ALLAIN 2010_2] [ALLAIN 2010_2] [ALLAIN 2010_2] [ALLAIN 2010_2] .. ..    \n \nNous avon s retrouvé ce type de com portem ent dans de s alliages « cousins » des aciers \nausténitiques FeMnC TWIP, les aciers austé nitiques FeNiC TRIP α’. La Figure 55 tirée de \nDagbert et al.  [DAGBERT 1996] montre l’évolution  des limites d’él asticité et résistan ces \nmécaniques d’aciers FeN iC don t les ratios de com pos ition  e n Ni et C ont été choisis \njudicieuse m ent afin que tous les alliages présen ten t la mê m e tem pérature Ms de  88  tran sformation martensitique α’ (environ – 50°C). C ette dernière température est com me \nl’EDE un in dicateur de la stabilité d’un point de v ue th ermochimique des alliages. De manière \nanalogue aux aciers FeMn C, les aciers FeNiC peu cha rgés en  carbone (C< 0.3%) présentent \ndes écrouissages faibles (différen ce entre Rm et Re ) sans effet TRIP alors que les alliages \nfortement chargés (C>0.3%) présentent un effet TRIP  et donc des écrouissages importants, et \nce, en présence concomitante d’un mécanisme de viei llissement dynamique. \n \n \nFigure Figure Figure Figure 55 55 55 55    : Li mite : Li mite : Li mite : Li mites s ss d’élasticité  d’élasticité  d’élasticité  d’élasticité (YS) (YS) (YS) (YS) et résistances mécaniques et résistances mécaniques et résistances mécaniques et résistances mécaniques (UTS) (UTS) (UTS) (UTS) en fonction de la teneur en carbone d’ac iers en fonction de la teneur en carbone d’ac iers en fonction de la teneur en carbone d’ac iers en fonction de la teneur en carbone d’ac iers \nFeNiC prés FeNiC prés FeNiC prés FeNiC présentant la  même Ms (température de transfo rmation entant la  même Ms (température de transformation entant la  même Ms (température de transformation entant la  même Ms (température de transformation ma rtensitique martensitique martensitique martensitique α’). Domaine d’appa rition de s  α’). Domaine d’appa rition de s  α’). Domaine d’appa rition de s  α’). Domaine d’appa rition de s \nmécanismes d mécanismes d mécanismes d mécanismes d’écrouissa ge ’écrouissa ge ’écrouissa ge ’écrouissa ge (TR  (TR  (TR  (TRIP et vieillissement dynamiq ue) [DABGERT 1996] IP et vieillissement dynamiq ue) [DABGERT 1996] IP et vieillissement dynamiq ue) [DABGERT 1996] IP et vieillissement dynamiq ue) [DABGERT 1996]    \n \nPour des EDE faibles, la susceptibilité à macler se mble donc augmen ter avec le carbone. On \nnotera en revanche qu’un alliage Fe35Mn0.6C présent ant une EDE forte (40 mJ/m 2) ne macle \npas malgré une teneur en carbone importante. Pour e xpliquer ce résultat, plusieurs \nhypothèses et mécanismes peuven t être invoqués. \n \n2.4.3.3.  Activation thermique du glisse ment et maclage mécan ique  \n \nCertains auteurs comme le groupe de Koyam a et al.  [KOY AMA 2012] suggèrent que le \nvieillisse ment dyn amique explique la propension à m acler ou à la transfor mation \nmartensitique ε. Leur ex plication est basée sur un différen tiel de mobilité supposé en tre \ndislocations partielles de tête et de queue bordant  une faute d’empileme nt en présence d’un \nprocessus de vieillisse men t dynamique. Cette h ypoth èse pourrait paraître convaincante mais \nelle ne peut perm ettre d’expliquer que des alliages  FeMnSiAl, bie n étudiés par ailleurs, \nprésentent de fortes cin étiques de m aclage [GRASSEL  2010]. La seule caractéristique \ncommune entre alliages FeMnC et FeMnSiAl susceptibl e d’expliquer cette prop ension  au \nmaclage semble être fin alemen t une contrainte eff ec tive élevée due à un processus de \nglisse ment thermiquemen t activé, comm e noté aussi p ar [CHUML YAKOV 2002]. \n \nEn rendant à l’échelle locale le glissement diffici le, le maclage peut être vu comme un \nmécanisme supplétif au glissement localement car de  nombreuses dislocations mobiles sont \n 89  bloquées par des atomes de carbone et ne peuvent as surer la vitesse de glisse ment [ALLAIN \n2010_2] . Les macles assureraient donc localement des « bou ffées » de vitesses de déformation \nsuffisantes (rôle adoucissant des m acles mis en évi dence par N . Shiekhelshouk \n[SHI EKHELSHOUK 2006] par exemple ou les bandes de c isaillement de Gutierrez \n[GUTIERREZ 2010]). Ce scénario est assez cohérent a ussi avec les observations de macles \nprin cipalem e nt dans les grains de la fibre <111>//D T lors d’une déformation en traction. On \npourrait aussi e nvisager un modèle reposant sur un différen tiel de mobilité des dislocations \npartielles, thermiquement activé, de manière analog ue à la proposition de Farenc et al.  dans \nun alliage TiAl [FARENC 1993]. \nOn retrouve aussi cette idée fin ale men t dans les ca lculs de Meyers et al.  [MEYERS 2001] \nd’un e contrainte critique de m aclage macroscopique,  basée sur une compétition en tre \ncontrainte effective pour le glisseme n t et le macla ge. Pour expliquer et quantifier cette \nconstatation  expérimentale décisive, un m odèle d’in teraction locale à l’échelle des \ndislocations reste à construire. \n \n2.4.3.4.  Glisseme nt planaire et maclage m écanique  \n \nLe second élément d’explication possible est que le  carbone favorise un glisseme n t planaire. \nCette planéité du glisse m ent a pour conséquence des  empileme n ts plus grands et donc \nconduit à abaisser la contrainte critique de maclag e effective de l’acier. Cela entraîne \nnaturellement une propension plus im portante au m ac lage mécan ique, comme le mon tre \nl’équation (7) de Venables.  \n \nNous avons observé bien entendu les caractéristique s de ce glissem ent planaire lors de nos \nétudes en MET de la nuan ce de référence. La Figure 56 montre par exemple une comparaison \nde la structuration des dislocations dans l’acier d e référence Fe22Mn0.6C après 5% de \ndéformation  [BARBIER 2009_1]  et un acier Fe30Mn  binaire après 20 % de déformati on \n[HUANG 2011] . Dans le premier cas, les segments de dislocation s  s’organise nt selon des \nstructures très an isotropes (de type Taylor) caract éristiques d’un glisse ment planaire \n[GUTIERREZ 2012], alors que dans le second, la dist ribution en cellules est très isotrope. \n  90  \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 56 56 56 56    : Microgra phie : Microgra phie : Microgra phie : Microgra phies s ss MET en champ clair des structures de  MET en champ clair des structures de  MET en champ clair des structures de  MET en champ clair des structures de dislocations dislocations dislocations dislocations (a) dans la n uance de  référenc e  (a) dans la n uance de  référenc e  (a) dans la n uance de  référenc e  (a) dans la n uance de  référenc e \naprès 5% de déformation après 5% de déformation après 5% de déformation après 5% de déformation [BARBIER 2009] [BARBIER 2009] [BARBIER 2009] [BARBIER 2009]  (b) dans un e nuance  binaire Fe30M n après 20% de dé formation  (b) dans un e nuance  binaire Fe30M n après 20% de dé formation  (b) dans un e nuance  binaire Fe30M n après 20% de dé formation  (b) dans un e nuance  binaire Fe30M n après 20% de dé formation \n[[ [[HUANG 2011 HUANG 2011 HUANG 2011 HUANG 2011] ] ]].. ..    \n \nLe glissement plan aire dans les structures austénit iques s’explique par un e inhibition des \nmécanismes de glisse ment dévié. Cette inhibition pe ut avoir quatre causes distinctes : \n• Une EDE faible qui ren d énergétiquement défavorable  la constriction des fautes \nd’em pilement et la recom binaison des dislocations p artielles en segme nt vis avant \nglisse ment dévié (processus de constriction [PUSCHL  2002]). C’est bien e ntendu le cas \ndes aciers austénitiques TWIP. Par contre, comme l’ EDE augmente avec la teneur en \ncarbone, le glissem e nt dévié devrait être facilité par cet élémen t ! \n• un m écanisme de mise en  ordre à courte distance (« Short Range Orderin g : SRO »). \nC’est une hypothèse envisagée depuis peu par DeCoom an et al.  [DECOOMAN 2009] et \nsuggérée par les travaux de Gerold et al.  [GEROLD 1989] et les calculs d’in teractions \nsur les couples MnC de Owen et Grujicic [OWEN 1999] . Aucune évidence \nexpérimentale de ce mécanisme n’a toutefois été pro duite à ce jour dans les aciers \ntern aires FeMnC (appariement de dislocations). Guti errez et Raabe ont mêm e mon tré \nl’absence de ce m écanisme dan s des alliages encore plus complexes comme les \nFeMnAlC [GUTIERREZ 2012]. C’est donc une cause que nous n’envisageons pas. \n• Une friction de réseau élevée. La probabilité d’un évènem ent de glisse m ent dévié \ns’exprime souve nt en fon ction de la distance d’anni hilation entre segments vis notée \nys [GEROLD 1989] lors d’un processus de restauratio n dynam ique : \n \n ()( )interne app cross 0crossτ τS τ2πθsinμb ys ± − =  (46) \n \nAvec θ cross  et S cross  d eux facteurs constants dépen dant de l’orien tation  cristalline. τ o, τ app  \net τ in terne  son t respectivement la friction de réseau, la cont rain te appliquée et un terme  91  de contrainte interne. Cette relation mon tre que pl us la friction de réseau τ o est élevée, \nplus la distance ys est gran de, plus les évènements  de glisse ment dévié seron t rares. La \ncontrainte effective due à l’interaction  avec les a tomes de carbone aura un effet \nsimilaire. Dans certaines orientations cristallines , ce phénomène sera amplifié par les \ntrès fortes contrain tes internes développées dans c es aciers. \n• Processus de constriction  perturbé par les atomes e n solution solide [GUTIERREZ \n2012]. Andrews et al.  [A N DREWS 2000] a montré que l’interaction entre at omes de \ncarbone et dislocations vis conduisait à une augmen tation de l’énergie de constriction. \nLes atomes de carbone con tribueraient donc à inhibe r le processus de glissem ent dévié. \nParm i ces quatre explications possibles, les deux d ern ières suggèren t que l’ajout de carbone \npeut rendre le glissement planaire, en inhibant loc alem e nt le processus de glissem e nt dévié ; \net ce de façon antagoniste à l’augmentation d’EDE. \n \n2.4.3.5.  PLC, structuration du glissement et contraintes loc ales  \n \nOn ne peut pas n on plus totalement e xclure une cont ribution possible du vieillissement \ndynamique à l’activation du maclage, comme le suggè re les figures précédentes. En effet, \nHon g a montré qu’en présence de vieillissement dyn a mique, les dislocations mobiles, donc \ngéom étriquement nécessaires, s’accumulaient dans de s zones de fortes contraintes internes au \ncours de la déformation [H ONG 2000]. Ce mécanisme p ourrait expliquer l’observation de très \nnom breuses région s, mal inde xées en EBSD, sans pour  autant être des macles, dans les grains \naprès déformation et correspondant à des zones de f orts gradients de déform ation. Ces régions \nressemblent donc à des sous-joints mais apparaissen t très tôt au cours de la déformation (20% \nde déformation dans le cas présenté sur la Figure 5 7). Ces régions pourraie nt à elles seules \nconstituer des zones de concentration de contrainte s n écessaires pour l’activation du maclage, \net qui ne pourraient pas être relaxées par l’ém issi on continue de dislocations parfaites, mais \nnécessiteraient une émission coordonnée (germ e de m acle). \n \n \nFigure Figure Figure Figure 57 57 57 57    : : : : CC CCa rtographie en contraste de bandes. Le grain  princ ipa l compo a rtographie en contraste de bandes. Le grain  princi pa l compo a rtographie en contraste de bandes. Le grain  princi pa l compo a rtographie en contraste de bandes. Le grain  princi pa l comporte un système de macl age rte un système de macl age rte un système de macl age rte un système de macl age. En rouge . En rouge . En rouge . En rouge : : : : \ndésorienta tion désorienta tion désorienta tion désorienta tion de 60° a utour de <111>.  de 60° a utour de <111>.  de 60° a utour de <111>.  de 60° a utour de <111>.    Les ca rrés blancs indiqu ent la présence de zones m on Les ca rrés blancs indiqu ent la présence de zones mo n Les ca rrés blancs indiqu ent la présence de zones mo n Les ca rrés blancs indiqu ent la présence de zones mo ntrant une forte trant une forte trant une forte trant une forte \ndésorientation et mal index ées. désorientation et mal index ées. désorientation et mal index ées. désorientation et mal index ées. Représentation du g radient de dé sorientation au sein du grain et tra cé  du pro fil Représentation du gradient de dé sorientation au sei n du grain et tra cé du pro fil Représentation du gradient de dé sorientation au sei n du grain et tra cé du pro fil Représentation du gradient de dé sorientation au sei n du grain et tra cé du pro fil \nde désorienta tion suivant la ligne noir.  de désorienta tion suivant la ligne noir.  de désorienta tion suivant la ligne noir.  de désorienta tion suivant la ligne noir.  La couleur  bleu r eprésente l’orientation majoritaire du grain  La couleur bleu r eprésente l’orientation majoritair e du grain  La couleur bleu r eprésente l’orientation majoritair e du grain  La couleur bleu r eprésente l’orientation majoritair e du grain  [[ [[BARBI ER BARBI ER BARBI ER BARBI ER \n2008 2008 2008 2008_1 _1 _1 _1]] ]].. ..     92   \n2.4.3.6.  Discussion  \n \nIl est possible que ces trois m écanismes stimule n t simultané ment le maclage, comme \nprocessus locale ment substitutif au glissement. Le premier contribue à rendre le glissem ent \nplus difficile alors que les deux  seconds contribue nt à diminuer la contrainte critique de \nmaclage. On ne peut don c réduire ce processus à une  simple transition en tre glissement et \nmaclage comme le propose Meyers et al.  [MEY ERS 2001], car le maclage n’est jamais observé  \nseul, sans glissemen t préalable.  \nTous les mécanism es décrits ci-dessus restent en l’ état des hypothèses, qu’il serait pertin ent \nd’approfondir à la fois en terme de m écanism es loca ux (à l’échelle des interactions en tre \ndislocations) et d’effet quantifié sur la cin étique  de maclage. \n \n2.5. 2.5. 2.5. 2.5.  Conclusions et Perspectives Conclusions et Perspectives Conclusions et Perspectives Conclusions et Perspectives    \n \nLes aciers TWIP sont des matériaux passionnan ts qui  sont loin d’avoir livré tous leurs secrets \nmalgré une intense recherche académique et in dustri elle. Ils couplen t les difficultés des aciers \ninox ydables austé n itiques, des aciers au carbon e et  des alliages d’aluminium (vieillissem ent \ndynamique). En partie grâce à nos travaux, l’effet TWIP est compris dans les gran des lignes \nmais les challenges scie ntifiques et techn iques res tent n ombreux : \n• Axe de recherche 1 : Mesure expérimen tale de la fra ction de phase maclée (cinétique \nde m aclage) par un e tech nique statistique et reprod uctible. C’est la source principale \nd’ambigüités dans l’interprétation  du comportement de ces aciers. Le couplage entre la \nmicroscopie à force atomique et la technique d’inde xation en  EBSD devrait permettre \ncette quantification  sans biais. \n• Axe de rech erche 2 : Iden tification et quantificati on des processus d’interactions en tre \ncarbone et dislocations (processus de vieillissem en t dyn amique atypique). Des essais de \ntraction à basse températures et à différentes vite sses devraient permettre de fournir \ndes infor mations originales pour modéliser ce proce ssus. \n• Axe de recherche 3 : Calcul de l’EDE. Les m éthodes thermodynamiques actuelles \nreposent sur de nombreux  param ètres d’ajustem e nt. D es progrès son t attendus dans le \nfutur grâce aux techniques ab initio . \n• Axe de recherche 4 : Définition d’un modèle pour la  cinétique de m aclage, basée sur \nune contrainte critique de maclage intégrant les no tions d’orientations mais aussi les \ncaractéristiques du glissem ent (planaire et therm iq uem e nt activé) \n• Axe de recherche 5 : Modélisation micromécan ique de s chan gements de trajet (trajet \nBauschinger et changement de trajets dur), capable de prédire la surface de charge et \nson évolution (écrouissage cinématique et isotrope) . A term e, le fo rmalism e devrait \npouvoir expliquer le comportement en fatigue.  93   \nCes différents travaux ne pourront qu’être en richis  par des comparaisons avec des alliages \nausténitiques « cousins » comme les FeN iC , NiC ou simple m ent FeMnSi, qui présentent des \ncomportem ents m écaniques et des m écanism es de défor m ation similaires, et sont très \nlargement étudiés dans la littérature. \n \n2.6. 2.6. 2.6. 2.6.  Pour le plaisir des yeux Pour le plaisir des yeux Pour le plaisir des yeux Pour le plaisir des yeux    \n \nPour conclure cette partie, je n e résiste au plaisi r de proposer sur la Figure 58 quelques \nmicrograph ies particulièrement « esthétiques » obte nues grâce à ces aciers FeMnC TWIP \n \n(a) \n (b) \n(c) (d) \nFigure Figure Figure Figure 58 58 58 58    : : : : (a ) Artefact d’attaque Nital sur (a ) Artefact d’attaque Nital sur (a ) Artefact d’attaque Nital sur (a ) Artefact d’attaque Nital sur la nuance de référ ence  la nuance de référence  la nuance de référence  la nuance de référence  – – –– les fig ures  les fig ures  les fig ures  les fig ures linéa ires sont prises parfois à tort  linéa ires sont prises parfois à tort  linéa ires sont prises parfois à tort  linéa ires sont prises parfois à tort \npour des macles par certains auteurs pour des macles par certains auteurs pour des macles par certains auteurs pour des macles par certains auteurs    !  !  !  !  [DUM AY 2008] [DUM AY 2008] [DUM AY 2008] [DUM AY 2008]     (b) Phénomène d’infiltration de cuivre liquide dan s la (b) Phénomène d’infiltration de cuivre liquide dans  la (b) Phénomène d’infiltration de cuivre liquide dans  la (b) Phénomène d’infiltration de cuivre liquide dans  la \nnua nce de ré férence ( nua nce de ré férence ( nua nce de ré férence ( nua nce de ré férence (fragilisa tion par métal liquid e, du cuivre en  l’occurence fragilisa tion par métal liquide, du cuivre en  l’occ urence fragilisa tion par métal liquide, du cuivre en  l’occ urence fragilisa tion par métal liquide, du cuivre en  l’occ urence) ) ) ) [DUM AY 2008] [DUM AY 2008] [DUM AY 2008] [DUM AY 2008] [BEAL 2011] [BEAL 2011] [BEAL 2011] [BEAL 2011]    (c) (c) (c) (c) \nSS SSol ol ol olidifica tion eutectique de TiC idifica tion eutectique de TiC idifica tion eutectique de TiC idifica tion eutectique de TiC (sur (sur (sur (sur-- --réseau  hex agonal) réseau  hex agonal) réseau  hex agonal) réseau  hex agonal) dans la nuanc e de référence dans la nuanc e de référence dans la nuanc e de référence dans la nuanc e de référence [DUM AY 2008] [DUM AY 2008] [DUM AY 2008] [DUM AY 2008]  (d)  (d)  (d)  (d) \nTransformation martensitique ε intense  apr Transformation martensitique ε intense  apr Transformation martensitique ε intense  apr Transformation martensitique ε intense  après déform ation d’un alliage à mémoire de forme Fe30Mn6Si, ob tenue ès déformation d’un alliage à mémoire de forme Fe30 Mn6Si, obtenue ès déformation d’un alliage à mémoire de forme Fe30 Mn6Si, obtenue ès déformation d’un alliage à mémoire de forme Fe30 Mn6Si, obtenue \ndans le  cadre du stage de  master de Roney Lino. dans le  cadre du stage de  master de Roney Lino. dans le  cadre du stage de  master de Roney Lino. dans le  cadre du stage de  master de Roney Lino.    \n 94  3. 3. 3. 3.  Comportement des aciers Dual Comportement des aciers Dual Comportement des aciers Dual Comportement des aciers Dual- - --Phase Phase Phase Phase    ; ; ; ; des des des des composite  composite  composite  composites s ss modèle  modèle  modèle  modèle    ?? ??    \n \n« En scien ce, la phrase la plus excitante que l’on peut entendre, celle qui annonce des \nnouvelles découvertes, ce n’est pas Eureka mais c’e st « drôle ». » \nIsaac Asim ov \n \n3.1. 3.1. 3.1. 3.1.  Introduction Introduction Introduction Introduction    \n3.1.1. 3.1.1. 3.1.1. 3.1.1.  Con texte historique et technique Con texte historique et technique Con texte historique et technique Con texte historique et technique    \n \nCon trairem ent aux aciers TWI P FeMnC, les aciers Dua l-Phase (DP) sont maintenant \nindustrialisés depuis les années 1990-1995 et large ment utilisés dans le domaine de la \nconstruction autom obile. Ils constituen t la m ajorit é des aciers TH R dit modernes ou de \nprem ière g énération. Ces aciers présentent générale men t des niveaux de résistan ces \nmécaniques compris e ntre 600 et 1200 MPa (cf. Figur e 1 page 12), et présen tent d’excelle n tes \nqualités de mise en  forme (emboutissabilité ou plia bilité). \n \nLe terme Dual-Phase signifie que ces aciers son t co nstitués principalement de deux phases, de \nla ferrite de structure cubique cen trée (CC) de m or phologie généralement polygonale et de la \nmartensite α’ CC ou de st ructure tetragonale centré e (TC), riche en carbon e, en fraction très \nvariable. De façon  caricaturale, la ferrite est con sidérée alors comm e une phase m olle, et la \nmartensite comme une phase dure. C’est cet aspect c om posite qui leur con fère leurs \nexcellentes propriétés mécaniques, compromis entre résistance et form abilité. \n \nLa Figure 59 montre des exe mples de pièces automobi les réalisées grâce à des aciers DP de \ndifférents grades et épaisseurs : pare-ch oc en acie r DP 1180 et pied milieu en acier DP 980 \nfine épaisseur pour des applications anti-intrusion  et une roue de style en DP600 forte \népaisseur. \n \n(a) (b) \n (c) \nFigure Figure Figure Figure 59 59 59 59    : Exemple d’application des aciers Dual : Exemple d’application des aciers Dual : Exemple d’application des aciers Dual : Exemple d’application des aciers Dual- - --Pha s Pha s Pha s Pha se dans le domain e de la c onstruction automobile  (a ) e dans le domain e de la c onstruction automobile (a )  e dans le domain e de la c onstruction automobile (a )  e dans le domain e de la c onstruction automobile (a )  \nPoutre de pare Poutre de pare Poutre de pare Poutre de pare- - --choc  en DP1180, (a) Pied choc  en DP1180, (a) Pied choc  en DP1180, (a) Pied choc  en DP1180, (a) Pied- - --mili eu en DP980 fine épaisseur et (c) roue de styl e en DP600 forte mili eu en DP980 fine épaisseur et (c) roue de style  en DP600 forte mili eu en DP980 fine épaisseur et (c) roue de style  en DP600 forte mili eu en DP980 fine épaisseur et (c) roue de style  en DP600 forte \népaisseur. Les va leurs épaisseur. Les va leurs épaisseur. Les va leurs épaisseur. Les va leurs indiquées indiquées indiquées indiquées se rapportent traditionnelle ment au nivea u de résista nce méca niq se rapportent traditionnelle ment au niveau de résis ta nce méca niq se rapportent traditionnelle ment au niveau de résis ta nce méca niq se rapportent traditionnelle ment au niveau de résis ta nce méca nique ue ue ue    \n(www.arcelor mittal.com) (www.arcelor mittal.com) (www.arcelor mittal.com) (www.arcelor mittal.com). . ..    \n 95   \nLes aciers DP industriels contiennent généralement d’autres phases comm e la perlite ou la \nbain ite. La quantité de ces secon des ph ases peut ex céder la fraction  de martensite mais ces \naciers restent par abus de langage commercialisés s ous le nom de DP. Dans la suite de ce \nmém oire, on ne s’intéressa bie n en tendu qu’aux  micr ostructures réellemen t Ferrite-\nMartensite. \n \nCes microstructures particulières peuvent être obte nues dans les plupart des aciers FeC \nhypoeutectoides, à partir soit d’une décomposition de l’austénite (germination de la ferrite au \nrefroidissem ent et transformation martensitique de l’austén ite résiduelle sous Ms) comme \ndans le cas d’un procédé de lamin age à chaud, soit dans le cas d’un recuit intercritique comme \ndans le cas d’un procédé de production de bandes de  fin e épaisseur laminées à froid. \n \nLa Figure 60 mon tre par exemple dans des diagrammes  des transformations te m pérature-\ntemps des chemins de production d’aciers DP en prod uits lam inés à chaud ou à froid (recuit \nde recristallisation). \n \n(a) (b) \nFig ure Fig ure Fig ure Fig ure 60 60 60 60    : Traiteme nts thermomécaniques typiques pour : Traiteme nts thermomécaniques typiques pour : Traiteme nts thermomécaniques typiques pour : Traiteme nts thermomécaniques typiques pour la pro duction industrielle la production industrielle la production industrielle la production industrielle    (en rouge) (en rouge) (en rouge) (en rouge) d’aciers Dual d’aciers Dual d’aciers Dual d’aciers Dual- - --\nPhase (a) s Phase (a) s Phase (a) s Phase (a) sous  forme ous  forme ous  forme ous  forme la minée la minée la minée la minée à  chaud et (b) sous forme  à  chaud et (b) sous forme  à  chaud et (b) sous forme  à  chaud et (b) sous forme la minée la minée la minée la minée à  froid rec uits représentés dans des diagra mmes  à  froid rec uits représentés dans des diagrammes  à  froid rec uits représentés dans des diagrammes  à  froid rec uits représentés dans des diagrammes \ntemps temps temps temps – – –– température. Sont indiqués les domaines typiques d’appa rition des différents produ its de décompositi on  température. Sont indiqués les domaines typiques d ’appa rition des différents produ its de décompositio n  température. Sont indiqués les domaines typiques d ’appa rition des différents produ its de décompositio n  température. Sont indiqués les domaines typiques d ’appa rition des différents produ its de décompositio n \nde l’austénite au refroidiss ement. de l’austénite au refroidiss ement. de l’austénite au refroidiss ement. de l’austénite au refroidiss ement.    \n \nDan s la suite de cet exposé, nous ne reviendrons qu e peu sur les procédés d’obtention des \nmicrostructures et les processus de morphogén èse qu i font encore l’objet de recherches dans \nle domaine des transformations de phase. Le lecteur  pourra se reporter par ex emple aux \ntravaux récents de thèse dédiée à ces questions [KR EBS 2009][V IARDIN 2008]. On se limitera \naussi à l’étude du comporteme nt mécanique des acier s Ferrite-Marten site « fraîche », bien que \nles aciers DP dit « revenu » aient aussi une grande  importance pratique et industrielle \n[PUSHKAREVA 2009][BOUAZIZ 2011_3]. \n \n 96  3.1.2. 3.1.2. 3.1.2. 3.1.2.  Bibliograph ie succincte et p Bibliograph ie succincte et p Bibliograph ie succincte et p Bibliograph ie succincte et problématique roblématique roblématique roblématique     \n \n3.1.2.1.  Lien  microstructure-propriétés  \n \nComme la m ontre la Figure 3 page 12, les aciers DP présenten t un comportement m écanique \nen traction caractérisé par une faible limite d’éla sticité sans palier de Lüders Re et un taux \nd’écrouissage initial important. Cette combinaison de propriétés leur perm et d’atteindre des \nallon gemen ts répartis élevés, et donc de grandes ré sistances mécaniques Rm. Leur rapport \nRe/Rm est généralement proche de 0.5. \n \nCette limite d’élasticité basse est due à la faible  contrainte d’écouleme nt de la ferrite 14  et se \nvisualise très bien  en l’absence de palier de Lüder s. L’absen ce de palier est surprenante en \ncomparaison des aciers H SLA (H igh Strength Low Allo yed) et en raison de leur teneur en  \ncarbone élevé. On l’attribue à la présence d’une de nsité importante de dislocations mobiles à \nl’interface m artensite/ferrite. Celle-ci s’explique  par la transformation martensitique à basse \ntempérature, par n ature displacive accompagn é d’une  importante dilatation volumique. La \nFigure 61 m ontre par e xem ple un e micrographie MET p rès de l’interface F/M d’un acier DP à \nl’état non vieilli. \n \n(a) \n (b) \nFigure Figure Figure Figure 61 61 61 61    : Microgra phie M ET des structures de dislocations dans la ferrite autour des ilots de martensite d’un  : Microgra phie M ET des structures de dislocations d ans la ferrite autour des ilots de martensite d’un : Microgra phie M ET des structures de dislocations d ans la ferrite autour des ilots de martensite d’un : Microgra phie M ET des structures de dislocations d ans la ferrite autour des ilots de martensite d’un \nacier D ual acier D ual acier D ual acier D ual- - --Pha se (a) dans l’état initial brut de fabrication et (b) après 10% de déformation , d’après Pha se (a) dans l’état initial brut de fabrication e t (b) après 10% de déformation , d’après Pha se (a) dans l’état initial brut de fabrication e t (b) après 10% de déformation , d’après Pha se (a) dans l’état initial brut de fabrication e t (b) après 10% de déformation , d’après \n[TIM [TIM [TIM [TIMOKH INA 2007] OKH INA 2007] OKH INA 2007] OKH INA 2007]    –– –– M = M artensite et PF = Ferrite.  M = M artensite et PF = Ferrite.  M = M artensite et PF = Ferrite.  M = M artensite et PF = Ferrite.    \n \nPar analogie avec les com posites à matrice métalliq ue, les taux d’écrouissage des aciers DP \ns’expliquent par la présence de phases dures dans u ne m atrice molle (l’effet DP). On  considère \nsouvent, à tort, que la m artensite est suffisan te «  dure » pour rester élastique au cours du \nchargement. Nous montrerons que cette idée est géné ralemen t fausse et que le com portem ent \nplastique de la martensite joue un grand rôle sur l e comportement m acroscopique des aciers \n                                                 \n14  Certains a uteurs attribuent la  faible limite d’éla sticité des ac iers DP aux contraintes internes dues  à c ette \ntransformation. Toute fois, ces contraintes internes  n’ont ja mais pu être mises en évidence par des ani sotropies de \ncomportement traction/compression. Il est par contr e, plus raisonnable de penser comme nous le  montrer ons \nque la  contrainte d’écoulement de la fer rite augmen te avec ces densités importantes de dislocations.  97  DP. De cette vision com posite, découle naturellemen t que l’écrouissage des aciers DP est \ncontrôlé au premier ordre par la fraction  de marten site. La Figure 62 montre par exemple les \ncourbes de traction  de 3 aciers DP avec des fractio ns variables de martensite [LIEDEL 2002].  \n \n(a) \n (b) \nFigure Figure Figure Figure 62 62 62 62    : (a ) Courbes de traction rationnelles d’un : (a ) Courbes de traction rationnelles d’un : (a ) Courbes de traction rationnelles d’un : (a ) Courbes de traction rationnelles d’une nua nce  Fe0.09C1.4Mn0.1Si0.7Cr avec  différentes fraction e nua nce Fe0.09C1.4Mn0.1Si0.7Cr avec  différentes fr action e nua nce Fe0.09C1.4Mn0.1Si0.7Cr avec  différentes fr action e nua nce Fe0.09C1.4Mn0.1Si0.7Cr avec  différentes fr actions s ss    \nde martensite , d’après de martensite , d’après de martensite , d’après de martensite , d’après [LIEDL  [LIEDL  [LIEDL  [LIEDL 2002 2002 2002 2002] ] ]] (b) Courbes de traction ration  (b) Courbes de traction ration  (b) Courbes de traction ration  (b) Courbes de traction rationnelles d’une nua nce Fe0.2C0.47Mn0.3Si0.1Cr nelles d’une nua nce Fe0.2C0.47Mn0.3Si0.1Cr nelles d’une nua nce Fe0.2C0.47Mn0.3Si0.1Cr nelles d’une nua nce Fe0.2C0.47Mn0.3Si0.1Cr \navec différe ntes frac tion avec différe ntes frac tion avec différe ntes frac tion avec différe ntes frac tions s ss de martensite et taille  de martensite et taille  de martensite et taille  de martensite et tailles s ss de str ucture  de str ucture  de str ucture  de str ucture – – ––  grains et ilots de martensite, d’après  grains et ilots de martensite, d’après  grains et ilots de martensite, d’après  grains et ilots de martensite, d’après [ [ [[JIANG JIANG JIANG JIANG \n1992]. 1992]. 1992]. 1992].     \n \nComme toutes les structures composites, les aciers DP vont présenter un composante \nd’écrouissage cinématique forte  due aux différence s de con traintes d’écoule ment entre les \nphases [ASARO 1975] [ALLAIN 2012] . Ce simple effet  composite est en fait très ex acer bé par \nle contraste très important de comportement entre l es phases, qui vont générer des \nincompatibilités importantes de déform ation entre l es phases [BOUAZIZ \n2005][KADKHODAPOUR 2011][KIM  2012]. Les gradients d e déformation résultant sont \nprin cipalem e nt localisés dans la ferrite, la phase molle et sont révélés par l’observation  en \nMET de fortes den sités de Dislocations Géométriquem e nt Nécessaires (DGN ) autour des ilots \nde marten site, comme le m ontre la Figure 61 [TIMOKH IN A 2007][KORZEKWA \n1984][GAR DEY 2005]. \n \nL’ex istence de ces gradien ts localisés rend le comp ortement des aciers DP très sensible aux \neffets de taille, en particulier à la taille de gra in  de la ferrite r ecristallisé et à la taille des i lots \nmartensitiques, bien qu’il soit difficile de découp ler ces effets expérim entalem ent des effets de \nfraction. La Figure 61 montre les courbes de tracti on d’aciers DP avec des fractions de phases \nconstantes (33% ou 40%) mais des tailles variables [JIANG 1992]. Compte tenu des tailles de \ngrain s ferritiques, les limites d’élasticité sont g lobalement con stantes mais les taux \nd’écrouissage évolue nt significativement. Nous revi en drons dans la suite de cette exposé sur \nles m ultiples sources de contraintes in ternes à l’é ch elle de ces m icrostructures (joints de \ngrain s ferritiques, interfaces ferrite/martensite, gradien ts de déformation dan s la martensite). \n \nParadoxale m ent, le comportement e n traction  des aci ers DP dépend peu par contre de la \nmorphologie (form e des structures martensitiques) e t de la topologie (distribution spatiale, \nconn ectivité) des phases. La Figure 63 m ontre les c ourbes de deux aciers élaborés à partir de la \nmêm e composition  mais ayant subis des traitem e nts t h ermiques de recuit différents.   98   \n(a) \n (b) (c) \nFigure Figure Figure Figure 63 63 63 63    : M icrographie optique après attaque : M icrographie optique après attaque : M icrographie optique après attaque : M icrographie optique après attaque N N NNital de deux  a ciers Dual ital de deux  a ciers Dual ital de deux  a ciers Dual ital de deux  a ciers Dual- - --Phase Fe0.09C 2Mn présentant la Phase Fe0.09C 2Mn présentant la Phase Fe0.09C 2Mn présentant la Phase Fe0.09C 2Mn présentant la \nmême fraction de ma rtensite (28% et 27% respectivem ent) mais des même fraction de ma rtensite (28% et 27% respectivem ent) mais des même fraction de ma rtensite (28% et 27% respectivem ent) mais des même fraction de ma rtensite (28% et 27% respectivem ent) mais des morphologies et topologies  morphologies et topologies  morphologies et topologies  morphologies et topologies différentes (a) différentes (a) différentes (a) différentes (a) \ndistribution isotrope distribution isotrope distribution isotrope distribution isotrope d’ilots d’ilots d’ilots d’ilots et (b) et (b) et (b) et (b) structur e structur e structur e structur e en « en « en « en «     ba ba ba bandes ndes ndes ndes    ». ». ». ». (c) Courbes de trac tion conventionn elles  (c) Courbes de trac tion conventionn elles  (c) Courbes de trac tion conventionn elles  (c) Courbes de trac tion conventionn elles \ncorrespondantes correspondantes correspondantes correspondantes (données de L. Durrenberger / M. Go uné  (données de L. Durrenberger / M. Gouné  (données de L. Durrenberger / M. Gouné  (données de L. Durrenberger / M. Gouné d’AM  d’AM  d’AM  d’AM ) ) )).. ..    \n \nLes morphologies et topologies résultantes de la ma rtensite sont très différentes mais les \nfractions de phase sensible men t équivalentes. La di fférence de comportement est faible \nmalgré une morphologie en bande ou très dispersé de  la martensite. Par con tre, cette \nmorphologie joue un rôle majeur dan s les processus d’endommagement comm e nous le \nverrons au chapitre 3.4 page 137. \n \n3.1.2.2.  Modélisation du comportement  \n \nDan s la littérature scientifique, on retrouve trois  gran des familles de modélisation \nmicromécanique du comportement des aciers Dual-phas e. \n• Les approch es monophasées à champs moye n s : le comp ortement du com posite est \nréduit au comportemen t de la ferrite durcie par la présence de martensite \n[SUGIMOTO 1997][BOUAZIZ 2001_2][JIANG 1992][MA 1989 ]. Ces modèles sont \nbasés sur l’h ypothèse implicite que la m artensite r este élastique lors du chargement et \nperm ettent de ren dre compte des effets de taille et  de fraction mais son t valables \nunique men t aux faibles fraction s de m artensite (pas  de percolation du réseau de \nmartensite, teneur en carbone local dans la marten s ite élevée). Ces modèles sont \ngénéralement développés pour décrire et prédire le comportement des matériaux lors \nde sollicitations sim ples (traction  monotone ou ess ai alterné). Elles peuven t toutefois \nêtre étendues en plasticité polycr istalline pour dé crire des com portem ents sous \nsollicitation s complexes (chargem ent biaxiés, chang ements de trajets) com me nous \nl’avons réalisé dans le cadre de la thèse de S. Dil lie n. \n• les approch es biph asées à champs moye n s : le compor tement de l’acier DP résulte de la \nmoyenne pondérée du comporteme nt des deux ph ases Fe rrite et Martensite. Ils \nreposent souvent dans la littérature sur des m odèle s d’homogénéisation spécifiques : \nsimples à un paramètre [LIAN 1991], type auto-coh ér ent avec des comportements \nlocaux elasto-viscoplastiques [BERBENNI 2004], type  Mori-Tanaka avec des \ncomportem ents elasto-plastiques [LANI 2007][ JACQUE S 2007], type Tomota avec des  99  comportem ents elasto-plastiques empiriques [TOMOTA 1992]. Ces approches \nsouffrent généralem ent de deux défauts majeurs : le s lois de comportements locales de \nchacune des phases sont empiriques, voire réductric es pour la martensite, et les effets \ninduits par les gradients de déformation s aux inter faces ferrite/martensite n e sont pas \npris en compte. Ces approches ne sont sensibles ni aux  effets de taille, ni aux effets de \nteneur en carbone dans la marten site. Delincé et al.  [DELINCE 2010] ont bien proposé \nune approche se nsible aux effets de tailles de la f errite, mais le com portem ent de la \nmartensite est décrit très sommaireme nt (elasto-par faite ment plastique). \n• les approch es locales par Elémen ts Finis (EF) sur V olume Elémentaire Représentatif \n(VER) de la microstructure : Ces modélisations réal isées sur des microstructures \nmodèles [H UPPER 1999] [AL-ABBASI 2007] [PRAHL 2007]  [LIEDL 2002] \n[UTHA ISANGSUK 2011] ou num érisées [LI 1990_2][LI 19 90_1] [PAUL 2012] [CH OI \n2009] [KUMAR 2007] [SODJIT 2012][PAUL 2013] permett en t de décrire les effets de \nfraction, de dispersion et de morph ologie de la mar tensite. Leurs principales \nlimitations réside n t dans l’absen ce com plète d’effe ts de taille (pas d’influence des \ngradients de déformation). Kadkhodapour et al.  ont cependan t proposé une approche \npar EF originale à coques conce n triques autour d’il ots de martensite pour reproduire \nun gradien t de comporteme nt dans la ferrite et ains i un effet de taille d’ilots \n[KADKHODAPOUR 2011]. Cette dém arche est très simila ire à celle envisagée par \nPipard et al [PIPARD 2009] pour décrire les effets de tailles dans la ferrite. Ces \nmodèles par EF nécessitent en outre des mises en do n nées généralement longues, en \nparticulier les modèles à m icrostructure numérisée.  \nDe façon surpren ante on retrouve peu dans la littér ature de modèle de plasticité \npolycristalline dédié à l’étude des DP [DILLIEN  2010_2] . \n \n3.1.3. 3.1.3. 3.1.3. 3.1.3.  Mise en perspective des travaux Mise en perspective des travaux Mise en perspective des travaux Mise en perspective des travaux     \n \nCon nues de longue date, ces microstructures ont fai t l’objet de très nombreuses études et de \nrevues dans le domaine académique dans les années 1 970-1980 soutenues par des intérêts \nsidérurgiques et industriels importants dans le cad re de la construction automobile, le \ntran sport ou le domaine des tôles fortes. L’engouem ent pour ces structures s’est ensuite \nattén ué au profit dans les années 1990 de l’étude d es aciers ferritiques TRIP. Cela étant, ce \ndern ier effet micromécan ique dans les aciers ferrit iques au carbone peut être vu comme un \neffet DP dynamique [PRAHL 2007][PERLADE 2003][LANI 2007]. \n \nMes travaux sur le comportemen t des aciers Dual-Pha se ont débuté en 2005 pour le groupe \nArcelormittal et on t principalem e nt été de nature t héorique (modélisation microm écanique). \nCes travaux intern es, et à ce jour partiellemen t co nfide ntiels [ALLAIN 2005] , ont été mené \nprin cipalem e nt en collaboration avec NSC et on t fai t l’objet de peu de publications.   100  D’un  point de vue collaboration  scie ntifique, j’ai co-encadré la thèse de S. Dillien au KUL \n(directeurs de thèse : P. VanHoutte, M. Seefeld) su r cette th ématique, intitulée : « Bridging \nthe Physics-Engineering Gap in Dual Ph ase Steel For m ability »,  \nJ’ai aussi participé directement à la thèse de B. K rebs au LETAM (directeurs de thèse : A. \nHazotte, L. Germain) en réalisant des calculs EF av ec l’étudiant, et plus indirectem ent à celle \nd’I. Pushkareva au LSGS (directeur de thèse : A. Re djaimia) sur leur endommagem e nt et à la \nthèse de A. Aouafi au LPMTM (directrices de thèse : M. Gaspérini, S. Bouvier) sur la \nmodélisation des effets de taille dans les aciers f erritiques. \n \nMes travaux ont eu plusieurs finalités dans ce doma ine, que n ous détaillerons dans la suite de \nce m émoire : \n \n• Axe 1 : Ex tension  d’un modèle monophasé en plastici té polycristalline pour des \napplications en rhéologie appliquée (prévision des surfaces de charges sous \nsollicitation s complexes). Ces travaux ont perm is d e fournir aux « m écaniciens » une \nextension du modèle de Teodosiu-Hu pour les aciers DP. Ce ch apitre sera aussi \nl’occasion d’introduire les différents mécanismes d ’écrouissages de ces aciers composite \n(de n ature isotrope et cinématique) constituant la base du modèle biphasé. \n \n• Axe 2 : Développement d’un modèle biphasé quasi-ana lytique pour des utilisations en \n« alloy-design » métallurgique. Ce modèle à vocatio n d’être « générique » en  décrivant \nnon seulem ent le comportemen t des aciers DP, mais a ussi les aciers IF (Interstitial \nFree) 100% ferritiques et les aciers 100% martensit iques, en tenant compte des effets \nde fraction (fraction et teneur en carbone dan s la martensite) et de tailles (taille de \ngrain s ferritiques, des ilots martensitiques). Cet outil a été développé à destination des \n« métallurgistes » et est basé sur une com préhensio n fine des mécanismes de plasticité \nde ces deux  phases et de leurs interactions. Après un e présentation  détaillée de ces \nmécanismes, des ex emples concrets d’utilisation  de l’approche seront discutés. \n \n• Axe 3 : Approfondisseme nt de n os connaissances dans  le domaine de \nl’endommagement des structures Dual-Phase. J’ai pil oté depuis 2008 un  projet de \ndéveloppem ent d’une ch aine de simulation numérique par EF du comportement \nmécanique et d’en domm agemen t des aciers DP (projet de recherche Arcelormittal \nDPN DI 187). Le synoptique de cette démarche est rep résentée sur la Figure 64. Elle \ns’étend de la num érisation de microstructures aux c alculs sur VER sen sibles aux \ngradients de déformation  et intégrant des mécanism e s simulant l’endommagem ent \n(élém ents cohésifs). Les objectifs scientifiques de  cette démarche son t de quantifier et \nde m ieux comprendre les mécanismes d’endommagement de ces structures DP et à \nterm e leurs actionneurs métallurgiques. Ceci n écess ite non seulem e nt une \nconn aissance fine et juste des états de contraintes  et déformation s des différen tes \nphases au cours de la déformation macroscopique m ai s aussi de pouvoir gérer la  101  compétition  entre les m écanism es d’en domm agemen t (r elaxation). Cette troisième \npartie sera consacrée à l’étude grâce à ces outils numériques d’un cas particulier, celui \nde l’impact néfaste des structures morph ologiques d ites « en ban des » sur \nl’endommagement des aciers DP. Les perspectives d’a mélioration de la dé marche \nseront enfin  exposées.  \n \n \nFigure Figure Figure Figure 64 64 64 64    : Synoptique de la dé marche de m : Synoptique de la dé marche de m : Synoptique de la dé marche de m : Synoptique de la dé marche de m odélisation par EF sur VER du  comportement et de odélisation par EF sur VER du  comportement et de odélisation par EF sur VER du  comportement et de odélisation par EF sur VER du  comportement et de \nl’endommagement des  aciers D ual l’endommagement des  aciers D ual l’endommagement des  aciers D ual l’endommagement des  aciers D ual- - --Phase Phase Phase Phase (Projet  (Projet  (Projet  (Projet de de de de Know ledge Building Know ledge Building Know ledge Building Know ledge Building AM AM AM AM )) )).. ..    \n \n3.2. 3.2. 3.2. 3.2.  Extension  d’un  modèle monophasé en plasticité polyc ristalline pour des Extension  d’un  modèle monophasé en plasticité polyc ristalline pour des Extension  d’un  modèle monophasé en plasticité polyc ristalline pour des Extension  d’un  modèle monophasé en plasticité polyc ristalline pour des \napplications en rhéologie appliquée applications en rhéologie appliquée applications en rhéologie appliquée applications en rhéologie appliquée    \n \nDan s cette première partie, nous reviendrons sur le s modèles mon ophasés analytiques du \ncomportem ent des aciers DP. Ces travaux issus de la  métallurgie physique mettent \ngénéralement en valeur les effets de fraction et de  tailles. Ils sont don c une base indispensable \npour notre plateforme générique de m odélisation dét aillée au chapitre 3.3. Malgré leurs \ndomaines d’utilisation limités en term es de fractio n de martensite, il s’agit d’outils de \nmodélisation d’un grand intérêt pratique. Ils sont sim ples m ais permetten t quand même la \ndescription de com posantes d’écrouissage ciné m atiqu es pour des calculs de m ise e n  forme par \nexem ple. Nous montreron s en outre qu’ils peuve nt êt re étendus dan s le cadre de la plasticité \npolycristalline et donc fournir un cadre th éorique pour des applications e n r héologie \nappliquée. \n \n \n  102  3.2.1. 3.2.1. 3.2.1. 3.2.1.  Com posante isotrope Com posante isotrope Com posante isotrope Com posante isotrope de l’effet DP  de l’effet DP  de l’effet DP  de l’effet DP    \n \nCette courte présentation bibliographique ne vise p as l’exhaustivité m ais permet d’évoquer les \ndifférents m écanism es à la base de l’écrouissage de  ces aciers et la nature de cet écrouissage. \n \nEn 2001, Bouaziz et al.  [BOUAZIZ 2001_2][PERLADE 2003] proposent de décrir e que la \ncontrainte d’écoulement ΣDP  d’un acier DP correspon d à celle de la ferrite pou r de faibles \nfraction de marten site F m. En suivant un  modèle de type Taylor (écrouissage par la forêt des \ndislocations), il vient alors : \n \n ()()DP DSS 111α\n0α\npα DP\npDPρ ρ αMμb σ εσ EΣ + +==  (47) \n \navec ρ DSS  la de nsité de dislocations statistiquement stockée s dans la ferrite et dont l’évolution \nva suivre une loi de type Mecking-Kocks-Estrin et ρ DP  une densité de dislocations \nadditionnelles dues à la présence de martensite. E pDP  est la déformation plastique \nmacroscopique et σ α et ε pα la contrainte et déformation plastique dans la fer rite. σ 0 est une \nfriction de réseau qui n e dépend que de la composit ion chimique. α est facteur de \ndurcissemen t de la forêt, M le facteur de Taylor, µ  le module d’élasticité en cisaille ment et b 111  \nle vecteur de Burgers des dislocations parfaites da ns la ferrite. Le terme ρ DP  représente les \ndensités de Dislocations Géométriquement Nécessaire s (DGN) réparties un iformément dans \nles grains ferritiques. Paradoxale ment, il correspo n d à un mécanisme de durcissement \nisotrope et est estimé comme les DGN au sens d’Ashb y en présence d’une particule dure de \ntaille d m [JIANG 1992][BOUAZIZ 2001][BOUAZIZ 2005] : \n \n p\nα\nm 111DP ε\ndb8MFρ=  (48) \n \nIl vient alors : \n \n ( )() ( ) () ( )\nrε E r.M. exp1.\ndF8.\nf.dε E f.M. exp1.b αMμ σ EΣ0DP\np\nmm 0DP\np\n111α\n0DP\npDP+ −−++ −−+=  (49) \n \nLe premier terme ρ SSD  est bien en tendu se nsible à la taille de grain fer ritique d et dépend d’un \nprocessus de restauration  dynamique paramétré par f  (le terme d’écrouissage latent est \nconsidéré n égligeable k = 0 en  référence à l’équati on (16)). Une pré-déformation ε 0 est \nintroduite pour reproduire les effets de la transfo rmation  martensitique qui génère de \nnom breuses dislocations m obiles (cf. Figure 61) dan s la ferrite. Le terme relevant des DGN  au \nsens d’Ashby est considéré comm e saturant, avec un paramètre r.  \n  103  Nous avons étudié par EBSD ces gradients de déforma tion dans la ferrite dans le cadre de la \nthèse de S. Dillien [D ILL I EN 2010_1][DILL IEN  2010_2] . L’acier étudié est un DP 600 (lam iné \nà ch aud) avec une structure très isotrope. La Figur e 65 montre des observation s en MEB \nEBSD avan t et après 15% de déformation en  traction.  La marten site est mal indexée et \napparait e n  noir. Le code couleur représente les an gles de désorien tations par rapport à la \nmoyenne du grain et les grains son t défin is grâce à  la méth ode MMC (Maximized \nMisorientation Contrast). Les désorientations sont mesurées le long des lign es et les résultats \nsont représentés sur les figures jointes. La figure  (a) montre que mê m e à l’état brut de trem pe, \nla m icrostructure contien t déjà de nombreuses dislo cations géom étriquement n écessaires \ndistordant la maille. Grâce aux gradie nts de désori entations, n ous avons évalué cette densité à \n6 10 13  m -2, correspondan t à une contribution  au durcisse m ent de 20 MPa environ. Ces \ndensités augmentent considérable ment après déformat ion (1.5 10 14  m -2 mesuré sur la figure \n(d)) et apportent une con tribution significative de  50-100 MPa environ. Cette contribution \nn’est pas suffisan te pour expliquer totalement l’ef fet DP, c’est pourquoi l’ajout d’une \ncontribution d’origine cin é matique est in dispensabl e pour une description cohérente. \n \n \n(a)    (b) \n \n(c)    (d) \nFig ure Fig ure Fig ure Fig ure 65 65 65 65    : : : : Cartogra phie EBSD d’une structure  DP600 (laminé à  chaud) (a) à l’état initial et (c) après 15% d e Cartogra phie EBSD d’une structure  DP600 (laminé à  c haud) (a) à l’état initial et (c) après 15% d e Cartogra phie EBSD d’une structure  DP600 (laminé à  c haud) (a) à l’état initial et (c) après 15% d e Cartogra phie EBSD d’une structure  DP600 (laminé à  c haud) (a) à l’état initial et (c) après 15% d e \ndéformation. Le code couleur représente la désorien tation par ra pport à la  moyenn e du gra in considéré selon la déformation. Le code couleur représente la désorien tation par ra pport à la  moyenn e du gra in considéré selon la déformation. Le code couleur représente la désorien tation par ra pport à la  moyenn e du gra in considéré selon la déformation. Le code couleur représente la désorien tation par ra pport à la  moyenn e du gra in considéré selon la \nméthode du MM C (Ma x imized M isorie ntation Contra méthode du MM C (Ma x imized M isorie ntation Contra méthode du MM C (Ma x imized M isorie ntation Contra méthode du MM C (Ma x imized M isorie ntation Contrast). st). st). st). Les zones non index ées (en noir) correspondent  à des  Les zones non index ées (en noir) correspondent à d es  Les zones non index ées (en noir) correspondent à d es  Les zones non index ées (en noir) correspondent à d es \nzones martensitiques. Les gradients de désorientati ons le long de  certaines lignes sont repr ésentés e n  (b) et ( zones martensitiques. Les gradients de désorientati ons le long de  certaines lignes sont repr ésentés e n  (b) et ( zones martensitiques. Les gradients de désorientati ons le long de  certaines lignes sont repr ésentés e n  (b) et ( zones martensitiques. Les gradients de désorientati ons le long de  certaines lignes sont repr ésentés e n  (b) et (d d dd) ) ) ) \nrespective ment respective ment respective ment respective ment    [DILLIEN 2010_2] [DILLIEN 2010_2] [DILLIEN 2010_2] [DILLIEN 2010_2]. . ..    \n \n 104  3.2.2. 3.2.2. 3.2.2. 3.2.2.  Introduction d’une composante ciné matique Introduction d’une composante ciné matique Introduction d’une composante ciné matique Introduction d’une composante ciné matique     \n \nLe m odèle de nature pureme nt isotrope présenté ci-d essus a été complété par Bouaziz en 2005 \n[BOUAZIZ 2005] grâce à un term e de nature cinématiq ue. Cette contribution correspond à la \ncontrainte en retour exercée par les particules dur es sur la matrice en  présence d’une \nincompatibilité de déform ation. Brown , Stobbs et At kinson [BROWN 1971][ATKINSON \n1974] exprim e nt cette con trainte supplém entaire com m e : \n \n * p\nα m DPX εF μ ξ 2 σ=  (50) \n \navec ξ un facteur d’accommodation proche de ½ et ε αp*  la déformation plastique n on relaxée \ndans la matrice. Ce terme est similaire à celui asso cié au durcissemen t par le maclage \nmécanique dans une approche type composite (cf. 2.3. 5 page 52). Bouaziz propose alors \nd’écrire :  \n \n ( )() ( ) () ( )\nX0p\nDP\nmm 0p\nDP\n111 0p\nDP σ\nrε E r.M. exp1.\ndF8.\nf.dε E f.M. exp1.b αMμ σ E Σ ++ −−++ −−+=α\n (51) \n ()\nr'r'.M.E exp1\ndF.b Mμ σp\nDP\nmm\n111 X−−=  (52) \n \nσX correspond à un écrouissage cinématique dérivé de l’équation (50) et saturant avec un seul \nparamètre r’. \n \nCette équation en fait très similaire dan s sa struc ture et ses dépendances au modèle proposé \npar Sugimoto et al.  en 1997 [SUGIMOTO 1997].  \n \n ( )* p\nα\nmp\nα m 111\nferrite matrice 0 μFε\nν1 55ν7ζ\n2dεFbμ σ σσ\n−−+ + +=  (53) \n \nLa seule différence réside dans le découplage entre  la contrain te d’écoule ment de la ferrite et \nl’écrouissage dû aux dislocations d’Ashby (troisièm e terme de l’égalité). \n \nDan s cette famille d’approche m onophasée, la modéli sation du com portem ent DP revien t à \najouter au comportement de la ferrite deux contribu tions : \n• la première est de nature cinématique et a pour ori gine les DGN restant au contact des \nilots de martensite. Ces dislocations in duise nt un champ de contrainte en retour à \nlongue distance, qui est directem ent proportionnel à la fraction de m artensite et peut \ndépendre de la taille des ilots.  105  • La seconde est de nature isotrope, et correspond à des DGN é m ises à grandes distan ces \ndes particules dans les grains de ferrite (de l’ord re du micron toutefois). Les cham ps \nqui en résultent sont non polarisés à grande distan ce et ces dislocations con tribuen t à \nl’écrouissage selon un mécanisme de type forêt. Ce terme est proportionnel à la racine \ncarrée de la fraction de m artensite et n on nul en d ébut de déform ation (effet de la \ntran sformation displacive avec ch angem ent de volume  à basse température). \nCes deux contributions vont naturelle m ent saturer a u cours de la déformation grâce à des \nmécanismes de relaxation , comm e l’ém ission de boucl es secondaires par ex e mple [D ILL I EN \n2010_2] . \n \n3.2.3. 3.2.3. 3.2.3. 3.2.3.  Extension polycristalline Extension polycristalline Extension polycristalline Extension polycristalline à destin ation des rhéolog ues  à destin ation des rhéologues  à destin ation des rhéologues  à destin ation des rhéologues    \n \nL’approche décrite ci-dessus au chapitre 3.2.2 a ét é étendue dans le cadre de la thèse de S. \nDillie n à une modélisation d’agrégat polycristallin  en  3D. La finalité de ce travail était la \nprédiction des surfaces de charges sous sollicitati ons complexes (trajets non  monotones) des \naciers DP, sur des bases de métallurgie physique. \n \nEn pratique, les travaux ont visé l’exte n sion aux a ciers DP du modèle micro-macro pour le \ncomportement de la ferrite développé par Peeters et al.  [PETEERS 2001] 15 . Ce m odèle très \nélaboré gère de multiples effets comme les change me nts de trajets durs grâce à une \ndescription des cellules de dislocations, mais malh eureusemen t pas les effets de taille de grains. \nLes cissions locales τ sc pour chaque système de glisse ment s ont été modifi é de la façon \nsuivante : \n \n ()( )BS\ns mD\nmBP\ns minit 0 c\ns τF τF τF1 ττ τ ++ −++=  (54) \n \navec τ init  un e cission due au dislocations initialement présent es dans la ferrite autour des ilots \nde martensite (zon e 3 sur la Figure 66) τBP  le terme de Bart Peeters (zone 1), τ D la cission due \nau DGN d’Ashby (zone 3) et τ BS  une contrainte de « back-stress » modélisée de manièr e \nempirique par une loi de type Voce (zon e 2). Cette é quation correspond à une extension des \néquations (52) ou (53) pour chaque système de glisse ment.  \n                                                 \n15  Ce modèle, rappelons-le, est la version issue de l a métallurgie physique du modèle de  rhéologie célèb re de \nTeodosiu-Hu.  106  \n \nFigure Figure Figure Figure 66 66 66 66    : Représentation schématique des mécanismes  : Représentation schématique des mécanismes  : Représentation schématique des mécanismes  : Représentation schématique des mécanismes  de durc issement considér és de durcissement considér és de durcissement considér és de durcissement considér és et de leur localisation dans  et de leur localisation dans  et de leur localisation dans  et de leur localisation dans \nune microstructure DP modèle. 1 = S une microstructure DP modèle. 1 = S une microstructure DP modèle. 1 = S une microstructure DP modèle. 1 = Stru tru tru tructure de  déforma tion en cellule cture de  déforma tion en cellule cture de  déforma tion en cellule cture de  déforma tion en cellule dans la ferrite (go uvernée par les loi s de  dans la ferrite (gouvernée par les loi s de  dans la ferrite (gouvernée par les loi s de  dans la ferrite (gouvernée par les loi s de \nPeteers Peteers Peteers Peteers et al. et al. et al. et al. )) )), 2 = Contrainte de back , 2 = Contrainte de back , 2 = Contrainte de back , 2 = Contrainte de back- - --stress du e a ux  ilots de ma rtensite, 3 = Nuag e de d islocations autour des stress du e a ux  ilots de ma rtensite, 3 = Nuag e de di slocations autour des stress du e a ux  ilots de ma rtensite, 3 = Nuag e de di slocations autour des stress du e a ux  ilots de ma rtensite, 3 = Nuag e de di slocations autour des \ngra ins de martensite gra ins de martensite gra ins de martensite gra ins de martensite    [DILLIEN 2010_2] [DILLIEN 2010_2] [DILLIEN 2010_2] [DILLIEN 2010_2] .. ..    \n \nLa Figure 67 mon tre les prédictions du modèle, une fois ajusté, du comportement lors de \nchan geme n ts de trajet durs (cross) ou alternés (Bau schinger) d’un  acier DP 600. Com me \natten du on notera que le changement de trajet dur n ’induit pas de durcisse ment alors que \nl’acier présente un fort effet Bauschinger. Ce r ésu ltat est à comparer à celui de la Figure 35 \npage 65. \n \n \nFig ure Fig ure Fig ure Fig ure 67 67 67 67    : Prédiction du comport : Prédiction du comport : Prédiction du comport : Prédiction du comportement par le modèle selon de s change ments de trajets typique ement par le modèle selon de s change ments de trajet s typique ement par le modèle selon de s change ments de trajet s typique ement par le modèle selon de s change ments de trajet s typiques s s s [DILLIEN [DILLIEN [DILLIEN [DILLIEN \n2010_2] 2010_2] 2010_2] 2010_2] .. ..     \n \nCe m odèle 3D a donc un e grande importance pratique pour définir la forme spécifique des \nsurfaces de charges de ces aciers. Toutefois, il so uffre encore de lacunes importantes pour \n 107  décrire tenir compte des effets de fraction ou de t aille, inhérente au modèle de Peeters et al. , \ny compris pour décrire le comporteme nt de la ferrit e. \n \n3.3. 3.3. 3.3. 3.3.  Plateforme de modéli sation  « Plateforme de modéli sation  « Plateforme de modéli sation  « Plateforme de modéli sation  «    générique générique générique générique    » des aciers Ferrite » des aciers Ferrite » des aciers Ferrite » des aciers Ferrite- - --Martensite Martensite Martensite Martensite    \n \nDepuis 2005, avec l’aide de mon collègue Olivier Bo uaziz, nous développons une \nmodélisation à ch amps m oyens du com portem ent des ac iers DP « générique », c'est-à-dire \nsensibles aux effets de tailles, de fraction et de composition , pour tous les aciers Ferrite-\nMartensite, des aciers IF aux aciers m artensitiques . Elle fait suite aux  développe ments \nanalytiques monophasés de mon collègue en 2001 et 2 005 rappelés au chapitre précédent.  \n \nCe n ouveau développement répond non  seulem e nt à l’a ttente des « m écaniciens », c'est-à-dire \nqu’il produit des lois de comporteme nt faisant la d istin ction en tre les écrouissages isotropes et \ncinématiques sur des bases m icrostructurales. Il a aussi pour vocation d’aider les \nmétallurgistes dan s des phases de conception produi t générique, en  étant un outil « d’alloy \ndesign » ou d’ingénierie des microstructures. En  ef fet, il outrepasse le domaine de validité des \nmodèles monophasés (restreint aux faibles fractions  de martensite) et est applicable à tous les \naciers Ferrito-Martensitique. \n \nCette modélisation  repose sur un e approche bi-phasé e à champs moye ns, c'est-à-dire que l’on \nconsidère séparém ent le comportement de la ferrite et de la marten site. Par con tre, \ncontrairement aux  travaux antérieurs de la littérat ure, nous avons fait l’effort d’introduire des \nlois à base microstructurale pour ces deux  constitu ants et pris le parti d’hypothèses \nd’homogénéisation  plus simples. Cette nouvelle ph il osoph ie permet de considérer des \nfractions de martensite élevées en tenan t compte de  la plasticité de cette phase, m ais pas des \ngradients de déformation aux interfaces ferrite/mar tensite. Nous avons donc introduit a \nposteriori des term es de couplage originaux entre l es phases pour décrire les dislocations \nd’Ashby.  \n \nLe m odèle est suffisamment simple pour fonctionner sur un simple tableur mais permet de \ntraiter de cas d’application s comple xes comme n ous le montrerons dans une dernière partie. \n \nDan s la suite de cet exposé, les indices ou exposan ts α et m  renverront respectivement aux \nterm es de la ferrite et de la m artensite, les lettr es capitales (Σ ou E par exemple) aux \ncomportem ents macroscopiques alors que R et X indiq ueron t des composantes isotropes ou \ncinématiques (à toutes les échelles). \n \n \n  108  3.3.1. 3.3.1. 3.3.1. 3.3.1.  CC CCom portem ent de la ferrite: Effet de la taille de g rain om portem ent de la ferrite: Effet de la taille de gr ain om portem ent de la ferrite: Effet de la taille de gr ain om portem ent de la ferrite: Effet de la taille de gr ain    \n \nLe m odèle de comportement de la ferrite a été dével oppé dans le cadre de la thèse d’A. Aoufi \n[BOUAZIZ 2008_2]  et est très similaire au modèle de description du comportement de \nl’austé nite dans le cas d’un  effet TWIP (cf. chapit re 2.3.4 page 49). Ce cas est cependant plus \nsimple car le libre parcours moyen des dislocations  reste constant (taille de grain).  \n \nOn suppose que la contrainte d’écoule m ent d’un acie r ferritique se décompose de la manière \nsuivante : \n \n ()α\nXα\nRα\n0 p σ σ σ εσ ++=αα (55) \n \navec  \n• σ0 un e contrainte de friction du au durcissement en s olution  solide et qui ne dépend \nque de la composition chimique : \n \n 80%Mo 60%Cr 80%Si 33%Mn 52 σα\n0 +++ +=  (56) \n \n• σR un terme de durcissement isotropes de type Taylor (durcissement de la forêt)  \n• σX un terme de durcissement cinématique. \n• εp est la déformation plastique (fonction  de la défor mation totale et de la déformation \nélastique ε el ) \n \n α\nelα α\np εεε−=  (57) \n \nLe terme de durcisseme n t isotrope s’écrit classique ment avec la densité de dislocations \nstatistique ment stockées (DSS) ρ α \n \n ()α 111α\npα\nR ρ αMμb εσ=  (58) \n \nPar contre, l’évolution de cette densité ne suit pl us une loi de type Mecking-Kocks-Estrin \n(MKE) simplifiée mais un e loi d’évolution inspirée de Sinclair et al.  [S INCLA IR 2006], déjà \nutilisée pour la modélisation des aciers TWIP. Cett e approche a l’avantage de pouvoir décrire \nles effets d e taille de grains sur un domaine beauc oup plus étendu que l’approche MKE \nconve ntionnelle, en particulier les tailles de grai ns de l’ordre du micron souve n t observés \ndans les aciers DP. \n  109  L’évolution de la densité de DSS avec la déform atio n plastique résulte de la compétition en tre \nun processus d’accumulation (sur les joints de grai ns et par écrouissage latent) et de \nrestauration  dynam ique. \n \n \n\n\n\n\n− +\n\n\n\n=α α α α\nα 111α\n0α\nα\npαρ f ρkdbnn- 1\nM.dεdρ (59) \n \navec d α la taille de grain ferritique, k α et f α deux constantes. n α et n 0 représentent le nom bre \nmoyen et m aximum de dislocations stockés sur les jo in ts de grains respectiveme nt (DGN). Ce \nterm e correspond donc à un effet d’écrantage des jo ints de grains par la présence de DGN. Ce \nflux de DGN  aux joints arrivant par bande de glisse men t s’écrit empirique ment : \n \n \n\n\n=α\n0α\n111α\nα\npα\nnn- 1\nbλ\ndεdn (60) \n \navec λ α l’espace ment moyen e ntre les bandes de glisse m ent arrivant sur les joints de grains.  \n \nLes DGN stockées sur les obstacles forts (seulement  les joints de grains dan s ce cas, \ncontrairement aux  aciers TWIP) génèrent un champ de  contraintes à longue distan ce dan s la \nmicrostructure, contribution additive à la contrain te d’écoulem e nt isotrope : \n \n α\nα111 α\nX n\ndbMμσ=  (61) \n \nCe terme est très sensible à la taille de grains (e n fait, à la densité d’interface). Dan s le cadre \nde la thèse, nous avons ajusté ce modèle sur les co urbes de traction  d’une très large gamme \nd’aciers IF (sans in terstitiels) avec des tailles d e grains très variables de 3.5 µm  à 20 µm, \nrepr ésentan t bien les microstructures ferritiques a tten dus des aciers DP [BOUAZIZ 2008_2] . \nCe m odèle permet de prédire en outre avec précision  les niveaux de contrain tes internes dans \ncertains de ces aciers (mesurés par essais Bauschin ger), comm e le montre la Figure 68 et aux \nréserves près discutés précédemm ent. \n  110  (a) \n (b) \nFig ure Fig ure Fig ure Fig ure 68 68 68 68    : (a) courbes de trac tion rationnelles ex pér : (a) courbes de trac tion rationnelles ex pér : (a) courbes de trac tion rationnelles ex pér : (a) courbes de trac tion rationnelles ex périmental es et simulées d’aciers ferritiques avec différente s imentales et simulées d’aciers ferritiques avec dif férentes imentales et simulées d’aciers ferritiques avec dif férentes imentales et simulées d’aciers ferritiques avec dif férentes \ntaille taille taille tailles s ss de grains (b) Composantes d’écrouissage c inématiq ue ex périmentales et simulées pou r 3 aciers  de grains (b) Composantes d’écrouissage c inématiqu e ex périmentales et simulées pou r 3 aciers  de grains (b) Composantes d’écrouissage c inématiqu e ex périmentales et simulées pou r 3 aciers  de grains (b) Composantes d’écrouissage c inématiqu e ex périmentales et simulées pou r 3 aciers ferritiq ues ferritiques ferritiques ferritiques    \navec différe ntes taille de grains en fonction de la  déformation équivalente (mesur e avec différe ntes taille de grains en fonction de la  déformation équivalente (mesur e avec différe ntes taille de grains en fonction de la  déformation équivalente (mesur e avec différe ntes taille de grains en fonction de la  déformation équivalente (mesur es par essais Bausch inger en s par essais Bauschinger en s par essais Bauschinger en s par essais Bauschinger en \nc isaillement) c isaillement) c isaillement) c isaillement) [BOUAZIZ 2008_2] [BOUAZIZ 2008_2] [BOUAZIZ 2008_2] [BOUAZIZ 2008_2] .. ..    \n \nLes valeurs de con stantes physiques et paramètres a justables du modèle sont repris dans le \ntableau suivant : \n \n    Valeur Valeur Valeur Valeur    signification physique signification physique signification physique signification physique    \nα 0.38 durcissement de la forêt \nM 2.77 m odule de cisaille men t \nµ 80 GPa facteur de Taylor \nb111 2.5 10 -10  m  vecteur de Burgers \nλα/b 111 90 espacem ent moyen en tre ban des de glissement \nnα0 6.2 n ombre maximum de dislocations stockées aux joints de \ngrains \nkα 0.007 écrouissage latent \nfα 1.3 restauration dynamique \nTableau Tableau Tableau Tableau 3 3 33    : Valeurs des paramètres du modèle de comportement  de  la ferrite : Valeurs des paramètres du modèle de comportement de  la ferrite : Valeurs des paramètres du modèle de comportement de  la ferrite : Valeurs des paramètres du modèle de comportement de  la ferrite [BOUAZIZ 2008_2] [BOUAZIZ 2008_2] [BOUAZIZ 2008_2] [BOUAZIZ 2008_2]     \n \nPour traiter de la transition élasto-plastique dans  cette phase, nous avons retenu un schéma \nexplicite et simplifié pour éviter le recours à des  boucles de con vergen ce (résolution du \nsystème d’équation  e n déformation totale) : \n \n Si Yσεα\n0≥α, alors Yσε εα\n0 α α\np−=  sinon 0 dεα\np=and αα\nelε ε=  (62) \net \n()()( )α αα ααYε,εσ min εσp =  \n \nAvec Y le m odule d’Youn g. \nCe choix ne se posera pas dans le cas du modèle de martensite car la transition entre élasticité \net plasticité est décrite naturelle m ent et explicit ement. \n  111  3.3.2. 3.3.2. 3.3.2. 3.3.2.  Com p Com p Com p Com portem ent de la martensite: Effet de la teneur e n carbone ortem ent de la martensite: Effet de la teneur en ca rbone ortem ent de la martensite: Effet de la teneur en ca rbone ortem ent de la martensite: Effet de la teneur en ca rbone    \n \n3.3.2.1.  Bilan  de la littérature  \n \nLa principale difficulté concernant ce constituant est qu’il n’existait pas dan s la littérature de \nloi de comportement à base microstructurale convain cante! \n \nC’est pourtant une phase connue de longue date et b eaucoup de travaux lui ont été consacrés \npour comprendre l’origin e de ses fortes contraintes  d’écoulem ent et de sa dureté. Elle a fait \nl’objet de nombreuses revues dans la littérature sc ie ntifique (transform ation de phase, \ncristallographie, résistance…) [KRAUSS 1999][OLSON 1992]. Ces travaux  montrent que le \nparamètre « microstructural » clef qui semble contr ôler le com portem ent de la martensite est \nsa teneur en  carbone. La dépendance de la dureté de  la marten site en fonction de la teneur en \ncarbone est certainemen t la relation microstructure /propriétés la mieux acceptée de la \nlittérature, mais la moins bien e x pliquée ! Les cor rélations avec d’autres défauts \nmicrostructuraux et leurs dime n sions associées (tai lle de lattes, de paquets, ancien joint de \ngrain  austénitique) et les limites d’élasticité de ces aciers sont largement discutées sans \nconsensus.  \nLes mécanismes d’écrouissage et leur modélisation o nt par contre tout bonnement été \nnégligés, ce qui est très surprenan t compte tenu de  l’im portan ce industrielle croissante de ces \naciers 16 . On retrouve bien dans la littérature quelques mod èles ph énoménologiques \npolynomiaux ou simplifiés de comporteme n t, mais pas  de modèle réellement à base \nmicrostructurale. \n \nLa première version de notre plateforme de modélisa tion  était basée sur le modèle de \ncomportem ent de marten site inspiré de celui des aci ers ferritiques à grains ultrafins [COBO \n2008]. Il était ajusté empiriquem ent et souffraient  de graves lacun es (description  des taux \nd’écrouissage initiaux en particulier). Depuis 2009 , nous avons décidé de revisiter les \nmécanismes fondame ntaux du comportement de la marte n site, en  collaboration  avec M. \nTakahashi de NSC, F. Danoix du GPM et dan s le cadre  de la thèse de G. Badinier à l’UBC \n(directeur de thèse : C. Sinclair). Ces travaux  ont  donné lieu à des publication s récen tes \n[BADINIER 2011][ALLAIN 2013] , en particulier notre nouveau m odèle de com portem e nt \npour la martensite [ALLAIN 2012] . \n \n                                                 \n16  Certaines universités japonaises restent to utefois  très actives dans ce do maine [NAKASHIM A 2007] \n[LHUISSIER 2011] [NAM BU 2009] avec  des tra vaux  de n ature principale ment expérimentale (M ET, RX) sur la  \nplasticité des aciers ma rtensitiq ues.  112  Dan s la suite de cet exposé, nous reviendrons sur l es caractéristiques principales de ce modèle, \nses perform ances et ses limites. I l ouvre un angle d’in vestigation original et des perspectives \nnouvelles pour l’analyse de ces structures martensi tiques. \n \nCette nouvelle approche a été suggérée par l’analys e préalable de nombreuses courbes de \ntraction et d’essais mécaniques d’aciers marten siti ques. Ces constatations se limitent au cas \ndes aciers faible m ent alliés présentant une structu re principale men t e n lattes [SHERMAN \n1983]. Les constatations principales sont les suiva ntes : \n• Tous les aciers m artensitiques, à l’état brut de tr em pe, présentent une limite de \nmicroplasticité faible et constante, de l’ordre de 400 MPa (en traction  ou com pression). \nCeci suggère que la martensite contie nt une fractio n significative de « zones molles » \nayan t une contrain te d’écouleme n t faible et indépen dante de la teneur en carbone. \n• Tous les aciers présenten t un fort taux d’écrouissa ge initial après cette limite de \nmicroplasticité, croissante avec la teneur e n carbo ne de l’acier considéré. Ce résultat \nexplique pourquoi les lim ites d’élasticité conventi onn elle augmentent dans ces aciers \navec la teneur en carbone, malgré une limite de mic roplasticité constante. \n• Cette dépen dance du taux  d’écrouissage à la ten eur en carbon e est m aintenue après de \ngran des déformations. \n• L’écrouissage des aciers marten sitique est principa lement d’origine cin ématique, \ncomme le suggère les rares essais Bausch inger dispo nibles dans la littérature. \n \nCe résultat suggère que le comportemen t des aciers martensitiques ne peut s’interpréter par \nun simple m écanisme de stockage des dislocations. L es taux d’écrouissage sont en effet très \nlargement supérieurs à Y /100 – Y le m odule d’Young [SEEGER 1963][KEH 1963][ANSELL \n1963]. Le m écanism e de déplétion (e xhaustion) e n di slocations mobiles suggéré par [NAMBU \n2009][ [NAKASHI MA 2007] ne peut n on plus expliquer les forts niveaux de contrain tes \ninternes. Par contre, toutes ces caractéristiques s uggèrent que la martensite ne doit pas être \nconsidérée comme une phase homogène, mais plutôt co mme un  composite h étérogène \ncontinu, constitué d’un mélan ge de phases « molles » et « dures », dont les fractions \nrespectives varient avec la teneur en carbone. Cett e idée a été suggérée récemmen t aussi par \nHutchinson  et al.  [HUTCHINSON 2011]. Les phases « m olles » contrôlen t le seuil de \nmicroplasticité alors que les phases « dures » en r estant élastique assurent le taux d’écrouissage \nmacroscopique (un e fraction élevée du module d’Youn g). Le comportemen t de la marten site \npeut donc se résum er à un e transition élasto-plasti que très étendue.  \n \n3.3.2.2.  Approche Composite Continue  \n \nLe comportement de la m artensite a donc été décrit grâce à un  modèle de Masing généralisé à \nun continuum de phases élastiques-parfaite m ent plas tiques en in teraction. Ch acune des \nphases du composite est décrite par son  module d’Yo ung Y (le mêm e pour toutes les phases)  113  et sa contrainte d’écoulem ent respective. Le compos ite est donc défini de m anière univoque \npar sa fonction con tinue de distribution  de den sité  de probabilité f(σ) de trouver une ph ase \nayan t une contrain te d’écoulement σ. Cette distribu tion est appelée dans la suite spectre de \ncontrainte et représente finale m ent la distribution  des contraintes d’écoulemen t dans la \nmicrostructure. On  définit alors sa fonction cumulé e F(σ) com me : \n \n ( ) ( ) ζ d ζ f σFσ\n∫\n∞ −=  (63) \n \nLa Figure 69 montre un e x emple de spectre de contra in tes et de sa fon ction cumulée. \n \n \nFigure Figure Figure Figure 69 69 69 69    : E xemple d e « : E xemple d e « : E xemple d e « : E xemple d e «    spectre de contraintes spectre de contraintes spectre de contraintes spectre de contraintes     » f et sa » f et sa » f et sa » f et sa fonction cumulée  F.  fonction cumulée  F.  fonction cumulée  F.  fonction cumulée  F.    \n \nPour des raisons de cohérence, ces fonctions doive n t respecter les con ditions suivan tes : \n \n ()\n( ) ( ) 1ζ d ζ f F0 σ f σ,\n= =∞+≥∀\n∫∞ +\n∞ − (64) \n \nSi on  appelle σ min  la contrainte d’écoulement de la phase la plus mol le de ce composite, on a \nalors  \n \n ()()0 σFet 0 σ f , σσmin = = ≤∀  (65) \n \nElle correspond à la phase du composite qui va cont rôler le seuil de microplasticité \nmacroscopique (environ 400-500 MPa). Une fois cette  distribution connue, on peut estimer la  114  contrainte macroscopique σ m de ce com posite en fonction de la déformation macr oscopique \nappliquée ε m : \n \n ( ) ( ) dσσ f σ σdσσ f σL\nmin Lσ\nσ σLm∫ ∫+∞ \n+ =  (66) \n \nLe premier terme de l’in tégrale correspond aux phas es déjà plastifiées et le second à celles \nrestant élastiques sous une contrainte limite σ L qui dépend du chargement et de l’interaction \nentre les ph ases. Dans le cas général, cette équati on ne peut être résolue an alytiquement. Par \ncontre, il est possible d’estimer le module tangent  de cette loi, indépendamm ent de la \nfonction F : \n \n ( ) ( ) ( )\n( )( ) ( ) ( )\n( )( )( )L\nLmmm m\nL L LL\nσL L L L L Lm\nσF1\nβσF\nY11\nddσYβ1βd dσσF1 dσ σF1dσdσσ f dσσσ f dσσσ f dσ\nL\n−\n+=⇒++×−=×−=\n\n\n\n\n\n\n\n+ −+ = ∫∞ +\nεεavec \nYβ1βε σσm m\nL\n++=  (67) \n \nAvec β, un paramètre con stant contrôlan t l’interact ion  entre les phases.  \n \n \nε εσ σβmm\n−−−=  (68) \n \nCe paramètre permet de gérer des scénarios de local isation caricaturaux allan t de l’iso-\ndéformation  entre toutes les phases du composite (β  = +∞ - hypoth èse de localisation type \nTaylor) à l’iso-con trainte (β = 0 - hypothèse de lo calisation type Sachs). Pour tous autres \ndétails calculatoires on se reportera à [ALLAIN 2012] .  \n \nDan s les condition s d’iso-déformation (β >> Y), l’é quation précédente se simplifie : \n \n ( )( )m\nmm\nYεF1 Y\ndεdσ−=  (69) \n \nLe taux d’écrouissage devient une sim ple fonction d es ph ases restant élastiques dans le \ncomposite et du m odule d’Youn g. Dans ce cas, la for mulation est trop rig ide pour ren dre \ncompte du comportement expérimental (contraintes in ternes trop élevées donc surestimation  115  du taux d’écrouissage). Par contre, cette formule s imple est largemen t utilisé dans le cas de la \nmodélisation de la fatigue [POLAK 1982][HOLSTE 1980 ]. Dans la suite, on retiendra une \nvaleur de β intermédiaire de Y/4 (50GPa).  \n \nOn notera que la for mulation retenue pour le module  tangent reste aussi valable dans le \ndomaine élastique. Si σm ≤ σ min  alors σ L ≤ σ min  et donc F(σ L) = 0. On retrouve alors le module \nd’Young Y : \n \n Y\ndεdσ\nmm\n= (70) \n \nCe modèle a ensuite été appliqué pour décrire le co mportement mécanique de 6 aciers \nmartensitiques. La fonction de distribution cumulée  F est décrite par une loi type Avrami à \ntrois param ètres : \n \n ( ) ( )( )\n\n\n\n\n\n\n\n +−−−= = +<n\n0friction min\nfriction minσσ σ σexp1 σF sinon0 σF alors σ σσ si  (71) \n \navec σ friction  la contrainte de friction due aux élém e nts en solu tion solide. \n \n σ friction  = 60 + 33 %Mn  + 81 %Si + 48 % Cr + 48 % Mo (72) \n \nIl ressort de la procédure d’ajustement que les par amètres n et σ min  peuve nt prendre des \nvaleurs con stantes (n = 1.82 et σ min  = 300 MPa respectivemen t) pour tous les aciers alo rs que \nσ0 caractéristique de la largeur de la distribution d e contrainte d’écoule men t dans le \ncomposite varie significations avec la ten eur en ca rbon e %C m : \n \n σ 0 = 645 + 5053 × (%C m)1.34  (73)  \n \nOn relèvera que l’ordre de grandeur de la somme σ min  + σ fric tion  correspond bien au 400-500 \nMPa attendu pour le seuil de m icroplasticité et pou rrait correspondre avec la contrainte \nd’écoulement d’un e structure ferritiques haute m ent déformée sans carbone en solution solide. \nOn rapporte dans la littérature des valeurs de dens ités de dislocation s de l’ordre de 1 à 5  10 14  \nm-2 [SHIBATA 2009] donc des valeurs de σ min  comprises entre 240 et 540 MPa environ, \nvaleurs très cohérentes à celles déduites de cette analyse. \n \nLa Figure 70 montre que le modèle, une fois ajusté,  permet de reproduire de façon précise les \ncourbes de comportement des 6 aciers con sidérés, ma is aussi la form e de leur taux \nd’écrouissage. \n  116  \n(a) \n (b) \nFigure  Figure  Figure  Figure  70 70 70 70    : (a) C ourbes de traction et (b) taux d’éc rouissag e  en fonc tion de la contrainte de 6 aciers : (a) C ourbes de traction et (b) taux d’éc rouissage  en fonc tion de la contrainte de 6 aciers : (a) C ourbes de traction et (b) taux d’éc rouissage  en fonc tion de la contrainte de 6 aciers : (a) C ourbes de traction et (b) taux d’éc rouissage  en fonc tion de la contrainte de 6 aciers \nmartensitiques avec des ten eurs en c arbone nominale s différentes (ex p érience et modèle  ajusté). La flè che martensitiques avec des ten eurs en c arbone nominale s différentes (ex p érience et modèle  ajusté). La flè che martensitiques avec des ten eurs en c arbone nominale s différentes (ex p érience et modèle  ajusté). La flè che martensitiques avec des ten eurs en c arbone nominale s différentes (ex p érience et modèle  ajusté). La flè che \nin in in indique le  seuil de microplasticité c onsta dique le  seuil de microplasticité c onsta dique le  seuil de microplasticité c onsta dique le  seuil de microplasticité c onstant pour tou s les acier s étudiés nt pour tous les acier s étudiés nt pour tous les acier s étudiés nt pour tous les acier s étudiés    [ALLAI N 20 [ALLAI N 20 [ALLAI N 20 [ALLAI N 201 1 112] 2] 2] 2].. ..    \n \nComme suggéré par Asaro [ASARO 1975], ce type d’app roche micromécanique offre \nd’intéressan tes possibilités pour évaluer des contr aintes internes (contribution cinétique de \ntype I selon  la classification d’Asaro) et de prédi ction du com portem ent sur des ch angements \nde trajet alternés. La Figure 71 m ontre par exemple  les prédictions du modèle lors d’un essai \nde Bauschin ger san s aucun  param ètre d’ajuste m ent su pplémen taire 17 . Peu de modèles peuvent \nse prévaloir d’une aussi bonne description (à la ch arge et la décharge) lors de chargem ent \nalternée san s param ètres d’ajustem ents supplé m entai res.  \n \n                                                 \n17  Comme le laisse entrevoir la  Figure 71, le modèle s urévalue sen siblement les contraintes  internes. On attribue \ncette trop grande rigidité à l’absence  d’écrouissag e des phase s du co mposite (comportement parfaitemen t \nplastique). Le  taux d’écrouissage est donc uniqu eme nt du ressort des pha ses non plastifiées, dont la p roportion \nest a lors surévaluée. C ette sur évaluation conduit n écessairement à  des contraintes internes plus impor tantes. \nUne façon de  résoudr e ce problème a  été proposée ré c emment par G. Badinier, qui consiste à introduire  un \nécrouissage linéaire da ns chacune des phases (modul e k’). La loi précédente devient alors : \n \n( )\n( ) ( )LL\nmm\nσF1Yk'βk'Y1σF 1Yk'ββk'\ndεdσ\n−−+−++\n=  \nQuand toutes les phases du composite ont plastifié,  F(σ L) \u0001 ∞ alors k'\ndεdσlimmm\nE=\n∞ →  117  \n \nFigure Figure Figure Figure 71 71 71 71    : Courbes de comport : Courbes de comport : Courbes de comport : Courbes de comportement ex périmental et simulé d’ un ac ier martensitique  lors d’un  trajet ement ex périmental et simulé d’un ac ier martensitiq ue  lors d’un  trajet ement ex périmental et simulé d’un ac ier martensitiq ue  lors d’un  trajet ement ex périmental et simulé d’un ac ier martensitiq ue  lors d’un  trajet \nBauschinger (chargement alter Bauschinger (chargement alter Bauschinger (chargement alter Bauschinger (chargement alterné). La phase de charg ement alternée (« né). La phase de chargement alternée (« né). La phase de chargement alternée (« né). La phase de chargement alternée («     ba ckw ard loading ba ckw ard loading ba ckw ard loading ba ckw ard loading    ») est ») est ») est ») est modélisée modélisée modélisée modélisée sa ns  sa ns  sa ns  sa ns \nparamètre supplémentaire par ra pport au chargemen paramètre supplémentaire par ra pport au chargemen paramètre supplémentaire par ra pport au chargemen paramètre supplémentaire par ra pport au chargement initial (« t initial (« t initial (« t initial («     forward loading forward loading forward loading forward loading    ») ») ») »)    [ALLAIN 2012] [ALLAIN 2012] [ALLAIN 2012] [ALLAIN 2012]. . ..    \n \nCette possibilité du modèle composite présente en s oi un gran d intérêt pour l’étude des aciers \nmartensitiques. Cependan t, cette description précis e des chemins alternés est trop complexe \npour être in tégrée dans le modèle biphasé, en parti culier car la description de la ferrite n ’est \npas aussi détaillée. En con séque nce, la composante d’écrouissage cinématique de la m artensite \na été évaluée de façon différente. En inspirant de nos travaux  préalables sur l’écrouissage des \naciers ferrite-perlitiques, on se propose d’écrire :  \n \n ( ) σ d Σ σσ f\n21σ\nminσm\nX∫+∞ \n− =  (74) \n \nOn relie l’écrouissage cinétique à la moye nne des d ifférences de comportement des phases par \nrapport à la moye n ne. De même que l’équation précéd ente, cette composante ciné m atique ne \npeut être in tégrée analytiqueme n t. Par contre, sa d érivée s’exprime aussi très simpleme nt : \n \n ()dΣΣF dσm\nX=  (75) \n \nLa Figure 72 montre les évolutions de ce terme σmX en  fonction de la déformation pour les 6 \naciers considérés sur la Figure 70. Les ordres de g randeurs prévus par ce term e sont cohérents \navec les mesures disponibles d’écrouissage ciné m ati que [COBO 2008]. \n  118  \n \nFigure Figure Figure Figure 72 72 72 72    : : : : Composante cinématique  de l’écrouissage d’après l ’é Composante cinématique  de l’écrouissage d’après l’éComposante cinématique  de l’écrouissage d’après l’éComposante cinématique  de l’écrouissage d’après l’é quation (7 quation (7 quation (7 quation (75 5 55) pour les aciers de ) pour les aciers de ) pour les aciers de ) pour les aciers de la  la  la  la Figure Figure Figure Figure 70 70 70 70 .. ..    \n \n3.3.2.3.  Nouvelles perspectives  \n \nCette nouvelle approche du com portem ent de la marte nsite est donc une donnée d’entrée \nidéale pour le modèle biphasé. En effet, elle décri t parfaitement la transition élasto-plastique \ndans la martensite et son très fort taux d’écrouiss age initiale mais aussi perm et d’estimer une \ncomposante cinématique intrinsèque à cette phase. C e modèle est en outre sen sible à la \nteneur en carbone de la m artensite. \n \nPar contre, l’origin e de cette dispersion de con tra intes d’écoulement (spectre de contrain tes) \nn’est pas clairemen t iden tifiée et reliée à des car actéristiques microstructurales (comme les \nlattes, blocs ou paquets ou bien comme les zones na nom aclées [KELLY 1963]).  \n \nNous travaillons donc sur différentes h ypothèses po ur pouvoir valider cette approche. Mon \naxe de travail principal reste l’identification des  éch elles d’hétérogénéités de déformation \nrésultantes. Elle nous permettra d’orienter nos rec herch es soit vers des explications \nstructurales (i.e. en relation avec la microstructu re) ou des explications d’ordre chimique \n(distribution de la teneur en carbone). A cette fin , je coordon ne et participe actuelle ment à 3 \nétudes à différentes échelles (MEB, MET, SAT) sur l e même acier m artensitique modèle de \ncomposition  Fe5Ni0.15C. \n• MEB EBSD et essai de traction in situ  en collaboration avec D. Barbier d’AM et C. \nTasan et S. Zaeffer er du MPI. Les prem iers résultat s m ontren t que la déformation  est \ntrès hétérogène, n on seulement à l’échelle des anci ens grains austénitiques, mais aussi \nà l’échelle des lattes. La Figure 73 mon tre un exe m ple de cliché MEB avan t et après \n4% de déformation  d’un paquet de lattes (mise en év idence par une légère attaque) très \ndéformé. L’analyse EBSD révèle que la désorientatio n évolue peu à l’intérieur des \nlattes mais se conce ntre près des interfaces, comme  observée aussi par [LHUISSI ER \n2011][NAMBU 2009]. Par contre, d’autres paquets son t très peu déformés, comme \natten dus par l’approche composite.  119   \n(a) \n(b) \nFig ure Fig ure Fig ure Fig ure 73 73 73 73    : : : : Microgra phie M EB après a ttaque d’un acier martens iti Microgra phie M EB après a ttaque d’un acier martensit i Microgra phie M EB après a ttaque d’un acier martensit i Microgra phie M EB après a ttaque d’un acier martensit ique (a) dans l’état initial que (a) dans l’état initial que (a) dans l’état initial que (a) dans l’état initial – – –– mise en évidence  mise en évidence  mise en évidence  mise en évidence \nd’un pa quet de lattes et (b) après déforma tion d’un pa quet de lattes et (b) après déforma tion d’un pa quet de lattes et (b) après déforma tion d’un pa quet de lattes et (b) après déforma tion in situ in situ in situ in situ     –– –– mise en évidence de bandes de  déforma tions  mise en évidence de bandes de  déforma tions  mise en évidence de bandes de  déforma tions  mise en évidence de bandes de  déforma tions \n(( ((mic rographies mic rographies mic rographies mic rographies de D. Barbier et  de D. Barbier et  de D. Barbier et  de D. Barbier et C. Tasa n)  C. Tasa n)  C. Tasa n)  C. Tasa n). . ..    \n \n• MET en collaboration avec P. Barges d’AM. Cette étu de a permis de montrer que la \nstructure en  lattes ne con tient qu’une faible fract ion de zones nanomaclées (environ \n15%). Cette fraction est très faible en  comparaison  de la fraction  de ph ases dures \nnécessaire pour expliquer le comporteme nt du com pos ite. L’hypothèse de Kelly \n[KELLY 1963] justifiant la dureté de la martensite sur cette seule fraction  est donc \ninsuffisante. \n \n \nFigure Figure Figure Figure 74 74 74 74    : : : : Micrographie M ET  en Micrographie M ET  en Micrographie M ET  en Micrographie M ET  en champ sombr e champ sombr e champ sombr e champ sombr e mettant en évidence des zones micromac lées dans les  mettant en évidence des zones micromaclées dans le s  mettant en évidence des zones micromaclées dans le s  mettant en évidence des zones micromaclées dans le s lattes de lattes de lattes de lattes de \nma rtensite ( fraction faible) ( ma rtensite ( fraction faible) ( ma rtensite ( fraction faible) ( ma rtensite ( fraction faible) (microgra phies de microgra phies de microgra phies de microgra phies de P. Barges). P. Barges). P. Barges). P. Barges).    \n \n• SAT (Sonde Atom ique Tomographique 3D) en collaborat ion avec F. Danoix du GPM \net M. Goun é de l’ICMCB. 10 pointes ont été analysée s pour donner un aperçu de la \ndistribution statistique du carbon e dans cet acier martensitique aux échelles les plus \n 120  fines. Contrairement aux observations sur des marte nsites vierges [ALLAIN  2013] , les \natom es de carbone ne sont jamais répartis de façon homogène, mais sont \nsystématiquement ségrégés soit sur des défauts plan s (joints de lattes ?), soit sur des \nrubans (dislocations) ou de façon  isotropes (décomp osition spinodale ?). La prochaine \nétape est de pouvoir extraire une pointe directemen t dans un e zone identifiée comme \ndure (non déformée) par les essais de traction in  situ  sous MEB. \n(a) (b) \nFigure Figure Figure Figure 75 75 75 75    : Distribution spatiale des atomes de c arbone obse rvée en sonde atomique tomographique 3D sur deux  : Distribution spatiale des atomes de c arbone obser vée en sonde atomique tomographique 3D sur deux  : Distribution spatiale des atomes de c arbone obser vée en sonde atomique tomographique 3D sur deux  : Distribution spatiale des atomes de c arbone obser vée en sonde atomique tomographique 3D sur deux  \npointes pointes pointes pointes (donn ées  (donn ées  (donn ées  (donn ées de de de de F. Danoix ) F. Danoix ) F. Danoix ) F. Danoix ) (a  (a  (a  (a) ) ) ) 65x 65x 290 nm3 65x 65x 290 nm3 65x 65x 290 nm3 65x 65x 290 nm3    –– –– ségrégation des atomes de carbone sur des défa uts  linéa ires  ségrégation des atomes de carbone sur des défa uts linéa ires  ségrégation des atomes de carbone sur des défa uts linéa ires  ségrégation des atomes de carbone sur des défa uts linéa ires \net plans (b) et plans (b) et plans (b) et plans (b) 34x 34 x 180 nm3 34x 34 x 180 nm3 34x 34 x 180 nm3 34x 34 x 180 nm3    –– –– ségrégation isotrope.  ségrégation isotrope.  ségrégation isotrope.  ségrégation isotrope.    \n \n3.3.3. 3.3.3. 3.3.3. 3.3.3.  Com portem ent du composite et ajustem ent du modèle Com portem ent du composite et ajustem ent du modèle Com portem ent du composite et ajustem ent du modèle Com portem ent du composite et ajustem ent du modèle    \n \n3.3.3.1.  Modélisation des in teractions entre Ferrite et Mart ensite  \n \nDan s la première et seconde partie de cette section , ont été présentés respectivement un \nmodèle de comportemen t de la ferrite sensible à la taille de grain et un m odèle de \ncomportem ent de martensite sen sible à la teneur en carbone. Afin de décrire le composé bi-\nphasé Ferrite-Martensite, il est n écessaire de suiv re un schém a dit d’homogénéisation (celui \nde H ill dan s ce cas) et d’établir des lois d’in tera ctions entre les phases (loi de transition \nd’échelle type IsoW et lois de couplage spécifique) . \n \nLa contrain te d’écoulement macroscopique du composi te Ferrite-Martensite peut s’exprim er \nen fonction  de la contrain te dans chacune des phase s pondérée par leurs fractions relatives. \nDe la même manière, on peut calculer la déform ation  m acroscopique : \n \n ()()()()ααεσF1 εσF EΣmm m\nmDP DP−+ =  (76) \n \n ()αεF1 εF Emm\nmDP−+=  (77) \n \n 121  On admet de manière im plicite par cette form ulation  que le comportemen t de la marten site \net de la ferrite son t les m êmes quelles que soient les échelles considérées. Nous reviendrons \ndans la partie discussion sur la validité de cette approximation  pour la martensite. \n \nNous avon s montré que la transition  élasto-plastiqu e était un processus clef dans le \ncomportem ent de la martensite. Il est donc importan t de distinguer les composantes de \ndéformation s élastique et plastique et ce à toutes les échelles. La déformation macroscopique \npeut donc se décom poser de la façon suivante : \n \n DP\npDP\nDP\npDP\nelDPEYΣE E E +=+=  (78) \n \nEDP p la déformation plastique macroscopique, E DP el  la déform ation élastique, fonction de la \ncontrainte m acroscopique et du m odule d’Youn g comm u n des deux phases. \n \nNous avon s choisi comm e loi de transition d’échelle  l’h ypothèse IsoW, qui s’exprime \nsimple ment sous la forme ; \n \n α α m mdεσ dεσ=  (79) \n \nCette loi permet de gérer de façon continue la tran sition élasto-plastique.  \n \nComme discuté au chapitre 3.3.2.1 page 111, le modè le en l’état ne traduirait l’effet DP que \npar un durcissement cinématique lié à la différence  de comportement entre la ferrite et la \nmartensite. Nous avons donc introduit un terme de c ouplage original pour une approche \nbiph asée qui reproduit un sur-durcissement isotrope  dans la ferrite. Ce terme correspond à \nl’introduction d’une densité de DGN d’Ashby dans l’ équation (58).  \n \n ()DGN\nα 111α\npα\nR ρ ρ αMμb εσ + =  (80) \n \nCette densité suppléme ntaire de dislocations s’expr ime de la façon suivante en s’in spirant de \nl’équation (49) : \n \n ( ) ( ) ( )\nDGN0m α\nDGN\nα 111 mm DGN\nβε εε β exp1\nLb1\nF1Fρ+− −−\n−=  (81) \n \navec L α une longueur caractéristique de la dispersion des dislocations d’Ashby dans la ferrite à \npartir des in terfaces ferrite/martensite et β un pa ramètre gérant un mécanisme de saturation \nde l’effet. La Figure 76 illustre le mécanisme décr it par l’équation (81) et observé  122  expérimentaleme n t sur les Figure 65 et Figure 61. L a distance L α doit être in férieure à la taille \nde grain ferritique et de l’ordre de grandeur de la  taille des ilots martensitiques. \n \n \nFigure Figure Figure Figure 76 76 76 76    : : : : Définition Définition Définition Définition de  de  de  de la  la  la  la    longueur longueur longueur longueur    LL LLαα αα    dans la ferrite (F) autour  d’une pa rticule de mart en site (M ),  de la taille dans la ferrite (F) autour  d’une pa rticule de marte n site (M ),  de la taille dans la ferrite (F) autour  d’une pa rticule de marte n site (M ),  de la taille dans la ferrite (F) autour  d’une pa rticule de marte n site (M ),  de la taille \nde grain ferritique d de grain ferritique d de grain ferritique d de grain ferritique d αα αα et taille d’ilot martensitique d  et taille d’ilot martensitique d  et taille d’ilot martensitique d  et taille d’ilot martensitique d mm mm    [ALLAI N 2 [ALLAI N 2 [ALLAI N 2 [ALLAI N 2005] 005] 005] 005]     \n \nDe m anière analogue à l’équation (49), on définit u ne densité initiale de dislocations due la \ndéformation  induite par la transfor mation martensit iques grâce à la pré-déformation ε 0. Cette \ndéformation  initiale contribue aussi à la densité d e dislocations statistiqueme n t stockées \n(DSS): \n \n ()0\nα 111mα\np α ε\ndbMF0 ερ ==  (82) \n \nDan s la mesure où ces dislocation s interagissen t da ns les grain s ferritiques, l’équation (59) est \nmodifiée et devien t : \n \n \n\n\n\n\n−+ +\n\n\n\n=α αDGN\nα α\nαα\n0α\nα\npαρ f ρ ρkbdnn- 1\nM.dεdρ (83) \n \n3.3.3.2.  Contraintes internes et durcissem ent cinématique  \n \nLes bases de cette n ouvelle approche biphasée étant  posées, le comportemen t des aciers DP va \ns’expliquer globaleme nt selon un processus classiqu e, décrit par de nombreux auteurs \n[TOMOTA 1992][LI 1990_1][LIAN 1991] [HUPPER 1999] e t sché matisé sur la Figure 77. \n FF FF    MM MM    \ndm Lα \ndα ⊥ \n ⊥ \n \n⊥ \n┴ ⊥ \n⊥ \n ⊥ \n ⊥ \n ⊥ \n⊥ \n ⊥ \n ⊥ \n ⊥ \n ⊥ \n ⊥ \n ⊥ \n \n⊥ \n  123  • Stade 1 : Si la contrainte macroscopique appliquée est in férieure à la contrainte \nd’écoulement de la ferrite, les deux phases se comp ortent élastique ment.  \n• Stade 2 : La courbe de com portem ent s’éloigne de la  lin éarité à partir du mome nt où la \nferrite plastifie. Le taux d’écrouissage macroscopi que est très important de l’ordre de \nFxY dans la mesure où la martensite reste élastique . Les gradients de déformations \nentre les phases sont im portan ts, gén érant beaucoup  de DGN aux interfaces. La \ncontribution cinématique du durcisseme nt est donc i mportan te. Cette ph ase \ncaractéristique de l’effet DP est souvent qualifiée  de « continous yielding ».  \n• Stade 3 : Toutefois, ce stade élasto-plastique est de courte durée, car la limite de \nmicroplasticité de la martensite est rapidement att einte. Le durcisse m ent cinématique \ndû à la différence de comportement entre ferrite et  martensite comm e nce à saturer. Le \ntaux  d’écrouissage diminue en conséquence. \n \nLe comportement macroscopique d’un acier DP résulte  donc d’une loi de m élange entre les \nétats de la ferrite (durcit par les DGN d’Ashby) et  de la martensite. La pente de la loi de \nmélange dépend du modèle de transition d’échelle re tenu et peut évoluer au cours de la \ndéformation . \n \n \nFigure Figure Figure Figure 77 77 77 77    :: :: Le  comportement macroscopique d’un acier DP résul te d’une loi de mélange (sy mbolisée  par la ligne  Le  comportement macroscopique d’un acier DP résult e d’une loi de mélange (sy mbolisée  par la ligne  Le  comportement macroscopique d’un acier DP résult e d’une loi de mélange (sy mbolisée  par la ligne  Le  comportement macroscopique d’un acier DP résult e d’une loi de mélange (sy mbolisée  par la ligne \npointillée) entre pointillée) entre pointillée) entre pointillée) entre les états de la ferrite (du  les états de la ferrite (du  les états de la ferrite (du  les états de la ferrite (durcie rcie rcie rcie par les DGN d’Ashby) et de la martensite. Il  s e décompose en  3  par les DGN d’Ashby) et de la martensite. Il  se dé compose en  3  par les DGN d’Ashby) et de la martensite. Il  se dé compose en  3  par les DGN d’Ashby) et de la martensite. Il  se dé compose en  3 \nstades dépendant de l’état de déformation des de ux  phases constitutives. stades dépendant de l’état de déformation des de ux  phases constitutives. stades dépendant de l’état de déformation des de ux  phases constitutives. stades dépendant de l’état de déformation des de ux  phases constitutives. La pente de la loi de mélange  dépend du  La pente de la loi de mélange dépend du  La pente de la loi de mélange dépend du  La pente de la loi de mélange dépend du \nmodèle de tran sition d’échelle retenu et peut évolu er au cours de la  déform modèle de tran sition d’échelle retenu et peut évolu er au cours de la  déform modèle de tran sition d’échelle retenu et peut évolu er au cours de la  déform modèle de tran sition d’échelle retenu et peut évolu er au cours de la  déform a tion. a tion. a tion. a tion.    \n Ferrite + DGN d’Ashby M artensi te \nC omposite \nFerrite-Martensite \n= comportement \nmacroscopique \nSTADE 3 \nLes deux p hases se \ndéforment pl astiquemen t  STADE 1 \nFerrite et \nMartensi te \nsont \nélastiques STADE 2 \nMar tensi te  est \nélastique, Ferrite \nest  plastique \nDéformati on Contrainte \nF errite    124  La contribution cinématique au durcissement X DP  est estimée a posteriori, selon  la \nméth odologie évoquée dans le chapitre précédent (cf . 2.3.5.2 page 55), sans paramètres \nsupplémentaires [ALLAIN  2008] . Elle tient com pte de deux contributions : \n• Les écrouissages cinématiques des phases respective s prises indépendamm ent mais \npondérés par les fractions de phase, \n• Un terme de mélange qui tient compte du gradient de  contraintes entre les deux \nphases. \n \n () ()()()α m\nm mα\nX mm\nX mDP DPσ σF1F σF1 σF E X − −+−+=  (84) \n \n3.3.3.3.  Ajustement du modèle : base de données et paramètre s  \n \nCette démarche de modélisation a été validée grâce à de nombreux résultats obtenus dans le \ncadre de cette étude (aciers DP modèles) et de la l ittérature. Seuls ont été retenus des aciers \nnon microalliés (précipitation), sans bainite et no n revenu (n on auto-reven u dans la mesure \ndu possible). Malh eureusement, la plupart des carac térisations microstructurale disponibles \nétaient lacunaires (fraction de marten site, taille de grain ferritique, taille des ilots de \nmartensite, teneur en carbone de la martensite). Sa uf si mesurées ou reportées par les auteurs \nrespectifs, les fractions de martensite et tailles de grains ferritiques ont été estimées à partir \ndes micrographies disponibles.  \n \nLa teneur en carbone moyenne dans la marten site est  calculée par un simple bilan carbone, \nen supposan t que la ferrite ne con tie nt pas de carb one : \n \n \nmmF%C%C=  (85) \n \nAvec %C la teneur nominale de l’acier. Par soucis d e cohérence, cette valeur est bornée à \n0.6% dans le modèle de comportement de la m artensit e. Cette valeur correspond à la ten eur \nen carbone seuil partir de laquelle on observe expé rim e ntale m ent un e saturation de la dureté \nde la martensite [KRAUSS 1999]. Cette saturation es t attribuée dans la littérature à une \nstabilisation  significative d’austén ite résiduelle dans la marten site. \n \nLe Tableau 4 mon tre les caractéristiques des aciers  DP ayant servies à ajuster le modèle \n(sources, composition chim ique, caractéristiques mi crostructurale mesurées \nexpérimentaleme n t et caractéristiques microstructur ales utilisées pour le calcul). La Figure 78 \nprésente la diversité des couples fraction de marte nsite / teneur n ominale en carbone des \naciers de cette base de données. Ils couvrent un e l arge gamme sauf le coin gauche en haut de \nla figure car ces couples sont inaccessibles du poi nt de vue des transformations de phases.  125   \n  \nTableau Tableau Tableau Tableau 4 4 44    : Ré férence,  composition, don nées microstructurale s disponibles et estimées des a ciers DP décrits dan s : Ré férence,  composition, don nées microstructurales  disponibles et estimées des a ciers DP décrits dans  : Ré férence,  composition, don nées microstructurales  disponibles et estimées des a ciers DP décrits dans  : Ré férence,  composition, don nées microstructurales  disponibles et estimées des a ciers DP décrits dans  \nla la la la littéra ture littéra ture littéra ture littéra ture aya nt servi à l’ajustement des paramètr es du modèle.  aya nt servi à l’ajustement des paramètres du modèl e.  aya nt servi à l’ajustement des paramètres du modèl e.  aya nt servi à l’ajustement des paramètres du modèl e.    \n \n  126  \n \nFigure Figure Figure Figure 78 78 78 78    : : : : Fraction de martensi Fraction de martensi Fraction de martensi Fraction de martensite et teneu r en carbone nominal e des  aciers décrits dans le te et teneu r en carbone nominale des  aciers décrits  dans le te et teneu r en carbone nominale des  aciers décrits  dans le te et teneu r en carbone nominale des  aciers décrits  dans le Tableau Tableau Tableau Tableau 4 4 44.. ..    \n \nDan s la mesure où les lois de comportement de la fe rrite et de la m artensite ont été ajustées \nsur des aciers mon ophasés de manière indépendante, les seuls paramètres d’ajustement du \nmodèle spécifique aux aciers DP sont les param ètres  βD GN , L α et ε 0. Dans une première version \ndu m odèle, nous avions trouvé une dépendance empiri que linéaire entre L α et la taille des \nilots d m. Ce résultat avait l’avantage de donner au m odèle une sen sibilité naturelle à cette \ndimension microstructurale, à l’in star des équation s du monophasé (cf. équations (51) et (52)). \nToutefois, les résultats étaient dispersés et ce te rme devenait n égligeable aux  grandes fractions \nde F m. Com me nous le verrons, il se pose m ême une questi on sur la définition de cette \ndimension quand la structure m artensitique est perc olée, voire englobante par rapport à la \nferrite. \n \nDan s sa version la plus récen te avec le modèle de m artensite composite, n ous avons \ndéterminé que les trois paramètres pouvaient être c hoisis constants pour tous les aciers. Les \nvaleurs retenues sont respectivem ent : \n• βDGN  = 18 (valeur cohérente avec les valeurs généraleme nt utilisées pour l’équation \n(52)) \n• ε0 = 0.05 (valeur cohérente avec les valeurs générale ment utilisées pour l’équation (51)) \n• Lα = 0.8 µm – cette valeur a le même ordre de grandeu r que l’épaisseur des zones dures \n(coques) déterminées par EF par Kadkh odapour et al.  pour décrire les effets de taille \ndans ces aciers [KADKHODAPOUR 2011].  \n \nLa Figure 79 montre une comparaison e n tre les contr aintes e x périmentales et modélisées pour \nles aciers de la base après 4 niveaux de déformatio n (ou moins si l’allongement réparti est \nfaible). Le m odèle permet donc de prédire de manièr e satisfaisante, sans biais, les tendan ces \nde cette base de données, des aciers ferritiques au x aciers martensitiques à haute teneur en  127  carbone. On peut considérer ce résultat comme très satisfaisant compte tenu de la qualité des \ndonn ées d’entrée (disparates et souvent fragme n tair es). \n \n \nFigure Figure Figure Figure 79 79 79 79    : Compa raison des contra intes vra ies ex pér imentale s et modélisées après diffé rents niv eaux  de : Compa raison des contra intes vra ies ex pér imentales  et modélisées après diffé rents niv eaux  de : Compa raison des contra intes vra ies ex pér imentales  et modélisées après diffé rents niv eaux  de : Compa raison des contra intes vra ies ex pér imentales  et modélisées après diffé rents niv eaux  de \ndéformation vraie pour les ac iers de la base de  don nées . déformation vraie pour les ac iers de la base de  don nées . déformation vraie pour les ac iers de la base de  don nées . déformation vraie pour les ac iers de la base de  don nées .    \n \n3.3.4. 3.3.4. 3.3.4. 3.3.4.  Vers une démarch e « Vers une démarch e « Vers une démarch e « Vers une démarch e «    d’alloy d’alloy d’alloy d’alloy- - --design design design design    »» »»     \n \n3.3.4.1.  Application  au DP 600  \n \nLe premier ex emple d’application du modèle concern e  un acier DP600 \n(Fe0.09C2Mn0.2Si0.3Cr), présen tant un e fraction de marten site faible (8%), une taille de \ngrain  ferritique de 5 µm environ  et une taille d’il ots martensitiques de 1µm . La Figure 80(a) \nmon tre une micrographie après attaque de cet acier (confirm ant l’absence de bainite) et la \nfigure (b) la construction de la courbe de traction  de cet acier à partir du com portem ent de ses \nconstituants, ferrite et m artensite. Pour  quelques couples contrain te-déformation de cette \ncourbe, son t indiquées par les lignes pointillées b leues les contraintes et déformations \nrespectives dans les deux constituants définissant la loi de mélange. Un modèle d’iso-\ndéformation  aurait donn é des lignes systématiquem en t verticales par ex emple, mais une \nréponse très rigide. A con trario, avec l’IsoW, la p ente de ces lignes évolue significativem ent \nau cours de la déformation  à partir d’un scé nario q uasim ent iso-déformation. \n  128  \n(a) \n (b) \n \n(c) \n (d) \nFigure Figure Figure Figure 80 80 80 80    : (a : (a : (a : (a) M icrographie optique après attaque de l’ac ier  DP600 consid éré (b) Détail de la modéli sation de  c et ) M icrographie optique après attaque de l’ac ier DP6 00 consid éré (b) Détail de la modéli sation de  cet ) M icrographie optique après attaque de l’ac ier DP6 00 consid éré (b) Détail de la modéli sation de  cet ) M icrographie optique après attaque de l’ac ier DP6 00 consid éré (b) Détail de la modéli sation de  cet \nacier, en distinguant le comportement de sa ferrite  et de  sa martensite. La  loi de mélange permettant de décrire acier, en distinguant le comportement de sa ferrite  et de  sa martensite. La  loi de mélange permettant de décrire acier, en distinguant le comportement de sa ferrite  et de  sa martensite. La  loi de mélange permettant de décrire acier, en distinguant le comportement de sa ferrite  et de  sa martensite. La  loi de mélange permettant de décrire \nle composite est représentée pa r des ligne le composite est représentée pa r des ligne le composite est représentée pa r des ligne le composite est représentée pa r des lignes pointil lées et évolue au cours de la déformation.  (c) et ( d) courbes  de s pointillées et évolue au cours de la déformation.  (c) et (d) courbes  de s pointillées et évolue au cours de la déformation.  (c) et (d) courbes  de s pointillées et évolue au cours de la déformation.  (c) et (d) courbes  de \ntraction et d’écrouissage expérimentales et modélis ées respectivement. traction et d’écrouissage expérimentales et modélis ées respectivement. traction et d’écrouissage expérimentales et modélis ées respectivement. traction et d’écrouissage expérimentales et modélis ées respectivement.    \n \nL’écrouissage ciném atique X DP  modélisé (équation (84)) n’a pas pu être comparé à  des donn ées \nexpérimentales sur cet acier en particulier. Par co ntre, les valeurs ont été comparées sur la \nFigure 81 à des données disponibles sur 2 aciers de  m ême grade (DP 600) et présentant des \ncourbes de comportement similaires [SADAGOPAN 2003] . L’écrouissage cinématique prédit \nest très coh érent, e n particulier pour l’acier DP60 0 (2), en termes de valeurs absolues et \nd’évolution  avec la déform ation (comportemen t satur ant après 5-10% de déformation comme \natten due par les modèles analytiques monophasés – c f. équation (52)).  \n  129  \n \nFig ure Fig ure Fig ure Fig ure 81 81 81 81    : Comportement en traction et valeur d’écrouissage  cinématique de deu x  aciers DP 600 décrits pa r : Comportement en traction et valeur d’écrouissage cinématique de deu x  aciers DP 600 décrits pa r : Comportement en traction et valeur d’écrouissage cinématique de deu x  aciers DP 600 décrits pa r : Comportement en traction et valeur d’écrouissage cinématique de deu x  aciers DP 600 décrits pa r \n[SADAGOPAN 2003 [SADAGOPAN 2003 [SADAGOPAN 2003 [SADAGOPAN 2003] ] ]]-- -- compara ison avec l’acier DP 600 décrit sur  la  compara ison avec l’acier DP 600 décrit sur  la  compara ison avec l’acier DP 600 décrit sur  la  compara ison avec l’acier DP 600 décrit sur  la Figu re  Figure  Figure  Figure  80 80 80 80.. ..    \n \nCette capacité à prédire non seulement les comporte ments sur trajets monotones avec une \ngran de précision m ais aussi les contrain tes interne s générées au cours de la déform ation (de \nnature inter- et intra-ph ases) est une preuve indir ecte supplé mentaire de la pertinence \nglobale de l’approche et un élément discriminan t fo rt pour le choix entre des modèles. \n \n3.3.4.2.  Effet de fraction et de dilution !  \n \nLa Figure 82 mon tre l’évolution des propriétés de t raction atten dues par le m odèle en \nfonction de la fraction de martensite, sans modifie r la chimie et la taille de grain ferritique. \nSont reportés les résistances m écaniques, limites d ’élasticité, allongem ents répartis et \nécrouissages ciné m atiques à saturation calculés.  \n \nCette investigation  virtuelle révèle deux domain es de comportement distincts : \n• Dom aine E : Aux faibles fractions, pour une teneur n ominale en carbone donné, la \nteneur en carbone dans la martensite est élevée. Le  comporteme nt de l’acier DP est \ncontrôlé par l’élasticité de la m artensite. La rési stance mécanique et l’écrouissage \ncinématique augmente rapidement avec la fraction. L e taux d’écrouissage reste élevé et \nconduit à de forts allonge m ents répartis. \n• Dom aine P : Au-delà d’un e valeur critique (en viron 20%), la sensibilité de résistance \nmécanique à la fraction diminue, sign e que le compo rtem ent est contrôlé par la \nplasticité de la martensite. La ten eur en carbon e d ans cette ph ase est alors in férieure à \n0.4%. Dans la mesure où le gradient de contraintes entre les phases se réduit, \nl’écrouissage ciném atique diminue ainsi que le taux  d’écrouissage, ce qui a pour \nconséquence une chute rapide des allonge ments répar tis.  130   \n \nFigure Figure Figure Figure 82 82 82 82    : Evolution simulée des propriétés en traction d’u n acier DP 600 avec : Evolution simulée des propriétés en traction d’un  acier DP 600 avec : Evolution simulée des propriétés en traction d’un  acier DP 600 avec : Evolution simulée des propriétés en traction d’un  acier DP 600 avec la fraction de martensite (taill e la fraction de martensite (taille la fraction de martensite (taille la fraction de martensite (taille \nde grain ferritique supposée constante). Rm = Ré sis tance mécaniq ue, Re = Limite d’élasticité, Ag = de grain ferritique supposée constante). Rm = Ré sis tance mécaniq ue, Re = Limite d’élasticité, Ag = de grain ferritique supposée constante). Rm = Ré sis tance mécaniq ue, Re = Limite d’élasticité, Ag = de grain ferritique supposée constante). Rm = Ré sis tance mécaniq ue, Re = Limite d’élasticité, Ag = A A AAllongement llongement llongement llongement \nréparti, X réparti, X réparti, X réparti, X m ax m ax m ax m ax  = C  = C  = C  = Contribution cinéma tique à  l’écrouissage à satur ation. ontribution cinéma tique à  l’écrouissage à saturatio n. ontribution cinéma tique à  l’écrouissage à saturatio n. ontribution cinéma tique à  l’écrouissage à saturatio n.    \n \nLa limite d’élasticité augm ente quasiment linéairem ent sur les deux domaines, car elle est \névaluée dan s un domaine où la marten site est systé m atiquem e nt élastique. On ne distin gue \ndonc pas les deux régimes. Dans la pratique, la sit uation  est plus comple xe car il est difficile de \ntravailler à taille de grain ferritique constan te. Ces domaines de comportemen t ont été \nobservés bien entendu ex périmentalem ent, comme le m ontre la Figure 83. Le m odèle est \nappliqué avec un bon accord aux donn ées internes su r un acier à \nFe0.09C1.9Mn0.1Si0.3Cr0.15Mo, avec des fractions de  martensite variables. Les tailles de \ngrain s ferritiques ont été estimées en microscopiqu e optique. \n \n \nFigure Figure Figure Figure 83 83 83 83    : Propriétés  en traction d’un acier DP ( : Propriétés  en traction d’un acier DP ( : Propriétés  en traction d’un acier DP ( : Propriétés  en traction d’un acier DP (Fe Fe Fe Fe0.09C 1.9Mn0.1Si0.3Cr0.15M o) 0.09C 1.9Mn0.1Si0.3Cr0.15M o) 0.09C 1.9Mn0.1Si0.3Cr0.15M o) 0.09C 1.9Mn0.1Si0.3Cr0.15M o) en fo en fo en fo en fonction de nction de nction de nction de la  fraction de la  fraction de la  fraction de la  fraction de \nmartensite (taille martensite (taille martensite (taille martensite (tailles s ss de gra  de gra  de gra  de grain ferritique mesurée in ferritique mesurée in ferritique mesurée in ferritique mesurées s ss). Les points corre spondent à ). Les points corre spondent à ). Les points corre spondent à ). Les points corre spondent à des données e x périmen tales et les des données e x périmentales et les des données e x périmentales et les des données e x périmentales et les    \nligne ligne ligne lignes continue s s continue s s continue s s continue s  aux  résultats du modèle. Rm = Résistan c e mécanique, Ag = allongement réparti.  aux  résultats du modèle. Rm = Résistanc e mécanique , Ag = allongement réparti.  aux  résultats du modèle. Rm = Résistanc e mécanique , Ag = allongement réparti.  aux  résultats du modèle. Rm = Résistanc e mécanique , Ag = allongement réparti.     131   \nLe domaine E, celui con trôlé par l’élasticité de la  m artensite, correspon d au domaine de \nvalidité des modèles analytique monophasés. La Figu re 84 démontre par e xemple que les \néquations (51) et (52) permetten t de reproduire de manière très satisfaisan te les courbes de \ntraction mais aussi la composan te cin ématique de l’ écrouissage de l’acier DP 600 décrit \nprécédemm ent. Le modèle monophasé décrit correcteme nt ce premier domaine mais s’avère \ntrop rigide pour capter la transition vers le régim e contrôlé par son comporteme nt plastique \n(au-delà de 20% de marten site sur la Figure 84 (b)) . \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 84 84 84 84    :: :: (a) Prédiction du modèle monopha sé (équa tions (51 ) et (52)) pour déc rire la courbe de traction et  (a) Prédiction du modèle monopha sé (équa tions (51)  et (52)) pour déc rire la courbe de traction et  (a) Prédiction du modèle monopha sé (équa tions (51)  et (52)) pour déc rire la courbe de traction et  (a) Prédiction du modèle monopha sé (équa tions (51)  et (52)) pour déc rire la courbe de traction et \nl’éc l’éc l’éc l’écrouissage cinématique de l’acier DP 600 de la  rouissage cinématique de l’acier DP 600 de la  rouissage cinématique de l’acier DP 600 de la  rouissage cinématique de l’acier DP 600 de la  Figur e Figure Figure Figure 80 80 80 80. (b) Compa raison des modèl es . (b) Compa raison des modèl es . (b) Compa raison des modèl es . (b) Compa raison des modèl es monophasés et  monophasés et  monophasés et  monophasés et    \nbiphasés sur une large gamme de fraction de martens ite. biphasés sur une large gamme de fraction de martens ite. biphasés sur une large gamme de fraction de martens ite. biphasés sur une large gamme de fraction de martens ite.    \n \nCette transition en tre domaines est bien enten du du e à la réduction de la teneur locale en \ncarbone dans la marten site (effet de dilution). En conséquence, la fraction critique \ncorrespondant à la transition augmente avec la te ne ur nom inale en carbone. La Figure 85 \nprésente un  exe mple de décalage de la transition  en tre les dom aines avec la teneur en carbone. \nSur la figure (b), les faibles fractions n’ont pas été con sidérées car in accessibles d’un point de \nvue métallurgique. \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 85 85 85 85    : Propriétés de tracti : Propriétés de tracti : Propriétés de tracti : Propriétés de traction simulées d’ac iers DP en fo nction de la  fraction de martensite (taille de gra i n on simulées d’ac iers DP en fonction de la  fraction de martensite (taille de gra in on simulées d’ac iers DP en fonction de la  fraction de martensite (taille de gra in on simulées d’ac iers DP en fonction de la  fraction de martensite (taille de gra in \nconstante) et transition entre régime constante) et transition entre régime constante) et transition entre régime constante) et transition entre régimes s ss contrôlé  contrôlé  contrôlé  contrôlés s ss par l’éla sticité  par l’éla sticité  par l’éla sticité  par l’éla sticité (domaine E) (domaine E) (domaine E) (domaine E) et par la plastic ité de la  martensite et par la plastic ité de la  martensite et par la plastic ité de la  martensite et par la plastic ité de la  martensite    \n(domain e P) (domain e P) (domain e P) (domain e P)    pour des pour des pour des pour des acier  acier  acier  aciers virtuels s virtuels s virtuels s virtuels    (a) (a) (a) (a) à 0.09%C et (b) à à 0.09%C et (b) à à 0.09%C et (b) à à 0.09%C et (b) à 0.25%C.  0.25%C.  0.25%C.  0.25%C.     \n \n  132  Pour finir cette discussion  sur les effets de dilut ion, la Figure 86 présente simultan ément les \neffets de fractions et de teneur en carbone nominal e sur les résistances mécaniques, limites \nd’élasticité et allon gements répartis prédits par l e modèle. Les calculs ont été réalisés pour une \ntaille de grain ferritique constante pour des raiso ns de simplicité. La transition e ntre domain es \nE et P est représentée par une ligne continue sur l es différentes figures et com parée en  \nparticulier sur la figure (b) aux calculs de la ten eur en  carbon e dans la martensite C m sur un \nmêm e plan. On s’aperçoit alors que la transition co rrespond très exacte ment à l’iso-C m égale à \n0.4%. Autrement dit, si un acier DP présente une te neur locale en  martensite in férieure à \n0.4%C, alors son  comportement macroscopique sera co ntrôlé principale ment par le \ncomportem ent plastique de la m artensite. A contrari o, si elle est supérieure, alors on peut \nconsidérer que la marten site se comporte comme un e phase élastique (approximation des \nmodèles m onoph asés). Dans ce domaine, les propriété s mécaniques dépendent \nprin cipalem e nt de la fraction de marten site. A titr e purement illustratif, les positionne ments \ntypiques d’aciers DP ou m artensitiques industriels sont repérés sur la figure (a). \n \n(a) \n (b) \n(c)\n (d) \nFig ure Fig ure Fig ure Fig ure 86 86 86 86    : Cartographie dans le plan fraction de martensite  et teneur no minale en carbone des propriétés de : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés de : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés de : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés de \ntr action modélisé es (a) résistance mécanique, tr action modélisé es (a) résistance mécanique, tr action modélisé es (a) résistance mécanique, tr action modélisé es (a) résistance mécanique, (b) t eneur loca l en carbone dans la (b) teneur loca l en carbone dans la (b) teneur loca l en carbone dans la (b) teneur loca l en carbone dans la martensite,  martensite,  martensite,  martensite, ( ( ((cc cc) limite ) limite ) limite ) limite \nd’éla sticité et ( d’éla sticité et ( d’éla sticité et ( d’éla sticité et (d d dd) allongement réparti. ) allongement réparti. ) allongement réparti. ) allongement réparti. La transition entre  les doma ines La transition entre  les domaines La transition entre  les domaines La transition entre  les domaines E et P E et P E et P E et P de comportement des a ciers DP est de comportement des a ciers DP est de comportement des a ciers DP est de comportement des a ciers DP est \nrepérée par une ligne  continue noire. S ur la figure  (a) s repérée par une ligne  continue noire. S ur la figure  (a) s repérée par une ligne  continue noire. S ur la figure  (a) s repérée par une ligne  continue noire. S ur la figure  (a) sont repéré s des positionnements typiques d’ac iers DP ou ont repéré s des positionnements typiques d’aciers D P ou ont repéré s des positionnements typiques d’aciers D P ou ont repéré s des positionnements typiques d’aciers D P ou \nmart mart mart martensitique s industriels. ensitique s industriels. ensitique s industriels. ensitique s industriels.     \n \nC’est ici l’occasion de clarifier une idée reçue ch ez les « métallurgistes » qui consiste à \nconsidérer que la résistance mécanique d’un acier D P est systématiqueme nt proportionnelle à \nla fraction de martensite sur toute la gamme. Cette  idée est  bien entendu fausse comme le \nmon tre les figures précédentes mais était soutenue par la représentation graphique des  133  donn ées ex périmentales de Davies (reprise sur la Fi gure 87) [DAVIES 1978]. En  fait, cette \nfigure très populaire est biaisée par l’accumulatio n de données (des aciers avec des te neurs en \ncarbone nominales différentes) et un parti pris sub jectif. La figure (b) reprend en fait les \nrésultats pour un seul acier de cette base de donné es, qui montrent bien une tran sition vers \n40% de martensite et un comportement saturan t. \n \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 87 87 87 87    : (a) Evolution de la résistance méca nique et la l imite d’élastic ité de différentes aciers DP avec de s : (a) Evolution de la résistance méca nique et la li mite d’élastic ité de différentes aciers DP avec des  : (a) Evolution de la résistance méca nique et la li mite d’élastic ité de différentes aciers DP avec des  : (a) Evolution de la résistance méca nique et la li mite d’élastic ité de différentes aciers DP avec des  \nteneurs no minales en carbone différentes, d’ teneurs no minales en carbone différentes, d’ teneurs no minales en carbone différentes, d’ teneurs no minales en carbone différentes, d’a près [ DAVIES 1978] (b) E x traction des don nées de résistan c e a près [DAVIES 1978] (b) E x traction des don nées de r ésistanc e a près [DAVIES 1978] (b) E x traction des don nées de r ésistanc e a près [DAVIES 1978] (b) E x traction des don nées de r ésistanc e \nmécanique pour un seul acier, montrant une transiti on de c omporte ment vers 40 % de martensite. mécanique pour un seul acier, montrant une transiti on de c omporte ment vers 40 % de martensite. mécanique pour un seul acier, montrant une transiti on de c omporte ment vers 40 % de martensite. mécanique pour un seul acier, montrant une transiti on de c omporte ment vers 40 % de martensite.    \n \n3.3.4.3.  Effet de la taille de grain ferritique  \n \nLe pendant de cette analyse en termes de fractions de martensite concerne la sensibilité du \nmodèle aux  tailles de grain ferritique. La Figure 8 8 m ontre des exem ples de performance du \nmodèle pour capter ces effets, con firman t la pertin ence de l’approche. \n \n(a) \n (b) \nFigure Figure Figure Figure 88 88 88 88    : : : : Comparaison entre résultats ex périmentaux (points ) et modèle Comparaison entre résultats ex périmentaux (points) et modèle Comparaison entre résultats ex périmentaux (points) et modèle Comparaison entre résultats ex périmentaux (points) et modèles s ss  (ligne  (ligne  (ligne  (lignes s ss c ontinue  c ontinue  c ontinue  c ontinues s ss) ) ) ) (a) (a) (a) (a) Effet de la ta ille Effet de la ta ille Effet de la ta ille Effet de la ta ille \nde grains de grains de grains de grains sur la contrainte d’écoulement pour diffé rentes valeurs de déformation (acier Fe0.2C0.47M n0. 3Si,  sur la contrainte d’écoulement pour différentes va leurs de déformation (acier Fe0.2C0.47M n0.3Si,  sur la contrainte d’écoulement pour différentes va leurs de déformation (acier Fe0.2C0.47M n0.3Si,  sur la contrainte d’écoulement pour différentes va leurs de déformation (acier Fe0.2C0.47M n0.3Si, \ndonnées données données données de  de  de  de    [[ [[JIANG 1992]) JIANG 1992]) JIANG 1992]) JIANG 1992]) (b)  (b)  (b)  (b) Cour Cour Cour Courbes de traction  de différents a ciers avec des fra ctions de martensite et taille bes de traction  de différents a ciers avec des fra c tions de martensite et taille bes de traction  de différents a ciers avec des fra c tions de martensite et taille bes de traction  de différents a ciers avec des fra c tions de martensite et tailles s ss    \nde grain ferriti de grain ferriti de grain ferriti de grain ferritique différent que différent que différent que différents (données de s (données de s (données de s (données de    [M ATL OCK 2005] [M ATL OCK 2005] [M ATL OCK 2005] [M ATL OCK 2005]). ). ). ).    \n \n \n  134  3.3.4.4.  Vers l’utilisation en  ingénierie des microstr ucture s  \n \nCes différen ts élém ents de discussion sur le rôle r espectif des différen tes caractéristiques de la \nmicrostructure des aciers DP nous amè n ent à considé rer l’un des intérêts in dustriels majeurs \nde cette approche « générique », c'est-à-dire d’êtr e un  puissant outil de « d’alloy design » ou \nd’ingénierie des microstructures. Il permet d’ident ifier par e xemple des solutions \nmicrostructurales non triviales pour répondre à un cah ier des charges donné.  \n \nSi l’on se fixe une double contrainte technique sur  deux paramètres de traction (Rm > 980 & \nAg > 8 % par ex e mple), la Figure 89(c) montre les m icrostr uctures des aciers pouvant \ncorrespondre à ces deux critères simultanément (déc rits indépendamm ent sur les figures (a) et \n(b)). Cette figure conduit à la conclusion que si l ’acier doit présenter, de plus, de faibles \nteneurs en  carbone pour des raison s de soudabilité par exem ple, alors la solution \nmétallurgique est unique (mélange de 50/50 de Ferri te et Martensite). Cet exercice \nd’application volontairem ent simpliste doit être ad apté en fonction  des faisabilités \nindustrielles (taille de grains ferritiques, présen ce de bainite) pour être pleinem ent \nopérationnel. \n(a) \n (b) \n(c) \nFig ure Fig ure Fig ure Fig ure 8 8 8899 99    : Cartographie dans le plan fraction de martensite  et teneur no minale en carbone des propriétés d : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés d : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés d : Cartographie dans le plan fraction de martensite et teneur no minale en carbone des propriétés de e e e \ntr action (a) résistance mécanique et tr action (a) résistance mécanique et tr action (a) résistance mécanique et tr action (a) résistance mécanique et    (b) allongement répa rti (b) allongement répa rti (b) allongement répa rti (b) allongement répa rti. Les zones vertes correspon dent aux aciers vérifiant . Les zones vertes correspondent aux aciers vérifia nt . Les zones vertes correspondent aux aciers vérifia nt . Les zones vertes correspondent aux aciers vérifia nt \nle cahier des charges (Rm > 980 le cahier des charges (Rm > 980 le cahier des charges (Rm > 980 le cahier des charges (Rm > 980 &  &  &  & A g > 8%) A g > 8%) A g > 8%) A g > 8%). . ..(c) superposition  (c) superposition  (c) superposition  (c) superposition  des zones répondant aux  deux cont raintes des zones répondant aux  deux contraintes des zones répondant aux  deux contraintes des zones répondant aux  deux contraintes    \ntechniques techniques techniques techniques simultanément  simultanément  simultanément  simultanément. . ..    \n \n \n  135  3.3.5. 3.3.5. 3.3.5. 3.3.5.  Lim ites e Lim ites e Lim ites e Lim ites et perspectives t perspectives t perspectives t perspectives    \n \nLe développement présen té est capable de comprendre  et de quan tifier les variations non \ntriviales des propriétés de traction des aciers Fer rito-Martensitiques, en fon ction des \nparamètres microstructuraux pertine nts d’après la l ittérature (fraction de martensite, taille de \ngrain s, compositions). I l repose sur une compréhens ion des différents m écanism es \nd’écrouissage de chacune des phases constituantes e t de leurs interactions (gradient de \ncontraintes et de déform ations entre les phases). C ette démarche permet une excellente \ndescription des con traintes intern es au cours de la  déformation et conduit à un faible nom bre \nde paramètres d’ajustemen t (3 pour les biphasés). C ontrairement aux approches monophasées, \nle comportement plastique de la martensite joue un rôle clef et permet des démarches \n« d’alloy design » et d’études de sensibilité sur t oute la gamme de fraction de marten site. \n \nUne différence im portante est apparue toutefois ent re les modèles monophasé et biph asé \nconcernant leur sensibilité à la taille d’ilot mart ensitique. En  suivant la dé marche \nd’ajuste men t proposée sur des données de la littéra ture disparates, il se mble difficile \nd’identifier une tendance claire reliant par ex e mpl e le paramètre L α et d m. Une meilleure \ndescription de cette sen sibilité dans la nouvelle a pproche semble un axe de recherche \nintéressant, avec une difficulté que nous évoqueron s dans la section  suivante con cernan t la \ndéfin ition de ce paramètre une fois la phase marten sitique percolée. \n \nLa Figure 90 illustre une autre lim ite du modèle. E lle représen te les écarts en  contrainte en tre \nles prédictions du modèle et l’ex périence pour deux  n iveaux  de déformation. Bie n  entendu, \naux fractions e xtrêmes (0% et 100% de martensite), les prédiction s présentent un accord \nexcellent. Par con tre, aux  fractions intermédiaires , le modèle surestime statistiquement les \ntaux  d’écrouissage. Une explication possible est li ée à la dispersion des teneurs e n carbone des \nilots de martensite observés expérimentalement [GAR CIA 2007]. Ces travaux expérime ntaux \nposent la question de l’utilisation  comm e donn ée d’ en trée du modèle de m artensite. Celui-ci \nest basé sur une distribution de propriétés supposé e de contraintes d’écoulem ent sur des aciers \nmartensitiques massifs. D’un point de vue métallurg ique, il n ’est absolumen t pas garanti que \ncette distribution soit aussi représentative de cel le des aciers DP. Cette question ce ntrale est \ndonc liée à l’imbrication de deux modèles de transi tion d’échelle. Les travaux engagés sur \nl’origine du spectre de con traintes et des échelles  en jeu sur le comportemen t de la marten site \nsont un préalable important avan t d’envisager une a daptation du modèle.  \n  136  \n \nFigure Figure Figure Figure 90 90 90 90    : Ecart en : Ecart en : Ecart en : Ecart entre modèle et ex périence en contrainte po ur deux niv eaux  de déformation (ou déformation tre modèle et ex périence en contrainte pour deux ni v eaux  de déformation (ou déformation tre modèle et ex périence en contrainte pour deux ni v eaux  de déformation (ou déformation tre modèle et ex périence en contrainte pour deux ni v eaux  de déformation (ou déformation \nmaximum) en fonction d e la frac tion de martensite p our le s différents aciers de la base de données. maximum) en fonction d e la frac tion de martensite p our le s différents aciers de la base de données. maximum) en fonction d e la frac tion de martensite p our le s différents aciers de la base de données. maximum) en fonction d e la frac tion de martensite p our le s différents aciers de la base de données.        \n \nA l’instar de tous les m odèles à cham ps moyens, une  limitation  du modèle importante \nconcerne ses capacités à prédire les processus d’en dommagement. Ces processus sont de \ndifférentes natures dans ces aciers (germ inations d e cavités aux interfaces ferrite/m artensite, \nruptures fragile et ductile de la martensite …) et très locaux. Dans ces structures composites, \nils in tervien ne nt très tôt au cours de la déformati on à cause des forts gradients de contrain tes \net de déformation s aux interfaces ferrite-martensit e et conduisent à une ruine rapide. Ce \ncomportem ent spécifique est caractéristique des aci ers DP et limite fortemen t leur application \nindustrielle.  \n \nUn modèle de décohésion des interfaces Ferrite/Mart ensite a été proposé récem ment par \nLandron et al.  [LANDRON 2010]. Ce critère in spiré par le critère d’Argon  tient compte des \ncontraintes intern es (gradients de contraintes aux  interfaces), force m otrices pour la \ndécohésion : \n \n C\neqeq\neq σX σσT1σχ >\n\n\n\n\n\n\n\n−+=  (86) \n \nAvec σ eq  la contrainte d’écoulement équivale nt, T la triaxi alité du chargement extérieur et X \nla contribution cin ématique au durcissement. Cette fonction a été évaluée par exemple sur la \nFigure 91 en fonction de la ten eur en carbon e et de  la fraction de martensite (T = 2). La \ncontrainte d’écoulement est prise égale à Rm et X à  sa valeur à saturation (Xmax).  \n  137  \n(a) \n (b) \nFigure Figure Figure Figure 91 91 91 91    : Cartographie da ns le plan fraction de martensite  et teneur nominale en carbone : Cartographie da ns le plan fraction de martensite et teneur nominale en carbone : Cartographie da ns le plan fraction de martensite et teneur nominale en carbone : Cartographie da ns le plan fraction de martensite et teneur nominale en carbone des propriétés  des propriétés  des propriétés  des propriétés \nmodélisés modélisés modélisés modélisés d’endommagement décrite s par l’éq uation ( 87) (a) Xmax d’endommagement décrite s par l’éq uation (87) (a) Xm ax d’endommagement décrite s par l’éq uation (87) (a) Xm ax d’endommagement décrite s par l’éq uation (87) (a) Xm ax, la  valeur de l’écrouissage ciné matique à , la  valeur de l’écrouissage ciné matique à , la  valeur de l’écrouissage ciné matique à , la  valeur de l’écrouissage ciné matique à \nsaturation saturation saturation saturation et (b) χ calc ul  et (b) χ calc ul  et (b) χ calc ul  et (b) χ calc ulé pour une tria xialité macroscopiqu e de T  = 2. é pour une tria xialité macroscopique de T  = 2. é pour une tria xialité macroscopique de T  = 2. é pour une tria xialité macroscopique de T  = 2. Posi tion de la transition entre  domaines E  Position de la transition entre  domaines E  Position de la transition entre  domaines E  Position de la transition entre  domaines E \net P reprise de  la et P reprise de  la et P reprise de  la et P reprise de  la Figure  Figure  Figure  Figure  86 86 86 86.. ..    \n \nCe modèle ouvre des perspectives in téressan tes e n  d écrivant l’impact d’un m écanisme \nd’en dommagement (décoh ésion e ntre ferrite et marten site), mais n e peut être appliqué de \nfaçon générique car un seul mécanisme est décrit. D e plus, ce comportement étan t contrôlé \npar la triaxialité des contraintes, la prise en com pte de la morphologie des phases se mblerait \nindispensable.  \nEn parallèle de ce développemen t d’un modèle à cham ps moyens, j’ai engagé une action  de \nrech erche pour mieux appréhen der les effets de la d ispersion et de la m orphologie de la \nmartensite sur le comportement des aciers DP grâce à des modélisations micromécaniques par \nElém ents Finis (EF) de Volume Elé mentaire Représent atif (VER) de m icrostructures. \n \n3.4. 3.4. 3.4. 3.4.  Modélisation par EF Modélisation par EF Modélisation par EF Modélisation par EF de VER  de VER  de VER  de VER : Effet de la morphologie : Effet de la morphologie : Effet de la morphologie : Effet de la morphologie de la martensite  de la martensite  de la martensite  de la martensite    \n \nDan s cette dernière partie, nous mon trerons un exem ple concret d’application sur des \nmicrostructures Dual-Phase des outils que j’ai déve loppés pour réaliser des calculs par EF \n(Elém ents Finis) sur VER (Volum e élém entaire Représ entatif) de microstructures. L’objectif \nultim e de ce projet est de m ieux comprendre la comp étition  entre les m écanism es \nd’en dommagement dans ces aciers et optimiser leur m icrostructures en conséquence, non \nseulement en term e de fraction et tailles mais auss i e n terme de morphologie. L’exem ple \ndétaillé dans la suite concerne les effets de la st ructure en bandes sur les propriétés \nd’en dommagement. Ces travaux ont été réalisés princ ipaleme n t dans le cadre de la thèse de B. \nKrebs au LETAM. \n3.4.1. 3.4.1. 3.4.1. 3.4.1.  Problématique Problématique Problématique Problématique    : Effet de la structure en bande : Effet de la structure en bande : Effet de la structure en bande : Effet de la structure en bande     \n \nA l’instar de [LI 1990_2][L I 1990_1][PAUL 2012][CHO I 2009][KUMAR 2007][SODJIT \n2012][PAUL 2013], nous avons m odélisé le com portem e nt d’aciers DP par la méthode des EF  138  en sollicitant num ériquem ent des VER de leur micros tructure. La démarche de simulation \nn’est pas nouvelle en soi, mais nous a permis d’inv estiguer des questions originales : \n• l’effet de la structuration dite « en bande » des D P sur les propriétés \nd’en dommagement des aciers DP \n• Effet du passage 3D / 2D sur des structures n uméris ées (issues de donn ées \nexpérimentales)  \n \nLa structure dite « en bandes » des aciers DP se ca ractérise par une distribution spatiale \nanisotrope et une m orphologie très allon gée des ilo ts de martensite. Elle est due à la présence \nde m icroségrégations de m angan èse issue du processu s de solidification (en  coulée continue \nou en lingots) mais n’apparaît que dans certaines c onditions de traitement thermique (vitesse \nde refroidissement et de taille grain austé nitique)  [KREBS 2009][V I ARDIN  2008]. La figure \nsuivante montre par e xem ple deux micrographies d’ac iers DP obte n us à partir d’une même \ncomposition  (Fe0.15C1.5Mn) mais ayant subi ou non u n traitem ent thermique \nd’homogénéisation  préalable.  \n \n(a) (b) \nFigure Figure Figure Figure 92 92 92 92    : M icrographie optique après attaque de deu x  micro structures DP : M icrographie optique après attaque de deu x  micros tructures DP : M icrographie optique après attaque de deu x  micros tructures DP : M icrographie optique après attaque de deu x  micros tructures DP obtenu  obtenu  obtenu  obtenue e ees à  partir d’une même  s à  partir d’une même  s à  partir d’une même  s à  partir d’une même  \ncomposition ( composition ( composition ( composition (Fe Fe Fe Fe0.15C1.5Mn) (a) structures isotropes  après tra ite ment thermique d’homogénéisa tion des 0.15C1.5Mn) (a) structures isotropes  après tra iteme nt thermique d’homogénéisa tion des 0.15C1.5Mn) (a) structures isotropes  après tra iteme nt thermique d’homogénéisa tion des 0.15C1.5Mn) (a) structures isotropes  après tra iteme nt thermique d’homogénéisa tion des \nmicroségréga tions de manganèse (b) str ucture dite «microségréga tions de manganèse (b) str ucture dite «microségréga tions de manganèse (b) str ucture dite «microségréga tions de manganèse (b) str ucture dite «     en bande en bande en bande en bande    » » » » -- -- Les deux aciers présentent la même fraction de  Les deux aciers présentent la même fraction de  Les deux aciers présentent la même fraction de  Les deux aciers présentent la même fraction de \nmartensite (e martensite (e martensite (e martensite (environ 30%) nviron 30%) nviron 30%) nviron 30%) [KREBS 2009]  [KREBS 2009]  [KREBS 2009]  [KREBS 2009]. . ..    \n \nCes structures en bandes sont très communes dans le s produits industriels et connues dan s la \nlittérature pour avoir un effet néfaste sur les pro priétés d’endommagemen t, i.e. les \nmécanismes de rupture et de cavitation à l’échelle de la microstructure con duisant à la ruine \nmacroscopique du matériau. Toutefois, la corrélatio n n ’est pas systématique car la rupture des \naciers DP résulte de la compétition de plusieurs mé canismes [PUSHKAREVA 2009], illustrés \ndans le cas d’un acier DP 600 sur la Figure 93 (rep rise d’Avramovic et al.  [AVRAMOVIC \n2009]) : \n• Germination de cavités sur des in clusion s, \n• Rupture fragile ou ductile de la martensite, \n• Germination de cavités par décohésion des interface s ferrite-m artensite (ou à \nprox imité des interfaces dans la ferrite), \n• Puis, coalescence des cavités par cisaillem ent. \n 139   \nLa Figure 93(g) montre un exemple de processus d’en dommagement d’un acier DP présentant \nune forte structure en bande. On iden tifie sur cett e figure plusieurs mécanismes opérant \nsimultaném ent (rupture fragile d’ilots, décoh ésion et coalescence par cisaille m ent). Les \nbandes de martensite seront donc plus ou moins des sites privilégiés de rupture en fonction  de \nleur morph ologie (continuité des bandes par ex emple ), de leur dureté et ténacité (teneur en \ncarbone) ou de leur facteur de forme (triaxialité d u tenseur des contraintes en pointe).  \n \n  (g) \nFig ure Fig ure Fig ure Fig ure 93 93 93 93    :: ::    Microgra phies M EB après attaque Microgra phies M EB après attaque Microgra phies M EB après attaque Microgra phies M EB après attaque montrant différents  proces sus d’endommage ment da ns u montrant différents proces sus d’endommage ment da ns u montrant différents proces sus d’endommage ment da ns u montrant différents proces sus d’endommage ment da ns unn nn acier  acier  acier  acier \nDual Dual Dual Dual- - --Phase (a ) germination sur des inc lusions (b), (c) rupture fragile de la  martensite, (d) et (e) germin ation des Phase (a ) germination sur des inc lusions (b), (c) r upture fragile de la  martensite, (d) et (e) germina tion des Phase (a ) germination sur des inc lusions (b), (c) r upture fragile de la  martensite, (d) et (e) germina tion des Phase (a ) germination sur des inc lusions (b), (c) r upture fragile de la  martensite, (d) et (e) germina tion des \nca vités par décohésion des interfaces  ferrite ca vités par décohésion des interfaces  ferrite ca vités par décohésion des interfaces  ferrite ca vités par décohésion des interfaces  ferrite- - --martensite, (f) coa lescence  des ca martensite, (f) coa lescence  des ca martensite, (f) coa lescence  des ca martensite, (f) coa lescence  des cavités par cisaill ement vités par cisaillement vités par cisaillement vités par cisaillement    d’après d’après d’après d’après \n[[ [[AA AAVRAM O VIC 2009 VRAM O VIC 2009 VRAM O VIC 2009 VRAM O VIC 2009] ] ]], (g) Micrographies M EB après a ttaque d’un acier D P montrant différents mécanisme s , (g) Micrographies M EB après a ttaque d’un acier DP  montrant différents mécanisme s , (g) Micrographies M EB après a ttaque d’un acier DP  montrant différents mécanisme s , (g) Micrographies M EB après a ttaque d’un acier DP  montrant différents mécanisme s \nd’endo mmagement en relation a vec une forte structur e en  bande (données AM ). d’endo mmagement en relation a vec une forte structur e en  bande (données AM ). d’endo mmagement en relation a vec une forte structur e en  bande (données AM ). d’endo mmagement en relation a vec une forte structur e en  bande (données AM ).    \n \nNotre objectif était donc d’étudier num ériquement l a susceptibilité à l’endommage ment de \ndifférentes microstructures selon  leur topologie (e n bandes ou visuellemen t isotropes). Nous \nnous somm es intéressés dans cette discussion à la d ifférence entre des approches 3D et 2D sur \ndes structures num érisées (à iso-fraction de marten site). \n3.4.2. 3.4.2. 3.4.2. 3.4.2.  Caractéristiques principales des simulations Caractéristiques principales des simulations Caractéristiques principales des simulations Caractéristiques principales des simulations    \n \nLes simulations ont été réalisées sur des volumes é lé m entaires reconstruits à partir \nd’observations microstructurales (microstructures n umérisées). Ce type d’approche est \nmain te nant assez couramment utilisé en 2D [CHOI 199 9][KUMAR 2007][SODJIT \n2012][PAUL 2013] (analyse d’un e seule coupe métallo graphique) mais à notre connaissance, \nce type de calcul n ’a jamais été réalisé en  3D. Dan s le cadre de cette étude, les microstructures \nont été reconstruites grâce à la technique des coup es sériées (repositionnées et calées en  \népaisseur grâce à des indentations), comm e le m ontr e le schéma de principe sur la Figure 94: \n \n 140   \nFigure Figure Figure Figure 94 94 94 94    : : : : Sc héma Sc héma Sc héma Sc héma de principe  de la tec hnique de coupe s sériée s utilisée pour reconstruire les structures DP en  de principe  de la tec hnique de coupe s sériées util isée pour reconstruire les structures DP en  de principe  de la tec hnique de coupe s sériées util isée pour reconstruire les structures DP en  de principe  de la tec hnique de coupe s sériées util isée pour reconstruire les structures DP en \n3D [KREBS 2009] 3D [KREBS 2009] 3D [KREBS 2009] 3D [KREBS 2009]    \n \nDan s ce domaine, on attend don c des progrès importa nts des techniques d’observation EBSD \ncouplées à un FIB (Field I on Beam), qui permettront  de recon struire en 3D non seule men t la \nstructure m artensitique (en utilisant l’in dice de q ualité pour distinguer ferrite et m artensite) \nmais aussi la structure et l’orientation cristallog raphique des grains ferritiques.  \n \nLa Figure 95 montre les coupes 2D  obtenues sur les deux aciers DP étudiés, présentant des \nfractions volumiques de m artensite similaire (30% e nviron) m ais une distribution topologique \ndes « ilots » différente (mêmes aciers que la Figur e 92). Ces coupes m étallographiques ont été \nsegm entées pour faire apparaître la marten site en b lan c et la ferrite en noir. Les \nreconstructions en  3D des réseaux de m artensite de ces deux  aciers sont représentées sur les \nfigures (c) et (d). \n \n(a) \n (b) \n(c) \n (d) \nFigure Figure Figure Figure 95 95 95 95    : (a) et (b) coupes 2D binarisées des structures D P en ba ndes et isotropes étudiées,  la ferrite appar aît en : (a) et (b) coupes 2D binarisées des structures DP  en ba ndes et isotropes étudiées,  la ferrite appara ît en : (a) et (b) coupes 2D binarisées des structures DP  en ba ndes et isotropes étudiées,  la ferrite appara ît en : (a) et (b) coupes 2D binarisées des structures DP  en ba ndes et isotropes étudiées,  la ferrite appara ît en \nnoir noir noir noir et la martensite en blanc et la martensite en blanc et la martensite en blanc et la martensite en blanc    ; (c) ; (c) ; (c) ; (c) et (d) reconstruction 3D des mêmes a ciers DP  et (d) reconstruction 3D des mêmes a ciers DP  et (d) reconstruction 3D des mêmes a ciers DP  et (d) reconstruction 3D des mêmes a ciers DP  – – –– seule la ma  seule la ma  seule la ma  seule la martensite a été rtensite a été rtensite a été rtensite a été \nreprésentée e n rouge pour représentée e n rouge pour représentée e n rouge pour représentée e n rouge pour qua ntifier la  percolation  de cette phase qua ntifier la  percolation de cette phase qua ntifier la  percolation de cette phase qua ntifier la  percolation de cette phase [KREBS  2009] [KREBS  2009] [KREBS  2009] [KREBS  2009] .. .. Les volumes reconstruits en  (c)  Les volumes reconstruits en  (c)  Les volumes reconstruits en  (c)  Les volumes reconstruits en  (c) \net (d) sont de 40 x 13 x 7.7  µm et (d) sont de 40 x 13 x 7.7  µm et (d) sont de 40 x 13 x 7.7  µm et (d) sont de 40 x 13 x 7.7  µm 33 33 et 33 x 31 x 6.5 µm  et 33 x 31 x 6.5 µm  et 33 x 31 x 6.5 µm  et 33 x 31 x 6.5 µm 33 33 respectivement.  respectivement.  respectivement.  respectivement.    \n 141   \nDan s les deux cas étudiés, la m artensite form e de f açon très surprenan te une structure \nconn exe. Ce résultat confirmé par des m esures de no m bres de connexités n égatifs malgré les \nobservations en 2D qui laissent penser que tous les  « ilots » sont disjoints. Dans le cas de la \nstructure en  bandes, il ex iste de claires jonctions  e ntre les bandes. Certains grains de ferrite \nsont même enchâssés dans un e matrice martensitique.  A contrario, dans la structure \nhom ogène, les ilots semblent connectés un par un. C ette dern ière configuration illustre bien \nla difficulté de définir une taille d’ilot représen tative d m, dans la mesure où le réseau de \nmartensite percole même aux faibles fractions.  \n \nLes microstructures DP sont le fruit de processus d e transformation de phase complex es, \nstructurés et multi-échelle. Elles ne sont pas rédu ctibles à des microstructures simples, \nrepr ésentables en 3D par des distributions aléatoir es inspirées des coupes 2D. Cette \nobservation justifie donc pleinem e nt l’intérêt des simulations e n 3D sur des microstructures \nréelles malgré les difficultés techniques de numéri sation.  \n \nOn pourra toutefois regretter que les volumes des s tructures reconstruites soient \ncomparativement faibles et qu’ils ne con stitue n t pa s des volumes représentatifs stricto-se n su. \nPour palier à ce manque de représen tativité, des ca lculs ont aussi été conduits sur des \nmicrostructures 2D virtuelles reproduisant des volu mes analysés plus importants. Ces \nmicrostructures ont été « simulées » numérique men t pour reproduire des fractions de \nmartensite de l’ordre de 30% mais des in dices de st ructures en  bandes différents (croissant sur \nla Figure 96). \n \n(a) (b) (c) \nFigure Figure Figure Figure 96 96 96 96    : M icrostructure s DP virtuelles en 2D reproduisant  des volu mes ana lysés plus grands que les : M icrostructure s DP virtuelles en 2D reproduisant des volu mes ana lysés plus grands que les : M icrostructure s DP virtuelles en 2D reproduisant des volu mes ana lysés plus grands que les : M icrostructure s DP virtuelles en 2D reproduisant des volu mes ana lysés plus grands que les \nnumérisations 3D (a) structures numérisations 3D (a) structures numérisations 3D (a) structures numérisations 3D (a) structures isotropes, (b) faib lement en bandes et (c) fort  isotropes, (b) faiblement en bandes et (c) fort  isotropes, (b) faiblement en bandes et (c) fort  isotropes, (b) faiblement en bandes et (c) forteme nt en bandes eme nt en bandes eme nt en bandes eme nt en bandes    [KR EBS 2009] [KR EBS 2009] [KR EBS 2009] [KR EBS 2009] .. ..    \n \nLe m aillage des microstructures en 2D et 3D a été r éalisé après segm entation sur un logiciel \ncommercial SIMPLEWARE, détourné de son dom aine d’ap plication originel, l’analyse \nd’image en  tomographie médicale et la micromécaniqu e biomédicale. Le résultat de la \nprocédure est un m aillage libre 3D adaptatif (m aill age plus grossier loin des interfaces) avec \ndes élémen ts quadratiques et tétraédriques comme le  montre les deux exem ples de la Figure \n97. \n  142  \n(a) \n (b) \nFigure Figure Figure Figure 97 97 97 97    : E xemple d e : E xemple d e : E xemple d e : E xemple d e mailla ge libre a daptatif réalisé mailla ge libre a daptatif réalisé mailla ge libre a daptatif réalisé mailla ge libre a daptatif réalisé sous SIM PLEWARE (a ) en 2D  sous SIM PLEWARE (a) en 2D  sous SIM PLEWARE (a) en 2D  sous SIM PLEWARE (a) en 2D (après  cal cul ABAQUS) (après  cal cul ABAQUS) (après  cal cul ABAQUS) (après  cal cul ABAQUS) et et et et \n(b) en 3D. (b) en 3D. (b) en 3D. (b) en 3D. La taille  caractéristique des éléme nts e st plus p La taille  caractéristique des éléme nts est plus p La taille  caractéristique des éléme nts est plus p La taille  caractéristique des éléme nts est plus pet ite que les etite que les etite que les etite que les rayons de courbures rayons de courbures rayons de courbures rayons de courbures d  d  d  dee eess ss structures  structures  structures  structures \nmartensitiques. martensitiques. martensitiques. martensitiques.    \n \nLa taille caractéristique des élém e nts est bien inf érieure à celle des rayons de courbures des \nstructures martensitiques. Cette finesse du maillag e permet d’investiguer le comportemen t à \nl’intérieur de cette phase. \n \nDan s cette première étape, les deux phases du compo site sont considérées comme des milieux \ncontinus m algré les échelles considérés (corps de V on Mises élasto-plastiques). On néglige de \nce fait la plasticité intra et polycristalline, ain si que l’anisotropie élastique des phases. Les \ngrain s de ferrite ne sont donc pas définis. Les cou rbes de comportement des deux constituants \nsont représentées sur la Figure 98: \n \n(a) (b) \nFigure Figure Figure Figure 98 98 98 98    : Lois de compor : Lois de compor : Lois de compor : Lois de comportement (a) de la ferrite et (b) de la martens tement (a) de la ferrite et (b) de la martens tement (a) de la ferrite et (b) de la martens tement (a) de la ferrite et (b) de la martensite ut ilisé pou r les calc uls EF ite utilisé pou r les calc uls EF ite utilisé pou r les calc uls EF ite utilisé pou r les calc uls EF [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..    \n \nCes courbes ont été déduites de calculs préalables avec le modèle biphasé afin d’introduire un \nsur-durcissement isotrope dans la ferrite dû aux  DG N et extrapolées linéaireme nt aux gran des \ndéformation s pour éviter des instabilités num érique s lors de la résolution. Les volum es \nélém entaires ont été soumis à des chargem ents de ty pe déformation plane, avec des \nconditions aux limites anti-périodiques. Les calcul s ont été résolus implicite ment en \ndéformation  imposée sous ABAQUS STANDARD. \n \n \n  143  3.4.3. 3.4.3. 3.4.3. 3.4.3.  Résultats Résultats Résultats Résultats    \n \n3.4.3.1.  Incidence de la topologie sur les propriétés macros copiques  \n \nLa Figure 99 montre les courbes de comportem ent con trainte – déformation  vraie déduites de \nla sollicitation des différents VER décrits ci-dess us ; les 3 microstructures virtuelles 2D \n(appelées m odèle homogène, faibleme n t et fortemen t en bandes) et les 2 microstructures \nnum érisées 3D (appelées réelle homogèn e ou ségrégée ). Les déformations son t représentées de \n0 à 100% de la déformation imposée (10% d’allongeme nt).  \n \n \nFigure Figure Figure Figure 99 99 99 99    : Courbes de comportement déduites des c al culs sur  VER (3 configu rations 2D virtuelles et : Courbes de comportement déduites des c al culs sur VER (3 configu rations 2D virtuelles et : Courbes de comportement déduites des c al culs sur VER (3 configu rations 2D virtuelles et : Courbes de comportement déduites des c al culs sur VER (3 configu rations 2D virtuelles et 2  2  2  2 \nconfig urations 3D numéris config urations 3D numéris config urations 3D numéris config urations 3D numérisées ées ées ées    [K REBS 2009] [K REBS 2009] [K REBS 2009] [K REBS 2009] .. ..    \n \nMalgré des fractions de m artensite ide n tiques, les microstructures en bandes présentent des \ntaux  d’écrouissages initiaux légèr ement supérieurs aux  microstructures hom ogènes, en 2D ou \n3D. Ce résultat numérique est très coh érent par rap port aux résultats expérime n taux de la \nFigure 63 page 98. L’effet de la topologie de la st ructure martensitique reste faible en inten sité \npar rapport aux effets de fraction  mê me dans le cas  d’une structuration très marquée, ce qui \njustifie plein e ment l’approche développée au ch apit re 3.3 page 107. \n \n3.4.3.2.  Hétérogénéités de la déformation  plastique  \n \nLa Figure 100 montre les résultats des simulations pour 2 VER 2D virtuels (hom ogène et \nfortement en bande) après 10% d’allongement ; les c ontraintes équivale ntes au sens de Von \nMises dans la ferrite et la marten site ainsi que la  déformation plastique équivale n te dans les \ndeux  phases. \n  144    \n  (a) \n  (b) \n \n  \n  (c) \n  (d) \n \n  \n  (e) \n  (f) \nFigure Figure Figure Figure 100 100 100 100    : Résultats des c alculs EF sur aprè s 10% d’allonge ment sur structures 2D virt uelles : Résultats des c alculs EF sur aprè s 10% d’allongem ent sur structures 2D virt uelles : Résultats des c alculs EF sur aprè s 10% d’allongem ent sur structures 2D virt uelles : Résultats des c alculs EF sur aprè s 10% d’allongem ent sur structures 2D virt uelles isotropes et  isotropes et  isotropes et  isotropes et  \nforte ment en bande forte ment en bande forte ment en bande forte ment en bande, (a) et (b) c ontraintes équivale ntes de Von Mises dans la , (a) et (b) c ontraintes équivalentes de Von Mises dans la , (a) et (b) c ontraintes équivalentes de Von Mises dans la , (a) et (b) c ontraintes équivalentes de Von Mises dans la ferrite , (c) et (d) dans la  martensite,  ferrite , (c) et (d) dans la  martensite,  ferrite , (c) et (d) dans la  martensite,  ferrite , (c) et (d) dans la  martensite, \n(e) et (f) déformation plastique é (e) et (f) déformation plastique é (e) et (f) déformation plastique é (e) et (f) déformation plastique équivalente dans l es deux phases  quivalente dans l es deux phases  quivalente dans l es deux phases  quivalente dans l es deux phases  [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..    \n \nLe comportement plastique de la ferrite est hauteme nt hétérogène. Des bandes de \ndéformation  plastiques intenses apparaissent rapide men t dans les deux types de VER (niveaux \nde déformation jusqu’à 35% locale ment). Ces ban des de déform ation ont souvent été \nobservées dans toutes les simulations de VER réalis tes d’aciers DP et corresponde nt à des \ncisaillements très in tenses comme le précise la Fig ure 101. \n  145  \n(a) \n (b) \nFig ure Fig ure Fig ure Fig ure 101 101 101 101    : Résultats des calc uls EF : Résultats des calc uls EF : Résultats des calc uls EF : Résultats des calc uls EF a près 10% d’allongement su r  structures 2D virtuelles  (a)  isot a près 10% d’allongement su r  structures 2D virtuell es  (a)  isot a près 10% d’allongement su r  structures 2D virtuell es  (a)  isot a près 10% d’allongement su r  structures 2D virtuell es  (a)  isotropes et (b) ropes et (b) ropes et (b) ropes et (b) \nforte ment en bande forte ment en bande forte ment en bande forte ment en bande    –– –– visualisa tion du c hamp de déforma tion pl  visualisa tion du c hamp de déforma tion pl  visualisa tion du c hamp de déforma tion pl  visualisa tion du c hamp de déforma tion pla stiq a stiq a stiq a stique en cisaillement (PE12). ue en cisaillement (PE12). ue en cisaillement (PE12). ue en cisaillement (PE12).    \n \nLe comportement de la m artensite sem ble bien plus h omogène comme attendu dans le cas \nd’un e phase en inclusion (inclusion type Eshelby). Cependan t statistiquement, la Figure 102 \nmon tre que la structuration en bandes conduit à une  structure martensitique en  moyenne \nplus chargée en terme de contraintes équivalentes. Ceci qui cohérent avec une contrainte \nd’écoulement du composite plus importante (cf. Figu re 99). La distribution en termes de \ndéformation  plastique confirme aussi que plus d’ilo ts m artensitiques seront fortem ent \ndéformés. \n \n(a) (b) \nFigure Figure Figure Figure 102 102 102 102    : Distribution en fréquence : Distribution en fréquence : Distribution en fréquence : Distribution en fréquence (a) (a) (a) (a) des contra intes équivalente s des contra intes équivalente s des contra intes équivalente s des contra intes équivalente s de Von Mises de Von Mises de Von Mises de Von Mises et ( et ( et ( et (b) défo rmation b) défo rmation b) défo rmation b) défo rmation \npl astique équivalente dans la martensite  dans les  3  config urations  2D virtu elles pl astique équivalente dans la martensite  dans les  3  config urations  2D virtu elles pl astique équivalente dans la martensite  dans les  3  config urations  2D virtu elles pl astique équivalente dans la martensite  dans les  3  config urations  2D virtu elles    [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..    \n \nUn comportement stricte ment équivale nt a été retrou vé lors des calculs sur VER 3D \nnum érisés, à la fois en terme de déform ation plasti que de la ferrite ou de chargem ent de la \nstructure m artensitique percolée. Les bandes de déf ormation n’apparaissent pas stricto sensu \ncomme des plans dans l’espace m ais sont plus proche s de ligame nts. La Figure 103(b) et la \nFigure 103(c) sont deux coupes sagittales qui m ontr en t l’extin ction de certaines bandes dans \nl’épaisseur. Des calculs sur des volumes plus im por tan ts seraient nécessaires pour confirm er \ncette tendance. Dans la suite, pour faciliter les r eprésentations, seul le cas des VER 2D sera \nétudié en détail. \n  146  \n(a) \n (b) \n (c) \nFigure Figure Figure Figure 103 103 103 103    : Résultats des calculs EF après 10% d’allonge ment  su r  structures 3D n umérisées (a) isotropes et (b)  : Résultats des calculs EF après 10% d’allonge ment su r  structures 3D n umérisées (a) isotropes et (b) : Résultats des calculs EF après 10% d’allonge ment su r  structures 3D n umérisées (a) isotropes et (b) : Résultats des calculs EF après 10% d’allonge ment su r  structures 3D n umérisées (a) isotropes et (b) \net (c) en bandes et (c) en bandes et (c) en bandes et (c) en bandes (2 plans de coupes)  (2 plans de coupes)  (2 plans de coupes)  (2 plans de coupes) [KREBS  2009 [KREBS  2009 [KREBS  2009 [KREBS  2009] ] ]].. ..    \n \nCes bandes de déformation ne sont pas un « artefact  » de simulation num érique et ont été \nmises en évidence récemm ent grâce au progrès de la corrélation d’im ages numériques in situ  \nen MEB [TASAN 2010][GHADBEIG I 2010]. La Figure 104 montre un bel ex emple de mesure \nde champs de déformation sur un  acier DP 1000 après  différen tes déformations \nmacroscopiques. Des ban des de déform ation dont l’éc helle est bien supérieure à la taille \ncaractéristique des structures martensitique son t o bservées comme dans le cas des simulations \nnum ériques. Par contre cette structuration se m ble a pparaitre plus tardivement (après 20% \nd’allongement), ce qui est peut-être un indice que les gradie nts de déformation aux interfaces, \nnon pris en compte dans ce calcul par EF, peuvent j ouer un rôle im portan t dans la \nstabilisation  des écoulements entre particules. \n  147  \n \nFigure Figure Figure Figure 104 104 104 104    : Mesures de champs de déformation plastiqu e par u ne  techniq ue de : Mesures de champs de déformation plastiqu e par un e  techniq ue de : Mesures de champs de déformation plastiqu e par un e  techniq ue de : Mesures de champs de déformation plastiqu e par un e  techniq ue de corrélation d’image s en MEB  corrélation d’image s en MEB  corrélation d’image s en MEB  corrélation d’image s en MEB in in in in \nsitu situ situ situ  sur un acier DP  sur un acier DP  sur un acier DP  sur un acier DP 1000 d’après  1000 d’après  1000 d’après  1000 d’après [GHADBEIGI 2010] [GHADBEIGI 2010] [GHADBEIGI 2010] [GHADBEIGI 2010]. . ..    \n \n3.4.3.3.  Vers les mécanismes d’endommageme nt  \n \nDan s la littérature, on  rapporte t rois m écanism es p rincipaux de germination de \nl’endommagement primaire dans ces structures : \n• l’endommagement ductile de la m artensite \n• la rupture fragile de la martensite \n• la décohésion entre ferrite et martensite \nCes trois m écanismes sont associés à des critères d ’apparition de différentes natures et \nrelevant de valeurs critiques très variables selon l’état métallurgique. Le premier m écanisme \npeut être associé à un critère de déform ation plast ique maximum à rupture ; le second à des \ncontraintes principales de traction critiques (crit ère type Griffith ) et le troisième à des \ncontraintes critiques nor m ales aux d’interfaces (cr itère type Argon) [PUSHKAREVA 2009]. \n \nNous avons pu vérifier sur la Figure 102(b) que la structure en bandes conduisait à des \ndéformation s plastiques plus élevées dans certains ilots de marten site (en considérant en \nparticulier la queue de distribution). La Figure 10 5 montre qu’elle est aussi néfaste en termes \nde contrain tes critiques m aximum dans la martensite  (cf. Figure 107(a)). Elle tend don c à \nfavoriser la rupture fragile des ilots de martensit e, mécan isme souven t observé \nexpérimentaleme n t. Il est assez difficile numérique ment d’évaluer son impact sur le troisième \ncritère. Par contre, nous avons aussi mis en évide n ce que la structuration en bande allait avoir \nun rôle défavorable sur les mécan ismes de croissanc e des cavités. La Figure 106 mon tre qu’elle \ncontribue à augmenter les valeurs de triaxialité du  te nseur des contraintes au n iveau des \npoin tes des ilots constitutifs des bandes (cf. Figu re 107(b)). Ces zon es correspondent en fait \naux intersections des bandes de cisaillem e nts in ten ses discutées précédemment.  148   \nLa topologie de la martensite contrôle donc les méc an ismes de germ ination  et de croissance \ndes cavités dans les processus d’endomm age ment. Tou tefois, nos résultats n e perm ettent pas \nde quantifier la compétition entre ces m écanismes e t leur prévalence, dans la mesure où les \ncritères de germination ne sont pas bien docum e ntés  dans la littérature et que ces m écanism es \ninteragisse n t certaine men t entre eux (relaxation ou  concentrations de contraintes par \nexem ple). Un modèle micromécanique com plet à cette échelle pourrait permettre de \nprogresser significative m ent sur ces questions et e nvisager des solution s métallurgiques \noptimisées pour réduire la sensibilité à l’endommag ent de ces structures (revenu, m écanism es \nde transform ation de phase, ..). \n \n(a) \n (b) \n (c) \nFig ure Fig ure Fig ure Fig ure 105 105 105 105    : Résultats des calc uls : Résultats des calc uls : Résultats des calc uls : Résultats des calc uls EF a près 10% d’allongement sur EF a près 10% d’allongement sur EF a près 10% d’allongement sur EF a près 10% d’allongement sur structures 2D virtue lles (a) isotr structures 2D virtuelles (a) isotr structures 2D virtuelles (a) isotr structures 2D virtuelles (a) isotropes et (b) opes et (b) opes et (b) opes et (b) \nfaiblement en bande faiblement en bande faiblement en bande faiblement en bande et (c)  et (c)  et (c)  et (c) fortemen  fortemen  fortemen  fortement en bande t en bande t en bande t en bande    –– –– visualisation d  visualisation d  visualisation d  visualisation de s e s e s e s champ  champ  champ  champs s ss de norme de la c ontrainte principale  de norme de la c ontrainte principale  de norme de la c ontrainte principale  de norme de la c ontrainte principale \nmaximum maximum maximum maximum     [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..    \n \n(a) \n (b) \n (c) \nFig ure Fig ure Fig ure Fig ure 106 106 106 106    : Résultats des calc uls EF a près 10% d’allongement  su r  structures 2D virtuelles  (a) isotropes et ( : Résultats des calc uls EF a près 10% d’allongement su r  structures 2D virtuelles  (a) isotropes et ( : Résultats des calc uls EF a près 10% d’allongement su r  structures 2D virtuelles  (a) isotropes et ( : Résultats des calc uls EF a près 10% d’allongement su r  structures 2D virtuelles  (a) isotropes et (b) b) b) b) \nfaiblement en band faiblement en band faiblement en band faiblement en bande et (c) fortement e n bande e et (c) fortement e n bande e et (c) fortement e n bande e et (c) fortement e n bande    –– –– visualisation d  visualisation d  visualisation d  visualisation de s e s e s e s champ  champ  champ  champs scal aires s scal aires s scal aires s scal aires de triaxialité du tenseur des  de triaxialité du tenseur des  de triaxialité du tenseur des  de triaxialité du tenseur des \ncontraintes contraintes contraintes contraintes    [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..    \n \n(a) (b) \nFig ure Fig ure Fig ure Fig ure 107 107 107 107    : Distribution en fréquence  (a) de la  contrainte p rinc ipale ma x imum : Distribution en fréquence  (a) de la  contrainte pr inc ipale ma x imum : Distribution en fréquence  (a) de la  contrainte pr inc ipale ma x imum : Distribution en fréquence  (a) de la  contrainte pr inc ipale ma x imum dans la martensite et (b) de  la dans la martensite et (b) de  la dans la martensite et (b) de  la dans la martensite et (b) de  la \ntriax ialité du tense ur des contraintes dans la ferr ite po ur les 3 configurations 2D  virtuelles triax ialité du tense ur des contraintes dans la ferr ite po ur les 3 configurations 2D  virtuelles triax ialité du tense ur des contraintes dans la ferr ite po ur les 3 configurations 2D  virtuelles triax ialité du tense ur des contraintes dans la ferr ite po ur les 3 configurations 2D  virtuelles [KREBS 2009] [KREBS 2009] [KREBS 2009] [KREBS 2009] .. ..     149   \n3.4.4. 3.4.4. 3.4.4. 3.4.4.  Perspectives de cette dém arche Perspectives de cette dém arche Perspectives de cette dém arche Perspectives de cette dém arche     \n \nCes travaux  de modélisation s’inscrive nt e n fait da ns une démarche plus gén érale évoquée en \nintroduction. Elle vise à développer une chain e de sim ulation numérique du com portem ent \nmécanique et d’endommagement des aciers DP. \n \nLes objectifs scientifiques de cette démarche sont de quantifier et de mieux comprendre les \nmécanismes d’endommage ment de ces structures DP et à terme leurs actionneurs \nmétallurgiques. Ceci nécessite n on seuleme nt une co nnaissance fine et juste des états de \ncontraintes et déformation des différen tes phases a u cours de la déformation mais aussi de \npouvoir gérer la compétition e ntre les m écanism es d ’en dommagemen t (relax ation). \n \nPour répon dre à ce besoin, les calculs par EF simpl ifiés décrits ci-dessus sont donc \ndoublement insuffisants. Les champs locaux sont cal culés san s te nir compte des m écanism es \nde plasticité cristalline et la susceptibilité à ce rtains processus d’endommagement calculée a \nposteriori. N ous travaillons donc actuellement sur deux  axes principaux : \n• le développeme nt d’une VMAT ABAQUS en collaboration  avec le CEIT (D. Gonzales, \nJM Esnaola, J. Gil Sevillano) intégrant la plastici té cristallin e. La spécificité de cette \napproche serait un e sensibilité directe aux gradien ts de déform ations (couplage simple). \nCet outil est en train d’être validé sur des cas si mples (acier ferritique mono- et \npolycristallin) [GONZALES 2011_1][GONZALES 2011_2]  avant d’être appliqué au cas \nplus complexe des aciers DP. Les premiers résultats  permettent de reproduire un effet \n« Hall et Petch » tout à fait satisfaisant et les d istributions statistiques de DGN dans un \npolycristal de fer. Ce développem ent permettra de c onfir mer l’influence respective des \ntailles de grains et d’ilots sur le com portem ent de s aciers DP et don c de faire \nprogresser aussi les modèles à champs m oyens. \n• La m odélisation m icromécanique de l’endommagemen t p ar l’introduction d’éléments \ncohésifs. La Figure 108 m ontre un exemple d’implé me ntation  de ces éléments autour \nd’un  ilot de martensite dans le cas du VER 2D virtu el homogène. Ils perm ettent de \nsimuler un  processus de décohésion en fonction de c ontraintes nor males \n(com portem ent de type pseudo-2D). La Figure 109 mon tre un  e xemple de simulations \noù les interfaces de deux des trois particules reco uvertes de ces éléments \ns’endommagent. I l apparaît alors des bandes de cisa illem e nt in te nses entre ces \nparticules (processus de localisation par cisaille m ent de type Brown-Embury).  \n  150  \n(a) \n (b) \n (c) \nFigure Figure Figure Figure 108 108 108 108    : Illustration de l’utilisation : Illustration de l’utilisation : Illustration de l’utilisation : Illustration de l’utilisation d’éléments cohésifs d’éléments cohésifs d’éléments cohésifs d’éléments cohésifs (en  gris) pour rep  (en  gris) pour rep  (en  gris) pour rep  (en  gris) pour reprod rod rod roduire le c omporte ment de uire le c omporte ment de uire le c omporte ment de uire le c omporte ment de \nl’inter l’inter l’inter l’interface ferrite face ferrite face ferrite face ferrite    (en rouge) (en rouge) (en rouge) (en rouge)- - --martensit e martensit e martensit e martensit e    (en vert (en vert (en vert (en vert et bleu  et bleu  et bleu  et bleu) dans le c as d ) dans le c as d ) dans le c as d ) dans le c as d’une configuration 2D virtuelle ’une configuration 2D virtuelle ’une configuration 2D virtuelle ’une configuration 2D virtuelle. (a), (b) et . (a), (b) et . (a), (b) et . (a), (b) et \n(c) agra ndissements successifs d’une même  zone. (c) agra ndissements successifs d’une même  zone. (c) agra ndissements successifs d’une même  zone. (c) agra ndissements successifs d’une même  zone.    \n \n(a) \n (b) \n (c) \nFigure Figure Figure Figure 109 109 109 109    : Résultats des calculs EF après 10% d’allonge m : Résultats des calculs EF après 10% d’allonge m : Résultats des calculs EF après 10% d’allonge m : Résultats des calculs EF après 10% d’allonge m en en en ent sur t sur t sur t sur structure 2D virt uelle isotrope (a) Contraint es structure 2D virt uelle isotrope (a) Contraintes structure 2D virt uelle isotrope (a) Contraintes structure 2D virt uelle isotrope (a) Contraintes \néquivalentes de Von M ise s (b) équivalentes de Von M ise s (b) équivalentes de Von M ise s (b) équivalentes de Von M ise s (b) D D DDé formation plasti é formation plasti é formation plasti é formation plastiqu e équiva lente et ( qu e équiva lente et ( qu e équiva lente et ( qu e équiva lente et (c c cc) E ) E ) E ) Endommagement (les cavités ndommagement (les cavités ndommagement (les cavités ndommagement (les cavités \nappar aissent en rouge). 3 particules sont recouvert es d’éléme nts cohésifs pour  simuler un proce ssus de appar aissent en rouge). 3 particules sont recouvert es d’éléme nts cohésifs pour  simuler un proce ssus de appar aissent en rouge). 3 particules sont recouvert es d’éléme nts cohésifs pour  simuler un proce ssus de appar aissent en rouge). 3 particules sont recouvert es d’éléme nts cohésifs pour  simuler un proce ssus de     \ndécohésion de l’interfa ce ferrite décohésion de l’interfa ce ferrite décohésion de l’interfa ce ferrite décohésion de l’interfa ce ferrite - - --martensite. martensite. martensite. martensite.    \n \n3.5. 3.5. 3.5. 3.5.  Conclusions et perspectives Conclusions et perspectives Conclusions et perspectives Conclusions et perspectives    \n \nMes travaux sur le comportement des aciers DP ont p orté sur différents types de modélisation \nayan t des finalités spécifiques ; \n• l’extension e n plasticité polycristalline d’un modè le m onophasé pour des applications \nen rhéologie appliquée (prévision  des surfaces de c harges sous sollicitations comple x es).  \n• le développement d’un modèle biphasé quasi-analytiq ue pour des utilisations en \n« alloy-design » métallurgique. Le modèle a été aju sté sur  un e large base de donn ées \nissues de la littérature et permet de capter avec j ustesse les effets de fraction  (dilution) \net de taille sur le comporteme nt des aciers DP. \n• le développement d’une chain e de simulation en  EF, de la numérisation de \nmicrostructures aux calculs sur VER sensible aux  gr adients de déformation  et \nintégrant des mécanismes simulant l’en dommagemen t ( élém ents cohésifs) en vue de \nparfaire les connaissances sur les effets de taille s, de morphologie et de topologie sur le \ncomportem ent et la rupture de ces aciers composites . L’approche est encore \nincomplète mais permet de traiter des  questions au premier ordre comm e l’aspect \nnéfaste d’un e structure en  bandes sur l’endomm age me n t. \n \nCes travaux ouvrent de nombreuses perspectives stim ulantes que nous avon s pu déjà \nlargement évoquer. Par ordre de difficulté croissan te :  151  • Axe de recherche 1 : Effet des éléments substitutio nnels sur com portem ent de la \nferrite. Les effets de taille dans cette phase semb lent largemen t compris et le \ncomportem ent suffisamm e nt bien reproduit à la fois par des approches analytiques \nmon ophasées ou les modèles de plasticité cristallin e par EF. Toutefois, l’influence sur \nl’écrouissage de certains éléments d’alliages « cla ssiques » en solution  solide, comme le \nsilicium ou même le manganèse, n’est ni comprise ni  décrite. \n• Axe de recherche 2 : com portem ent de la martensite « fraiche ». N os travaux sur le \ncomportem ent de la martensite ont ouvert un  angle d e recherche original sur cette \nphase. Sa n ature composite est à préciser expérimen talement e n termes d’échelle. Se \npose aussi naturelleme nt la question de la validité  de cette description dans le cas des \naciers DP (imbrication de deux  approches de transit ion d’échelle). Ces résultats \ndevraient naturellement conduire à une meilleure co mpréhension aussi du \ncomportem ent de la martensite « revenue ». \n• Axe de recherche 3 : Effet de taille et de morpholo gie dans les modèles biphasés. Ces \nquestions n e pourront progresser qu’avec le dévelop pement de la chaine de simulation \npar EF, qui permettra de découpler num ériquem e nt le s différents effets. \n• Axe de recherche 4 : investiguer la compétition ent re les m écanism es \nd’en dommagement. Ce travail reposera sur la chaine de simulation par EF mais devra \nêtre validé par des résultats ex périmentaux sur des  structures m odèles (interface \nferrite-martensite modèle) et des structures « r eve nus ». Ce dernier traitement chan ge \nles conditions de transfert de charges en tre les ph ases, les critères d’endommagem ent \nsans modifier la morphologie et les tailles de stru ctures. \n  152  4. 4. 4. 4.  Conclusion personnelle Conclusion personnelle Conclusion personnelle Conclusion personnelle    \n \nCe m anuscrit détaille mes résultats de recherch e su r le comportement mécanique des aciers \npour la construction automobile. Ils couvrent une l arge gamme de m écanism es et de \nmicrostructures, des aciers austénitiques TWIP aux  différents types de microstructures \nferritiques. Ces systèmes présentent toutefois de s urprenantes ressem blances, et je pense avoir \nmon tré, du point de vue de la démarche, la transver salité de certains concepts comme \nl’écrouissage par effet com posite, applicable aux a ciers TWIP aussi bie n qu’aux aciers DP ou \nles effets de tailles de microstructures sur le com portement (taille de grain s ou espacem ent \nentre micromacles). Ces analogies m e son t particuli èrement utiles pour traiter du \ncomportem ent des structures bainitiques sans carbur es auxquelles je m’in téresse beaucoup \nactuelle men t. Elles présen tent en effet simultanéme n t une structuration en latte à l’échelle \nnanométrique et un fort effet TRIP (transformation de l’austén ite en martensite α’). \n \nJe pense avoir aussi illustré les possibles applica tions industrielles et technologiques actuelles \nde ces travaux. Toutefois, ce documen t est loin d’ê tre un aboutissement et propose de \nnouvelles pistes de recherches dans ces domain es dé duites de l’analyse de la littérature et de \nmes contributions person nelles. Elles permettront à  terme de mieux comprendre et prévoir \nles comportements complexes de ces aciers, donc fav oriser leurs implémentations.  \n \nJ’espère enfin avoir  démontré par ce document, illu strant ma dém arche scientifique, m es \ncollaborations et encadrem ents, m es principales réu ssites et question s ouvertes, ma capacité à \ndiriger des recherches. \n \n \n \n \n \n  153  5. 5. 5. 5.  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Annexe Annexe Annexe Annexe    \n \nLa question  de la germination des macles nous a ame nés à considérer l’influence d’une \nprécipitation fine sur ce processus. Les interactio ns possibles entre précipitations et maclage \nont été ainsi étudiés dans le cadre de la thèse d’A . Dumay, sujet très peu documen té dans la \nlittérature scientifique. A titre purement illustra tif, on notera que la revue pourtant très \ncomplète de Christian et Mahajan dédiée au m aclage mécanique, ne discute de ces possibles \ninteractions que sur un paragraph e de 10 lignes sur  120 pages [CHRISTIAN 1995]. \n \nCette étude a été en partie consacrée à l’étude d’u ne fin e précipitation de carbures de \nvanadium sur le comportement mécanique et la micros tructure de maclage de la nuance de \nréférence (Fe22Mn 0.6C+0.2% atomique de VC). Une étu de détaillée en MET des précipités a \nperm is d’estimer leur taille moyenne (de l’ordre de  7 nm ) mais surtout leur degré de \ncohérence (orientation cube-cube entre matrice et p récipités {111} γ//{111} VC  et un faible degré \nde cohérence) (cf. Figure 111(a)).  \n \nLa comparaison en tre les courbes de traction d’un a lliage avec précipités et d’un acier de \ncomposition  équivalente sans précipités et une m ême  taille de grain  montre que la \nprécipitation ne modifie pas l’écrouissage de l’all iage m algré des fractions précipitées \nimportantes. Elle n’induit qu’un simple durcissemen t de type de Orowan (contribution \nadditive à la contrainte d’écoule ment) (cf. Figure 110). La précipitation  des carbures de \nvanadium n ’a donc pas d’impact sur la m icrostructur e de maclage (cinétique et structuration \nspatiale). Cette vision est confortée par les obser vations en MET (cf. Figure 111) révélant \nl’absence d’interactions fortes en tre micromacles e t précipités (simple contour nement, pas de \nblocage fort et systématique). Cette étude n ’a pas non plus permis d’observer que les \nprécipités pouvaie n t s’avérer être des sites de ger mination privilégiés pour le maclage. \n \n \n \n \n  164  \n \nFigure Figure Figure Figure 110 110 110 110    : Courbes de traction rationnelles de la  nuance de  référence et de sa version mic roallié a u vanadium : Courbes de traction rationnelles de la  nuance de référence et de sa version mic roallié a u vanadium : Courbes de traction rationnelles de la  nuance de référence et de sa version mic roallié a u vanadium : Courbes de traction rationnelles de la  nuance de référence et de sa version mic roallié a u vanadium \n[DUM AY  2009] [DUM AY  2009] [DUM AY  2009] [DUM AY  2009] .. ..    \n \n(a) \n (b) \nFigure Figure Figure Figure 111 111 111 111    : M icrographies en  M ET : M icrographies en  M ET : M icrographies en  M ET : M icrographies en  M ET – – –– c hamp clair (a)  c hamp clair (a)  c hamp clair (a)  c hamp clair (a) observation des précipités de carb ure  de vana dium ava nt observation des précipités de carbure  de vana dium a va nt observation des précipités de carbure  de vana dium a va nt observation des précipités de carbure  de vana dium a va nt \ndéformation déformation déformation déformation – – –– contraste en grain de café dû à  la cohérence rési duelle avec la ma trice (b) interactions entre   contraste en grain de café dû à  la cohérence résid uelle avec la ma trice (b) interactions entre   contraste en grain de café dû à  la cohérence résid uelle avec la ma trice (b) interactions entre   contraste en grain de café dû à  la cohérence résid uelle avec la ma trice (b) interactions entre  \nnanomacles e nanomacles e nanomacles e nanomacles et précipités après déforma tion t précipités après déforma tion t précipités après déforma tion t précipités après déforma tion [DUM AY 2009] [DUM AY 2009] [DUM AY 2009] [DUM AY 2009]. . ..    \n  165  Curriculum vitae Curriculum vitae Curriculum vitae Curriculum vitae    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  166   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  167  Rapport de soutenance Rapport de soutenance Rapport de soutenance Rapport de soutenance    UNIVERSITE DE LORRAINE - INSTITUT NATIONAL POLYTECH NIQUE DE LORRAI NE \n \n \nMon activité de recherche scientifique concerne pri ncipalement la compréhension et la modélisation du comportement mécaniques \ndes aciers ; des mécanismes fondamentaux à la défor mation macroscopique. Ce mémoire est consacré en pa rticulier à l’effet TWIP \n(TWinning Induced Plasticity) des aciers austénitiq ues FeMnC à haute teneur en manganèse et l’effet Du al-Phase des aciers Ferrite-\nMartensite. \nL’effet TWIP est un mécanisme d’écrouissage spécifi que des aciers austénitiques FeMnC lié à un process us de maclage mécanique, \nmécanisme de déformation compétitif au glissement d es dislocations. L’accumulation de ces macles, défa uts plans d’épaisseur \nnanométrique, crée au cours de la déformation une m icrostructure enchevêtrée et difficilement franchis sable par les dislocations \nmobiles à l’intérieur des grains austénitiques. Au cours de nos travaux, ces microstructures ont été e xpliquées et quantifiées à \ndifférentes échelles. Nous avons ainsi pu modéliser  la double contribution du maclage à l’écrouissage grâce à une augmentation de \ndensité de dislocations statistiquement stockées et  à une contribution de nature cinématique, associée  à l’incompatibilité de \ndéformation entre macles et matrice. L’influence de  ce mécanisme a en conséquence été mieux comprise l ors de trajets de mise en \nforme complexes. Afin de pouvoir optimiser le compo rtement mécanique de ces aciers TWIP, notre second axe de recherche a porté \nsur l’effet de leurs compositions chimiques sur ces  mécanismes d’écrouissage, en particulier au traver s de la relation entre maclage \nmécanique et énergie de défaut d’empilement (EDE). Ces travaux ont débouchés sur l’identification d’un  « paradoxe carbone » que \nnous sommes en passe de résoudre. \nMes travaux ultérieurs de modélisation du comportem ent des aciers Dual-Phase (DP) Ferrite-Martensite s e sont aussi attachés à \ndécrire systématiquement les effets de fraction et de tailles des microstructures. Ils ont eu différen tes finalités : \n• l’extension en plasticité polycristalline d’un modè le monophasé analytique pour des applications en rh éologie appliquée \n(prévision des surfaces de charges sous sollicitati ons complexes).  \n• le développement d’un modèle biphasé générique pour  des utilisations en « alloy-design » métallurgique . Le modèle intègre \nen outre nos travaux les plus récents sur les acier s martensitiques (Approche Composite Continu) et a été ajusté sur une \nlarge base de données issues de la littérature. \n• l’approfondissement de nos connaissances sur les ef fets de morphologie et de topologie de la microstru cture DP sur le \ncomportement et la rupture de ces aciers composites . Il passe par le développement d’une chaîne de sim ulation à champs \nlocaux par Eléments Finis (EF), allant de la numéri sation aux calculs sur Volume Elémentaire Représent atif (VER) de la \nmicrostructure, sensibles aux gradients de déformat ion, et intégrant les mécanismes d’endommagements p ertinents. \nL’approche est encore incomplète mais permet de tra iter des questions au premier ordre comme l’aspect néfaste d’une \nstructure en bandes sur l’endommagement. \n \nMy scientific activities aim at understand and pred ict the mechanical behaviour of steels, from fundam ental mechanisms to \nmacroscopic deformation. This manuscript is dedicat ed to the TWIP effect (TWinning Induced Plasticity)  of high manganese \naustenitic steels and DP effect (Dual-Phase) in mor e conventional Ferrite-Martensite steels. \nTWIP effect is a specific work-hardening mechanism of high manganese austenitic steels. It is related to a specific deformation \nprocess which enters in competition to dislocation gliding, the mechanical twinning. The accumulation of twins, which are \nnanometre thick planar defects, generates along wit h the deformation an intricate microstructure which  can hardly be crossed and \novercame by mobile dislocations inside austenitic g rains. In our works and publications, these evolvin g microstructures have been \nquantified at different scales and explained. We th us have modelled the TWIP effect as a double contri bution to work-hardening: an \nisotropic one related to an increase in statistical ly stored dislocations debris due to the decrease i n the mean free paths of mobile \ndislocations (Mecking-Kocks-Estrin process) and a k inematical contribution associated to a strain inco mpatibility between the twins \nand the matrix. The influence of this mechanism has  been better understood along with complex forming operations (change in \nloading directions). In order to optimize the behav iour of these typical steels, our second research a xis is about the effect of alloying \nelements on work-hardening mechanisms, in particula r throughout the relationship between mechanical tw inning and Stacking Fault \nEnergy (SFE). These works has permitted to highligh t a “carbon paradox” that we are quite confident to  solve in a next future. \nMy subsequent works about the modelling of the DP s teels behaviour has permitted to capture systematic ally fraction and size \neffects in these microstructures. Three different a pproaches have been developed with varying objectiv es: \n• Extension in a polycristalline plasticity framework  of one dimension analytical models for application s in rheology (load \nsurface predictions under complex loadings). \n• Development of a generic multiphase mean field mode l for metallurgical alloy-design applications. The model integers our \nmore recent development about martensitic steels (C CA approach) and has been adjusted on a large datab ase from literature. \nThe model is thus validated from IF steels to fully  martensitic steels. \n• Improvement of our knowledge about the morphology a nd topology effect on DP microstructures on behavio ur and \ndamaging processes. It has required developing a Fi nite Element simulation chain, from 3D reconstructi on and meshing of \nRepresentative Volume Element of the microstructure , sensitive to strain gradient, and integrating loc al damaging \nmechanisms (cohesive elements). This local field ap proach is still incomplete but permits to handle fi rst order questions as \nthe detrimental effect of band structure on propert ies. \n \nArcelorm ittal Maizières Research SA, Voie Romaine, BP 30320 F-57283 Maizières les Metz "    },    {        "title": "1909.03275v1.Designing_rare_earth_free_permanent_magnets_in_Heusler_alloys_via_interstitial_doping.pdf",        "content": "arXiv:1909.03275v1  [cond-mat.mtrl-sci]  7 Sep 2019Designing rare-earth free permanent magnets in Heusler\nalloys via interstitial doping\nQiang Gaoa, Ingo Opahlea, Oliver Gutﬂeischa,b, Hongbin Zhanga\naInstitut f¨ ur Materialwissenschaft, Technische Universi t¨ at Darmstadt, 64287, Darmstadt,\nGermany\nbFraunhofer-Research Institution Materials Recycling and Resource Strategies IWKS,\n63457, Hanau, Germany\nAbstract\nBased on high-throughput density functional theory calculations , we investi-\ngated the eﬀects of light interstitial H, B, C, and N atoms on the mag netic\nproperties of cubic Heusler alloys, with the aim to design new rare-ea rth free\npermanent magnets. It is observed that the interstitial atoms ind uce signiﬁcant\ntetragonal distortions, leading to 32 candidates with large ( >0.4 MJ/m3) uni-\naxial magneto-crystalline anisotropy energies (MAEs) and 10 case s with large\nin-plane MAEs. Detailed analysis following the the perturbation theor y and\nchemical bonding reveals the strong MAE originates from the local c rystalline\ndistortions and thus the changes of the chemical bonding around t he intersti-\ntials. This provides a valuable way to tailor the MAEs to obtain competit ive\npermanent magnets, ﬁlling the gap between high performance Sm-C o/Nd-Fe-B\nand widely used ferrite/AlNiCo materials.\nKey words: Permanent magnets, Interstitial, Tetragonal distortion,\nMagneto-crystalline anisotropy energy\n2010 MSC: 00-01, 99-00\n∗Corresponding author\nEmail addresses: hzhang@tmm.tu-darmstadt.de (Hongbin Zhang)\nPreprint submitted to Elsevier September 10, 20191. Introduction\nPermanentmagnetsareofgreattechnicalimportanceformanyk eytechnolo-\ngies such as electric vehicles, wind turbines, and automatisation and robotics to\nnameonlyafew [1]. Lookingatthe intrinsicmagneticproperties, such materials\ndemand a large magneto-crystalline anisotropy energy (MAE), a siz able satura-\ntion magnetization, and a high Curie temperature. The MAE originate s from\nthe spin-orbit coupling (SOC) and sets an upper limit for the microstr ucturede-\npendent coercivityofpermanent magnets. At present, rare-ea rthmagnets based\non Sm-Co (MAE: 17.0 MJ/m3, Magnetization (M s): 910 kA/m) and Nd-Fe-B\n(MAE: 5.0 MJ/m3, Ms: 720 kA/m) are prototypes of high performance perma-\nnent magnets, with a substantial cost and performance gap to ot her classes of\ncommerciallyavailablepermanentmagnets such asAlNiCo (MAE: 0.04MJ /m3,\nMs: 50kA/m) and ferrites(MAE: 0.03MJ/m3, Ms: 125kA/m) [2]. Thus, there\nis a great interest to develop novel permanent magnets so that th e full spectra of\napplicationscan be achieved, ideally without criticalelements such as rare-earth\nelements [3, 4].\nAn enlightening idea was proposedto achievegiant MAE in tetragonally dis-\ntorted FeCo alloys [5], where both the tetragonal distortion and ﬁn e tuning of\nthenumber ofelectronsbyalloyingarecrucialforthe enhancedMA E.Follow-up\nexperimental studies on FeCo alloys deposited on various substrat es conﬁrmed\nthe theoretical prediction [6]. Nevertheless, due to the strong te ndency for the\nFeCo alloys to relax, it is diﬃcult to maintain the tetragonal distortion induced\nby the underlying substrates for thin ﬁlms thicker than 2 nm [6, 7, 8]. Recently,\nfollowing the prediction based on DFT calculations [9, 10], systematic s tudies\nhave been performed on FeCo+X (X= C and B), where spontaneous tetragonal\ndistortions with c/a=1.04 can be induced by a few atomic percent inte rstitial\ndoping ofC or B atomsoccupying the octahedralinterstitial sites. The resulting\nMAE can be as largeas 0.5 MJ/m3with B concentration up to 4 at%, where the\ntetragonal strain reaches 5%. For Fe 0.38Co0.62, a large interstitial concentration\nof 9.6 at% B was achieved. [10] The eﬀect of light interstitials on the ma gnetic\n2properties of body-centered cubic (BCC) iron has also been well st udied.α-\nFe with 12.5 at% content of nitrogen interstitial has been grown by s puttering\non the MgO (100) substrates, leading to about 10% tetragonal dis tortion and\nsigniﬁcant enhancement of magnetization and MAE [11]. First-princip le cal-\nculations and experimental results show that Fe with nitrogen inter stitial has\nsizable MAE, favoring perpendicular magnetization [11]. Using the mole cular\nbeam epitaxy, boron has been incorporated into bcc Fe as interstit ial dopants,\nwhich give rise to tetragonal distortions but the resulting MAE still f avors in-\nplanemagnetizationdue totendency forBatomstobe agglomerate d[12], where\nthe interstitial content of B atoms can be as high as 14 at%.\nConsidering only the crystal structure, the austenite phase of H eusler alloys\nwith the conventional cubic cell can be regarded as a 2 ×2×2 supercell of the\nbcc lattice. In this regard, light interstitials such as H, B, C, and N ca n also\nbe promising to induce signiﬁcant tetragonal distortions and thus s ubstantial\nMAE to Heusler alloys, like the FeCo alloys and bcc Fe. It is noted that t he\nHeusler alloys in the tetragonal martensitic phase do show signiﬁcan t MAE. For\ninstance, among 286 Heusler compounds, a systematic high throug hput (HTP)\nscreening suggests 19 potential tetragonal systems with large o ut-of-plane MAE\n(as large as 0.9 MJ/m3) [13]. Matsushita et al.found 15 Heusler compounds\nhave tetragonal distortions of which the MAEs ranges from -12 MJ /m3to 5.19\nMJ/m3[14]. Focus on Ni based full Heusler compounds, Herper et.al.[15] found\ntetragonal Ni 2FeGe has an MAE of 0.95 MJ/m3, which can be further increased\nto 1 to 2 MJ/m3by non-magnetic doping. Furthermore, imposing strain by\nproper substrates is helpful to engineer a large MAE out of the cub ic Heusler\nalloys. It is found that the out-of-plane MAE of epitaxial Co 2MnGa (001) ﬁlms\ncan be remarkably enhanced from 0.11 MJ/m3to 0.33 MJ/m3by changing\nthe substrate from ErAs/InGaAs/InP to ScErAs/GaAs [16]. Last ly, previous\nexperiments have already demonstrated that interstitials can be in corporated\nintoHeusleralloys,leadingtoenhancedmechanicalstabilityandmagn etocaloric\neﬀect [17, 18]. For Ni 43Mn46Sn11Cx, when the interstitial content x is increased\nfrom 0 to 8 the martensitic phase transformation temperature is in creased from\n3196to249K,whilearemarkableincreaseofMAEisobservedwhenxisin creased\nfrom 0 to 2 [17]. Due to large loss of manganese in content of x=8, the re is even\na distortion of crystal structure from Hg 2CuTi-type to the Cu 2MnAl-type [17].\nSimilar eﬀect has also been observed in Ni 50Mn34.8In14.2, Ni43Mn46Sn11and\nNi50Mn38Sb12doped with B interstitial [18, 19, 20].\nIn this work, focusing on developing rare-earth free permanent m agnets,\nwe have performed high-throughput ﬁrst-principles calculations t o investigate\nthe eﬀects of light interstitials ( e.g., H, B, C, and N) on cubic Heusler alloys.\nAfter identifying the most favorable site preference of the inters titial atoms, the\nMAE of compounds with negative formation energy was evaluated to select the\nmost promising candidates. Apart from thermodynamically stable cr iteria, the\ndisorder eﬀect should also be considered, which is however beyond t he scope of\nthe present paper and saved for future study. We observed tha t the induced\nMAE can be as large as 2.4 MJ/m3, and there are 32 systems with a sizable\nout-of-planeMAE (¿ 0.4 MJ/m3). Detailed analysisbased on the Bain path and\nthe atom-resolvedMAE revealthat not only the globaltetragona ldistortionbut\nalso the associated local chemical bonding are crucial for the inter stitial induced\nmagnetic anisotropy.\n2. Computational details\nStarting with 128 full Heusler alloys with space group Fm ¯3m including at\nleast one of magnetic atoms Cr, Mn, Fe, Co, and Ni from the Inorga nic Crystal\nStructure Database (ICSD) [21] (cf. Table A.1 in Appendix A), we pe rformed\ndensity functional theory (DFT) calculations ﬁrstly to identify the energetically\nmost favored interstitial sites for H, B, C, and N atoms. There are four types\nof interstitial sites based on the symmetries, as shown in Fig 1(a). T he DFT\ncalculations are managed with our in-house developed high-through put envi-\nronment (HTE) [22, 23], using both the Vienna ab initio Simulation Packa ge\n(VASP) [24, 25] and full-potential local-orbital (FPLO) [26, 27] cod es. The\nstructure optimization is performed in a two step manner. Firstly, u ltrasoft\n4pseudopotentials (US-PP) [28] are used in combination with the PW91 [29] ex-\nchange correlation functional, where the cutoﬀ energy for the pla ne wave basis\nis set to 250 eV and and a k-mesh density of 30 ˚A−1. Secondly, the structure\nis relaxed using the projector augmented plane wave (PAW) method with the\nexchange-correlation functional under the generalized gradient approximation\n(GGA) parameterized by Perdew, Burke, and Ernzerhof (PBE) [30 ] with in-\ncreasing plane wave expansion as 350 eV and k-mesh density as 40 ˚A−1to\nachieve good convergence. After obtaining the energy lowest con ﬁguration, the\nMAEs of candidates with negative formation energy are calculated b y using\nFPLO with a k-mesh density of 120 ˚A−1to guarantee ﬁne convergence. For the\nMAE calculations of Ni 2FeGa with C interstitial, the resulting k−mesh is set\nas 24×24×17. The bonding analysis is done in terms of the crystal orbital\nHamilton population (COHP) evaluated using the LOBSTER code [31].\n3. Results and discussions\nAs shown in Fig. 1(a), the systems we considered correspond to do ping\n6.25 at% interstitial atoms (I) into the full Heusler alloys (X 2YZ), leading to\na general chemical formula X 2YZI1/4. This is in accordance with the typical\ndoping concentrations experimentally accomplishable, e.g., 12.5 at% content of\nNin Fe and 9.6at%ofB in Fe 0.38Co0.62. [10,11]Like Fe-Coalloys, weﬁnd light\ninterstitials can indeed cause stable tetragonal distortion to cubic full Heusler\nalloys, which is quantizated by the c/a ratio between the c-axis and in -plane\nlattice constants. As shown in Table 1, with N interstitials, Fe 2NiAl has the a\ntetragonal distortion as large as c/a=1.57. Such strong tetrago nal distortions\nprevail in the other Heuslers with the other types of interstitial at oms, which\nbreak the cubic symmetry and hence lead to possible signiﬁcant MAE. From\nthe theoretical point of view, the MAE is deﬁned as the total energ y diﬀerence\nbetween the magnetization directions parallel to [100] (in-plane) an d [001] (out-\nof-plane) directions as\nMAE =E[100]−E[001] (1)\n5whereEαisthetotalenergywhenmagnetizationdirectionisparallelto α. When\nthe MAE value is positive (negative), the spontaneous magnetizatio n will lie in\nthe out-of-plane (in-plane) direction. Nevertheless, not all the in terstitials are\nthermodynamicallystable,asindicatedbytheformationenergy. Th ecandidates\nwith an MAE more than 0.4 MJ/m3and a negative formation energy are listed\nin Table 1.\nWe notice all the parent Heusler compounds listed in Table 1 are ferro mag-\nnetic apart from Mn 2VGa and Rh 2NiSn. In our high throughput calculations,\nfor convenience, all Heusler compounds are assumed to be ferrom agnetic (FM).\nPrevious studies [32, 33] have shown Rh 2MnAl is an antiferromagnet with Mn\nare antiferromagnetic coupling between nearest neighbors in the ( 111) plane,\nwhich is still in the same antiferromagnetic phase after incorporatin g C or N\ninterstitials. As to Mn 2VGa, experimental research [34] has shown it is a half-\nmetallic ferrimagnet with antiferromagnetic coupling between Mn and V with\na total net saturation magnetization per formula unit as 1.88 µBat 5 K. After\ninducing interstitial (C, B or N), Mn 2VGa is still ferrimagnettic with antifer-\nromagnetic coupling between Mn and V, although initial spin conﬁgura tion is\nferromagnetic. Mn 2VGa have large MAE values as 1.82 MJ/m3, 1.50 MJ/m3\nand 1.26 MJ/m3with B, C and N interstitial, respectively. However, due to the\nferrimagnettic phase, the resulting magnetization densities for Mn 2VGa with\nB, C and N interstitial are as weak as about 0.04-0.05 µB/˚A3. Among all\nlisted compounds in Table 1, Rh 2NiSn is weak ferromagnetic as experimental\nstudy [35] suggests it has a magnetic moment 0.6 µBper formula unit. Our cal-\nculations demonstrate that H interstitials can induce a tetragonal distortion of\nc/a=1.26 and a sizable MAE value as 0.82 MJ/m3, whereas the magnetization\nis only about 0.02 µB/˚A3.\nAs shown in Table 1, we found 32 compounds with a largeout-of-plane MAE\n(¿ 0.4MJ/m3)aswellas10compoundswith largein-planeMAE(absolutevalue\nlarger than 0.4 MJ/m3). In general, the interstitial atoms prefer to be located\nat the octahedral centers (including both the 24f and 24g sites) e xcept for the\nH interstitials in Au 2MnAl which is stable at the tetrahedral center. For the\n6cases of octahedral center, the interstitials mostly prefers 24f sites (1\n4,0,0) where\nthere are the same atoms in the plane perpendicular to the c-axis. O n the\nother hand, for Co 2FeAl with N, Au 2MnAl with N and C, Ni 2MnSn with B,\nC and N, interstitials prefer 24g sites (1\n2,1\n4,1\n4). We note that Fe 3Ge with H\ninterstitial has the largest magnetization density as 0.13 µB/˚A3as well as quite\nlarge MAE value (1.50 MJ/m3), indicating it is a promising permanent magnet.\nFurthermore, comparing with the magnetization and MAE of experim entally\nrealizedpermanentmagnets[1,36,37,38,39,40]Heusleralloyswit hinterstitials\ncan ﬁll the gap between the low performance magnets (such as AlNiC o and\nferrite) and high performance magnets (such as Sm-Co and Nd-Fe -B) in terms\nof MAE and magnetization, which can spread a wide spectrum of applic ations.\nInterestingly, Au 2MnAl with H is the only candidate where the interstitials\nprefer the tetrahedral center (16e site). However, for cases of Au2MnAl with N\nand and C, interstitial prefers to be located in the octahedral cen ters with in-\nplane MAE. Such special interstitial behaviors can be easily underst ood based\non the chemical bonding. Intuitively, due to the large atomic sphere s of Au\natoms, there is more space between the tetrahedron edge bound than the other\nHeusler compound. For instance, in Au 2MnAl with H, the bond length of H-Au\npair in tetrahedral center (1.83 ˚A) is comparable with that in octahedral center\n(1.93˚A). On the other hand, the bond length of Cu-H pair for Cu 2MnAl with\nH in tetrahedral center is just 1.62 ˚A, of which the value is obviously smaller\nthan the H-Au pair for H interstitial in the tetrahedral center of A u2MnAl (1.83\n˚A). This suggests Au atom can really provide more space for interst itials in the\ntetrahedral site. It should be noticed that Cu-H pair in octahedra l center also\nhas a bit larger bond length (1.70 ˚A) than that in tetrahedral center. However,\nin the tetrahedral center case, the bond length is too small to pro vide enough\nspace for the interstitials. Thus, the H interstitials prefer the oct ahedral centers\nin Cu2MnAl. On the other hand, for Au 2MnAl with C and N, it is observed\nthat the interstitial atoms still prefer the octahedral center be cause of the larger\natomic radii of C and N atoms compared to that of H. Therefore, in o rder\nto get the interstitials incorporated at the tetrahedral center, two conditions\n7should be satisﬁed: (a) The interstitial atoms should be small; (b) Th ere should\nbe large atoms in the parent compound, providing more space. Diﬀer ent site\npreference of the H and C/N interstitials induces signiﬁcant change s on the\nMAE of Au 2MnAl,e.g., H-interstitials favor out-of-plane magnetization while\nC/N interstitials lead to in-plane magnetization.\nAccordingtoTable.1, Fe 2CoGawithinterstitialsisapromisingcandidatefor\npermanent magnets. However, in the ICSD database [21], Fe 2CoGa (ICSD ID:\n102385and197615)andFe 2CoGe(ICSDID: 52954)areinthefull Heuslerstruc-\nture, while early M¨ ossbauer measurements have shown Fe 2CoGa and Fe 2CoGe\nare energetically favored in the inverse Heusler structure [41, 42 ]. Previous the-\noretical study [43] found that full Heusler Fe 2CoGa have a martensitic phase\ntransition with a c/a ratio as 1.4, which is also conﬁrmed by our calculat ion\n(cf. Fig. 2(a)). According to our Bain-path calculations, the inver se Heusler\nstructure is still more energetically favored for Fe 2CoGa, even after considering\nH, B, C, and N interstitials. Nevertheless, after introducing inters titials, for\nthe full Heusler structure, the c/a ratio is near to 1.4; whereas fo r the inverse\nHeusler structure, the c/a ratio of Fe 2CoGa with interstitials is just from 1.1-1.2\ndue to there is no metastable phase (Fig. 2(a)). The MAE values of t he inverse\nFe2CoGa with B, C, N, and H interstitial are 0.1026 MJ/m3, 0.2148 MJ/m3,\n0.3798 MJ/m3, and 0.1925 MJ/m3, respectively. Such lower MAE values are\npartially due to that interstitials induce much weaker tetragonal dis tortion to\nFe2CoGa for inverse Heusler structure (1 .1≤c/a≤1.2) than that for full\nHeusler structure (1 .45≤c/a≤1.5).\nMore interestingly, C, N, and H interstitials induce signiﬁcant MAEs to\nNi2FeGa. Experimental studies suggest that Ni 2FeGa can be grown by melt-\nspinning technique [44] or glass-purify method [45], transforming f rom high\nchemical ordering L2 1structure (full Heusler) to martensitic structure at 142\nK with a high Curie temperature of 430 K[44]. Further experiments s howed\npolycrystalline alloys Ni 53+xFe20−xGa27have smaller but comparable entropy\nchangesasclassicalmagnetocaloricHeusleralloysystemsNi-Mn-Ga andNi-Mn-\nSn[46]. DFT calculationssuggestthatNi 2FeGahasatetragonal(corresponding\n8to the martensitic phase) structure of c/a=1.35 [47, 15] with an MA E as 0.318\nMJ/m3[15]. We also found that Ni 2FeGa is stable in the tetragonal structure\nwith a c/a ratio as 1.35 (Fig. 2(b)) and a comparable MAE as 0.2334 MJ/ m3\n(0.0698meV per chemical formula cell). However, the energy diﬀere nce between\ntetragonal and cubic structures is as small as 2.80 meV/atom. As p roposed by\nBarman, the martensite phase transition temperature is proport ional to the\nenergy diﬀerence between cubic and martensite phases [48], as man ifested by\nthe experimental martensitic transition at 142 K [44]. After inducing intersti-\ntial C, H, or N, Ni 2FeGa is stable in the tetragonal phase with c/a≈1.40.\nCorrespondingly, the MAEs have been enhanced to 1.43 MJ/m3, 0.94 MJ/m3,\nand 0.56 MJ/m3for Heusler Ni 2FeGa with N, C, and H interstitials, respec-\ntively. Obviously, C and N interstitials cause more signiﬁcant enhance ment on\nthe MAE than the H interstitials, though the resulting c/a ratios are compara-\nble. Therefore, we suspect that both the tetragonal distortion and the chemical\nbonding environmentwill inﬂuence the MAE values for Heusler with inte rstitial,\nwhich will be discussed in detail below.\nTurning now to the origin of the induced MAE by interstitials, from the\ntheoretical perspective, beside the shape anisotropy due to the magnetic dipole-\ndipole interaction, the magneto-crystalline anisotropy (MCA) can b e attributed\nto the spin-orbit coupling (SOC), which is the dominant contribution t o MAE\nand hence coercivity for PMs. Based on the perturbation theory, Bruno [49]\npointed out that the MCA can be formulated as\nMCA =−/summationdisplay\niξi\n4µB∆µi, (2)\nwhereξidenotes the atomic SOC constant and ∆ µiis the orbital moment\ndiﬀerence between the magnetization directions parallel to [001] an d [100] for\nthei-th atom. We note that such a model is best applicable for strong ma gnets\nwhere the majority spin channel is almost fully occupied, whereas th ere is a\nmore general formula considering the spin-ﬂip and quadruple terms [50]. Taking\nNi2FeGa as an example, Table 2 shows the atom-resolved orbital momen ts and\nthe resultingcontributionstothe MCAusingBruno’sformula, where the atomic\n9SOC constants for Ni and Fe are 630 cm−1(corresponding to 78.1100meV) and\n400 cm−1(corresponding to 49.5937 meV) taken from Ref. [51]. The resulting\nMCA for Ni 2FeGa with C, N, and H interstitials based on the Eq. (2) are 0.316\nmeV/f.u., 0.528 meV/f.u. and 0.330 meV/f.u., respectively. Correspo ndingly,\nthe MAEs based on Eq. (1) are 0.296 meV/f.u., 0.439 meV/f.u. and 0.16 7\nmeV/f.u., respectively. The relative MAE diﬀerences of Bruno’s mode l to that\nof Eq. (1) are 17.17%, 23.83% and 79.61%. Nevertheless, the trend is correctly\nreproduced and we believe the atomic-resolved contributions evalu ated based\non Eq. 2 are still valuable to elucidate the origin of MCA. It is notewort hy that\nthe tetragonal distortion ratios for Ni 2FeGa with H, C and N interstitial are\n1.39, 1.40 and 1.40, respectively (cf. Table 1). To make a direct comp arison to\nthe pristine Ni 2FeGa, we evaluated the MCA and orbital moments for Ni 2FeGa\nwithout interstitials but with imposed c/a=1.40, resulting in an MCA of 0 .066\nmeV and 0.170 meV per chemical formula by using Eq. (1) and Bruno’s m odel\nEq. (2), respectively. Again, the MAEs obtained from the Bruno’s m odel can\nbe well compared with that from Eq. (1) for Ni 2FeGa with C and N interstitial,\nbut not for H interstitial case and parent compound.\nTheremarkablevariationoftheorbitalmomentsandtheresultings igniﬁcant\nenhancement of MCAs can be attributed to the atoms surrounding the intersti-\ntial atoms. It is noted that C and N interstitials can give rise a signiﬁca nt MCA\nto Ni2FeGa, while the eﬀect of H interstitial is rather weaker. Following Ta-\nble 2, it is clear that without interstitials (c/a=1.40), Fe atoms have t he leading\ncontribution to the MCA of 0.26 meV per atom, while the contribution f rom Ni\n(about -0.039 meV per atom) is an order of magnitude lower with oppo site sign.\nThe change in c/a from 1.35 to 1.40 has minor inﬂuence on the MCA and o rbit\nmoment. After considering interstitial N (H), the contribution for Ni-i atoms\nwithin the same plane is enhanced to 0.250 meV (0.097 meV) per atom. A s to\nC interstitial atoms, the MCA of Ni-i atoms is slightly (Fig. 1(b)) incre ased to\n0.039 meV per atom. That is, all types of the interstitial atoms lead t o a sign\nchange of the contribution to MCA for Ni-i. On the other hand, the o rbital\nmoments and thus the resulting MCA contribution are very compara ble for the\n10Ni-ii atoms with and without interstitials, because the Ni-ii atoms are far away\nfrom the interstitials. Furthermore, for the H interstitial case, b oth the MCA\nand orbital moments of all Fe (including iii, iv and v) atoms change only s lightly\ncomparing to those in the pristine compound with imposed c/a=1.40, w hereas\nthe N and C interstitials lead to signiﬁcant enhancement of contribut ion for Fe-\niii atoms to MCA. For instance, the MCA contributions of Fe-iii atoms below\nthe interstitials are increased to 0.681 meV and 0.508 meV per atom wit h N\nand C interstitials, more than two times larger that (0.260 meV) in the parent\ncompound. Meanwhile the contributions from Fe-iv and Fe-v atoms a re slightly\nreduced. Therefore, the interstitial atoms have very strong inﬂ uence on the\nMCA of the local surrounding atoms, while the global tetragonal dis tortion has\nrelatively marginal eﬀects.\nThe eﬀects of interstitials on MCAs and orbital magnetizations can b e fur-\nther understood based on the chemical bonding pairs between the interstitials\nand surrounding magnetic atoms. For instance, the octahedral c enter (intersti-\ntial) H, C and N to octahedral planar corner Ni-i almost have the sam e bond\nlengths as 1.85 ˚A, 1.88˚A and 1.88 ˚A. However, the integrated COHP of H-Ni\nbond is just -0.63 eV, which is much weaker than that of the compara ble bond\nstrength of C-(Ni-i) (-2.12 eV) and N-(Ni-i) (-1.96 eV) bonds. On th e other\nhand, the octahedral below corner Fe-iii to the interstitial H, C an d N have\nsimilar bond lengths as 1.65 ˚A, 1.83˚A and 1.83 ˚A, while the bond integrated\nCOHP of H-Fe (iii) (-1.24 eV) can be comparable to that of C-(Fe-iii) (- 2.88 eV)\nand N-(Fe-iii) (-2.38 eV) bonds. Obviously, the bond strengths of C interstitial\nto Fe-iii) and Ni-i are the strongest. This explains the signiﬁcant cha nge of\norbital moments of Ni 2FeGa with C interstitial comparing to Ni 2FeGa at the\nsame tetragonal distortion ratio without interstitial. In this view, t he H inter-\nstitial just induce tetragonal distortion to Ni 2FeGa, while C and N interstitial\nnot only induce tetragonal distortion but also change the chemical environment\nby forming strong bonds. We notice that the interstitial to planer N i-i atom\nhave comparable bond lengths as to lower Fe-iii atoms but weaker int egrated\nCOHP for each interstitial cases. Such bonding behaviors explains t he eﬀect of\n11interstitial on the magnetization and the MCA for Fe-iii atoms is stro nger than\nthat for Ni-i atoms.\n4. Conclusion\nBased on high-throughput DFT calculations, we investigated the eﬀ ects of\n(H, B, C, and N) interstitials on the magnetic properties of cubic full Heusler\ncompounds. We identiﬁed 32 compounds with substantial uniaxial MA E. De-\ntailed analysis reveals that in addition to the breaking of the cubic sym metry,\nthe changes in the local crystalline environment can induce signiﬁcan t contri-\nbution to the MAE, which can be attributed to the chemical bonding b etween\nthe interstitial and surrounding magnetic atoms. 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(b) The cr ystal structure for the tetragonal full\nHeusler compound Ni 2FeGa with interstitial (int.) at the most stable octahedral sites.\nFigure 2: Total energy as a function of tetragonal distortion ratio (c /a) for Fe 2CoGa\nwith and without interstitials. The reference energy is the energy of the compound in cubic\ninverse Heusler structure for Fe 2CoGa with each interstitial as well as the parent compound.\nThe opened and ﬁlled symbols represent the results Fe 2CoGa in inverse and full Heusler\nstructures, respectively. (b) The total energy as a functio n of tetragonal distortion ratio for\nNi2FeGa in full Heusler structure with and without interstitia ls. Here the reference energy is\nthe energy of the compound in cubic full Heusler structure.\nTable captions:\nTable 1: The basic information of the most promising candidates of He usler compounds\nwith interstitials, where “site” marks the energetically p referred interstitial site, ∆ Hindicate\nthe formation energy in unit of eV/atom, c/a ratio of resulti ng lattice constants along c-axis\nand in-plane, MAE in MJ/m3and meV/f.u. (in parenthesis), total magnetic moment M tot\nin the unit of µB/f.u., and the magnetization M/V in the unit of µB/˚A3. It should be notices\nthe general chemical formula for Heusler compound with inte rstitial is X 2YZI1/4, where I is\nthe interstitial.\nTable 2: The orbital moment ( µl, in unit of µB) and the magneto-crystalline anisotropy\nenergy (MCA, in unit of meV) energy values for Ni 2FeGa with and without interstitials.\nHere the MCA is evaluated from Bruno’s formula. µland/summationtext(in unit of meV) denote the\ndiﬀerence of orbital moment between two magnetization dire ctions ([001] and [100]) and the\nsummation of MCA energy, respectively. The general chemica l formula for Heusler Ni 2FeGa\nwith interstitial is Ni 2FeGaI1/4, where I is the interstitial.\n19Figure 1:\nFigure 2:\nFigures\n20Tables\n21Table 1:\nParent int. site ∆ H c/a MAE M totM/V\nFe2CoGa B 24f -0.0616 1.45 1.4949 (0.4998) 5.56 0.1120\nC 24f -0.0486 1.48 1.3017 (0.4072) 5.37 0.1092\nN 24f -0.1088 1.50 1.3180 (0.4295) 5.36 0.1089\nH 24f -0.0922 1.48 2.3677 (0.6863) 5.95 0.1227\nNi2FeGa C 24f -0.1207 1.40 0.9636 (0.2961) 2.93 0.0595\nN 24f -0.1620 1.40 1.4292 (0.4386) 2.97 0.0602\nH 24f -0.1916 1.39 0.5582 (0.1668) 3.13 0.0654\nFe2CoGe H 24f -0.0627 1.51 0.6291 (0.2080) 5.48 0.1142\nN 24f -0.0576 1.56 0.4047 (0.1368) 5.01 0.1025\nFe2NiAl H 24f -0.2545 1.53 0.4947 (0.1298) 4.56 0.0955\nN 24f -0.2842 1.57 0.5270 (0.1368) 4.40 0.0902\nFe2NiGa H 24f -0.1915 1.55 0.5670 (0.1298) 4.67 0.0977\nB 24f -0.1219 1.51 0.7853 (0.1758) 4.42 0.0893\nC 24f -0.1208 1.53 0.9217 (0.1863) 4.31 0.0874\nN 24f -0.1620 1.53 1.3295 (0.2429) 4.45 0.0939\nCo2MnGa C 24f -0.1289 1.13 0.5267 (0.6303) 4.52 0.0920\nN 24f -0.1836 1.12 0.4755 (0.1576) 4.76 0.0922\nCo2MnGe C 24f -0.0788 1.25 0.5388 (0.1226) 4.06 0.0831\nN 24f -0.1226 1.29 0.5476 (0.1325) 4.13 0.0841\nCo2MnSi C 24f -0.2571 1.21 0.5384 (0.1464) 4.16 0.0898\nRh2MnAl C 24f -0.5088 1.10 0.9501 (0.3336) 4.25 0.0742\nN 24f -0.5457 1.06 1.1675 (0.4487) 4.49 0.0784\nRh2NiSn H 24f -0.2288 1.26 0.8236 (0.3063) 0.99 0.0166\nMn2VGa C 24f -0.1533 1.20 1.5038 (0.4874) 2.26 0.0435\nB 24f -0.1474 1.23 1.8263 (0.5987) 2.48 0.0472\nN 24f -0.2377 1.21 1.2674 (0.4087) 2.34 0.0451\nCo2FeAl N 24g -0.2770 1.08 0.4881 (0.3009) 5.01 0.1038\nAu2MnAl H 16e -0.1835 0.92 0.7732 (0.6489) 3.82 0.0582\nN 24g -0.1975 1.27 -0.4091 (-0.2412) 3.66 0.0476\nC 24g -0.2770 1.21 -0.5271 (-0.4923) 3.82 0.0574\nNi2MnIn C 24f -0.0057 1.21 -1.0288 (-0.3513) 3.96 0.0685\nNi2MnGa H 24f -0.2519 1.27 -1.3278 (-0.4898) 4.20 0.0852\nB 24f -0.2204 1.28 -0.5822 (-0.1850) 4.02 0.0788\nC 24f -0.1780 1.29 -0.9573 (-0.3031) 3.91 0.0771\nFe3Ge H 24f -0.0563 1.42 1.5018 (0.4655) 6.44 0.1319\nB 24f -0.0254 1.16 0.5868 (0.1812) 5.54 0.1211\nFe3Ga B 24f -0.0710 1.21 0.6896 (0.2142) 6.16 0.1229\nN 24f -0.1101 1.19 0.5184 (0.1610) 5.92 0.1022\nNi2MnSn B 24g -0.0959 1.06 -0.6747 (-0.2064) 3.75 0.0645\nC 24g -0.0480 1.17 -0.4261 (-0.1421) 3.70 0.0653\nN 24g -0.0892 1.17 -0.4756 (-0.1613) 3.74 0.0697\nRh2MnSn C 24f -0.2679 1.26 -0.8846 (-0.3529) 3.66 0.0572\n22Table 2:\nint. Ni-iNi-ii Fe-iiiFe-ivFe-v/summationtextwith int.H[001]0.0240.020 0.0680.0630.066\n[100]0.0190.022 0.0430.0400.046\n∆0.005-0.002 0.0250.0230.020\nMCA0.097-0.039 0.3090.2840.2470.330\nN[001]0.0250.020 0.0680.0670.072\n[100]0.0120.024 0.0130.0530.049\n∆0.013-0.004 0.0550.0140.023\nMCA0.250-0.078 0.6810.1730.2850.528\nC[001]0.0130.022 0.0580.0670.073\n[100]0.0110.024 0.0170.0480.051\n∆0.002-0.002 0.0410.0190.021\nMCA0.039-0.039 0.5080.2350.2600.316\nc/a Ni Fe/summationtextw/o int.1.35[001]0.022 0.065 -\n[100]0.024 0.044 -\n∆-0.002 0.021 -\nMCA-0.039 0.260 0.180\n1.40[001]0.021 0.061 -\n[100]0.023 0.041 -\n∆-0.002 0.020\nMCA-0.039 0.248 0.170\n23Highlights\n•Rare earth free permanent magnets can be realized in tetrago nally distorted fullHeusler\nalloys induced by light interstitial atoms.\n•Bain path calculations reveal that interstitials cause sta ble tetragonal distortion to full\nHeusler alloys.\n•Analysis based on the perturbation theory and chemical bond ing suggests that the\nuniaxial anisotropy can be attributed to change in the local crystalline environments\naround the interstitials.\n•We postulate that this provides a universal way to tailor the magnetic properties of\nprospective permanent magnets.\n24A. Appendix\n25Table A.1: All the considered Heusler compounds (Com.) toge ther with the ICSD ID number.\nCom. ID Com. ID Com. ID Com. ID\nAu2MnAl 57504 Co2CrAl 57600 Co2FeAl 57607 Co2HfAl 110809\nCo2MnAl 606611 Co2NbAl 57620 Co2TaAl 606667 Co2TiAl 606680\nCo2VAl 57643 Co2ZrAl 57648 Co2CrGa 102318 Co2CrIn 416260\nCo2FeGa 102392 Co2FeGe 247268 Co2FeIn 102392 Co2FeSi 622985\nCo2HfGa 102433 Co2MnGa 623116 Co2NbGa 623126 Co2TaGa 102451\nCo2TiGa 102453 Co2VGa 623228 Co2LiGe 53673 Co2MnGe 52971\nCo2TiGe 169469 Co2ZnGe 52994 Co2HfSn 102483 Co2MnSb 53002\nCo2MnSi 106484 Co2MnSn 102332 Co2NbSn 102554 Co2ScSn 102646\nCo2TiSi 53080 Co2VSi 53086 Co2TiSn 102583 Co2VSn 102684\nCo2ZrSn 102687 Cu2CrAl 57653 Cu2MnAl 607012 Cu2CoSn 103057\nCu2FeSn 151205 Cu2MnIn 102996 Cu2MnSb 53312 Cu2MnSn 103057\nCu2NiSn 103069 Fe2CrAl 184446 Fe2MnAl 57806 Fe2MoAl 57807\nFe2NiAl 57808 Fe2TiAl 57827 Fe2VAl 57832 Fe2CoGa 103473\nFe2CoGe 52954 Fe2CrGa 102755 Fe2NiGa 103460 Fe2TiGa 103469\nFe2VGa 103473 Fe2MnSi 632569 Fe2VSi 53555 Fe2TiSn 103641\nFe2VSn 103644 Mn2VAl 57994 Mn2RhGa 247951 Mn2VGa 103813\nMn2RuGe 247950 Mn2RuSn 247949 Mn2WSn 104980 Ni2CrAl 57662\nNi2HfAl 57901 Ni2MnAl 57976 Ni2NbAl 58016 Ni2ScAl 58050\nNi2TaAl 58055 Ni2TiAl 58063 Ni2VAl 58071 Ni2ZrAl 58081\nNi2CuSb 53320 Ni2CuSn 103068 Ni2HfGa 103734 Ni2MnGa 103803\nNi2NbGa 103839 Ni2ScGa 103874 Ni2TaGa 103881 Ni2TiGa 103886\nNi2VGa 103892 Ni2ZrGa 103902 Ni2LiGe 53673 Ni2MnGe 192566\nNi2ZnGe 53865 Ni2HfIn 54595 Ni2HfSn 104250 Ni2MgIn 51982\nNi2MnIn 639954 Ni2ScIn 59446 Ni2TiIn 59451 Ni2ZrIn 59460\nNi2LiSi 44819 Ni2LiSn 25325 Ni2MgSb 104841 Ni2MgSn 104842\nNi2TiSb 76700 Ni2ZrSb 76703 Ni2ScSn 105339 Ni2TiSn 105369\nNi2VSn 105376 Ni2ZrSn 105383 Pd2MnAl 57981 Pd2MnAs 107955\nNi2NbSn 105181 Pd2MnGe 53705 Rh2NiSn 105327 Pd2MnIn 51990\nPd2MnSb 643312 Pd2MnSn 104945 Rh2MnAl 57986 Rh2MnGe 53706\nRh2MnPb 104936 Rh2MnSn 104964 Ru2FeSi 53525 Ru2FeSn 103615\nFe3Al 57793 Fe3Ga 108436 Fe3Ge 53462 Fe3Si 53545\nMn3Si 76227 Ni3Al 58038 Ni3Sb 76693 Ni3Sn 105354\n26"    },    {        "title": "1202.6166v1.Gallium_Substituted__114__YBaFe4O7__From_a_ferrimagnetic_cluster_glass_to_a_cationic_disordered_spin_glass.pdf",        "content": " 1 Gallium Substituted “114”  YBaFe 4O7: From a ferrimagnetic cluster glass to \na cationic disordered spin glass  \n \nTapati  Sarkar *, V. Caignaert, V. Pralong and B. Raveau  \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal Juin, 14050 CAEN, France  \n \nAbstra ct \n \n The study of the ferrites  YBaFe 4-xGaxO7 shows that the substitution of Ga for Fe in \nYBaFe 4O7 stabilizes the hexagonal symmetry for 0.40 \n x \n 0.70, at the expense of the cubic \none. Using combined measurements of a. c. and d. c. magnetization, we estab lish that Ga \nsubstitution for Fe in YBaFe 4O7 leads to an evolution from a geometrically frustrated spin \nglass (for x = 0) to a cationic disorder induced spin glass (x = 0.7 0). We also find  an \nintermediate narrow range of doping where the samples are clearl y phase separated having \nsmall ferrimagnetic clusters embedded in a spin glass matrix. The origin of the ferrimagnet ic \nclusters  lies in the change in symmetry  of the samples  from cubic to hexagonal (and a \nconsequent lifting of the geometric al frustration) as a result of  Ga doping.  We also show the \npresence of exchange bias and domain wall pinning in these samples. The cause of both these \neffects can be traced back to the inherent ph ase separation present in the  samples.  \n \n \n \n \n \n \n \n \nKeywords : “114” oxides, magne tic frustration , phase separation .   \n \n \n \n \n* Corresponding author: Dr. Tapati Sarkar  \ne-mail: tapati.sarkar @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 Introduction  \n \n The recent studies of the “114” cobaltites (Ln,Ca) 1BaCo 4O7 [1 – 5] and ferrit es \n(Ln,Ca) 1BaFe 4O7 [6 – 8] have generated a lot of interest in the scientific community because \nof their complex magnetic, electronic  and thermoelectric properties  [9]. These cobaltites and \nferrites have the same basic structure, and are closely related to  spinels and barium \nhexaferrites by their close packing of “O 4” and “BaO 3” layers.  This close packing forms a 3 -\ndimensional framework [Fe 4O7]\n (or [Co 4O7]\n) consisting of corner -sharing FeO 4 (or CoO 4) \ntetrahedra , with the lanthanide elements occupying the octahedral sites of this framework.  The \ntriangular geometry of the cobalt (or iron) sublattices ( Fig. 1 ) plays a dominant role in their \nmagnetic properties. It was indeed shown that for hexagonal LnBaCo 4O7 cobaltites ( Fig. 1 \n(a)), there exists a strong com petition between the 1 D magnetic ordering along the \n  \ndirection in the “Co 5” trigonal bipyramids, and the magnetic frustration in the (001) plane \nbuilt up of “Co 3” triangles [10, 11]. In fact, the magnetic frustration can be lifted by an \northorhombic dist ortion of the structure. This is illustrated by the concomitant structural and \nmagnetic transitions that appear at low temperature in these cobaltites [1, 2, 12], and by the \nferrimagnetic structure of CaBaCo 4O7 [13]. Similarly, the “114” ferrites exhibit a  competition \nbetween 1 D magnetic ordering and 2 D magnetic frustration, as has been shown for the \nhexagonal phases CaBaFe 4O7 [6], and for CaBa Fe4-xLixO7 [14]. But importantly, the “114” \nferrites differ from the “114” cobaltites by the fact that the LnBaFe 4O7 oxides exhibit a cubic \nstructure [7]. Though the latter is closely related to the hexagonal structure, the iron sublattice \nis very different ( Fig. 1 (b) ), consisting of “Fe 4” tetrahedra instead of “Fe 5” bipyramids and \n“Fe 3” triangles. No structural tra nsition appears at low temperature, and consequently, the \ncubic ferrites exhibit a spin glass behaviour due to a perfect  geometrical frustration.  Further, \nthe LnBaFe 4O7 series exhibit s an oxidation state disorder. Unlike the case of CaBaCo 4O7 [13], \nno char ge ordering is observed in LnBaFe 4O7, and this disorder is also important for the \nobserved glassiness.  \nRecently, we showed that the substitution of a divalent cation, Zn2+, for iron in \nYBaFe 4O7, allowed the hexagonal symmetry to be stabilized at the detrim ent of the cubic one \n[15]. Paradoxically, it was observed that the substitution of this diamagnetic cation for Fe2+ \ninduces ferrimagnetism, in contrast to the spin glass behaviour of the undoped phase \nYBaFe 4O7. In fact, a competition between ferrimagnetism  and magnetic frustration was \nobserved for the hexagonal phase YBaFe 4-xZnxO7. This was interpreted as the effect of two  3 antagonist phenomena: the partial lifting of the geometrical frustration due to the appearance of \nthe hexagonal symmetry inducing a 1 D magnetic ordering, and the existence of cationic \ndisordering favouring the glassy state.  \n Bearing in mind that the Fe2+:Fe3+ ratio is a crucial factor governing  the magnetic \nproperties of iron oxides, it must be emphasized that the substitution of Zn2+ for Fe2+ increases \nthe average valence of iron , i.e. the Fe2+:Fe3+ ratio decreases from 3 in the spin glass phase \nYBaFe 4O7 to 2.6 – 1.5 in the solid solution YBaFe 4-xZnxO7 when x changes from 0.4 to 1.5 \n[15]. In order to further understand the role of the ave rage valence of iron in the magnetic \nproperties of these ferrites, we have investigated the possibility of substitution of a diamagnetic \ncation such as gallium for Fe3+ in the YBaFe 4O7 structure. In the present study of the ferrite \nYBaFe 4-xGaxO7, we show t hat the introduction of gallium in the structure stabilizes the \nhexagonal symmetry, similar to the zinc substitution, but differently from the latter, the lifting \nof the geometrical frustration induces the formation of ferrimagnetic clusters embedded in a \nspin glass matrix, which tend to disappear as the gallium content increases, leading to a pure \nspin glass for higher Ga content, with a higher T g compared to YBaFe 4O7. \n \nExperimental  \n \nPhase -pure samples of YBaFe 4-xGaxO7  [x = 0.4 0 – 0.70] were prepared by solid state \nreaction technique. The precursors used were Y 2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe \npowder. First, the precursor BaFe 2O4 was prepared from a stoichiometric mixture of BaCO 3 \nand Fe 2O3 annealed at 1200°C for 12 hrs in air. In a second step, a stoichiometric mixture of \nY2O3, BaFe 2O4, Ga 2O3, Fe 2O3 and metallic Fe powder was intimately ground and pressed in \nthe form of rectangular bars. The bars were then kept in an alumina finger, sealed in silica \ntubes under vacuum and annealed at 1100°C for 1 2 hrs. Finally, the samples were quenched to \nroom temperature in order to stabilize the “114” phase.  \nThe X -ray diffraction patterns were registered with a Panalytical X’Pert Pro \ndiffractometer under a continuous scanning mode in the 2\n  range 10° - 120° an d step size \n 2\n \n= 0.017°.  The cationic composition was confirmed by means of Energy Dispersive X -Ray \nSpectroscopy  (EDS) technique using a Scanning Electron Microscope (ZEISS Supra 55). The \nd. c. magnetization measurements were performed using a superconduc ting quantum \ninterference device (SQUID) magnetometer with variable temperature cryostat (Quantum \nDesign, San Diego, USA). The a. c. susceptibility, \n ac(T) was measured with a PPMS from \nQuantum Design with the frequency ranging from 10 Hz to 10 kHz. H ac was kept fixed at 10  4 Oe, while H dc was varied from 0 Oe to 2000 Oe. All the magnetic properties were registered on \ndense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \nResults and discussion  \n \n \n Similar  to Zn substitution, Ga substitution also favours the forma tion of the hexagonal \nphase at the expense of the cubic one. Nevertheless, the homogeneity range of the hexagonal \nYBaFe 4-xGaxO7 solid solu tion is significantly different  (0.40 \n  x \n 0.70) vis – à – vis that of \nYBaFe 4-xZnxO7 [15]. The cubic symmetry of YBaF e4O7 is retained for 0 \n  x \n 0.20, whereas \nthe domain 0.20 < x < 0.40 is biphasic, corresponding to a mixture of the cubic and hexagonal \nphases.  On the other hand,  for x > 0.70 , several impurity phases appear , namely Y 2O3 and \nGa2O3. The cationic compositio n of the single phase obtained for the range 0.40 \n  x \n 0.70 \nusing EDS analysis are also shown in Table 1 .  \n  \nStructural characterization  \n \n In Fig. 2, we show the X -ray diffraction (XRD) pattern s of the two end members, (a) \nYBaFe 3.6Ga0.4O7 and (b) YBaFe 3.3Ga0.7O7 as representative example s. As stated before, t he \nsample s are seen to stabilize in the hexagonal symmetry with the space group P63mc. The \nRietveld analysis of the lattice structure was done using the FULLPROF refinement program \n[16] and the fit s are also shown in Fig. 2. All the samples in the range x = 0.4 0 – 0.70 were \nseen to stabilize in the same hexagonal symmetry.  \nThe extracted cell parameters have been tabulated in Table 1 . The ionic radius of Fe3+ (0.49 \nÅ) is very similar to that of Ga3+ (0.47 Å). As can be seen from the extracted cell parameters \nshown in Table 1, a increases very slightly as x increases (an increase of only ~ 0.08 % as x \nincreases from 0.4 to 0.7), while c shows a slight decrease (~ 0.11 %). This causes the cell \nvolume to re main practically unchanged as a function of doping in accordance with the \nsimilar ionic radii of Fe3+ and Ga3+. \n \nD. C. magnetization studies  \n \n In the “114” ferrites, it has been established earlier [8, 15] that ferrimagnetism is \ninherently linked with the cross -over from cubic to hexagonal symmetry. The doping -induced \ntransition to the hexagonal symmetry involves a partial lifting of the 3D geometrical  5 frustration, which is the root cause of the appearance of ferrimagnetism. Thus, we restrict our \ndiscussion  of the magnetic data to the YBaFe 4-xGaxO7 samples exhibiting hexagonal symmetry \n(0.4 \n  x \n 0.7). We note here that the cubic samples (x < 0.2) are spin glasses similar to the \nundoped YBaFe 4O7, and will not be discussed further.  \n The temperature dependence  of d. c. magneti c susceptibility (\n dc = M/H)  was registered \naccording to the standard zero field cooled (ZFC) and field cooled (FC) procedures. A \nmagnetic field of 0.3 T was applied during the measurements. The measurements were done in \na temperature rang e of 5 K to 300 K. The \n ZFC(T) and \n FC(T) curves of all the samples are \nshown in Fig. 3. \n Undoped YBaFe 4O7 is a spin glass with T g = 50 K [ 7]. The ZFC \n dc versus T curve for \nYBaFe 4O7 shows a pure cusp -like shape [ 7], typical of canonical spin glasses.  A close look at \nFig. 3 reveals that the YBaFe 4-xGaxO7 series of samples shows two different kinds of low \ntemperature M ZFC(T) curves vis -à-vis the shape of the curves. While the \n dc(T) curve of the \nhighest substituted  sample (x = 0.7 0) is very similar to that o f canonical spin glasses (with a \npeak at ~ 50 K and a gradual decrease of the magnetization value below 50 K),  for the lowest \ndoped sample (x = 0.4 0), there is a sharp drop in the susceptibility  value below the temperature \nat which \n ZFC reaches its maximum  value (75 K). The susceptibility  value drops sharply till ~ \n50 K (marked by a black arrow in Fig. 3 (a)), below which the decrease in \n ZFC is more \ngradual.  We note here that the measuring field that we have chosen (0.3 T) is smaller than the \ncoercive fiel d of the YBaFe 3.6Ga0.4O7 sample at T = 5 K (data shown later in Fig. 5 ). Thus, it is \nquite possible that the sharp drop in the susceptibility value occurs at the temperature where the \ncoercive field of the sample becomes smaller than 0.3 T. However, follow ing this argument, \nwe should have obtained similar sharp drops in the \n ZFC(T) curves for the x = 0.5 and x = 0.6 \nsamples also, as the coercive fields of the x = 0.5 and x = 0.6 samples at T = 5 K are also larger \nthan 0.3 T. Instead , it is observed that the  sharp drop in the \n ZFC(T) curve seen in the x = 0.40 \nsample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.50 and x = 0.60 samples.  \nMore importantly, a study of the temperature dependence of the coercive field (H C) of the x = \n0.4  sample (d ata not shown here) reveals that H C becomes smaller than 0.3 T  at ~ 17 K (i.e. at \na temperature much below 50 K).  This suggests  that this feature is not a simple effect of the \ncoercive  field, rather it may have  a more complex origin.  \nAnother possibility is  that this sudden decrease in \n ZFC(T) is due to domain wall \npinning effects, which  has, in fact,  been observed  previously  in manganites [1 7 – 19]. Due to \npinning, the domains would not  freely rotate below the pinning temperature unless a high  6 enough extern al field is present to overcome the pinned state. Upon zero field cooling, the \ndomains would be  pinned into random orientations. Whe n a low field is applied (0.3 T  in this \ncase), the pinning effect still dominates over the effect of the applied magnetic fi eld, and the \nmagnetization is lower than what would be expected in the absence of pinning. However, the \npinned domain walls can be thermally activated by increasing the temperature. This could be \nthe cause of the  visible jump in the \n ZFC(T) curve at the te mperature where the pinning effects \nare overcome by temperature (~ 50 K).  As can be seen in Fig. 3, the pinning effect gradually \ndecreases as the doping concentration is increased (the sharp drop in the \n ZFC(T) curve seen in \nthe x = 0.4 0 sample is reduced to small kinks in the \n ZFC(T) curves for the x = 0.5 0 and x = \n0.60 samples, and completely vanishes for the x = 0.7 0 sample).  This indicates  that the domain \nwall pinning is more prominent for small doping and vanishes for higher doping. This is \ncounter -intuitive if the pinning is thought to arise due to the presence of Ga in the lattice. Thus, \nthe fact that the domain wall pinning decreases with an increase in the doping concentration \nleads us to believe that this pinning does not arise from the disorder in  the system. Rather, it \nhas a more complex origin, which we discuss later.  \n Before we proceed further, we perform some additional measurements to make sure \nthat the sharp drop in the \n ZFC(T) curve observed in the lowest doped sample is indeed due to \ndomain  wall effects , and not arising from some additional (antiferro) magnetic transition in the \nsample.  Thus, we subject the x = 0.4 0 sample to a degaussing experiment [1 7], wherein the \nsample was initially cooled from 300 K down to 5 K in a zero external magne tic field. At 5 K, \na large magnetic field (5 T) was applied. The magnetic field was then reduced to zero, and the \nsample was degaussed at 5 K by cycling a field of reducing intensity so that the remanent \nmagnetization of the sample was reduced to zero. A m agnetic field of 0.3 T was then applied, \nand the \n dc(T) curve was recorded while warming the sample, in the same wa y as ZFC \nmagnetization is recorded. The results are shown in Fig. 4. We find that the sudden sharp drop \nobserved in the normal ly obtained  ZFC curve ( Fig. 4 (a)) vanishes when the sample is \nsubjected to a high enough magnetic field, and then degaussed  (Fig. 4 (b)). This experiment, \nthus, provides supplementary evidence that the sudden drop in magnetization seen below 75 K \nis not due to any kind of (antiferro) magnetic transition  in the sample , but is probably  \nassociated with domain wall pinning effects. The high magnetic field (5 T) to which the sample \nwas subjected was sufficient for domain wall displacements thereby destroying  the pinning. In \nfact, a ZFC magnetization recorded under a high enough field of 5 T (see inset of Fig. 4 (a)) \ndoes not show any sharp drop in the magnetization of the sample.   7 The d. c. magnetization M(H) curves of all the samples  registered at T = 5 K  are shown \nin Fig. 5. The virgin curves of the M(H) loops are represented by black circles while the rest of \nthe M(H) loops are shown by red lines. The first notable point is that for higher Ga substitution \n(x = 0.7 0), the M(H) loop is narrow and S – shaped ( Fig. 5 (d)), which is quite typical of spin \nglasses  and superparamagnets . On the other hand, for lower Ga substitution, the samples have \nlarger loops with higher values of the c oercivity and remanent magnetization, which keep \nincreasing as the doping concentration is decreas ed. This indicates the presence of a higher \ndegree of magnetic ordering in the lower doped samples as compared to the higher doped ones.  \n Another feature  which strongly supports the presence of domain wall pinning  is that the \nvirgin curve of the x = 0.4 0 sample lies slightly outside the main M(H) loop ( Fig. 5 (a)). This \nunusual feature of the virgin curve lying outside the main hysteresis loop has earlier been \nassociated with irreversible domain wall motion in spinel oxides [ 20]. We also note that the \nvirgi n curve starts to shift inside the main M(H) loop as the doping concentration (x) is \nincreased, and for the x = 0.7 0 sample, the entire virgin curve lies inside the main M(H) loop \n(Fig. 5 (d)). We again note that the domain wall pinning is more prominent i n the samples with \nlower doping concentration.  \n In Fig. 6, we once again show the d. c. magnetization M(H) curves of all the samples \nregistered at T = 5 K, but in three different modes: (i) normal ZFC mode, (ii) FC mode with a \nmagnetic field of 2 T  and (iii) FC mode with a magnetic field of H = - 2 T. In the ZFC mode, \nthe samples were cooled from 300 K to 5 K in zero external magnetic field, following which M \nversus H curves were registered. In the FC mode, on the other hand, the samples were cooled \nfrom 30 0 K to 5 K in the presence of an external magnetic field (H = 2 T  or – 2 T), and then M \nversus H curves were registered.  For the highest substituted  sample (x = 0.7 0), all three  M(H) \ncurves overlap each other ( Fig. 6 (d)). However, for lower doping concent ration, the field \ncooled M(H) loops exhibit shifts both in the field as well as in the magnetization axes.  This is \nthe exchange bias phenomenon [ 21, 22] that results from exchange interaction between \nferromagnetic and antiferromagnetic materials. In our YB aFe 4-xGaxO7 samples with low Ga \nconcentration, the observed exchange bias can be explained in terms of interfacial exchange \ncoupling between the coexisting ferrimagnetic cluster glass and the disordered spin glass -like \nphases.  This exchange bias effect ari sing from the inherent phase separation in the YBaFe 4-\nxGaxO7 samples is similar to that seen in some disordered manganites [ 23]. As can be seen \nfrom Fig. 6, the exchange bias effect keeps decreasing as the doping concentration is increased, \nand as stated b efore, it completely disappears for the doping concentration x = 0 .70. We can \nexplain this observation by considering that for lower Ga concentration, the samples consist of  8 coexisting ferrimagnetic clusters embedded in a spin glass -like matrix, but as the  doping \nconcentration by the diamagnetic cation (Ga) is increased, the ferrimagnetic clusters are \nprogressively reduced and we ultimately get a homogeneous spin glass (x = 0.7 0). The absence \nof phase separation in the x = 0.7 0 sample, thus, results in an a bsence of the exchange bias \neffect. The fact that the YBaFe 4-xGaxO7 samples with lower Ga concentration are intrinsically \nphase separated, while the x = 0.7 0 sample is not, also affords us an alternative explanation for \nthe domain wall pinning effects seen  in the lower doped samples. As stated previously, the fact \nthat the domain wall pinning is seen in the lower doped samples and not in th e x = 0.7 0 sample \nmeans that it cannot arise from the disorder in the system. Rather, we believe that the pinning \narise s from an interplay between the two magnetic phases in the phase separated samples. Such \na domain wall pinning process arising from the interplay between two coexisting magnetic \nphases has been seen earlier in intermetallic alloys [ 24]. We also note that a part from the \nexchange bias effect in the lower doped samples, field cooling also results in an overall \nincrease in the coercivity and remanence magnetization values. This can be interpreted as an \nincrease in the volume fraction of the magnetically ordered  phase when the samples are cooled \nin the presence of an external magnetic field.  Since field cooling improves the remanence in \nthe lower doped samples, hence it was important to register M -H curves after field cooling  \nwith positive as well as negative coo ling fields and check whether the M -H loops shift in \nopposite directions in order to confirm that there is indeed a genuine exchange bias effect in \nthe lower doped samples.  \n \nA. C. magnetic susceptibility studies  \n \n The measurements of the a. c. magnetic sus ceptibility \n ac(T, f, H) were performed at \ndifferent frequencies ranging from 10 Hz to 10 kHz, and different external magnetic field s \n(Hdc) ranging from 0 T to 0.2 T using a PPMS facility. The amplitude of the a.  c. magnetic \nfield was ~ 0.001 T In Fig. 7 and Fig. 8, we show the temperature dependence of the real (in -\nphase) component of the a. c. susceptibility  in the temperature range 10 K – 160 K  of the \nlowest doped sample (x = 0.4 0) and the highest doped sample (x = 0.7 0) respectively, with a  \nmeasuring f requency of 10 kHz and in zero magnetic field ( Hdc = 0).  \nFrom Fig. 7, it is clear that the \n '(T) curve of the x = 0.4 0 sample shows two features, \none at T = 86 K (marked by a black arrow), and the second at T = 44 K (marked by a red \narrow). While the high  temperature feature at 86 K is a clear peak, the low temperature one at \n44 K is a shoulder like feature and is more clearly evidenced in the imaginary (out -of-phase)  9 component of the a. c. susceptibity (shown in inset (a) of Fig. 7). Repeating the measure ments \nusing four measuring frequencies (ranging from 10 Hz – 10 kHz) reveals that both the features \nare frequency dependent (this is shown in inset (b) of Fig. 7). The x = 0.7 0 sample, on the other \nhand, shows only one feature (at T = 60 K) in the \n '(T) an d \n''(T) curves, shown in the main \npanel and inset (a) of Fig. 8 respectively. Inset (b) of Fig. 8 shows the \n '(T) curves measured \nusing four different frequencies, and reveals that this peak  at 60 K is also strongly frequency \ndependent.  We note that the i maginary part of \n  for YBaFe 3.6Ga0.4O7 is ~ 8 % of the real part \nof \n . This is commonly found for systems where the spin domains are relatively large. \nYBaFe 3.6Ga0.4O7 can thus be described as a cluster glass. On the other hand, the imaginary part \nof \n for YBaFe 3.3Ga0.7O7 is significantly smaller (about 2.4% of the real part of \n ) which \nindicates that YBaFe 3.3Ga0.7O7 is closer to a canonical spin glass.  \n Although all the features in the a. c. susceptibility curves of the two samples described \nabove have a si ngle commonality , in that they are all strongly frequency dependent, but the \nnature a nd origin of these peaks can be quite  different.  Specifically, we need to establish the \nnature of the low temperature shoulder in the x = 0.4 sample at T ~ 50 K. Since it occurs close \nto the temperature where we ha ve evidenced domain wall pinning from  the d. c. magnetic data, \nit is tempting to attribute this low temperature shoulder in the a. c. susceptibility data to the \nsame phenomenon. However, it is also possible that t his low temperature feature is a \nsuperparamagnetic effect of the ferrimagnetic clusters.  To investigate this , we perform further \nmeasurements of \n '(T) and \n ''(T) of the two limiting samples  (x = 0.4 0 and x = 0.7 0) in the \npresence of different external magn etic fields H dc ranging from 0 to 0.2 T. The results for the \nYBaFe 3.3Ga0.7O7 sample are shown in Fig. 9. It is seen that both \n ' and \n '' are strongly \nsuppressed by the magnetic field.  The peak also shows a continual shift towards lower \ntemperature as the e xternal magnetic field is increased  (see the black arrows in Fig. 9). This is \ntypical of the behaviour of a spin glass freezing temperature under the influence of magnetic \nfield. \n In Fig. 10, we show the results for the YBaFe 3.6Ga0.4O7 sample. It is seen t hat w hile the \nhigh temperature peak was significantly suppressed in the presence of external magnetic field, \nthe low temperature peak was largely unaffected relative to the case H dc = 0. This rules out the \nscenario of superparamagnetism being responsible f or this low temperature feature, and \nconfirms that the 50 K anomaly arises due  to enhanced domain wall pinning,  signatures of \nwhich have been observed and commented upon earlier in the d. c. magnetization \nmeasurements also.  We also note that the high tempe rature peak shifts towards higher  10 temperature as the external magnetic field is increased (see the black arrow in Fig. 10 (a)). \nThis is not expected for a pure  spin glass freezing transition, where the peak should shift \ntowards lower temperature as the mag netic field is increased. We believe that this anomaly \narises because the x = 0.4 0 sample is not a pure spin glass, rather it is a phase separated sample \nconsisting of ferrimagnetic clusters embedded in a spin glass matrix.  \n \nConclusion  \n \n These results show  that the substitution of Ga3+ for Fe3+ in YBaFe 4O7 induces a \nstructural transition from cubic to hexagonal, similar to the substitution of Zn2+ for Fe2+ in this \ncompound. Though the two types of substitutions induce a lifting of the geometrical frustratio n \nthrough a change of the structure, the effect of these diamagnetic cations upon the magnetic \nproperties is different. A strong ferrimagnetic component is induced by zinc substitution [15], \nwhereas Ga substitution leads to the formation of ferrimagnetic c lusters embedded in a spin \nglass matrix, essentially leading to phase separation in the samples. The difference originates \nfrom the opposite evolution of the Fe3+:Fe2+ ratio as the substitution rate increases in the two \ncases. Both Fe3+ and Fe2+ exhibit th e high spin configuration since they have a tetrahedral \ncoordination in these ferrites. Thus, the magnetic moment induced by the eg2t2g3 Fe3+ cations \nshould be much higher than that induced by the eg3t2g3 Fe2+ cations. Hence, an increase in the \nFe3+:Fe2+ ratio should favour stronger magnetic interactions.  In the case of Zn2+ doping, the \nFe3+:Fe2+ ratio increases , thereby  favouring the appearance of ferrimagnetism. On the other \nhand, for Ga3+ doping, the Fe3+:Fe2+ ratio decreases , thereby inducing only weak \nferrimagnetism and cluster formation . In both series, YBaFe 4-xGaxO7 and YBaFe 4-xZnxO7, a \ndilution effect is observed with an increase in the doping concentration. As a consequence, \nferrimagnetism is weakened for higher concentrations in the Zn – phase. In the Ga – phase, the \nferrimagnetic clusters are magnetically coupled by exchange interactions mediated through the \nsurrounding spin glass matrix. For higher Ga concentrations, the exchange coupling between \nthe ferrimagnetic clusters becomes less efficient, ultimately leading to the formation of a pure \nspin glass phase for x = 0.70, which is similar to the pristine sample (x = 0), but with a slightly \nhigher T g (60 K). The Ga – substituted phase also differs from the Zn – phase by the presence \nof exchange bias  and domain wall pinning. The cause of both these effects can be traced back \nto the inherent phase separation present in the samples.  \n \n  11 Acknowledgements  \n \nWe acknowledge the CNRS and the Conseil Regional of Basse Normandie for financial \nsupport in the frame  of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \nReferences  : \n \n      [1]     Martin Valldor and Magnus Andersson, Solid State Sciences , 2002 , 4, 923  \n      [2]     Martin Valldor, J. Phys.: Con dens. Matter ., 2004 , 16, 9209  \n      [3]     A. Huq, J. F. Mitchell, H. Zheng, L. C. Chapon, P. G. Radaelli, K. S. Knight and P.  \n               W. Stephens, J. Solid State Chem ., 2006 , 179, 1136  \n      [4]     D. D. Khalyavin, L. C. Chapon, P. G. Radaelli, H. Zheng and J. F. Mitchell, Phys.  \n                Rev. B , 2009 , 80, 144107  \n      [5]     V. Caignaert, V. Pralong, A. Maignan and B. Raveau, Solid State Communications ,  \n                2009 , 149, 453  \n      [6]     B. Raveau, V. Caignaert, V. Pralong, D.  Pelloquin and A. Maignan, Chem. Mater .,  \n                2008 , 20, 6295  \n      [7]     V. Caignaert, A. M. Abakumov, D. Pelloquin, V. Pralong, A. Maignan, G. Van  \n                Tendeloo and B. Raveau, Chem. 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Sarkar, V. Pralong , V. Caignaert and B. Raveau , Chem. Mater ., 2010, 22, 2885  \n      [16]  J. Rodriguez -Carvajal, An Introduction t o the Program FULLPROF 2000; Laboratoire  \n               Léon Brillouin, CEA -CNRS: Saclay, France (2001)  \n      [17]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 220, 106  \n      [18]   C. R. Sankar and P. A. Joy, Phys. Rev. B , 2005 , 72, 024405  \n      [19]   T. Gao, S. X. Cao, K. Liu, B. J. Kang, L. M. Yu, S. J. Yuan and J. C. Zhang, Journal  \n                of Phys: Conf. Series , 2009 , 150, 042038  \n      [20]   P. A. Joy and S. K. Date, J. Magn. Magn. Mater ., 2000 , 210, 31 \n      [21]   R. L. Stamps,  J. Phys. D, 2000 , 33, R247  \n      [22]   W. H. Meiklejohn and C. P. Bean , Phys. Rev ., 1956 , 102, 1413  \n      [23]   S. Karmakar, S. Taran, E. Bose, B. K. Chaudhuri, C. P. Sun, C. L. Huang and H. D.  \n                Yang, Phys. Rev. B , 2008 , 77, 144409  \n      [24]   A. Bracchi, K. Samwer, S. Schneider and J. F. Löffler, Appl. Phys. Lett ., 2003 , 82,  \n                721 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  13 Table caption s \n \nTable 1 : Cell parameters as obtained from the Rietveld refinement of X -ray powder diffraction \ndata.  \n \nFigure  captions  \n \nFigure 1:  Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, see text.  \n \nFigure 2:  X-ray diffraction pattern along with the fits for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \nFigure 3: Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected \naccording to zero field cooling (ZFC) and field cooling (FC) processes for YBaFe 4-xGaxO7 (a) \nx = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at B = 0.3 T.  \n \nFigur e 4: \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode without \ndegaussing, and (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see \ntext for details). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field o f 5 T.  \n \nFigure 5:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \nFigure 6:  M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field cooling (red open circles), field cooling in \na field of 2 T (black lines) and field cooling in a field of - 2 T (blue line s).  \n \nFigure 7 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.   14  \nFigure 8 : Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \nFigure 9 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed a t f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \nFigure 10 : The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 Table 1  \n \nDoping \nconcentration \n(x)  \nCrystal \nsystem \n(Space \ngroup)  \n  \nUnit cell parameters  \n c / a \n2 x as \nobtained \nfrom EDS \nanalysis  a (Å)  c (Å)  \n0.40 Hexagonal \n(P63mc)  \n6.320 (1) \n 10.383 (1) 1.6428 3.02 0.43 (2)  \n0.50 Hexagonal \n(P63mc)  \n6.322 (1) \n 10.376 (1) 1.6413 3.15 0.50 (1)  \n0.60 Hexagonal \n(P63mc)  \n6.323 (1) \n 10.37 4 (1) 1.6407 2.94 0.59 (1)  \n0.70 Hexagonal \n(P63mc)  \n6.325 (1)  \n 10.37 2 (1)  1.6398 3.46 0.74 (3)  \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 1 . Schematic representation of (a) hexagonal LnBaCo 4O7 and (b) cubic YBaFe 4O7 \n(adapted from Ref. 8). For details, s ee text.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 2. X-ray diffraction pattern along with the fit s for (a) YBaFe 3.6Ga0.4O7 and (b) \nYBaFe 3.3Ga0.7O7. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  18 \n \n \nFig. 3. Temperature dependence of the magnetic susceptibility (\n dc = M/H) collected according \nto zer o field cooling (ZFC) and field cooling (FC) processes for  YBaFe 4-xGaxO7 (a) x = 0.40, \n(b) x = 0.50, (c) x = 0.60 and (d) x = 0.70, measured at H = 0.3 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  19 \n \n \nFig. 4. \nZFC(T) curves of YBaFe 3.6Ga0.4O7 recorded (a) in the ZFC mode  without dega ussing , \nand (b) after applying a magnetic field of 5 T and degaussing the ZFC sample (see text for \ndetails). The inset in (a) shows \n ZFC(T) recorded under a magnetizing field of 5 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  20 \n \n \nFig. 5. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50, (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K . The virgin curves are shown in black circles, while the rest of the \nhysteresis loops are shown in red lines.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  21 \n \n \nFig. 6. M (H)  curves for YBaFe 4-xGaxO7 (a) x = 0.40, (b) x = 0.50,  (c) x = 0.60 and (d) x = \n0.70, registered at T = 5 K, measured after zero field  cooling (red open circles), field cooling in \na field of 2 T (black lines)  and field cooling in a field of - 2 T (blue lines) .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  22 \n \n \nFig. 7. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.6Ga0.4O7 as a function of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibi lity, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  23 \n \n \nFig. 8. Temperature dependence of the real (in -phase) component of a. c. susceptibility  for \nYBaFe 3.3Ga0.7O7 as a f unction of temperature measured in zero magnetic field (H dc = 0), using \na frequency of 10 kHz. Inset (a) shows the imaginary (out -of-phase) component of the a. c. \nsusceptibility, and inset (b) shows the real (in -phase) component of the a. c. susceptibility  \nmeasured using four different frequencies.  \n \n \n \n \n \n \n \n \n \n \n \n  24 \n \n \nFig. 9. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.3Ga0.7O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and  Hac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n \n \n \n \n \n \n \n \n  25 \n \n \nFig. 10. The (a) real (in -phase) and (b) imaginary (out -of-phase) component of a. c. \nsusceptibility for YBaFe 3.6Ga0.4O7 as a function of temperature. The driving frequency was \nfixed at f = 1 kHz and H ac = 10 Oe. Each curve was obtained under different applied static \nmagnetic field (H dc) ranging from 0 T to 0.2 T.  \n \n \n \n \n "    },    {        "title": "2301.11277v1.Room_temperature_spin_glass_behavior_in_zinc_ferrite_epitaxial_thin_films.pdf",        "content": "arXiv:2301.11277v1  [cond-mat.other]  25 Jan 2023Room-temperature spin glass behavior in zinc ferrite epita xial thin ﬁlms\nJulia Lumetzberger,1Verena Ney,1Anna Zhakarova,2Nieli Daﬀe,2Daniel Primetzhofer,3and Andreas Ney1,∗\n1Johannes Kepler University Linz, Institute for Semiconduc tor and\nSolid State Physics, Altenberger Strasse 69, 4040 Linz, Aus tria\n2Swiss Light Source (SLS), Paul Scherrer Institut, 5232 Vill igen PSI, Switzerland\n3Department of Physics and Astronomy, ˙Angstr¨ om Laboratory,\nUppsala University, Box 516, SE-751 20 Uppsala, Sweden\n(Dated: January 27, 2023)\nZinc ferrite (ZnFe 2O4) epitaxial thin ﬁlms were grown by reactive magnetron sputt ering on\nMgAl 2O4and Al 2O3substrates varying a range of preparation parameters. The r esulting structural\nand magnetic properties were investigated using a range of e xperimental techniques conﬁrming epi-\ntaxial growth of ZnFe 2O4with the nominal stoichiometric composition and long range magnetic\norder at and above room temperature. The main preparation pa rameter inﬂuencing the tempera-\ntureTfof the bifurcation between M(T) curves under ﬁeld cooled and zero-ﬁeld cooled conditions\nwas found to be the growth rate of the ﬁlms, while growth tempe rature or the Ar:O 2ratio did not\nsystematically inﬂuence Tf. Furthermore Tfwas found to be systematically higher for MgAl 2O4as\nsubstrate and Tfextends to above room temperature. While in some samples Tfseems to be more\nlikely correlated with superparamagentism, the highest Tfoccurs in ZnFe 2O4epitaxial ﬁlms where\nexperimental signatures of magnetic glassiness can be foun d. Element-selective X-ray magnetic cir-\ncular dichroism measurements aim at associating the magnet ic glassiness with the occurrence of a\ndiﬀerent valence state and lattice site incorporation of Fe pointing to a complex interplay of various\ncompeting magnetic interactions in ZnFe 2O4.\nI. INTRODUCTION\nZinc ferrite (ZnFe 2O4) belongs to the crystallographic\ngroup of normal spinels of the form AB 2O4where in the\nideal case the A-cation (Zn2+) exclusively occupies the\ntetrahedral (Td) lattice sites as Zn2+\nTdwhile the B-cation\n(Fe3+) is found on the octahedral (Oh) sites as Fe3+\nOh.\nThe magnetic properties of zinc ferrite have been un-\nder investigation for quite some time revealing a rather\ncomplex situation. Early studies of the bulk material re-\nportantiferromagnetic(AFM) orderwith averylowN´ eel\ntemperature of 9K [1, 2]. Later-on it was demonstrated\nby magnetic neutron scattering experiments on ZnFe 2O4\nsingle crystals that even in perfect crystals geometrical\nfrustration leads to an unusual magnetic behavior [3]. In\nparticular, it was pointed out, that the Fe3+\nOhsublattice\ncan be regarded to be similar to various pyrochlores or\nLaves phases which are known for their intrinsic geomet-\nrical frustration [3]. The situation becomes even more\ncomplex when defects such as inversion are considered\nwhich are expected to occur, in particular, in thin ﬁlms\nof ZnFe 2O4. A partial inversion in ZnFe 2O4has the\nstoichiometric formula of [Zn 1−δFeδ]Td[ZnδFe2−δ]OhO4,\nwhereδdenotes the degree of inversion. In the ideal case\nofδ= 0, i.e., no inversion, there is only the weak AFM\nsuperexchangeinteraction between Fe3+\nOhwhich is usually\ndenoted as J BB[4] plus the geometrical frustration men-\ntioned before [3]. J BBcan be held responsible for the\nAFM order at low temperatures in bulk single crystals.\nFor a ﬁnite degree of inversion there is an additional,\n∗Electronic address: andreas.ney@jku.atmuch stronger AFM superexchange interaction J ABbe-\ntween Fe3+on Td (also called A-site) and Oh (or B-site)\nsites, i.e., Fe3+\nTdand Fe3+\nOh[4], which leads to ferrimag-\nnetism for incomplete inversion [5]. The additional J AA\nexchange between the Fe3+\nTdis the weakest [6]. However,\nif the additional Fe3+\nTdis not compensated by Zn2+\nOh, i.e.,\nif there is some degree of deviation from the ideal stoi-\nchiometryofZnFe 2O4, someﬁnite amount ofFe2+\nOhhasto\nform because of charge neutrality. This results in an ad-\nditional double exchange (DE) over the Oh (or B-) sites\nJDE\nBBbetween Fe3+\nOhand Fe2+\nOh, which results in spin cant-\ning in magnetite [4] or in non-stoichiometricZnFe 2O4[5].\nIn many cases there are reports on some ﬁnite degree of\ninversion in ZnFe 2O4and an upper limit of δ= 0.6 has\nbeen found by the analysis of the magnetic moment of\nthe Fe [7], X-ray absorption spectroscopy (XAS) [8] or\nvia Rietveld reﬁnement of X-ray diﬀraction (XRD) data\n[9]. Therefore, the magnetic order in ZnFe 2O4can be\nexpected to be highly complex, especially for thin ﬁlms,\nwhere the presence of various kinds of defects such inver-\nsion and/or oﬀ-stoichiometry can be expected.\nThe growth of zinc ferrite in thin ﬁlm form is moti-\nvated by a range of possible applications such as gas sen-\nsors, photo catalytic disinfection or other photo-catalytic\napplications, see [9, 10] and Refs. therein. Besides\nthat, zinc ferrite thin ﬁlms can also be considered as an\ninteresting semiconducting material in spintronics with\ntunable magnetic properties [4, 6, 11–13]. A variety\nof reports of diﬀerent types of magnetic order in zinc\nferrite can be found throughout the literature ranging\nfromferro(i)magnetic[4,6,13–17], tosuperparamagnetic\n[8, 11, 18, 20, 21] and to spin glass behavior [7, 9, 19, 22].\nNote, that in some cases superparamagnetismwith inter-\nparticleinteractionsisassociatedwith aso-calledcluster-2\nglass behavior [7, 8, 19]. However, only a few studies\nreport on characteristic experimental signatures of mag-\nnetic glassiness [7, 9, 19, 22], while others report only\ntemperature-dependent magnetization [ M(T)] measure-\nments under diﬀerent ﬁeld-cooling conditions which how-\never could also be associated with superparamagnetism,\ne.g. [8]. Finally, the control of defects and thus the\nmagnetic properties was reported to be experimentally\nachievable by varying diﬀerent preparation parameters,\ni.e., oxygen partial pressure [6, 12, 17, 18, 21], stoichio-\nmetric composition [4], post-growth thermal treatment\n[19, 20] or deposition rate [7].\nSimilar to the reported types of magnetism in zinc fer-\nrite, also the techniques for sample preparation span a\nwide range from pulsed laser deposition (PLD) [4, 6, 7,\n11–13, 16, 17, 20], over reactive magnetron sputtering\n(RMS) [8, 14, 15, 18, 19, 21] for thin ﬁlm growth, to ball\nmilling [9] and solid state reaction [22] for bulklike sam-\nples. Likewise, a range of diﬀerent substrates has been\nused for thin ﬁlm growth amongst them are MgAl 2O4\n[12], c-plane Al 2O3[7, 11], a-plane Al 2O3[16], SrTiO 3\n[13, 17, 20], MgO [4, 6, 11], Si(001) and Si(111) [14, 21]\nand glass substrates [8, 15, 18, 19]. It is remarkable,\nthat spin glass behavior has mainly been reported for\nbulk ZnFe 2O4[3] or bulklike nanopowders [9, 22] while\nfor thin ﬁlm samples mostly a cluster glass is inferred\n[7, 8, 19]. Among the reports of cluster glass behavior\nonly one is based on thin ﬁlm growth by PLD on single\ncrystalline substrates [7], while the others rely on sput-\ntered polycrystalline ZnFe 2O4samples [8, 19] making a\nlarger amount of defects expectable, e.g., due to an in-\ntrinsicallylargenumberofgrainboundaries. Finally, also\nthe temperature range, where magnetic glassiness is ob-\nserved ranges from below 20 K for the single crystalline\nZnFe2O4in [3, 22], over around 100 K for the annealed\nZnFe2O4nanopowder in [9] up to 300 K for the PLD\ngrown ZnFe 2O4epitaxial ﬁlms at deposition rates above\n3 nm/s which drops down to below 100 K for rates below\n2 nm/s [7].\nHere, we report on epitaxial thin ﬁlm growth of\nZnFe2O4by RMS on two diﬀerent substrates namely\nMgAl2O4and c-plane Al 2O3. Various preparation pa-\nrameters have been varied in order to control the for-\nmation of defects in a systematic way for epitaxial thin\nﬁlm samples. In agreement with [7] the most relevant\npreparation parameteris found to be the deposition rate.\nFor high deposition rates ZnFe 2O4ﬁlms exhibit spin-\nglass like behavior up to rather high temperatures on\nMgAl2O4substrates which is shifted to lower tempera-\ntures on Al 2O3substrates. In contrast, the stoichiom-\netry on ZnFe 2O4is maintained throughout the sample\nseries and also the oxygen partial pressure were found to\nplay a minor role in the resulting magnetic properties.\nThe spin-glass behavior is associated with a signiﬁcant\namount of inversion up to δ∼0.3, corroborating earlier\nreports [8, 9]. In addition, a signiﬁcant magnetic polar-\nization of the Zn cation is found at room temperature\nby means of element selective magnetometry indicatingthat the microscopic origin of the magnetic properties of\nZnFe2O4is even more complex.\nII. EXPERIMENTAL DETAILS\nZinc ferrite was fabricated using reactive magnetron\nsputtering (RMS) from an oxide target having the\nnominal composition of ZnFe 2O4. The epitaxial thin\nﬁlms were grown on doubleside polished single crys-\ntalline spinel [MgAl 2O4(001)] and c-plane sapphire\n[Al2O3(0001)] substrates in an ultrahigh vacuum (UHV)\nchamber with a base pressure of 4 ×10−8mbar and a\nworking pressure of 4 ×10−3mbar. To determine the\nideal growthparameters, the deposition temperature was\nvaried from room temperature (RT) to 550 °C, the Ar:O 2\nratiofrom10:0to 10:0 .5, andthe sputtering powerfrom\n20W to 100W, which corresponds to a growth rate from\n0.36 to 3.69 nm/min. The nominal thickness is kept at\n40 nm and is controlledvia a quartz crystal microbalance\nwhichisatroomtemperaturesothattheactualthickness\nof most of the ﬁlms is by 10-20% lower because of the el-\nevated temperature of the substrate during growth. The\nstructural properties of the ﬁlms were investigated by X-\nray diﬀraction (XRD) measurements with a Pananalyti-\ncal X’Pert MRD recording ω−2θscans and symmetric\nas well as asymmetric reciprocal space maps (RSM). The\nchemical composition was determined by ion beam anal-\nysis, i.e. Rutherford backscattering spectrometry (RBS)\nusinga2MeVHe+primarybeamattheTandemLabora-\ntory at Uppsala University. To disentangle the element\nspeciﬁc contributions, the spectra were analyzed using\nthe SIMNRA software [23]. Details of the experimental\nsetup are described elsewhere [24]. Furthermore, Elec-\ntron Recoil Detection (ERDA) with a primary ion beam\nof 36MeV iodine ions was employed to rule out contam-\ninations with light elements like H or C.\nThe magnetic properties were measured by integral\nsuperconducting quantum interference device (SQUID)\nmagnetometryusinga QuantumDesign MPMS-XL5 sys-\ntem applying the magnetic ﬁeld in the ﬁlm plane. The\nM(H) curves were recorded in range of ±5T at 300K\nand 2K and M(T) curves have been recorded from 2K\nup to 395K at 10 mT while warming after a cool down\nin 5 T (FH), under nominally zero-ﬁeld cooled conditions\n(ZFC) as well while cooling down in 10 mT (ﬁeld cooled,\nFC). Additionally, waiting time experiments were per-\nformed analogous to the ones in [25] by cooling down\nthe sample in zero ﬁeld and introducing a waiting time\ntwaitat various waiting temperatures Twaitwhich is typ-\nically 10000s. Then a M(T) curve identical to a ZFC\ncurve without waiting time was subsequently recorded.\nSubtracting these two curves represents a typical waiting\ntime experiment for spin glasses where a so-called ZFC\nmemory—or hole-burning—eﬀect can be seen by a dip\nin the diﬀerence of the magnetization with and without\nwaiting time around Twait[25]. A second memory ex-\nperiment already used before for ZnFe 2O4in [9] was per-3\nformed in addition, in which a M(T) curve is recorded\nunder FC conditionswith 10mT and a waitingtime twait\nat nominally zero ﬁeld is inserted at several Twaitbefore\nthe FC curve is resumed at 10 mT. Then a subsequent\nM(T) is recorded in 10 mT while warming. The typical\nsignature of a spin glass in these so-called FC memory\nexperiments is a relaxation of the magnetization at Twait\nin the FC curve and the presence of an inﬂection point\naroundTwaitin the subsequent M(T) curve while warm-\ning [9]. Note, however, that also superparamagnets ex-\nhibit similar signatures in the FC memory experiments\n[26]. All M(H) andM(T) data were corrected for the\ndiamagnetic background of the substrate which was de-\ntermined from the M(H) curves a high magnetic ﬁeld at\n300 K [27]. The M(H) curves at 2 K for samples grown\nonMgAl 2O4hadtobecorrectedforanadditionalparam-\nagnetic contribution which was determined form a M(H)\nmeasurement of bare MgAl 2O4from the same batch of\nsamples. In general, zero ﬁeld conditions are referred to\nnominally 0.0 mT after the superconducting magnet had\nbeen reset (magnet reset option of the MPMS) and any\napplied magnetic ﬁeld is afterwards limited to 10 mT;\nthis assures a residual pinned magnetic ﬁeld of typi-\ncally 0.1 mT or less [28]. A cooling and heating rate\nof 1 K/min is used for all M(T) measurements in both\nFC and ZFC memory experiments. Note, that the typ-\nical frequency-dependent ac-susceptibility measurements\nlike in [7, 19, 22] were not available for the used SQUID.\nThe element speciﬁc magnetic properties have been\ninvestigated by x-ray absorption near edge spec-\ntroscopy (XANES) and x-ray magnetic circular dichro-\nism (XMCD) measurementswhich were performedat the\nXtreme beamline at the Swiss Light Source (SLS) [29].\nThe XMCD spectrawererecordedat the Fe L3/2- andZn\nL3/2-edges at 300K under 20 °grazing incidence in total\nelectron yield. For the Fe-edges the magnetic ﬁeld was\nset to 5 T and only the circular polarization has been\nswitched to obtain the XMCD. For the Zn-edges the di-\nrection of the magnetic ﬁeld has been reversed as well\nto minimize artifacts. The XMCD spectra at the Fe L3\nedge are compared to simulations carried out by mul-\ntiplet ligand ﬁeld theory using the CTM4XAS package\n[30]. These simulations have been used before to deter-\nmine the site occupancy and formal oxidation state of\nFe and Ni in nickel ferrites [31] and Zn/Al doped nickel\nferrites [32]. For the present work the simulation param-\neters for Fe2+\nOh, Fe3+\nOh, and Fe3+\nTdare identical to those in\n[32] and details on the simulations can be found there.\nIII. EXPERIMENTAL RESULTS\nThe structural properties of the ZnFe 2O4thin ﬁlms\nwere analyzed by symmetric ω−2θscans using XRD.\nFigure1(a) shows a comparison of the diﬀractograms of\nzinc ferrite grown on MgAl 2O4and Al 2O3where the lat-\nter is shifted upward for clarity. The samples were grown\nwith a nominal thickness of 40nm at a substrate tem-/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s49/s48/s53/s49/s48/s54/s51/s53 /s52/s48 /s52/s53\n/s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48 /s49/s54/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s32/s45/s32/s50 /s32/s40/s176/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s32/s56/s48/s87 /s32/s111/s110/s32/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s56/s48/s87 /s32/s111/s110/s32/s65/s108\n/s50/s79\n/s51\n/s90/s110/s70/s101\n/s50/s79\n/s52/s40/s51/s49/s49/s41/s65/s108\n/s50/s79\n/s51/s40/s48/s48/s54/s41\n/s90/s110/s70/s101\n/s50/s79\n/s52/s40/s48/s48/s52/s41/s77/s103/s65/s108\n/s50/s79\n/s52/s40/s48/s48/s52/s41/s40/s97/s41\n/s40/s98/s41/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s99/s111/s117/s110/s116/s115/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s107/s101/s86/s41/s32/s56/s48/s87 /s32/s111/s110/s32/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s56/s48/s87 /s32/s111/s110/s32/s65/s108\n/s50/s79\n/s51\n/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110\n/s32/s70/s101\n/s32/s90/s110\n/s50/s32/s77/s101/s86/s32/s72/s101/s43\nFIG. 1: (a) Structural characterization by X-ray diﬀractio n:\ncomparison of the symmetric ω−2θscans of ZnFe 2O4ﬁlms\ngrown at 80 W on MgAl 2O4(black) and Al 2O3(red). (b)\nChemical composition determined by means of RBS spectra\nrecorded with a 2 MeV He+primary ion beam of ZnFe 2O4\nﬁlms grown at 80 W on MgAl 2O4(black) and Al 2O3(red).\nperature of TS=450 °C with an Ar:O 2ratio of 10:0 .5\nand a sputtering power of 80W. The XRD scan of the\nsamples grown on MgAl 2O4exhibits a (004) reﬂex at\n41.79±0.09°withafullwidthathalfmaximum(FWHM)\nof 0.6°, corresponding to a perpendicular lattice param-\netera⊥= (8.64±0.02)/RingA. The sample grown on Al 2O3\nexhibits the (311) reﬂex at 35 .61°correspondingto a per-\npendicular lattice parameter of a⊥= (7.28±0.02)/RingA. For\nthis sample weak Laue oscillations can be seen indicat-\ning a smoother growth compared to the sample grown\non MgAl 2O4. In addition, an asymmetric RSM along\nthe (¯1¯15) plane has been recorded for a 100nm sam-\nple grown on MgAl 2O4with a sputtering power of 60W\n(not shown). It reveals an in-plane lattice parameter of\na/bardbl= (8.32±0.05)/RingAwhichprovidesevidencethatthe ﬁlm\nisrelaxed,sincetheﬁlmpeakdoesnotalignwiththesub-\nstrate peak. The reﬂection from the MgAl 2O4substrate\ncorresponds to a lattice parameter of asub= 8.08/RingA,\nwhich implies a lattice mismatch of ∼4.3% with re-4\nspect to bulk ZnFe 2O4(a0= 8.441/RingA) (JCPDS card No.\n82-1049). A comparison between various ﬁlms grown\non MgAl 2O4and Al 2O3indicates no signiﬁcant diﬀer-\nence in crystalline quality with similar FWHM despite\nthe change in texture from (004) on MgAl 2O4to (311)\non Al2O3. No other reﬂexes can be found underlining\nhighly textured growth of ZnFe 2O4for both substrates\nand the samples are devoid of other crystalline phases.\nFurthermore, the chemical composition of ZnFe 2O4on\nboth substrates is determined using RBS. In Fig. 1(b)\nthe RBS data of the 80W sample grown on MgAl 2O4\n(black squares) is shown in comparison to the sample\ngrown on Al 2O3(red circles). Both ZnFe 2O4ﬁlms have\nno deviation from the nominal stoichiometry within the\nuncertainties of the measurement technique. Finally, the\nsamples are investigated using ERDA to check for con-\ntaminations of light elements like H or C but neither\nelement could be detected (not shown). We can there-\nfore conclude that our ZnFe 2O4samples have the nom-\ninal stoichiometric composition, grow epitaxially on ei-\nther substrate and are devoid of a signiﬁcant amount of\nsecondary phases or contaminants within the detection\nlimits of XRD and ERDA.\nIn a ﬁrst step, the magnetic properties are investigated\nusing standard M(H) curves which are shown in Fig. 2\nrecorded at (a) 300K and (b) 2K for ZnFe 2O4grown\nat 60W on MgAl 2O4(black squares) and Al 2O3(red\ncircles). ZnFe 2O4grown on Al 2O3has a higher magne-\ntization of MS= 130±15kA/m compared to growth\non MgAl 2O4whereMS= 110±15kA/m at 300K.\nFor both samples MSincreases to above 200kA /m at\n2K. For the M(H) curves recorded at 2K for ZnFe 2O4\ngrownon MgAl 2O4the full squaresdenote the datawhen\nonly the diamagnetic contribution has been subtracted.\nNote, that in a previous publication on ZnFe 2O4grown\non MgAl 2O4this apparently paramagnetic behavior has\nbeen attributed to cationic disorder of the Fe3+in [12].\nHowever, if a bare MgAl 2O4substrate is measured, one\nalsomeasuresa net-paramagneticbehavioraftersubtrac-\ntion ofthe diamagnetismsothat onehas to attribute this\nparamagnetic contribution to the MgAl 2O4substrate it-\nself. The open squares are the data where also the mea-\nsuredparamagneticbackgroundofthebareMgAl 2O4has\nbeen subtracted and no obvious paramagnetic contribu-\ntion of the ZnFe 2O4is visible any more. The insets in\nFig.2enlarge the low-ﬁeld behavior of the M(H) curves.\nWhiletheyarevirtuallyanhystereticat300Kaclearhys-\nteresis with a coercive ﬁeld of Hc= 80±10mT is found\nfor ZnFe 2O4ﬁlms on either substrate. This behavior at\n2 K is consistent with most of the ZnFe 2O4ﬁlms grown\nonarangeofdiﬀerentsubstratesreportedthroughoutthe\nliterature reporting ferro(i)magnetism or superparamag-\nnetism both in terms of magnetization as well as coercive\nﬁeld at low temperatures and clearly rules out the pure\nantiferromagnetic behavior of bulk ZnFe 2O4.\nFigure3shows the M(T) behavior recorded at 10mT\nunder FC conditions (full symbols) as well as after ZFC\nconditions (open symbols) for the ZnFe 2O4grown at/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s52/s48/s48/s45/s49/s53/s48/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s45/s52 /s45/s50 /s48 /s50 /s52\n/s45/s53/s48/s48/s53/s48/s45/s48/s46/s48/s52 /s48/s46/s48/s48 /s48/s46/s48/s52\n/s45/s48/s46/s50 /s48/s46/s48 /s48/s46/s50/s45/s50/s48/s48/s48/s50/s48/s48\n/s32/s32/s77 /s32/s40/s107/s65/s47/s109/s41\n/s48/s72/s32/s40/s84/s41/s32/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s99/s111/s114/s114/s101/s99/s116/s101/s100\n/s32/s65/s108\n/s50/s79\n/s51/s40/s98/s41\n/s84/s32/s61/s32/s50/s32/s75/s32\n/s32/s32\n/s48/s72/s32/s40/s84/s41/s77 /s32/s40/s107/s65/s47/s109/s41/s54/s48/s32/s87/s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47\n/s32/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s65/s108\n/s50/s79\n/s51/s40/s97/s41\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s32/s32\n/s32\n/s32/s32\n/s32\n/s32/s32\nFIG. 2: SQUID measurements of M(H) curves shown for the\n60 W ZnFe 2O4grown on MgAl 2O4(black squares) and Al 2O3\n(red circles) at (a) 300 K and (b) at 2 K. (a) shows an M(H)\ncurve at RT (b). At 2 K the paramagnetic contribution of the\nMgAl 2O4substrate has been subtracted (open squares). The\ninsets enlarge the measurements at low ﬁelds.\n60W on MgAl 2O4(a) and Al 2O3(b), i.e., the identi-\ncal pair of samples as in Fig. 2. Both samples exhibit\na clear bifurcation between the FC and ZFC curves in-\ndicating a blocking or spin-freezing peak at a temper-\nature of Tf= 290K for ZnFe 2O4/MgAl 2O4(a) and at\nTf= 190K for ZnFe 2O4/Al2O3(b). Both 60W sam-\nples together with the two 80W samples shown in Fig.\n1are part of a sample series where only the sputtering\npower and thus the deposition rate has been changed\nwhile all other growth parameters have been kept con-\nstant. In terms of magnetization as well as coercivity\nall samples from these series show comparable magnetic\nbehavior. The only systematic dependency on the sput-\ntering power is an increase of the measured Tfwith in-\ncreasing sputtering power. The insets in Fig. 3show\nthe measured Tfas a function of the sputtering power\nfor ZnFe 2O4/MgAl 2O4(a) as well as for ZnFe 2O4/Al2O3\n(b). Irrespectiveofthecomparableincreasewithsputter-\ning powerthe overallvalues of Tfaresystematicallylower\nfor the Al 2O3substrate by about 100K. Note that the\nobtained Tffor the samples grown on either substrate,\nare well above the values usually reported for zinc ferrite5\n/s50/s48/s52/s48/s54/s48/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s50/s48 /s52/s48 /s54/s48 /s56/s48/s49/s55/s48/s49/s56/s48/s49/s57/s48/s50/s48/s48\nµ\n/s48/s72/s32/s61/s32/s49/s48/s32/s109/s84/s40/s97/s41/s84/s32/s40/s75/s41/s77 /s32/s40/s107/s65/s47/s109/s41/s54/s48/s32/s87/s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s70/s67\n/s32/s90/s70/s67\n/s84\n/s102µ\n/s48/s72/s32/s61/s32/s49/s48/s32/s109/s84/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s50/s54/s48/s50/s56/s48/s51/s48/s48/s51/s50/s48/s84\n/s102/s32/s40/s75 /s41\n/s115/s112/s117/s116/s116/s101/s114/s32/s112/s111/s119/s101/s114/s32/s40/s87/s41/s77/s103/s65/s108\n/s50/s79\n/s52\n/s40/s98/s41\n/s84/s32/s40/s75/s41/s77 /s32/s40/s107/s65/s47/s109/s41\n/s54/s48/s32/s87/s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s65/s108\n/s50/s79\n/s51\n/s32/s70/s67\n/s32/s90/s70/s67/s84\n/s102/s65/s108\n/s50/s79\n/s51/s84\n/s102/s32/s40/s75 /s41\n/s115/s112/s117/s116/s116/s101/s114/s32/s112/s111/s119/s101/s114/s32/s40/s87/s41\nFIG. 3: M(T) curves recorded at 10 mT under ﬁeld cooled\n(FC, full symbols) andafter zero-ﬁeld cooled (ZFC, opensym -\nbols) conditions shown for the 60 W ZnFe 2O4grown on (a)\nMgAl 2O4(black squares) and (b) Al 2O3(red circles) sub-\nstrates. The insets show the dependence of Tfon the sput-\ntering power for both substrates, respectively.\n[9, 17, 20, 22]. Only in few cases the Tfis found at such\nelevated temperatures [7, 19] and a controllable shift of\nTfis only reported in [7] so far; however, the drop in Tf\nwith decreasingdeposition rate in [7] is by a factor of two\nmore pronounced compared with the present case. Note,\nthat the power series was grown by varying the sputter\npower nonmonotonously so that a dependence of Tfon\nthe growth sequence—and thus target degradation—can\nbe ruled out.\nIn a second step, the dependence of Tfon other growth\nparameters shall be brieﬂy summarized. Figure 4(a)\ncompiles the sample series as a function of the growth\ntemperature Tgrowthfrom RT up to 550 °C for ZnFe 2O4\ngrown on MgAl 2O4at a sputtering power of 60W and\nan Ar:O 2ratio of 10:0 .5. The XRD shows no signiﬁ-\ncant changes for Tgrowth≥300°C while the sample at\nRT appears to be virtually amorphous. The inset shows\nMSat 300 K and Tfas determined from SQUID mea-\nsurements analogous to Figs. 2and3.Tfis found to\nbe about constant around 250K with a slight tendency\nto decrease for higher Tgrowth; the only exception is the/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s49/s48/s53/s49/s48/s54/s49/s48/s55/s52/s49 /s52/s50 /s52/s51 /s52/s52 /s52/s53\n/s52/s49 /s52/s50 /s52/s51 /s52/s52 /s52/s53/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52/s49/s48/s53/s49/s48/s54/s49/s48/s55\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s48 /s50/s48/s48 /s52/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s32/s45/s32/s50 /s32/s40/s176/s41\n/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s84\n/s103/s114/s111/s119 /s116/s104\n/s32/s53/s53/s48/s176/s67\n/s32/s53/s48/s48/s176/s67\n/s32/s52/s53/s48/s176/s67\n/s32/s52/s48/s48/s176/s67\n/s32/s51/s48/s48/s176/s67\n/s32/s50/s48/s48/s176/s67\n/s32/s82/s84/s77/s103/s65/s108\n/s50/s79\n/s52/s40/s48/s48/s52/s41\n/s90/s110/s70/s101\n/s50/s79\n/s52/s40/s48/s48/s52/s41/s40/s97/s41\n/s40/s98/s41/s77/s103/s65/s108\n/s50/s79\n/s52/s40/s48/s48/s52/s41\n/s90/s110/s70/s101\n/s50/s79\n/s52/s40/s48/s48/s52/s41/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s32/s45/s32/s50 /s32/s40/s176/s41/s65/s114/s58/s79\n/s50/s32/s40/s115/s99/s99/s109/s41\n/s32/s49/s48/s58/s48/s46/s53\n/s32/s49/s48/s58/s48/s46/s51/s53\n/s32/s49/s48/s58/s48/s46/s50/s53\n/s32/s49/s48/s58/s48/s46/s49/s53\n/s32/s49/s48/s58/s48/s32\n/s79\n/s50/s32/s102/s108/s111/s119/s32/s40/s115/s99/s99/s109/s41/s77 \n/s115/s32/s40/s107/s65 /s47/s109 /s41/s32/s47/s32 /s84\n/s102/s32/s40/s75 /s41\n/s77 \n/s115/s32/s40/s107/s65 /s47/s109 /s41/s32/s47/s32 /s84\n/s102/s32/s40/s75 /s41\n/s84\n/s103/s114/s111/s119/s116/s104/s32/s40/s176/s67/s41\nFIG. 4: (a) Structural and resulting magnetic properties as\na function of the growth temperature Tgrowthfor ZnFe 2O4\ngrown on MgAl 2O4. The dependence of the identical param-\neters as a function of the Ar:O 2ratio is shown in (b). Ms\n(black squares) and Tf(red circles) shown in the insets were\nextracted from SQUID measurements.\namorphous sample at RT where Tfis clearly reduced. In\ncontrast, MSsteadily increases with increasing Tgrowth\nwhich can be taken as an indication for an increasing\namount of inversion, i.e., of Fe3+\nTdin analogy with [7, 9].\nHowever; for the present sample series this increasing MS\nwithTgrowthhas no obvious inﬂuence on the observed\nTf. This trend is opposite to the annealing series in [9],\nwhere the amount of inversion and thus resulting MS\nis decreasing with increasing annealing temperatures for\nnanopowdered ZnFe 2O4. Note, that increasing Tgrowth\nin epitaxial growth typically leads to a decrease in ac-\ntual thickness compared to the nominal one which would\nlead to a decrease in MSwhich was calculated from the\nnominal thickness. On the other hand, this increase in\nMScan also be associated with an increase in the order\ntemperature which is in all cases above 400 K and thus\nbeyond the accessible temperature range of the SQUID\nmagnetometer and thus unknown.\nA second ZnFe 2O4sample series was grown on\nMgAl2O4at a sputtering power of 60W and a ﬁxed6\nTgrowthof 450 °C while varying the Ar:O 2ratio which\nis compiled in Fig. 4(b). The XRD of all samples does\nnot show any signiﬁcant changes with increasing oxygen\ncontent. In the inset the resulting MSat 300 K and\nTfare shown. While MSis independent on the Ar:O 2\nratio within error bars, Tfdoes not show a conclusive\ntrend, mostly because of a rather high Tffor the sam-\nple at Ar:O 2ratio of 10:0 .15. Disregarding this, a faint\nincrease within errorbars may be inferred but there is\nno pronounced dependence of Tfon the Ar:O 2ratio, es-\npecially if this is compared with the dependence on the\ngrowth rate shown in Fig. 3(a). This ﬁnding is rather in-\nteresting, since in [7] the growth rate has been associated\nwith a deﬁciency in oxygen leading to an increase in Tf.\nHowever, all samples were found to be highly resistive\nabove the GΩ-range (not shown). Therefore, a signif-\nicant amount of oxygen vacancies can be ruled out for\nthe entire series because the two samples grown without\nand with maximum oxygen partial pressure have virtu-\nally identical physical properties where the high oxygen\npartialpressureinRMSshouldsafelyruleoutanyoxygen\ndeﬁciency. In turn, the dependence of Tfon the growth\nrate, which is consistently found in [7] and Fig. 3(a), can-\nnot depend on the existence of oxygenvacancies for RMS\ngrown ZnFe 2O4. To summarize this part, the two sample\nseries shown in Fig. 4underline, that the relevant prepa-\nration parameter to control Tfis the sputtering power\nand thus growth rate, while Tgrowthand the Ar:O 2ratio\nplay a minor role in the resulting magnetic properties, in\nparticular, Tfis not directly controllable via oxygen va-\ncancies. Therefore, in the following only samples grown\nat Ar:O 2ratio of 10:0 .5 andTgrowth= 450 °C, as those\nin Figs.1to3, will be discussed further.\nIn a next step, the actual type of magnetic order shall\nbe determined because the reports in the literature range\nfrom ferro(i)magnetism, over superparamagnetism, to a\ncluster glass or spin-glass behavior. As pointed out\nabove,MSfor the present set of samples is found to\nbe consistent with most of the reports found through-\nout the literature, while the bifurcation at Tfis found\nat rather elevated temperatures. Figure 5(a) shows the\nM(T) behavior recorded under FC conditions (full sym-\nbols) as well as after ZFC conditions (open symbols) for\nthe ZnFe 2O4grown at 80W on MgAl 2O4for an external\nﬁeld of 5mT (black squares) and 10mT (red circles). As\nexpected Tfincreases with decreasing external magnetic\nﬁeld which is consistent with both spin-freezingas well as\nsuperparamagnetism. Reducingthe externalﬁeld further\nto ﬁelds of 0.2 to 0.5 mT), where most of the spin glass\nexperiments are typically carried out [25], Tfgets very\nclose to the maximum attainable temperature of 400 K\nof the SQUID magnetometer (not shown). Therefore, all\nsubsequent experiments are only carried out at ﬁelds of\n5 mT or 10 mT to keep the maximum achievable temper-\nature well above Tf. Note, that unfortunately the SQUID\ndoes not allow to go above the magnetic order tempera-\nture which is in all cases above 400 K.\nTo get a ﬁrst estimate on the existence of magnetic/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48\n/s84\n/s119 /s97/s105/s116\n/s84\n/s119 /s97/s105/s116/s84\n/s119 /s97/s105/s116\n/s84\n/s119 /s97/s105/s116/s40/s98/s41\n/s32/s77 /s32/s40 /s101/s109/s117/s41\n/s84/s32/s40/s75/s41/s32/s70/s67/s32/s40/s49/s48/s32/s109/s84/s41/s32/s73/s83\n/s32/s70/s72/s32/s40/s49/s48/s32/s109/s84/s41\n/s116\n/s119/s97/s105/s116/s32/s61/s32/s49/s48/s52\n/s32/s115/s32/s40/s48/s32/s109/s84/s41/s56/s48/s32/s87/s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52/s84\n/s102/s84/s32/s40/s75/s41\n/s32/s77 /s32/s40 /s101/s109/s117/s41/s32/s70/s67/s32/s40/s53/s32/s109/s84/s41\n/s32/s90/s70/s67/s32/s40/s53/s32/s109/s84/s41\n/s32/s70/s67/s32/s40/s49/s48/s32/s109/s84/s41\n/s32/s90/s70/s67/s32/s40/s49/s48/s32/s109/s84/s41/s40/s97/s41\n/s56/s48/s32/s87/s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52\n/s32/s32/s32 /s32FIG. 5: (a) M(T) curves of the 80 W ZnFe 2O4grown on\nMgAl 2O4recorded under ﬁeld cooled (FC, full symbols) and\nafter zero-ﬁeld cooled (ZFC, open symbols) conditions at\n5 mT (black squares) and 10 mT (red circles). (b) M(T)\ncurves recorded while cooling at 10 mT (FC) with intermit-\ntent stops (IS, open squares) with various Twaitwithtwaitof\n10,000 s marked by arrows. The M(T) curve while warming\nis subsequently recorded at 10 mT (red line).\nglassiness, the FC memory sequence used in [9] was car-\nried out. Figure 5(b) shows the FC M(T) curve recorded\nat 10 mT with intermittent stops (IS, open symbols) at\nvariousTwaitwhich are marked with arrows. Here the\nﬁeld was reduced to 0 mT for a waiting time twaitof\n10,000 s. Then the ﬁeld was set to 10 mT again and\nthe cooling down is resumed. Subsequently M(T) is\nmeasured at 10 mT while heating at the same rate as\nduring FC without any IS (FH, full line). In the FC\ncurve clear steps can be seen for most Twaitwhich are\nmost pronounced just below the maximum of the ZFC\ncurve in Fig. 5(a), i.e., also below Tf, while they are vir-\ntually absent above Tf. Note, that in Fig. 5we show\nthe magnetic data in emu to demonstrate the absolute\nsize of the steps in comparison to the detection limit of\nthe SQUID of 2 −4·10−7emu [27, 28]. These steps\ndemonstrate magnetic relaxation during twait. More im-\nportant, the subsequent FH curve shows clear inﬂection\npoints around Twait, and the inset enlarges the two most7\nprominent ones. Therefore, there is a ﬁrst experimen-\ntal evidence for magnetic glassiness in epitaxial ZnFe 2O4\nanalogous to ZnFe 2O4nanopowders in [9]; however, this\nglassiness extents to rather high temperatures which are\nonlycomparabletothosereportedin[7]. Acaveatisstill,\nthat such a behavior can also be observed and modeled\nin superparamagnetic samples as discussed in detail in\n[26] where only subtleties in these type of FC memory\nsequences allow to distinguish a superparamagnet from\na superspin-glass. Therefore ZFC memory experiments\nare needed in addition.\nFigure6provides additional experimental evidence for\nmagnetic glassiness of the same sample by ZFC memory\nexperimentsadoptedafter[25]. Herethe sampleiscooled\ndown under ZFC conditions once without any waiting\ntime and once cooling is stopped at Twait= 270 K for\nvarying waiting times twaitfrom 500 s to 50,000 s. Sub-\nsequently, an M(T) curve is measured at 5 mT while\nwarming (FH). In Fig. 6(a) the diﬀerence ∆ Mbetween\nthe FH without and with Twaitis plotted for all twait.\nNote, that ∆ Mis provided in emu and the visible scat-\nter in the diﬀerence data is around 1 −2·10−8emu,\nwhich demonstrates the high reproducibility of the data\nrecorded with the SQUID magnetometer. It should be\nstressed that this is only possible if the magnet is reset\nbeforethe measurementtoeliminateanytrappedﬂux. In\nall subsequent measurements one has to avoid magnetic\nﬁelds larger than 10 mT so that the nominal and actual\nﬁeld are identical for all measurementswithin 0.1mT be-\ntween which ∆ Mis taken. For twaitof 500 s and 1,000 s\nan increase of ∆ MbelowTfis visible with a maximum\naround the maximum of the ZFC M(T) curve, i.e., ∆ M\nfollows the shape of the ZFC curve. However, the max-\nimum of the ZFC curve for the given experimental con-\nditions is around 300 K while the maximum of the ∆ M-\ncurve is around 225K, i.e., shifted to lowertemperatures\nand does not go back to zero. This low temperature in-\ncrease of ∆ Mis diﬃcult to be explained in a straight-\nforward manner, because the nominally ZFC conditions\nonly correspond to less than 0.1 mT [28]. Therefore the\ndiﬀerence between the ZFC with an without twaitof 500s\nat270K,i..e., below Tfimpliesthatthesystemisallowed\nto spend additional 500 s close to the freezing tempera-\nture in a tiny, but ﬁnite ﬁeld. If one considers a super-\nparamagnetic ensemble close to its blocking temperature\nthis implies more time for thermally activated switching\ninatinyﬁeldwhichimprintsatinyadditionalmagnetiza-\ntion because the residual ﬁeld induces a slight imbalance\nin the probabilityof switching paralleland antiparallelto\nit. A superparamagnetic ensemble would further imply\nrelatively fast characteristic time-scales for the switching\nattempts. This would be in accordance that ∆ Mwith\ntwaitof 500 s and 1.000 s are virtually identical, because\nall the switching events are already done, while without\ntwaitthe system is ramped through the blocking temper-\nature with a rate of 60s/K, so much less switching events\ncan take place around the blocking temperature, where\nthe tiny residual ﬁeld is suﬃcient to aid the thermally/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48\n/s40/s98/s41\n/s116\n/s119/s97/s105/s116/s32/s61/s32/s49/s48/s44/s48/s48/s48/s32/s115\n/s32/s77 /s32/s40 /s101/s109/s117/s41\n/s84/s32/s40/s75/s41/s32/s51/s50/s48/s32/s75/s32 /s32/s51/s48/s48/s32/s75\n/s32/s50/s56/s48/s32/s75/s32 /s32/s50/s54/s48/s32/s75\n/s32/s50/s52/s48/s32/s75/s32 /s32/s50/s50/s48/s32/s75\n/s32/s50/s48/s48/s32/s75/s32 /s32/s49/s54/s48/s32/s75/s84\n/s119 /s97/s105/s116/s32/s61/s56/s48/s32/s87\n/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52/s116\n/s119 /s97/s105/s116/s32/s61\n/s32/s84/s32/s40/s75/s41\n/s70/s72/s32/s40/s53/s32/s109/s84/s41\n/s97/s102/s116/s101/s114/s32/s90/s70/s67/s32/s105/s110/s32/s48/s32/s109/s84\n/s32/s77 /s32/s40 /s101/s109/s117/s41/s32/s53/s48/s48/s32/s115\n/s32/s49/s44/s48/s48/s48/s32/s115\n/s32/s53/s44/s48/s48/s48/s32/s115\n/s32/s49/s48/s44/s48/s48/s48/s32/s115\n/s32/s53/s48/s44/s48/s48/s48/s32/s115\n/s84\n/s119/s97/s105/s116/s32/s61/s32/s50/s55/s48/s32/s75/s40/s97/s41\nFIG. 6: Characteristic hole-burning experiment for the 80 W\nZnFe2O4sample grown on MgAl 2O4. (a) shows the depen-\ndence on the waiting time twaitfor a ﬁxed waiting tempera-\ntureTwaitof 270 K while (b) shows the dependence on Twait\nfor a ﬁxed twaitof 10,000 s.\nactivated switching events. The remaining low tempera-\ntureincreaseisthusthefrozen-inresultofmoreswitching\nevents close to Tfresulting in a waiting-time imprinted\nadditional magnetization. This is further corroborated\nby the fact, that the low-temperature increase is found\nto decrease with decreasing Twait, i.e., a waiting further\nbelowTfand thus in a region with potentially slower\ndynamics, see Fig. 6(b). We thus infer that this low tem-\nperature increase of ∆ Mis most likely to be indicative\nof a superparamagnetic-like behavior with relatively fast\ndynamics rather than classical rejuvenation eﬀects in su-\nperferromagnets as discussed in [25].\nBeyond this low-temperature increase of ∆ Mseen for\nalltwaitin Fig.6(a), there is a minimum evolving with\nincreasing twaitbecoming clearly visible at 5,000 s and\nbeingmostpronouncedat50,000s. Thisisatypicalchar-\nacteristic of a (super)spin glass as discussed in [25, 26];\nhowever, the minimum is shifted to lower temperatures\ncompared to Twaitby about 20 K. This shift is also seen,\nirrespective of the actual Twait, see also Fig 7(a) further\nbelow. Figure 6(b) shows ∆ Mcurves for a ﬁxed twaitof8\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48 /s51/s53/s48\n/s40/s98/s41\n/s116\n/s119/s97/s105/s116/s32/s61/s32/s49/s48/s44/s48/s48/s48/s32/s115\n/s32/s77 /s32/s40 /s101/s109/s117/s41\n/s84/s32/s40/s75/s41/s32/s51/s48/s48/s32/s75\n/s32/s50/s56/s48/s32/s75\n/s32/s50/s54/s48/s32/s75\n/s32/s50/s52/s48/s32/s75\n/s32/s50/s50/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s49/s54/s48/s32/s75/s84\n/s119 /s97/s105/s116/s32/s61/s56/s48/s32/s87\n/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52/s116\n/s119 /s97/s105/s116/s32/s61\n/s32/s84/s32/s40/s75/s41\n/s70/s72/s32/s40/s53/s32/s109/s84/s41\n/s97/s102/s116/s101/s114/s32/s90/s70/s67/s32/s105/s110/s32/s48/s32/s109/s84\n/s32/s77 /s32/s40 /s101/s109/s117/s41\n/s32/s49/s44/s48/s48/s48/s32/s115\n/s32/s53/s44/s48/s48/s48/s32/s115\n/s32/s49/s48/s44/s48/s48/s48/s32/s115\n/s32/s53/s48/s44/s48/s48/s48/s32/s115\n/s84\n/s119/s97/s105/s116/s32/s61/s32/s50/s55/s48/s32/s75/s40/s97/s41\nFIG. 7: Identical set of data as in Fig. 6 for the 80 W\nZnFe2O4sample grown on MgAl 2O4for (a) varying twaitfor\nTwait= 270 K (magenta dahed line) and (b) varying Twaitfor\ntwait=10,000 s; however, ∆ Mis taken diﬀerently (see text).\n10,000sforvarious Twait. ForTwaitaboveTfnominimum\nis visible and only the low temperature increase can be\nseen. In contrast, a clear minimum is observable which\nis strongest for Twaitof 280 K, i.e., close to Tf. For lower\nTwaitis becomes less pronounced and the minimum shifts\nto lower temperatures, which are however always below\nthe respective Twait, e.g., the minimum in ∆ MforTwait\nof 160 K is at 150 K (orange pentagons). Therefore, the\n80 W ZnFe 2O4sample grown on MgAl 2O4shows all the\nexperimental characteristics of magnetic glassiness. On\nthe one hand a FC memory eﬀect characteristic of su-\nperparamagnets and (super)spin glasses [26] which have\nbeen reported for ZnFe 2O4before [9], see Fig. 5(b). On\nthe other hand, a twait-dependent minimum is observed,\nwhich is known as hole-burning experiment [25] and is\nabsent in superparamagnets but is seen in (super)spin-\nglasses [26]. On the other hand, the low-temperature\nincrease of ∆ Mfor short twaitorTwaitaboveTfresemble\nmore of superparamagnetic-like behavior. However, we\nwill show in the following that superparamagnetic-like\nbehavior and magnetic glassiness coexist.\nFigure7provides an alternative way of presenting theidentical results as in Fig. 6. In this case ∆ Mis taken\nas the diﬀerence between all data with respect to (a)\ntwait=500 s and (b) Twait= 320 K. In other words,\nhere ∆M(T) should only contain the magnetic glassiness\nsincethesuperparamagneticbehavior—whichisreﬂected\nby the low temperature increase of ∆ Mseen for short\ntwaitin Fig.6(a), ortwaitwell above Tfin Fig.6(b)—\nis subtracted and thus only the slow, glassy dynamics\ncan be seen. Figure 7(a) reveals that ∆ M(T) oftwait\nof 500 s and 1,000 s are virtually identical, since only a\nzero line is visible. In other words, the fast dynamics of\nthe superparamagnet are over while the slow dynamics\nof the glassiness have not yet set in, both referring to\nthe experimental accuracy. Having thus subtracted the\nfast dynamics the evolving dip in ∆ M(T) withtwaitof\n5,000 s and higher nicely represents the remaining mag-\nnetic glassiness with its characteristic slow dynamics and\nZFC memory eﬀect. It is furthermore visible that the\nminimum in ∆ M(T) does not align with Twaitwhich is\nindicated by the dashed magenta line in Fig. 7(a).Twait\nmerely appears to align with the inﬂection point of the\nhigh-temperature side of ∆ M(T) which is also seen in\ntheTwait-dependence of ∆ M(T) in Fig. 7(b). Also here,\nthe low-temperature increase of ∆ Mseen in Fig. 6(b)\nis fully removed by taking ∆ Malways with respect to\nTwait= 320 K. Note, that in Fig. 7(b)Twait= 300 K\nis not nicely visible but the dip in ∆ M(T) is very weak\nand clearly less pronounced compared to the others and\nin fact may only reﬂect the limits of reproducibility of\nthese types of SQUID experiments; one should keep in\nmind that twoZFC M(T)curveslikeinFig. 5(a) aresub-\ntracted from each other, i.e., the signal size and thus the\nrelative accuracy of each data point varies (slightly) over\nthe entire T-range which can easily aﬀect diﬀerence sig-\nnalsoftheorderof1 ·10−7emu. Nevertheless, Figs. 6and\n7nicelydemonstratethatinZnFe 2O4superparamagnetic\nand glassy behavior coexist and can be separated from\neach other. This is quite remarkable, since an epitaxial\nﬁlm of ZnFe 2O4is structurally quite distinct from a su-\nperparamagnetic ensemble like horse-spleen ferritin or a\nsuperspin-glass like a dense ensemble like Fe 3N nanopar-\nticles which were both investigated in [26]. Yet, ZnFe 2O4\nepitaxial thin ﬁlms exhibit both types of magnetic or-\nder at the same time. Therefore, the observed magnetic\nglassiness appears to be better described in terms of a\ncluster glass like in [7], i.e., a superparamagnetic-like en-\nsemble with (frustrated) intercluster interactions. How-\never, these interactions have to be inhomogeneous and\ndisordered throughout the sample and in contrast to the\nnanopowderin [9] they have no obvious structural origin.\nOne has to therefore conclude that they stem from local\nvariations of the cation distribution, i.e., from chemical\nor A/B disorder and thus they crucially depend on a ﬁ-\nnite amount of inversion. This in turn also explains why\nhighly crystalline bulk ZnFe 2O4samples in [1–3] exhibit\nquite distinct magnetic properties.\nSince we have seen that Tfis a function of the growth\npower during the sputtering process, the two power se-9\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48\n/s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s48/s46/s50/s48/s46/s48\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s49/s48/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s53/s116\n/s119/s97/s105/s116/s32/s61/s32/s49/s48/s44/s48/s48/s48/s32/s115/s84/s32/s40/s75/s41/s77 /s32/s40/s107/s65/s47/s109/s41/s115/s112/s117/s116/s116/s101/s114/s32/s112/s111/s119/s101/s114\n/s32/s49/s48/s48/s32/s87 \n/s32/s56/s48/s32/s87 \n/s32/s54/s48/s32/s87 \n/s32/s52/s48/s32/s87 \n/s32/s50/s48/s32/s87 \n/s77/s103/s65/s108\n/s50/s79\n/s52/s40/s97/s41/s32\n/s32\n/s116\n/s119/s97/s105/s116/s32/s61/s32/s49/s48/s44/s48/s48/s48/s32/s115\n/s40/s98/s41/s77 /s32/s40/s107/s65/s47/s109/s41\n/s84/s32/s40/s75/s41/s115/s112/s117/s116/s116/s101/s114/s32/s112/s111/s119/s101/s114\n/s32/s56/s48/s32/s87 \n/s32/s54/s48/s32/s87 \n/s32/s52/s48/s32/s87 \n/s32/s50/s48/s32/s87 /s65/s108\n/s50/s79\n/s51/s32\n/s32/s32/s32\nFIG. 8: Comparison of the hole-burning experiments for\nZnFe2O4grown on MgAl 2O4(a) and Al 2O3(b) as a func-\ntion of sputter power (details see text). The insets show the\nhigh sputter power samples only.\nries of ZnFe 2O4samples grown on MgAl 2O4and Al 2O3\nshall be directly compared. For that we have chosen\nto perform the hole-burning ZFC waiting experiments\nof Fig.6(b) on the identical relative temperature scale\nfor each sample. In other words, the highest and lowest\ntemperature of the M(T) curves as well as Twaithave\nbeen chosen to be a the same relative temperature with\nrespect to Tfto assure that the samples spent compa-\nrable time-spans in regions with comparable magnetiza-\ntion dynamics. Note, that in addition the full experi-\nment for all Twaitof Fig.6(b) on an absolute temper-\nature scale have also been performed (not shown), but\nthe direct comparison in essence reveals the identical re-\nsult. Figure 8shows the ∆ Mcurves for the power-series\nof ZnFe 2O4grown on MgAl 2O4(a) and Al 2O3(b) for\ntwaitof 10,000 s; the insets enlarge the samples grown at\nhigh sputtering powers. Irrespective of the substrate the\nsamples grown at sputtering powers of 20 W and 40 W\ndo only show the low-temperature increase of ∆ M, i.e.,\nmostly superparamagnetic-like behavior; for ZnFe 2O4on\nAl2O3a faint and broad minimum is visible which how-\never does not show a clear shift with Twaitor a pro-\nnounced dependence with twait. Therefore, we consider/s48/s49/s55/s48/s48 /s55/s48/s53 /s55/s49/s48 /s55/s49/s53 /s55/s50/s48 /s55/s50/s53 /s55/s51/s48\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s55/s48/s48 /s55/s48/s53 /s55/s49/s48 /s55/s49/s53 /s55/s50/s48 /s55/s50/s53 /s55/s51/s48/s48/s49\n/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s70/s101/s32/s76\n/s51/s47/s50/s45/s101/s100/s103/s101/s115\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s40/s97/s41/s112/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s110/s111/s114/s109/s46/s32/s88/s65/s78/s69/s83/s59\n/s59/s32/s56/s48/s32/s87 \n/s32/s50/s48/s32/s87 /s88/s77 /s67/s68/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52\n/s51/s54/s37/s32/s70/s101/s50/s43\n/s79/s104 /s47/s51/s50/s37/s32/s70/s101/s51/s43\n/s79/s104 /s47/s51/s51/s37/s32/s70/s101/s51/s43\n/s84/s100\n/s51/s50/s37/s32/s70/s101/s50/s43\n/s79/s104 /s47/s51/s56/s37/s32/s70/s101/s51/s43\n/s79/s104 /s47/s51/s48/s37/s32/s70/s101/s51/s43\n/s84/s100\n/s70/s101/s32/s76\n/s51/s47/s50/s45/s101/s100/s103/s101/s115\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103\n/s59\n/s59/s40/s98/s41/s110/s111/s114/s109/s46/s32/s88/s65/s78/s69/s83\n/s112/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s56/s48/s32/s87 \n/s32/s50/s48/s32/s87 /s88/s77 /s67/s68\n/s51/s52/s37/s32/s70/s101/s50/s43\n/s79/s104 /s47/s52/s56/s37/s32/s70/s101/s51/s43\n/s79/s104 /s47/s49/s56/s37/s32/s70/s101/s51/s43\n/s84/s100\n/s50/s56/s37/s32/s70/s101/s50/s43\n/s79/s104 /s47/s50/s56/s37/s32/s70/s101/s51/s43\n/s79/s104 /s47/s52/s52/s37/s32/s70/s101/s51/s43\n/s84/s100/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s65/s108\n/s50/s79\n/s51\nFIG. 9: Normalized XANES and XMCD spectra recorded at\nthe FeL3/2-edges under grazing incidence at 300 K for the\n80 W and 20 W ZnFe 2O4samples grown on (a) MgAl 2O4and\n(b) Al 2O3, respectively. The XMCD spectra have also been\nsimulated to determine the relative amount of the individua l\nFe species (see text).\nthis part as inconclusive, i.e., not as clear experimental\nevidence for glassiness. In contrast, the samples grown\nat 60 W and higher all show a hole-burning behavior in\nthe ZFC memory experiments which is pronounced for\nZnFe2O4on MgAl 2O4, see inset of Fig. 8(a), but rather\nweak for ZnFe 2O4on Al2O3, for which only a faint min-\nimum can be seen, see inset of Fig. 8(b). Therefore, in\nZnFe2O4on Al2O3only superparamagnetic-like can be\ninferred and signatures of magnetic glassiness are faint\nand limited to high sputtering powers. This goes hand-\nin-hand with a more pronounced maximum in the ZFC\ncurves, see Fig. 3(b) and an increased magnetization, see\nFig.2(a). In contrast, ZnFe 2O4on MgAl 2O4exhibits\na clear transition from superparamagnetic-like behavior\nat low sputtering powers with clear signs of magnetic\nglassinessexisting at high sputtering powers, i.e., growth\nrates. To ultimately clarify what causes the discrep-\nancy in the magnetic properties for ZnFe 2O4grown on\nMgAl2O4andAl 2O3aswellasatlowandhighsputtering\npowers, the 20 W and the 80 W samples were subjected\nto an element-selective magnetic characterization using\nXMCD.10\nFigure9showsthemeasuredXANESandXMCDspec-\ntra at the Fe L3/2-edges for ZnFe 2O4grown on MgAl 2O4\n(a) and Al 2O3(b) for the samples grown at 20 W and\n80 W, respectively. The XMCD at the Fe L 3-edge has\nbeen also simulated by respective multiplet ligand ﬁeld\ntheory using the CTM4XAS code using the identical\nparameters as in [32]. In brief, the negative peaks in\nthe XMCD spectrum are stemming from the octahedral\ncontributions Fe Oh, where Fe2+\nOhis mostly seen at lower\n(706.6 eV) and Fe3+\nOhat higher (708.5 eV) photon ener-\ngies; the positive peak at 707.8 eV can be assigned to\nFe3+\nTd. The experimental XMCD can be reproduced by\nadjusting the relative concentrations of Fe3+\nOh, Fe2+\nOh, and\nFe3+\nTdto match the experimental XMCD; the results of\nthis are given in Fig. 9. It can be seen in Fig. 9(a), that\nthere are no pronounced diﬀerences between the exper-\nimental XMCD spectra of the Fe L3/2-edge XMCD for\nZnFe2O4/MgAl 2O4grownat either 20 Wor80 Was well\nas for the respective results of the simulation. About one\nthird ofthe Fe is located on tetrahedralsites, i.e., the de-\ngree of inversion δis around 0.3 for both sputtering pow-\ners. Also a signiﬁcant amount of Fe2+\nOhis found, which\nwould suggest a strong contribution from a JDE\nBBdouble\nexchange interaction which appears to be slightly larger\nfor the 80 W sample which exhibits the magnetic glassi-\nnessin comparisontothe 20Wsample, whichonly shows\nthe superparamagnetic-like low temperature increase of\n∆M. The ZnFe 2O4/Al2O3samples in Fig. 9(b) exhibit a\ndiﬀerent behavior. Here the 80 W sample has a strongly\nreducedcontributionofFe3+\nTdcomparedtothe FeOhcom-\npared to the 20 W sample which has the highest relative\ncontent of Fe3+\nTd. On the other hand, the actual positive\npeak in the XMCD is of identical size in both samples. It\ntherefore appears, the XMCD intensity for the FeOhis\nreducedwhiletheamountofFe3+\nTdremainsconstant. This\nmay appear as contradiction at ﬁrst sight, since the rel-\native contents may suggest diﬀerent degrees of inversion\nfor the two samples. However, one should keep in mind\nthat the magneticsuperexchangeinteractionon the octa-\nhedralsites JBBis weaklyantiferromagneticwhile double\nexchange leads to spin canting [4, 5]. Since the magnetic\norder is observed up to above room temperature for all\nsamples in this work, the JABsuperexchange mechanism\nhasto playasigniﬁcantrole, whichis consistentwith a ﬁ-\nnite degree of inversion of the order of 0.3. In that light,\nthe presence of a ﬁnite amount of inversion giving rise\nto Fe3+\nTdis a prerequisite for magnetic order at elevated\ntemperatures but does not play a decisive role for the\npresence of magnetic glassiness, since Fe3+\nTdis found in all\nfour samples while glassiness is only found in the 80 W\nsamples, in particular in those grown on MgAl 2O4. In\naddition, the presence of Fe2+\nOhin all samples further sug-\ngests the presence of an additional JDE\nBBdouble exchange\nmechanism associated with spin canting. Here the rela-\ntiveamountofFe2+\nOhincreasesonlyslightlyfromthe20W\nsampleon Al 2O3over20Won MgAl 2O4, 80WonAl 2O3\nto 80 W on MgAl 2O4, i.e., it follows the trend of in-\ncreasing glassiness of the samples. However, the changes/s49/s48/s49/s48 /s49/s48/s50/s48 /s49/s48/s51/s48 /s49/s48/s52/s48 /s49/s48/s53/s48 /s49/s48/s54/s48/s48/s49\n/s45/s50/s48/s50/s52/s54/s48/s49/s49/s48/s49/s48 /s49/s48/s50/s48 /s49/s48/s51/s48 /s49/s48/s52/s48 /s49/s48/s53/s48 /s49/s48/s54/s48\n/s45/s50/s48/s50/s52/s54/s110/s111/s114/s109/s46/s32/s88/s65/s78/s69/s83\n/s112/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s50/s48/s87 /s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s65/s108\n/s50/s79\n/s51/s56/s48/s87 /s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s65/s108\n/s50/s79\n/s51/s88/s77 /s67/s68/s32/s40/s37/s41/s50/s48/s87 /s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52\n/s90/s110/s32/s76\n/s51/s47/s50/s45/s101/s100/s103/s101/s115\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s56/s48/s87 /s32/s90/s110/s70/s101\n/s50/s79\n/s52/s47/s77/s103/s65/s108\n/s50/s79\n/s52\n/s40/s98/s41/s112/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s110/s111/s114/s109/s46/s32/s88/s65/s78/s69/s83/s40/s97/s41\n/s90/s110/s32/s76\n/s51/s47/s50/s45/s101/s100/s103/s101/s115\n/s84/s32/s61/s32/s51/s48/s48/s32/s75\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s88/s77 /s67/s68/s32/s40/s37/s41\nFIG. 10: Normalized XANES and XMCD spectra recorded\nat the Zn L3/2-edges under grazing incidence at 300 K for the\n80 W and 20 W ZnFe 2O4samples grown on (a) MgAl 2O4and\n(b) Al 2O3, respectively.\nare rather small and the signiﬁcance of determining such\nsmall changes with multiplet ligand ﬁeld simulations is\nlimited. Obviously, there is no straightforward mech-\nanism for the occurrence of magnetic glassiness which\ncan be derived form the XMCD spectra at the Fe L3/2-\nedges. A ﬁnite degree of inversion has to play a role but\nmostly for the high order temperatures observed. The\nexistence of glassiness appears to be linked to a delicate\nbalance of the various competing exchange interactions\nas well as the local cationic conﬁguration which has to\nbe inhomogeneous throughout the sample as discussed\nabove. Obviously the growth rate inﬂuences mostly the\nlatter where the highest growth rates favor glassiness,\nmost likely via increased local cationic disorder, in par-\nticlular in ZnFe 2O4on MgAl 2O4substrates.\nFigure10shows the measured XANES and XMCD\nspectra at the Zn L3/2-edges of the identical set of sam-\nples as in Fig. 9. The XANES for ZnFe 2O4on MgAl 2O4\nin Fig.10(a) is rather similar to the one of ZnFe 2O4on\nAl2O3in (b). All four samples exhibit a ﬁnite XMCD\nwith comparable spectral shape; all XMCD spectra were\nderived by reversing both, helicity of the light as well\nas the magnetic ﬁeld and it was veriﬁed that the XMCD\nspectrumnicelyreverseswithreversingexternalﬁeld(not11\nshown). The size of the Zn L3/2-edge XMCD follows the\namount of Fe3+\nTdas seen in Fig. 9, i.e., the Zn XMCD\nis largest for the 20 W sample on Al 2O3, which has the\nhighestrelativeFe3+\nTdcontentandit islowestforthe20W\nsample on Al 2O3which has the lowest relative Fe3+\nTdcon-\ntent. It is thus reasonable to assume that the magnetic\npolarization of Zn in ZnFe 2O4is mostly associated with\nZn2+\nOh. Inturn, thisimpliesthataweaklypolarizedcation\nsubstitutes for a strongly polarized one thus reducing the\neﬀective exchange. This is consistent with the exper-\nimental observation that the 80 W ZnFe 2O4/MgAl 2O4\nhas the highest Tfand the lowest magnetic polarization\nof Zn while the highest Zn polarization in the 20 W\nZnFe2O4/Al2O3sample is associated with the lowest Tf.\nTo verify this hypothesis, more sophisticated theoretical\ncalculations beyond the multiplet ligand ﬁeld codes is re-\nquired where the individual spectroscopic signatures in\nthe ZnL3/2-edge XANES and XMCD can be associated\nwith the actual Zn species which however goes beyond\nthescopeofthispaper. Nevertheless,itisalreadyevident\nthat a too high degreeof inversionas seen bystrongmag-\nnetic polarization of the Zn together with a high relative\ncontent of Fe3+\nTdis unfavorable for both, high Tfas well as\nmagnetic glassiness and high growth rates appear to be\nan experimental means to control/limit excessive inver-\nsion but at the same time assure suﬃcient local cationic\ndisorder to induce magnetic glassiness.\nIV. DISCUSSION AND CONCLUSION\nZnFe2O4epitaxial thin ﬁlms have been grown on\nMgAl2O4and Al 2O3substrateswith varyingpreparation\nconditions. All samples were investigated with respect to\ntheir basic structural and magnetic properties and long\nrange magnetic order was found above room tempera-\nture for all samples. The stoichiometric composition of\nthe samples was veriﬁed using RBS. A clear bifurcation\nbetween M(T) curves under FC and ZFC conditions is\nfound at Tf, which is systematically higher for ZnFe 2O4\non MgAl 2O4by about 100 K. Tfis found to systemati-\ncally increase with increasing the sputtering power and\nthus growth rate in agreement with [7]. The Ar:O 2ratio\nwas not found to inﬂuence neither TfnorMsin a system-\natic manner; increasing Tgrowthincreases only Mswhile\nTfexhibits no systematic changes.\nAn in-depth study of the magnetic properties us-\ning FC as well as ZFC memory experiments reveals\nmagnetic glassiness for samples grown at high sput-\nter powers. The glassiness is more pronounced for\nZnFe2O4/MgAl 2O4compared to ZnFe 2O4/Al2O3, where\nthe signatures of magnetic glassiness beyond those in\nFC memory experiments are generally weak. At lower\ngrowth rates the signatures of glassiness are absent and a\nlow-temperatureincreaseof∆ Misobservedwhichpoints\ntowards superparamagnetic-like behavior and the signa-\ntures in FC memory experiments are weak and the ZFC\nmemory experiments shows no hole burning eﬀect in ac-cordance with the expectations for superparamagnetic\nsamples [26]. In contrast, at high growth rates, in par-\nticular for ZnFe 2O4/MgAl 2O4ZFC memory experiments\nshow an additional hole burning eﬀect which is charac-\nteristic for spin glasses [25] and superspin glasses [26].\nSince the glassiness coexists with superparamagnetic-like\nsignatures, in particular, the low temperature increase of\n∆M, the structural properties of the ZnFe 2O4epitaxial\nﬁlms are quite diﬀerent from the nanoparticle ensembles\nin [26] the observed magnetic properties are described\nbest as cluster glass in analogy to comparable observa-\ntions for epitaxial ZnFe 2O4in [7].\nAn in-depth characterization based on XANES and\nXMCD reveals that a ﬁnite magnetic polarization at the\nZnL3/2edges exists in all ZnFe 2O4samples which adds\nmore complexity to the magnetic interactionsbeyond the\nusually discussed Fe-based exchange. At the Fe L3/2the\nXMCD is used to extract the relative concentrations of\nFe3+\nTd, Fe3+\nOh, and Fe2+\nOhby means of multiplet ligand ﬁeld\nsimulations as done before [31, 32]. The abundance of\nFe3+\nTdcorrelates well with the size of the magnetic polar-\nizationofZnand thusboth canserveasameasureforthe\ndegreeofinversion. Forhighestinversion Tfisfoundtobe\nlowest and signatures of magnetic glassiness are absent.\nIn contrast, the sample with the strongest signatures of\nmagnetic glassiness, the 80 W ZnFe 2O4/MgAl 2O4, ex-\nhibits no signiﬁcant changes in the Fe L3/2-edge XMCD\ncompared to the superparamagnetic-like 20 W sample.\nThe most prominent tendency appears to be the rel-\native amount of Fe2+\nOhwhich can be associated with a\ndouble-exchange mechanism which was held responsible\nfor spin canting [4, 5]. Since the diﬀerences in XMCD be-\ntween the respective samples are small, this correlation\nbetween glassiness and Fe2+\nOhcannot be taken as signiﬁ-\ncant but merely a starting point for more elaborate the-\noretical work to understand the details of the obtained\nXMCD spectra beyond the multiplet ligand ﬁeld simula-\ntions. Most likely the cluster glass behavior in epitaxial\nZnFe2O4cannot be assigned to the actual structure of\nthe materials like in common superparamagnets or dense\nnanoparticle ensembles [26] but due to local variations\nof the stoichiometry, leading to an inhomogeneous local\ncation distribution throughout the sample. As a con-\nsequence, the local magnetic moments in ZnFe 2O4are\ndisordered due to partial inversion and partially canted\ndue to the presence of Fe2+\nOh, which leads to characteristic\nsignatures of a cluster glass at rather high temperatures\nwhich is mostly controllable by the growth rate as re-\nported for ZnFe 2O4before [7].\nAcknowledgments\nJ. L. gratefully acknowledges funding by FWF project\nORD-49 at the initial stage of this work. A.Z. acknowl-\nedges the ﬁnancial support by the Swiss National Science\nFoundation (SNSF) under Project No. 200021-169467.\nThe X-ray absorption measurements were performed on12\nthe EPFL/PSI X-Treme beamline at the Swiss Light\nSource, Paul Scherrer Institut, Villigen, Switzerland. In\naddition, support by VR-RFI (ContractsNo. 2017-00646\n9 and No. 2019-00191) and the Swedish Foundation for\nStrategicResearch(SSF, ContractNo. RIF14-0053)sup-porting accelerator operation at Uppsala University is\ngratefully acknowledged. 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B 102, 054402 (2020)."    },    {        "title": "0804.3292v1.Unusual_magnetic_behavior_in_ferrite_hollow_nanospheres.pdf",        "content": "Unusual magnetic behavior in ferrite hollow nanospheres\nE. Lima Jr.\u0003, J. M. Vargas, R. D. Zyslery\nCentro At\u0013 omico Bariloche and Instituto Balseiro, 8400 S. C. de Bariloche, RN, Argentina\nH. R. Rechenberg\nInstituto de F\u0013 \u0010sica, Universidade de S~ ao Paulo, 05315-970, S~ ao Paulo, SP, Brazil.\nJ. Arbiol\nTEM-MAT Serveis Cienti\fco-t\u0013 ecnicos, Universidad de Barcelona, 08028, Barcelona, Spain.\nG. F. Goyaz, A. Ibarra, M. R. Ibarra\nInstituto de Nanociencia de Arag\u0013 on, Universidad de Zaragoza, 50009, Zaragoza, Spain.\n(Dated: December 4, 2021)\nWe report unusual magnetic behavior in iron oxide hollow nanospheres of 9.3 nmin diameter.\nThe large fraction of atoms existing at the inner and outer surfaces gives rise to a high magnetic\ndisorder. The overall magnetic behavior can be explained considering the coexistence of a soft\nsuperparamagnetic phase and a hard phase corresponding to the highly frustrated cluster-glass like\nphase at the surface regions.\nPACS numbers: 75.50.Tt, 75.50.Gg, 75.50.LK, 75.60.Ej\nKeywords: Fine Particles, Superparamagnetism, Magnetic ordering, Magnetic Freezing\nFinite size-e\u000bect occurs in nanostructured materials as\nthin \flms, nanoparticles and nanowires. The control of\ntheir morphology and functionalities at the nanoscale is\na prerequisite for some biomedical applications that use\nnanoparticles as nanovectors for drug delivery [1]. Spher-\nical empty nanocapsules are appealing for these applica-\ntions because they could store larger amounts of drug\nthan solid NPs of the same size. The unique magnetic\nphenomena reported for core-shell nanoparticles along\nthe last years have been usually assigned to the complex\nsurface microstructure and/or exchange interactions at\nthe core/surface interface [2, 3]. The magnetic behavior\nof the surface atoms is characterized by the existence of\nbroken symmetry and exchange bonds which introduce\nstructural and magnetic disorder and originate an en-\nhancement of the magnetic anisotropy and the coercive\n\feld [4]. On the bases of the huge surface/bulk atomic\nratio, Hollow NanoSpheres (HNS) provide an excellent\nscenario to study the competition between surface and\nbulk magnetism at nanoscale level and open up new per-\nspectives for theoretical developments. The synthesis of\nHNS have been recently reported [5], using controlled ox-\nidization of FeorFe3O4nanoparticles after synthesis, at\ntemperatures of 473-473 K. In this work we present a dif-\nferent, low-temperature synthesis method for obtaining\nmonodisperse ferrite HNS without need of subsequent ox-\nidization process. The unusual magnetic behavior found\n\u0003Presently at: Instituto de F\u0013 \u0010sica, Universidade de S~ ao Paulo,\nBrazil\nycorresponding author, e-mail: zysler@cab.cnea.gov.ar\nzcorresponding author, e-mail: goya@unizar.esin these particles can be interpreted on the basis of soft\nand hard phases and the large surface/bulk atomic ratio\ndue to both inner and outer surfaces of a hollow sphere.\nFerrite HNS were prepared by modi\fcations at the\nhigh-temperature organic-phase synthesis from the pre-\ncursorFe(acac)3at phenylether (boiling point \u0018533-\n543K) in the presence of a long-chain alcohol (1,2-\nhexadecanediol) and oleic acid and oleylamine surfac-\ntants [6], using a molar ratio precursor:surfactant of 1:9\nto control the \fnal diameter of the particles [7]. The\nsynthesis lasted 30 minutes in argon \rux ( \u00180.5L=min: ),\nbut di\u000berently from those described in the literature, a\n(noncontrolled) temperature reduction was induced dur-\ning the synthesis procedure. Final HNS were coated\nby surfactant molecules, avoiding agglomeration and in-\ncreasing the chemical stability of the surface. They were\nfurther dispersed by dilution in toluene and alcohol di-\nlutes polyethylemine (PEI). After that the solvents were\nleft to evaporate, being stirred from time to time. At\nthe end, HNS dispersed to 5 % wtin PEI were obtained;\nthis dilution is su\u000ecient to ensure a negligible dipolar\ninteraction. The morphology, structure and composition\nof the particles were studied using High-resolution TEM\n(HRTEM) combined with Energy Electron Loss Spec-\ntroscopy (EELS) and Energy Filtered TEM (EFTEM)\nas well as high angular annular dark \feld (HAADF) and\nBright Field TEM (BFTEM). The samples were prepared\nby dropping a colloidal solution of HNS onto a carbon-\ncoated copper grid. Fig. 1-a shows a general view of the\nsample in which 9.3 nmin diameter nanoparticles are\nobserved. The iron oxide spinel structure obtained from\nthe Fourier transformation of the HRTEM (1-b) images\nis consistent with that obtained from X-ray di\u000braction.arXiv:0804.3292v1  [cond-mat.other]  21 Apr 20082\nFIG. 1: a) TEM micrograph of the sample where the pro-\njection of nanoparticles shows toroidal-like structures of 9.3\nnm in diameter, b) HRTEM obtained on one of the nanostruc-\ntures c) HAADF STEM images and d) STEM pro\fle through\none single nanoparticle indicated in c) showing an enhanced\ncontrast at the particle edge.\nFurthermore, broadening of the X-ray patterns re\rects\nthe existence of crystallite sizes of 2 nm, much lower than\nthe 9.3nmobserved from HRTEM. Detailed analysis of\nmagni\fed HRTEM revealed the polycrystalline nature of\nthe nanostructures with the absence of any preferential\norientation. It is important to point out that depending\non the defocus, crystal plains with d-spacing correspond-\ning to spinel structure were observed either on the exter-\nnal part or on the top of the inner part. This discards the\nexistence of toroidal-like structures. EELS and EFTEM\nanalysis (not shown) have revealed that the only elements\ncomposing the nanospheres were Fe and O. HAADF (Fig.\n1-c) and STEM (Fig. 1-d) analysis denotes a increase of\nthe density at the outer part. All these data reveal that\nthe observed nanoparticles are HNS. Moreover, BFTEM\nmicrograph at high tilt angle (35o) do not show changes\non the observed morphology denoting once more that\nour nanoparticles are HNS. Figure 2 displays the M(T)\ncurves (H= 20 Oe) measured in the ZFC and FC (cooling\n\feldH= 20Oe). The ZFC results exhibit a sharp peak\nat\u001836K. This temperature coincides with irreversibil-\nity temperature (i.e. the temperature above which the\nZFC and FC curves coincide). This is a indication of the\nexistence of a very narrow size distribution as observed\nin Fig. 1-a. Unexpectedly, the ZFC magnetization as the\nFC magnetization turns up below 20 K. This anomaly\noccurs at the same temperature range where we observe\nFIG. 2: Magnetization as a function of temperature. All\ncurves are collected from 2 Kup to 300Kwith applied \feld\nofH= 20Oe. ZFC (lower branch) and FC (upper branch)\ndata are shown. Inset: Thermal and frequency dependence of\nthe out-of-phase \u001f00component of the ac susceptibility.\nFIG. 3: Low-\feld section of the magnetization isotherms\n(in ZFC and FC modes) at T= 2K. Top inset: thermal\ndependence of the bias Hb(T) obtained from the FC M(H)\nresults. Bottom inset: M(H) measured up to 20 kOe. (a),\n(b), (c) and (d) refer to Fig. 4 (see text).\nthe rise inMFC(T). The inset of Fig. 2 displays the out-\nof-phase component \u001f00of ac susceptibility as a function\nof temperature under a magnetic \feld of 2 Oeand at fre-\nquenciesf= 0.5, 1, 3 and 10 kHz. The results exhibit\ntwo maxima located at TBB= 45-55KandTf\u001812K.\nThe maximum at TBBwas associated to the blocking pro-\ncess of the superparamagnetic (SPM) magnetic moments\nin inner part of the HNS, hereafter (BULK). TBBhas a\nlarge dependence on frequency, which is an indication of\nthe existence of a thermally activated process, with an\nenergy barrier Ea= 1:8\u000210\u000013erg. Assuming that Ea\nis the product KeffVBULK (VBULK is estimated from\nM ossbauer experiments as will be discussed later on), we\nobtainKeff= 1:3\u0002106erg=cm3for this phase, slightly\nhigher than bulk magnetite. The second maximum at\nTfhardly change with frequency and we associated it to3\nthe freezing of a cluster-glass like phase (CGP) structure\nin the disordered uncompensated surface regions (outer\nsurfaceS1and inner surface S2). The rise of the ZFC\nand FC magnetization showed in Fig. 2 could be asso-\nciated to the uncompensated magnetic moment at the\nsurface, which will provide an increase of the ferrimag-\nnetic moment below Tf. Fig. 3 shows a detail of the\nmagnetization loops measured in the ZFC and FC (cool-\ning \feldH= 10kOe). We observed a large \"loop shift\"\nin FC cycle when measured at temperatures below the\nCGP freezing temperature ( Tf<20K). The bias \feld,\nHb, is de\fned as the center \feld of the shifted magnetic\nloop. Usually, in \"core-shell\" nanoparticles, Hbis asso-\nciated to the bias anisotropy induced by the \"exchange\ninteraction\" between the magnetic microstructures in the\nfrustrated ordered shell pinned by a large surface mag-\nnetic anisotropy and the soft ferromagnetically ordered\ncore of the particle [2, 8]. In our case, we should un-\nderstand the origin of this bias \feld considering that the\nnanoparticles are not \"core-shell\" but hollow nanospheres\nwith two surface layers ( S1andS2) and the BULK in-\nner region, as represented in \fgure 4. In principle we\ncan assume that S1andS2regions are highly frustrated\nmagnetic layers of magnetic clusters with a large surface\nmagnetic anisotropy (responsible for the low temperature\nfreezing of the surface magnetic moments, Tf). The inner\nBULK region shows a SPM behavior with low magnetic\nanisotropy. When cooling the sample down to 2 Kun-\nder an external applied magnetic \feld H0, the resultant\nmagnetization along the applied \feld direction will be the\nsum of the contribution of the surface regions ( ~MS) and\nthe BULK ( ~MB):~M=~MS+~MBin the HNS (a). Once\nthe \feld is removed (b) at 2 K(belowTf), the freeze mag-\nnetic moments will keep a remanent magnetization ( ~Mr\nS,\nsee hysteresis loop of Fig. 3). We will consider that ~MS\nis contributed by ~Mr\nS, which is pinned during the whole\nhysteresis cycle, and ~Mup\nS, which is the unpinned \feld\ninduced component at the surface ( ~MS=~Mr\nS+~Mup\nS).\nWhen we apply an external magnetic \feld opposite to the\nmagnetization direction (c), the magnetization within the\nBULK will rotate at low \feld values (soft phase) but the\nfreeze magnetic moments at S1andS2will retain the\nmagnetic state in which were frozen Mr\nS. This process\nwill reduce ~MreachingM= 0 at~H0=\u0000~Hb, where\n~MB=\u0000~MS. The \feld necessary to compensate Mr\nSis\nthe responsible of the \"bias \feld\" (see Figure 4). In-\ncreasingH0an increase of the magnetization is obtained\nfavoring an alignment of the magnetic moments along\ntheH0in opposite direction to the (a) and (b) situation.\nIncreasing H0in the opposite direction of the cooling\n\feld, we can reach the situation depicted in (d), which is\nalmost symmetric to (a). However, ~Mr\nScontribution re-\nmains and is the responsible for the shift of the hysteresis\nloop toward positive values of magnetization when H0is\nagain reduced. ~Mr\nSis originated in the FC process be-\nFIG. 4: Schematic representation of the magnetization pro-\ncess of the sample after FC below Tf. (See explanation in the\ntext).\nlowTfand it is absent when the sample is ZFC. Thus,\nHbdisappears for temperatures above Tfor ZFC due to\nrandom alignment in the CGP ( Mr\nS).\nM ossbauer spectra (MS) were taken at 4.2-300 Kin\na liquid He \row cryostat with a conventional constant-\nacceleration spectrometer in transmission geometry us-\ning a57Co/Rh source. For in-\feld measurements, the\npowder sample was mounted in a vertical source-sample-\ndetector setup in the bore of a 140 kOe superconduct-\ning magnet, such that the direction of \r-ray is parallel\nto the applied \feld. The spectra were \ftted by using\nLorentzian line shapes, and a foil of \u000b\u0000Fewas used\nto calibrate the velocity scale. The room-temperature\nMS spectrum is a doublet with narrow lines (line width\nw=0.65mm=s ),IS=0.36mm=s and quadrupolar split-\ntingQS=0.98mm=s . TheISvalue is similar to what\nis commonly observed in nanostructured ferrites in SPM\nregime [9]. However, the QSvalue, which originates in\nthe local charge density symmetry, is much larger than\nthe expected for these materials, re\recting a local sym-\nmetry lower than cubic, which in turn will break the Fe-\nFe superexchange paths and/or oxygen vacancies located\nat both inner and outer surfaces. This is consistent with\nthe picture of a magnetically disturbed spin con\fguration\nat the surface [10]. At 4.2 K (Fig. 5-a) the relaxation\ntime is slow enough to ensure a static hyper\fne splitting,\nand the spectrum could be \ftted with two sextets associ-\nated to sites A and B in the spinel-type crystalline lattice.\nThe obtained hyper\fne \feld values ( Bhf= 491 and 455\nkOefor sites A and B, respectively) are smaller than bulk\nvalues [11], an e\u000bect usually observed in core-shell struc-4\ntures and assigned to the small Ea(and the associated\nsoftening) for the collective magnetic excitations which\nact to reduce the hyper\fne \felds with respect to their\nvalues atT= 0K[12]. However, in our case the Ea\nvalue is of the same order than observed for crystalline\nmagnetite nanoparticles. Thus, the origin of reduced Bhf\nis the surface disorder. The large linewidth values of the\nmagnetic sextets (Fig. 5-a) also indicate a locally disor-\ndered environment of Fe ions. In MS experiments under\napplied \feld, the e\u000bective hyper\fne \feld Beffwill be\nthe vector sum of the applied \feld Happand the hyper-\n\fne \feldBhf. Because of the strong antiferromagnetic\ninteraction between sublattices A and B in the ferrites,\nBeffof the sub-lattice A increases while Beffof the sub-\nlattice B decreases. Moreover, the relative intensities of\nthe six-line MS spectra are given by: 3 : p: 1 : 1 :p: 3,\np= 4 sin2\u000b=(1 + cos2\u000b), where\u000bis the angle between\nthe spin and the gamma-ray direction [13]. Therefore,\nlines 2 and 5 vanish when the magnetic moments of the\nparticles align to the applied \feld. Fig. 5-b shows the\nMS spectra measured at 4.2 K under Happ= 120kOe,\nwhich is composed of very broad, strongly overlapping\nlines. Considering our TEM and magnetic data, we pro-\nposed a \ftting procedure based on the combination of\ntwo crystalline sextets with narrow lines plus a continu-\nousP(Beff) distribution consistent with the existence of\nthe (CGP). The crystalline sextets were assumed to cor-\nrespond to spins in A and B sites aligned with applied\n\feld (red and blue subspectra, respectively). In addition,\nthe intensities of lines 2 and 5 are \fxed at 0 for these com-\nponents. The relative area of these crystalline subspec-\ntra is 6-7 %, showing that only a small fraction of spins,\nprobably located in the BULK region, are aligned to the\nexternal \feld. The hyper\fne \feld distribution resulting\nfrom the \ftting is displayed in the inset of Fig. 5-b, and\nit is associated to the sites not aligned with Happ. Its\ncontribution amounts to 87 %, re\recting a high fraction\nof misaligned moments. If we consider that the surface\nmoments are the only one disordered at high \feld, we\ncan estimate a thickness of 0.9 nmforS1andS2regions\nand an inner diameter of 4 nm(empty region) compati-\nble with STEM pro\fle (see Fig. 1-d). The e\u000bective \feld\ndistribution is very broad, with an equivalent probability\nfor allBeffvalues between those obtained for the two\ncrystalline sextets. In addition, the ratio between the\nline 2 and 3 ( I23) for this component is very close to 2,\nthe same as for a randomly oriented sample. This result\nsupports the picture of a spatially disordered freezing of\na large amount (87%) of spins residing in the morpholog-\nically disordered surface areas.\nSummarizing, we propose a new synthesis method to\nobtain ferrite hollow nanospheres. The magnetic char-\nacterization of this type of nanoestructures brings about\nrelevant phenomena which are explained within a sim-\nple model based on the coexistence of a SPM soft phase\n(BULK) and the CGP hard phase ( S1andS2).\nFIG. 5: M ossbauer spectra: a) Low temperature 4.2 Kand b)\nunder applied \feld ( Happ= 120kOe). Solid line is the \ftted\nspectrum and dashed red and blue lines are the subspectra\nreferent to sites A and B, respectively. Inset of \fgure 5-b\nis the hyper\fne \feld distribution obtained from the \ftting\nprocedure.\nWe acknowledge the critical reading and discussion\nwith Prof. P. A. Algarabel. This work received \fnancial\nsupported from Argentinian ANPCyT, CONICET and\nUNCuyo, Spanish MEC under grants: NAN2006-26646-\nE and Consolider CSD2006-12, and Brazilian FAPESP\nand CNPq. G. F. G. acknowledges support from the\nSpanish MEC through the Ramon y Cajal program. E.\nL. Jr. acknowledges fellowship by FAPESP.\n[1] M. Arruebo, R. Fernandez-Pacheco, M. R. Ibarra, J. San-\ntamar\u0013 \u0010a, Nano Today 2, 23 (2007). G. F. Goya, V. Graz\u0013 u,\nM. R. Ibarra, Current Nanoscience 4, 1 (2008).\n[2] B. Martinez, X. Obradors, L. I. Balcells, A. Rouanet, C.\nMonty, Phys. Rev. Lett. 80, 181 (1998).\n[3] X. Battle, A. Labarta, J. Phys. D: Appl. Phys. 35, R15\n(2002).\n[4] R. H. Kodama, A. E. Berkowitz, Phys. Rev. B 59, 6321\n(1999).\n[5] S. Peng, S. Sun, Angew. Chem. Int. Ed. 46, 4155 (2007).\nA. Cabot, V. F. Puntes, E. Shevchenko, Y. Yin, L. Bal-\ncells, M. A. Marcus, S. M. Hughes, A. P. Alivisatos, J.\nAm. Chem. Soc. 129, 10358 (2007).\n[6] S. Sun, H. Zeng, D. B. Robinson, S. Raoux, P. M. Rice,\nS. X. Wang, G. Li, J. Am. Chem. Soc. 126, 273 (2004).\n[7] J. M. Vargas, R. D. Zysler, Nanotechnology 16, 1474\n(2005).\n[8] R. D. Zysler, M. Vasquez Mansilla, D. Fiorani, Eur. Phys.\nJ. B 41, 171 (2004).\n[9] G. F. Goya, T. S. Berqu\u0013 o, F. C. Fonseca, M. P. Morales,\nJ. Appl. Phys. 94, 9520 (2003).\n[10] J. M. D. Coey, Phys. Rev. Lett. 27, 1140 (1971).\n[11] A. D. Arelaro, E. Lima Jr., L. M. Rossi, P. K. Ky-\nohara, H. R. Rechenberg, J. Magn. Magn. Mater,\ndoi:10.1016/j.jmmm.2008.02.066\n[12] S. M\u001crup, J. Magn. Magn. Mater. 37, 39 (1983).\n[13] D. P. E. Dickson, F. J. Berry (Eds.), \"M ossbauer Spec-\ntroscopy\", Cambridge University Press (1986), Chap. 4."    },    {        "title": "1309.5690v1.Remarkably_high_value_of_Capacitance_in_BiFeO3_Nanorod.pdf",        "content": "1\n \n Remarkably high value of Capacitance in BiFeO 3 Nanorod \n \nNabanita Dutta, S.K. Bandyopadhyay*, Subhas is Rana, Pintu Sen and A.K. Himanshu \nVariable Energy Cyclotron Centre, \n1/AF, Bidhan Nagar, Kolkata-700 064, India. \n*corresponding author. E-mail: skband@vecc.gov.in  \n  \nAbstract: \nA remarkably high value of specific capac itance of 450 F/g has been observed through \nelectrochemical measurements in the electrode made of multiferroic Bismuth Ferrite (BFO) in the form \nof nanorods protruding out. These BFO nanorods were developed on porous Anodised Alumina (AAO) \ntemplates using wet chemical technique. Diameters of nanorods were in th e range of 20-100 nm. The \nhigh capacitance is attributed to the nanostructure . The active surface charge has been evaluated \nelectrochemically by cyclic voltamme try (CV) at different scanning ra tes and charge-discharge studies. \nThe specific capacitances we re constant after several cycles of ch arge-discharge leading to their useful \napplication in devices. The mechanism of accumula tion of charge on the el ectrode surface has been \nstudied. \n \nKeywords: Bismuth Ferrite, Nanorod, Pseudoca pacitance, Electrochemical Double Layer  \nCapacitance. \n 2\nIntroduction: \nElectrochemical capacitors (EC) , popularly known as supercapacitors provide high power and \nlong cycle life, essential for ener gy storage devices. They are cate gorized as electrochemical double \nlayer capacitors (EDLC) and pseudocapacitors. In  EDLC, capacitance originates in the charge \nseparation at the electrode-electroly te interface, whereas pseudocapacita nce arises from fast, reversible \nfaradaic redox reactions taking place on or near th e surface of the electrode  [1]. Electrochemical \nperformances of a material as electrode can be  assayed by cyclic voltammetry and galvanostatic \ncharge-discharge studies of speci fic capacitance. An electrode is judged by its capac itance value and \nthe number of charge-discharge cy cles it withstands, maintaining th e constancy of capacitance. This \nbrings forth the search for a wide variety of mate rials. In general, transition metal oxides like RuO 2, \nMnO 2 etc. show high specific capacita nce with their redox behavior. \nIn this context, it is quite interesting to study the capacitance of mu ltifunctional materials like \nthe multiferroic Bismuth ferrite (BFO) showing coexistence of ferroelectricity as well as \nantiferromagnetism with wide applications [2]. Lokha nde et al. [3] observed specific capacitance value \nof 81F/gm in BFO films. Attempts have been ma de to obtain BFO in 1-Dimensional nanostructure \nforms like nanowire, nanorod, etc. [4,5]. The nanostr uctured forms can be expected to offer better \nefficiency owing to large surface ar ea giving  rise to a high value of sp ecific capacitance, for example  \nin α-MnMoO 4 nanorods [6]. This prompted us to study th e capacitance of BFO na nostructure with its \nredox behaviour. \nIn this letter, we report the development of BFO nanorod by the wet chemical template assisted \nmethod and its capacitance studies by electrochemical means. We ha ve evaluated specific capacitance \nboth through cyclic voltammetry at different scanning rates and charge-discharge studies 3\ngalvanostatically employing different  currents. To our knowledge, ther e has not been any study of BFO \nnanorod in this respect so far. \nExperimental details:  \nAAO templates of 60 µm thickness and 13 mm diam eter with pore size di stribution ranges from \n20-100 nm were employed. 0.1M solutions of Bi(NO 3)3 and Fe(NO 3)3 were prepared with \nstoichiometric amounts of the nitrates using methoxymeth anol as solvent with pH adjusted to 2-3.  The \nfilling of nanopores was achieved by th e directional flow of the ions  in the template adopting the \ncontrolled vacuum technique.  The templates with pores containing solution we re sintered for 3 hours \nat 7500C to get the required phase without unwanted grain growth. We attemp ted controlled etching \nprocess with 1M NaOH and the bun dle of nanowires and nanorods em erged. Weights of BFO nanorods \nmeasured using a very sensitive microbalance (reso lution of 1µgm) by subtracting the weights of \nrespective AAO were in the range 80-100µg. \nThe nanowires and nanorods were examined by FE I Scanning Electron Microscope (SEM) with a \nresolution of 6nm aided by Energy Dispersive X-Ray (EDX) for comp ositional analysis. Transmission \nelectron microscopy (TEM) was done by high resolution TEM (Model: FEI T20 with applied voltage of 200KV). Selective Area Electron Diffraction (SAED) was also unde rtaken to ascertain crystal \nstructure of the nanorods. \nCyclic Voltammetry (CV) and galvanostatic char ge-discharge were studied with AUTOLAB-30 \npotentiostat/galvanostat for both BF O on AAO and AAO blank templates being used as electrodes. The \nhalf etched templates used in SEM were employed for this purpose with the protrusion length of 1µm \n(as visible by SEM) over the template of thickne ss 60µm. Electrodes were prepared by connecting Cu \nlids with AAO/BFO templates through conducting silver paste. All the el ectrochemical experiments (i.e 4\nCV, Charge-discharge) were perfor med with two-electrode system having identical electrodes with \nrespect to shape, size and made of same ac tive electrode materials (i.e. Type-I symmetric \nsupercapacitor) using an electrolyte containing 1M Na 2SO 4 in water. A platinum electrode and a \nsaturated Ag/AgCl electrode were used as counte r and reference electrodes respectively. All the CVs \nwere measured between -0.6 to +0.6 V (i.e. oper ating window of 1.2V) with  respect to reference \nelectrode at different scan rate s (5mV/s to 50mV/s). Constant cu rrents ranging from 15 to 30 µAmp \nhave been employed for charging/discharging the cel l in the voltage range from -0.6 to +0.6 V.  \nIn a symmetrical system where the active material  weight is the same for the two electrodes,  \n                                              mCCs2=                                                    (1) \nwhere m is the active mass of the single electrode, C is the discharge capacitance and Cs is the specific \ncapacitance of the  electrode [7]. The charge a ccumulated on BFO was assayed by subtracting the \ncharge on blank AAO from that on BF O/AAO electrode. We have applied several cycles of CV as well \nas charge- discharge to study the stab ility of the system with cycling. \nResults and Discussions: \nThere are two kinds of 1D nanos tructure in the form of nanow ires as well as nanorods as \napparent from figs. 1a and 1b similar to earlier gr oups [4,5]. Fig. 1a shows the bundles of nanowires \nand nanorods which have come out after etching the template with NaOH. Basically they are nanorods \nbut we observed them in the form of a bundle after etching. There is a distri bution in diameters of \nnanorods as revealed in fig. 1a. Cros s sectional view is showing development of nanorods in Fig. 1b- a \nrepresentative case. It de monstrates the structures of several na norods (around 20 in a region of 5 µm x \n5 µm) with high aspect ratio protruding out of the pores after partial etching. The compositional \nanalysis was performed by EDX analysis at different na norods and they reflected Bi:Fe atomic ratios of  5\nslightly more than 1:1 reflecting a little more Bi c ontent with a fluctuation within 3  per cent. Fig. 1c \nshows the TEM picture of nanorods of  high density with intact struct ure. The inset indicates a clear \nSAED pattern with prominent rings signifying th e development of polycrystalline BFO.    \nTypical cyclic voltammograms (C V) of different samples at a scan rate of 50 and 10mV/s \nbetween -0.6 and 0.6V in aqueous solution of 1M Na 2SO 4 are shown in Fig. 2a. Cyclic voltammograms \nof different samples are quite sy mmetrical with a mirror image of the current response from voltage, \nindicating ideal pseudocapacitative be havior and excellent reversibility  in charging and discharging at \na constant rate over the voltage  range of –0.6 to 0.6V [8]. Voltamm etric charges (q*)  at different \npotential scan rate v (mV/s) were obtained by integration of the voltammetric curves followed by \ndivision with the geometric surface area of the sa mples without correction for background capacitative \ncurrent. The charge accumulation on BFO/AAO and AAO individually as sayed by CV established that \nthe contribution from AAO was sign ificantly less than BFO/AAO for same weight. Typical values \nwere 2.270x10-3 Coulombs for BFO/AAO and 6.45x10-4 Coulombs for AAO at the scanning rate of \n20mv/sec. \nThe applicability of the supercapacitor can also  be evaluated by means of the galvanostatic \ncharge–discharge studies.  Charge–discharge profiles of differen t samples are shown in Fig. 2b. They \nexhibit a pseudocapacitative characteristic [9]. In general, the specific capacitance decreases gradually \nwith increasing discharge current density due to increasing IR drops leading to higher d v/dt. We have \nobserved the charge-discharge cycles at different cu rrents and over quite substantial number of times – \nmore than 100 and the times for charging and disc harging are constant over long cycles for same \ncurrent. 6\nIn cyclic voltammetry, the accumulation of charge on the electrode dominates on the surface. \nTotal surface charge (q* total) related to the electrochemical active surface area of th e electrode can be \ndivided into two parts–  q* out and q* in. \n                                                     q* total = q* out + q* in                                      (2)                                               \nqכout is the contribution of charge from the outer region of the electrodes directly exposed to the \nelectrolyte and qכin  correlates with that from the inner part  of the electrodes hi dden in pores, grain \nboundary, etc. and reflects the regions  of difficult accessibility for the ionic species assisting the \nsurface redox reaction, essential for enhancing pse udocapacitance. Voltammetric charge q* depends on \nthe potential scan rate v [10]. The dependence is given by the relation: \n                                                 q*( v) =  q* ∞  + c v-1/2                                            (3) \nwith q* ∞  corresponding to q* out in eqn. (2) and c is a constant of proportionali ty. At faster scan, the \ndiffusion of ions is limited only to  the more accessible sites, i.e. the outer surface of the electrode. \nTherefore, extrapolation of q* to scan rate v = ן( i.e. v−1/2 = 0) from the linear por tion of the q* versus \nv−1/2 plot can provide the accumulation of outer charge qכout  (from eqn. 3) related to the more easily \naccessible sites. On the other hand, q* can be extrapolated the other way round to  v =0 to extract q tot. \nSince q* varies as v-1/2, 1/q* is expected to behave linearly as v1/2. Thus, the extrapolation of q* to the \nscan rate v = 0 in the plot 1/q* versus v1/2 gives the total charge q*\ntotal, which is related to both inner and \nouter active sites of the electrode [1 0]. One can thus easily calculate the charge related to the inner sites \n(i.e. less accessible sites) q*\n in of eqn. (2). \n                                                   1/q*( v) = (1/q 0*) + c* v1/2                                             (4) \nThe outer charge q*\nout only will be relevant for extractin g the specific capacitance of BFO \nnanorods. Fig. 3a shows q* versus v-1/2 plot with linear fit. Extrapolation of q* to v-1/2=0 gives the 7\nintercept 1.5 x 10-4 coulombs as q out. Total charge obtained by extrapolating the plot of 1/q* vs. v1/2 to \nv1/2 =0 in Fig. 3b is 3.673 x 10-3 coulombs. Thus, q*\nin = 3.523x10-3 coulombs.  The low value of q* out \ncompared to total charge is not su rprising considering the fact that q* out is due to the c ontribution solely \nfrom BFO nanorods protruding out of nanopores. \nLet us now try to understand the charge distribu tion in BFO/AAO template network. Our system \nconsists of BFO nanorods some of them protrudi ng out of pores along with porous AAO template with \npore sizes varying from 20-100nm. The combined netw ork is schematically presented in fig. 4. Total \ncharge will be distributed as inner charge in th e pores (some of which cont ains BFO) and as outer \ncharge on the BFO nanorods protruding 1 µm on th e average above the surface of the template. Around \nhalf of the pores filled by BFO have protruded nanor ods; others are inside the pores. The depth of the \npores is 60µm– the thickness of the template. Thus, the protruded portion of BF O– solely responsible \nfor the contribution to qכout is 1/60 of the wt. of protruded BFO nanorod which itself is half of the total \nwt. 80µgms of BFO. Rest is  embedded in pores.  qכout (obtained by extrapolati ng the plot of charge q* \nvs. v-1/2 to v-1/2 =0 and taking the intercept) is 1.50x10-4 coulombs. On this basis, the specific \ncapacitance of BFO nanorod structure comes out to  be 450 F/gm. This large value of specific \ncapacitance can be at tributed to the nanostructu re form of BFO nanorod.  \nThe pseudocapacitative behavior of BFO stems from its red ox reaction. BiFeO 3 is more readily \nreduced than oxidised, with the creation of oxygen vacancies and Fe2+ species. The highly \nunfavourable energy (> 4 eV) estim ated for disproportionation of Fe3+ (to Fe2+ and Fe4+) suggests that \ntetravalent iron ions are unlikely to form in this material through this process.  \n \n 8\nConclusion: \nWe have studied capacitance of BFO nanorods de veloped on AAO templates by electrochemical \nmeans. Capacitance of AAO as evaluated by accumulate d charge is significantly less as compared to \nBFO. BFO nanorods protruded from the templa te surface demonstrated a very high specific \ncapacitance of 450F/gm as measured from the char ge accumulated on the outer surface.  The high \nspecific capacitance is due to the particular nanostructure in the fo rm of the rod. There is a close \nresemblance between the capacitance assayed by CV  and charge-discharge methods. The system is \nquite stable with respect to rep eated cycling. BFO system undergoe s a redox process with O vacancies \nbeing generated giving rise to pseudocapacitative behavior with high specific capacitance. The high \nspecific capacitance in the nanorod structure coupled with stability at long cycles brings forth its \napplication as electrodes in batteries. \nAcknowledgements:  \nAuthors gratefully acknowledge  Dr. P.K.Mukhopadhyay and Mr. Sa kti Nath Das of S.N.Bose \nCenter for basic Sciences for SE M studies and Mr. Pulak Kumar Roy of Saha Instititute of Nuclear \nPhysics for TEM studies. \nReferences:  \n[1] Zhang, Y.; Feng, H.; Wu, X.; Wang, L.; Zhang, A.;  Xia, T.;  Dong, H.; Li, X.; Linsen. Z. Int. \n“Progress of electrochemical capacito r electrode materials: A review”  Jl. Hydrogen Energy 2009, 34, \n4889-4899.  \n[2] Zhang, X. Y.; Lai, C. W.; Zh ao, X.;Wang. D. Y.; Dai, J. Y. “Synthesis and ferroelectric properties \nof multiferroic BiFeO3 nanotube arrays” Appl. Phys. Lett. 2005, 87, 143102-143104. 9\n[3] Lokhande, C.D.;  Gujar, T.P.  ; Shinde ,  U.R.; Mane,  R.S. ;  Han.  S.H.  “ Electrochemical \nsupercapacitor application of  pervoskite thin films”  Electrochem. Comm.  2007,  9, 1805-1807. \n[4] Gao, F.;   Yuan. Y.; Wing, K.F.; Chen, X.Y.; Chen,  F; Liu, J.M.; Ren, Z.F;  “Preparation and \nphotoabsorption characte rization of BiFeO 3 nanowires”  Appl. Phys. Lett. 2006, 89 ,102506- 102508”. \n[5] Xie, S.H.; Li, J.Y.; Proksch, R.; Liu. Y.M.;  ; Zhou, Y.C. ; Liu. Y. Y.; Pan.  L.N. ; Qiao, Y. \n“Nanocrystalline multiferroic BiFeO 3 ultrafine fibers by sol-gel based electrospinning ” Appl. Phys. \nLett. 2008, 93  222904-222906. \n[6] Purusothaman,  K.K.;  Cuba, M.;  Muralidharan,  “ Supercapacitor behavior of α-\nMnMoO 4 nanorods on different electrolytes” G. Mat. Res. Bull.  2012, 47, 3348–3351.  \n[7] Portet, C.; Taberna.  P.L.; Simon,  P. ; Laberty-Robert, C. “ Modification of Al current collector \nsurface by sol–gel deposit for carb on–carbon supercapacitor applications”    Electrochim. Acta   2004 , \n49, 905-912. \n [8] Reddy, R. N ; Reddy, R.  G.,   “ Electrochemical Double Layer capac itance properties of carbon in \naqueous and nonaqueous electrolytes”   J. Power Source  2003 , 124,  330-337.  \n [9] Burke, L.D., Murphy, O.J.  “ Cyclic voltammetry as a technique fo r determining the surface area of \nRuO 2 electrodes”    Jl. Electroanal. Chem. 1979, 96, 19-25. \n [10] Ardizzone, S.;  Fregonara,  G.;  Trasat ti, S.  ““Inner” and “ outer” active surface of \nRuO 2 electrodes”,  Electrochem. Acta    1990, 35, 263-267. \n \n    \n 10\n\n(a) \n(b)\n(c)\n-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6-0.20-0.16-0.12-0.08-0.040.000.040.080.120.160.20\n(a)Current (m-Amp)\nVoltage (V) 50 mV\n 10 mV\n0 200 400 600 800 1000-0.6-0.4-0.20.00.20.40.6Voltage (Volts)  \nTime (sec)(b) \n \n \nFig. 1. (a)  SEM showing bundles of nanorod, (b)  SE M of nanorod protruding fr om pores and (c) TEM \nof BFO nanorods and corresponding SAED  pattern shown in the inset.   \n  \n \n \n \n \n \n   \nFig. 2. (a) Cyclic voltammogram of BFO on AAO at different scan rates and (b) Charge-discharge \ncycle of BFO on AAO template at 30µAmp current. \n \n \n 11\n0.0 0.1 0.2 0.3 0.40.00000.00050.00100.00150.00200.00250.00300.0035 (a)q (coulombs)\n ν-1/2 (volts/sec)-1/22345670100200300400500600700800 \n (b)1/q (coulombs)-1\nν1/2 (volts/sec)1/2 \n   \n \n     \n                    Fig. 3. (a) q* versus v\n-1/2 plot to extract q out (b) 1/q* vs. v1/2 to extract q* total. \n \n   \n \n      \n      \nFig. 4. Schematic diagram of BFO nanorod devel oped on AAO template. 1 µm protrusion from the \npores is there with the depth of the pores being 60µm.   \n AAO template \nBFO nanorod \n60µm1µm"    },    {        "title": "0901.3031v1.Phase_sensitive_Brillouin_scattering_measurements_with_a_novel_magneto_optic_modulator.pdf",        "content": "arXiv:0901.3031v1  [cond-mat.other]  20 Jan 2009Phase sensitive Brillouin scattering measurements with a n ovel magneto-optic\nmodulator\nF. Fohr, A.A. Serga, T. Schneider, J. Hamrle, and B. Hillebrands\nFachbereich Physik and Forschungszentrum OPTIMAS, Techni sche Universit¨ at Kaiserslautern,\nErwin-Schr¨ odinger-Straße 56, D-67663 Kaiserslautern, G ermany\n(Dated: June 7, 2018)\nA recently reported phase sensitive Brillouin light scatte ring technique is improved by use of a\nmagnetic modulator. This modulator is based on Brillouin li ght scattering in a thin ferrite ﬁlm.\nUsing this magnetic modulator in time- and space Brillouin l ight scattering measurements we have\nincreased phase contrast and excluded inﬂuence of optical i nhomogeneities in the sample. We\nalso demonstrate that the quality of the resulting interfer ence patterns can be improved by data\npostprocessing using the simultaneously recorded informa tion about the reference light.\nPACS numbers:\nI. INTRODUCTION\nSpace- and time-resolved Brillouin light scattering\n(BLS) spectroscopy is a well established technique to in-\nvestigate the spin-wave dynamics in thin magnetic ﬁlms\n[1] . However, this method is based on a simple count-\ning of inelastically scattered photons. Thus it only al-\nlows for the spatial and temporal mapping of spin-wave\nintensities. No phase information about magnetic excita-\ntions is accessibleby conventionalBLS. At the same time\nthe phase information is crucial to answer questions con-\ncerning problems such as formation of coherent states\nin a magnon gas and evolution of nonlinear spin-wave\neigenmodes [2] as well as nonlinear phase accumulation\n[3], peculiarities of spin-wave excitation process [4], 2-\ndimensional phase structure of spin-wave beams in mag-\nnetically anisotropic media, etc.\nThe ﬁrst realization of a phase-sensitive BLS setup\nas well as the results obtained by means of phase-\nsensitive BLS spectroscopy have been presented by us\nin Ref. [3, 4, 5]. It was shown that the implementation of\nphaseresolutioninto Brillouinlight spectroscopyleads to\nSample□channel\nFabry-Perot-\ninterferometer\nMagnetSNMagnet\nReference□channel\nPhoto-\ndetectorLaser\nSN\nE MagnetMagnet\nsrEE\nModulator\nPhase\nShifterAttenuator\nFIG. 1: (Color online) Experimental setup: phase resolved\nBLS. Phase sensitivity is created by interference between t he\nlight inelastically scattered by the spin waves (dashed lin e)\nand acoherent reference beam (dotted line), frequencyshif ted\nand turned in polarization by the magneto-optic modulator.a complete picture of the underlying physical processes\nby combining space-, time- and phase-resolution into the\nmeasurement.\nHere we report on further improvement of this technique\nbyimplementationofanew typeofmagneto-opticmodu-\nlator, based on Brillouin light scattering in a thin ferrite\nﬁlm. Furthermore we demonstrate that the use of the\nmagneto-optic modulator also improves the BLS dynam-\nical range.\nII. PRINCIPLE OF OPERATION\nTo understand the principle of phase sensitivity it is\nimportant to notice that the inelastically scattered light,\nwhich is detected in the BLS measurements, is deter-\nmined by both amplitude and phase of the scattering\nspin wave. However, the phase information is lost when\nthe photon is received by a photodetector, it only can\nregister the arrival of a photon, i.e. quantum of energy.\nTo access phase information we use interference between\ninelasticallyscatteredlightcreatedintheprocessofprop-\nagation of the probing laser beam through the sample\nwith reference scattered light provided by the magneto-\noptic modulation.\nWithout a ﬁxed phase correlation, the intensity of the\nsample beam E2\nsand the reference beam E2\nrcombine to\nE2\ns+E2\nr. (1)\nIf conditions for interference are fulﬁlled, the sample\nbeam and the reference beam combine to\nE2\ns+2EsErcos(ϕ)+E2\nr. (2)\nwith an additional phase-term.\nThe magneto-optic modulator used here was produced\non the base of an yttrium-iron-garnet (YIG) ferrite ﬁlm.\nAs one can see in Fig. 1 the laser beam is initially focused\non a 10µmthick in-plane magnetized YIG ﬁlm stripe in\nthe magneto-optic modulator. The bias magnetic ﬁeld2\nis oriented parallel to the spin-wave propagation direc-\ntion, and thus a backward volume magnetostatic spin\nwave (BVMSW) [6] is excited in the ﬁlm. As a result\nof nonelastic light scattering by this wave, a part of the\nbeam is now frequency shifted and simultaneously ro-\ntated in polarization by 90◦forming the reference beam.\nFocussing of the laser beam is necessary to enhance the\neﬃciency of the scattering process. Firstly, the beam is\nfocussed near to the exciting antenna, where the inten-\nsity of the spin wave and scattering-probabilityis highest\nand secondly, as we have a propagating and not a stand-\ning spin wave, the phase of the scattered light is well\ndeﬁned only if the scattering process occurs in an area\nwhich length is small compared to the wavelength of the\nspin wave.\nNote that a great part of the laser light passes the mod-\nulator unchanged in frequency and polarization. Some\npart of this undisturbed light undergoes a second scat-\ntering process in the sample and forms the sample beam.\nHaving the same polarization direction and the same fre-\nquency, the reference beam and the sample beam now\nfulﬁll the condition for interference at the photodetector.\nThe modulator is driven by the same microwave signal\nwhich is used to excite the spin waves. This guarantees\nnot only that the reference light has exactly the same\nfrequency as the inelastically scattered light but also the\nnecessary phase coherence between both signals. The\namplitude of the reference signal and the phase diﬀer-\nenceϕcan be adjusted by using a microwave attenuator\nand a phase shifter respectively (see Fig. 1). Since the\nlight in the signal and the reference channnels share the\nsame spatial path, thermal and mechanical stability is\nensured.\nThe resulting interference picture allows us to visualize\nthe phase fronts and to calculate the phase proﬁles (i.e.\nthe time-dependent phasediﬀerence between the exciting\nmicrowave signal and the spin wave at any given point)\nof the investigated spin waves from measured interfer-\nence maps with diﬀerent additional phase shifts between\nreference and signal. For a more detailed description of\ntheunderlyinganalysisprocedureandthephase-sensitive\nBLS setup see Refs. [3, 4].\nBefore the implementation of the magneto-optic modula-\ntor, the necessary frequency shift for the reference beam\nwas created by using an electro-optic modulator based on\na lithium-niobate crystal [5, 7]. However phase-resolved\nBLS spectroscopy of magnetic excitations using the pro-\nposed magneto-optic modulation has several advantages\ncompared to the previous electro-optic-modulation.\n(i)In order to increase the conversion eﬃciency, the\nlithium-niobate-crystal in our electro-optic modulator\nwas placed inside a microwave cavity. Due to the strong\ndistortion of the electric ﬁeld inside the cavity by inﬂu-\nence of this dielectric ( ε= 40) material, the adjustment\nof the modulator was very complicated and as a result\nonly a few ﬁxed frequencies were accessible. This re-\nstriction does not exist for a magneto-optic modulator\nwhich can be tuned continuously in frequency by an ex-/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48 /s49/s52/s48/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s91/s97/s46/s117/s46/s93\n/s84/s105/s109/s101/s32/s91/s110/s115/s93(E +E )s r12\nEr22\nEs2Er12\nFIG. 2: (Color online) Time proﬁle example of the interfer-\nence between sample pulse E2\nsand reference pulse E2\nr1. An\nadditional reference pulse E2\nr2is recorded with a time delay\nof 900 ns.\nternal magnetic ﬁeld. Thus, the RF-frequency can be\nimplemented as an additional free parameter during a\nmeasurement.\n(ii)Another important advantage of a magneto-optic\nmodulator, which is basedon Brillouinlight scattering, is\nthat the polarization of the reference beam and the sam-\nplebeamisthesame,i.e. bothbeamschangepolarization\nwith respect to the light source. On contrary, the electro-\noptic modulator does not change the polarization of the\nfrequency-shifted light and as a result no interference\nwith non-elastically scattered light occurs without spe-\ncial disalignment of the polarization ﬁlter placed before\nthe interferometer. In conventional BLS spectroscopy\nthis ﬁlter is used to block the elastically-scattered light\nand thus consequently decreases the noise. In this way\nthe disalignment of the ﬁlter, necessary when using an\nelectro-optic modulator, allows interference but also sig-\nniﬁcantly increases the noise level. This disalignment is\nnot necessaryforthe proposedmagneto-opticmodulator,\nwhere the polarization condition for the best interference\nis fulﬁlled automatically.\nIII. EXPERIMENTAL RESULTS\nIn the following we present example measurements\nwith the new setup. Spin-wave pulses are excited in a\n4,5 mm long YIG-ﬁlm in BVMSW-conﬁguration. Fig-\nure 2 shows the comparison between the time proﬁles\nof the sample pulse ( E2\ns, dashed line) and the construc-\ntive interference (( Es+Er1)2, solid line) between sample\npulse and reference pulse E2\nr1. The data for the inten-\nsity of the sample pulse is obtained by switching oﬀ the\nexciting microwave-current in the modulator. The dura-\ntion is 300 ns for the reference pulse and 100 ns for the\nsample pulse, respectively. The pulse carrier frequency is\n7 GHz at a bias magnetic ﬁeld H0of 1815 Oe. To obtain\nthe information about spatial and temporal distribution3\nFIG. 3: (Color online) Phase resolved measurement of a prop-\nagating spin-wave packet using a magneto-optic modulator.\nThe ﬁrst column is obtained with the conventional space- and\ntime-resolved Brillouin light scattering setup and shows t wo-\ndimensional maps of the spin-wave intensity distribution f or\ngiven times. The right column shows the corresponding inter -\nference patterns. Each proﬁle is normalized to the respecti ve\nmaximum value at 217 ns.\nof the background, an additional reference pulse E2\nr2was\nrecorded with a time delay of 900 ns.\nThe spatial distribution of the propagating spin-wave\npacket is shown in Fig. 3. The ﬁrst column shows the\ndecaying intensity of the spin-wave packet propagating\nin the sample. In the second column the interference\npattern of the sample-beam and reference beam is de-\npicted. It is clearly observable that, for the long propa-\ngation times, the spin wave is much more visible in the\ninterference patterns in comparison to the intensity pro-\nﬁles. Thus an increase in the dynamic range of the BLS\nsetup is obtained.\nThe sample beam in Eq. (2) contains all information\nabout the spin wave and the pure reference component\nE2\nrcan be regarded as a background, including defects\nand optic nonuniformities in the YIG-ﬁlm. The resulting\npicture can be improved by just dividing the interference\nproﬁle by the intensity proﬁle of the time delayed refer-\nence beam.\nE2\ns\nE2r+2Es\nErcos(ϕ)+1. (3)Since reference beam and sample beam share the same\nspatial path, both are inﬂuenced by the same distur-\nbances. Thus by dividing both signals, the noise can\nbe signiﬁcantly reduced. The result of this calculation is\npresented in Fig. 4. Spurious signals, e.g. the spot at the\nupper right corner, are completely erased from the in-\nterference pattern and the distribution of the spin wave\nitself can be distinguished better from the homogenous\nwhite noise.\nIn conclusion we have improved phase sensitive Bril-\nlouin light scattering spectroscopy by implementation of\na new kind of magneto-optic modulator.\nFIG. 4: (Color online) Phase resolved measurement of a spin-\nwave packet using a magneto-optic modulator. The ﬁrst two\npanels show the intensity map of the interference pattern, d e-\nscribed by Eq. (2) at 286 ns and the time delayed reference\npulseE2\nr2. In the third panel the interference pattern is di-\nvided by the intensity proﬁle of the time delayed reference\nbeam, resulting in Eq. (3). Thus signal variations caused by\noptic nonuniformities are removed (white arrows). Each pro -\nﬁle is normalized to the respective maximum value at 286 ns.\nAcknowledgments\nSupport by the DFG (SFB/TRR 49, JST-DFG\nHi380/21-1 and Graduiertenkolleg 792) is gratefully ac-\nknowleged.\n[1] S.O. Demokritov, B. Hillebrands andA. N.Slavin, Physics\nReports348, 159 (2001)\n[2] S.O. Demokritov, A.A. Serga , V.E. Demidov, B. Hille-\nbrands, M.P. Kostylev and B.A. Kalinikos, nature426,\n159 (2003)\n[3] T. Schneider, A.A. Serga , T. Neumann, M.P. Kostylev\nand B. Hillebrands, EPL77, 57002 (2007)\n[4] T. Schneider, A.A.Serga, T. Neumann, M.P. KostylevandB. Hillebrands, Phys. Rev. B77, 214411 (2008)\n[5] A.A.Serga, T. Schneider, S.O.Demokritov, M.P. Kostyle v\nand B. Hillebrands, Appl. Phys. Lett. 89, 063506 (2006)\n[6] R. W. Damon and J. R. Eshbach, J. Phys. Chem. Solids\n19, 308 (1961)\n[7] S. Caponi, M. Dionigi, D. Fioretto et al., Rev. Sci. In-\nstrum.,72, 198 (2001)"    },    {        "title": "1502.01121v1.Novel_microwave_near_field_sensors_for_material_characterization__biology__and_nanotechnology.pdf",        "content": "J. Appl. Phys. 113, 063912 (2013) \n \nNovel microwave near-field sensors for material characterization, biology, \nand nanotechnology \n \nR. Joffe, E.O. Kamenetskii, and R. Shavit \n \nMicrowave Magnetic Laboratory \nDepartment of Electrical and Computer Engineering \nBen Gurion University of the Negev, Beer Sheva, Israel \n \nNovember 28, 2012 \n \nAbstract \n \nThe wide range of interesting electromagnetic be havior of contemporary materials requires that \nexperimentalists working in this field master many diverse measurement techniques and have a \nbroad understanding of condensed matter ph ysics and biophysics. Measurement of the \nelectromagnetic response of materials at mi crowave frequencies is important for both \nfundamental and practical reasons. In this pa per, we propose a novel near-field microwave \nsensor with application to mate rial characterization,  biology, and nanotechnology. The sensor is \nbased on a subwavelength ferrite-disk reso nator with magnetic-dipolar-mode (MDM) \noscillations. Strong energy concentration and unique  topological structures of the near fields \noriginated from the MDM resonators allow e ffective measuring material parameters in \nmicrowaves, both for ordinary structures and objects with chiral properties. \n \nPACS number(s): 84.40.-x; 76.50.+g; 78.70.Gq; 87.50.S \n I. Introduction \n \nImplementation of imaging in microwave frequencies gives the opportunity for \nelectrodynamics experiments with natural materials and artificial structures. In many \nmicrowave sensing devices, resonant structures are essential elements because they allow \nlocalization of high field areas. They are very efficient in the frequency band for which they were designed, since the signal-to-noise ratio in  a resonator structure increases with resonator \nquality factor Q. This increase in sensitivity and field strength is accompanied by a narrower \nfrequency band, with the drop in amplitude depending on Q, which results from the shift in resonant frequency with different environmen ts. Consequently, resonance frequency and \namplitude tracking are employed. Typical devices for imaging in microwaves use conventional \nresonant structures. There are, for example, narrow resonant slots in a rectangular hollow \nwaveguide, strip- and microstrip-line resonators, coaxial-cavity and coaxial-line resonators. A \ndetailed description of these structures can be  found in review articles [1, 2]. Recently, open \nplanar LC resonators have been suggested [3] for microwave sensing.  \n    For subwavelength characterization of microwave material parameters, special metallic \nprobes are mostly used. The near fields of su ch metallic probes are well known evanescent-\nmode fields [1, 2]. There is a trade-off between the quality factor of a resonator and the \ncoupling between an object and a tip. To improve the resolution and bring it well below the free-space electromagnetic wave length, different  ways for optimization of the scanner tip \nstructures had been suggested. It becomes clear that new perfect lenses that can focus beyond \nthe diffraction limit could revolutionize near-fiel d microwave microscopy. In the present study,  2we propose a novel microwave near-field sensor w ith application to mate rial characterization, \nbiology, and nanotechnology. This sensor is re alized based on a small ferrite-disk resonator \nwith magnetic-dipolar-mode (MDM) oscillati ons [4 – 7]. The wavelength of the MDM \noscillations in ferrite resonators  is two-four orders of the magnitude less than the free-space \nelectromagnetic-wave wavelength at the same microwave frequency [8]. Application of these \nproperties in near-field microwave microscopy allows achieving submicron resolution much \neasier than in the existing microwave microscopes with standard resonant structures [1, 2]. \nMoreover, since a MDM ferrite resonator is a multiresonance structure, a complete-set mode \nspectrum of MDM oscillations can be used to get a complete Fourier image (in the frequency \nor k\n-space domain) in a localized region of a sample [4 – 7, 9]. \n    Due to the growing interaction between biological sciences and electrical engineering \ndisciplines, effective sensing and monitoring of biological samples becomes an important \nsubject. This, especially, conc erns the near-field microwave microscopy  of chemical and \nbiological structures [3, 10 – 13]. One of the si gnificant questions, both for fundamental studies \nand applications, is biophysical modeling of microwave-induced nonthermal biological effects \n[14, 15]. Despite the fact that reports of nonthermal microwave effects date back to the 1970s, \nthere is a great deal of renewed interest. The rapid rate of adoption of mobile phones and \nmobile wireless communications into society ha s resulted in public concern about the health \nhazards of microwave fields emitted by such de vices. Direct detection of biological structures \nin microwave frequencies and understanding of the molecular mechanisms of nonthermal \nmicrowave effects is a problem of a great importance. Nowadays, however, microwave \nbiosensing is represented by standard microw ave techniques for measuring dielectric and \nconductivity properties of materials [3, 10 – 13]. Proper correlation of these material \nparameters with structural characteristics of chemical and biological objects in microwaves appears as a serious problem.  \n    Compared to microwave biosensing, optical biosensing is represented in a larger variety of \neffective tools. Examining the structure of proteins by circular dichroism is an effective and \nwell known optical technique [16]. Another techni que is based on the resonant subwavelength \ninteractions between plasmon (or electrostati c) oscillations of me tal nanoparticles and \nelectromagnetric fields. Localized surface plasmon resonance spectroscopy of metallic \nnanoparticles is a powerful tool for chemical  and biological optical sensing [17]. For \nbiomedical diagnostics and pathogen detection,  special plasmonic structures with left- and \nright-handed optical superchiral fields have been recently proposed. These structures \neffectively interact with large biomolecules, in particular, and chiral materials in general [18]. \n    Chirality is of fundamental interest for chemistry and biology not only in optics, but, \ncertainly, also in microwaves. However, th e near-field patterns of nowadays microwave \nsensors do not have symmetry breakings and so cannot be used for microwave characterization \nof chemical and biological objects with chiral properties as well as chiral metamaterials. Can \none use the main ideas and results of the optical  subwavelength photonics to realize microwave \nstructures with subwavelength confinement and chirality of the near fields? Since resonance \nfrequencies of electrostatic (plasmon) oscillati ons in small particles are very far from \nmicrowave frequencies, an answer to this question is negative. It becomes sufficiently apparent that in microwaves, the problem of effective characterization of chemical and biological \nobjects will be solved when one develops special sensing devices with microwave chiral \nprobing fields. Recently, it has been shown that the MDM ferrite particles may create \nmicrowave superchiral fields with strong subwavelength localization of electromagnetic \nenergy. There are the particles with magnetosta tic (MS) (or MS-magnon) oscillations, but not \nwith electrostatic (plasmon) oscillations [19 – 22]. \n    In this paper, we propose a novel near-fiel d microwave sensor with application to material \ncharacterization, biology, and nanotechnology. Th e sensor is based on a small ferrite-disk  3resonator. The MDM oscillations in this resonator are characterized by time and space \nsymmetry breakings. Development of such subwavelength sensors for direct microwave \ncharacterization of microscopic material structure with application to biology and \nnanotechnology is a subject of high importance. In microwave near-field sensing with \nsuperchiral fields, we have a completely new mechanism of the material-field interaction. \n\"New truths become evident when new tools become available”; these words of Nobel Laureate Rosalyn Yalow may concern also the proposed near-field microwave sensors. \n II. Microwave superchiral fields \n The near fields originated from a normally magnetized ferrite disk with MDM oscillations have intrinsic chiral topology. They are characterized by strong subwavelength localization of energy \nand vortices of the power-flow density. In the vi cinity of a ferrite disk, one can observe the \npower-flow whirlpool. Fig. 1 illustrates the fiel d structure at the resonance frequency of MDM \noscillation. No such confinement and topology of the fields are observed at nonresonance \nfrequencies. An electric field inside a ferrite di sk has both orbital and spin angular momentums \n[23]. This results in helical-mode resonances [7, 20]. When an electrically polarized dielectric \nsample is placed above a ferrite disk, every sepa rate dipole in a sample will precess around its \nown axis. For all the precessing dipoles, there is an orbital phase running [see Fig. 2].  \n         \n    The mechanical torque exerted on a given el ectric dipole is defined as a cross product of the \nMDM electric field E\n and the electric moment of the dipole p[21, 22]: \n \n                                                                 pEN .                                                                 (1) \n \nThe dipole p appears because of the electric polarizatio n of a dielectric by the RF electric field \nof a microwave system. The torque exerting on th e electric polarization due to the MDM electric \nfield should be equal to reaction torque exerting on the magnetization in a ferrite disk. Because \nof this reaction torque, the precessing magnetic moment density of the ferromagnet will be under additional mechanical rotation at a certain frequency \n. For the magnetic moment density of the \nferromagnet, M\n, the motion equation acquires the following form [21, 22]: \n \n                                                              dMMHdt   \n,                                                 (2) \n \nThe frequency  is defined based on both, spin and orbital, momentums of the fields of MDM \noscillations. One can see that at dielectric loadings, the magnetization motion in a ferrite disk is \ncharacterized by an effective magnetic field \n \n                                                                     effHH\n.                                                         (3) \n \nSo, the Larmor frequency of a ferrite structure w ith a dielectric loading should be lower than \nsuch a frequency in an unloaded ferrite disk. \n    The near fields originated from MDM ferrite particles are characterized by subwavelength confinement of energy and chirality properties. In optics, a parameter ch aracterizing superchiral \nfields is a so-called optical chirality density [18]. For quantitative characterization of microwave \nsuperchiral fields created by MDM ferrite part icles, we use the helicity density factor \nF. For \ntime-harmonic fields, parameter F is expressed as [21, 22]:  4 \n                                                                 *\n0Im4F EE   \n| .                                            (4) \n \nFrom this equation one can establish the essent ial property of local microwave fields which \npossess chiral symmetry: the components of complex vectors E\n and E\n must be parallel and \nphase shifted giving rise a nonzero imaginary comp onent to their dot product. It is worth noting \nthat a sign of the helicity factor of a microwav e superchiral field originated from a MS-magnon \nmode depends on a direction of a bias magnetic field 0H\n. This is one of the main points in our \nwork.  \n \nIII. Material characterization  \n \nThe MDM spectral characteristics in thin fe rrite disks are well observed in microwave \nexperiments [24 – 27], can be studied analytically [4 – 7], and are well illustrated by numerical \nresults based on the HFSS Ansoft solver [19, 20, 23]. Fig. 3 shows the numerically simulated \nMDM spectral characteristic for a thin ferrite disk placed in a 10TE -mode rectangular \nwaveguide. An insert in the figure shows geometry of the structure. The yttrium iron garnet \n(YIG) disk has a diameter of 3D mm and a thickness of 05.0t mm. The saturation \nmagnetization of a ferrite is  1880  4sM G. The disk is normally magnetized by a bias \nmagnetic field  49000H Oe. For better understanding the field structures, in numerical \nstudies we use a ferrite disk with very small losses: the linewidth of a ferrite is 0.1 OeH . \nThe waveguide walls are made of a perfect electric  conductor (PEC). In the spectrum in Fig. 3 \none sees the module of the reflection (the 11S scattering-matrix parameter) coefficient. The \nresonance modes are designated  in succession by numbers n = 1, 2, 3… It is evident that \nstarting from the second mode, the coupled states of the electromagnetic fields with MDM vortices are split-resonance states. The properties of these coalescent resonances, denoted in \nFig. 3 by single and double primes, were analyzed in details in Ref. [20]. The near fields at the \nMDM resonances are characterized by the helicity properties. Fig. 4 gives evidence for distinct \nparameters of helicity density of the fields, calculated based on Eq. (4).  \n    Topological effects appearing in the matte r-field interaction for th e microwave superchiral \nfields open a perspective for uni que near-field characterization of  material parameters. Due to \nstrong energy localization and the field helicity one has novel tools for effective measuring \nmaterial parameters in microwaves. As an example of such measuring, we use a structure \ncomposed by a MDM ferrite disk and two dielectric cylinders loading symmetrically a ferrite \ndisk. Fig. 5 (a) shows this structure placed inside a \n10TE -mode rectangular waveguide. The \ndielectric cylinders (every cylinder is with the diameter of 3 mm and the height of 2 mm) are \nelectrically polarized by the RF electric field of the 10TE  mode propagating in a waveguide. \nThe dielectric loadings do not destroy the entire MDM spectrum, but cause, however, the \nfrequency shifts of the resonance peaks. The frequency characteristics of a module of a \nreflection coefficient for the 1st MDM resonance at different di electric parameters of the \ncylinders are shown in Fig. 5 (b). One can see that at dielectric loadings, the 1st MDM \nresonance appears as coalescent resonances (the resonances 1 and 1). One of the main \nfeatures of the frequency characteristics of a stru cture with the symmetrical dielectric loadings, \nis the fact that the resonances of the 1st MDM become shifted not only to the lower frequencies, \nbut appear to the left of the Larmor frequency of an unloaded ferrite disk. For a normally \nmagnetized ferrite disk with the pointed above quantities of the bias magnetic field and the  5saturation magnetization, this Larmor frequency (calculated as 1\n2H i f H , where  is the \ngyromagnetic ratio  and iH is the internal DC magnetic field) is equal to 8,456 GHzHf . \nWhen a ferrite disk is without dielectric loadings, the entire spectrum of MDM oscillations is \nsituated to the right of the Larmor frequency Hf. Since dielectrics do not destroy the entire \nMDM spectrum, one can suppose that the Larmor  frequency of a structure with dielectric \nloadings ()D\nHf is lower than the Larmor frequency of an unloaded ferrite disk ()D\nH H f f . This \nstatement is well clarified by the above analysis with use of Eqs. (1) – (3). It is worth noting \nthat dielectric loadings change distribution of the helicity density. The helicity-parameter \ndistributions calculated based on Eq. (4) are shown in Fig. 6 for different dielectric constants of loading cylinders. The frequencies correspond to the resonance-peak positions in Fig. 6. As one \ncan see, the dielectric loadings not only reduce the quantity of the helicity factor, but result in \nstrong modification of the near-field structure.     For practical purposes in the microwave charact erization of materials, it is more preferable \nto use an open-access microstrip structure with a ferrite-disk sensor, instead of a closed \nwaveguide structure studied above. Such a microstrip structure is shown in Fig. 7. Fig. 8 \nrepresents the frequency characteristic of a module of the transmission (the \n21S scattering-\nmatrix parameter) coefficient for a microstrip stru cture with a thin-film ferrite disk. While in a \ndiscussed above waveguide structure with an enclosed ferrite disk, the main features of the \nMDM spectra are evident from the reflection characteristics, in the shown microstrip structure, \nthe most interesting are the transmission characteristics. Classification of the resonances shown in Fig. 8 is made based on analytical studies in Ref. [5]. There are resonances corresponding to \nMDMs with radial and azimuth variations of th e magnetostatic-potential wave functions in a \nferrite disk. One can see that the frequencies of the 1\nst and 2nd radial-variation resonances \nshown in Fig. 8 are in a good correspondence with the frequencies of the 1st and 2nd resonances \nshown in Fig. 3 for a waveguide structure. Between the 1st and 2nd resonances of the radial \nvariations one can see the resonance of the azimuth-variation mode. This resonance appears \nbecause of the azimuth nonhomogeneity of a microstrip structure. For experimental studies, we \nuse a ferrite disk with the same parameters as pointed above for numerical analyses. The only \ndifference that in an experimental disk the linewidth of a ferrite is 0.8 Oe H . An \nexperimental microstrip structure is realized on a dielectric substrate (Taconic RF-35, \n3.52r , thickness of 1.52 mm). Characteristic impe dance of a microstrip line is 50 Ohm. For \ndielectric loadings, we used cylinders of co mmercial microwave dielectric (non magnetic) \nmaterials with the dielectric permittivity parameters of 30r  (K-30; TCI Ceramics Inc) and \n50r  (K-50; TCI Ceramics Inc). A transmission coefficient was measured with use of a \nnetwork analyzer. With use of a current supply we established a quanti ty of a normal bias \nmagnetic field 0H\n, necessary to get the MDM spectrum in a required frequency range. It is \nevident that there is a sufficiently good correspondence between the numerical and \nexperimental results of transformation of the MDM spectra due to dielectric loadings. It is \nnecessary to note, hewever, that instead of a bias magnetic field used in numerical studies \n(04900 H Oe), in the experiments we applied lowe r quantity of a bias magnetic field: \n04708 H Oe. Use of such a lower quantity (giving us the same positions of the non-loading-\nferrite resonance peaks in the numerical studies and in the experiments) is necessary because of \nnon-homogeneity of an internal DC magnetic fi eld in a real ferrite disk. A more detailed \ndiscussion on a role of non-homogeneity of an internal DC magnetic field in the MDM spectral \ncharacteristics can be found in Refs. [5, 25].  6    For effective localization of energy of MD M oscillations at micron and submicron near-field \nregions, special field concentrators should be used. In particular, there can be a thin metal wire \nplaced on a surface of a ferrite disk. Fig. 9 shows a microstrip MDM sensor with a wire \nconcentrator. A bias magnetic field 0H\n is directed normally to a disk plane. The wire electrode \nhas diameter of 100 um. In a shown structure of a ferrite disk with a wire electrode, the helical \nwaves localized in a ferrite disk are transmitted to the end of a wire electrode. The electric field of a microstrip structure causes a linear displ acement of charge when interacting with a short \npiece of a wire, whereas the magnetic field of a MDM vortex causes a circulation of charge. \nThese two motions combined cause an excitation of an electron in a helical motion, which \nincludes translation and rotation. Fig. 10 illustrate s this effect. The electr ic field distributions on \na wire electrode, shown for two different time phases, give evidence for helical waves. Due to such helical waves, at a butt end of a wire one has electric and magnetic fields with mutually \nparallel components [see Fig. 11 (\na), (b)] and a chiral surface electric current [see Fig. 11 ( c)]. \nAll this results in appearing of the power-flow-density vortex [see Fig. 12 ( a)] and nonzero \nhelicity density F [see Fig. 12 ( b)] at the butt end of a wire electrode. Fig. 13 ( a) shows a \nmicrostrip structure with a field concentrat or for localized material characterization. \nExperimental results of the MDM spectra transformations for different dielectric samples are \nshown in Fig. 13 ( b).  \n    \nIV. Microwave chirality discrimination \n A crucial point on the proposed microwave sensor is the fact that a sign of the helicity factor of a \nmicrowave superchiral field originated from a MD M ferrite particle depends on a direction of a \nbias magnetic field \n0H\n. Fig. 14 shows distribution of the helicity density of the MDM near fields \nat two opposite orientations  of a normal bias magnetic field. This distribution is obtained for a \nferrite disk placed in a microstrip structure. The disk is without a dielectric loading. Because of \ndependence of a sign of the helicity parameter on orientation of a bias magnetic field, one can propose microwave chirality discrimination using a ferrite-disk sensor  loaded by “right” and \n“left” enantiomeric structures. Fig. 15 illustrates discrimination of enantiomeric structures by the \nnumerically simulated MDM spectra at two opposite orientations of a bias magnetic field \n0H\n. As \na test structure, mimicking an object with chiral properties, a small dielectric sample with \nsymmetry-breaking geometry of surface metallizati on is used. Investigations are made based on \na microstrip structure shown in Fig. 7. For the “r ight” and “left” structures loading a ferrite disk, \nthere are clearly observed frequency shifts between high-order resonance peaks in the reflection \nspectra.  \n    For discrimination of enantiomeric structur es at localized regions, a microstrip MDM sensor \nwith a wire concentrator can be used. The field properties shown in Fig. 12 give evidence for the \nsymmetry breakings. When one changes oppositely an orientation of a bias magnetic field 0H\n, \none has an opposite rotation of the power flow  and an opposite sign of the helicity density F (red \ncolored instead of blue colored). This allows pr ediction of localized sensing samples with “right” \nand “left” handedness. As a locali zed test structures mimicking objects with chiral properties, we \nused small metallic helices. Fig. 16 shows the setup of the measurement system with a helical \ntest structure for local determination of material chirality. A wire concentrator is placed near a metallic helix without an electric contact with it. Detailed pictures of the sensor with small left- \nand right-handed helix particles are shown in Fig. 17. In Fig. 18, one can see numerical results of \nthe transmission coefficients for the left-ha nded and right-handed helical test structure, \nrespectively. The spectral characteristics are ob tained for two opposite orientations of a bias \nmagnetic field \n0H\n. It is worth noting that spectral recognition of enantiomeric structures at  7opposite orientations of a bias magn etic field shown in Fig. 15 (a) is different from the results of \nspectral discrimination shown in Fig. 18. While in  the former case loading of a ferrite disk by \nenantiomeric samples leads to frequency shifts in the reflection spectrum, in the last case, one \nobserves amplitude differences in the resonance peaks of the transmission spectrum. A helical \ntest sample shown in Fig 16 very slightly loads a ferrite disk and no frequency transformations in \nthe transmission and reflection spectra are observed in this case.      It is also worth noting that the results in  Fig. 18 exhibit very specific symmetry properties. \nFollowing these results one observes restoration of an entire transmission spectrum when \nhandedness of a sample is changed together with ch ange of direction of a bias magnetic field. As \nit was shown in Refs [7, 20, 21], helicity of the near field is originated from double-helix \nresonances of MDM oscillations in a quasi-2D ferrite disk. For two helical modes, giving a \ndouble-helix resonance in a ferrite disk, there is no parity ( \n ) and time-reversal (  ) invariance \n– the  -invariance. Such -symmetry breaking does not guarantee real-eigenvalue spectra. It \nwas shown, however [7], that by virtue of quasi-two dimensionality of the MDM spectral \nproblem, one can reduce solutions from helical to cylindrical coordinates and such a cylindrical-\ncoordinate problem has the  -invariant solutions. On the other hand, change of handedness of \na helical test structure cannot be considered as the space reflection operation. Certainly, electric \ncurrents induced in enantiometric structures (metallic helices) by MDM near fields are not  -\ninvariant quantities. It is evident that specific symmetries (sample handedness  bias magnetic \nfield) of the spectra in Fig. 18 are not related to the -symmetry. At the same time, these \nsymmetries highlight unique topological properties of microwave superchiral fields.    \n \nV. Discussion on credibility of the numerical results    \n \nOne of the main questions for discussions may co ncern credibility of the results, obtained based \non the classical-electrodynamics HFSS program,  for description of the shown non-trivial \ntopological effects originated from the MDM ferrite disks. The properties of the MDM \noscillations inside a ferrite disk are well observed numerically due to the ANSOFT HFSS \nprogram [19, 20, 23]. From the HFSS studies of microwave structures with thin-film ferrite \ndisks, one has strong regularity of the results. One can see consistent pattern of both the spectral-\npeak positions and specific topological structure of the fields of the oscillating modes. One \nobserves the same eigenresonances and the same field topology of the resonance eigenstates in \ndifferent guiding structures [when, for example,  a ferrite disk is placed in a rectangular \nwaveguide and when it is placed in a microstr ip structure]. There is a very good correspondence \nof the HFSS results with the analytical and experimental results.  \n    It is worth noting that with use of the HFSS program for our studies we are faced with \nnumerical solutions of a non-integrable (path-de pendent) electromagnetic problem. In a case of \nferrite inclusions in microwave structures, acting in the proximity of the ferromagnetic resonance, the phase of the electromagnetic wave reflected from a ferrite boundary depends on \nthe direction of the incident wave. For a given orientation of a bias magnetic field, one can \ndistinguish the right-hand and left-hand rays of  electromagnetic waves. This fact, arising from \npeculiar boundary conditions of the fields on the dielectric-ferrite interfaces, leads to the time-\nreversal symmetry breaking effect in microwave resonators with inserted ferrite samples. A \nnonreciprocal phase behavior for el ectromagnetic fields on a surface of a ferrite disk results in \nthe problem of path-dependent numerical integration. For the spectral-problem solutions, the \nHFSS program, in fact, composes the field structures from interferences of multiple plane EM \nwaves inside and outside a ferrite particle. In such a numerical analysis one obtains the pictures \nof the field structures (resulting in the real-spa ce integration) based on integration in the \nk-space  8of the EM fields. In the k-space integration, the fields are expanded from very low wavenumbers \n(free-space EM waves) to very high wavenumbers (the region of MS oscillations). It is also \nworth noting that inside (and nearly outside) a ferrite disk, the HFSS numerical results can be \nwell applicable for confirmation of the analytically-derived quantum-like models based on the \nmagnetostatic-potential scalar wave function . There exist many examples in physics when \nnon-classical effects are well modeled by a combined  effect of numerous classical sub-elements.  \n \nVI. Conclusion \n Presently subwavelength characterization of microw ave material parameters is considering as a \nvery topical subject. An evident progress in scanning near-field microwave microscopy allows \nmake precise measurements of different material structures on submicron scales. Among these material structures there are, in particular, bi ological systems. One can state, however, that \nnowadays for biological structure characterizations, optical techniques are considered as the most \neffective. Together with the well known technique s of circular dichroism, there are different \nplasmonic methods, including recently develope d optical measurements based on so-called \nsuperchiral near fields. All these techni ques are not applicable to microwaves. \n    The problem of effective characterization of  chemical and biological objects in microwaves \ncan be solved when one develops special sensing devices with microwave chiral probing fields. \nIn this paper, we showed that small ferrite-d isk resonators with magnetic-dipolar-mode (MDM) \noscillations may create microwave superchiral fi elds with strong subwavelength localization of \nelectromagnetic energy. Based on such properties  of the fields, we propose a novel near-field \nmicrowave sensor with application to material characterization, biology, and nanotechnology. \nWith use of these sensors one opens unique pe rspective for effectiv e measuring material \nparameters in microwaves, both for ordinary st ructures and objects with chiral properties. \n    \nReferences \n [1] S. M. Anlage \net al, e-print cond-mat/0001075 (2000). \n[2] B. T. Rosner and D. W. van der Weide, Rev. Sci. Instrum. 73, 2505 (2002). \n[3] H.-J. Lee, J.-H. Lee, and H.-I. Jung, Appl. Phys. Lett. 99, 163703 (2011).  \n[4] E. O. Kamenetskii, Phys. Rev. E, 63, 066612 (2001). \n[5] E. O. Kamenetskii, M. Sigalov, an d R. Shavit, J. Phys.: Condens. Matter 17, 2211 (2005). \n[6] E. O. Kamenetskii, J. Phys. A: Mathematical and Theoretical 40, 6539 (2007). \n[7] E. O. Kamenetskii, J. Phys.: Condens. Matter 22, 486005 (2010). \n[8] A. Gurevich and G. Melkov, Magnetic Oscillations and Waves  (CRC Press, New York, \n1996). \n[9] R. Ioffe, E. O. Kamenetskii, and R. Shavit, In Proc. of the 41st European Microwave \nConference, pp. 385-388. Manchester, UK, Oct. 9-14, 2011. \n[10] M. Dragoman et al, Appl. Phys. Lett. 99, 253106 (2011). \n[11] T. Yilmaz and Y. Hao, In Proc. of 2011 Loughborough Antennas & Propagation \nConference, pp. 1 – 4  (14-15 November 2011, Loughborough, UK). \n[12] Y.-I. Kim, Y. Park, and H. K. Baik , Sens. Actuators A: Phys. 143, 279 (2008). \n[13] C. Dalmay et al, Sens. Actuators A: Phys. 162, 189 (2010). \n[14] H. Bohr and J. Bohr, Phys. Rev. E 61, 4310 (2000). \n[15] I. V. Belyaev et al, Bioelectromagnetics  30, 129 (2009). \n[16] S. M. Kelly, T. J. Jess, and N. C. Price, Biochim. Biophys. Acta 1751 , 119 (2005).  \n[17] K. A. Willets and R. P. Van Duyne, Annual Rev. Phys. Chem. 58, 267 (2007). \n[18] E. Hendry et al, Nature Nanotechnol. 5, 783 (2010). \n[19] E.O. Kamenetskii, M. Sigalov, and R. Shavit, Phys. Rev. A 81, 053823 (2010).  9[20] E.O. Kamenetskii, R. Joffe, and R. Shavit, Phys. Rev. A 84, 023836 (2011). \n[21] E. O. Kamenetskii, arXiv:1111.4359 (2011). \n[22] E. O. Kamenetskii, R. Joffe, and R. Shavit, arXiv:1111.4361 (2011). \n[23] M. Sigalov, E. O. Kamenetskii, and R. Shavit, J. Phys.: Condens. Matter 21, 016003 \n(2009). \n[24]. J.F. Dillon, J. Appl. Phys. 31, 1605, (1960).  \n[25] T. Yukawa and K. Abe, J. Appl. Phys. 45, 3146 (1974). \n[26] E.O. Kamenetskii, A.K. Saha, and I. Awai, Phys. Lett. A. 332, 303 (2004).     \n[27] M. Sigalov, E.O. Kamenetskii, and R. Shavit, J. Appl. Phys. 104, 053901 (2008) \n \n  \nFigure captions \n \nFig. 1. ( a) Intensity and geometry of the power-fl ow distribution: strong subwavelength \nlocalization of energy and vortex behavior of microwave near fields. ( b) Schematic picture of the \npower-flow whirlpool in the vi cinity of a ferrite disk. \n \nFig. 2. An electric field inside a ferrite disk has both orbital and spin angular momentums. When \nan electrically polarized dielectric sample is placed above a ferrite disk, electric dipoles in a \ndielectric sample precess and accomplish an orbital geometric-phase rotation. A bias magnetic \nfield 0H\n is directed normally to a disk plane. For an opposite direction of 0H\n, one has an \nopposite rotation of an electric field and an opposi te direction of precession of electric dipoles.  \n \nFig. 3. Frequency characteristics of a module of the reflection coeffi cient for a rectangular \nwaveguide with an enclosed thin-film ferrite disk. The resonance modes are designated in \nsuccession by numbers n = 1, 2, 3… The coalescent resonances are denoted by single and double \nprimes. An insert shows geometry of a structure. \n Fig. 4. The near-field helicity\n parameters. ( a) Near-field helicity for the 1st MDM. ( b) Near-field \nhelicity for the 2nd (the resonance 2''). ( c) Absence of the near-field helicity for non-resonance \nfrequencies. \n \nFig. 5. (a) A sample of a ferrite disk with two loading dielectric cylinders placed inside a 10TE -\nmode rectangular waveguide. (b) Frequency ch aracteristics of a mo dule of the reflection \ncoefficient for the 1st MDM at different parameters of a symmetrical dielectric loading. \nFrequency 8,456 GHzHf  is the Larmor frequency of an unloaded ferrite disk. \n \nFig. 6. Numerically calculated helicity-parameter distributions for the 1st MDM at different \ndielectric constants of loading cylinders. The di stributions are shown on the cross-section plane \nwhich passes through the diameter a nd the axis of the ferrite disk. \n \nFig. 7. A microwave microstrip structure (sensor) with a MDM ferrite disk and a sample under \ninvestigation. \n Fig. 8. Transformation of the MDM spectrum due to a dielectric loading in a microstrip structure. \n(\na) Numerical results; ( b) Experimental results. \n  10Fig 9. A sensor with a wire concentrator localized material characterization. ( a) Geometry of a \nmicrostrip structure; ( b) A magnified picture of a MDM ferrite disk with a wire electrode. \n \nFig. 10. Evidence for helical waves. The electr ic field distributions on a wire electrode for \ndifferent time phases. ( a) Time phase 1t; (b) Time phase 2t. \n \nFig 11. The fields and currents at a butt end of a wire electrode. ( a) An electric field; ( b) A \nmagnetic field; ( c) A chiral surface electric current. \n   \nFig. 12.  Field structure on a butt end of a wire concentrator: ( a) the power-flow-density vortex \nand ( b) the helicity density F. When one changes oppositely an orientation of a bias magnetic \nfield 0H\n, one has an opposite rotation of the power flow and an opposite sign of the helicity \ndensity F (red colored instead of blue colored).  \n  Fig. 13. A microstrip structure for localized material characterization in a dielectric sample. (\na) \nGeometry of a structure; ( b) MDM spectra of the reflection coefficient at a dielectric loading \n(experimental results). \n \nFig 14.  Helicity density F of the MDM near fields at two orie ntations of a normal bias magnetic \nfield 0H\n. A ferrite disk is without a dielectric load ing and is placed in a microstrip structure \nshown in Fig. 7.   \n Fig. 15. (\na) Discrimination of an enantiomeric structure by the MDM spectra at two opposite \norientations of a bias magnetic field 0H\n. (b) Test structure, mimicking an object with chiral \nproperties: a small dielectric sample with symme try-breaking geometry of surface metallization.  \n \nFig. 16. Setup of the measurement system with a helical test structure for local determination of \nmaterial chirality. \n \nFig. 17. Sensor with small helix particles. ( a) left-handed helix particle; ( b) right-handed helix \nparticle. \n \nFig. 18. Transmission coefficients for small helix particles. ( a) left-handed helix particle; ( b) \nright-handed helix particle. \n  \n \n \n \n  \n \n  11\n                           \n                                       \n                          Ferrite disk \n                                         ( a)                                                                            ( b) \n \nFig. 1. ( a) Intensity and geometry of the power-fl ow distribution: strong subwavelength \nlocalization of energy and vortex behavior of microwave near fields. ( b) Schematic picture of the \npower-flow whirlpool in the vi cinity of a ferrite disk. \n \n \n \n \nFig. 2. An electric field inside a ferrite disk has both orbital and spin angular momentums. When \nan electrically polarized dielectric sample is placed above a ferrite disk, electric dipoles in a \ndielectric sample precess and accomplish an or bital geometric-phase ro tation. A bias magnetic \nfield 0H\n is directed normally to a disk plane. For an opposite direction of 0H\n, one has an \nopposite rotation of an electric field and an opposi te direction of precession of electric dipoles.  \n  12\n \n \nFig. 3. Frequency characteristics of a module of the reflection coeffi cient for a rectangular \nwaveguide with an enclosed thin-film ferrite disk. The resonance modes are designated in \nsuccession by numbers n = 1, 2, 3… The coalescent resonances are denoted by single and double \nprimes. An insert shows geometry of a structure. \n \n  \n   \n  \n                        ( a)                                              ( b)                                               ( c) \n Fig. 4. The near-field helicity\n parameters. ( a) Near-field helicity for the 1st MDM. ( b) Near-field \nhelicity for the 2nd (the resonance 2''). ( c) Absence of the near-field helicity for non-resonance \nfrequencies.  \n  \n    \n  \n                                          ( a)                                                                       ( b)       \n \nFig. 5. ( a) A sample of a ferrite disk with two loading dielectric cylinders placed inside a 10TE -\nmode rectangular waveguide. ( b) Frequency characteristics of a module of the reflection \ncoefficient for the 1st MDM at different parameters of a symmetrical dielectric loading. \nFrequency 8,456 GHzHf  is the Larmor frequency of an unloaded ferrite disk. \n                                                                                             13\n \n                             8.52256  resf GHz       8.451485resf GHz      8.38578resf GHz   \n \nFig. 6. Numerically calculated helicity-parameter distributions for the 1st MDM at different \ndielectric constants of loading cylinders. The d istributions are shown on the cross-section plane \nwhich passes through the diameter a nd the axis of the ferrite disk. \n \n                       \n \nFig. 7. A microwave microstrip structure (sensor) with a MDM ferrite disk and a sample under \ninvestigation. \n \n   \n   \n                                          ( a)                                                                       ( b)       \n Fig. 8. Transformation of the MDM spectrum due to  a dielectric loading in a microstrip structure. \n(\na) Numerical results; ( b) Experimental results. \n  14\n   \n      \n                                                 ( a)                                                                      ( b) \n Fig. 9.  A sensor with a wire concentrator  for localized material characterization. (\na) Geometry \nof a microstrip structure; ( b) A magnified picture picture of a MDM ferrite disk with a wire \nelectrode. \n \n \n       \n  \n                                       ( a)                                                                          ( b) \n \nFig. 10. Evidence for helical waves. The electric field distributions on a wire electrode for \ndifferent time phases. ( a) Time phase 1t; (b) Time phase 2t. \n \n      \n            \n                 \n  \n               ( a)                                                      ( b)                                                    ( c) \n \nFig 11. The fields and currents at a butt end of a wire electrode. ( a) An electric field; ( b) A \nmagnetic field; ( c) A chiral surface electric current. \n \n       \n                                        \n     \n                                 ( a)                                                                              ( b)         \n \nFig. 12.  Field structure on a butt end of a wire concentrator: ( a) the power-flow-density vortex \nand ( b) the helicity density F. When one changes oppositely an orientation of a bias magnetic  15field 0H\n, one has an opposite rotation of the power flow and an opposite sign of the helicity \ndensity F (red colored instead of blue colored).  \n  \n   \n              \n  \n                                   ( a)                                                                                ( b) \n \nFig. 13. A microstrip structure for localized material characterization in a dielectric sample. ( a) \nGeometry of a structure; ( b) MDM spectra of the reflection coefficient at a dielectric loading \n(experimental results). \n \n \n \n \nFig 14.  Helicity density F of the MDM near fields at two orie ntations of a normal bias magnetic \nfield 0H\n. A ferrite disk is without a dielectric load ing and is placed in a microstrip structure \nshown in Fig. 7.   \n  \n   \n  \n                                        (a)                                                                        ( b)                            \nFig. 15. ( a) Discrimination of an enantiomeric structure by the MDM spectra at two opposite \norientations of a bias magnetic field 0H\n. (b) Test structure, mimicking an object with chiral \nproperties: a small dielectric sample with symme try-breaking geometry of surface metallization.  \n  16\n \n \nFig. 16. Setup of the measurement system with a helical test structure for local determination of \nmaterial chirality. \n \n \n \n                                       ( a)                                                                        ( b)                            \nFig. 17. Sensor with small helix particles. ( a) left-handed helix particle; ( b) right-handed helix \nparticle. \n \n \n  \n                                        ( a)                                                                        ( b)                            \n \nFig. 18. Transmission coefficients for small helix particles. ( a) left-handed helix particle; ( b) \nright-handed helix particle.  \n \n "    },    {        "title": "2102.12385v1.Gamma_irradiated_nanostructured_NiFe2O4__Effect_of_gamma_photon_on_morphological__structural__optical_and_magnetic_properties.pdf",        "content": "Gamma irradiated nanostructu red NiFe 2O4: Effect of γ -photon on \nmorphological, structural, optical and magnetic properties  \nSapan Kumar Sena,*, Majibul Haque Babub, Tapash Chandra Paulc, Md. Sazzad  Hossaind, \nMongur  Hossaine, Supria  Duttaf, M. R. Hasang, M. N. Hossainh, M. A. Matinh, M. A. Hakimh, \nParimal  Balac \naInstitute of Electronics, Atomic Energy Research Establishment, Bangladesh Atomic Energy \nCommission, Dhaka -1349 , Bangladesh  \nbBasic Science Division, World University of Bangladesh, Dhaka -1205, Bangladesh . \ncDepartment of Physics, Jagannath University, Dhaka -1100, Bangladesh . \ndDepartment of Physics, University of Dhaka, Dhaka -1000, Bangladesh . \neHunan Key Laboratory of Two\n Dimensional Materials, State Key Laboratory for \nChemo/Biosensing and Chemometrics, College of Chemistry and Chemical Engineering, Hunan \nUniversity, Changsha, 410082 China . \nfMinistry of Education, Government of the People's Republic of Bangladesh, Dhaka, \nBangladesh.  \ngMaterials Science Division, Atomic En ergy Center, Dhaka, Bangladesh Atomic Energy \nCommission, Dhaka -1000 , Bangladesh . \nhDepartment of Glass & Ceramic Engineering, Bangladesh University of Engineering & \nTechnology, Dhaka -1000 , Bangladesh . \n \nAbstract  \nThe current manuscript highlights the preparat ion of NiFe 2O4 nanoparticles by adopting sol -gel \nauto combustion route. The prime focus of this study is to investigate the impact of γ -irradiation \non the microstructural, morphological, functional, optical and magnetic characteristics. The \nresulted NiFe 2O4 product s hav e been characterized employing numerous instrumental equipment s \nsuch as FESEM , XRD , UV–visible spectroscopy, FTIR  and PPMS  for a variety of γ -ray doses  (0 \nkGy, 25 kGy and 100 kGy) . FESEM micrographs illustrate the aggregatio n of ferrite nanoparticles \nin pristine  NiFe 2O4 product having an average particle size of 168 nm and the surface morphology \nis altered after exposure to γ -irradiation. XRD spectra have been analyzed employing Rietveld \nmethod  and t he results of the XRD investigation reveal the desired phases (cubic s pinel phases) of \nNiFe2O4 with observing other  transitional phase s. Several microstructural parameters such as  bond \nlength, bond angle , hopping length etc. have been determined from the analysis of Rietveld \nmethod.  This stu dy reports  that the γ -irradiations demonstrate a great influence on optical bandgap \nenergy  and it varies from 1.80 and 1.89 eV evaluated via K -M function . FTIR measurement depicts \na proof for the persistence of Ni -O and Fe -O stretching vibrations within the respecti ve products \nand thus indicating the successful development of NiFe 2O4. The saturation magnetization (M S) of \npristine  Ni ferrite product is noticed to be 28.08 emu/g. A considerable increase in M S is observed \nin case of low γ -dose (25 kGy) and a decrement n ature is disclosed  after the result of high dose of \nγ-irradiation (100kGy) . Keywords:  Sol-gel auto combustion; NiFe 2O4; Gamma-irradiation; Rietveld refinement; \nSaturation magnetization; Optical bandgap  \n*Corresponding author:  sapansenphy181@gmail.com ; ORCID ID: 0000 -0001 -5086 -2758  \n1 Introduction  \nIn recent years, nanocrystalline spinel ferrites have been investigated immensely owing to \npotential applications in chemical sensors, microwave absorbers, permanent magnets, high \ndensity recording systems, ferrofluid technology, biomedical,  imaging and high -frequency \ndevice  applications  [1][2][3]. Importantly, they possess excellent both electrical and magnetic \nproperties, which are extremely sensitive to the ma ny factors such as chemical composition, \nsynthesis method, annealing temperature or temperature treatment, and cation distribution at \ntetrahedral (A) and octahedral [B] site s [3]. Among the various spinel ferrites, however, NiFe 2O4 \n(nickel ferrite) has drawn an optimum attention recently due to various applications in gas \nsensors  [4], spintronics  [5], microwave absorption  [6], catalyst  [7], lithium -ion batteries  [8], \nhydrogen production  [9], even in bi omedicine  [10] etc.. As more and more considerations have \nbeen devoted keenly to the nano -sized magnetic materials for inherent their unique properties \ncompared to their bulk counterparts, so the scientific engrossment on nano -sized NiFe 2O4 is on \nthe expanding  in the rese arch community . In this direction, the magnetism of NiFe 2O4 is \npredominantly intriguing due to its substantial saturation magnetization and unique magnetic \nstructures. In general, NiFe 2O4 belongs to an inverse spinel structure with Ni2+ ions on octahedral \nB sites (denoted as O h-sites) and Fe3+ ions on both of the tetrahedral A (denoted as T d-site and O h-\nsites) sites equally  [11]. This is typically maintained by the formation energy in favor of the \nreverse spinel rather than spinel structure  [12]. It is also found to have the mixed spinel structure \nwith the inverse one, i.e., some Ni2+ ions may occupy the Td- site [11][13][14]. \nA general formula for a nickel ferrite structure is (Ni 1-xFex) [Ni xFe2-x]O4, where x is the degree of \ninversion. According to the crystal field theory (CFT), magnetic moments are rising from the \nlocal moments of the Ni2+ with 3d8 as well as Fe3+ with 3d5 electrons. Significantly, the net \nmagnetization comes from the Ni2+ (Oh-sites) cations alone (~ 2μB), while Fe3+ moments (~ 5μB) \nin a high spin state for both Oh- and T d-sites are antiparallel and abandon with each other  [11]. \nHowever, the modification in physical properties (i.e structural and magnetic properties) of \nnanoferrites can be justifie d by instigating radiation damage by means of swift heavy ions [15], \nlaser beam  [16], proton  [17] and gamma radiation  [18]. Lately, several irradiation systems are \nimplemented in the cutting -edge world because it is a striking tool for modifying the physical \nproperties of  nanoparticles in the research and development in commercial applications and \nindustrial technologies like aeronautical and satellite communication, pollution control, material \ndevelopment, security systems and so on [15][19][20]. Upon electromagnetic nature, penetrating \npowe r, very short wavelength and all medium propagation property, the gamma ray ( 𝛾) is \nconsidered as ionizing radiation  [21]. So, the present inquiry deals with an interaction of 𝛾 with \nNiFe 2O4 which can tune its physical properties since it is known to generate controlled defects of various types such as point, cluster, and columnar defects in the material s [22]. Besides, \nirradiation with 𝛾 rays contain the plausibility of dislocation of Fe3+ ions from tetrahedral A sites \nto Fe2+ ions at octahedral B sites  [19]. \nSeveral research works have  been focused on different preparation route s to synthesize NiFe 2O4 \nnanoparticles, such as co-precipitation  [23], sol -gel [24], spray pyrolysis  [25], mechanical \nactivation  [26], hydrothermal method  [27], high energy ball milling [28], etc.. Concurrently, to \nimprove the physical properties of nanoferrites using 𝛾 rays, various methods have been \nemployed to synth esize ferrite nanoparticles. Recently, Raut et al.  [19] synthesized ZnFe 2O4 \nusing sol –gel auto -combustion technique and reported that saturation magnetization and \nmagnet on number increased by 𝛾radiation dose of 50 and 100 kGy. Raut et al. [29] also prepared \nCoFe 2O4 by sol -gel auto combustion with total radiation d oses of 50 and 100 kGy and showed  \nthat lattice parameters decline  with Coercivi ty (H c), Remanence  magnetization (M r) and \nanisotropy field (H k) shrinkages as a function of 𝛾 radiation  doses . Based on the above survey,  \ntill now, there is no study on the imp act of 𝛾 radiations  on NiFe 2O4 nanoparticles synthesized by \nsol-gel auto combustion method . Therefore, the aim of the research work is to synthesize and \ninvestigate some of the physical parameters of NiFe 2O4 before and after 𝛾-irradiation . \n2 Experimental sections  \n2.1 Materials  \nIn this experiment , Nickel Nitrate Hexahydrate (Ni(NO 3)2.6H 2O), Ferric Nitrate Nanohydrate \n(Fe(NO 3)3.9H 2O), Citric Acid (C 6H8O7.H2O), Ammonium Hydroxide (NH 4OH) w ere used for \nthe sample preparation. The samples were made  by mixing the compositions ( Ni(NO 3)2.6H 2O, \nFe(NO 3)3.9H 2O, C6H8O7.H2O and NH 4OH) together. The required materials were employed as \nreceived.  \n2.2 Synthesis route  \nSol-Gel Auto -Combustion route  has been employed to prepare the NiFe2O4 nano particles  as a \nfunction of γ-irradiation for 0, 25 and 100 kGy doses involving the metal nitrates of the \ningredient components as raw materials in citric ac id matrix. The synthesis rout e has been \nexplained as follows: the metal nitrates and citric acid were dissolved in an appropriate amount \nto keep the molar ratio of metal  ions and used citric acid to1:1. A little quantity of ammonia was \ngradually dissolved into the starting solution to balance pH = 7 and stabilize the nitrate -citrate \nsol. The obtained precursor aqueous solution was stirred vigorously with a magnetic stirre r at 60 \n°C. After that the sol was placed into a tray and heated gradually to 120 °C to change into an \nextremely viscous brown gel. The gel was slowly heated to 250 °C in order to attain dried gel \nand after few minutes, the gel started to completely burnt to form a crisp powder. The probable \nchemical rea ction during the synthesis of Ni Fe2O4 products is given as:  \n𝑁𝑖(𝑁𝑂3)2+2𝐹𝑒(𝑁𝑂3)3+20\n9𝐶6𝐻0𝑂7=𝑁𝑖𝐹𝑒2𝑂4+40\n3𝐶𝑂 2+80\n9𝐻2𝑂+4𝑁2 2.3 Co -60 G amma irradiation \nThe synthesi zed NiFe 2O4 products have been  subjected to γ -irradiation origin ated from a 60Co \nsource  with various  doses (25 kGy  and 10 0 kGy ) at Institute of Food and Radiation Biology \n(IFRB), Atomic Energy  Research Establishment , Bangladesh Atomic Energy Commission, \nDhaka, Bangladesh.  The activity of 60Co  source at the time of exposure is 12.17 kGy/h  and \nliquid phase dosimetry arrangement  (Cericcerous) has been employ ed to enumerate  the γ-dose \nrate. \n2.4 Characterization techniques  \nThe morphological investigation of the products has been acquired by F ield Emission Scanning \nElectron Microscopy  (FESEM)  (JEOL JSM -7600F, USA) . X-ray Diffraction  (XRD) (model \nX′PertPRO XRD Philips PW3040, Netherlands) has been  applied on the resulted NiFe 2O4 and γ-\nirradiated NiFe 2O4 to identify the phases and crystallinity.  XRD structure is equipped with a Cu -\nKα radiation (λ = 1.5404 Å)  maintained in the range of 2θ from 15 ⁰ to 90⁰. XRD spectra have \nbeen investigated through the Rietveld technique  as implemented in Fullporf software.  To \nexamine the functional groups associated with ferrite samples , Fourier Transform Infrared \nSpectroscopy  (FTIR ) equipment  has been employed during this experiment.  Physical  Properties \nMeasurement S ystem (PPMS), Quantum Design Dyna Cool at ambient conditions  has been used  \nto measure the magnetic hysteresis loop on the obtained products as well as different magnetic \nparameters such as saturation magnetization (M s), remnant magnetization (M r) and coercive field \n(Hc). DRS  informations  has been recorded  by a UV –Vis device  (Model: PerkinElmer UV –Vis–\nNIR Spectrometer Lambda 1050)  at a wavelength of 200 –800 nm . \n3 Results and discussion  \n3.1 Surface morphology properties  \nFig. 1 reveals t he FESEM images of pristine  and γ-irradiated NiFe 2O4 with comp aratively low \n(25kGy)  and high (100 kGy)  doses of 𝛾 radiations.  It is clearly seen in the FESEM images  that \nthe surface morphology of NiFe 2O4 is being altered with  total dose of  𝛾 radiation . The same \noutcomes have been detected in the  previous literature  [30][31][32]. According  to the FESEM \nimages  [Fig. 1(a,d)], the highest average  particle size is pronounced 168 nm for pristine  sample. \nIt is seen that result of employing a low γ dose, the surface disrupts and making smaller particles \ndue to the lattice vibration, atomic displacement and local heating effects [33]. The disintegration \nof particle s can also be attributed to induced compressive stress which causes intrinsic defect \nrecombination or reordering of initially disordered phase and aids the particles to regain their \nshapes [34][35]. In addition, at a low dose, γ photon has a higher possibility of interaction with \nthe materials and then photoelectric absorption dominates, re sulting in the de crement  of average \nparticle size of NiFe 2O4 from 168 to 135 nm [36]. Conversely, in case of high dose  (100 kGy) , \nthe average particle size is slightly increased from 135 to 140 nm as shown in Fig. 1(c,f).  The \nswelling of average particle size may be occurred due to the decrease of photoelectric  absorption  \n[37]. It is well known that particles amalga mate at higher γ dose owing to no electron transfe r, thereby, to make sure incre ment of average particle size [38]. A hypothetical growth mechanism \nof NiFe 2O4 particle s in pristine  and 𝛾-irradiated  samples with different γ doses is depicted in Fig. \n2. \n \n \n \n \n \n \n \n \n \n \nFig. 1 The FESEM images  and particle size distribution  of NiFe 2O4 samples : (a, d) for 0 kGy , (b, \ne) for 25 kGy, and (c, f) for 100 kGy doses . \n \n \n \n \n \n \n \nFig. 2 A plausible growth mechanism of pristine  and 𝛾-irradiated  NiFe 2O4 samples.  \n3.2 Structural properties  \n \nFig. 3 depicts the Rietveld refinement (R.R) of the structure by XRD data is processed using \nFullprof suite software. By a meticulous R.R analysis, a duel common inverse spinel  cubic \nstructure (Fd3̅m) and hematite phase (R 3̅c) is firmly confirmed for the pristine  and irradiated \nsamples. It can be evidently indexed to the face centered cubic (fcc) structure of NiFe 2O4 \n(JCPDS Card  No.10 -0325)  [39] as shown in Fig. 3. On the o ther hand, a nickel phase (F m3̅m) \nhas appeared for only the  irradiated samples. From Fig. 3, it can be seen that the most prominent \nintense peak is manifested at 2𝜃 = 35.74° corresponding to the (31 1) plane of NiFe 2O4 for \n(a) (b) (c) \n(d) (e) (f) pristine  sampl e. After that, the place of (31 1) plane orientation has been changed from 2𝜃 = \n35.75° to 35.79° by applying gamma radiation. This may be attributed to the formation of Fe3+ \nions in the place of Fe2+ ions. It is also noticeable that the intensity of (3  1 1) peak increases \nwhile (40 0) peak intensity decline for applying 25 kGy gamma radiation. Conversely, owing to \n100 kGy gamma r adiation, the intensity of (400) peak rises sharply when (31 1) peak falls \ndrastically. Such  results, the shift of the peak  intensity is in good agreement with that reported by \nseveral authors and they ascribed  that to the lattice distortion arisen after irradiation  [40][41]. \nTo know the lattice parameters, we have done the R.R analysis using the FullProf software.  The \nquality of the R.R is determined by mea ns of a set of conventional statistical parameters.  \nTypically, the quality of the R.R can be determined using several statistical parameters such as \ngoodness of fit ( 𝜒2), weight Profile R -factor (R wp) and expected R -Factor (R exp), whereas  Rwp \ncomparations the adjusted data with the experimental data, R exp estimates the quality of the \nexperimental data and  𝜒2=(𝑅𝑤𝑝\n𝑅𝑒𝑥𝑝⁄ )2\n[42]. During the R.R process, 𝜒2 initiates with a \nmaximum value when the goodness of fit model is poor and decreases as the fit data matches the \nexperimental data.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3 The Rietveld refinement of XRD pattern with different γ irradiations and a face centered \ncubic (fcc) structure of NiFe 2O4 \nF d -3 m Table 1 Several structural parame ters of pristine and γ-irradiated samples  \n Parameters  0 kGy  25 kGy  100 kGy  \n \n \nLattice \nconstants (Å)  \nNiFe 2O4 (𝑎=𝑏=𝑐)  \n8.3304   \n8.3268   \n8.3189  \nFe2O3 (𝑎=𝑏,𝑐) 5.4033, 5.2724  5.4022, 5.3045  5.0309, 13.7998  \nNi (𝑎=𝑏=𝑐) --- 3.5980  3.3129  \n \n \nVolume ( Å3) NiFe 2O4 578.1011  577.3437  575.7062  \nFe2O3  133.3125  134.0625  302.4775  \nNi  --- 46.5782  36.3614  \n \n \nCrystallite size \n(nm)  DD-S 34.04  34.71  34.12  \nDave. 24.55  51.16  42.61  \nDW-H 71.23  83.23  77.78  \nDR-R 70.12  77.78  75.01  \n \n \nRietveld \nRefinement  Rwp(%) 8.10 8.04 8.07 \nRexp (%) 6.95 7.04 6.98 \nG.O.F ( 𝜒) 1.69 1.88 1.56 \nOxygen position, (u)  0.22920  0.25356  0.39745  \n \n \n \n \nBond length \n(Å)  \n \n \nNiFe 2O4 Fe-O 1.92552  2.11176  3.29151  \nFe-Fe 2.94526  2.94397  2.94120  \nNi1-O 2.10304  1.75146  --- \nNi2-O 1.50415  1.85415  --- \nNi1- Ni2 4.16522  4.16340  4.15953  \nNi1-Fe 3.45362  3.45211  3.44882  \nNi2-Fe 1.80359  1.80281  1.80110  \n \nFe2O3 Fe-O 1.4709, 1,5088, \n1.5270,  2.2582  1.8016, 2.0098  1.8154, 2.1871  \nFe-Fe 1.5270  2.1372  3.4068, 2.9507  \nNi2 Ni-Ni ---- 2.5442, 3.5981  2.3426, 3.3129  \n \n \n \n \nBond angle \n(deg.)   \n \n \nNiFe 2O4 O-Fe-O 79.28, 100.72, \n100.75, 100.73  88.40, 91.60,  9.20,  \nO-Ni1-O 109.45, 109.47, \n109. 48  109.4712  109.4712  \nO-Ni2-O 109.51, 109.48, \n109.45  109.4712  --- \nFe-O-Ni1 117.96  126.40  116.67  \nFe-O-Ni2 62.03  53.60  O-Fe-Ni2=98.60  \n \n Fe2O3 Fe-O-Fe 61.6417  67.9588  71.4986  \nO-Fe-O 85.9783  78.7610  85.4653  \n The R.R process continues until convergence is reached with values less than 2.0 or close to 1.0, \nwhich designates a correlation between the experimental data and adjustment model used. The \nvalues of R wp, Rexp, and 𝜒2 are presented in Table 1. In addition, the estimated Rietveld refined \nlattice parameter and volume values are also tabulated in Table 1. It can be seen (Table 1) that \nboth the  lattice parameter and volume are decreasing simultaneously with increasing the gam ma \nradiations which signifies that atomic position is displaced significantly. The shrinking of lattice \nparameter and volume due to the lattice vacancies produced after 𝛾 radiation dose  reasons \ndistortion and deviation from the spinel cubic structure. Bes ides, γ radiation typically formed the \ncompressive strain and then generated some disorder into the NiFe 2O4 lattice structure [43].    \nHowever, the crystallite size of the samples is calculated from the line width of the (3 1 1) peak \nusing Debye -Scherrer's (D -S) equation [44]:DD−S=kλ\nβcosθ, where ′DD−S′ is the crystallite size,  ′𝜆′ \nis the wavelength of CuK α  radiation (1.54 04Å), ‘β’ is the full width half maxima (FWHM), and \n‘θ’ is the diffraction angle of the strongest characteristic peak.  After that, we have also estimated \nthe average crystallite  size ( Dave .) using the Lorentz function by Origin pro -2018. From Table 1, \nit is clearly evident that in both cases the DD−S and Dave . increases drastically at 0 and 25 kGy \nsamples, and decrea ses at the 100 kGy sample . To get the more accurate crystallogra phic \nparameters such as  crystallite size, strain, atomic structure, bond length and bond angle, the \nWilliamson –Hall (W -H) method and Rietveld refinement (R.R) is performed precisely. Initially, \nthe W -H method gives the values of crystallite size and strain  concurrently.  \nThe equation of W -H method is  [45]: 𝛽𝑐𝑜𝑠𝜃 =4𝜀𝑊𝐻𝑠𝑖𝑛𝜃 +𝐾𝜆\n𝐷𝑊𝐻, where  is an intercept in W -H \nplot that corresponds to the crystallite size (DW−H) and slope  𝜀𝑊−𝐻 corresponds to the strain as \nshown in Fig. 4. It can be seen that D W−H lies between 71.23 to 83.23 nm which emulates the \nDD−S and D ave trend. Additionally, the Rietveld refined (R -R) crystallite size is found from 70.12 \nto 77.78 nm. It is observed that the crystallite size increases after 25 kGy γ radiation doses, \nwhich is ascrib ed to the following: γ radiation interacts with the particles or grains and ionizes  \nFe3+ ions to Fe2+ ions [46] as follows : \n𝐹𝑒3++𝛾→𝐹𝑒2++𝑒−          (1) \nBut the crystallite size decreases at 100 kGy γ radiation dose due to the modif ication of the ratio \nof Fe2+/ Fe3+ ions in octahedral -B site that can be expressed by the following equation [47]:  \n𝐹𝑒2++𝛾→𝐹𝑒3++𝑒−           (2) \n \nThrough the R -R analysis, we have determined the bond length and bond angle to know the \ngamma radiation effect into the NiFe 2O4 structure. Firstly, in the case of NiFe 2O4, the Fe -O and \nNi2-O bond length is grown up abruptly while Fe -Fe, Ni 1-O, Ni 1- Ni2, Ni1-Fe and Ni 2-Fe bond \ndistance declining after using gamma radiations as shown in Table 1 and Fig. 5. \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4 Williamson –Hall plot for NiFe 2O4 with different γ irradiations. A comparison crystallite \nsize studies among various methods.  \n \nInterestingly, there is no bond length appeared between Ni1-O and Ni 2-O at 100 kGy sample that \nmay be attributed to the development of Fe -O bond distance. Further, multiple bond distance of \nFe-O, Fe -Fe and Ni -Ni is detected for Fe 2O3 and Ni struc tures as shown Table 1. Likewise, in the \nevent of a bond angle, we have found multiple bond angles in the  NiFe 2O4 and Fe 2O4 structures.  \nThe O -Ni1-O and O -Ni2-O bond angle almost analogous, on the contrary,  Fe-O-Ni1 and Fe -O-Ni2 \nbond angle completely different in the  NiFe 2O4 structure . What’s more, an interest point has \nbuild  that Ni 2 atom does not make bond length and bond angle as well at 100 kGy sample. It can \nbe speculated that owing to the higher gamma radiation Ni 2-O bond s plit up and then make a \nnew bond only with the Fe atom as shown in Fig. 5. \nHowever, the Hopping length (distance between magnetic ions in tetrahedral site -LA and \noctahedral sites -LB) is another important structural parameter which estimated by these equations  \n[48]: 𝐿𝐴=1\n4𝑎√3 and 𝐿𝐵=1\n4𝑎√2, where ′𝑎′ is the lattice parameter of NiFe 2O4 structure. The \ncalculated values of the L A and L B are presented in the Table 2. It is seen (Table 2) that the \nhopping lengths in L A and L B sites decrease concurrently with changing gamma radiations. But \nother groups showed that in the case of ZnFe 2O4 and CoFe 2O4, hopping lengths increased after \nusing gamm a radiations as they have found greater values of lattice parameters [19][29]. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5 Several Rietveld refined structures with denoting bond lengths and bond angles.  \n \nIn the present study , the lattice parameter, volume, and bond length are shrunk with altering \ngamma radiations which are firmly confirmed that these structural parameters control the \nHopping lengths.  Further, the inter -ionic distances (average bond lengths) at tetrahedral -RA and \noctahedral sites -RB is calculate d using the following relations  [49]:  \n𝑅𝐴=𝑎√3(𝑢−1\n4)                                            (3) \n \n𝑅𝐵=𝑎 √3𝑢2−2𝑢+1\n16                                   (4) NiFe 2O4 NiFe 2O4 NiFe 2O4 \nFe2O3 Fe2O3 \nFe2O3 Ni2 Ni2 Where ‘u’ represents the Rietveld refined oxygen positional parameter which values are 0.22920, \n0.25356 and 0.39745 for 0, 25 and 100 kGy gamma radiation samples, respectively. Our refined \n‘u’ values are analogous wit h previous reported data  [50]. But other groups have been taken \ntheoretical value of oxygen position parameter (u=0.381 Å). Patange et al. has chosen ‘u’ value \nfor Al3+ substituted NiFe 2O4 nanoparticles despite the Rietveld refinement analysis  [51]. It is \nobserved that ‘u’ is increased with rising gamma radiations, and R A is increased while declining \nRB as shown in Table 2.  \n \nTable 2  The estimated Hopping lengths and average bond length of NiFe 2O4 for various γ \nirradiations  \n \nParameters  0 kGy  25 kGy  100 kGy  \n \nHopping \nlength (Å) La 3.6072  3.6056  3.6022  \nLb 2.9445  2.9440  2.9412  \n \nAverage bond \nlength  (Å) Tetrahedral site  (RA) 1.5035  1.8542  3.9257  \nOctahedral sites (R B) \n 2.2691  2.0525  1.9331  \n \nThe change of R A and R B can be explained on the basis of the drastic movement of oxygen \nposition parameters.  Since oxygen position is developed considerably with rising gamma \nradiations, so the NiFe 2O4 structure becomes slowly close to the fcc structure and oxygen ions \nare moving towards octahedral coordinated at the end. However, several studies are investigated \non spinel ferrite by gamma doses. Here, we have reported on the effect of gamma dose  over \nseveral spinel ferrites in Table 3.  \n \nTable 3  A comparison study for various spinel ferrites under γ irradiations  \n \nSample  Rietveld  \nrefinement  No. of  \nphases  Lattice \nconstant  Determined bond \nlength and bond angle  Determined \nRA and R B Hopping \nlength  Ref \nZnFe 2O4 No Single  Increase  No No Increase  [19] \nCoFe 2O4 No Single  Increase  No No Increase  [29] \nNiFe 2O4 Yes Three  Decrease  Yes Yes Decrease  present  \n \n3.3 Optical properties  \nThe optical band gap (E g) of the 𝛾pristine  and γ-irradiated NiFe 2O4 samples has  been calculated \nfrom the Kubelka –Munk (K –M) function through the following equation  [52]: \n𝐹(𝑅)∞𝛼=(ℎ𝜗−𝐸𝑔)2\nℎ𝜗                                               (5) \nWhere, ‘R’ is the reflectance, ′𝛼′ is the absorption coefficient, ′ℎ𝜗′ is the incident light energy \nand ′𝐸𝑔′ is the optical band gap. Fig.  6 shows the Eg plots of the NiFe 2O4 treated with different γ \nirradiation doses.  The Eg is calculated by plotting a graph between  (𝐹(𝑅)×ℎ𝜗)2 versus  ℎ𝜗. In \nthe pristine  NiFe 2O4, the E g is 1.85 eV. Subsequently, for 25 and 100 kGy irradiation doses, the \nEg is found to be 1.80 and 1.89 eV, respectively. Thus, we observe that the E g shrinkages when low doses of γ irradiation are used, while with  high dose γ irradiation the E g is increased. The \nchange in E g can be expounded by the following ways: (i)  quantum -size effect may be \naccountable for altering the E g. Because it is resulting from the strong interaction  between the \nsurface of NiFe 2O4 and γ photons  [53].  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 6 The Eg of NiFe 2O4 with various γ irradiation doses and with that a comparison of E g study  \n(ii) decrease of E g may be ascribed to the creation of localized states into the NiFe 2O4 structure \ndue to structural defects  [54]. (iii) enhancement of  Eg due to the optical scattering at the particle \nor grain boundaries  and intrinsic absorption  [21]. Besides, it is anticipated that the  𝑉𝑂 ̈is created \nwithin NiFe 2O4 structure during high er dose of γ irradiation that will play a significant role to \nincrease the  Eg [55]. However, it is manifested that the  γ irradiation significantly  affects the \ncrystallite size and hence alter the optical band gap of  NiFe 2O4. \n3.4 FTIR  analysis \nFig. 7  displays the FTIR diagram  of pristine NiFe 2O4 and NiFe 2O4 irradiated with γ -doses for \nanalysis of chemical bonds within the resulted products.  The prominent bands with sharp peaks \nhave been observed at 365 cm-1 and 547 cm-1 which are assigned to stretching vibrations of Ni -O \nbond in octahedral complexes  and tetrahedral Fe -O vibration s, respectively  [56][57]. This \nanalysis of FTIR presents the consistency with published results in numerous bo dies of literature  [58][59][60] and ensures the construction  of the spinel NiFe 2O4 nano particles  identifying  \ncharacteristic  peaks in FTIR diagram . The FTIR studies clearly support the formation of Ni \nferrite as observed in XRD analysis.  The presence of band positions in the ranges of 1000 –1300 \ncm−1 and 2000 –3000 cm−1 demonstrates  the survival of O –H, C –O, and C=H stretching mode of \norganic compounds [61]. Two peaks appear  at 1381 cm-1 and 3754 cm-1 in case of the pristine  \nand γ -irradiated products , which belongs  to stretching vibration of C –H and t he contribution of \nN–H bonds , respectively  [61]. \n \nFig. 7 FTIR spectra of pristine  and γ -irradiated NiFe 2O4 \n3.5 Magnetic properties  \nFig. 8  shows the room temperature magnetization (M) for the applied field (H) of as -prepared \nNiFe 2O4 nanoparticles before (0 kGy) and after irradiation (25 kGy and 100 kGy). The applied \nfield was ranged from -20 kOe to +20 kOe, and it is apparent that the magnet ization is not \nsaturated at 20kOe. Low coercivity  and low hysteresis indicates the ferrimagnetic behavior of the \nsamples. The m agnetic p arameters such as saturation magnetization (M s), Remanence ( Mr) and \ncoercivity ( Hc) are summarized in Table 4 . The saturation magnetization (M s) was found for the \npristine  sample was ~ 28 emu/g, which is less than the bulk NiFe 2O4. It can be attributed to the \nfact that NiFe 2O4 becomes mixed spinel from the inverse spinel structure at the nanoscale  [62]. \nThe tetrahedral A -site consists  of ferromagnetically ordered Fe3+ ions, and the octahedral B -site \nconsists of Fe3+ and Ni2+ ions. The magnetic contribution in the inverse spinel arises from the \nNi2+ in the octahedral B -site. Due to antiferromagnetic ordering, the Fe3+moments from both the \nA-site and B -site cancel each other. Considering the magnetic moments of Fe3+ and Ni2+ ions are \n5 𝜇𝐵 and 2 𝜇𝐵, the net magnetic moment is 2 𝜇𝐵by Neel’s two sublattice model  [63]. Therefore, \nthe proposed change in the cation distribution at the nanoscale can be interpreted as (Ni xFe1-\nx)A[Ni 1-xFe1+x]B. However, the presence of Ni2+ might not be that dominant in the tetrahedral A -\nsite for the pristine  samples. Also, the synthesis methods significantly impact the magnetic \nproperties on spinel ferrites  [64]. \nThere is an increment in the  Ms from 28 emu/g to 41 emu/g after the 𝛾−irradiation of 25 kGy.  \nAfter the 𝛾−irradiation, the degree of mixed -phase may have increased, and more Ni2+ has \ntransferred to the A -site. Also, the irradiation releases the pinned domains on the surface, and the \nsignificant increment of magnetization was observed  [65][66]. Furthermore, the onset of \ncrystallite size growth was observed after the irradiation. Crystallite size growth may accompany \nthe migrati on of Fe3+ ions to the B -site [2][67]. The decreasing trend in the lattice parameter also \nestablished the emergence of strong ionic interactions among the lattice sites. Therefore, the \nenhanced saturation magnetization was observed.  \n \n \n \n \n \n \n \n \n \n \nFig. 8 M-H curves of the pristine  and 𝛾−irradiated samples of NiFe 2O4. \nConsequently, it will be interesting to investigate the magnetic properties on exposing the \nsamples at higher 𝛾−radiation. However, a strong 𝛾−irradiation of 100 kGy  decreased  the Ms \ndrastically to 20 emu/g. The strong 𝛾−irradiation causes the adverse effect on magnetic ordering \nby the ion -induced disorder. Also, high energy ions may penetrate the sample. The host atoms \nand molecules interact with the irradiated gamma photons  via inelastic collision.  \nTable 4 Magnetic p arameters of pristine and γ-irradiated samples of NiFe 2O4 \n \n \n \n \nThe interaction causes energy loss and introduces defects or partial amorphization depending on \nthe amount of energy lost [68]. Hence, a de crease in crystallite size was observed. The reduction \nof crystallite size introduces spin canting, magnetic dead layers and weakening of super -Radiation \n(kGy)  Ms(Ex) \n(emu/g)  Remanence, \nMr(emu/g)  Coercivity, H c \n(Oe) \n0 28.08  4.87 0.0022  \n25  40.63  8.44 0.0303  \n100 20.00  4.87 0.0216  exchange interaction. Consequently, a substantial reduction in saturation magnetization has \nemerged.  \n4 Conclusi ons \nNanoparticles NiFe 2O4 ferrites in spinel cubic structure have been synthesized via sol -gel auto -\ncombustion route. We have investigated the structural, morphological, magnetic and optical  \nproperties o f the resultant ferrite nanop article s with the variation of γ -doses. FESEM \nmicrographs clearly indicate the variation in morphology and aggregation nature within the  bare \nand γ -irradiated Ni ferrite  products . According to the XRD investigation of the conducted \nresearch, it is concluded that an irregular variation in crystallite size leads from the results of \nScherrer method. In addition, analysis of the W -H plot and R -R exhibit  the identical trend of \nvariation upon incorporation of γ -irradiation into the NiFe 2O4 crystal network. The crystalli te \nsize of pristine  Ni ferrite  sample is observed as 24.55  nm-71.23  nm. The result after γ -irradiation \nwith 25 kGy shows the increment n ature of crystallite size and then decre ase after exposure to \nhigh γ -dose (100 kGy) . From the Rietveld analysis numerous structural para meters such as bond \nlength,  bond angle, hopping length etc . are estimated in this current study.  The optic al bandgap \nenergy is estimated at 1.85 eV  in the pristine  product and consequently , is the c ase of  low \n(25kGy) and high ( 100 kGy ) γ-doses it corresponds to be 1.80 and 1.89 eV, respectively. 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Nanotechnol. 8 (2017) \n045002. doi:10.1088/2043 -6254/aa853a.  \n[68] B. Parvatheeswara Rao, K.H. Rao, P.S.V. Subba Rao, A. Mahesh Kumar, Y.L.N. Murthy, K. \nAsokan, V. V. Siva Kumar, R. Kumar, N.S. Gajbhiye, O.F. Caltun, Swift heavy ions  irradiation \nstudies on some ferrite nanoparticles, Nucl. Instruments Methods Phys. Res. Sect. B Beam \nInteract. with Mater. Atoms. 244 (2006) 27 –30. doi:10.1016/j.nimb.2005.11.009.  \n "    },    {        "title": "1908.08629v2.Damping_enhancement_in_coherent_ferrite_insulating_paramagnet_bilayers.pdf",        "content": "Damping enhancement in coherent ferrite/insulating-paramagnet bilayers\nJacob J. Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4\nChristoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6,\u0003\n1Department of Applied Physics, Stanford University, Stanford, CA, USA\n2NIST Center for Neutron Research, Gaithersburg, MD, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA\n4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA\n5Cornell High Energy Synchrotron Source, Ithaca, NY, USA\n6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: October 29, 2019)\nHigh-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising\nnanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the\nimpact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here,\nwe investigate the spin dynamics and chemical and magnetic depth pro\fles of 15-nm-thick MAFO\ncoherently interfaced with an isostructural \u00191-8-nm-thick overlayer of paramagnetic CoCr 2O4\n(CCO) as an all-oxide model system. Compared to MAFO without an overlayer, e\u000bective Gilbert\ndamping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness.\nWe attribute this damping enhancement to spin scattering at the \u00181-nm-thick chemically disordered\nlayer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our\nresults indicate that damping in ferrite-based heterostructures is strongly in\ruenced by interfacial\nchemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite\nthickness.\nI. INTRODUCTION\nEmerging spintronic device schemes leverage magnon\nspin currents in electrically insulating magnetic oxides\n(e.g., ferrites), unaccompanied by dissipative motion\nof electrons, for computing and communications\napplications1,2. Low-dissipation spintronic devices\nbecome particularly attractive if insulating ferrite thin\n\flms with low magnetic damping can serve as sources\nof magnon spin currents. Such low-damping ferrites\ninclude not only epitaxial garnet ferrites (e.g., YIG)3{11\nthat have been widely used in studies of insulating\nspintronics2{4,12{15, but also coherently strained epitaxial\nspinel ferrites16{18with crucial technical advantages over\ngarnets, such as lower thermal budget for crystallization,\nhigher magnon resonance frequencies, and potential to be\nintegrated coherently with other spinels and perovskites\nwith various functionalities19{22.\nIn general, low-damping ferrite thin \flms must be\ninterfaced with other materials to realize spintronic\ndevices. It is therefore essential to understand whether\nand how damping in the ferrite is impacted by the\nproximity to another material. For instance, to convert\nbetween electronic and magnonic signals through direct\nand inverse spin Hall or Rashba-Edelstein e\u000bects23,\nthe low-damping ferrite needs to be interfaced with\na nonmagnetic metal with strong spin-orbit coupling.\nSpin transport and enhanced damping through spin\npumping24in ferrite/spin-orbit-metal structures has\nalready been extensively studied3,4,12{15,25. Moreover,\nthe low-damping ferrite can be interfaced with an\ninsulating antiferromagnetic or paramagnetic oxide, in\nwhich signals can be transmitted as a pure magnon\nspin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled\nprototypes of magnon spin valves37{39, the fundamental\nimpact of insulating oxide interfaces on spin dynamics\nhas remained mostly unexplored. In particular, it is an\nopen question whether or how damping of the ferrite is\nenhanced from spin dissipation within the bulk of the\nadjacent anti/paramagnetic oxide or from spin scattering\nat the oxide interface.\nHere, we investigate how room-temperature magnetic\ndamping in epitaxial ferrimagnetic spinel MgAl-ferrite\n(MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced\nwith an overlayer of insulating paramagnetic spinel\nCoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO\nbilayer is an isostructural model system, possessing\na coherent interface with continuous crystal lattices\nbetween the spinel ferrite and paramagnet. We \fnd that\nthe presence of MAFO/CCO interface increases damping\nby more than a factor of >3 compared to MAFO without\nan overlayer. We attribute this damping enhancement {\nwhich is comparable to or greater than spin pumping\ne\u000bects reported for ferrite/spin-orbit-metal bilayers { to\nspin scattering by the ultrathin ( \u00181 nm) chemically\ndisordered layer at the MAFO/CCO interface. Our\n\fndings show that spin scattering at oxide interfaces\nhas a profound in\ruence on damping, even when the\nchemically disordered layer is a small fraction of the total\nmagnetic layer thickness.\nII. FILM GROWTH AND STRUCTURAL\nCHARACTERIZATION\nEpitaxial thin \flms of 15-nm-thick MAFO interfaced\nwith 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2  [cond-mat.mtrl-sci]  26 Oct 20192\n40 42 44 46MAO (004)\nMAFO/CCO\n      (004)\nMAFO (004)log10(Intensity) (arb. units)\n2q (deg)CCO (004)\n-0.180 -0.175 -0.1700.570.580.590.600.610.620.63-\n \n   (115)-(a) (c)\nqip(Å-1)qop(Å-1)\nCCO\n(25 nm)MAO(b)\n-0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units)\nDw004 (deg)MAFO/CCO\nFigure 1. (a) 2 \u0012-!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are o\u000bset for\nclarity. (b) Rocking curve scans about the (004) \flm peak for the \flms shown in (a). (c) Reciprocal space map of epitaxial\nCCO(25 nm) coherently strained to the MAO substrate.\nreceived single-crystal MgAl 2O4(MAO) substrates via\npulsed laser deposition. A KrF 248 nm laser was\nincident on stoichiometric targets of MAFO and CCO\nwith \ruences of \u00191.5 J/cm2and\u00191.3 J/cm2,\nrespectively. Both \flms were grown in 10 mTorr (1.3\nPa) O 2and were cooled in 100 Torr (13 kPa) O 2.\nMAFO \flms were grown at 450\u000eC, whereas CCO \flms\nwere deposited at 300\u000eC in an attempt to minimize\nintermixing between the MAFO and CCO layers. These\ngrowth temperatures, much lower than >700\u000eC typically\nrequired for epitaxial garnets3{11, are su\u000ecient to fully\ncrystallize MAFO and CCO. The low crystallization\ntemperatures of the spinels o\u000ber an advantage over\nthe oft-studied garnets, with more opportunities for\nisostructural integration with coherent interfaces. The\nMAFO \flms exhibit a room-temperature saturation\nmagnetization of\u0019100 kA/m and a Curie temperature of\n\u0019400 K18. To obtain consistent ferromagnetic resonance\nresults, MAFO \flms were grown and subsequently\ncharacterized by ferromagnetic resonance (FMR) ex-situ;\nafter surface cleaning with ultrasonication in isopropanol,\nCCO overlayers were then deposited as described above.\nGrowth rates were calibrated via X-ray re\rectivity.\nOur structural characterization of MAFO and\nCCO shows high-quality, coherently strained \flms.\nIn symmetric 2 \u0012-!X-ray di\u000braction scans, only\npeaks corresponding to the (00 `) re\rections are\nobserved, indicating that the \flms are highly epitaxial.\nAdditionally, as seen in Fig. 1(a), Laue oscillations\naround the (004) Bragg re\rections in both single-layer\nMAFO and CCO layers as well as MAFO/CCO bilayers\ndenote smooth interfaces. Furthermore, MAFO, CCO,\nand MAFO/CCO samples all exhibit essentially the\nsame \flm-peak rocking curve widths (FWHM) of \u00190.06\u000e\n(Fig. 1(b)). Reciprocal space mapping of the ( \u00161\u001615)\nre\rection in 25-nm-thick single-layer CCO on MAO\n(Fig. 1(c)) reveals that the in-plane lattice parameter of\nthe \flm coincides with that of the substrate, indicating\nCCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between\nCCO and MAO of \u00193 %, coherently strained growth of\nCCO of up to 40 nm has been previously reported on\nMAO substrates41. For our CCO \flm, we calculate an\nout-of-plane lattice constant c\u00198:534\u0017A from the 2 \u0012-!\nscan; taking the in-plane lattice parameter a= 8:083\u0017A of\nthe MAO substrate, the resulting tetragonal distortion of\ncoherently strained CCO is c=a\u00191:055, similar to that\nfor coherently strained MAFO18.\nStructural characterization results underscore the\nquality of these epitaxial \flms grown as single layers and\nbilayers. Considering the comparable high crystalline\nquality for MAFO, CCO, and MAFO/CCO { as\nevidenced by the presence of Laue oscillations and narrow\n\flm-peak rocking curves { we conclude that MAFO/CCO\nbilayers (with the total thickness limited to \u001423 nm) are\ncoherently strained to the substrate. In these samples\nwhere the substrate and \flm layers are isostructural, we\nalso do not expect antiphase boundaries43{46. Indeed,\nwe \fnd no evidence for frustrated magnetism, i.e., high\nsaturation \feld and coercivity, that would arise from\nantiphase boundaries in spinel ferrites43{46; MAFO/CCO\nbilayers studied here instead exhibit soft magnetism, i.e.,\nsquare hysteresis loops with low coercivity <0.5 mT,\nsimilar to our previous report on epitaxial MAFO thin\n\flms18. Thus, MAFO/CCO is a high-quality all-oxide\nmodel system, which permits the evaluation of how spin\ndynamics are impacted by a structurally clean, coherent\ninterface.\nIII. FERROMAGNETIC RESONANCE\nCHARACTERIZATION OF DAMPING\nTo quantify e\u000bective damping in coherently strained\nMAFO(/CCO) thin \flms, we performed broadband\nFMR measurements at room temperature in a coplanar\nwaveguide setup using the same procedure as our prior\nwork16,18. We show FMR results with external bias3\nmagnetic \feld applied in the \flm plane along the [100]\ndirection of MAFO(/CCO); essentially identical damping\nresults were obtained with in-plane \feld applied along\n[110]47. Figure 2(a) shows the frequency fdependence of\nhalf-width-at-half-maximum (HWHM) linewidth \u0001 Hfor\na single-layer MAFO sample and a MAFO/CCO bilayer\nwith a CCO overlayer thickness of just 1.3 nm, i.e., less\nthan 2 unit cells. The linewidth is related to the e\u000bective\nGilbert damping parameter \u000beffvia the linear equation:\n\u0001H= \u0001H0+h\u000beff\ng\u00160\u0016Bf (1)\nwhere \u0001H0is the zero-frequency linewidth, his Planck's\nconstant,g\u00192:05 is the Land\u0013 e g-factor derived from the\nfrequency dependence of resonance \feld HFMR ,\u00160is the\npermeability of free space, and \u0016Bis the Bohr magneton.\nIt is easily seen from Fig. 2(a) that with the addition\nof ultrathin CCO, the damping parameter is drastically\nincreased, i.e., >3 times its value in bare MAFO.\nFigure 2(b) shows that the damping enhancement\nseen in MAFO/CCO is essentially independent of\nthe CCO thickness. This trend suggests that\nthe damping enhancement is purely due to the\nMAFO/CCO interface, rather than spin dissipation in\nthe bulk of CCO akin to the absorption of di\u000busive\nspin current reported in antiferromagnetic NiO26,35,48.\nWe note that other bulk magnetic properties of\nMAFO (e.g., e\u000bective magnetization, Land\u0013 e g-factor,\nmagnetocrystalline anisotropy) are not modi\fed by the\nCCO overlayer in a detectable way. We also rule\nout e\u000bects from solvent cleaning prior to CCO growth\nor thermal cycling in the deposition chamber up to\n300\u000eC, as subjecting bare MAFO to the same ex-\nsitu cleaning and in-situ heating/cooling processes as\ndescribed in Section II, but without CCO deposition,\nresults in no measurable change in damping. The\ndamping enhancement therefore evidently arises from the\nproximity of MAFO to the CCO overlayer.\nWe consider two possible mechanisms at the\nMAFO/CCO interface for the observed damping\nenhancement:\n(1) Spin current excited by FMR in MAFO\nmay be absorbed via spin transfer in an interfacial\nproximity-magnetized layer49of CCO, whose magnetic\nmoments may not be completely aligned with those of\nMAFO. While CCO by itself is paramagnetic at room\ntemperature, prior studies have shown that Co2+and\nCr3+cations in epitaxial CCO interfaced with a spinel\nferrite (e.g., Fe 3O4) can develop measurable magnetic\norder50. Such damping enhancement due to interfacial\nmagnetic layer is analogous to spin dephasing reported\nfor ferromagnets interfaced directly with proximity-\nmagnetized paramagnetic metal (e.g., Pt, Pd)49.\n(2) Even if CCO does not develop proximity-induced\nmagnetism, chemical disorder at the MAFO/CCO\ninterface may enhance spin scattering. For instance,\nchemical disorder may lead to an increase of Fe2+\n0 10 20 300246810HWHM Linewidth (mT)\nFrequency (GHz)(a)\n(b)MAFO/CCO\neff≈ 0.007\nMAFO\neff≈ 0.002\n0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff\nCCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency\nfor MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The\ne\u000bective Gilbert damping parameter \u000beffis derived from\nthe linear \ft. (b) \u000beffplotted against the CCO overlayer\nthickness. The dashed horizontal line indicates the average of\n\u000befffor MAFO without an overlayer.)\ncations at the MAFO surface, thereby increasing\nthe spin-orbit spin scattering contribution to Gilbert\ndamping in MAFO compared to its intrinsic composition\ndominated by Fe3+with weak spin-orbit coupling18,51.\nAnother possibility is that chemical disorder at the\nMAFO/CCO interface introduces magnetic roughness\nthat gives rise to additional spin scattering, perhaps\nsimilar to two-magnon scattering recently reported for\nferromagnet/spin-orbit-metal systems52.\nIn the following section, we directly examine interfacial\nproximity magnetism and chemical disorder to gain\ninsight into the physical origin of the observed damping\nenhancement in MAFO/CCO.\nIV. CHARACTERIZATION OF INTERFACE\nCHEMISTRY AND MAGNETISM\nTo evaluate the potential formation of a magnetized\nlayer in the interfacial CCO through the magnetic\nproximity e\u000bect, we performed depth-resolved\nand element-speci\fc magnetic characterization\nof MAFO/CCO bilayers using polarized neutron\nre\rectometry (PNR) and soft magnetic X-ray\nspectroscopy. PNR measurements were performed\nusing the PBR instrument at the NIST Center for\nNeutron Research on nominally 15-nm-thick MAFO\nlayers capped with either thick (5 nm) or thin (3 nm)4\nCCO overlayers. PNR measurements were performed in\nan in-plane applied \feld of 3 T at temperatures of 300\nK and 115 K, the latter case being slightly above the\nnominal 97 K Curie temperature of CCO41,42. Incident\nneutrons were spin-polarized parallel or anti-parallel to\nthe applied \feld both before and after scattering from\nthe sample, and the re\rected intensity was measured\nas a function of the perpendicular momentum transfer\nvector Q. The incident spin state of measured neutrons\nwere retained after scattering, corresponding to the\ntwo non-spin-\rip re\rectivity cross sections ( \"\"and##).\nSince all layers of the \flm are expected to saturate well\nbelow the applied \feld of 3 T, no spin-\rip re\rectivity is\nexpected and these cross sections were not measured.\nSince PNR is sensitive to the depth pro\fles of the\nnuclear and magnetic scattering length density (SLD),\nthe data can be \ftted to extract the chemical and\nmagnetic depth pro\fles of the heterostructure. In this\ncase, we used the Re\r1D software package for this\npurpose53. Figure 3(a,b) shows the 300 K re\rectivities\nand spin asymmetry curves of a nominal MAFO (15\nnm)/CCO (5 nm) sample alongside the depth pro\fle\n(Fig. 3(c)) used to generate the \fts shown. The\nbest \ft pro\fle (Fig. 3(c)) provides no evidence of a\nlayer with proximity-induced magnetization in the CCO.\nRather, we note that there appears to be a layer of\nmagnetization suppression near both the MAO/MAFO\nand MAFO/CCO interfaces. Further, the interfacial\nroughnesses of both the MAO/MAFO and MAFO/CCO,\n0.9(1) nm and 1.35(5) nm respectively, are signi\fcantly\nlarger than the CCO surface roughness of 0.27(3) nm\nand the bare MAFO surface roughness of <\u00180.5 nm54.\nThe interfacial roughnesses are signatures of chemical\nintermixing at the spinel-spinel interface leading to\ninterfacial suppression of the magnetization and/or Curie\ntemperature. Thus, we \fnd that the MAFO/CCO\ninterface, although structurally coherent, exhibits a\nchemically intermixed region on the order of one spinel\nunit cell thick on either side.\nTo obtain an upper limit of the proximity-induced\ninterfacial magnetization in CCO, we performed Markov-\nchain Monte-carlo simulations as implemented in the\nDREAM algorithm of the BUMPS python package.\nThese simulations suggest an upper limit (95% con\fdence\ninterval of) 7 emu/cc in the 1.5 nm of the CCO closest\nto the interface. In this case, the model evaluated the\nMAFO as a uniform structural slab but allowed for total\nor partial magnetization suppression at both interfaces,\nwhile the CCO layer was treated as a uniform slab with\nan allowed magnetization layer of variable thickness at\nthe interface.\nHowever, we note that equivalently good \fts are\nobtained using simpler models that \ft a single MAFO\nlayer with magnetically dead layers at the interfaces and\na completely nonmagnetic CCO layer. Equivalent results\nwere obtained for the thick CCO sample at 115 K and\nfor the thin CCO sample. We therefore conclude that the\nPNR results strongly favor a physical picture in which the\nFigure 3. (a) Spin-polarized neutron re\rectivity and (b)\nspin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer\nalongside theoretical \fts. (c) Nuclear and magnetic scattering\n(scaled \u000210) length density pro\fle used to generate the \fts\nshown. Error bars represent \u00061 standard deviation.\nCCO is notmagnetized through the magnetic proximity\ne\u000bect.\nTo con\frm the PNR results and examine the e\u000bect\nof a CCO overlayer on the local environment of Fe\ncations in MAFO, we performed temperature-dependent\nX-ray absorption (XA) spectroscopy and X-ray magnetic\ncircular dichroism (XMCD) measurements at Beamline\n4.0.2 of the Advanced Light Source at Lawrence Berkeley\nNational Laboratory. We note that the detection\nmode (total electron yield) used here for XA/XMCD\nis sensitive to the top \u00195 nm of the sample, such that\nFe L edge signals from CCO-capped MAFO primarily\ncapture the cation chemistry near the MAFO/CCO\ninterface. Measurements were performed in an applied\n\feld of 400 mT along the circularly polarized X-ray beam,\nincident at 30\u000egrazing from the \flm plane. To minimize\ndrift e\u000bects during the measurement, multiple successive\nenergy scans were taken and averaged, switching both\napplied \feld direction and photon helicity so that all\nfour possible combinations of \feld direction and helicity\nwere captured at least once. XA and XMCD intensities\nwere normalized such that the pre-edge is zero and\nthe maximum value of the average of the (+) and\n(\u0000) intensities is unity. In the case of the Co L-\nedge, measurements were taken with energy sweeps\ncovering both Fe and Co edges, and for consistency\nboth edges were normalized to the highest XAS signal,\ncorresponding to the Fe L 3-edge.\nFigure 4(a) compares the XA of a bare MAFO \flm5\nFigure 4. (a) 300 K X-ray absorption spectra of MAFO and\nMAFO/CCO (3 nm) grown on MAO. (b) Photon helicity-\ndependent XA spectra and XMCD of the Fe L-edge for a\nMAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr\nL-edge XA and XMCD of the same bilayer.\nwith one capped by 3 nm of CCO. The two XA lineshapes\nare nearly identical, indicating the same average Fe\noxidation state and site-distribution in CCO-capped\nand uncapped MAFO \flms. It is therefore likely that\nthe reduced interfacial magnetization observed through\nPNR is a result of a defect-induced Curie temperature\nreduction, rather than preferential site-occupation of Co\nand Cr that might increase the Fe2+content in the\nintermixed interfacial region.\nWe further note that although a large XMCD signal\nis observed on the Fe-edge at 300 K (Fig. 4(b)), neither\nthe Co nor Cr L edges exhibit any signi\fcant magnetic\ndichroism, as shown in Figs. 4(c)-(d). Similar results\nare obtained on the Cr L edge at 120 K. Consistent\nwith the PNR results, we thus \fnd no evidence for\na net magnetization induced in the CCO through the\ninterfacial magnetic proximity e\u000bect.\nOur \fnding of suppressed interfacial magnetism\nin MAFO/CCO is reminiscent of earlier reports\nof magnetic dead layers in epitaxially-grown ferrite-\nbased heterostructures55{57. For example, prior\nPNR experiments have revealed magnetic dead layers\nat the interfaces of ferrimagnetic spinel Fe 3O4and\nantiferromagnetic rock-salt NiO or CoO, even when the\ninterfacial roughness is small (e.g., only 0.3 nm)55,56.\nA magnetic dead layer of 1 spinel unit cell has also\nbeen reported at the interface of Fe 3O4and diamagnetic\nrock-salt MgO grown by molecular beam epitaxy57.\nWe note that in these prior studies, the spinel ferrite\flms interfaced with the rock salts (NiO, CoO, MgO)\npossess antiphase boundaries. Suppressed magnetism\nis known to result from antiphase boundaries, as they\nfrustrate the long-range magnetic order and reduce\nthe net magnetization of the ferrite44. By contrast,\nthere is no evidence for antiphase boundaries in all-\nspinel MAFO/CCO grown on spinel MAO; therefore,\nthe suppressed magnetism at the MAFO/CCO interface\ncannot be attributed to antiphase-boundary-induced\nmagnetic frustration.\nAnother possible scenario is that magnetic dead layer\nformation is a fundamental consequence of the charge\nimbalance between di\u000berent lattice planes, as recently\nshown in a recent report of (polar) Fe 3O4undergoing\natomic reconstruction to avoid \\polar catastrophe\" when\ngrown on (nonpolar) MgO58. In our study on all-\nspinel heterostructures, there may also be some degree of\ncharge mismatch depending on the relative populations\nof cations on the tetrahedrally- and octahedrally-\ncoordinated sites at the MAFO/CCO interface, although\nthe charge mismatch is expected to be only \u0019\u00061, i.e.,\na factor of\u00195-6 smaller than that in MgO/Fe 3O458.\nThus, atomic reconstruction driven by charge imbalance\nappears unlikely as a dominant source of the magnetic\ndead layer in MAFO/CCO. We instead tentatively\nattribute the dead layer to atomic intermixing driven by\ndi\u000busion across the MAFO/CCO interface during CCO\noverlayer deposition.\nV. DISCUSSION\nOur PNR and XA/XMCD results (Section IV) indicate\nthat the damping enhancement observed in Section III\narises from chemical disorder, rather than proximity-\ninduced magnetism, at the MAFO/CCO interface.\nWe emphasize that this interfacial disordered layer\nis con\fned to within \u00192 spinel unit cells. We\nalso note that this interfacial disorder is due to\natomic intermixing, but not structural defects (e.g.,\ndislocations, antiphase boundaries), in this coherent\nbilayer system of MAFO/CCO. Nevertheless, this\nultrathin chemically disordered layer alone is evidently\nsu\u000ecient to signi\fcantly increase spin scattering.\nConsidering that the cation chemistry of Fe in MAFO\ndoes not change substantially (Fig. 4(a)), the interfacial\nspin scattering is likely driven by magnetic roughness,\nleading to a mechanism similar to two-magnon scattering\nthat accounts for a large fraction of e\u000bective damping in\nmetallic ferromagnet/Pt bilayers52.\nWe now put in context the magnitude of the damping\nenhancement \u0001 \u000beff, i.e., the di\u000berence in the e\u000bective\nGilbert damping parameter between CCO-capped and\nbare MAFO,\n\u0001\u000beff=\u000bbilayer\neff\u0000\u000bferrite\neff; (2)\nby comparing it with ferrite/spin-orbit-metal systems\nwhere spin pumping is often considered as the source6\n0.0000.0020.0040.0060.008\n MAFO/CCO\n   [this study] MAFO/W\n [Riddiford] MAFO/Pt\n [Riddiford]YIG/Pt\n[Wang]Daeff\nYIG/Pt\n [Sun]\nFigure 5. Comparison of the enhancement of the e\u000bective\nGilbert damping parameter \u0001 \u000befffor MAFO/CCO and\nferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt\n[Wang], and MAFO/Pt(W) [Riddiford] are adapted from\nRefs.59,60, and61respectively. The values of \u0001 \u000befffrom the\nliterature are normalized for the saturation magnetization\nof 100 kA/m and magnetic thickness of 15 nm for direct\ncomparison with our MAFO/CCO result.\nof damping enhancement. Since damping enhancement\nfrom spin pumping or interfacial scattering scales\ninversely with the product of the saturation of\nmagnetization Msand the magnetic layer thickness tm,\nthe values of \u0001 \u000befftaken from the literature59{61are\nnormalized for direct comparison with the MAFO \flms\nstudied here with Ms= 100 kA/m and tm= 15 nm.\nAs summarized in Fig. 5, \u0001 \u000befffor MAFO/CCO\nis comparable to { or even greater than { \u0001 \u000beff\nfor ferrite/metal bilayers. This \fnding highlights that\nthe strength of increased spin scattering in a ferrite\ndue to interfacial chemical disorder can be on par\nwith spin dissipation due to spin pumping in metallic\nspin sinks. More generally, this \fnding suggests that\nspecial care may be required in directly relating \u0001 \u000beff\nto spin pumping across bilayer interfaces (i.e., spin-\nmixing conductance52), particularly when the FMR-\ndriven magnetic layer is directly interfaced with a spin\nscatterer.\nFurthermore, the strong interfacial spin scattering {\neven when the oxide interface is structurally coherent\nand the chemically disordered layer is kept to just <\u00182\nunit cells { poses a signi\fcant challenge for maintaining\nlow damping in ferrite/insulator heterostructures. This\nchallenge is partially analogous to the problem of reduced\nspin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also\nlikely due to interfacial chemical disorder and magnetic\ndead layers. However, we emphasize that the problems of\nantiphase boundaries43{46and charge-imbalance-driven\natomic reconstruction58, which have posed intrinsic\nchallenges for devices with MgO/Fe 3O4interfaces, are\nlikely not applicable to all-spinel MAFO/CCO. It is\ntherefore possible that deposition schemes that yield\nsharper interfaces, e.g., molecular beam epitaxy, can be\nemployed to reduce interfacial imperfections and hence\nspin scattering at MAFO/CCO for low-loss all-oxide\ndevice structures.\nVI. CONCLUSIONS\nWe have shown that e\u000bective damping in epitaxial\nspinel MgAl-ferrite (MAFO) increases more than\nthreefold when interfaced coherently with an insulating\nparamagnetic spinel of CoCr 2O4(CCO). This damping\nenhancement is not due to spin pumping into the\nbulk of CCO. Our depth-resolved characterization of\nMAFO/CCO bilayers also reveals no proximity-induced\nmagnetization in CCO or signi\fcant change in the\ncation chemistry of MAFO. We attribute the giant\ndamping enhancement to spin scattering in an ultrathin\nchemically disordered layer, con\fned to within 2 spinel\nunit cells across the MAFO/CCO interface. Our results\ndemonstrate that spin dynamics in ferrite thin \flms are\nstrongly impacted by interfacial disorder.\nAcknowledgements - This work was supported in\npart by the Vannevar Bush Faculty Fellowship program\nsponsored by the Basic Research O\u000ece of the Assistant\nSecretary of Defense for Research and Engineering and\nfunded by the O\u000ece of Naval Research through grant\nno. N00014-15-1-0045. 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Li thium ferrite and intermetal compound MnSb  \ncharacterized by the absence of single-pulse echo in them – belong to the second type.   \nFor signals of two-pulse stimulated echo in the materials of the first type a short time \nand a long time of relaxations are observed. The short time is about the order of value shorter less than the spin-spin relaxation time.  The long time is close to the transverse \nrelaxation time of single-pulse echo formed by the distortion mechanism. The \nmechanisms that provide the possible interpreta tions of the peculiarities of the processes \nof nuclear magnetic relaxation are discussed. \n Key words:  stimulated echo, magnets, relaxation processes. \n  \n  \n 2INTRODUCTION \n \nOne of the main ways to measure of the longitudinal (spin-lattice) relaxation 1T and \ninvestigation of spin-diffusion processes is th e method of stimulated spin echo [1]. In the \ntraditional Hahn mechanism of spin-echo formati on for the observation of stimulated echo it is \nnecessary to have at least three resonant RF pulses [1]. The possibility to observe the stimulated \necho is related as it is known with fact that the z-component of magnetization of j -th \nisochromate after the termination of the second RF  pulse depends on the phase of its transverse \ncomponent at the start of the second pulse 12τωϕ⋅Δ=j j , where jωΔ is the resonant frequency \nof j-th isochromate in the system of coordinates rotating with the RF pulse filling frequency \nRFω, 12τ.- is the interval between the first and the second RF pulses. \nIn work [2] was developed the SPE theory for Hahn spin systems was developed, i.e. the pulse \nresponse of spin-system occurring at the action of one wide RF pulse at time approximately \nequal to the duration of the exciting RF pulse. \nIn following the theory was developed in a number of works, among which let us note [3-7] \nones. It was shown that using the SPE method on e could obtain practically the same information \non spin-system as by two-pulse echo (TPE) Hahn method. But it was also noted [2] that the \ntransverse relaxation time ST2, defined using the TPE method due to the difference of effective \nmagnetic fields in the rotating coordinate system (RCS) in the dephasing and rephasing \nprocesses of nuclear isochromates, as  result of which angular rates of j-th isochromate \nprecession during the RF action and after it s termination are different and equal to, \ncorrespondingly, ()2/12\n12 'ωωω +Δ=Δj j  and RF j jωωω−=Δ ,  jω is the NMR frequency of j-\nth isochromate, 1 1 Hnηγω=  is the pulse amplitude in the frequency units, η is the enhancement \nfactor and nγ is the gyromagnetic ratio. Due to the '\nj jωωΔ≠Δ ′ inequality the SPE signal could \nbe observed only at fulfillment of specific condit ions, pointed in [2]. Besides it, the transverse \nrelaxation time ST2 could turn out to be shorter than 2T measured by the TPE method. \nThe formation mechanism of SPE makes it possi ble also the observation of TPSE in case of \napplication of a short additional (r eading) RF pulse [2,8]. The point is in the fact that during the \nRF pulse action the inclination of isochromate precession axis (that causes the trajectory in the \nxy plane to be ellipsoidal and produces the SPE signal) makes also the z-component of the \nisochromate dependent on the phase of its transver se component at the install of termination of \nthe first pulse 1' 'τωϕ ⋅Δ=j j . Thus, the difference of inclinations of effective magnetic fields in \nthe PCS during the action of after the termination of RF pulse causes the formation of the SPE \nand the possibility to observe the TPSE using onl y two, wide and short, exciting RF pulses. \nFollowing to [2, 6], it is easy to find out the expression for the TPSE intensity. \nAssume system of nuclear spins in an equilibriu m state is excited by the RF pulse of duration \n1τ, amplitude 1H and frequency RFω. The motion of j-th isochromate magnetization vector \njmrin RCS constitutes then prec ession around an effective field \n()j k Hj n err r\n11ωωγ +Δ=−,                                                       (1)  \n 3which is described by a system of equations (f or simplicity, the relaxation processes are not \ntaken into account): \nxj zj xjj yj zj yjj xj m m m m m m m1 1 , , ω ω ωω = Δ−= −Δ= & & & .                              (2) \njkrr\n,  are unit vectors of the RCS along z and y axes, correspondingly. \nFrom the solution of system (2) under the equilibrium initial conditions m m m mzj yj xj === ,0  \n[2] it follows that the longitudinal nuclear magnetization after the pulse is equal to \n( )j j j zj t m m θ ωθ2 ' 2cos cos sin +Δ = ,                                                (3) \nwhere m is the equilibrium value of the nuclear magnetization, jθ is the angle between eHr\n (1) \nand z axes and j jωωθ ′Δ=/ sin1 . \nLet us consider the effect of the second (reading) RF pulse in conditions \n1 12 2 2 T T T <<<<∗τ , \nwhere 2T is the transverse irreversible relaxation time and ∗\n2T characterizes the transverse \nreversible relaxation time (∗ ∗Δ/1~2T , where ∗Δ is the half-width of half-maximum of the \ninhomogeneously broadened NMR line). At implementation of these conditions after elapsing \ntime 12τ before the application of reading RF pulse it remains only the longitudinal component \nof nuclear magnetization \n()( )j j j zj m m θτωθ τ2\n1' 2\n12 cos cos sin +Δ = . \nA sufficiently short second RF pulse, during whic h the relaxation processes and inhomogeneous \nbroadening do not play role, turns the longitudinal magnetization around the x axis on angle α \nso that just after the application of reading pulse the transverse component of magnetization is \nequal to \n() ( )j j j yj m m θτωθα τ2\n1' 2\n12 cos cos sin sin +Δ = .                                      (4) \nThe further calculation of the TPSE intensity is similar one carried out in [2]. The contribution \nto the TPSE intensity gives the first term reflecting the dependence of z-component of j-th \nisochromate magnetization on the phase of its tr ansverse component in the time moment, of the \nextinction of the first pulse 1' 'τωϕ ⋅Δ=j j . Allowing for the fact that the SPE formation \nmechanism is the most effective at the nonresonant excitation of spin system when the condition \n1 0ωω>>Δ  [2] is fulfilled one could obtain for the TPSE intensity the expression \n⎭⎬⎫\n⎩⎨⎧−\n⎭⎬⎫\n⎩⎨⎧−⋅Δ≈112\n212\n01exp 2 exp sin21\nT Tcm Istτ τ\nωωαη ,                              (5) \nwhere c is a coefficient that takes into account the geometry of experiment, the time of TPSE \nappearance, if time is reckoned from the end of th e auxiliary pulse, is approximately equal to the \nduration of the first RF pulse 1τ.  \n 4Thus, optimum conditions of observation of TPSE of the reading RF pulse should be resonant \nand of 90о, similar to Hahn mechanism [1]. \nFor the intensities of TPSE and SPE are valid the following simple assessments [2]: \n2\n01~ωω\nΔTPSEI  and 3\n01~ωω\nΔSPEI , correspondingly. \nEarlier the TPSE for the considered ease was observed for 151Eu  nuclei in a polycrystal sample \nof ferrite-garnet 125 3OFeEu  at temperature 1.7 К [8]. The calculation carried out in frames of \nnonresonant mechanism [2] showed that for a suffici ently short reading RF pulse of the TPSE it \ncould be observed three more spin echo signals in time instants 12τ and 1 12ττ+, formed by the \ntraditional Hahn mechanism. The echo signals intensity in instants 12τ and 1 12ττ+ are \nproportional to \n01\nωω\nΔ, but the echo intensity in instant 1 12ττ− is proportional 2\n01\nωω\nΔ, small \nand for this reason was not observed in work [8]. \nThe TPSE signals were observed in powdered NiNiFeFe61 57,  and VFe51 samples [9]. The \nTPSE signal at 2 1ττ≤ was observed at the distance 1τ from the back front of the second pulse \nand its position was independent of the time interval 12τ between the exciting pulses. In case \nfrom the front of the second pulse at 2 1ττ> two components were observed at the distance 1τ \nfrom its back front. The SPE signal in this work was not observed. In work [10] in the same \nsystems it was studied also the signal of main  or multiple echo possessing the multicomponent \nstructure and linear dependence of the appearing time of one of its main components (2C \ncomponent in designations of work [10]) on the duration of exciting pulse. It was noted \nunusually short relaxation time of this component as compared with other one (1C component) \nof multiple echo, which coincides with the signal of usual TPE in the limit of short pulses. The \nnature of multiple echo components was studied in work [11] on the example of Fe57 NMR in \nlithium ferrite where it was simultaneously intensive signals of SPE, TPSE and multiple echoes \n(1E, 2E, 1C and 2C signals [10]). In work [11] it was revealed the synchronous change of 1E, \n2E, and 2C echo signals intensities depending on τ1, pointing to the related mechanisms of \ntheir formation. Besides it, it was established the nature of short relaxation time for 2C \ncomponent. This time appeared to be close to the relaxation time of 1E component which was, \nas it was earlier established, almost on two orders of value shorter as compared with transverse \nrelaxation time 2T, determined from the Hahn TPE method (ST2=60 μs and 2T=1200 μs, \ncorrespondingly [12]). Further, in work [13] it was studied the dynamics of 1E and 1C \ncomponents in polycrystalline sample of ferromagnetic Co where the ratio of relaxation times \nof this components appeared to be different and was defined by the longer relaxation time of \nSPE in cobalt, formed by the distortion mechanism: namely, ST2 = 0.4 ÷ 0.7 2T, where 2T = 60 \nμs. \nIn this work we present the results of the similar investigations of TPSE nature in polycrystal \nsamples of lithium ferrite, cobalt, intermetal MnSb  and half metal MnSiCo2  where signals of \nSPE could be formed by nonresonant or distortion mechanisms of by the both mechanisms \nsimultaneously.  \n 5It will be studied in more details the formation mechanisms of SPE and TPSE in systems where \nthe investigated SPE signals where formed by the distortion mechanism. \nIn work [14] it was studied experimentally and theoretically the peculiarities of the distortion \nmechanism at formation of SPE in polycrystalline films of Co at helium temperatures. The \ninfluence of transition processes on the fronts of RF pulse was visualized directly by the \nsupplying signal from the spectrometer resonato r to the input of the high-frequency pulsed \noscilloscope S-75. But in work [13] it was suggested other, more effective method for \nvisualization of the influence of transition proc esses on the RF pulse fronts by application of \nadditional magnetic video-pulse (MVP) field, allowing more detailed study of RF pulse \ndistortion structure. The method consists in the comparison of timing influence of MVP on SPE \nand TPSE signals and is calculated in Fig. 4 in work [13] on the example of lithium ferrite. \nThe analysis of timing diagrams shows the definite analogy between the MVP influence on the \nSPE and TPSE signals in lithium ferrite. \nThe corresponding timing diagrams for the MVP influence for Co and MnSiCo2  are \ninvestigated in work [15] and confirm conclusions of work [13] for magnets where the SPE \nsignals are formed by the distortion mechanism.  \nOn the basis of obtained picture of the MVP influence one could make conclusion on the \ndefinite duration of transition proc ess regions in the resonant circuit of spectrometer and assume \nthat at decreasing of the PF pulse duration it is possible their overlapping resulting in the \nsimplification of pulse structure, approaching it to rectangular shape when disappeared the edge \nstructure of RF pulse revealing at application of MVP [15]. In this range of RF pulse duration it \ncould take place the change of one SPE format ion mechanism on other being characteristic for \nsystems without the observable RF pulse distortion where the SPE signal is absent at resonant excitation as it is the case in lithium ferrite [6]. \nWith this aim it was carried out corresponding measurements of SPE and TPSE intensities in \nthe range of RF pulse duration down to short one, close to the recovery time of the spectrometer \ncharacterizing the transient loss of its sensitivity following the RF burst which in our case \nconstitutes value of the order of ~ 1 \nμs [13]. \n \nEXPERIMENTAL RESULTS AND DISCUSSIONS \n \nA standard phase-incoherent spin-echo spectrometer was employed for measurements in \nfrequency range 40-400 МHz at temperature 77 К. In frequency range 40-220 МHz a standard \nself-excitation RF oscillator has been used. Th e frequency of oscillator could be gradually \nretuned by using a number of circuits with diffe rent inductance coils and adjustable capacities. \nIn the frequency range 200-400 МHz it was used a commercial manufactured oscillator based \non the two-wire Lekher-type line including two coils with different numbers of turns. For pulse \nlengths ranging between 0.1 and 50 μs a maximum RF field produced of the sample was \nestimated to be about 3.0 Oe, while the rise and fa ll times of RF pulse fronts were no more than \n0.15 μs. The recovery time of the spectrometer characterizing the transient loss of its sensitivity \nfollowing the RF burst, was about ~ 1 μs. For investigation of lit hium ferrite the resonant \nsystem of spectrometer was modified similar to described in work [12] allowing us to increase \nsharply its sensitivity as compared with one in work [13].  \n 6It was used the circular discs of dielectric li thium ferrite and its solid solutions with zinc \n4 5.2 5.0 OZn FeLix x−  (0≤χ≤0.25) enriched by isotope Fe57 (96.8 %) and also hexagonal \npolycrystalline cobalt and half metallic MnSiCo2  for NMR investigations of Co59 and Mn55 \nnuclei. Half metals are interesting for applicatio ns in spintronics [16]. The properties of all \nabove noted samples are described in details in [13]. \n \nFig. 1.  Oscilloscope signals of SPE, TPSE and multiple echo (1E, 2E and 1C, 2C, correspondingly) in lithium \nferrite (а) and cobalt (b). Lower beams show the RF pulse shape, amplitude and duration at: \na) 1τ=14 μs, 2τ=1.2 μs, 12τ=40 μs, NMRf =71.6 МHz, T=77К; \nb) 1τ=12 μs, 2τ=2.5 μs, 12τ=22 μs, NMRf  = 217 МHz, T=77К. \nIn Fig.1 it is presented oscilloscope signals of  SPE, TPSE and multiple echo following the \napplications of two pulse of optimal power in li thium ferrite and cobalt, but in Fig. 2-6 it is \nshown the corresponding relaxation dependences for TPSE, SPE and TPE in lithium ferrite, \ncobalt, MnSb  and MnSiCo2 , correspondingly, obtained by changing the duration of the first \nwide RF pulse and at fixed value of 12τ. It should be noted also that SPE signals in lithium \nferrite (Fig. 7) and intermetal MnSb  were not observed in the limit of single-pulse excitation by \na single RF pulse (when pulse repetition period is long as compared with 1T) in the conditions \nof our experiment. \n \n \nFig. 2 . а) Intensities of TPSE intensity at single excitation by a pair of RF pulses (1), and also the TPSE (2) and \nTPE (3) at the repetition rate of pair and ,correspondingly single RF pulses pf =100 Hz depending on the duration \nof the first RF pulse 1τ (at 2τ=1 μs) in lithium ferrite. The frequency of Fe57 NMR NMRf  =71.7 МHz and T  \n 7=77 К. b). The intensity of TPE depending on the duration of time interval between RF pulses. The repetition rate \nof pairs of RF pulses pf =100 Hz and ==2 1ττ 1μs, NMRf =71.7 МHz and T =77 К. \n \nFig. 3 . Intensities of TPSE (1) and SPE (2) depending on the duration of the first RF pulse 1τ (at 2τ=1 μs), and \nalso TPE (3) on the interval between pulses 12τ (at ==2 1ττ 1 μs) in cobalt at NMRf  =217 МHz, T =77 K \n \nFig. 4 . Intensities of TPSE of Mn55 nuclei in MnSb  depending on the duration of the first RF pulse 1τ (at 2τ=1 \nμs, 12τ=15 μs) (1) and the TPE depending on the interval between RF pulses 12τ (at ==2 1ττ 1 μs) (2) at \nNMRf =225 МHz and T =77 K. \n  \n 8\n \nFig. 5 . Intensities of TPSE (1) and SPE (2) depending on the duration of the first RF pulse 1τ (at 2τ=1 μs, 12τ=27 \nμs) and also TPE (3) on the interval between RF pulses 12τ (==2 1ττ 1.2 μs) in MnSiCo2  for Co59 nuclei, at \nNMRf  =142 МHz, T =77 K. \n \nFig. 6 . Intensities of TPSE (1) and SPE (2) depending on the duration of the first RF pulse 1τ (at 2τ=1 μs, 12τ=30 \nμs), and also TPE (3) on interval between RF pulses 12τ (at ==2 1ττ 1.2 μs) at repetition rate of RF pulse pair: \na) 5 Hz and b) 50 Hz in MnSiCo2  for Mn55 nuclei, NMRf  =353 МHz, T =77 К. \n \nFig. 7 . Intensities of SPE (1) and TPSE (2) depending on the repetition rate F of pulse train in lithium ferrite. For \nSPE - NMRf  = 71 МHz and pτ = 8.5 μs (pττ=1 – RF pulse duration), and for TPSE - NMRf  =71 МHz, 1τ= 15 \nμs, 2τ= 1.2 μs and 12τ = 60 μs [11].   \n 9Thus, the observed in the present work fast relaxation processes for TPSE are apparently. As \nseen from the corresponding graphs (Fig. 3, 5), the assumption on the possibility of changing \nthe SPE formation mechanisms at decreasing the duration of first RF pulse could turn out right, as the experimental dependences of the TPSE intensity on the duration of the first RF pulse do \nnot have any peculiarities in the range of \n1τ when the SPE intensity starts to reduce in the case \nof spin echo of Co59 nuclei. Besides it, the observed relaxation dependences could be \napproximated by the two relaxation processes: the first one being related with the distortion \nmechanism, and the second one characterized by the much shorter relaxation time most \nprobably could be related with the nonresonant mechanism [2]. Let us note, that the characteristic hump-like shape of SPE signals is observed only at recording echo signals from \nCo59 nuclei in Co and MnSiCo2 , but it is absent in the case of SPE signals from Mn55 and \nFe57 nuclei what could be caused by the contri bution of anisotropic part of HF interaction \nwhich is much stronger in the case of Co59 nuclei. \nSimilar ones firstly experimentally revealed in  work [11] and they could be understood in the \nframes of the nonresonant mechanism [2, 12]. \nThe suggested approach could turn out to be useful for the clearing out the SPE formation \nmechanism in systems where the distortion mechanism is effective. Besides it, this approach could help to advance in the understanding of the problem in what degree the SPE properties in \nsuch systems are defined by the physical properties of systems under study as example, are there \nmetals of magnetic dielectrics – ferrites), or by  the residual RF pulse distortions caused by \ntransition processes in spectrometers. \nIn conclusion, let us note that we studied the T PSE in different magnets, such as lithium ferrite \n4 5.2 5.0 OZn FeLix x−  and intermetal MnSb  when the SPE signals are not observed in conditions of \nsingle-pulse excitation, and also on such ones where the SPE signals are observed in the same \nconditions and are formed by the distortion mechanisms – in ferrometal cobalt and in the half \nmetal MnSiCo2 . \nThe TPSE relaxation processes in these magnets have two-component character. One of \ncomponents has relaxation rate approximately on  the order of value exceeding the transverse \nrelaxation rate measured by TPE, but another one has the relaxation value close to the transverse \nrelaxation rate of SPE in systems where the SPE signal is formed by the distortion mechanism. \nBesides it, the experimental results point to the possible role of the HF anisotropy in the \nformation of the characteristic hu mp-like dependence of SPE from Co59 nuclei in Co and \nMnSiCo2  on the duration of RF pulse. \n \n \n \n \n \n \n  \n 10REFERENCES \n \n1. A. Lösche, Kerninduktion (Deutscher Verlag der Wissenschaften, Berlin, 1957; Inostrannaya \nLiteratura, Moscow, 1963). \n2. V.P. Chekmarev, M.I. Kurkin, S.I. Goloshapov. Sov. Phys. JETP. 49, 851-856 (1979). \n3. R. Kaiser. J. Magn. Res. 42,103-109 (1981). \n4. A. Ponti Molecular Physics. 95, 943-955 (1998). \n5. L.N. Shakhmurato et al., Phys. Rev. A. 55, 2955-2967(1997). \n6. A.M. Akhalkatsi et al., Phys. Met. Metallogr. 9, 10–46(2002). \n7. L.L. Buishvili et al., Sov. Phys. Solid State 24, 1808-1810(1982).  \n8. G.I. Mamniashvili and V.P. Chekmarev Bullet. of the Georgian Academy of Sciences. 103, \n285-287(1981). \n9. D.K. Fowler et al., Phys. Status Solidi. 92, 545-553(1985). \n10. R.W.N Kinnear et al., Phys. Lett. A. 76, 311-314(1980). \n11. A.M. Akhalkatsi et al., J. Appl. Phys. 105, 07D303-1 - 07D303-3(2009). \n12. A.M. Akhalkatsi et al., Phys. Lett. A 291, 34–38(2001). \n13. I.G.Kiliptari and V.I. Tsifrinovich, Phys. Rev. B. 57, 11554–1154(1998). \n14. V.I. Tsifrinovich et al., Sov. Phys. JETP. 61, 886-694(1985). \n15. A.M. Akhalkatsi et al., Physics of Metals and Metallography. 105,351-355(2008). \n16. J.S. Moodera, Phys. Today. 54, 39-44(2001). \n \n \n \n \n \n "    },    {        "title": "0807.4281v1.Magnetic_dipolar_and_electromagnetic_vortices_in_quasi_2D_ferrite_disks.pdf",        "content": "Magnetic-dipolar and electromagnetic vortices in quasi-2D ferrite disks \n \nM. Sigalov, E.O. Kamenetskii, and R. Shavit \n \nBen-Gurion University of the Negev, Beer Sheva 84105, Israel \n \nJune 19, 2008 \n \nAbstract  \nMagnetic-dipolar-mode (MDM) oscillations in a quasi-2D ferrite disk show unique dynamical \nsymmetry properties resulting in appearance of topologically distinct structures. Based on the \nmagnetostatic (MS) spectral problem solutions, in this paper we give an evidence for eigen MS \npower-flow-density vortices in a fe rrite disk. Due to these circular  eigen power flows, the MDMs are \ncharacterized by MS energy eigen states. It becomes ev ident that the reason of stability of the vortex \nconfigurations in saturated ferrite samples is completely different from the nature of stability in magnetically soft cylindrical do ts. We found a clear correspondence between analytically derived \nMDM vortex states and numerically  modeled electromagnetic vortices  in quasi-2D ferrite disks.  \n \nPACS numbers: 76.50.+g, 68.65.-k, 03.65.Vf \n \nI. INTRODUCTION \n \nIn ferromagnetic systems, one can clearly distinguish three characteristic scal es. There are the scales \nof the spin (exchange interaction)  fields, the magnetostatic (dipole- dipole interaction) fields, and the \nelectromagnetic fields. These fields  may define different vortex st ates, which are ch aracterized by \ndifferent kinds of physical phenomena and are obs erved in ferromagnetic samples with different \nsizes. Together with magnetizat ion vortices in magnetically so ft samples [1 – 3] and the \nmagnetostatic (MS) vortex behaviors in saturated fe rrite disks [4], one can observe electromagnetic \nvortices originated from ferrite samples in microwav e cavities [5 – 8]. In the last case, vortices \nappear because of the time-reversal -symmetry-breaking (TRSB) effects resulting in a rich variety of \nthe electromagnetic wave topological phenomena [5  – 8]. In spite of the fact that vortices are \nobserved in different kinds of physic al phenomena, yet such \"swirling\"  entities seem to elude an all-\ninclusive definition. In quite a number of problems one  can define a vortex as a circular flow which \nis attributed with a certain phase factor and a circular integral of a gradient of the phase gives a non-\nzero quantity. This quantity is multiple to a number of full rotations.      In a magnetically soft cylindr ical dot of micrometer or submic rometer size, the vortex structures \nare stable because of competition between the exchange and dipole interactions [1 – 3]. These stable vortex configurations are consider ed as very attractive objects fo r fundamental physics studies and \nfor potential use as a unit cell fo r high density magnetic storage a nd magnetic logic devices. The \nvortex structures in magnetically saturated cylindri cal samples are characterized by negligibly small \nexchange fluctuations [4]. It becomes evident that the reason of stability of the vortex configurations \nin saturated dots should be completely different from the nature of stability in magnetically soft cylindrical dots. The magnetic di pole interaction provides us wi th a long-range mechanism of \ninteraction, where a magnetic medium is consid ered as a continuum. It is well known that \nmagnetostatic (MS) ferromagnetism has a char acter essentially diffe rent from exchange \nferromagnetism [9, 10]. This statement finds a st rong confirmation in confinement phenomena of \nmagnetic-dipolar-mode (MDM) (or MS) oscillations. MDM problems border with the exchange- 2interaction magnetization dynamics, on one hand, and with the electromagnetic  behaviors, on the \nother hand. The feature of MDM osci llations in small samples with strong temporal dispersion of the \npermeability tensor: )( ωµµtt= , is the fact that one neglects va riation of the exchange energy and \nvariations of the electric energy [11]. In the attempts to consider magnetic-dipolar interactions in \nferrite samples with the dipolar and exchange ener gy competition, one usually  solves \"pure static\" \nmagnetostatic (MS) equations for the dipolar fi eld [12 – 14]. It should be taken into account, \nhowever, that the magnetic-dipolar interaction provides us with a l ong-range mechanism of \ninteraction, where a magnetic medium is considered  as a continuum and to get correct solutions for \nMS dynamical processes inside a ferrite particle  one has to presuppose existence of certain \nretardation effects  for the MS fields. The magnetic samp les which exhibit the MS resonance \nbehaviors in microwaves are described by the MS-potential wave functions  ) ,(trrψ .  \n    A recently published spectral theory of MDMs in quasi-2D ferrite disks [4, 15, 16] gives a deep \ninsight into an explanation of the experimental  MS multiresonance absorption spectra shown both in \nwell known previous [17, 18] and new [19, 20] studies. The theory suggests an existence of \nothogonality relations for the MS-pot ential eigen modes. In a quasi -2D ferrite disk, a differential \nequation for this scalar complex wave function rese mbles the Schrödinger equation. It follows that \nthe MDM spectral problem in a normally magneti zed ferrite disk is characterized by energy \neigenstates [15, 16].  The energy orthogonality for MDMs in a ferrite  disk is accompanied with the \ndynamical symmetry breaking effects resulting in appearance of the vortex structures [4]. This \nshows that the vortex states can be created not onl y in magnetically soft \"small\" (with the dipolar \nand exchange energy competition) cylindrical dots, but also in magnetically saturated \"big\" (when the exchange fluctuations can be neglected) cylindrical samples.      Together with the exchange-interaction and magnetic-dipolar vortex be haviors, electromagnetic \n(Poynting-vector) vortices can be observed in fe rromagnetic systems. A ferrite is a magnetic \ndielectric with low losses. This may allow for electromagnetic waves to penetrate the ferrite and results in an effective interaction between the electromagnetic waves and magnetization within the ferrite. At the ferromagnetic resonance conditions (F MR), such an interaction can demonstrate very \ninteresting physical features. Because of inserti ng a piece of a magnetized ferrite into the resonator \ndomain, a microwave resonator behaves under odd-time -reversal symmetry. Solutions of the fields \nin microwave systems with enclos ed ferrite samples are considered  as solutions of non-integrable \nMaxwellian problems. In this case a ferrite sample  may act as a topological  defect causing induced \nelectromagnetic vortices. It was sh own numerically [6] that the pow er-flow lines of the microwave-\ncavity field interacting with a ferrite sample, in  the proximity of its FMR, forms whirlpool-like \nelectromagnetic vortices. It was found that in su ch non-integrable structures, magnetic gyrotropy and \ngeometrical factors lead to spec ial effects of dynamical symmetry breaking resulting in effective \nchiral magnetic currents and topol ogical magnetic charges [7, 8].  \n    Based on the magnetostatic (MS) spectral problem solutions, in this paper we show the presence of eigen MS power-flow-density vortices in a ferr ite disk. Such circular eigen power flows give \nevidence for the MDM energy eigen states. This resu lts in stability of the vortex configurations in \nmagnetically saturated dots. We analyze the MDM fi elds in a ferrite sample. Based on a numerical \nanalysis we show that for certain geometry of a ferrite disk and certain  FMR frequency and bias-\nmagnetic-field regions one  has strong pronounced eigenfunction patterns  of the topologically \ndistinct electromagnetic vortex stru ctures. The spectral properties of MDM oscillations in a ferrite \ndisk cannot be analytically described based on the complete-set Maxw ell equations. So the \ndynamical processes for the MS fields arising fr om the MS-potential-wave-function description \ncannot be formally considered as (and reduced to) the effects obtained from the complete-set \nMaxwell equations. The Maxwell-e quation mode eigenstates in a ferrite disk are numerically \nmodeled objects in the ray phase space. The fact th at such non-integrable-electromagnetic-problem \nsolutions are in clear correspondence with the an alytical spectral soluti ons for the MDM vortex \nstates in quasi-2D ferrite disks is one of the most interesting phenomena shown in this paper.  3II. MDM VORTICES IN Q UASI-2D FERRITE DISKS \n A. MDM spectral problem in a quasi-2D ferrite disk  \nGenerally, in classical electrom agnetic problem solutions for time-varying fields, there are no \ndifferences between the methods of solutions : based on the electric- and magnetic-field \nrepresentation or based on the potential represen tation of the Maxwell equations. For the wave \nprocesses, in the field representation we solve a system of first-order part ial differential equations \n(for the electric and magnetic vect or fields) while in the potential representation we have a smaller \nnumber of second-order differential equations (for the scalar electric  or vector magne tic potentials). \nThe potentials are introduced as formal quantities  for a more convenient way to solve the problem \nand a set of equations for potentials are equivalent in all respects to the Maxwell equations for fields \n[21]. The situation can become co mpletely different if one supposes  to solve the electromagnetic \nboundary problem for wave processes in small samples of a strongly temporally dispersive magnetic \nmedium [11]. For such magnetic samples with th e MS resonance behaviors in microwaves, the \nproblem cannot be formally reduced to the comple te-set Maxwell-equation re presentation [4, 15, 16] \nand one becomes faced with a special role  of the MS-potential wave function ) ,(trrψ . This scalar \nwave function acquires a physical meaning in the MD M spectral problem and results in experimental \nobservation of energy shifts of oscillating m odes and eigen electric moments [17 – 20].  \n    For a quasi-2D ferrite disk  with cylindrical coordinates θ,,rz ,  one uses separation of variables \nfor MS-potential wave function and a spectral problem is formulated  with respect to membrane MS \nfunctions (described by coordinates θ,r). In solving the spectral probl em for MDM oscillations, one \ndeals with two types of differential-ope rator equations. The first type is  \n \n                                                               () 0~~\n ˆ ˆ =\n\n\n\n−⊥ϕβBRi Lr\n,                                                             (1) \n \nwhere ϕ~ is a dimensionless membrane MS-potential wave function, B~r\n is a dimensionless \nmembrane function of a magnetic flux density. For  a normally magnetized disk characterizing by a \ndiagonal, µ, and off-diagonal, aµ, components of the permeability tensor µt, the components of \nvector B~r\n are: θϕµϕµ∂∂+∂∂=~ ~~\na r irB   and ri Ba∂∂−∂∂=ϕµθϕµθ~ ~~. In Eq. (1), ⊥Lˆ is a differential-matrix \noperator: \n \n                                                              ()\n\n\n\n\n⋅∇−∇≡\n⊥⊥−\n⊥\n⊥0ˆ1µt\nL ,                                                             (2) \n \n(subscript ⊥ means correspondence with the in-plane, θ,r, coordinates), β is the MS-wave \npropagation constant along z axis )  ~( zieβϕϕ−≡ , and Rˆ is a matrix: \n \n                                                                 \n\n\n\n−≡00ˆ\nzz\neeRrr\n,                                                                 (3) \n                                                            4where zer is a unit vector along z axis. Operator ⊥Lˆ becomes self-adjoint for homogeneous boundary \nconditions (continuity of ϕ~ and rB~) on a lateral surface of a ferrite disk.     \n    The second type of a di fferential-operator equation is  \n \n                                                             () 0~ )(ˆ 2=−⊥ηβ G ,                                                              (4)                    \n \nwhere η~ is a dimensionless membrane MS-potential  wave function (diffe rent from function ϕ~), \n \n                                                                     2 ˆ\n⊥ ⊥∇≡µ G ,                                                                    (5)                    \n \nand 2\n⊥∇ is the two-dimensional (with respect to in-plane coordinate s) Laplace opera tor. Operator ⊥Gˆ \nis positive definite for negative quantities µ. Outside a ferrite region Eq. (4) becomes the Laplace \nequation ( 1=µ ). Double integration by parts on square S – an in-plane cross-section of a disk \nstructure – of the integral ∫⊥\nSdS G*~ )~ˆ(ηη  gives the boundary conditions for self-adjointess of operator \n⊥Gˆ.  \n    MS-potential wave functions η~ and ϕ~ are different one from another.  In a cylindrical coordinate \nsystem we can write ) (~)(~~θηηη r=  and ) (~)(~~θϕϕϕ r= . For angular parts we have θνθη ~)(~ ie− and \nθνθϕ ~)(~ ie−,  where .... 3,2,1±±±=ν  For self-adjointess of operator ⊥Gˆ one has the homogeneous \nboundary condition for a disk of radius ℜ:  \n \n                                                              0~ ~\n =\n\n\n∂∂−\n\n\n∂∂\n+ −ℜ= ℜ= r r r rη ηµ .                                                (6) \n \nAt the same time, for self-adjointess of operator ⊥Lˆ the homogeneous boundary condition is: \n \n                                                      () −\n+ −ℜ=\nℜ= ℜ= ℜ−=\n\n\n∂∂−\n\n\n∂∂\nra\nr r r rϕνµ ϕ ϕµ~~ ~\n.                                     (7) \n \n \nFunctions ) (~rη  and ) (~rϕ  are the Bessel functions and the bounda ry conditions (6) and (7) can be \nrepresented, respectively, as  \n \n                                                                       0 )(21\n=′+′−\nνν\nννµKK\nJJ                                                      (8) \n \nand  \n                                                              \n0||)(21\n=ℜ−′+′−βνµµ\nνν\nνν a\nKK\nJJ.                                                (9) \n                                                            5Here ν ννν K KJJ ′ ′  and , , ,  are the values of the Bessel functions of order ν and their derivatives (with \nrespect to the argument) on a lateral cylindrical surface ( ℜ=r ). From boundary condition (7) [or \n(9)] it evidently follows that for an integer azimuth number ν and a given sign of parameter aµ, \nthere are different functions, +ϕ~ and −ϕ~, for positive and negative direc tions of an angle coordinate \nwhen πθ2 0≤≤ . So function ϕ~ is not a single-valued function. It changes a sign when θ is rotated \nby π2. At the same time, function η~ is a single-valued function. \n    For a given magnetic-dipolar mode, function ϕ~ is normalized on a powe r flow density along z \naxis. It is well known that in an axially magnetized infinite ferrite rod, ther e exist two magnetostatic \nwaves propagating along a positive direction of z axis and corresponding to the +ϕ~ and −ϕ~ solutions \n[22].  These two waves are degenera te with respect to an average density of an accumulated energy, \nwhich is a norm for function η~. This fact has a formal aspect for MS-wave propagation in an \ninfinite ferrite rod, but acquires, however, a peculia r physical meaning in a ferrite disk [4, 15, 16]. In \na spectral problem of MDMs in a normally magnetized quasi-2D ferrite disk, Eq. (6) [or Eq. (8)] describe so-called essential boundary conditions  while Eq. (7) [or Eq. (9)] describe so-called natural \nboundary conditions  [16, 23]. A specific boundary relation on a lateral surface of a ferrite disk gives \na clear correspondence between double-valued functions \nϕ~ and single-valued functions η~[4]: \n \n                                                                  ()()ℜ=±ℜ=±=r rηδϕ~ ~,                                                         (10) \n \nwhere  \n        \n                                                                       θδ±−\n±±≡iqef                                                               (11) \n \nis a double-valued edge (at ℜ=r ) function. The azimuth number ±q is equal to 21± and for \namplitudes we have − +−=f f  and ±f = 1. Because of the boundary relation (10), the topological \neffects are manifested by the generation of re lative phases which accumulate on the edge wave \nfunction ±δ. These topological effects become apparent  through the integral  fluxes of the pseudo-\nelectric fields [4]. There should be th e positive and negative fluxes corresponding to \ncounterclockwise and clockwise edge-function ch iral rotation. The di fferent-sign fluxes are \ninequivalent to avoid cancellation.  Every MDM in a thin ferrite di sk is characterized by a certain \nenergy eigenstate and two different -sign fluxes of the pseudo-electric  fields which are energetically \ndegenerate.    \nB. Power-flow-density vortices and energy eige nstates in a quasi-2D ferrite disk with MDM \noscillations \n \nFor MS wave propagating in a ferrite rod, membrane function ϕ~ is normalized on a power flow \ndensity along z axis. This is an observable quantity. For MDM oscillations in a ferrite disk, power \nflow density along z axis is evidently equal to zero. One of th e unique features of such oscillations is \nthe presence of azimuth power flow densities. \n    In a general representation, for monochr omatic MS-wave processes with time variation ~tie ω, the \npower flow density (in Gaussian units) is expressed as [4]:  \n \n                                                           ()* * 16B Biprr rψψπω− = .                                                           (12)  6 \nThe proof that this is really a power flow  density arises from the following equation: \n \n                  () ()() ()() [ ]B B B BiB Bi rtrr tr rr\n⋅⋅−⋅⋅−=−⋅∇−− − 1 * *1* * *\n16  16ωµ ωµπωψψπω.                      (13) \n       \nThis is, in fact, the energy ba lance equation for monochromatic MS waves in lossy magnetic media. \nIn the right-hand side of this e quation we have the aver age density of magnetic losses taken with an \nopposite sign. Thus the term in the left-hand side  is the divergence of th e power flow density.  \n    The power flow density fo r a certain magnetic-dipolar mode n is  \n \n                                                           ( )* *\n16nn nn n B Biprr rψψπω− = .                                                      (14) \n                        \nFor MS-potential wave function for mode n one has [4, 15, 16] \n \n                                                                ) ,(~)(θϕξψ r z Cn nn n= ,                                                      (15)  \n \nwhere ) (znξ  is an amplitude factor, nC is a dimensional coefficient, and ) ,(~θϕrn  is a dimensionless \nmembrane function. For magnetic flux density (n nB ψµ∇⋅−=rtr\n ), we can write  \n \n                                                               ()⊥+= eBeB Bn zzn nrr r\n ~,                                                         (15) \n \nwhere    \n                                                             \n() ),(~)(θϕξrzzC Bnn\nn zn∂∂−=                                                 (16) \n \nand   \n                                                           [ ]⊥ ⊥⊥ ⋅ ∇⋅ −= e r z C Bn nn nr rt),(~)(~θϕµξ .                                       (17)  \n \nSubscript ⊥ corresponds to transversal (with respect to z axis) components. \n    For oscillating MDMs in a quasi-2D ferrite disk not only the z component of the power flow \ndensity is equal to zero. It is easy to show that the r component of the power fl ow density is equal to \nzero as well. There is, however,  non-zero real azimuth componen t of the power flow density: \n \n                     () () \n\n\n\n\n\n\n\n∂∂+∂∂+\n\n\n\n∂∂−∂∂− =r rirz Cizpn\nnn\nn an\nnn\nn n n*\n**\n* 2 2~~~~~~~~1)(16)(ϕϕϕϕµθϕϕθϕϕµξπω\nθ.        (18) \n \nWith use of representation ) (~)(~ ~θϕϕϕn n n r= , where θνθϕni\nn e−~)(~ and  nν is an integer, one has \n \n                                  () ()\n\n∂∂− − =rr\nrr z Crzrpn\na n n n nn\nn)(~\n)()(~)(  8)(~\n),(2 2 ϕµνµϕξωπϕ\nθ .                     (19) \n  7This is a non-zero circula tion quantity around a circle rπ2. An amplitude of a MS-potential function \nis equal to zero at 0=r . For a scalar wave function, this  presumes the Nye and Berry phase \nsingularity [24]. Circ ulating quantities ()θ),(zrpn  are the MDM power-flow- density vortices with \ncores at the disk center. At  a vortex center amplitude of ()θnp is equal to zero. It follows from Eq. \n(19) that for a given mode number n characterizing by a certain function ) (~rnϕ  there will be \ndifferent power flow densities  ()θ),(zrpn  for different signs of the azimuth number nν.  \n    For calculation of the power  flow density, the wave number βand functions )(zξ and )(~rϕ  for a \ngiven mode can be found based on solution of a sy stem of two equations: a characteristic equation \nfor MS waves in an axially magnetized ferrite r od [Eq. (9)] and a characteristic equation for MS \nwaves in a normally magnetized ferrite slab:  \n                                                              \n()µµβ+−−=12 tan h ,                                                           (20) \n \nwhere h is a disk thickness [15, 16]. \n    Stable vortex structures of the power fl ow density (PFD) for ever y oscillating MDM give an \nevidence for energy eigenstates in a quasi-2D ferrite disk. Since in a lossless fe rrite the divergence of \nthe PFD vortex is equal to zero, the only possibility  to change the PFD topol ogical structure is via \ndiscrete energy transition betw een the MDM energy levels. The statement that confinement \nphenomena for MS oscillations in a normally magnetiz ed ferrite disk demonstrate typical atomic-like \nproperties of discrete energy levels becomes eviden t from an analysis of experimental absorption \nspectra. The main feature of mu lti-resonance line spectra observe d in well known experiments [17, \n18] is the fact that high-order peaks correspond to lowe r quantities of the bias DC magnetic field. \nPhysically, the situation looks as follows. Let )(\n0)(\n0   and B AH H  be, respectively, the upper and lower \nvalues of a bias magnetic field corresponding to the bord ers of a region. We can estimate a total \ndepth of a “potential well” as: ())(\n0)(\n0 0 4B AH HM U − =∆π , where 0M is the saturation magnetization. \nLet )1(\n0H be a bias magnetic field, co rresponding to the main absorptio n peak in the experimental \nspectrum ()(\n0)1(\n0)(\n0A BH H H << ). When we put a ferrite sample into  this field, we supply it with the \nenergy: )1(\n0 0 4 HMπ . To some extent, this is a pumping-up en ergy. Starting from this level, we can \nexcite the entire spectrum from the main mode to the high-order modes. As a value of a bias \nmagnetic field decreases, the “particle” obtains th e higher levels of negative energy. One can \nestimate the negative energies necessary for transitions from the main level to upper levels. For \nexample, to have a transition from the first level )1(\n0H to the second level )2(\n0H \n()(\n0)1(\n0)2(\n0)(\n0A BH H H H <<< ) we need the density energy surplus: ())2(\n0)1(\n0 0 124 H HM U − =∆π . The \nsituation is very resembling the increasing a negativ e energy of the hole in semiconductors when it \n“moves” from the top of a valence band. In a cla ssical theory, negative-energy solutions are rejected \nbecause they cannot be reached by a continuous loss of energy. But in quantum theory, a system can \njump from one energy level to a discretely lowe r one; so the negative-energy solutions cannot be \nrejected, out of hand. When one continuous ly varies the quantity of the DC field 0H, for a given \nquantity of frequency ω, one sees a discrete se t of absorption peaks. It means that one has the \ndiscrete-set levels of potential energy. The line spectra appear due to th e quantum-like transitions \nbetween energy levels of a ferrite disk-form particle.     The statement of stable vortex structures of the PFD and energy eige nstates of MDMs reveals \nsome discrepancies for MS-potenti al functions on a lateral surface of  a ferrite disk. To settle the \nproblem, one should impose a certai n phase factor [4]. The solutions for MS-potential functions \nϕ~  8with additional phase factors satisfy the mode orthogonality relations. Such boundary phase factors \ncause appearance of effective surface circular magnetic  currents on a lateral surf ace of a disk [4]. It \nappears that circular eigen power  flows are accompanied with th e fluxes of the pseudo-electric \nfields. These fluxes are induced so  that the edge wave function δ provides necessary boundary \nconditions to achieve singlevaluedness of function η~ [see Eq. (10)]. This guaranties energy \neigenstates of MDMs in a ferrite disk.  \n    All the energy relations have  physical meaning when are written for a single-valued membrane \nfunction ) ,(~θηr. For MDM oscillations one has energy eigenstates which are characterized by a \ntwo-dimensional (“in-plane”) di fferential operator [4, 15, 16] \n \n                                                                    2 16ˆ\n⊥ ⊥ ∇=µπqgF ,                                                            (21)                    \n \nwhere qg is a dimensional normalization coefficient for mode q. The normalized average (on the RF \nperiod) density of accumulated magnetic energy of mode q is determined as \n \n                                                                    ()2\n16nq\nngE βπ= .                                                           (22) \n \nThe energy eigenvalue problem is de fined by the differential equation: \n \n                                                                    nn nE Fηη~ ~=⊥),                                                                (23) \n \nFor MDMs in a ferrite disk at a constant frequency one has the energy orthonormality:   \n                                                              0~~) ( = −\n∫∗\n′ ′\nSnn n n dS E E ηη ,                                                     (24) \n    \nwhere S is a cylindrical cross section of an open disk. Different mode energies one has at different \nquantities of a bias magnetic field. From the princi ple of superposition of states, it follows that wave \nfunctions nη~ ( ,...2,1=n ), describing our \"quantum\" system, are \"v ectors\" in an abstract space of an \ninfinite number of dimensions – the Hilber t space. The scalar-wave membrane function η~ can be \nrepresented as  \n \n                                                                     ∑=\nnnnaηη~ ~                                                                (25)              \n \nand the probability to find a system in a certain state q is defined as \n \n                                                                 2\n* 2 ~ ~∫=\nSn n dS aηη .                                                          (26)      \n \nIt is very important to note that fo r the energy eigenvalue problem, wavenumber nβ in Eq. (22) and \nthe mode structure for nη~ are defined based on solutions of a system of two equations: Eq. (8) and \nEq. (20).  9C. The field structures in a quasi-2D ferrite disk with MDM vortices \n \nThe MDM vortices in a quasi-2D ferrite disk are ch aracterized by specific field structures. For non-\nuniform ferromagnetic resonances in small ferrite samples there are no electromagnetic retardation \neffects since one neglects electric displacement cu rrents [11]. It means that  the electric and magnetic \nfields in our case of a quasi-2D ferrite disk ca nnot be obtained analyti cally from the Maxwell-\nequation spectral  problem. At the same time, the fi elds can be derived from the MDM spectral  \nproblem solved for the MS- potential wave functions ) ,(trrψ . So in the ferrite-disk boundary value \nproblem, the MS-potential wave function is a prim ary notion while the electric and magnetic fields \ncan be related to a secondary concept.  \n    Inside a ferrite disk ( hz r ≤≤ℜ≤ 0  , ) one has: \n \n                                         ()νθ\nν β\nµβ\nµβθψjez zrJzr−\n\n\n\n\n−+\n\n\n\n−=  sin1 cos ,, .                             (27)                     \n \nThis function satisfies characteristic equations (9 ) and (20). Based on such a MS-potential function \none defines the magnetic field ( ψ∇−=rr\nH ) inside a ferrite disk as  \n \n                                 ()νθ\nν β\nµβ\nµβ\nµβθi\nr ez zrJ zrH−\n\n\n\n\n−+\n\n\n\n−′\n−=  sin1 cos ,,  ,                         (28) \n   \n                                 ()νθ\nν θ β\nµβ\nµβθθiez zrJrizrH−\n\n\n\n\n−+\n\n\n\n−−=  sin1 cos ,, ,                            (29)    \n \n                                 ()νθ\nν β\nµβ\nµββθi\nz ez zrJ zrH−\n\n\n\n\n−+−\n\n\n\n−=  cos1 sin  ,, .                             (30)                     \n \n    In a small sample of a material with a st rong temporal dispersion of  permeability, differential \nequations for the magnetic field and for the magne tic flux density are the same as the corresponding \nequations for \"pure\" magnetostatics. The only (and substantial) difference is that the permeability is not a constant scalar quantity but is a tensor with  the components strongly dependent on frequency. \nIn this case, a role of the electric field in the wa ve process is not clearly determined. Formally, one \ncan suppose that for the monochromatic MS-wave process there exists a curl electric field \nEr\ndefined \nby the Faraday law. One can represen t the electric field as follows: \n \n                                                    mciHciBciEr r r rr\n 4   πωωω −−=−=×∇ ,                                          (31) \n \nwhere mr is RF magnetization. With the ×∇r\n differential operation for the left-hand and right-hand \nsides of Eq. (31) and taking into account that 0=×∇Hrr\n, one obtains: \n \n                                                                 mciEr r\n×∇=∇  42πω .                                                       (32) \n  10Here we used the relation 0=⋅∇Er\n. This follows from the fact th at in the MS-wave description no \nelectric polarization effects ar e taken into consideration.  \n    The electric field can be formally represen ted as being originated fr om an effective electric \ncurrent:   \n                                                                   \n)(\n22 4 ejciEr rπω=∇ ,                                                         (33) \n \nwhere   \n                                                                     m c j\nerrr\n×∇≡)(.                                                             (34) \n   \nTaking into account that H mrtr⋅=χ , where the magnetic susceptibility [26] \n  \n                                                              \n\n\n\n\n−=\n00 000χχχχ\nχaa\nii\nt,                                                        (35)                    \n \nwe have the following components of  an effective electric current: \n \n                    νθ\nν ν β\nµβ\nµβχνβ\nµβ\nµβχi a e\nr ez zrJrrJ ic j−\n\n\n\n\n−−\n\n\n\n\n\n\n\n−+\n\n\n\n−′\n−=  cos1 sin   2\n)( ,                         (36) \n \n                    νθ\nν ν θ β\nµβ\nµβνχβ\nµβ\nµχβi a eez zrJrrJ c j−\n\n\n\n\n−−\n\n\n\n\n\n\n\n−+\n\n\n\n−′\n−=  cos1 sin   2\n)( ,                         (37) \n                 \n                    νθ\nν β\nµβ\nµβ\nµµ\nπi a e\nz ez zrJ ic j−\n\n\n\n\n−+\n\n\n\n−−=  sin1 cos \n41)( .                                                    (38) \n \n    In the MDM vortex structure, a magnetic fiel d described by Eqs. (28) – (30) and an effective \nelectric current described by Eqs. (36) – (37) [a nd therefore an electric fi eld described by Eq. (33)] \nshow the running azimuth wave behaviors. By multiplying these equations at tie ω and taking real \nparts, one has the real-time azimuth waves.   \n    The electric field determined from the abov e consideration has an auxiliary character for MS-\nwave processes. For a curl electric field one can introduce a magnetic vector potential: mA Er r\n×−∇≡ . \nBased on the Faraday law and taking into account that ψωµ∇⋅−= )(tr\nB , we have \n \n                                                0 )( ) (2=∇⋅ −⋅∇∇−∇ ψωµωtr r\nciA Am m.                                            (39) \n                                                                                           \nThis equation shows that formally two types of gauges are possible. In  the first type of a gauge we \nhave: \n                                                                       0=⋅∇mAr\n                                                                 (40)  \n                                              \nand, therefore,  11 \n                                                           ψωµω∇⋅ =∇ )(2 tr\nciAm.                                                         (41)   \n                                                    \nThe second type of a gauge is written as  \n                                                    \n0 )( ) ( =∇⋅ +⋅∇∇ ψωµωt r\nciAm                                                      (42) \n                                               \nand, therefore,  \n                                                                    \n02=∇mAr\n.                                                                    (43) \n \nThe last equation shows that any sources of the el ectric field are not defined and thus the electric \nfield is not defined at all. So only the first type of a gauge, givi ng Eq. (41), should be taken into \naccount. The main point, however, is  that the considered above ga uge transformation does not fall \nunder the known gauge transformati ons, neither the Lorentz gauge nor  the Coulomb gauge [21], and \ncannot formally lead to the wave  equation. Moreover, to have a wave process one should assume \nthat there exists a certain  non-physical mechanism describing the e ffect of transforma tion of the curl \n(mA Er r\n×−∇= ) electric field to  the potential ( ψ−∇=Hr\n) magnetic field.  \n    Analytically described MD M spectral properties are well verifi ed in microwave experiments. \nWhen a thin-film ferrite disk is placed in a microwave cavity, one observes experimentally the \nenergy shifts and eigen electric moments of oscilla ting MDMs [17 – 20]. An analysis of excitation of \nthe MDM vortices in a ferrite disk by external electromagnetic fields is, however, beyond the frames \nof analytical solutions. Electromagn etically, a microwave cav ity with an inserted ferrite disk is a \nnon-integrable system. It appears that for a very thin  ferrite disk and for a certain region of external \nparameters (a working frequency and a bias magne tic field), numerical si mulation shows a set of \nelectromagnetic modes inside a disk. These modes are characterized by the PFD  vortex states. It is \nvery surprisingly to find that th e structures of the PFD vortices  and the field pa tterns for these \nnumerically modeled electromagnetic modes are in excellent correspondence with the analytically \nderived PFD vortices and the field patt erns of MDMs in a ferrite disk. \n \nIII. ELECTROMAGNETIC VORTICES AND TO POLOGICAL RESONANT STATES IN  \n     CAVITIES WITH THIN-FILM FERRITE DISKS \n \nIn solving Maxwell's equations, one of the powerful approaches is  the short-wavelength \napproximation leading to the ray pict ure. Rays are the solutions of th e Fermat's variational principle, \nwhich in particular implies the laws of reflection and refraction at interfaces. As soon as one makes \nthe transition from wave physics to the classical ray dynamics, concepts such as \"trajectory\" and \n\"phase space\" become meaningful. For microwave and optical cavities one may use a model of the \ngeometrical rays within the boundaries – the bill iard model. By changing shape parameters in \nbilliard models, it is possible to describe systems with classical motion ranging from integrable to fully chaotic. For example, the\n dynamics of a classical particle (a  classical ray) bouncing on a plane \nbetween hard walls depends in a characteristic way on a shape of the billiard. Its motion in a \nrectangular or circular billiard is  regular and the system is integrab le. However, if a circular obstacle \nis put inside the rectangle the system becomes chaotic . In a case of a metal ci rcular obstacle one has \nthe Sinai billiard. On the other hand , if an integrable syst em of a rectangular or circular billiard is \ntransformed into a stadium, one has a chao tic system of the Bunimovich billiard.   12    Integrable systems are characterized by a comple te set of \"good quantum numbers\". If, however, \nthe ray system is not integrable then the quantiza tion procedure fails. It is not known in this case \nwhat the correspondence could be between normal m odes of the wave system  and objects in the ray \nphase space, if indeed there is  any correspondence at a ll. The distinction be tween integrable and \nnon-integrable classical dynamical systems has the qualitative imp lication of regular motion vs. \nchaotic motion. A chaotic system cannot be deco mposed; the motion along one coordinate axis is \nalways coupled to what happens al ong the other axis. In a case of classical non- integrable problems, \nthe corresponding wave equation ca nnot be reduced to a set of mu tually independent ordinary \ndifferential equations. Their couplin g makes it impossible to label the wave solutions by a complete \nset of eigen numbers. At the same time, becau se of a clear correspondence between the two-\ndimensional Helmholtz equation and the Schrödinger e quation, an analysis of non-integrable planar \nmicrowave and optical cavities can be made ba sed on the theories of quantum chaos. A \nsuperposition of random plane waves can be used to describe the statistics of chaotic wave functions \n(the so-called \"wave chaos\") in  electromagnetic cavities [25]. \n    Microwave cavities with ferrite inclusions ca n be modeled as billiards with the time reversal \nsymmetry breaking effects. Because of the time-reve rsal symmetry breaking effects, a system of a \nrectangular-waveguide cavity  with an embedded inside ferrite di sk (even having sizes much small \ncompared with the cavity sizes) is not a weakly perturbed integr able system, but a non-integrable \nsystem [5 – 8]. At the same time, electromagnetic wave processes inside a ferrite sample for a constant bias field can be ap proximated as chaotically propaga ting plane waves. When a bias \nmagnetic field is directed along z axis, a ferrite material is described by the permeability tensor  \n \n                                                             \n\n\n\n\n−=\n1 0 000\nµµµµ\nµaa\njj\nt,                                                        (44)  \n \nand the plane-wave propagation in an infinite ferrite medium is characterized by the following scalar \neffective permeability parameter [26]:  \n                         \n) cos (sin2cos4)1 ( sin )1 ( sin2)(2 22 2 2 2 4 2\nµθθµµθ µθ µθθµ\nk ka k k k\nk eff++− ±− +=⊥ ⊥,                    (45) \n \nwhere kθ is an angle between direc tions of the wave vector kr\n and  the bias magnetic field 0Hr\n, and  \n \n                                                                 µµµµ2 2\na−≡\n⊥.                                                                 (46) \n \nTwo limit cases have to be taken into account: for 2πθ=k  one has an effective permeability \nparameter ⊥=µµeff and for 0 =kθ  there are two effective permeability parameters \n \n                                                a effµµµ+=\n+     and      a effµµµ−=\n−,                                           (47) \n \ncharacterizing, respectively, the plane waves with  right-hand and left-hand circular polarizations. \nOnly for the right-hand circularly polarized  wave, one can disti nguish two behaviors: 0>\n+effµ  and  130<\n+effµ . The quantity \n−effµis always positive [26]. Quantitie s and signs of components of the \npermeability tensor (and therefore quantities and signs of effective permeabilities ⊥µ and \n±effµ) are \ndependent on a frequency, a bias magnetic fiel d and on a saturation magnetization parameter of a \nferrite.      Let us consider separately plane electrom agnetic waves propagating perpendicular to a bias \nmagnetic field and parallel to a bias magnetic field in  a lossless ferrite medium. In our studies of the \nelectromagnetic fields inside a normally magneti zed ferrite disk with positive scalar permittivity \nε, \nthe most interesting case corr esponds to posi tive quantities ⊥µ when wavenumbers \n \n                                                                    ⊥ ⊥=εµω\nck                                                                 (48) \n \nare real quantities. For given bias magnetic field 0H and saturation magnetization 0M, 0>⊥µ  if \n⊥<ωω , where ) (0 0 Mωωωω +=⊥ , 0 0Hγω= , 0 4MMπγω= , and γis the gyromagnetic ratio.     \n    For frequencies ⊥<ωω  we consider now two separate regions: (a) 0ωω<  and (b) ⊥<<ωωω0 . \nWhen 0ωω< , one has 0 >\n+effµ  and so there are real wavenumbers  \n \n                                                                   \n± ±=effckεµω\n|| .                                                             (49) \n \nFor ⊥<<ωωω0 , one has 0 <\n+effµ . In this case  \n \n                                                                    \n− −=effckεµω\n||                                                              (50) \n \nare real wavenumbers, while   \n                                                                   \n+ +=effci k µεω\n||                                                            (51) \n \nare imaginary wavenumbers.     A ferrite sample analyzed in Ref. [8] has the following material parameters: saturation \nmagnetization  1880  4=\nsMπ G, the linewidth Oe 8.0=∆H .  permittivity 15=rε . For such material \nparameters, we will take a bias magnetic field  49000=H Oe and choose the cavity resonance \nfrequency 328 .8=f GHz. This corresponds to the region where 0ωω< . In neglect of lossless, a \nsimple calculation gives the following quantit ies of wavenumbers in a ferrite medium: =⊥ 0kk 6.2, \n=\n+ 0 ||kk 25.3, and =\n− 0 ||kk 4.41, where c kω=0 . A disk analyzed in Ref. [8] has \ndiameter 6=D mm and thickness 5.0=t mm. For such geometry, it is evident that for the right-hand-\ncircular-polarization wave s propagating along a normal z axis (with the wavenumber \n+||k) there is \nalmost the same order of the phase variation on a scale of the disk thickness as for the waves \npropagating along any direction in the xy plane (with the wavenumber ⊥k)  on a scale of a disk \nradius.  \n    When a normally magnetized ferrite disk is pl aced in a rectangular waveguide cavity so that the \ndisk axis is perpendicular to a wide wall of a wave guide (see Fig. 1), the phase velocities for plane  14waves propagating inside a disk are much less than outside a disk. An incoming wave propagates \nfreely outside the ferrite, but can be  trapped inside for some time. The mean \"escape\" time of a ray \ninside a ferrite disk into a waveguide vacuum sp ace is much bigger than the time for the ray pass \nacross the disk in a waveguide vacuum space. This cl early presumes a chaotic nature of the classical \nray motion inside a sample with the time-reversal  symmetry breaking effects. A ray is scattered \nseveral times from the disk's boundaries before exiting the disk. The electromagnetic fields \napproximately obey a statistical cond ition. This leads to special topol ogical structures for the field \nand power flow distributions insi de a ferrite disk. Fig. 2 shows a typical vortex picture for the \nPoynting vector distribution inside a ferrite disk when a disk is placed in a maximal cavity electric \nfield. This picture was obtained based on numerical experiments with  use of the HFSS (the software \nbased on FEM method produced  by ANSOFT Company) CAD si mulation programs for 3D \nnumerical modeling of Maxwell eq uations [27]. The program determ ines both modulus and phase of \nthe fields. There is an evident pict ure of the Poynting-vector vortex in  Fig. 2. A detailed study of the \nelectromagnetic vortices inside a ferrite disk at frequencies0ωω<  is given in recent publications [7, \n8].  \n    Now let us consider the same disk with the same bias magnetic field Oe  49000=H , but for the \ncavity frequency  52.8=f GHz. This frequency is within the region ⊥<<ωωω0 . For a lossless \nferrite, there are real quantities =⊥ 0kk 6.33 and =\n− 0 ||kk 4.43, but an imaginary quantity \n=\n+ 0 ||kk j34.5. So in a direction of a bias magnetic field there are evanescent plane waves. Fig. 3 \nshows the Poynting vector distribu tion inside a disk for frequency  52.8=f GHz, when a disk is \nplaced in a maximal cavity electric field. There is a picture with a very chaotic Poynting-vector \ndistribution, without any r obust topological structure. \n    We will aim our further studies to an analysis of  the role of the disk geometry. We consider a disk \nwith the same diameter, but with a very small thic kness, ten times less than in the previous case. So \nnow the disk parameters are: diameter 6=D mm and thickness 05.0=t mm. From numerical \nexperiments with such a thin disk, we found that for the frequency region0ωω<  there are no \nqualitative differences with the pictures of the Po ynting-vector and field dist ributions shown in Refs. \n[7, 8]. The situation, however, becomes dras tically different for the frequency region ⊥<<ωωω0 . \nFor a very thin disk in such a frequency region, one can observe now topological resonant states .  \n    To a certain extent, electromagnetic wave proce sses inside a normally magnetized thin ferrite disk \nfor a constant bias field and at frequencies ⊥<<ωωω0  can be approximated as chaotically \npropagating plane waves in a vertica lly thin electromagnetic resonator with certain material filling. \nThis allows reducing a problem to the quas-2D b illiard model which mathematical properties are \ngenerally well studied for classical chaos and its quantum manifestation.  For a 2D billiard system the \nrandom wave model implies that a typical chaotic wave function may be written locally as a random \nsuperposition of plane waves inside a ferrite at  fixed frequency. The properties of membrane \nfunctions of individual ferrite-dis k modes should be extracted from th e behavior of whole families of \nray trajectories in the xy plane. In the limit of small wavelengt h compared to the resonator size, one \ncan use a well known Weyl formula for an average ei genvalue density [28]. On the other hand, the \nwave function in a disk may be expanded in circ ular waves with good angul ar moments and random \namplitude coefficients [29].  \n     \nIV. CORRESPONDENCE BETWEEN ANALI TICALLY DERIVED MDM VORTEX \nSTATES AND NUMERICALLY MODELE D ELECTROMAGNETIC VORTICES \n \nQualitatively, the field structures of  topological resonance states in a thin ferrite disk at frequencies \n⊥<<ωωω0  are not strongly dependent on a disk diam eter. So for our studies of the observed  15eigenfunction patterns we will use the disk having diameter 3=D mm and thickness 05.0=t mm. \nSuch disk geometry is very interesting for us sin ce it corresponds to the geom etry of a ferrite sample \nused in our recent experiments [20]. Based on th e HFSS-program numerical studies, we analyze \nexcitation of the topological resonant states in a ferrite disk placed in a microwave cavity at a bias \nmagnetic field Oe  49000=H . For our numerical studies we used  a short-wall rectangular waveguide \nsection. The disk axis was oriented along the E-field of a waveguide 10TE  mode (see Fig. 1). The \nferrite material parameters are the same as for the above studies . Fig. 4 (a) shows numerically \nobtained frequency characteristic of an absorption coefficient for a ferrite disk in a waveguide \ncavity. One clearly sees the multiresonance regular absorption spectra. As we will show, for every absorption peak one has a robust picture of a topol ogical resonant state. Mo reover, we find a very \ngood correspondence of these topological  states with the an alytically derived PFD vortices and the \nfield patterns of MDMs in a ferrite disk.     The resonance peak positions obtained from an analytical solution of Eqs. (8) and (20) are shown in Fig. 4 (b). The numerical absorption peaks a nd the analytical peaks ca lculated based on the \nessential boundary conditions are, in fact, at the same positions [see Figs. 4 (a) and 4 (b)]. Figs. 5 \nand 6 show the power flow density distributions respectively for the 1\nst (f = 8.52 GHz) and 2nd (f = \n8.66 GHz) modes in a quasi-2D ferrite disk. Th ere is an excellent correspondence between a \nnumerically modeled (HFSS-program) electromagneti c and analytically derived MDM power-flow-\ndensity vortices. Analytical results for the power flow density distributions  were obtained based on \nEq. (19) for the fundamental-thickness and first-order-azimuth MDMs. Numbers n in Eq. (19), being \nthe numbers of zeros in the Bessel function, corres pond to different radial va riations. We analyzed \ndistributions of ()θnp for first two modes ( n = 1, 2) at 1 +=nν  when a bias magnetic  field is directed \nalong z axis.   \n    A vortex can be defined as a circular flow which is attributed with a cert ain phase factor and a \ncircular integral of a gradient of the phase give s a non-zero quantity. This quantity is multiple to a \nnumber of full rotations. In ou r case, the phase factor of the mode vortex appears from the \ntopological properties of the azimuth ally rotating magnetic field. Figs . 7 and 9 represent galleries of \nthe numerically modeled magnetic field distribut ions on the upper plane of a ferrite disk, \nrespectively, for the 1st and 2nd topological resonance states at di fferent time phases. Every resonant \nstate is characterized by a strong pronounced eigenf unction pattern.  Similar distributions for the 1st \nand 2nd magnetic-dipolar modes analytically derive d based on Eqs. (28) – (30) one can see, \nrespectively, in Figs. 8 and 10. The observed pictures  for magnetic fields of different modes have the \nsame azimuth variations but are distinguished by th e radius variations. A ve ry peculiar property of \nthese pictures is the fact of the azimuthal rotation  of the mode magnetic field for an observer being \nsituated outside a ferrite disk. Si nce for an observer being situated inside a ferrite disk, these modes \nhave π2 azimuth variations, there is an evident π4counterclockwise mode azi muthal rotation for an \noutside observer. The phase of th e final mode state differs from that of the initial state by \ng dφφφ+= , where dφ and gφ are the dynamical and geometrical phases, respectively [30].  \n    At the same time, contrary to the magne tic field azimuth variations, there are no azimuth \nvariations for the mode electric fields. This essent ially differs from well know n field distributions in \nintegrable electromagnetic cylindrical structur es. Really, in any integrable electromagnetic \ncylindrical structure (such as a cylindrical wa veguide or resonator) one  has from Maxwell's \nequations the same order of the azimuth variation for the electric and magnetic fields (see e.g. Refs. \n[31, 32]). Figs. 11 and 13 represent galleries of the numerically mode led electric field distributions \non the upper plane of a ferrite disk, respectively, for the 1st and 2nd topological resonance states at \ndifferent time phases. Figs. 12 and 14 show in-plane effective-electric-current distributions for the 1st \nand 2nd magnetic-dipolar modes analytica lly derived based on Eqs. (36) – (38). From these analytical \nresults for effective-electric-current distributions one can obtain qualitative pictures for the electric \nfields. As it follows from Eq. ( 33), the electric field should be o90  shifted with respect to the  16effective electric current. This  gives a good correspondence w ith the numerically modeled \ndistributions for the electric fields.      It is very important to note that the in-plane  electric fields on the u pper and lower planes of a \nferrite disk are in opposite  directions at any time phase. Since th e disk thickness is much, much less \nthan the rectangular waveguide height, the disk in  a cavity can be clearly replaced by a sheet with \nlinear surface magnetic currents. A surface density of the effective ma gnetic current is expressed as \n \n                                                         \nm\nlower upper icE Enr rrr π4) ( −=−× ,                                                 (52) \n \nwhere upperEr\n and lowerEr\n are, respectively, in-plane electric fields on the upper and lower planes of a \nferrite disk and nr is a normal to a disk plane directed along a bias magnetic field. Evidently, \nlower upper E Err\n−= . Following pictures in Figs. 11 and 13,  one can conclude that there are rotating  linear \nsurface magnetic currents.  \n \nCONCLUSION \n In confined magnetically ordere d structures one can observe vo rtices of magnetization and \nelectromagnetic power flow vortices . There are topologically distinct and robust states. In this paper \nwe showed that in a normally magnetized quasi-2D ferrite disk there exist eigen power-flow-density vortices of magnetic-dipolar-mode osci llations. Because of such circul ar power flows, the oscillating \nmodes are characterized by stable magnetostatic energy states and di screte angular moments of the \nwave fields. Stable MDM vortex co nfigurations are very attractiv e objects for fundamental physics \nstudies of confined magne tic structures with dynamical symmetry breaking effects. The elements \nwith such topologically di stinct and robust states may find pote ntial use as a unit cell for magnetic \nstorage and magnetic logic devices. They can be al so considered as very interesting objects for \ndifferent microwave applications.       The observed MDM eigenstates and vortex structur es in a thin ferrite disk can be analyzed in \ncorrespondence with numerically modeled Maxwell-equation object s in the ray phase space. \nMicrowave systems with enclosed ferrite sample s are considered as non- integrable Maxwellian \nproblems. In a case of very thin ferrite disks one  can observe topologically distinct and robust \nstructures – the electromagnetic vorti ces – with eigenmode field structur es. It is very interesting that \nin a quasi-2D ferrite disk one  can find a clear co rrespondence between such pure numerical \nMaxwell-equation solutions of th e vortex states and the analyti cally derived MDM vortex states \nobtained as a result of the MS-poten tial-wave-function spectral solutions. \n  \nReferences \n [1] A. Hubert and R. Schäfer, Magnetic Domains  (Springer, Berlin, 1998).   \n[2] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T. Ono, Science \n289, 930 (2000). \n[3] K. Yu. Guslienko, V. Novosad, Y. Otani, H. Shima, and K. Fukamichi, Phys. Rev. B 65, 024414 \n(2002). \n[4] E. O. Kamenetskii, J. Phys. A: Math. Theor. 40, 6539 (2007).  \n[5] M. Vrani čar, M. Barth, G. Veble, M. Robnik, and H.-J. Stöckmann, J. Phys. A: Math. Gen. 35, \n4929 (2002). \n[6] E. O. Kamenetskii, M. Sigal ov, and R. Shavit, Phys. Rev. E 74, 036620 (2006). \n[7] M. Sigalov, E. O. Kamenetsk ii, and R. Shavit, Phys. Lett. A 372, 91 (2008). \n[8] M. Sigalov, E. O. Kamenetskii, and R. Shavit, J. Appl. Phys. 103, 013904 (2008). \n[9] J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 (1946).  17[10] H. Puszkarski, M. Krawczyk, and  J.-C. S. Levy, Phys. Rev. B 71, 014421 (2005). \n[11] L.D. Landau and E.M. Lifshitz, Electrodynamics of Continuous Media , 2nd ed. (Pergamon, \nOxford, 1984). \n[12] P. H. Bryant, J. F. Smyth, S. Sc hultz, and D. R. Fredkin, Phys. Rev. B 47, 11255 (1993). \n[13] K. Yu. Guslienko, S. O. Demokrit ov, B. Hillebrands, and A. N. Slavin , Phys. Rev. B 66, \n132402 (2002). \n[14] C. E. Zaspel, B. A. Ivanov, J. P.  Park, and P. A. Crowell, Phys. Rev. B 72, 024427 (2005). \n[15] E. O. Kamenetskii, Phys. Rev. E 63, 066612 (2001). \n[16] E. O. Kamenetskii, M. Sigalov, and R. Shavit, J. Phys.: Condens. Matter 17, 2211 (2005).  \n[17] J. F. Dillon Jr., J. Appl. Phys. 31, 1605 (1960).  \n[18] T. Yukawa and K. Abe, J. Appl. Phys. 45, 3146 (1974). \n[19] E. O. Kamenetskii, A.K. Saha, and I. Awai, Phys. Lett. A 332, 303 (2004). \n[20] M. Sigalov, E.O. Kamenetskii, a nd R. Shavit, arXiv:0708.2678; arXiv:0712.2305. \n[21] J. D. Jackson, Classical Electrodynamics , 2nd ed. (Wiley, New York, 1975).  \n[22] R. I. Joseph and E. Schlömann, J. Appl. Phys. 32, 1001 (1961). \n[23]  S. G. Mikhlin, Variational Methods in Mathematical Physics  (New York, McMillan, 1964). \n[24] J. F. Nye and M. V. Berry , Proc. R. Soc. London Ser. A 336, 165 (1974).  \n[25] X. Zheng, T. M. Antonsen Jr., and E. Ott, arXiv:cond-mat/0408327. \n[26] A. Gurevich and G. Melkov, Magnetic Oscillations and Waves  (CRC Press, New York, 1996). \n[27] High Frequency Structure Simulator, www.hfss.com, Ansoft  Corporation, Pittsburgh, PA \n15219. \n[28] E. Ott, Chaos in Dynamical Systems , 2nd ed. (Cambridge University Press, 2002). \n[29] L. Kaplan and Y. Alhassid, arXiv:0802.2410 (2008). \n[30] E. J. Galvez, P. R. Crawford, H. I. Sztul, M.  J. Pysher, P. J. Haglin, and R. E. Williams, Phys \nRev. Lett. 90, 203901 (2003). \n[31] D. M. Pozar, Microwave Engineering , 2nd ed. (Wiley, New York, 1998). \n[32] A. W. Snyder and J. D. Love, Optical Waveguide Theory  (Chapman and Hall, London, 1983). \n    \nFigure captions  \n Fig. 1. A model of a short-wall  rectangular waveguide section w ith a normally magnetized ferrite \ndisk.   \nFig. 2.  The Poynting vector dist ribution inside a ferrite disk at \n0ωω< . A bias magnetic field \n 49000=H Oe and the cavity resonance frequency 328.8=f GHz. Disk diameter 6=D mm and \nthickness 5.0=t mm. \n \nFig. 3. The Poynting vector distri bution inside a ferrite disk at ⊥<<ωωω0 . A bias magnetic field \n 49000=H Oe and the cavity resonance frequency 52 .8=f GHz. Disk diameter 6=D mm and \nthickness 5.0=t mm. \n \nFig. 4. Spectral characteristics for a thin ferrite disk. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm. ( a) Numerically obtained absorption coefficients; ( b) \nanalytically calculated peak positions. \n  18Fig. 5. The power flow de nsity distribution for the 1st mode (f = 8.52 GHz) in  a quasi-2D ferrite disk. \nA bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. (a) \nNumerically modeled electromagnetic vortex; (b) an alytically derived MDM vortex. A black arrow \nin Fig. 5 (a) clarifies the powe r-flow direction inside a disk. \n Fig. 6. The power flow de nsity distribution for the 2\nnd mode ( f = 8.66 GHz) in a quasi-2D ferrite \ndisk. A bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. (a) \nNumerically modeled electromagnetic vortex; (b) anal ytically derived MDM vortex. Black arrows in \nFig. 6 (a) clarify the power-flo w directions inside a disk. \n Fig. 7. A perspective view for the numerically  modeled magnetic field distributions on the upper \nplane of a ferrite disk for the for the 1\nst topological resonance state ( f = 8.52 GHz) at different time \nphases. There are evident π4 azimuthal rotations. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm.  \n \nFig. 8. A gallery of the analytic ally derived in-plane magnetic fi eld distributions on the upper plane \nof a ferrite disk for the 1st magnetic-dipolar mode ( f = 8.52 GHz) at different time phases. A bias \nmagnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. There are evident \nπ4 azimuthal rotations. \n \nFig. 9. A top view for the numerically modeled ma gnetic field distributions on the upper plane of a \nferrite disk for the for the 2nd topological resonance state ( f = 8.66 GHz) at different time phases. \nThere are evident π4 azimuthal rotations. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm. \n \nFig. 10. A gallery of the analyti cally derived in-plane magnetic field distributions on the upper plane \nof a ferrite disk for the 2nd magnetic-dipolar mode ( f = 8.66 GHz) at some time phases. A bias \nmagnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. There are evident \nπ4 azimuthal rotations. \n \nFig. 11. A top view for the numeric ally modeled electric field dist ributions on the upper plane of a \nferrite disk for the for the 1st topological resonance state ( f = 8.52 GHz) at different time phases. A \nbias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n \nFig. 12. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferri te disk for the 1st magnetic-dipolar mode ( f = 8.52 GHz) at different time \nphases. A bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n \nFig. 13. A top view for the numeric ally modeled electric field dist ributions on the upper plane of a \nferrite disk for the for the 2nd topological re sonance state ( f = 8.66 GHz) at different time phases. A \nbias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n \nFig. 14. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferr ite disk for the 2nd magnetic-dipolar mode ( f = 8.66 GHz) at some time phases. \nA bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n \n  19 \n \nFig. 1. A model of a short-wall  rectangular waveguide section w ith a normally magnetized ferrite \ndisk.  \n \n  \n \n \nFig. 2.  The Poynting vector dist ribution inside a ferrite disk at 0ωω< . A bias magnetic field \n 49000=H Oe and the cavity resonance frequency 328.8=f GHz. Disk diameter 6=D mm and \nthickness 5.0=t mm. \n \nFerrite disk\nShort wall\nRectangular waveguide x\nyz  \n0Hr\n  \nx  y \nz 0Hr 20 \n  \n \n \nFig. 3. The Poynting vector distri bution inside a ferrite disk at ⊥<<ωωω0 . A bias magnetic field \n 49000=H Oe and the cavity resonance frequency 52.8=f GHz. Disk diameter 6=D mm and \nthickness 5.0=t mm. \n  \n \nFig. 4. Spectral characteristics for a thin ferrite disk. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm. ( a) Numerically obtained absorption coefficients; ( b) \nanalytically calculated peak positions. \n                                                    \n(a) \n(b)\nx  y \nz 0Hr 21  \n                       \n                                             \n                                                           ( a) \n  \n  \n \n      (b) \n  \nFig. 5. The power flow de nsity distribution for the 1st mode (f = 8.52 GHz) in  a quasi-2D ferrite disk. \nA bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. (a) \nNumerically modeled electromagnetic vortex; (b) an alytically derived MDM vortex. A black arrow \nin Fig. 5 (a) clarifies the powe r-flow direction inside a disk. \n  \n     x y \nz 0Hr 22 \n \n  \n                      \n \n                                                       ( a)  \n \n  \n                              \n  \n                                                        \n                                                                   ( b) \n    \nFig. 6. The power flow de nsity distribution for the 2nd mode ( f = 8.66 GHz) in a quasi-2D ferrite \ndisk. A bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. (a) \nNumerically modeled electromagnetic vortex; (b) anal ytically derived MDM vortex. Black arrows in \nFig. 6 (a) clarify the power-flo w directions inside a disk. \n \n  \n x y \nz 0Hr 23\n  \n  \n  \n                                                          \nFig. 7. A perspective view for the numerically  modeled magnetic field distributions on the upper \nplane of a ferrite disk for the for the 1st topological resonance state ( f = 8.52 GHz) at different time \nphases. There are evident π4 azimuthal rotations. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm.   \n    \n      \n                                      °=0 tω °=90 tω\n°=180 tω °=270 tω x y \nz 0Hr 24 \n      \n  \n  \n       \n  \n  \n     \nFig. 8. A gallery of the analytic ally derived in-plane magnetic fi eld distributions on the upper plane \nof a ferrite disk for the 1\nst magnetic-dipolar mode ( f = 8.52 GHz) at different time phases. A bias \nmagnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. There are evident \nπ4 azimuthal rotations. \n \n              °=0 tω  \n°=180 tω °=270 tω 25\n  \n  \n  \nFig. 9. A top view for the numerically modeled ma gnetic field distributions on the upper plane of a \nferrite disk for the for the 2nd topological resonance state ( f = 8.66 GHz) at different time phases. \nThere are evident π4 azimuthal rotations. A bias magnetic field  49000=H Oe. Disk \ndiameter 3=D mm and thickness 05.0=t mm.  \n \n   \n  \n  \n \n \nFig. 10. A gallery of the analyti cally derived in-plane magnetic field distributions on the upper plane \nof a ferrite disk for the 2nd magnetic-dipolar mode ( f = 8.66 GHz) at some time phases. A bias \nmagnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm. There are evident \nπ4 azimuthal rotations. °=0tω °=90tω\n°=180tω °=270tωx y \nz 0Hr\n°=0tω °=90tω 26 \n  \n \n  \n  \n  \n Fig. 11. A top view for the numeric ally modeled electric field dist ributions on the upper plane of a \nferrite disk for the for the 1\nst topological resonance state ( f = 8.52 GHz) at different time phases. A \nbias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n                          \n                                    °=0tω °=90tω\n°=180tω °=270tω x y \nz 0Hr 27  \n  \n  \n  \n  \n \nFig. 12. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferri te disk for the 1st magnetic-dipolar mode ( f = 8.52 GHz) at different time \nphases. A bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n °=0tω °=90tω\n°=180tω °=270tω 28\n  \n \n \nFig. 13. A top view for the numeric ally modeled electric field dist ributions on the upper plane of a \nferrite disk for the for the 2nd topological re sonance state ( f = 8.66 GHz) at different time phases. A \nbias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n \n \n \n                                 \nFig. 14. A gallery of the analytically derived in-p lane effective-electric-cu rrent distributions on the \nupper plane of a ferr ite disk for the 2\nnd magnetic-dipolar mode ( f = 8.66 GHz) at some time phases. \nA bias magnetic field  49000=H Oe. Disk diameter 3=D mm and thickness 05.0=t mm.  \n  \n °=0tω °=90tω\n°=180tω °=270tωx y \nz 0Hr\n°=0tω °=90tω"    },    {        "title": "1104.1535v1.Magnetic_nanocomposites_at_microwave_frequencies.pdf",        "content": "arXiv:1104.1535v1  [cond-mat.mtrl-sci]  8 Apr 2011Publishing information:\nJ.V.I. Timonen, R.H.A. Ras, O. Ikkala, M. Oksanen, E. Sepp¨ al¨ a, K. C halapat, J. Li, G.S. Paraoanu, Magnetic\nnanocomposites at microwave frequencies , inTrends in nanophysics: theory, experiment, technology , edited by V.\nBarsan and A. Aldea, Engineering Materials Series, Springer-Verlag , Berlin (ISBN: 978-3-642-12069-5), pp. 257-285\n(2010).\nDOI: 10.1007/978-3-642-12070-1 11Magnetic nanocomposites at microwave frequencies\nJaakko V. I. Timonen, Robin H. A. Ras, and Olli Ikkala\nMolecular Materials, Department of Applied Physics, Schoo l of Science and Technology,\nAalto University, P. O. Box 15100, FI-00076 AALTO, Finland.\nMarkku Oksanen and Eira Sepp¨ al¨ a\nNokia Research Center, It¨ amerenkatu 11-13, 00180 Helsink i, Finland.\nKhattiya Chalapat, Jian Li, and Gheorghe Sorin Paraoanu\nLow Temperature Laboratory, School of Science and Technolo gy,\nAalto University, P. O. Box 15100, FI-00076 AALTO, Finland.\n(Dated: April 11, 2011)\nMost conventional magnetic materials used in the electroni c devices are ferrites, which are com-\nposed of micrometer-size grains. But ferrites have small sa turation magnetization, therefore the\nperformance at GHz frequencies is rather poor. That is why fu nctionalized nanocomposites com-\nprising magnetic nanoparticles ( e.g.Fe, Co) with dimensions ranging from a few nm to 100 nm, and\nembedded in dielectric matrices ( e.g.silicon oxide, aluminium oxide) have a signiﬁcant potentia l\nfor the electronics industry. When the size of the nanoparti cles is smaller than the critical size\nfor multidomain formation, these nanocomposites can be reg arded as an ensemble of particles in\nsingle-domain states and the losses (due for example to eddy currents) are expected to be relatively\nsmall.\nHere we review the theory of magnetism in such materials, and we present a novel measurement\nmethod used for the characterization of the electromagneti c properties of composites with nano-\nmagnetic insertions. We also present a few experimental res ults obtained on composites consisting\nof iron nanoparticles in a dielectric matrix.\nI. INTRODUCTION\nFor a long time have ferrites been the best choice of material for va rious applications requiring magnetic response\nat radio frequencies (RF). In recent times, there has been a stro ng demand both from the developers and the end-\nusers side for decreasing the size of the modern-day portable com munication devices and to add new functionalities\nthat require access to broader communication bands or to other b ands than those commonly used in communication\nbetween such devices. All this should be achieved without increasing power consumption; rather, a decrease would\nbe desired. The antenna for example is a relatively large component o f modern-day communication devices. If the\nsize of the antenna is decreased by a certain factor, then the res onance frequency of the antenna is increased by the\nsame factor [1]. As a result, in order to compensate this increase in t he resonance frequency, the antenna cavity\nmay be ﬁlled with a material in which the wavelength of the external ra diation ﬁeld is reduced by the same factor.\nThe wavelength λinside a material of relative dielectric permittivity ǫand the relative magnetic permeability µis\ngiven by λ=λ0/√ǫµwhereλ0is the wavelength in vacuum. Hence, it is possible to decrease the wav elength inside\nthe antenna - and therefore also the size of the antenna - by incre asing the permittivity or the permeability or the\nboth. Once the size reduction is ﬁxed - that is, ǫµis ﬁxed - the relative strength between the permittivity and\nthe permeability needs to be decided. It is known that the balance be tween these two aﬀects the bandwidth of the\nantenna. Generally speaking, high- ǫand low- µmaterials decrease the bandwidth of the microstrip antenna while\nlow-ǫand high- µmaterials keep the bandwidth unchanged or even increase it [2].\nTypical high- µmaterials are magnetically soft metals, alloys, and oxides. Of these, metals and alloys are unsuitable\nfor high-frequency applications since they are conducting. On the other hand, non-conducting oxides - such as the\nferrites mentioned above - have been used and are still being used in many applications. Their usefulness originates\nfrom poor conductivity and the ferrimagnetic ordering. But ferrit es are limited by low saturation magnetization\nwhich results in a low ferromagnetic resonance frequency and a cut -oﬀ in permeability below the communication\nfrequencies [3]. The ferromagnetic resonance frequency has to b e well above the designed operation frequency to\navoid losses and to have signiﬁcant magnetic response. However, m odern standards such as the Global System for\nMobile communications (GSM), the Wireless Local Area Network (WLA N), and the Wireless Universal Serial Bus\n(Wireless USB) operate in the Super High Frequency (SHF) band or in its immediate vicinity [4]. The frequency\nrange covered by the SHF band is 3-30 GHz and it cannot be accesse d by the ferrites whose resonance frequency is\ntypically ofthe orderof hundreds of MHz [3]. Hence, other kinds of m aterialsneed to be developed for the applications\nmentioned.\nThe important issue related to the miniaturization by increasing the p ermittivity and/or the permeability is the2\nintroducedenergydissipation. Insomecontexts,lossesaregood inasensethattheyreducetheresonancequalityfactor\nand hence increase the bandwidth. The cost is increased energy co nsumption which goes to heating of the antenna\ncavity. In general, several processes contribute to losses in mag netic materials. At low frequencies, the dominant loss\nprocess is due to hysteresis: it becomes less important as the freq uency increases, due to the fact that the motion\nof the domain walls becomes dampened. The eddy current loss plays a dominant role in the higher-frequency range:\nthe power dissipated in this process scales quadratically with freque ncy. In this paper will have a closer look at this\nsource of dissipation, which can be reduced in principle by using nanop articles instead of bulk materials. Another\nimportant process which we will discuss is ferromagnetic resonance (due to rotation of the magnetization).\nAll these phenomena limit the applicability of standard materials for hig h-frequency electronics. However, the\nSHF band may be accessed by the so called magnetic granularmateria ls. A granular material is composed of a non-\nconducting matrix with small (metallic) magnetically soft inclusions. Su ch composites have both desired properties;\nthey are non-conducting and magnetically soft. Granular materials are of special interest at the moment since the\nsynthesis of extremely small magnetic nanoparticles has taken maj or leaps during the past decades. Especially the\nsynthesis of monodisperse FePt nanoparticles [5] and the synthes is of shape and size controlled cobalt nanoparticles\n[6] have generated interest because these particles can be produ ced with a narrow size distribution. In addition,\nsmall nanoparticles exhibit an intriguing magnetic phenomenon called s uperparamagnetism. Superparamagnetic\nnanoparticles are characterized by zero coercivity and zero rema nence which can lead to a decrease in loss in the\nmagnetization process [7].\nThere have been numerous studies investigating dielectric and magn etic responses of diﬀerent granular materials.\nFor example, an epoxy-based composite containing 20% (all percen tages in this article are deﬁned as volume per\nvolume) rod-shaped CrO 2nanoparticles has been demonstrated to have a ferromagnetic re sonance around 8 GHz\nand relative permeability of 1.2 [8]. Similarly, a multimillimetre-large self-as sembled superlattice of 15 nm FeCo\nnanoparticles has been shown to have a ferromagnetic resonance above 4 GHz [9].\nThis raises the interesting question of whether it would be possible in g eneral to design novel nanocomposite\nmaterials with speciﬁed RF and microwave electromagnetic propertie s, aiming for example at very large magnetic\npermeabilities and low loss at microwave frequencies. Such propertie s should arise from the interparticle exchange\ncoupling eﬀects which, for small enough interparticle separation, e xtends over near-neighbour particles, and from\nthe reduction of the eddy currents associated with the lower dimen sionality of the particles. In this paper, we aim\nat evaluating the feasibility of using magnetic polymer nanocomposite s as magnetically active materials in the SHF\nband.\nThe structure of the paper is the following: in Section II we review br ieﬂy the physics of ferromagnetism in\nnanoparticlee, namely the existence of single-domain states (Subs ection IIA), ferromagnetic resonance and the Snoek\nlimit (Subsection IIC), and eddy currents (Subsection IIC). In Se ction III we discuss theoretically issues such as the\nrequirements stated by thermodynamics on the possibility of disper sing nanoparticles in polymers (Subsection IIIA).\nA set of rules governing the eﬀective high-frequency magnetic res ponse in magnetic nanocomposites is developed in\nSubsection IIIB. Then we describe the experimental details and pr ocedures used to prepare and characterize the\nnanocomposites (Section IV). We continue to Section V where we ﬁr st discuss a measurement protocol which allow\nus to measure the electromagnetic properties of the iron nanocom posites (Subsection VA). Finally, as the main\nexperimental result of this paper, magnetic permeability and dielect ric permittivity spectra between 1-14 GHz are\nreported in Subsection VB for iron-based nanocomposites (conta ining Fe/FeO nanoparticles in a polystyrene matrix)\nas a function of the nanoparticle volume fraction. This paper ends w ith a discussion (Section VI) on how to improve\nthe magnetic performance in the SHF band.\nII. MAGNETISM IN NANOPARTICLES\nMagnetic behavior in ferromagnetic nanoparticles is brieﬂy reviewed in this section c.f.[10]-[12]. The focus is\nespeciallyinthesocalledsingle-domainmagneticnanoparticleswhichlac kthetypicalmulti-domainstructureobserved\nin bulk ferromagnetic materials. The topics to be discussed are: A) w hen does the single-domain state appear, B)\nwhat is its ferromagnetic resonance frequency, and C) what are t he sources of energy dissipation in single-domain\nnanoparticles.\nA. Existence criteria for the single-domain state\nA magnetic domain is a uniformly magnetized region within a piece of ferr omagnetic or ferrimagnetic material.\nMagnetic domains are separated by boundary regions called the dom ain walls (DW) in which the magnetization\ngradually rotates from the direction deﬁned by one of the domains t o the direction deﬁned by the other. The domain3\nTABLE I: The saturation magnetization ( MS) [13], the anisotropy energy density K[13] [14], the Q-factor Eq. (3), the single-\ndomain diameter in the hard material approximation dSD,HARDEq. (1), single-domain diameter in the isotropic material l imit\ndSD,SOFTEq. (4), and the domain wall width dDWEq. (2), for iron, cobalt, and nickel. The exchange stiﬀness es used in the\ncalculations are from [15].\nMS K QdSD,HARDdSD,SOFTdDW\n(emu/cm3)(erg/cm3) (nm) (nm)(nm)\nIron (BCC) 1707 4.8×1050.075 5 8963\nCobalt (HCP) 1440 4.5×1060.996 26 169 26\nNickel (FCC) 485−5.7×1040.110 13 173113\nwall thickness ( dDW), which depends on the material’s exchange stiﬀness coeﬃcient ( A) and the anisotropy energy\ndensity ( K), extends from 10 nm in high-anisotropy materials to 200 nm in low-an isotropy materials. The domain\nthickness, on the other hand, depends more on geometrical cons iderations. For example, in one square centimeter\niron ribbon, 10 µm thick, the domain wall spacing is of the order of 100 µm. The spacing increases if the thickness\nis reduced. Reducing the thickness over a critical value leads to the complete disappearance of the domain walls.\nThat state is called the single-domain (SD) state. Between multidoma in and single-domain states there may be a\nvortex state: this is not discussed however here. Similarly, the dom ains in spherical nanoparticles vanish below a\ncertain diameter which is of the order of few nanometers or few ten s of nanometers. In hard materials this diameter\n(dSD,HARD) can be estimated to be roughly ([11], p. 303):\ndSD,HARD≈18√\nAK\nµ0M2\nS, (1)\nwhereMSis the saturation magnetization and µ0is the vacuum permeability. The equation is based on the assump-\ntion that the magnetization follows the energetically favorable direc tions (easy axes or easy planes) deﬁned by the\nanisotropy. The single-domain diameter given by Eq. (1) should be alw ays compared to the domain wall thickness\ngiven by ([11], p. 283)\ndDW=π/radicalbigg\nA\nK. (2)\nIf the diameter of the particle is less than the wall thickness, it is obv ious that it cannot support the wall. The\ncondition dSD,HARD> dDW, leads to the criterion\nQdef=18\nπK\nµ0M2\nS>1. (3)\nOn the other hand, in magnetically soft nanoparticles the magnetiza tion does not necessary follow the easy directions.\nIn the perfectly isotropic case, that is K= 0, the surface spins are oriented along the spherical surface an d a vortex\ncore is formed in the center of the particle if the particle is above the single-domain limit. The single-domain diameter\n(dSD,SOFT) of a perfectly isotropic nanoparticle is given by ([11], p. 305),\ndSD,SOFT≈6/radicalBigg\nA\nµ0M2\nS/bracketleftbigg\nlndSD,SOFT\na−1/bracketrightbigg\n, (4)\nwhereais the lattice constant. This equation can be solved by the iteration m ethod. The single-domain diameter is\nmore diﬃcult to estimate if the anisotropy is non-zero but does not m eet the requirement of Eq. (3). In that case,\nthe single-domain diameter is likely to rest between the values predict ed by Eqs. (1) and (4).\nSingle-domain diameters, domain wall thicknesses and other relevan t physical quantities for selected ferromagnetic\nmetals are shown in Table I. The additional surface-induced anisotr opy has been neglected. The uniaxial hexagonal\nclose packed (HCP) cobalt is the only strongly anisotropic material w ithQ≈1. The body centered cubic (BCC) iron\nand the FCC nickel fall in between hard and soft behavior.4\nB. Ferromagnetic resonance and the Snoek limit\nThe two major processes contributing to the magnetization chang e are the domain wall motion and the domain\nrotation. The resonance frequency of the domain wall motion is typ ically less than the resonance frequency of the\ndomain rotation. Hence, the only process active in the highest freq uencies is the domain rotation which is associated\nwith the ferromagnetic resonance (FMR).\nThe natural [36] ferromagnetic resonance was ﬁrst explained by S noek to be the resonance of the magnetization\nvector (/vectorM) pivoting under the action of some energy anisotropy ﬁeld ( /vectorHA) [16]. The origin of the anisotropy is not\nrestricted. It can be induced, for example, by an external magne tic ﬁeld, magnetocrystalline anisotropy or shape\nanisotropy. It is common to treat any energy anisotropy as if it was due to an external magnetic ﬁeld.\nThe motion of the magnetization around in the anisotropy ﬁeld is desc ribed by the Landau-Lifshitz equation [17],\nd/vectorM\ndt=−ν(/vectorM×/vectorHA)−4πµ0ˆλ\nM2\nS(/vectorM×(/vectorM×/vectorHA)), (5)\nwhereˆλis the relaxation frequency (not the resonance frequency) and νis the gyromagnetic constant given by ([10],\np. 559)\nν=geµ0\n2m≈1.105×105g(mA−1s−1)≈2.2×105mA−1s−1, (6)\nwheregis the gyromagnetic factor (taken to be 2), eis the magnitude of the electron charge and mis the electron\nmass.\nIf the Landau-Lifshitz equation is solved, one obtains the resonan ce condition ([10] p. 559)\nfFMR= (2π)−1νHA, (7)\nwherefFMRis the resonance frequency and HAis the magnitude of the anisotropy ﬁeld.\nFor example, for HCP cobalt the magnetocrystalline anisotropy ene rgy density ( UA) is given by ([10], p. 264)\nUA=Ksin2θ≈K/parenleftbigg\nθ2−1\n3θ4+.../parenrightbigg\n, (8)\nwhereθis the angle between the easy axis and the magnetization. The energ y density due to an imaginary magnetic\nﬁeld is given by ([10], p. 264)\nUA=−µ0HAMScosθ≈ −µ0HAMS/parenleftbigg\n1−1\n2θ2+.../parenrightbigg\n. (9)\nBy comparing the exponents one obtains\nHA=2K\nµ0MS≈0.62 T\nµ0, (10)\nand from Eq. (7)\nfFMR= (2π)−1νHA≈17 GHz. (11)\nIt is tempting to use nanoparticleswith as high anisotropyas possible in orderto maximize the FMR frequency. Un-\nfortunately, the permeability decreases with the increasing anisot ropy; for uniaxial materials the relative permeability\nµis given by ([10], p. 493),\nµ= 1+µ0M2\nSsin2θ\n2K. (12)\nIt is easy to show that that Eqs. (7),(10), and (12) lead to\n∝angbracketleftµ∝angbracketright·fFMR=νMS\n3π, (13)\nwhere∝angbracketleftµ∝angbracketrightis the angular average of the relative permeability (which we assume m uch larger than the unit). This\nequation is known as the Snoek limit. It is an extremely important resu lt since it predicts the maximum permeability5\nTABLE II: Maximum relative permeability ( µ) Eq. (13) achievable in cubic and uniaxial materials with po sitive anisotropy as\na function of the saturation magnetization ( MS) and the FMR frequency ( fFMR).\nSaturation magnetization µ0MS\nfFMR\n(GHz)0.1 T0.3 T0.5 T1.0 T2.0 T\n0.119.757.794.3187.60374.2\n0.54.712.219.738.375.6\n1.02.96.610.319.738.3\n2.01.93.85.710.319.7\n5.01.42.12.94.78.5\nachievable with a given FMR frequency as a function of the saturatio n magnetization. It can be shown to be valid\nfor both the uniaxial and cubic materials (taken that K >0). Some values for the maximum relative permeability as\na function of the FMR frequency and the saturation magnetization are shown in Table II.\nIt has been found out that the Snoek limit can be exceeded in materia ls of negative uniaxial anisotropy [18]. In that\ncase, the magnetization can rotate in the easy plane perpendicular to the c-axis. Such materials obey the modiﬁed\nSnoek limit ([10], p. 561)\nµ·fFMR=νMS\n3π/radicalbigg\nHA1\nHA2, (14)\nwhereHA1is the anisotropy ﬁeld along the c-plane (small) and HA2is the anisotropy ﬁeld out of the c-plane (large).\nOne such material is the Ferroxplana [12].\nC. Eddy currents and other sources of loss\nMagnetic materials can dissipate energy through various processe s when magnetized. When the oscillation period\nof the external driving ﬁeld is long, the main sources of loss are the p rocesses that contribute to the hysteresis. The\nhysteresis loss is linearly proportionalto the frequency of the driv ing ﬁeld since the loss during one complete hysteresis\ncycle (B-Hloop) is proportional to the area within the cycle (assuming that the hysteresis loop does not change with\nthe frequency). The main contribution to the hysteresiscomes fr om the domain wall motion and pinning and a smaller\ncontribution is due to the magnetization rotation and domain nucleat ion. The domain wall motion is damped as the\nfrequency is increased over the domain wall resonance so that only the magnetization rotation persists to the highest\nfrequencies. In addition to the domain rotation hysteresis, the los s in the SHF band stems also from the electrical\ncurrents induced by the changing magnetic ﬁeld inside the particles\nA change in the magnetic ﬁeld ( B) inside a piece of material with ﬁnite resistivity ( ρ) induces an electric ﬁeld which\ngenerates an electric current as stated by the Faraday’s law. This current is called eddy current. It dissipates energy\ninto the sample through the electrical resistance. For example, th e averaged loss power ∝angbracketleftP∝angbracketrightin a spherical nanoparticle\nof radius rcan be calculated to be [37]\n∝angbracketleftP∝angbracketright=2π\n151\nρr5/angbracketleftBigg/parenleftbiggdB\ndt/parenrightbigg2/angbracketrightBigg\n=4π3\n15ρr5/parenleftBig\nfˆB/parenrightBig2\n, (15)\nwherefis the frequency of the driving ﬁeld and ˆBis the amplitude of the oscillating component of the total\nmagnetization. From Eq. (15) it is obvious that the loss power per un it volume increases as r2, indicating that the\nloss can be decreased by using ﬁner nanoparticles. Notice that the loss power will vanish above the FMR resonance\nsince there cannot be magnetic response above that frequency, that isˆB→0.\nFor example, the loss power per unit volume ( p) in cobalt nanoparticles can be calculated to be\np≈32/parenleftBigr\nnm/parenrightBig2/parenleftbiggf\nGHz/parenrightbigg2/parenleftBiggˆB\nT/parenrightBigg2\nW\ncm3. (16)6\nIf the volume of the magnetic element is 0 .1 cm3, the radius of the nanoparticles 5 nm, and ˆB= 1.8 T one obtains\n0.26 mW for loss power.\nIt has been shown that this simple approach is inadequate to describ e the eddy current loss in materials containing\ndomain walls [10]. The eddy currents in multidomain materials are localized at the domain walls, which leads to\na roughly four-times increase in the loss. However, since there are no walls present in single-domain nanoparticles\nand the magnetization reversal can take place by uniform rotation , this model is considered here to be adequate in\ndescribing the eddy current loss in single-domain nanoparticles.\nOne more matter to be addressed is the penetration depth of the m agnetic ﬁeld into the nanoparticles. Because\nthe eddy currents create a magnetic ﬁeld counteracting the magn etic ﬁeld that induced the eddy currents, the total\nmagnetic ﬁeld is reduced when moving from the nanoparticle surface towards its core. The depth ( s) at which the\nmagnetic ﬁeld is reduced by the factor 1/ eis called the skin-depth and it is given by ([10], p. 552),\ns=/radicalbigg2ρ\nωµµ0. (17)\nFor example, from Eq. (17) the skin-depth for cobalt ( ρ= 62 nΩm and µ= 10) at 1 GHz is 1.3 µm and at 10 GHz\n400 nm. Hence, cobalt nanoparticles that are less than 100 nm in diam eter would already be on the safe side. The\nsituation is rather diﬀerent in typical ferrites for which ρ≈104Ωm and µ=103, giving 5 cm for the skin depth.\nTherefore ferrites can be used in the bulk form in near-microwave a pplications.\nIII. MAGNETIC POLYMER NANOCOMPOSITES\nIn the simplest form a polymer nanocomposite is a blend of small partic les (the diameter is less than 100 nm)\nincorporated in a polymeric matrix. Polymer nanocomposites are cha racterized by the convergence of three diﬀerent\nlength scales: the average radius of gyration of the polymer molecu les (RG), the averagediameter of the nanoparticles\n(2r), and the average nearest-neighbor distance between the part icles (d), as shown in Fig. 1. In such composites,\nthe polymer chains may not adopt bulk-like conformations [19]. Assoc iated with this, there can be a change in the\npolymer dynamics which can lead to either an increase or a decrease in the glass transition temperature. Furthermore,\nthe nanoparticles bring their own ﬂavor to the nanocomposite - mag netism, in our particular case.\nd2r\nRG\nFIG. 1: A schematic illustration of a polymer nanocomposite . The average radius of the nanoparticles (2 r) (ﬁlled dark circles),\nthe average radius of gyration of the polymer molecules ( RG) (the thick black line inside the ﬁlled light-gray circle) a nd the\naverage nearest-neighbor distance ( d) between the nanoparticles are of the same magnitude.\nThe most severe problem faced in polymer nanocomposites is the agg regationof nanoparticles. The thermodynamic\nstability of the nanoparticle dispersion has been addressed in the re cent literature experimentally, theoretically and\nthrough computer simulations. The experiments have showed that nanoparticles aggregate even at small particle\nvolume fractions – less than 1% in many compositions [20]. Theoretica l considerations and computer simulations\nhave revealed that the quality of the nanoparticle dispersion depen ds delicately on the balance between the entropic\nand the enthalpic contributions – quite similarly as in polymer blends [21 ]. The solution for the dispersion dilemma\nhas been pursued by modifying the nanoparticle surface, changing the architecture and size of the polymer and by\napplying alternative processing conditions.7\nThe simulation results and the theoretical arguments presented in the literature are often diﬃcult to interpret.\nFurthermore, they do not take into account the magnetic interac tions in magnetic nanocomposites. The aim of\nsubsection IIIA is to analyze the factors aﬀecting the dispersion q uality of magnetic nanoparticles in non-magnetic\npolymers. Subsection IIIB discusses the eﬀective magnetic respo nse of such nanocomposites.\nA. Factors Aﬀecting the Nanoparticle Dispersion Quality\n1. Attractive Interparticle Interactions\nThere has been considerable interest in modifying chemically the nano particle surface towards being more com-\npatible with the polymer [22], [23]. Especially important surface modiﬁca tion techniques are the grafting-techniques.\nThey involve either a synthesis of polymer molecules onto nanopartic le surface (grafting-from) or attachment of func-\ntionalized polymers onto the the nanoparticle surface (grafting-t o). The advantage of the grafting-techniques is that\nthey can make the nanoparticle surface not only enthalpically compa tible with a polymer, but the grafted chains also\nexhibit similar entropic behavior as the surrounding polymer molecules . One disadvantage is that these techniques\nrequire precise knowledge of the chemistry involved.\nIt is well-established that a monolayer of small molecules attached to the nanoparticle surface is not enough to\nsigniﬁcantly enhance the quality of the dispersion even if the surfac e molecules were perfectly compatible with the\npolymer – that is, they were identical to the constitutional units of the polymer. This is due to the fact that the\nLondon dispersion force [38] acting between the nanoparticles is e ﬀective over a length which increases linearly with\nthe nanoparticle diameter. This is proven in the following.\nThe London dispersion energy ( ULONDON) between two identical spheres, diameters 2 r, separated by a distance d\nwas ﬁrst shown by Hamaker to be [24],\nULONDON =−A121\n6\n(2r)2\n2(2r+d)2+(2r)2\n2/parenleftBig\n(2r+d)2−(2r)2/parenrightBig+ln/parenleftBigg\n1−(2r)2\n(2r+d)2/parenrightBigg\n, (18)\nwhereA121is the eﬀective Hamaker for the nanoparticles (phase 1) immersed in the polymeric matrix (phase 2). The\nHamaker constants are typically listed for two objects of the same material in vacuum from which the eﬀective value\ncan be calculated by using the approximation [24]\nA121≈/parenleftBig/radicalbig\nA11−/radicalbig\nA22/parenrightBig2\n, (19)\nwhereA11isthe Hamakerconstantforthe nanoparticlesandA 22is theHamakerconstantforthe medium. The typical\neﬀective Hamaker constant for metal particles immersed in organic solvent or a polymer is approximately 25 ·10−20J.\nBy using this value, the London potential Eq. (18) is plotted for 5 nm metal particles in Figure 2A and for 15\nnm particles in Figure 2B. The distance ( dkBT,LONDON) over which the London dispersion force is eﬀective can be\nestimated by setting the interaction energy equal to the thermal energy and by solving for the distance. The result\nis [39]\ndkBT,LONDON = (α−1)·2r≈2r\n3, (20)\nwhereαis a constant in excess of unity and typically around 1.33 for metals imm ersed in organic medium. This linear\ndependence is shown in Figure 2C. Typically, nanoparticles are cover ed with a monolayer of alkyl chains ranging up\nto 20 carbon-carbon bonds in length. Even if the chains were totally extended and rigid, their length would be only\nroughly 2 nm. Such a shielding layer can protect only nanoparticles les s than 12 nm in diameter from aggregation.\nFortunately, the thermodynamic equilibrium is not solely dependent o n the enthalpy which alwaysdrivesthe system\ntowards the phase separation. The additional component is entro py which opposes the separation. The Gibbs free\nenergy (G) which determines the thermodynamic stability in the constant temp erature and the constant pressure is\ngiven by G=H−TS, whereHis enthalpy and Sis entropy. The entropic term per unit volume in a mixture of\nnanoparticles and small molecular weight solvent molecules can be est imated to be [40]\n−TS\nV=−kBT\nVS/bracketleftbigg\nln/parenleftbiggx\nx−φ/parenrightbigg\n+φ\nxln/parenleftbiggx−φ\nφ/parenrightbigg /bracketrightbigg\n≈ −kBT\nVSφ\nxlnx\nφ, (21)8\n0 5 10 -10 -8-6-4-20\nd (nm) U  (kB T)\n010 20 30 40 50 -50 -40 -30 -20 -10 0\nd (nm) U  (kB T)\n0 5 10 15 010 20 30 \n2r (nm) dk\nB T  (nm)A B\nC D\nd2r \nFIG. 2: Comparison between the London dispersion force and t he magnetic dipolar interaction between two identical meta l\nnanoparticles. A) The reduced London potential Eq. (18) (gr ey thin curve), the magnetic dipolar energy Eq. (24) (black\nthin curve) and the total interaction energy (black thick cu rve) between two 5 nm metal nanoparticles. B) The same for two\nmetal particles 15 nm in diameter. The magnetic dipolar ener gy curve is overlapping with the total interaction curve. C) The\ndistance between the particle surfaces as a function of the p article diameter when the interaction energy is comparable to the\nthermal energy. The black line corresponds to the magnetic i nteraction Eq. (25) and the grey to the London dispersion Eq.\n(20). D) Schematic illustration and deﬁnition of the used va riables.\nwhereVSis the volume of the solvent molecule, φis the volume fraction of the nanoparticles and xis the volume\nratio between a nanoparticle and a solvent molecule. In the case of x= 1 the equation properly reduces to\n−TS\nV=−kBT\nVS[−φlnφ−(1−φ)ln(1−φ)], (22)\nwhich corresponds to the entropy of mixing between two molecules o f the same size.\nFor example, the volume of a toluene molecule is approximately 0 .177 nm3and the volume of a 10 nm nanoparticle\nis 524 nm3. In that case x≈3000. Eq. (21) states that the entropy of mixing is reduced by a fa ctor 1/1300 in a 1%\nnanocomposite when compared to a situation in which both the nanop articles and the solvent molecules were of the\nsame size. Without a proof, it is suggested that the magnitude of th e entropy is even less when the nanoparticles are\nmixed with polymer molecules. The suggestion is justiﬁable due to the e ntropic restrictions introduced by covalent\nbonding between the monomer units.\nIf the nanoparticles are magnetic, they interact with each other m ore strongly than non-magnetic nanoparticles.\nThe magnetic dipolar interaction energy ( UM) between two particles, 2 rin diameter, is given by [10]\nUM=µ0\n4π(d+2r)3(3(m1·/hatwider)(m2·/hatwider)−m1·m2), (23)\nwhere/hatwideris the unit vector between the particles, dis the distance between the particle surfaces and m1andm2are\nthe magnetic moments of the particles. Assuming that the particles are magnetically single-domain, their saturation\nmagnetizationis MSandthatthemagnetizationvectorsareparalleltoeachotherand totheunitvector,theinteraction\nenergy is reduced to\nUM=−8π\n9r6\n(d+2r)3µ0M2\nS. (24)9\nSimilarly to the eﬀective distance of the London dispersion force, on e can derive the distance at which the magnetic\nenergy is comparable to the thermal energy. It is given by\ndkBT,MAGNETIC =/parenleftbigg8π\n9µ0M2\nS\nkBT/parenrightbigg1\n3\nr2−2r. (25)\nTo give an example, the magnetic interaction energy Eq. (24) is draw n for two pairs of cobalt particles, 5 nm and\n15 nm in diameter, in Figures 2A and 2B, respectively. The interaction between the 5 nm particles is dominated\nby the London dispersion potential and only weakly modiﬁed by the ma gnetic interaction. In the case of the 15 nm\nparticles, the magnetic interaction is eﬀective over a distance of 50 nm, rendering the London attraction negligible.\nIn order to shield magnetic nanoparticles from such a long-ranging in teraction with a protective shell is unpractical.\nFirst of all, the maximum achievable nanoparticle volume fraction/parenleftBig\nˆφMAGNETIC/parenrightBig\nis limited by the shielding. If the\nshielding layer volume is not taken to be a part of the nanoparticle volu me, the maximum achievable volume fraction\n(neglecting entropic considerations) is proportional to\nˆφMAGNETIC ∝r3\n(dkBT,MAGNETIC +2r)3∝r−3. (26)\nOn the other hand, the maximum volume fraction/parenleftBig\nˆφLONDON/parenrightBig\nlimited by shielding against the London attraction\ndoes not depend on the nanoparticle size:\nˆφLONDON ∝r3\n(dkBT,LONDON +2r)3= const. (27)\nSecond, the shielding against the magnetic dipolar attraction by usin g the conventional grafting techniques is diﬃcult\ndue to the enormous length required from the grafted chains.\nBased on the considerations presented in this Section, it is unlikely th at a uniform dispersion of magnetic nanopar-\nticles of decent size can be achieved by using the conventional shield ing strategy. The magnetic interaction starts to\ndominate the free energy when the magnetic nanoparticles are 10 n m in diameter or larger. Furthermore, the entropic\ncontribution decreases approximately as x−1wherexis the volume of the nanoparticle relative to the volume of the\nsolvent molecule. Hence, the dispersion dilemma needs to be approac hed from some other point of view than the\nconventional shielding strategy.\n2. Eﬀect of the polymer size, architecture, and functionaliz ation\nA general dispersion strategy proposed by Mackay et al.suggests that the quality of a nanoparticle dispersion is\nstrongly enhanced if the radius of gyration of the polymer is larger t han the average diameter of the nanoparticle\n[20]. The radius of gyration ( RG) for a polymer molecule which is interacting neutrally with its surround ings is given\nbyRG≈/radicalbig\nC/6√\nNa where N is the number of monomers, ais the length of a single monomer and C is the Flory\nratio. For the polystyrene that for example we use the equation yie lds 16 nm for the radius of gyration ( C≈9.9 ,\nN≈2400 and a≈0.25 nm ). It is based on the assumption that small particles can be inco rporated within polymer\nchains easily but large particles prevent chains from achieving their t rue bulk conformations. In other words, large\nparticles stretch the polymer molecules and hence introduce an ent ropic penalty. Pomposo et al. have veriﬁed the\nMackay’s proposition in a material consisting of polystyrene and cro sslinked polystyrene nanoparticles [25]. Such a\nsystem is ideal in a sense that the interaction between the polymer m atrix and the nanoparticles is approximately\nneutral. That emphasizes the entropic contribution to the free en ergy. However, if the main contribution to the free\nenergy is enthalpic, as it is in magnetic nanocomposites, one should us e the Mackay’s proposition with a considerable\ncare. The entropic enhancement is most likely much smaller than the e nthalpic term, rendering the improvement in\nthe dispersion quality negligible.\nOne other remedy for the dispersion dilemma is to replace the linear po lymer by a star-shaped one. It has been\nshown both theoretically [21] and experimentally [26] that it can lead t o a spontaneous exfoliation of a polymer-\nnanoclay composite. It has been also demonstrated that replacing polystyrene in a polystyrene-nanoclay composite\nby a telechelic hydroxyl-terminated polystyrene results in exfoliatio n. Since the polymer-nanoclay composites are\ngeometrically diﬀerent from the polymer-nanoparticle composites, one cannot directly state that these techniques\nwould also work with polymer-nanoparticles composites.10\nB. Eﬀective Magnetic Response\nThe eﬀective relative permeability of a nanocomposite containing sph erical magnetic inclusions can be determined\nfrom several diﬀerent eﬀective medium theories (EMT) [27]. The tw o most popular are the Maxwell-Garnett formula\nµ= 1+3φµNP−1\nµNP+2−φ(µNP−1), (28)\nand the symmetric Bruggeman formula\nµNP−µ\nµNP+2µφ+1−µ\n1+2µ(1−φ) = 0, (29)\nwhereµis the eﬀective relative permeability, µNPis the relative permeability of the nanoparticles and φis the\nnanoparticle volume fraction. The eﬀective relative permeability of a nanocomposite containing spherical particles\n(µNP= 10) is plotted in Fig. 3 according to both Eqs. (28) and (29). Below 2 0% ﬁlling, the dependence of the\npermeability on the volume fraction is approximately linear. However, the rate of the linear increase is not as high as\nwould be expected for homogeneous mixing. The Bruggeman theory has been shown to agree with the experiments\nwith similar materials as studied in this article [28]. Before using the Brug geman theory one needs to know what is\nthe permeability of the nanoparticles. For uniaxial single-grain part icles it is ([10], p. 439)\nµNP,UNIAXIAL = 1+µ0M2\nSsin2θ\n2K(30)\nand for cubic particles\nµNP,CUBIC=\n\n1+µ0M2\nSsin2θ\n2K, K >0\n1−3µ0M2\nSsin2θ\n4K, K <0. (31)\nwhereθis the angle between the easy axis and the external ﬁeld. Permeabilit ies of some ferromagnetic metals are\ncalculated in Table III. It should be pointed out that once again the s urface anisotropy has been neglected, and that\nit is most likely that the experimentally determined permeabilities are sm aller than those in Table III.\n0 0.2 0.4 0.6 0.8 112345678910 \nNanoparticle  volume  fraction    Effective  permeability   !\nFIG. 3: The eﬀective relative permeability of a nanocomposi te containing spherical magnetic inclusions ( µNP=10) as a function\nof the nanoparticle volume fraction. The Maxwell-Garnett t heory prediction (black line) was obtained from Eq. (28) and the\nBruggeman theory prediction (grey line) from Eq. (29).\nThe eﬀective magnetic response of a polymer nanocomposite conta ining single-domain nanoparticles can be deter-\nmined from the following rules:\n•The FMR frequency determines the high-frequency limit of the magn etic response. The FMR frequency is\ndetermined from the eﬀective anisotropy ﬁeld by using Eq. (7).11\nTABLE III: The saturation magnetization MS[13], the anisotropy energy density ( K) [13, 14], and the calculated relative\npermeabilities µNP(from Eq.(30) and Eq.(31)) for selected ferromagnetic meta ls./angbracketleftµNP/angbracketrightrefers to the calculation where the\npermeability has been averaged over the isotropic distribu tion of the easy axes.\nMS K µNP/angbracketleftµNP/angbracketright\n(emu/cm3)(erg/cm3)(θ=π/2)\nIron (BCC) 1707 4.8×10539 26\nCobalt (HCP) 1440 4.5×1064 3\nNickel (FCC) 485−5.7×10440 27\nTABLE IV: Compositions of the nanocomposites prepared for e lectromagnetic characterization.\nDesignation Nanoparticle type ϕ(%)\nPS/QS-Fe 5% Quantum Sphere Iron 5\nPS/QS-Fe 10% Quantum Sphere Iron 10\nPS/QS-Fe 15% Quantum Sphere Iron 14.7\n•The permeability below the FMR is dispersion-free since the only magne tization process taking place in single-\ndomain particles is the domain rotation (which is associated with the FM R).\n•The magnitude of the permeability is determined from the Bruggeman theory, Eq. (29).\n•The permeability of the nanoparticles - which is used in the Bruggeman theory - is determined from the eﬀective\nanisotropy energy density and the saturation magnetization acco rding to the Eqs. (30) and (31).\nTheanisotropyusedinthecalculationsshouldbethetruetotalanis otropy: thesumofthe(bulk) magnetocrystalline\nanisotropy, the surface anisotropy, and the anisotropy due to m agnetic ﬁeld. Especially if the bulk anisotropy is small,\nthe surface anisotropy can be the dominant term. Since the exper imental data on the surface anisotropy is scarce, it\nhas been neglected in the analysis so far.\nIV. PREPARATION AND CHARACTERIZATION\nIn this section we describe the experimental details and procedure s used to prepare and characterize the nanocom-\nposites.\nA. High volume fraction nanocomposites for high-frequenci es\nNanocompositescontainingironnanoparticlesfor the SHF band cha racterizationweremade accordingto the follow-\ning procedure. First, a desired amount of nanoparticles (provided by Quantum Sphere, from now on abbreviated QS)\nwere weighted and mixed with 15 ml of toluene (Fluka, purity better t han 99.7%). The desired amount of polystyrene\nwas added and allowed to dissolve before vigurously sonicating the so lution to break nanoparticle aggregates. The\ntoluene was allowed to evaporate, resulting in a dark polystyrene-lik e ﬁlm which was then collected.\nUsing this method, we have prepared nanocomposites of iron nanop articles, with three diﬀerent concentrations, 5%,\n10% and 15% (see Table IV).\nB. Transmission electron microscopy: structural analysis\nThe nanoparticles were imaged with a TEM (FEI Company model Tecna i G2 BioTwin) in bright ﬁeld at the\nacceleration voltage of 120 kV. Before imaging the alignment of the m icroscope was checked and corrected. The\nimage was recorded with a digital camera (Gatan model UltrascanTM 1000) and its contrast and brightness was\nadjusted after acquisition. An image of iron nanoparticles is shown in Figure 4.12\nFIG. 4: Bright ﬁeld TEM images of the Quantum Sphere iron (sca le bar is 50 nm).\nC. Magnetometry: low-frequency permeability\nStatic hysteresis loops (the magnetization versus the applied ﬁeld) of the nanoparticles were measured with a\nSuperconducting Quantum Interference Device (SQUID) magnet ometer (Quantum Design model MPMS XL) at 300\nK. Roughly 1 mg of the nanoparticles was encapsulated in a piece of alu minum foil (approximately 100 mg) and\nattached to the plastic straw sample holder with Kapton tape. The p ermeability was extracted from the measured\nmagnetization curve by ﬁtting a straight line to the low-ﬁeld part of t he curve. The demagnetizing factor was\napproximated to be zero because the nanoparticles were compres sed into ﬂat layers inside the aluminum wrap and\nthe layer surface was aligned along the external ﬁeld.\nD. X-Ray Diﬀraction: Structure of the Nanoparticles\nThe nanoparticle structure was analyzed with XRD. The diﬀraction in tensities of the nanoparticles were measured\nas a function of the diﬀraction angle 2 θwith a diﬀractometer (PANalytical model X’Pert PRO MRD) using Cu K α\nradiation (wavelength of 0.154056 nm) at room temperature. The X RD patterns of QS-Fe nanoparticles is shown in\nFigure 5. The samples were prepared by ﬁlling a circular cavity (35 mm in diameter and 0.7 mm high) bored into an\nacrylic glass plate with the particles. The powder was compressed an d smoothed with a piece of a silicon wafer. The\nadhesion between the powder and the plate was suﬃcient to hold the powder within the cavity even though the plate\nwas turned vertically for the measurement. The sample was scanne d from 30◦to 90◦for one hour. The resulting\ndata was processed by ﬁrst stripping oﬀ the peaks due to the Cu K α2radiation and by ﬁltering the background noise.\nThe data was smoothed if the signal-to-noise ratio was poor. Secon d, the Lorentzian function was ﬁtted to all peaks\nusing the (self-implemented) Gauss-Newton algorithm. The perfor mance of the algorithm was excellent in the case\nof well-deﬁned peaks, but vague peaks had to be ﬁtted manually. Fr om the ﬁtted peaks the angle, the FWHM and\nthe intensity (integrated over the peak area) were extracted. B ased on these values, the composition of nanoparticles\nwas determined. Furthermore, the coherently scattering domain size was estimated from broadening of the FWHM.\nThe natural width of a peak due to diﬀractometer was determined b y measuring an annealed silicon powder sample\nand assuming that the coherently scattering domains were so large that their contribution to the broadening of the\nFWHM was negligible. The broadening due to the lattice strain was assu med to be minimal.13\n30 40 50 60 70 80 90100101102\n2θ IntensityQS−Fe\nFIG. 5: XRD spectrum of iron nanoparticles.\nTABLE V: Summary of the nanoparticles and their properties. Particle diameters dwere estimated from the TEM images and\nthe crystalline composition and the average crystallite di ameters dCRYSTfrom the XRD measurement.\ndMSµCrystal ϕdcryst\nnm(emu/g) (nm) (%)(nm)\nQuantum Sphere Iron 20-30 12512.3Fe (BCC) 50±59±1\nFeO50±53±1\nE. Summary\nTo summarize our results (see Table V), we ﬁnd that The Quantum Sp here iron (QS-Fe) nanoparticles are roughly\n20-30 nm in diameter and most of the particles exhibit a core-shell st ructure. Based on the XRD analysis, the core is\nsuggestedtocomprise9nmBCCironcrystallitesandthe shell3nmFe Ocrystallites. Thecompositionwasdetermined\ntobea50%-50%balancebetweentheoxideandthemetalphases. T hemeasuredsaturationmagnetization(125emu/g)\nis in rough agreement with the metal volume fraction estimated from XRD and the saturation magnetization given\nin literature for pure iron (218 emu/g) [13].\nV. HIGH-FREQUENCY PROPERTIES\nA. Coaxial airline technique: permittivity and permeabili ty in the SHF band\nAbroadbandcoaxialairlinemethoddevelopedin[31]wasusedtomeas urethecomplexpermittivityandthecomplex\npermeability of magnetic composites in the superhigh-frequency ba nd (SHF). The technique involves measurement of\nthe reﬂection parameters S11andS22, the transmissionparameters S12andS21, and the groupdelay througha sample\ninserted inside a 7 mm precision coaxial airline. The measurement was d one by connecting the coaxial airline to a\nvector network analyzer (Rohde and Schwarz ZVA40) using a pair o f high-performance cables (Anritsu 3671K50-1).\nPrior to the measurement, the errors due to the loss and reﬂectio n in the cables, connectors and the network analyzer\nwere removed by performing a SOLT calibration up to both ends of th e RF cables.\nThe sample required in the coaxial waveguide measurement is a cylinde r, 7.00 mm in diameter, with a 3.04 mm hole\nin the middle. Its thickness can be adjusted between 4 mm and 10 mm in order to avoid the dimensional resonance.\nThe samples were made by hot-pressing each nanocomposite inside a polished 7 mm hole drilled through a steel\nplate. Prior to the pressing, the plate and the nanocomposites insid e the holes were sandwiched between two sheets\nof poly(ethylene terephthalate) and further between two solid st eel plates. The assembly was inserted into a hot-press\n(Fontijne model TP 400) at 160◦C and kept there for two minutes. After the nanocomposite had so ftened, a 400\nkN force was applied over the plates. After waiting for another two minutes, the pressure was released and the plate\nsystem was disassembled. The holes containing the softened and co mpressed nanocomposites were reﬁlled and the\npressing was done again. The ﬁlling was repeated one more time. Afte r the third pressing, the assembly was placed\nbetween two metal plates cooled with circulating water under a 400 k N force. After the plates had cooled down\nto room temperature, the pressure was released and the plates w ere disassembled. Before detaching the solidiﬁed\ncylindrical samples, the plate with the samples was sandwiched betwe en two 5 mm thick steel plates with 3.1 mm\nholes exactly above and under of each of the 7 mm holes in the centra l plate. All the three plates were aligned with\nrespect to each other and clamped together. The assembly was ﬁx ed under a vertical boring machine and holes were\ndrilled through the nanocomposites through the guiding 3.10 mm holes in the upper plate. The used drill bit was 2.9\nmm in diameter since it was found out that the drilling produced holes 0.1 —0.2 mm wider than the drill bit. After14\ndrilling, the construction was disassembled and the samples were det ached by gently pushing them out of the holes. If\nnecessary, the pellets were ﬁnished by carefully removing any imper fections with sandpaper. The sample dimensions\nwere measured with a caliper.\nBelow we brieﬂy present the measurement method; the mathematic al relations between the S-parameters and the\nmaterial parameters are given. More discussions, including detailed error analysis of the method we use, are presented\nin [31]. Basically, the method is developed based on the multiple reﬂectio n model, as shown schematically in Fig. 6.\nFIG. 6: The model of multiple reﬂection between two interfac es. Figure republished with permission ( c/circleco√yrtIEEE 2009) from [31].\nWhen the wave arrives at the ﬁrst interface at z= 0, the reﬂection and transmission occurs. This means part of the\nwave is reﬂected with a coeﬃcient Γ, and part of it is transmitted with a coeﬃcient T21. The transmitted wave then\ntravels through the second medium and gets reﬂected again at the second interface with a coeﬃcient Γ while part of\nit is transmitted through the second interface with a coeﬃcient T12. It can be seen from Fig. 6 that this transmission\nand reﬂection continuously occurs (ideally) an inﬁnite number of time s or until the wave has lost all of its energy.\nTo ﬁnd the total reﬂection coeﬃcient in this model, we need to sum up all the reﬂected waves. The superposition\nof waves can be calculated in the same way as the summation of vecto rs in which both amplitude and phase must be\nconsidered. We know that a wave traveling a distance Lthrough the second medium has a propagation factor given\nby\nP=e−γ2L, (32)\nwhereγ2=iω/v2=iωn2/c.\nThe total reﬂection coeﬃcient can then be expressed as follows\nΓtot= Γ+T21T12ΓP2+T21T12Γ3P4+...=Γ(1−P2)\n1−Γ2P2, (33)\nwhere\nΓ =1−/radicalBig\nǫ2µ1\nǫ1µ2\n1+/radicalBig\nǫ2µ1\nǫ1µ2, (34)\nand\nT12= 1+Γ =2\n1+/radicalBig\nǫ1µ2\nǫ2µ1=/radicalbiggǫ2µ1\nǫ1µ2T21. (35)\nSimilarly, the total transmission coeﬃcient in terms of Γ and Pis\nTtot=P/parenleftbig\n1−Γ2/parenrightbig\n1−Γ2P2. (36)\nIn practice, the study of discontinuities within a transmission line is do ne via the measurement of S-parameters.\nConsidering the measurement setup as illustrated schematically in Fig . 7, we can see that when a wave travels from15\n1                                                              2\nL L L\nPort 1\nCalibration PlanePort 2\nCalibration PlaneAir              Sample           Air\nFIG. 7: The diagram of a transmission line containing two int erfaces and the planes at which scattering parameters are\nmeasured. Figure republished with permission ( c/circleco√yrt2009 IEEE) from [31].\nthe ﬁrst port to the ﬁrst interface, it accumulates a phase chang e of−γ1L1, whereγ1=iωn1/c≈iω/c. Similarly,\nfrom the second interface to the second port, it will pick up anothe r phase change of −γ1L2. This means\nS21=S12=e−γ1(L1+L2)Ttot=e−γ1(L1+L2)P(1−Γ2)\n1−Γ2P2, (37)\nS11=e−2γ1L1Γtot=e−2γ1L1Γ/parenleftbig\n1−P2/parenrightbig\n1−Γ2P2, (38)\nand\nS22=e−2γ1L2Γtot=e−2γ1L2Γ/parenleftbig\n1−P2/parenrightbig\n1−Γ2P2. (39)\nIn principle, there are many ways to obtain the material parameter s based on the above equations. The method\npresented here is chosen because it does not require the measure ment ofL1andL2; as a result, material parameters\ncan be accurately determined.\nThe algorithm proceeds further by ﬁrst deﬁning two reference-p lane invariant quantities, namely\nA=S11S22\nS21S12=Γ2\n(1−Γ2)2(1−P2)2\nP2, (40)\nand\nB=e2γ1(Lair−L)(S21S12−S11S22) =P2−Γ2\n1−Γ2P2. (41)\nNext, Eq. (41) is solved for P2,\nP2=B+Γ2\n1+BΓ2. (42)\nThen, simply by substituting P2into (40), a new expression for Ais obtained,\nA=Γ2(1−B)2\n(B+Γ2)(1+BΓ2), (43)\nwhich gives us\nΓ2=−A(1+B2)+(1−B)2\n2AB±/radicalBig\n−4A2B2+[A(1+B2)−(1−B)2]2\n2AB, (44)\nwhere the sign in this equation is chosen so that |Γ| ≤1. As we can see, these expressions for P2and Γ2are\nreference-plane invariant.16\nIn the next step, another quantity is deﬁned, namely\nR=S21\nSo\n21=eγ1LP(1−Γ2)\n1−P2Γ2. (45)\nSubstituting P2from Eq. (42) into Eq. (45), we get a new expression for P,\nP=R1+Γ2\n1+BΓ2e−γ1L. (46)\nBy using Eqs. (44) and (46), we can determine the other constitut ive parameters of materials, for example the\ncomplex index of refraction,\nn=n′+in′′=√µrǫr=1\nγ1Lln/parenleftbigg1\nP/parenrightbigg\n. (47)\nSimilartotheNicolson-RossWeiralgorithm,[32]-[33], themethodrequir estheevaluationofgroupdelayforchoosing\nthe correct result. But, it should be noted that, only the real par t ofn, in Eq. (47), is multi-valued, the imaginary\npart is not, i.e.every root provides the same n′′. So measuring the imaginary part of the index of refraction does\nnot require the evaluation of the group delay. This concept could be practically useful for examples when energy loss\nresonances are studied.\nIn case of non-magnetic materials, determining the complex permitt ivity from ǫr=n2provides a better alternative\nrelative to the NRW method. This is because, this way, one does not n eed to calculate the relative wave impedance\nz= (1+Γ) /(1−Γ), which exhibits high errors at and around the Bragg resonance f requencies [34].\nAs discussed in [31], extrasteps must be done if this method is applied t o measurematerialswith unknown magnetic\nproperties. One way to do so, is to simply use one of the roots ±Γ of Eq. (44), and simultaneously plot the spectra\nof bothǫrandµr. Then, based on chemical analysis, the permeability spectra can be extracted. This algorithm is\nbased on the fact that the sign of Γ only swaps the values of permitt ivity and permeability.\nB. Nanocomposites: permittivity and permeability in the SH F band\nWe now present our experimental results corresponding to a nano composite comprising iron nanoparticles (20–30\nnm, BCC) in polystyrene (PS/QS-Fe) (Figures 9,8,10).\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s48/s46/s57/s49/s46/s48/s49/s46/s49/s49/s46/s50/s49/s46/s51/s49/s46/s52/s49/s46/s53\n/s32/s32/s53/s32/s37\n/s49/s48/s32/s37\n/s49/s53/s32/s37/s73/s114/s111/s110/s114/s39\n/s102/s32/s32/s91/s71/s72/s122/s93/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s114/s34\n/s102/s32/s32/s91/s71/s72/s122/s93/s73/s114/s111/s110/s32/s53/s32/s37\n/s49/s48/s32/s37\n/s49/s53/s32/s37\nFIG. 8: The complex relative permeability (real part in the l eft ﬁgure and imaginary part in the right ﬁgure) of PS/QS-Fe\nnanocomposites between 2 and 12 GHz. The black data points ar e from the 5% sample, the red data points from the 10%\nsample and the green data points from the 15% sample.\nAll the composites 5%, 10% and 15% exhibited mild ferromagnetic reso nances between 6 GHz and 8 GHz. These\nresonances correspond to the anisotropy ﬁelds between 0.20 T an d 0.27 T. The expected anisotropy ﬁeld calculated17\nfrom bulk BCC iron magnetocrystalline anisotropy is 56 mT. This large d iﬀerence may be explained by additional\nanisotropy components due to surface eﬀects or particle-partic le interactions. In the case of surface anisotropy the\nbroadening of the resonance peak would be due to the ﬁnite size dist ribution of the nanoparticles, and in the case\nof particle-particle interactions due to variations of the polarizing ﬁ eld due to irregular spatial arrangement and\norientation of the particles. The Snoek limit (Eq. 13) predicts that n o higher relative magnetic permeability than 8.5\ncan be achieved at 5 GHz in (positive) uniaxial and cubic materials which are either bulk or composites containing\nspherical inclusions. According to the Bruggeman theory (Eq. 29) the eﬀective relative permeability is at maximum\none sixth of 8.5 in a nanocomposite containing less than 15% magnetic in clusions. We ﬁnd a relative permeability\nof the order µ= 1.3 in the 15% nanocomposite, which is already pushing the Snoek limit. Th e exact determination\nof whether the Snoek limit has been exceeded depends on the natur e of the anisotropy which would require precise\nknowledge of the surface contribution.\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s50/s52/s54/s56\n/s32/s49/s53/s32/s37\n/s32/s49/s48/s32/s37/s73/s114/s111/s110/s114/s39\n/s102/s32/s32/s91/s71/s72/s122/s93/s32/s53/s32/s37\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s114/s34\n/s73/s114/s111/s110\n/s32/s32/s32/s53/s32/s37\n/s32/s49/s48/s32/s37\n/s32/s49/s53/s32/s37\n/s102/s32/s32/s91/s71/s72/s122/s93\nFIG. 9: The complex relative permittivity (real part in the l eft ﬁgure and imaginary part in the right ﬁgure) of PS/QS-Fe\nnanocomposites between 2 and 12 GHz. The black data points ar e from the 5% sample, the red data points from the 10%\nsample and the green data points from the 15% sample.\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s110 /s39\n/s102/s32/s32/s91/s71/s72/s122/s93/s73/s114/s111/s110\n/s53/s32/s37/s49/s48/s32/s37/s49/s53/s32/s37\n/s50 /s52 /s54 /s56 /s49/s48 /s49/s50 /s49/s52 /s49/s54 /s49/s56/s45/s48/s46/s54/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s110 /s34\n/s73/s114/s111/s110\n/s102/s32/s32/s91/s71/s72/s122/s93/s53/s32/s37\n/s49/s48/s32/s37\n/s49/s53/s32/s37\nFIG. 10: The complex index of refraction (real part in the lef t ﬁgure and imaginary part in the right ﬁgure) of PS/QS-Fe\nnanocomposites between 2 and 12 GHz. The black data points ar e from the 5% sample, the red data points from the 10%\nsample and the green data points from the 15% sample.18\nThe permeabilities (both real and imaginary) in all composites were ro ughly constants over the measurement range.\nWhile the real part increased roughly linearly with the volume fraction , there was much larger jump in the imaginary\npart from 10% to 15% than from 5% to 10%. This behavior could be att ributed to the increase in conductivity due\nto exceeding the percolation threshold.\nThe relative permittivities were observed to increase with the nanop article volume fraction from approximately 2.5,\nwhich is a typical value for polystyrene. The imaginary parts were ob served to be signiﬁcantly larger than 10−4(a\ntypical value for pure polystyrene). The increase in both the real part and the imaginary part is understood to be\ndue to the electrical polarizability of the nanoparticles in the electric ﬁeld. The dispersion for the 5% and the 10%\ncomposites was also comparatively lower than for the 15% sample (Fig ure 10).\nVI. CONCLUSIONS: HOW TO IMPROVE THE PERFORMANCE IN THE SHF BA ND\nThehigh-frequencymagneticperformanceisalwaysacompromiseb etweenthepermeabilityandtheFMRfrequency.\nIncreasing the magnetic anisotropy, no matter wherefrom it origin ates, decreases the permeability but increases the\nFMR frequency. Onlyincreasingthesaturationmagnetizationincre asesboththe permeabilityandtheFMR frequency\n(Eq. 13). Hence, the saturation magnetization should be maximized while a compromise needs to be done with the\nanisotropy. Further degree of freedom stems from the shape of the magnetic inclusions. It is well known that the\nresonance frequency of an arbitrary magnetic body with the dema gnetization factors Nx,NyandNzis given by the\nKittel formula [12]\nfFMR= (2π)−1ν/radicalBig\n[HA+(Nx−Nz)MS][HA+(Ny−Nz)MS], (48)\nfrom which the well known resonance formulas for bulk, ﬁlm, rod and sphere can be derived ( νis the gyromagnetic\nconstant, deﬁned as in Eq. (6)). In the cases of spheres ( Nx=Ny=Nz= 1/3) and bulk material ( Nx=Ny=\nNz= 0), the resonance frequency is directly proportional to the anis otropy ﬁeld (as was assumed in Subsection IIB).\nIn the inﬁnite rod limit ( Nx=Ny= 1/2,Nz= 0) the FMR frequency is linearly proportional to the saturation\nmagnetization (assuming that MS≫HA) and in the thin ﬁlm limit ( Nx=Ny= 0,Nz= 1) to the square root of\nthe saturation magnetization and the anisotropy ﬁeld (assuming th atMS≫HA). Even in the case of HCP cobalt,\nwhich has a high anisotropy ﬁeld of µ0HA≈0.63 T, the highest resonance frequency is obtained in the non-isotr opic\ngeometries, namely the inﬁnite rod and the thin ﬁlm. The same conclus ion is valid for the BCC iron ( µ0HA≈56 mT),\nand the FCC nickel ( µ0HA≈16 mT). However, due to the surface anisotropy, in the end the an isotropy ﬁeld in\nnanoscale rods, spheres and ﬁlms can be much larger than in bulk. It should be understood that the FMR frequency\ndepends on the shape of the magnetic inclusions, but obtaining qualit ative results from calculations without knowing\nthe surface anisotropy is not possible (as already pointed out in Sub section IIB). In addition, the FMR frequency\ndepends on particle-particle interactions.\nThe volume fraction of the inclusions has obviously an eﬀect on both t he resonance frequency and the permeability.\nThe permeability of a nanocomposite with spherical inclusions can be c alculated directly from the Bruggeman theory\n(Subsection IIIB) but in all the other cases the Bruggeman equat ion must be solved iteratively and self-consistently\nwith the Landau-Lifshitz equation [35]. The results from such calcula tions indicate that 1) the FMR frequency of a\nnanocomposite containing spherical inclusion does not depend on th e volume fraction and 2) in all other cases the\nFMR frequency smoothly varies from the single-inclusion limit to the ho mogeneous bulk limit. Hence, the set of\nrules for predicting the high-frequency magnetic performance st ated in Subsection IIIB are valid only for spherical\nnanoparticles. As argued above, the FMR resonance is the lowest in the bulk and spherical particle limits. The results\npresented in Subsection VB agree with the literature in a sense that the FMR frequency was found out to vary only a\nlittle with the nanoparticle volume fraction. The small variation might b e due to slight deviations from ideal spherical\nform or due to the aggregation of the nanoparticles.\nThe magnetic performancein nanocompositescould be improvedbyt aking all the aboveconsiderationsinto account\nwhen designingthe material. In addition, it is preferentialto use mon odisperse, single-crystalnanoparticlesin orderto\nobserve well-deﬁned resonancepeaks. Without such information, quantitative evaluation of the magnetic performance\nis diﬃcult. Also, the surface eﬀects such as the surface anisotrop y have to be taken into account since nanoparticles\nhave a huge surface-to-volume ratio compared to bulk materials. B ecause the magnitude and the symmetry of the\nsurface anisotropy is diﬃcult to calculate, it cannot be taken in prac tice into consideration before the measurement.\nInstead, it is the deviation of the observed FMR frequency from th e value expected from bulk magnetocrystalline\nanisotropy that indicates the magnitude of the surface anisotrop y. After having decided the target permeability and\nresonance frequency and having approximated the type and the v olume fraction of the magnetic inclusions, one still\nneeds to ﬁnd the appropriate processing route which can lead to su ch a nanocomposite. As argued in this article, and\nalso accepted in literature, homogeneous blends of plain nanopartic les in polymers are almost impossible to achieve19\neven at the lowest ﬁlling ratios. In the high-frequency applications t he role of the nanoparticles is not just an additive\nsince the practical volume fractions (from the application perspec tive) begin from 10%. Hence, the nanoparticles\nshould be a supporting part of the nanocomposite — not an additive.\nSuppressing the large permittivity and dielectric loss will be a diﬃcult ta sk. The imaginary losses can be reduced\nby using single-crystal nanoparticles in which the conduction electr on scattering is suppressed. Tackling the real part\nis much more diﬃcult, since all metallic nanoparticles are highly conduct ive and their polarizability should of the\nsame order.\nVII. ACKNOWLEDGEMENTS\nThis work was supported by the Finnish Funding Agency for Technolo gy and Innovation (TEKES). G.S.P. would\nlike to acknowledge also partial support from the Academy of Finland (Acad. Res. Fellowship 00857 and projects\n129896, 118122, and 135135). K.C. wishes to thank the Thailand Co mmission on Higher Education for ﬁnancial\nsupport.\n[1] Y. Shirakata, et al., ”High Permeability and Low Loss Ni-Fe Composite Material f or High-Frequency Applications”, IEEE\nTransactions on Magnetics, 44, 2100 (2008).\n[2] R. C. Hansen and M. Burke, ”Antennas with magneto-dielec trics”, Microwave and Optical Technology Letters 26, 75\n(2000).\n[3] R. A. Waldron, Ferrites: an introduction for microwave e ngineers, (Van Nostrand, London, 1961).\n[4] ”Super high frequency”, http://en.wikipedia.org/wik i/Super highfrequency (2.2.2009).\n[5] S. Sun, et al., ”Monodisperse FePt Nanoparticles and Ferromagnetic FePt Nanocrystal Superlattices”, Science 287, 1989\n(2000).\n[6] V. F. Puntes, et al., ”Colloidal Nanocrystal Shape and Size Control: The Case of Cobalt”, Science 291, 2115 (2001).\n[7] S. Moerup and M. F. Hansen, ”Fundamentals and Theory”, in Handbook of Magnetism and Advanced Magnetic Materials,\nedited by H. Kronmueller and S. Parkin, (John Wiley & Sons, 20 07).\n[8] L. Z. Wu, et al., ”Studies of high-frequency magnetic permeability of rod- shapes CrO2 nanoparticles”, Phys. Stat. Sol.\n204, 755 (2006).\n[9] C. Desvaux, et al., ”Multimillimetre-large superlattices of air-stable iro n-cobalt nanoparticles”, Nat. Mater. 4, 750 (2005).\n[10] S. Chikazumi, Physics of Ferromagnetism, (Oxford Univ ersity Press, New York, 1997).\n[11] R. C. O’Handley, Modern Magnetic Materials, (John Wile y & Sons, Inc., 2000).\n[12] C. Kittel, Introduction to Solid State Physics, (John W iley & Sons, New York, 1971).\n[13] C. M. Sorensen, Magnetism in Nanoscale Materials in Che mistry, edited by K. J. Klabunce, (Wiley-IEEE, 2001).\n[14] S. J. Steinmuller, et al., ”Eﬀect of substrate roughness on the magnetic properties o f thin fcc Co ﬁlms”, Physical Review\nB76, (2007).\n[15] P. K. George and E. D. Thompson, ”Exchange Stiﬀness in He xagonal Cobalt”, Physical Review Letters 24, 1431 (1970).\n[16] J. L. Snoek, ”Dispersion and absorption in magnetic fer rites at frequencies above one Mc/s”, Physica 15, 207 (1948).\n[17] L. Landau and E. Lifshitz, ”On the theory of the dispersi on of magnetic permeability in ferromagnetic bodies’”, Phy s. Z.\nSowjetunion 8, (1935).\n[18] G. H. Jonker, et al., Philips Tech. Rev. 18, 1956 (1956).\n[19] R. Krishnamoorti and R. A. Vaia, ”Polymer Nanocomposit es”, Journal of Polymer Science: Part B: Polymer Physics 45,\n3252 (2007).\n[20] M. E. Mackay, et al., ”General Strategies for Nanoparticle Dispersion”, Scien ce311, 1740 (2006).\n[21] A. C. Balazs and S. Chandralekha, ”Eﬀect of polymer arch itecture on the miscibility of polymer/clay mixtures”, Pol ymer\ninternational 49, 469 (2000).\n[22] E. Glogowski, et al., ”Functionalization of Nanoparticles for Dispersion in Po lymers and Assembly in Fluids”, Journal of\nPolymer Science: Part A: Polymer Chemistry 44, 5076 (2006).\n[23] A. H. Latham and M. E. Williams, ”Controlling Transport and Chemical Functionality of Magnetic Nanoparticles”, Ac c.\nChem. Res. 41, 411 (2008).\n[24] J. Israelachvili, ”Intermolecular and Surface Forces ”, (Academic Press, London, 1985).\n[25] J. A. Pomposo, et al., ”Key role of entropy in nanoparticle dispersion: polystyr ene-nanoparticle/linear-polystyrene\nnanocomposites as a model system”, Phys. Chem. Chem. Phys. 10, 650 (2007).\n[26] C. Barnes, et al., ”Spontaneous Formation of an Exfoliated Polystyrene-Cla y Nanocomposite Using a Star-Shaped Poly-\nmer”, J. Am. Chem. Soc. 126, 8118 (2004).\n[27] Sihvola, ”Electromagnetic mixing formulas and applic ations”, (The Institution of Electrical Engineers, 1999).\n[28] J. H. Paterson, et al., ”Complex permeability of soft magnetic ferrite/polyeste r resin composites at frequencies above 1\nMHz”, Journal of Magnetism and Magnetic Materials 196-197 , 394 (1999).\n[29] H. P. Klug and L. E. Alexander, X-Ray Diﬀraction Procedu res, (John Wiley and Sons, 1954).20\n[30]˚A. Bjrck, Numerical methods for least squares problems, (SI AM, Philadelphia, 1996).\n[31] K. Chalapat, K. Sarvala, J. Li, and G. S. Paraoanu, ”Wide band Reference-Plane Invariant Method for Measuring Elect ro-\nmagnetic Parameters of Materials”, IEEE Trans. Microw. The ory Tech. 57, 2257 (2009).\n[32] A. M. Nicolson and G. F. Ross, ”Measurement of the intrin sic properties of materials by time-domain techniques”, IE EE\nTrans. Instrum. Meas., IM-19, pp. 377-382, Nov. 1970.\n[33] William B. Weir, ”Automatic measurement of complex die lectric constant and permeability at microwave frequencie s”,\nProc. IEEE, 62, pp. 33-36, Jan. 1974.\n[34] A.-H. Boughriet, C. Legrand and A. Chapoton, ”Nonitera tive stable transmission/reﬂection method for low-loss ma terial\ncomplex permittivity determination”, IEEE Tran. Microw. T heory Tech., 45, pp. 52-57, Jan. 1997.\n[35] R. Ramprasad et al., ”Fundamental Limits of Soft Magnetic Particle Composites for High Frequency Applications”, Phys.\nStat. Sol. 233, 31 (2002).\n[36] ”Natural” is used so that this resonance is distinguish ed from the dimensional resonance associated with standing waves\nwithin a sample of ﬁnite size. The dimensional resonance tak es place when the end-to-end length ( L) of the sample times\ntwo is equal to an integer multiple of the wavelength of the ra diation ( λ) in the material. The same is mathematically\nexpressed as L=nλ/2 =n/2λ0/√ǫµwhereλ0is the wavelength in vacuum and nis a positive integer. The resonance\nfrequency ( fr) is given by fr=c/λ0=nc/2L√ǫµwherecis the speed of light. For example, the ﬁrst dimensional reso nance\nin a 10 mm long sample with ǫµ= 9 is approximately 5 GHz. The dimensional resonance can be a voided in experiments\nby carefully estimating the product ǫµand designing the sample length accordingly.\n[37] Assume thata spherical nanoparticle (radius r, resistivity ρ), is placed in an alternating magnetic ﬁeld so thatthe magne tic\nﬁeld inside the particle is B. According to the Faraday’s equation 2 πxE(x) =−πx2dB/dtwherexis the distance from\nthe particle center and E(x) is the electric ﬁeld. The diﬀerential current dIcirculating around the cylindrical shell at\nthe distance xis given by dI= 2πxE(x)/dRwhere the diﬀerential resistance dRis given by dR=ρ2πx/h(x)dxwhere\nh(x) is the height of the cylindrical shell and dxis the thickness of the shell. Now the dissipated power can be calculated\nfromP=/integraltext\n2πxE(x)dI= (π/ρ)(dB/dt)2/integraltextr\n0x3√\nr2−x2dx. Changing the integration variable from xtorsinϕsimpliﬁes\nthe integral to P= (π/ρ)(dB/dt)2r5/integraltextπ/2\n0/parenleftbig\nsin3ϕ−sin5ϕ/parenrightbig\ndϕwhere the integral part is equal to 2/15. Assuming that the\nmagnetic ﬁeld is given by B=ˆBsin2πfleads to the result given in Eq. (15).\n[38] Despite of its name it is an attractive force.\n[39] By setting ULONDON =−kBTand deﬁning a reduced variable α= 1 +dkBT,LONDON/2rone can rewrite Eq. (18) as\n6kBT/A121= 1/(2α2)+1/[2/parenleftbig\nα2−1/parenrightbig\n] +ln/parenleftbig\n1−1/α2/parenrightbig\n≡f(α) . This equation can be solved for αby plotting y=f(α)\nandy= 6kBT/A121and by locating the point of intersection. The eﬀective dist ance can be calculated by inserting the\nobtained intersection point into the equation deﬁning the r educed variable.\n[40] Assume that an arbitrary lattice of N sites is ﬁlled with N1nanoparticles and N2solvent molecules. Each nanoparticle\nincorporates x lattice sites and each solvent molecule one l attice site. Then the number of microstates (Ω) is approxima tely\nΩ≈N!/((N−N1)!N1!) . Notice that N−N1/negationslash=N2in contrast to the mixing theory of small molecules of same si ze. The\nEq. (23) is obtained from the deﬁnition of entropy S=kBlnΩ by simple algebraic manipulation and by assuming that\nthe density of the nanoparticles is low ( N1≪N)."    },    {        "title": "1111.4027v2.Field_induced_Magnetic_Transition_in_Cobalt_Ferrite.pdf",        "content": "arXiv:1111.4027v2  [cond-mat.mtrl-sci]  12 Jan 2012MMM2011/CU-10\nField-induced Magnetic Transition in Cobalt-Ferrite\nMartin Kriegisch,∗Reiko Sato-Turtelli, Herbert M¨ uller, and Roland Gr¨ ossinger\nInstitute of Solid State Physics, Vienna University of Tech nology\nWiedner Hauptstrasse 8-10/E138, A-1040 Vienna, Austria\nWeijun Ren and Zhidong Zhang\nShenyang National Laboratory for Materials Science,\nInstitute of Metal Research, and International Center for M aterials Physics,\nChinese Academy of Sciences, Shenyang 110016, People’s Rep ublic of China\n(Dated: November 9, 2018)\nWe present magnetostriction and magnetization measuremen ts of a cobalt ferrite (Co 0.8Fe2.2O4)\nsingle crystal. We observe unusual behaviour in the magneti c hard axis of the single crystal which\nmanifests in a jump of the magnetization curve at a critical ﬁ eld. This ﬁrst order magnetization\nprocess (FOMP) which is explained as an anisotropy driven tr ansition is visible at temperatures\nlower than 150 K. By analyzing the anisotropy constants we fo und that the higher order anisotropy\nconstant K2dominates the anisotropy energy. In the magnetostriction m easurements the FOMP is\nclearly visible as a huge jump in the [111] direction, which c an be explained by means of a geometric\nmodel.\nPACS numbers: 75.80.+q, 75.47.Lx, 75.30.Kz, 75.50.-y\nKeywords: Magnetostriction, Cobalt ferrite, FOMP, Anisot ropy\nINTRODUCTION\nCobalt Ferrite is under examination since more than\nsixty years, but still there is quite a lot of open prob-\nlems. In 1988 Guillot observed a jump in the ﬁeld depen-\ndence of the magnetization in pure Cobalt Ferrite with\nthe composition Co 1.04Fe1.96O4and also in Cd substi-\ntuted Cobalt Ferrite [1]. This jump was explained as\na spin-ﬂip. Because of the rather low critical ﬁeld this\nexplanation was not conclusive. Therefore within the\npresent work this transition was studied in more detail.\nSAMPLE PREPARATION\nThe single crystal with the composition Co 0.8Fe2.2O4\nwas grown by ﬂux method. The starting materials are\n18 g Na 2B4O7·10H2O (Borax), 2.3 g CoO (99.99%),\nand 6.7 g Fe 2O3(99.99%). After suﬃciently mixing, the\nmaterials were put in a tightly closed Pt crucible and\nheated from room temperature to 1370◦C at a rate of\n100◦C/h, and then held for a period of 6 h; slowlycooled\nfrom 1370 to 990◦C at 2◦C/h, followed by a furnace\ncooling by switching oﬀ the power supply [2]. The com-\nposition of the single crystal was checked by XRD and\nSEM investigation.\nEXPERIMENTAL PROCEDURES\nMagnetization was measured from 5 to 400 K in a\nvibrating sample magnetometer with a superconducting\n9 T coil. The single crystal was oriented by XRD in\na Laue setup and then transfered to the VSM sampleholder. The error from transferring the single crystal\nfrom one sample holder to the other was usually smaller\nthan 1.5◦.\nThe magnetostriction was measured with a miniature\ncapacitive dilatometer described in [3] using a cryostat\nwith a variable temperature insert (VTI) and a super-\nconducting 9 T coil.\nRESULTS AND DISCUSSION\nMagnetization\nThe degreeofinversionofthe cationdistribution ofthe\ninverse spinel (A2+)[B3+\n2]O4was calculated to i= 0.625\nwith (Co2+\n0.8−iFe3+\n0.2+i)[Co2+\niFe3+\n2−i]O4and the magnetic\nmoment of µ=µB−Sites−µA−Sites= 4.1µBatT= 5 K.\nThe value of saturation magnetization at 5 and 10 K is\npractically the same due to the the very high ordering\ntemperature.\nThe magnetization measurements revealed that the\n[100] axis is the easy axis of magnetization of Co-ferrite\nover the whole temperature range. Below 150 K a jump\nin the magnetization at ﬁelds occurs as it was also found\nin a similar material (Co 1.04Fe1.96O4) in Ref. [1]. In\nﬁgure 1 the normalized magnetization at T= 10 K is\nplotted versus the external magnetic ﬁeld. The jump\nis clearly visible at roughly 7 T in the [111] axis. The\ncritical ﬁeld diﬀers from that published by Guillot which\ncan be understood regarding the diﬀerent sample com-\nposition. A linear ﬁt between 1 and 6 T was performed\nshowing that the extrapolation to 0 T leads to a value of\n1√\n3, which indicates that the magnetization vector lies in2\nFIG. 1. Normalised Magnetization of Co 0.8Fe2.2O4single\ncrystal at T= 10 K\nthe[100]axisandonlytheprojection( cos(α)) ofthe[100]\nmagnetization vector into the [111] axis is measured. By\nincreasing the magnetic ﬁeld the vector starts to rotate\ntowards the [111] axis. However as soon as the measured\nmoment in the [111] axis achieves the value of1√\n2, the\nmeasured moment becomes equal to the [110] value mea-\nsured at 0 T. At this point the system needs more energy\n(ﬁeld) to overcomethe magnetic anisotropyenergycorre-\nsponding to the [110] axis. As a consequence the magne-\ntization vector rotates in the [110] axis to a local energy\nminimum in order to reduce energy. In ﬁgure 2 the oﬀset\nof the linear ﬁts assuming such a geometrical model be-\nlow the transition are shown. The error due to misalign-\nment is found to be 1.5◦for the [111] axis and 0.3◦for\nthe [110]axis. Obviouslyour geometric model works well\nfrom 5 K up to 250 K. Above 250 K the anisotropy along\nthe [110] and [111] axis decreases and the magnetization\nvector can rotate smoothly (without a discontinuity) out\nof the easy axis.\nBy this geometrical considerations one can see that the\norigin of the magnetization jump is not a spin-ﬂip, but\na rotation in the magnetization due to the higher or-\nder anisotropyconstantswhich cause amore complicated\nshape of the anisotropy energy EA.\nSuch anisotropy driven magnetization jumps are gener-\nally called FOMP (First Order Magnetization Process)\nand were also found in other complex systems and alloys\n(as e.g. PrCo5 - see [4], RE 2Fe17C [5], REFe 11Ti [6] or\nNd2Fe14B [7]).FIG. 2. Oﬀset of the linear ﬁts of normalized magnetization\nof Co 0.8Fe2.2O4as a function of temperature. The straight\nlines indicate the geometrical ﬁt\nMagnetic Anisotropy\nThe magnetic anisotropy was determined with the in-\ntegralmethod EA=/integraltextMS\n0HdMalongthe measuredcrys-\ntallographic axis of the single crystal. Applying the for-\nmulaEA=K0+K1(α2\n1α2\n2+α2\n2α2\n3+α2\n3α2\n1)+K2(α2\n1α2\n2α2\n3),\nwhere the αiare the direction cosines in polar coordi-\nnates, the anisotropy constants K0,K1andK2were\ndetermined. For using the integral method it is pre-\nrequisite to saturate the sample fully. It is reported that\nthe saturation along the intermediate and hard axis is\nreached at around 18 T at T= 4 K [1]. Our measure-\nments were only performed up to 9 T, therefore extrapo-\nlatingthe M(H) curvestosaturationcausesaratherlarge\nerror (±20% for K1and±30%K2), but they are still\nin the range of reported values of K1in literature.\nIn ﬁgure 3 the anisotropy constants K0,K1andK2of\nthe single crystal and K1of a polycrystalline sample are\nplotted against the temperature. For obtaining K1for\nthe polycrystalline sample we used the law of approach\nto saturation. Measuring the polycrystalline sample the\nmaximum magnetic ﬁeld was only 9 T and the sample\nnot fully saturated. Accordingly all reported values in\nliterature using the law of approach to saturation are\nnot giving the correct value of K1, because they estimate\nonlyK1and not higher order anisotropy constants.\nIt isinterestingto notethat novalues for K2arereported\nin literature. Above T= 150 K the value of K2is∼6\ntimes higher than K1and below the FOMP the factor\nis even much higher, yielding in an increased anisotropy\nalong the [111] and [110] axes.3\nFIG. 3. Anisotropy constants of Co 0.8Fe2.2O4single crystal\nand one polycrystalline specimen as afunction of temperatu re\nFIG. 4. Magnetostriction measurements of Co 0.8Fe2.2O4sin-\ngle crystal at T= 4.2K\nMagnetostriction\nWe are presenting the ﬁrst magnetostriction mea-\nsurement at T= 4.2 K demonstrating also the eﬀect\nof such a ﬁeld induced transition in the magnetoelastic\nbehaviour as shown in ﬁgure 4. Due to hysteresis eﬀects(remanence) the magnetostriction measurements at\nlow temperatures are very diﬃcult and need a special\nmeasuring procedure as will be published elsewhere. But\nwhen increasing the magnetic ﬁeld above the critical\nﬁeld, a huge jump in the magnetostriction occurs. At\nhigher ﬁelds due to rotational magnetization process no\nhysteresis eﬀects are observed.\nCONCLUSIONS\nWe have proved that the jump in the magnetization\ncan be explained as an anisotropy driven transition\nwhich is called “FOMP”. The transition is caused by a\nrotation of the magnetization vector jumping over an en-\nergy barrier. At low temperatures the second anisotropy\nconstant K2is increasing which strengthens the [100]\naxis as easy axis and underlines the [110] as intermediate\nand the [111] as hard axis. These assumptions are also\nsupported by geometric considerations explaining the\nvalues of the M(H) curves as measured in the diﬀerent\ncrystallographic directions. Additionally we present an\naccurate measurement of λ111at a temperature of 4.2 K\nshowing also this critical transition in the magnetoelastic\nbehaviour.\nThe ﬁnancial support by the FWF under the NFN-\nproject numbers S 10406 and S 10403 is gratefully ac-\nknowledged.\n∗kriegisch@ifp.tuwien.ac.at\n[1] M. Guillot, J. Ostorero and A. Marchand, Zeitschrift f¨ u r\nPhysik B Condensed Matter, 71, p.193, 1988\n[2] W. Wang and X. Ren, Journal of Crystal Growth, 289,\np.605, 2006\n[3] M. Rotter, H. M¨ uller, E. Gratz, M. Doerr and M. Loewen-\nhaupt, Review of Scientiﬁc Instruments, 69, p.2742, 1998\n[4] G. Asti and F. Bolzoni, Journal of Applied Physics, 50,\np.7725, 1979)\n[5] R. Gr¨ ossinger, X. C. Kou, T. H. Jacobs and\nK. H. J. Buschow, Journal of Applied Physics, 69, p.5596,\n1991\n[6] X. Kou, T. Zhao, R. Gr¨ ossinger, H. Kirchmayr, X. Li and\nF. de Boer, Physical Review B, 47, p.3231, 1993\n[7] X. C. Kou, M. Dahlgren, R. Gr¨ ossinger and G. Wiesinger,\nJournal of Applied Physics, 81, p.4428, 1997"    },    {        "title": "2011.03390v3.Phase_boundary_near_a_magnetic_percolation_transition.pdf",        "content": "European Physical Journal B manuscript No.\n(will be inserted by the editor)\nPhase boundary near a magnetic percolation transition\nGaurav Khairnara,1, Cameron Lerchb,1,2, Thomas Vojtac,1\n1Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA\n2Department of Mechanical Engineering and Materials Science, Yale University, New Haven, Connecticut 06520, USA\nReceived: date / Accepted: date\nAbstract Motivated by recent experimental observa-\ntions [Phys. Rev. 96, 020407 (2017)] on hexagonal fer-\nrites, we revisit the phase diagrams of diluted magnets\nclose to the lattice percolation threshold. We perform\nlarge-scale Monte Carlo simulations of XY and Heisen-\nberg models on both simple cubic lattices and lattices\nrepresenting the crystal structure of the hexagonal fer-\nrites. Close to the percolation threshold pc, we \fnd that\nthe magnetic ordering temperature Tcdepends on the\ndilutionpvia the power law Tc\u0018jp\u0000pcj\u001ewith expo-\nnent\u001e= 1:09, in agreement with classical percolation\ntheory. However, this asymptotic critical region is very\nnarrow,jp\u0000pcj.0:04. Outside of it, the shape of the\nphase boundary is well described, over a wide range\nof dilutions, by a nonuniversal power law with an ex-\nponent somewhat below unity. Nonetheless, the perco-\nlation scenario does not reproduce the experimentally\nobserved relation Tc\u0018(xc\u0000x)2=3in PbFe 12\u0000xGaxO19.\nWe discuss the generality of our \fndings as well as im-\nplications for the physics of diluted hexagonal ferrites.\n1 Introduction\nDisordered many-body systems feature three di\u000berent\ntypes of \ructuations, viz., static random \ructuations\ndue to the quenched disorder, thermal \ructuations, and\nquantum \ructuations. Their interplay can greatly a\u000bect\nthe properties of phase transitions, with possible con-\nsequences ranging from a simple change of universality\nclass [1] to exotic in\fnite-randomness criticality [2, 3],\nclassical [4] and quantum [5, 6] Gri\u000eths singularities,\nas well as the destruction of the transition by smearing\nae-mail: grktmk@mst.edu\nbe-mail: cameron.lerch@yale.edu\nce-mail: vojtat@mst.edu[7{10]. Recent reviews of some of these phenomena can\nbe found in Refs. [11{13]. Randomly diluted magnetic\nmaterials are a particularly interesting class of systems\nin which the above interplay is realized. Here, the dis-\norder \ructuations correspond to the geometric \ructua-\ntions of the underlying lattices which can undergo a ge-\nometric percolation transition between a disconnected\nphase and a connected (percolating) phase [14].\nRecently, the behavior of diluted magnets close to\nthe percolation transition has reattracted attention be-\ncause of the unexpected shape of the phase bound-\nary observed in the diluted hexagonal ferrite (hexafer-\nrite) PbFe 12\u0000xGaxO19[15]. Pure PbFe 12O19orders fer-\nrimagnetically at temperatures below about 720 K [16].\nThe ordering temperature Tccan be suppressed by ran-\ndomly substituting nonmagnetic Ga ions for Fe ions in\nPbFe 12\u0000xGaxO19. It vanishes when xreaches the crit-\nical valuexc\u00198:6. This value is very close the perco-\nlation threshold xp= 8:846 of the underlying lattice1,\nsuggesting that the transition at xcis of percolation\ntype [15]. Remarkably, the phase boundary follows the\npower lawTc(x) =Tc(0)(1\u0000x=xc)\u001ewith\u001e= 2=3 over\nthe entirex-range from 0 to xc. This disagrees with\nthe prediction from classical percolation theory [14, 17]\nwhich yields a crossover exponent of \u001e >1 for contin-\nuous symmetry magnets, at least for dilutions close to\nxc.\nIn this paper, we therefore reinvestigate the phase\nboundary close to the percolation transition of diluted\nclassical planar and Heisenberg magnets by means of\nlarge-scale Monte Carlo simulations. The purpose of\nthe paper is twofold. First, we wish to test and verify\nthe percolation theory predictions, focusing not only on\n1The lattice in question is the lattice of exchange interactions\nbetween the Fe ions.arXiv:2011.03390v3  [cond-mat.dis-nn]  5 Feb 20212\nthe asymptotic critical behavior but also on the width\nof the critical region and the preasymptotic properties.\nSecond, we wish to explore whether the classical per-\ncolation scenario can explain the experimental observa-\ntions in PbFe 12\u0000xGaxO19[15].\nOur paper is organized as follows. In Sec. 2, we in-\ntroduce the diluted XY and Heisenberg models and dis-\ncuss their qualitative behavior. Section 3 summarizes\nthe predictions of percolation theory. Our Monte Carlo\nsimulation method is described in Sec. 4. Sections 5.1\nand 5.2 report our results for model systems on cubic\nlattices and for systems de\fned on the hexagonal ferrite\nlattice, respectively. We conclude in Sec. 6.\n2 The Models\nConsistent with the dual purpose of studying the crit-\nical behavior of the phase boundary close to a mag-\nnetic percolation transition and of addressing the ex-\nperimental observations in diluted hexaferrites [15], we\nconsider two models, viz., (i) site-diluted classical XY\nand Heisenberg models on simple cubic lattices and (ii)\na classical Heisenberg Hamiltonian based on the hexa-\nferrite crystal structure using realistic exchange inter-\nactions. Comparing the results of these di\u000berent models\nwill also allow us to explore the universality of the crit-\nical behavior.\n2.1 Site-diluted XY and Heisenberg models on cubic\nlattices\nWe consider a simple cubic lattice of N=L3sites. Each\nsite is either occupied by a vacancy or by a classical\nspin, i.e., an n-component unit vector Si(n= 2 for\nthe XY model and n= 3 for the Heisenberg case). The\nHamiltonian reads\nH=\u0000JX\n<i;j>\u000fi\u000fjSi\u0001Sj: (1)\nHere, the sum is over pairs of nearest-neighbor sites,\nandJ > 0 denotes the ferromagnetic exchange inter-\naction. (In the following, we set Jto unity for the\ncubic lattice simulations.) The quenched independent\nrandom variables \u000fiimplement the site dilution. They\ntake the values 0 (vacancy) with probability pand 1\n(occupied site) with probability 1 \u0000p. We employ pe-\nriodic boundary conditions. Magnetic long-range order\ncan be characterized by the order parameter, the total\nmagnetization\nm=1\nNX\niSi: (2)\nFig. 1 Double unit cell of PbFe 12O19. 24 Fe3+ions are lo-\ncated on \fve distinct sublattices.\nThe qualitative behavior of this model as a function\nof temperature Tand dilution pis well understood (see,\ne.g., Ref. [18] for an overview). For su\u000eciently small di-\nlution, the system orders magnetically below a critical\ntemperature Tc(p). The critical temperature decreases\ncontinuously with puntil it reaches zero at the perco-\nlation threshold pcof the lattice. For dilutions beyond\nthe percolation threshold, magnetic long-range order is\nimpossible because the system breaks down into \fnite\nnoninteracting clusters. The point p=pc,T= 0 is a\nmulticritical point at which both the geometric \ructua-\ntions of the lattice and the thermal \ructuations become\nlong-ranged.\n2.2 Hexaferrite Heisenberg model\nPbFe 12O19crystallizes in the magnetoplumbite struc-\nture, as illustrated in Fig. 1. A double unit cell con-\ntains 24 Fe3+ions in \fve distinct sublattices; they are\nin the spin state S= 5=2. Below a temperature of about\n720K, the material orders ferrimagnetically, with 16 of\nthe Fe spins pointing up and the remaining 8 Fe ions\npointing down [16]. Note that the high critical temper-\nature and the high spin value suggest that a classical\ndescription should provide a good approximation.\nIn PbFe 12\u0000xGaxO19, the randomly substituted Ga\nions, which replace the Fe ions, act as quenched spinless\nimpurities. To model this system, we start from the\nhexaferrite crystal structure and randomly place either\na vacancy (with probability p) or a classical Heisenberg\nspinSi(with probability 1 \u0000p) at each Fe site. The\ndilutionpis related to the number xof Ga ions in the3\nunit cell by p=x=12. The Hamiltonian reads\nH=\u0000X\ni;jJij\u000fi\u000fjSiSj: (3)\nThe quenched random variables \u000fidistinguish vacan-\ncies and spins, as before. The values of the exchange\ninteractions Jijstem from the density functional cal-\nculation in Ref. [19]; they are scaled by a common fac-\ntor to approximately reproduce the critical temperature\nTc= 720Kof the undiluted material. In most of our\nMonte Carlo simulations, we include only the leading\n(strongest) interactions which are between the following\nsublattice pairs: 2a-4f IV, 2b-4fVI, 12k-4fIV, 12k-4fVI.\nThese interactions are non-frustrated and establish the\nferrimagnetic order. We also perform a few test calcula-\ntions to explore the e\u000bects of additional couplings which\nare signi\fcantly weaker but frustrate the ferrimagnetic\norder.\nThe qualitative features of the phase diagram of the\nmodel (3) are expected to be similar to those discussed\nin the previous section. With increasing dilution p, the\ncritical temperature Tc(p) is continuously suppressed\nand reaches zero at the site percolation threshold. The\nvalue of the percolation threshold of the lattice spanned\nby the leading non-frustrated interactions between the\nFe ions was determined in Ref. [15] by means of Monte\nCarlo simulations. They yielded pc= 0:7372(5), corre-\nsponding to xc= 8:846(6) Ga ions per unit cell. (The\nnumbers in brackets show the error estimate of the last\ndigit.)\n3 Predictions of Percolation Theory\nIn this section, we brie\ry summarize the predictions of\nclassical percolation theory for the shape of the phase\nboundaryTc(p) close to multicritical point p=pc;T=\n0 [14, 17, 20]. Close to this point, two length scales\nare at play, the percolation correlation length, \u0018pwhich\ncharacterizes the size of \fnite isolated clusters of lat-\ntice sites and the magnetic thermal correlation length\non the critical in\fnite percolating cluster at pcdenoted\nby\u0018T. The percolation correlation length \u0018pdiverges\nas\u0018p\u0018jp\u0000pcj\u0000\u0017pas the percolation threshold is ap-\nproached. The magnetic thermal correlation length be-\nhaves as\u0018T\u0018T\u0000\u0017Tfor continuous-symmetry magnets\ndescribed by the n-vector model with n>1.\nTo \fnd the phase boundary, consider the magneti-\nzation near the critical point. It ful\flls the scaling form,\nm(p\u0000pc;T) =jp\u0000pcj\fX(\u0018T=\u0018p): (4)\nForp < pc, the magnetic phase transition occurs at\na particular value xcof the argument of the scalingfunctionX. At the magnetic transition, we therefore\nhave\u0018T=xc\u0018p. This yields the power law relation\nTc(p)\u0018jp\u0000pcj\u001e: (5)\nThe crossover exponent \u001etakes the value \u001e=\u0017p=\u0017T.\n(In contrast, \u0018Tdiverges exponentially, \u0018T\u0018(e\u00002J=T)\u0000\u0017T,\nfor Ising magnets, leading to a logarithmic dependence\nTc(p)\u0018ln\u00001(1=jp\u0000pcj).)\nUsing a renormalization group calculation, Coniglio\n[17] established the relation \u0017T= 1=~\u0010R. Here, ~\u0010Rchar-\nacterizes the resistance Rof a random resistor network\non a critical percolation cluster of linear size Lvia\nR\u0018L~\u0010R.\nThe exponent ~\u0010Rcan be related to the well-known\nconductivity critical exponent twhich describes how\nthe conductivity \u001bof the resistor network depends on\nthe distance from the percolation threshold, \u001b\u0018jp\u0000\npcjt. To do so, consider a resistor network on a perco-\nlating lattice close to pcbut on the percolating side. Its\nbehavior is critical for clusters of size less than \u0018pand\nOhmic for sizes beyond \u0018p. For ad-dimensional system\nof linear size L\u001d\u0018p, we can employ Ohm's law to\ncombine blocks of size \u0018p, yielding\nR(L) =R(\u0018p)\u0012L\n\u0018p\u0013\u0012L\n\u0018p\u0013\u0000(d\u00001)\n\u0018\u0018~\u0010Rp\u0018d\u00002\npL2\u0000d:(6)\nThe conductivity on the percolating side thus behaves\nas\u001b\u0018\u0018\u0000(d\u00002+~\u0010R)\np\u0018jp\u0000pcj\u0017p(d\u00002+~\u0010R). Thus, we ob-\ntain the hyperscaling relation, t= (d\u00002 + ~\u0010R)\u0017por\n~\u0010R=t=\u0017p\u0000d+2. Using the numerical estimates t=\u0017p=\n2:28(2) and\u0017p= 0:876(2) [21, 22] for three-dimensional\nsystems yields ~\u0010R= 1:28(2), predicting a crossover ex-\nponent of\u001e=\u0017p=\u0017T=\u0017p~\u0010R= 1:12(2).2\n4 Numerical Simulations\n4.1 Monte Carlo method\nTo \fnd the critical temperature for a given dilution of\nthe system, we perform large-scale Monte Carlo (MC)\nsimulations. These simulations employ the Wol\u000b [25]\nand Metropolis [26] algorithms. Speci\fcally, a full MC\nsweep consists of a Wol\u000b sweep followed by a Metropolis\nsweep. The Wol\u000b algorithm is a cluster-\rip algorithm\nwhich is bene\fcial in reducing critical slowing down of\nthe system near criticality. The Metropolis algorithm\nis a single spin-\rip algorithm. It is required to achieve\n2The crossover exponent has also been computed within an\nexpansion in powers of \u000f= 6\u0000dyielding\u001e= 1 +\u000f=42 to\n\frst order in \u000f[23, 24]. The resulting value, \u001e= 1:071, is\nsurprisingly close to the best numerical estimate \u001e= 1:12(2).4\n0 20 40 60 80 100\n0.33\n0.32\n0.31\nE\nHot start\nCold start\n0 20 40 60 80 100\nnumber of MC sweeps0.050.100.150.200.25m\nFig. 2 Equilibration of the energy per site Eand the mag-\nnetization mfor a cubic lattice XY model of size L= 56,\ndilutionp= 0:66 , and temperature T= 0:156 averaged over\n20 disorder con\fgurations. The comparison of hot and cold\nstarts shows that the system equilibrates after roughly 50\nMonte Carlo sweeps despite being close to the multicritical\npoint.\nequilibration of small isolated clusters of lattice sites\nwhich might form as a result of dilution.\nFor the cubic lattice calculations, we consider sys-\ntem sizes ranging from L3= 103toL3= 1123. We\nhave simulated 4000 \u000040000 independent disorder con-\n\fgurations for each size. For the hexaferrite lattice, we\nsimulate systems consisting of 103to 403double unit\ncells (each double unit cell contains 24 Fe sites) us-\ning 100\u0000300 independent disorder con\fgurations for\neach size. All physical quantities of interest, such as\nenergy, magnetization, correlation length, etc. are aver-\naged over the disorder con\fgurations. Statistical errors\nare obtained from the variations of the results between\nthe con\fgurations.\nMeasurements of observables must be performed af-\nter the system reaches thermal equilibrium. We deter-\nmine the number of Monte Carlo sweeps required for\nthe system to equilibrate by comparing the results of\nruns with hot starts (for which the spins initially point\nin random directions) and with cold starts (for which\nall spins are initially aligned). An example of such a\ntest for a cubic lattice XY system close to multicrit-\nical point is shown in Fig. 2. The energy and order\nparameter attain their respective equilibrium values af-\nter roughly 50 Monte Carlo sweeps. Similar numerical\nchecks were performed for other parameter values as\nwell as for the cases of Heisenberg spins on cubic and\nhexaferrite lattices. Based on these tests, we have cho-\nsen 150 equilibration sweeps (using a hot start) and 500\nmeasurement sweeps per disorder con\fguration for the\ncubic lattice simulations. For the hexaferrite lattice, weperform 1000 equilibration sweeps and 2000 measure-\nment sweeps (using a hot start). Note that the combi-\nnation of relatively short Monte Carlo runs and a large\nnumber of disorder con\fgurations leads to an overall\nreduction of statistical error [27{29].\n4.2 Data analysis\nWe employ the Binder cumulant [30] to precisely esti-\nmate the critical temperature Tc. It is de\fned as\ng=\u0014\n1\u0000hjmj4i\n3hjmj2i2\u0015\ndis(7)\nwhereh:::idenotes the thermodynamic (Monte Carlo)\naverage and [ :::]disdenotes the disorder average. The\nBinder cumulant gis a dimensionless quantity, it there-\nfore ful\flls the \fnite-size scaling form\ng(t;L;u ) =g(t\u0015\u00001=\u0017;L\u0015;u\u0015\u000e): (8)\nHere,\u0015is an arbitrary scale factor, t= (T\u0000Tc)=Tcde-\nnotes the reduced temperature, and \u0017is the correlation\nlength exponent of the (magnetic) \fnite-temperature\nphase transition. We have included the irrelevant vari-\nableucharacterized by the exponent \u000e>0 to describe\nthe corrections from the leading scaling behavior ob-\nserved in our data. Setting the scale factor \u0015=L\u00001,\nwe obtaing(t;L;u ) =F(tL1=\u0017;uL\u0000\u000e) whereFis a di-\nmensionless scaling function. Expanding Fin its second\nargument yields\ng(t;L;u ) =\b(tL1\n\u0017) +uL\u0000\u000e\bu(tL1\n\u0017): (9)\nIn the absence of corrections to scaling ( u= 0), the\nBinder cumulants at t= 0 corresponding to di\u000berent\nsystem sizes have the universal value \b(0), i.e., the crit-\nical temperature is marked by a crossing of all Binder\ncumulant curves. If corrections to scaling cannot be ne-\nglected (u6= 0), this is not the case (see, e.g., Ref. [31])\nbecauseg(0;L;u ) is not independent of Lbut takes the\nvalueg(0;L;u ) =\b(0)+uL\u0000\u000e\bu(0). Instead, the cross-\ning point shifts with Land approaches t= 0 asL!1 .\nThe functional form of this shift can be worked out ex-\nplicitly by expanding the scaling functions \band\bu,\ng(t;L;u ) =\b(0) +tL1\n\u0017\b0(0) +uL\u0000\u000e\bu(0): (10)\nUsing this expression to evaluate the crossing temper-\natureT\u0003(L) between the Binder cumulant curves for\nsizesLandcL(wherecis a constant) yields\nT\u0003(L) =Tc+bL\u0000!with!=\u000e+1\n\u0017(11)\nwhereb\u0018uis a non-universal amplitude.5\nTo determine the crossing temperature, we \ft the g\nvsTdata sets corresponding to di\u000berent system sizes\nwith separate quartic polynomials.(Quartic polynomi-\nals provide reasonable \fts within the temperature range\nof interest while avoiding spurious oscillations.) The in-\ntersection point of these polynomials yields the crossing\ntemperature T\u0003. To estimate the errors of the crossing\ntemperature we use an ensemble method. For each g(T)\ncurve, we create an ensemble of arti\fcial data sets ga(T)\nby adding noise to the data\nga(T) =g(T) +\u0001g(T)r : (12)\nHere,ris a random number chosen from a normal dis-\ntribution of zero mean and unit variance, and \u0001g(T) is\nthe statistical error of the Monte Carlo data for g(T).\nNote that we use the same random number rfor the\nentireg(T) curve, leading to an upward or downward\nshift of the curve. This stems from the fact that the sta-\ntistical error \u0001g(T) is dominated by the disorder noise\nwhile the Monte Carlo noise is much weaker. This im-\nplies that the deviations at di\u000berent temperatures of the\nBinder cumulant from the true average are correlated.\nRepeating the crossing analysis with these ensembles\nof curves, we get ensembles of crossing temperatures.\nTheir mean and standard deviation yield T\u0003and the\nassociated error \u0001T\u0003, respectively.\n5 Results\nIn this section we report the results of our simulations\nfor cubic and hexaferrite lattices occupied by XY or\nHeisenberg spins.\n5.1 Cubic Lattices\nWe investigate the behavior of both XY and Heisenberg\nmodels on cubic lattices. To check the validity of our\nsimulations, we \frst consider clean (undiluted) lattices.\nWe \fnd critical temperatures of Tc= 2:2017(1) and\nTc= 1:44298(2) for XY and Heisenberg spins, respec-\ntively. They agree well with previously known numerical\nresults [32, 33].\nWe now turn to diluted systems, starting with the\nXY case. For reference, the site percolation threshold\nof the simple cubic lattice is at the vacancy probability\npc= 0:6883923(2) [22]. For low dilutions ( p<0:64), the\nBinder cumulant vs. temperature curves for all simu-\nlated system sizes cross through exactly the same point\nwithin their statistical errors, implying that corrections\nto the leading \fnite-size scaling behavior are not im-\nportant. Therefore, we determine Tcfrom the cross-\ning of theg(T) curves of the two largest system sizes,\n1.80 1.85 1.90 1.95 2.00 2.05 2.10\nT0.350.400.450.500.550.600.65g\nL=20\nL=28\nL=40\nL=56\nL=80\nL=112\n1.93 1.94 1.950.40.50.6\nFig. 3 Binder cumulant gvs temperature Tfor the cubic\nlattice XY model with dilution p= 0:10. The statistical errors\narising from the Monte Carlo simulation are smaller than the\nsymbol size. The inset show the intersection region of the\ncurves more closely. All curves cross at the same point within\ntheir statistical errors.\n0.185 0.190 0.195 0.200 0.205 0.210 0.215 0.220 0.225\nT0.400.450.500.550.600.65g\nL=20\nL=28\nL=40\nL=56\nL=80\nL=112\n0.2025 0.2050 0.20750.580.600.62\nFig. 4 Binder cumulant gvs temperature Tfor the XY model\non a cubic lattice for dilution p= 0:65, i.e. close to pc. The\ncurves do not all cross at the same temperature. Instead, the\ncrossing progressively shifts as Lincreases. The statistical\nerrors arising from the Monte Carlo simulation are smaller\nthan the symbol size.\nL3= 803andL3= 1123. The ensemble method is ap-\nplied to \fnd the error of Tc. Fig. 3 shows an example\nof this situation for dilution p= 0:1.\nFor higher dilutions ( p\u00150:64) in the vicinity of\nthe percolation threshold pc, the crossing of the Binder\ncumulant vs. temperature curves is less sharp. Specif-\nically, the crossing temperature T\u0003(L) of the curves\nfor linear system sizes Landp\n2Lshifts visibly to-\nwards higher temperatures as the system sizes are in-\ncreased. An example (for p= 0:65) is demonstrated in\nFig. 4. As shown in the previous section, this shift is\ncaused by corrections to the leading \fnite-size scaling6\n0.000 0.002 0.004 0.006 0.008 0.010\n1/L\n0.20250.20300.20350.20400.20450.20500.20550.20600.2065T*\np=0.65\nFig. 5 Extrapolation to in\fnite system size of the crossing\ntemperature T\u0003of the Binder cumulant curves for system\nsizesLandp2Lusing!= 1:5. The dilution is p= 0:65. A \ft\nto Eq. (11) gives Tc= 0:2064(4). The error bars of T\u0003have\nbeen determined using the ensemble method described in Sec.\n4.2.\n0.002 0.004 0.006 0.008 0.010\n1/L\n0.050.100.150.200.25T*\np=0.64\np=0.65\np=0.66\np=0.67\np=0.675\np=0.678\np=0.68\np=0.6815\np=0.6825\nFig. 6 Overview of the extrapolations of the crossing tem-\nperaturesT\u0003for several dilutions near pcusing!= 1:5. The\nerror bars \u0001T\u0003are smaller than the symbols.\nbehavior. According to Eq. (11), it can be modeled as\nT\u0003(L) =Tc+bL\u0000!. To \fnd the asymptotic (in\fnite\nsystem size) value of Tc, we thus \ft the crossing tem-\nperatureT\u0003(L) to Eq. (11). As !is expected to be uni-\nversal, i.e., to take the same value for all dilutions near\npc, we perform a combined \ft for all dilutions p\u00150:64\nand treat!as a \ftting parameter. This combined \ft\nproduces!= 1:5\u00060:4. An example of the resulting\nextrapolation is presented in Fig. 5 for p= 0:65. The\n\fgure shows that the \fnite-size shifts of the crossing\ntemperature are not very strong. This is further con-\n\frmed in Fig. 6 which presents an overview of the \fts\nfor all dilutions from p= 0:64 top= 0:6825.\n0.0 0.64 0.678p\n5\n 4\n 3\n 2\n 1\nln|ppc|\n3\n2\n1\n01ln(Tc)\n=0.80(1)\n=1.09(2)\n0.0 0.2 0.4 0.6\np012Tc\nFig. 7 Phase boundary of the site-diluted XY model on a\ncubic lattice. Main panel: Log-log plot of Tcvs.jp\u0000pcj. The\nstraight lines are power-law \fts, Tc\u0018jp\u0000pcj\u001e. They are shown\nas solid lines within the \ft range. The dotted and dash-dotted\nlines are extrapolations. For details see text. Inset: Overview\npresented as linear plot of Tcvs.p. All error bars of the data\npoints are smaller than the symbol size.\nThe resulting phase boundary Tc(p) of the site-diluted\nXY model on a cubic lattice is shown in Fig. 7. The\noverview given in the inset demonstrates that Tc(p) is\nindeed continuously suppressed with increasing pand\napproaches zero as p!pc. To analyze the functional\nform ofTc(p) close topc, the main panel of Fig. 7 shows\na log-log plot of Tcvs.jp\u0000pcj. We observe that the\nphase boundary follows two di\u000berent power laws, close\nto the percolation threshold pcand further away from\npc. The asymptotic value of \u001eis determined from a \ft of\nthe data closest to pc(viz.pbetween 0:678 to 0:6825),\nyielding a crossover exponent of \u001e= 1:09(2). Its error\nestimate is a combination of the statistical error from\nthe \ft and a systematic error estimated from the ro-\nbustness of the value against changes of the \ft interval.\nThe asymptotic value of \u001eagrees reasonably well with\nthe prediction of percolation theory. The asymptotic\npower law describes the data for dilutions above about\np= 0:65. The asymptotic critical region thus ranges\nfrom about p= 0:65 topc= 0:6883923.\nThe preasymptotic behavior of Tc(p) forpbetween\np= 0 top= 0:64 also follows a power law in good\napproximation. However, the exponent is signi\fcantly\nbelow unity, \u001e= 0:80(1).\nWe proceed in the same manner for the Heisenberg\nmodel on the cubic lattice. Starting from the clean case,\nwe gradually increase dilution and \fnd Tc(p). In the\ncase of Heisenberg spins, we \fnd that the corrections\nto \fnite-size scaling are weaker than in the XY case.\nEven in the vicinity of pc, all Binder cumulant curves\nintersect in a single point within their statistical errors.7\n0.096 0.098 0.100 0.102 0.104 0.106 0.108\nT0.580.600.620.640.66g\nL=20\nL=28\nL=40\nL=56\nL=80\nL=112\n0.102 0.103 0.1040.6200.6250.6300.635\nFig. 8 Binder cumulant gvs temperature Tfor dilution p=\n0:65 on cubic lattice and Heisenberg spins. All curves cross at\nthe same temperature. Error bars are smaller than the symbol\nsize.\n0.0 0.63 0.67p\n5\n 4\n 3\n 2\n 1\nln|ppc|\n4\n3\n2\n1\n01ln(Tc)\n=0.86(1)\n=1.08(2)\n0.0 0.2 0.4 0.6\np0.00.51.01.5Tc\nFig. 9 Phase boundary of the site-diluted Heisenberg model\non a cubic lattice. Main panel: Log-log plot of Tcvs.jp\u0000pcj.\nThe straight lines are power-law \fts, Tc\u0018jp\u0000pcj\u001e. They are\nshown as solid lines within the \ft range. The dotted and dash-\ndotted lines are extrapolations. For details see text. Inset:\nOverview presented as linear plot of Tcvs.p. All error bars\nof the data points are smaller than the symbol sizes.\nAs an example, the gvsTdata forp= 0:65 are shown\nin Fig. 8. The critical temperatures Tc(p) and its er-\nror are therefore determined from the Binder cumulant\ncrossing for system sizes L3= 803andL3= 1123, the\nlargest systems simulated.\nThe phase boundary of the site-diluted Heisenberg\nmodel on a cubic lattice is constructed from these data\nand shown in Fig. 9. Similar to the XY case, we ob-\nserve two separate power law exponents governing the\nphase boundary. The dilutions p&0:65 constitute the\nasymptotic critical region with crossover exponent \u001e=\n1:08(2), in agreement with the percolation theory pre-\n0.0 0.65 0.71p\n5\n 4\n 3\n 2\n 1\nln|ppc|\n234567ln(Tc/K)\n=0.88(2)\n=1.12(3)\n0.0 0.2 0.4 0.6 0.8\np0200400600Tc(K)\nFig. 10 Phase boundary for the Heisenberg model on a\nhexagonal ferrite lattice. The main panel shows the log-log\nplot ofTcvs.jp\u0000pcj. The statistical errors of the data (de-\ntermined by the ensemble method) are smaller than the sym-\nbol size. The straight lines are \fts to Tc\u0018jp\u0000pcj\u001e. They\nare shown as solid lines within the \ft range. The dotted and\ndash-dotted lines are extrapolations. For details see text. The\ninset shows a linear plot the complete phase boundary Tc(p).\ndiction. The nonuniversal preasymptotic crossover ex-\nponent obtained for dilutions p.0:62 is again smaller\nthan unity, \u001e= 0:86(1), but somewhat larger than in\nthe XY case.\n5.2 Hexagonal Ferrite Lattice\nWhereas the asymptotic critical behavior of the phase\nboundary close to the percolation threshold is expected\nto be universal, its behavior outside the asymptotic crit-\nical region does not have to be universal. For a bet-\nter quantitative understanding of the magnetic phase\nboundary of the diluted hexaferrites, we therefore also\nperform simulations of the Heisenberg model (3) using\nthe hexaferrite crystal structure and realistic exchange\ninteractions. In the calculations, we focus on the lead-\ning non-frustrated interactions, as outlined in Sec. 2.2.\nThe site percolation threshold for the lattice spanned\nby these interactions is pc= 0:7372(5) [15].\nAs before, the critical temperature Tcfor a given\ndilution is determined from the Binder cumulant cross-\nings. Corrections to the \fnite-size scaling were found\nto be negligible within the statistical errors. Thus, we\nused the Binder cumulant crossing of the two largest\nsystem sizes (283and 403double unit cells) to \fnd\nTc. The resulting phase boundary is shown in Fig. 10.\nThe behavior of this phase boundary is very similar\nto the cubic lattice results. High dilutions, p&0:68,\nfall into the asymptotic critical region with a crossover\nexponent of \u001e= 1:12(3), in excellent agreement with8\n0 2 4 6 8\nx0.02.55.07.510.012.515.017.520.0T3/2\nc/103 (K3/2)\nExperimental Data\nMonte Carlo, leading + frustrated interactions\nMonte Carlo, leading interactions only\nFig. 11 Comparison between the numerically determined\nphase boundary Tc(x) and the experimental data for\nPbFe 12\u0000xGaxO19[15]. The tuning parameter xis related to\nthe dilution by x=12 =p. The Monte Carlo simulations show\na more rapid suppression of Tcwithx. Including additional\nweak frustrated interactions increases the discrepancy.\nthe percolation theory predictions. This also con\frms\nthe universality of the asymptotic crossover exponent.\nThe preasymptotic exponent \u001e= 0:88(2) that governs\nthe behavior for dilutions below about 0.65 is smaller\nthan unity and takes roughly the same value as for the\nHeisenberg model on the cubic lattice.\nOur numerical results disagree with the experimen-\ntally observed 2/3 power law, Tc(x) =Tc(0)(1\u0000x=xc)2=3.\nIn the simulations, the transition temperature Tcis sup-\npressed more rapidly with xthan in the experimental\ndata (see Fig. 11). To explore possible reasons for this\ndiscrepancy, we also perform test simulations that in-\nclude additional weaker exchange interactions [19] that\nfrustrate the ferrimagnetic order. The results of these\nsimulations, which are included in Fig. 11, show that\nthese weaker frustrating interactions have little e\u000bect\nat low dilutions. At higher dilutions, when the ferri-\nmagnetic order is already weakened, the frustrating in-\nteractions further suppress the transition temperature.\nThey thus further increase the discrepancy between the\nexperimental data and the Monte Carlo results.\n6 Conclusion\nTo summarize, motivated by recent experimental ob-\nservations on hexagonal ferrites, we have studied clas-\nsical site-diluted XY and Heisenberg models by means\nof large-scale Monte Carlo simulations, focusing on the\nshape of the magnetic phase boundary. We have ob-\ntained two main results.First, for high dilutions close to the lattice percola-\ntion threshold, the critical temperature depends on the\ndilution via the power law Tc\u0018jp\u0000pcj\u001ein all studied\nsystems. In this asymptotic region, we have found the\nvalues\u001e= 1:09(2) and 1.08(2) for XY and Heisenberg\nspins on cubic lattices, respectively. For the Heisenberg\nmodel on the hexaferrite lattice, \u001e= 1:12(3). These val-\nues agree with each other and with the prediction \u001e=\n1:12(2) of classical percolation theory. The crossover\nexponent\u001ethus appears to be super-universal, i.e., it\ntakes the same value not just for di\u000berent lattices but\nalso for XY and Heisenberg symmetry.\nInterestingly, the asymptotic critical region of the\npercolation transition is very narrow, as the asymptotic\npower-laws only hold in the range jp\u0000pcj.0:04. At\nlower dilutions, the phase boundary still follows a power\nlaw injp\u0000pcj, but with an exponent that appears to\nbe non-universal and below unity (in the range between\n0.8 and 0.9).\nOur second main result concerns the origin of the\n2=3 power law, Tc(x) =Tc(0)(1\u0000x=xc)2=3, that was\nexperimentally observed in PbFe 12\u0000xGaxO19over the\nentire concentration range between 0 and close to the\npercolation threshold [15]. Neither the asymptotic nor\nthe preasymptotic power laws identi\fed in the simu-\nlations match the experimental result. In fact, in all\nsimulations, the critical temperature is suppressed more\nrapidly with increasing dilution than in the experiment.\nThe observed shape of the magnetic phase boundary in\nPbFe 12\u0000xGaxO19thus remains unexplained.\nPotential reasons for the unusual behavior may in-\nclude the interplay between magnetism and ferroelec-\ntricity in these materials [34] or the presence of quan-\ntum \ructuations (arising from the frustrated magnetic\ninteractions mentioned above), even though it is hard\nto imagine that these stay relevant at temperatures as\nhigh as 720 K. Another possible explanation could be\na statistically unequal occupation of the di\u000berent iron\nsites in the unit cell by Ga ions. Exploring these pos-\nsibilities remains a task for the future. Disentangling\nthese e\u000bects may also require additional experiments\nintroducing further tuning parameters such as pressure\nor magnetic \feld in addition to chemical composition.\nAcknowledgements We acknowledge support from the NSF\nunder Grant Nos. DMR-1506152, DMR-1828489, and OAC-\n1919789. The simulations were performed on the Pegasus and\nFoundry clusters at Missouri S&T. We also thank Martin\nPuschmann for helpful discussions.\nReferences\n1. G. Grinstein and A. Luther, Phys. Rev. B 13, 1329\n(1976).9\n2. D. S. Fisher, Phys. Rev. Lett. 69, 534 (1992).\n3. D. S. Fisher, Phys. Rev. B 51, 6411 (1995).\n4. R. B. Gri\u000eths, Phys. Rev. Lett. 23, 17 (1969).\n5. M. Thill and D. A. Huse, Physica A 214, 321 (1995).\n6. A. P. Young and H. Rieger, Phys. Rev. B 53, 8486 (1996).\n7. T. Vojta, Phys. Rev. Lett. 90, 107202 (2003).\n8. R. Sknepnek and T. Vojta, Phys. Rev. B 69, 174410\n(2004).\n9. G. Schehr and H. Rieger, Phys. Rev. Lett. 96, 227201\n(2006).\n10. J. A. Hoyos and T. 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Hasenbusch, Physica A: Statistical\nMechanics and its Applications 201, 593 (1993).\n33. R. G. Brown and M. Ciftan, Phys. Rev. B 74, 224413\n(2006).\n34. S. E. Rowley, Y.-S. Chai, S.-P. Shen, Y. Sun, A. T. Jones,\nB. E. Watts, and J. F. Scott, Scienti\fc Reports 6, 25724\n(2016)."    },    {        "title": "1309.4031v1.Epitaxial_Growth_of_Spinel_Cobalt_Ferrite_Films_on_MgAl__2_O__4__Substrates_by_Direct_Liquid_Injection_Chemical_Vapor_Deposition.pdf",        "content": "arXiv:1309.4031v1  [cond-mat.mtrl-sci]  16 Sep 2013Epitaxial Growthof Spinel Cobalt FerriteFilms on MgAl 2O4Substrates by\nDirect Liquid Injection Chemical VaporDeposition\nLiming Shen,1Matthias Althammer,1Neha Pachauri,1B. Loukya,2Ranjan Datta,2Milko Iliev,3Ningzhong\nBao,4and ArunavaGupta1,a)\n1)University of Alabama, Center for Materials for Informatio n Technology MINT and Dept. of Chem, Tuscaloosa,\nAL35487 USA\n2)International Centre for Materials Science, Jawaharlal Ne hru Centre for Advanced Scientiﬁc Research, Jakkur P.O.,\nBangalore 560064, India\n3)Texas Center for Superconductivity and Department of Physi cs, University of Houston, Houston, Texas 77204-5002,\nUSA\n4)State Key Laboratory of Material-Oriented Chemical Engine ering, College of Chemistry and Chemical Engineering,\nNanjingUniversity of Technology, Nanjing, Jiangsu 210009 , P.R.China\n(Dated: 29 October 2018)\nThe direct liquid injection chemical vapor deposition (DLI -CVD) technique has been used for the growth of cobalt\nferrite(CFO)ﬁlmson(100)-orientedMgAl 2O4(MAO)substrates. Smoothandhighlyepitaxialcobaltferri tethinﬁlms,\nwiththeepitaxialrelationshipMAO (100)[001]/bardblCFO(100)[001],areobtainedunderoptimizeddepositionconditions.\nThe ﬁlms exhibit bulk-like structural and magnetic propert ies with an out-of-plane lattice constant of 8 .370˚A and a\nsaturationmagnetizationof420kA /mat roomtemperature. TheRaman spectraofﬁlmson MgAl 2O4supportthe fact\nthat the Fe3+- and the Co2+-ions are distributed in an ordered fashion on the B-site of t he inverse spinel structure.\nThe DLI-CVD technique has been extended for the growth of smo oth and highly oriented cobalt ferrite thin ﬁlms on\na variety of other substrates, including MgO, and piezoelec tric lead magnesium niobate-lead titanate and lead zinc\nniobate-leadtitanatesubstrates.\nI. INTRODUCTION\nCobalt ferrite, CoFe 2O4(CFO), with the inverse spinel\nstructure is well known to have a relatively large magnetic\nanisotropy, high Curie temperature of around 800 K, mod-\nerate saturationmagnetization,remarkablechemicalstab ility,\nandgoodmechanicalhardness. Theseexcellentpropertieso f-\nferCFOnumeroustechnologicalapplicationsinareassucha s\nhardpinninglayers,pressuresensors,actuators,andtran sduc-\ners, spin ﬁlters, and drug delivery.1–3Recently, much atten-\ntion hasbeenfocusedon thegrowthof epitaxialspinel ferri te\nthinﬁlmsonpiezoelectricsubstrates-a typeofmagnetoele c-\ntric multiferroic composite structure proposed to overcom e\nthe limited choice of single-phase multiferroic materials ex-\nhibitingcoexistenceofstronglycoupledferro/ferrimagn etism\nand ferroelectricity.4,5The coexistence of ferroelectric and\nmagnetic phases in such multiferroic heterostructures pro -\nduces novel propertiesresulting from the interfacial coup ling\nof structural, electrical, and magnetic order parameters. Be-\ncause of its large magnetostrictive coefﬁcient, CFO has bee n\nconsidered as an important component for multiferroic com-\nposites that exhibit a strong coupling of the order paramete rs\nthrough the heteroepitaxy of the two lattices.6–8In order to\ncoupleviatheinterface,epitaxialgrowthofCFOwithchemi -\ncallysharp,smoothinterfaces,aswellasgoodmagneticpro p-\nerties, is critical. Up to date, epitaxial CFO ﬁlms have been\nsuccessfully fabricated by pulsed laser deposition (PLD),9\nmolecular beam epitaxy (MBE),10and metal organic chem-\nical vapor deposition (MOCVD).11These CFO ﬁlms have\nbeengrownprimarilyonsinglecrystalsubstratessuchaspe r-\na)Electronic mail: agupta@mint.ua.eduovskiteSrTiO 3(STO),isostructuralspinel MgAl 2O4(MAO),\nrocksaltMgO, andvariousbufferedsubstrates. The thickne ss\nof these ﬁlms is usually in the range of several nanometers\nup to 400 nm, but their magnetic properties are usually far\nfrom bulk properties and also the quality of ﬁlm microstruc-\nture decreases with increasing thickness. 3D island forma-\ntion at high growth temperature, large lattice mismatch and\nthe mismatch of thermal expansion coefﬁcients between the\nﬁlm and substrates have been considered as the major obsta-\nclestoachievehighqualityCFOthinﬁlms.9Thus,itremainsa\nchallengefordevelopingefﬁcientsynthetictechniquesto fab-\nricateepitaxial thickCFO ﬁlmswith near-perfectmicrostr uc-\ntureandexcellentmagneticpropertiesclosetobulkCFO.Di -\nrect liquidinjectionchemicalvapordeposition(DLI-CVD) is\nan attractive method for the fabrication of multi-componen t\nstoichiometric ﬁlms because of its accurate delivery of sev -\neral precursors dissolved in a common solvent.12This depo-\nsition method also has the advantages of relatively low cost ,\nhigh deposition rates, industrial compatibility, and the ﬂ exi-\nbility to fabricatemultilayerstructuresoverconvention althin\nﬁlm processes. DLI-CVD has already been utilized to pre-\npare atomicallysmooth NiFe 2O4(NFO) thin ﬁlms at temper-\natures of 600 −700◦C with saturation magnetization values\nonlyslightlylowerthanbulkNiFe 2O4.13AsCFOpossessesa\nsimilarspinelferritestructurelikeNiFe 2O4,DLI-CVDshould\nalso potentially enable the growth of high quality CFO thin\nﬁlms with bulk-like properties. In this work, we present the\nresults of our growth studies and the structural and magneti c\nproperties of high-quality thick CFO ﬁlms ( ≈1µm) epitax-\nially grown on (100)-oriented MAO substrates by DLI-CVD\nmethod. A narrow growth temperature window is found for\nthe growth of high quality CFO thin ﬁlm. Last but not least,\nweshowthattheoptimizeddepositionparametersaresuitab le2\nfor the growth of thick CFO ﬁlms on a variety of other sub-\nstrates [(MgO, lead magnesium niobate-lead titanate (PMN-\nPT), and lead zinc niobate-lead titanate (PZN-PT)] with ex-\ncellentstructuralquality.\nII. EXPERIMENTAL\nThe DLI-CVD setup used for the experiments has already\nbeen described in a previous report.13In our growth process,\nanhydrous Co(acac) 3and Fe(acac) 3(acac is an abbreviation\nfor acetylacetonate) in the molar ratio of 1:2 were dissolve d\nin N,N-dimethyl formamide (DMF) to form a clear homo-\ngeneous precursor solution. The precursor solution was the n\nintroduced into a Brooks Instrument DLI 200 vaporizer sys-\ntem, vaporized by heated Ar gas, followed by reaction with\noxygen (O 2) gas on heated substrates placed in a CVD reac-\ntor, resulting in the formation of a stoichiometric CFO ﬁlm\non the substrate. The ﬂow rates of Ar and O 2gas were op-\ntimized to be 400 and 150 sccm, respectively, with a precur-\nsor injection rate of 9 .5 g/h. The ﬁlm deposition was car-\nried out at temperaturesranging from 550◦C to 700◦C for 1\nh. After deposition, the ﬁlms were cooled to room tempera-\nture in the reactor under ﬂowing O 2. Film morphology was\ncharacterizedbyﬁeld emissionscanningelectronmicrosco py\n(FE-SEM, JEOL-7000) as well as atomic force microscopy\n(AFM, Veeco nanoscope-IV).To determine the crystal phase\nand epitaxy of the resulting ﬁlms, a standard 4 circle x-ray\ndiffraction (XRD) setup (Phillips X pert Pro) was used with\na CuKαsource. Magnetic propertiesof the CFO ﬁlms were\ndetermined via superconductingquantum interference devi ce\n(SQUID)magnetometry(MPMSbyQuantumDesign)attem-\nperatures 5 K ≤T≤350 K and in magnetic ﬁelds of up to\n5 T. Polarized Raman spectra were measured with XX,XY,\nX′X′, andX′Y′scattering conﬁgurations using a 488nm ex-\ncitation source at room temperature. In these notations the\nﬁrstandsecondlettersindicatethepolarizationoftheinc ident\nand scattered light, respectively, along the cubic X/bardbl[100],\nY/bardbl[010],X′/bardbl[110], orY′/bardbl[−110]directions of the MAO\nsubstrate. Transmission electron microscope (TEM) cross\nsectional samples were prepared by conventionalmechanica l\npolishingandArionmilling. Electrondiffractionandimag ing\nwere performedin a FEI TITAN3 TM80−300kV aberration\ncorrectedtransmissionelectronmicroscope.\nIII. RESULTSANDDISCUSSION\nWeﬁrstlookintothestructuralqualityofCFOﬁlmsgrown\nonMAO(100)substratesattemperaturesrangingfrom550to\n700◦C. All the other optimized growth parameters, such as\nﬂowrateandvaporizationtemperatureofprecursorsolutio ns,\nﬂowrateofO 2,andthetotalpressureinthereactor,werekept\nconstant for all ﬁlms. Figure 1(a) shows the normal Bragg\nXRD spectra near the (400) reﬂection of the MAO substrate\nofCFO ﬁlmsgrownatdifferentdepositiontemperatures. The\nCFO grown at 550◦C exhibits no ﬁlm reﬂections and thus is\namorphous,which isprobablydueto the low depositiontem-42° 44° 46°700°C\n670°C\n620°C\n600°CI (a.u.)\n2θ550°C\n20° 40° 60° 80° 100° 120°100101102103104105I (cps)\n2θ\n-0.3° 0.0° 0.3°0102030 I (kcps)\n∆ω0° 90° 180° 270°101102103I (cps)\nϕ(a)\n(c)(b)\n(d) CFO (220) CFO (400)CFO (400)\nCFO (400) MAO (400) MAO (400)MAO (600) MAO (200) \nMAO (800) CFO (800) \n0.23°\nFIG. 1. (Color online) (a) X-ray diffraction 2 θ−ωscans of CFO\nﬁlm grown on (100) oriented MAO at 550◦C (black), 600◦C (red),\n620◦C (green), 670◦C (blue), and 700◦C (magenta) . (b) wide-\nrange2θ−ωscan,(c)rockingcurveoftheCFO(400)reﬂection,and\n(d) phi (ϕ)-scan of the CFO (220) reﬂection for a CFO ﬁlm grown\nat 670◦C. The narrow rocking curve with a FWHM of 0 .23◦is in-\ndicative of the high structural quality. The phi-scan exhib its a 4-fold\nsymmetryindicating a cube-on-cube growth ofCFOon MAO.\nperature, which leads to a low crystallization rate of the ﬁl m\non the substrate. CFO ﬁlms grown between 600 and 670◦C\nexhibita single-phasespinelstructure,withoutanyaddit ional\npeaks from impurity phases. The characteristic (400) reﬂec -\ntion of the CFO ﬁlms become narrower and sharper as the\ndeposition temperature increases, indicating that the epi taxy\nand microstructure improves steadily. The appearance of a\nsharpKαpeak splitting and high intensity of the (400) ﬁlm\nreﬂection at 2 θ≈43.25◦for the CFO ﬁlm grown at 670◦C\n[Fig. 1(a)] indicatesa highlyepitaxial growthfor this dep osi-\ntion temperature. For higher growth temperaturesthe quali ty\nof the ﬁlms starts to degrade, as evident from the disappear-\nanceofthesharp Kαpeaksplittingforaﬁlmgrownat700◦C.\nThe ﬁlm grownat 700◦C showsa very roughcracked sur-\nfacewithsomelargeareaﬁlmpeeling. Theroughsurfacecan\nbe caused by a faster deposition rate at higher temperature\ncausing a higher density of defects. These ﬁlms can easily3\npeel off fromthe substrate surface duringthe coolingproce ss\nduetostresscausedbythedifferenceinthermalexpansionc o-\nefﬁcients of ﬁlm and substrate. Therefore, spinel CFO ﬁlms\nwith good epitaxy and texture can only be prepared within\na narrow growth temperature window around 670◦C in our\nDLI-CVD setup. The thickness of the ﬁlms can be directly\ncontrolledbythedepositiontime.\nFromthe XRD resultsin Fig. 1(a)we extractedthe follow-\ningout-of-planelattice constantsforCFO thinﬁlmsgrowna t\n600,620,and670◦C: 8.352˚A,8.360˚A,and8.370˚A,respec-\ntively. Although all the CFO ﬁlms are rather thick ( ≈1µm),\nthelatticeconstantsofthesesamplesaresmallerthantheb ulk\nvalue (8.377˚A), suggesting an in-plane tensile stress in the\nﬁlms. As the lattice constantof MAO (8 .09˚A) is smaller, the\noccurrence of an in-plane tensile stress is unexpected. Gao\net al.14found that the unexpected in-plane tensile stress in\nCFO/SrRuO 3heterostructure is strongly correlated with the\ndensity of the interface misﬁt dislocations, which are form ed\nat the deposition temperature to relieve the stress caused b y\nthe lattice mismatch. Besser et al.15proposed a stress model\nfor heteroepitaxial magnetic oxide ﬁlms based on the differ -\nentthermalexpansionratesbetweenthesubstrateandtheﬁl m\nduring the cooling process. The residual stress after cooli ng\nthe sample to room temperature is proportional to the differ -\nence in the thermal expansion coefﬁcient between the ﬁlm\nand substrate and is not related to the bulk lattice parame-\nters. A tensile stress on CFO can be created because CFO\n(14.9×10−6K−1)13has a larger thermal expansion coef-\nﬁcient than that of MAO (9 .5×10−6K−1),14resulting in a\nfastercontractionrateinCFO thanin MAOwhencoolingthe\nsamplesfromthedepositiontemperaturetoroomtemperatur e.\nFrom the reductionof the tensile stress in oursamples for in -\ncreasingtemperature,we concludethat the differencein th er-\nmal expansion is not the cause for stress, but more likely the\nformationofmisﬁt dislocationsis dominant.\nFigure 1(b) shows a 2 θ−ωXRD scan over a larger 2 θ\nrangefortheCFOsampledepositedat670◦C,exhibitingonly\n(h00) reﬂections of substrate and ﬁlm. No secondary phases\narevisibleinthisfullrangeXRD scan. Theseﬁndingsupport\nthatweareabletogrowsinglephaseCFOﬁlmsbyDLI-CVD.\nToanalyzetheﬁlmmosaicspreadweinvestigatedthewidth\noftherockingcurve( ω-scan)[Fig.1(c)]oftheCFO(400)re-\nﬂection for the sample grown at 670◦C. The measured full\nwidthathalfmaximum(FWHM)is0 .22◦,whichiscompara-\nbleto ourresultsonNFO ﬁlms.13\nTo determine the epitaxial relationship between ﬁlm and\nsubstrate, we carried out XRD phi( ϕ)-scans at the (220) re-\nﬂection of the CFO ﬁlm and MAO substrate. The results\nfor the CFO(220) reﬂection are shown in Fig. 1(d). The\nϕ)-scan exhibits four sharp reﬂections, 90◦apart, conﬁrm-\ning the cubic symmetry of the CFO ﬁlms grown at 670◦C.\nFurthermore, we conclude that the epitaxial relationship i s\nMAO(100)[001]/bardblCFO(100)[001].\nThe morphologyand surface roughness of the synthesized\nﬁlms has been characterized by SEM and AFM. Figure 2(a)\nshows the cross-section SEM image of a CFO ﬁlm deposited\nat 670◦C. The sample exhibits a clear and sharp interface\nbetween the CFO ﬁlm and MAO substrate. The thickness of\n(a) (b)\nMAOCFO\n1.04 µm \n10 nm\n0 \n10 \nµm 2468\n010 \nµm 8\n6\n4\n2\n0\nFIG. 2. (Color online) (a) Cross-sectional SEMimage and (b) AFM\nimageof aCFOﬁlmgrownon(100)-oriented MAOat670◦C. Both\nimages show that the ﬁlm has a very low surface roughness. Fro m\nthe AFMmeasurement we obtain a RMSroughness of 1 .1nm.\nthe ﬁlm is measured to be 1 .04µm. The ﬁlm appears very\nsmooth and dense, without any bulk cracks observedthrough\ntheentirecrosssection. Figure2(b)isthe3Dsurfaceploto fa\n10×10µm2AFM image of the sample surface, exhibiting a\nroot mean square (RMS) roughnessof 1 .1nm. No cracks are\nobservedovertheentireﬁlmsurface.\nThe ﬁlm cation stoichiometry was determined by energy-\ndispersive x-ray Spectroscopy (EDS). The EDS analysis re-\nsults yields a Fe/Co ratio close to the expected value of 2.0,\nindicatingastoichiometricCFO ﬁlmwithoutcontamination s.\n(b) (a)\nThreading\nDislocationsseparate spots\n200 nm CFO220\n400220\n400 000\n220 220\nMAO(c)\n(d)\nMAO\n CFO\nmisfit\ndislocation\ninterface \nFIG. 3. (Color online) Selected area electron diffraction p attern\n(a) from cobalt ferrite ﬁlm only; (b) from both the ﬁlm and the\nMAO substrate for a CFO ﬁlm grown on MAO(100) at 670◦C.\nThese images point to a relaxed growth of CFO on MAO consisten t\nwiththeepitaxialrelationshipdeterminedfromXRDmeasur ements:\nMAO(100)[001]/bardblCFO(100)[001]. (c) The bright ﬁeldTEMimage\nwithg=/angbracketleft400/angbracketrightreveals several misﬁt dislocations within the CFO\nthinﬁlm. (d) Fourier ﬁlteredimage showing formationof mis ﬁtdis-\nlocations at ﬁlm-substrate interface.\nTransmission electron microscopy (TEM) images and se-\nlected area electron diffraction (SAED) patterns have been\nrecorded to further examine the epitaxial growth and mi-\ncrostructureoftheCFOﬁlmgrownat670◦C. SAEDpatterns4\ntakenalongthe [001]directionanddiffractionspotsfromdif-\nferent lattice planes of CFO ﬁlm are indexed in Fig. 3(a). In\nFigure3(b),twosetsofdiffractionspots,originatingfro mthe\nCFO ﬁlm and MAO (100) substrate, overlap with each other\nat lower order diffractions and slightly separate at higher or-\nder diffractions. This conﬁrms the relaxed growth of CFO\non MAO under these deposition conditions. Moreover, these\nresultsconﬁrmtheepitaxialrelationshipfoundinXRDexpe r-\niments[c.f. Fig.1(d)]: MAO (100)[001]/bardblCFO(100)[001].\nThe bright ﬁeld TEM ( g=/angbracketleft400/angbracketright) image [Fig. 3(c),(d)] of\nthe sample reveals the microstructure of the ﬁlm in detail.\nThreading dislocations and dark diffused contrast areas ar e\nobserved in these images. These dislocations are likely gen -\neratedduringthethinﬁlmstrainrelaxationprocesscaused by\nlatticemismatchand/orthermalexpansiondifferencebetw een\ntheﬁlmandsubstrate. Somedarkdiffusedcontrastmayresul t\nfrom anti-phase domains or cation ordering, as suggested by\nDattaet al..16,17Figure 3(d) shows the Fourier ﬁltered image\nof a interface area. At the interface of CFO and MAO we\nobserve some misﬁt dislocations. Nevertheless, these resu lts\ndemonstratethatusingtheoptimizedsetofdepositionpara m-\neterswe obtainCFO ﬁlmswith alow densityofdislocations.\n200 300 400 500 600 700 800F2gF2gF2gEgEgA1g-463λexc= 488 nmCFO/MAO\n-657-616-473\n-577\n-694-309-208XY\nX'Y'X'X'XXIntensity (a.u.)\nRaman Shift (cm)-1\nFIG. 4. Polarized Raman spectra obtained for a CFO ﬁlm on MAO\nsubstrategrownat670◦Cusinga488nmexcitationatroomtemper-\nature. Additional lines of A1gsymmetry are observed at 473 cm−1\nand 616 cm−1. This suggests an ordered distribution of Co2+and\nFe3+onthe octahedral sites.\nFigure4 showsthe polarizedRamanspectra ofa CFO ﬁlm\ngrownon MAO(100)substrate at 670◦Cwith a 488nm exci-\ntation at roomtemperature. The numbersof Raman linesand\ntheir positions are consistent with those of previous studi es\nof CFO samples in the forms of crystals, powders, and thin\nﬁlms.18–21The apparent differences between XYandX′Y′\nspectraandbetween XXandX′X′spectraprovideunambigu-\nousevidencethattheﬁlmisofexcellentepitaxialquality. The\nnumberoftheRamanlinesexceedsthatexpectedforanormal\nspinel or an inverse spinel with Co/Fe disorder at the B-site s,\nwhere onlyﬁve Raman modes( A1g+Eg+3F2g) are allowed.\nWith a Co/Fe disorder, only one A1gmode is allowed for the\nXXandX′X′scattering conﬁgurations,but forbiddenfor XY.\nWhile the Egmodeis allowed in XX,X′X′, but forbiddenforXYandX′Y′scatteringconﬁgurations. OurCFOﬁlmexhibits\nexhibits additional Raman modes of A1gsymmetry centered\nat473cm−1and616cm−1. TheEgmodeisallowedwith XX,\nX′X′,andX′Y′,butforbiddenfor XY. TheF2gmodesarethe-\noretically allowed with XYandX′X′, but forbidden with XX\nandX′Y′. Here again we observe an additional F2gmode at\n657cm−1. The issue of additional Raman peaks in the spec-\ntra of inverse spinels has been discussed in detail for close ly\nrelated NiFe 2O4(NFO). The same argumentscan also be ap-\nplied to CFO as the structure and polarized Raman spectra\nof both materials are similar. Within the model proposed by\nIvanovetal.,22theadditionalpeaksreﬂectthefactthatatami-\ncroscopic level Co2+and Fe3+are not randomly distributed,\nbutorderedattheoctahedralsites. Theseresultsshowthat our\nCFO ﬁlms grown with the optimized deposition parameters\nexhibit the inverse spinel structure with indication of cat ion\norderingat theoctahedralsites.\n-4 -2 0 2 4\nµ0H (T)-0.8 -0.4 0.0 0.4 0.8-600-400-2000200400600\nT=300 KM (kA/m)\nµ0H (T)T=5 K\n0 50 100 150 200 250 3000200400600\nbulk CFOM (kA/m)\nT(K)CFO on MAO\nµ0H=3 T(c)(a) (b)\nCFO on MAO\nin plane\nFIG. 5. (Color online) SQUID magnetometry measurements of\na CFO ﬁlm grown on MAO substrate at 670◦C. (a) Hysteresis\nloops measured at 5K (black, closed squares) and 300K (red, o pen\nsquares) with magnetic ﬁeld applied along the [001] in-plan e direc-\ntion. At lower temperatures the coercive ﬁeld increases. (b ) Same\nhysteresis loops as in (a) at larger external magnetic ﬁelds . For 5K\nthe ﬁlmsaturates at ﬁelds ≥2T. (c) Temperature dependence of the\nsaturation magnetization measured at µ0H=3 T applied in plane\n(black circles). The dashed blue line represents the temper ature de-\npendent saturation magnetization for bulk CFO taken from 23 . The\nblackandredsquaresarethevaluesextractedfromthe M(H)loopsat\n5K and 300K, respectively. The diamagnetic signal from the M AO\nsubstrate has been subtracted for thisanalysis.\nWe analyzedthe magnetic propertiesof a 1 µm thick CFO\nﬁlm grown on MAO at 670◦C using SQUID-magnetometry.\nThe obtained data has been corrected by subtracting the\ntemperature independent, diamagnetic contribution from t he\nMAO substrate. The hysteresis loops ( M(H)) obtained at5\n5 K and 300 K are shown in Fig. 5(a) and (b). The coer-\ncive ﬁeld increases from 70 mT at 300 K to 300mT at 5 K.\nWeattributethischangetoanincreaseinmagneticanisotro py\nwith decreasing temperatures. As evident from Fig. 5(b) for\nµ0H≥3T, the magnetizationsaturates at both temperatures.\nWeextractasaturationmagnetizationof420kAat T=300K\nand 530kA at T=5K. The value for the saturation magne-\ntization at 300 K is slightly lower than the already reported\nvalue for CFO thin ﬁlms of 430 kA and the bulk value of\n450kA.24–26As evident from the temperature dependence in\nFig.5(c)andthe M(H)loops[Fig.5(b)]thesaturationmagne-\ntization increases with decreasing temperature. We note th at\nthe strongerincrease in saturationmagnetizationat T≤10K\nismost likelycausedbyparamagneticimpuritiesin theMAO\nsubstrate. This effect also inﬂuences the quantitative val ue\nextracted for the saturation magnetization at 5 K. Neverthe -\nless, the increase in saturation magnetization is also pres ent\natT≥10 K. Compared to the temperature dependence data\nof bulk CFO by Pauthenet23[dashed blue line in Fig. 5(c)],\ntheﬁlmexhibitsthesamequalitativetemperaturedependen ce.\nThe quantitative values for the ﬁlm are for T≥10K always\nslightly lower than the bulk data. The magnetic characteriz a-\ntion revealsthat ourCFO ﬁlms grownunderoptimizeddepo-\nsition conditionsare state-of-the-artwith a room tempera ture\nsaturationmagnetizationthat is93%ofthebulkvalue.\n10110210310410510640° 42° 44° 46° 48°2θI (cps)\n40° 42° 44° 46° 48°101102103104105106I (cps)\n2θ101102103104105106I (cps)(a)\n(c)(b)\nPZNPT (200)CFO (400)\nCFO (400)\nCFO (400)MgO(200)\nPMNPT (200)\nFIG. 6. XRD 2 θ−ωscans for CFO ﬁlms grown on (a) (100)-\noriented MgO, (b) (100)-oriented PZN-PT, and (c) (100)-ori ented\nPMN-PT using our optimized deposition conditions. All ﬁlms ex-\nhibit a (100) orientation.\nAdditionally,wehaveinvestigatedtheversatilityofouro p-\ntimizeddepositionconditionsbygrowingCFOthinﬁlmsona\nvariety of different substrates. For this study we deposite d1µm thick CFO ﬁlms on (100)-oriented MgO, along with\npiezoelectricleadmagnesiumniobate-leadtitanate(PMN- PT)\nand lead zinc niobate-lead titanate (PZN-PT) substrates. W e\nanalyzed the structural properties using XRD. The results o f\nthe 2θ−ωscans are shown in Fig. 6(a)-(c). For all of these\ndifferent substrates we observe a epitaxial growth with the\n(h00)planes parallel to the ﬁlm surface. In addition, wide\nrange 2θ−ωscans (not shown here) exhibit only reﬂec-\ntions of the (100)-oriented CFO ﬁlm and substrates. This\nexcludes the existence of any secondary phases in the sam-\nples. From the XRD measurements we extracted the follow-\ning CFO out-of-plane lattice constants: 8 .340˚A for MgO,\n8.370˚A for PZN-PT, and 8 .374˚A for PMN-PT; These dif-\nferences can be attributed to the difference in lattice mis-\nmatch for the different substrates. For MgO we expect a\ntensile in-plane stress, while for PMN-PT and PZN-PT we\nexpect an in-plane compressive stress. Only for CFO on\nMgO our experimental values and the expected stress state\nagree. AsdiscussedaboveforthegrowthonMAOsubstrates,\nthese results suggest that the density of misﬁt dislocation s\nis the dominating factor, which determines the out-of-plan e\nlattice constant. Moreover, we used ϕ-scans on the CFO\nﬁlm and substrate (220)-reﬂectionsto determine the in-pla ne\nepitaxial relationship (We here neglected the small rhombo -\nhedral distortion for PMN-PT and PZN-PT). For all three\nsubstrates we ﬁnd: MgO ,PZN−PT,PMN−PT(100)[001]/bardbl\nCFO(100)[001]. The XRD analysis of the growth of CFO\non different substrates using DLI-CVD shows that our op-\ntimized deposition parameters allow the growth of epitaxia l\n(100)-orientedﬁlmsindependentofthe substrate.\nThe surface roughness of the CFO ﬁlms on different sub-\nstrateshasbeenevaluatedbyAFMscansona10 µm×10µm\narea. From these measurements we obtain a RMS roughness\nof 3.9 nm, 6.0 nm, and 2 .4 nm for our CFO ﬁlms grown on\nMgO,PMN-PT,andPZN-PT,respectively.\nIV. SUMMARY\nIn summary, we have grown CFO ﬁlms on (100)-oriented\nMAO substrates by DLI-CVD. A narrow deposition tem-\nperature window is found for growing high quality, epi-\ntaxial CFO ﬁlms. XRD and TEM analysis of CFO ﬁlms\ngrown under these conditions yield the epitaxial relation:\nMAO(100)[001]/bardblCFO(100)[001]. AFMandSEMmeasure-\nments show that the ﬁlms exhibit a low RMS surface rough-\nnessof1.1nm. Ramanstudiesoftheoptimizedﬁlmsconﬁrm\nthe excellent structural qualityand suggest an orderingof the\nCo2+andFe3+ionsontheB-latticesites. Thesaturationmag-\nnetization of 420 kA /m at 300 K is very close to the CFO\nbulk value, and the temperature dependence of the saturatio n\nmagnetizationnicelyagreeswithmeasurementsonbulkCFO.\nMoreover, we have demonstrated that the DLI-CVD process\nis highly versatile and enables the growth of epitaxial and\nsmooth CFO ﬁlms on a variety of other substrates, including\nMgO, PMN-PT and PZN-PT. A comparison of the expected\nin-plane strain and the observed out-of-plane lattice spac ing\nsupport the fact that the density of misﬁt dislocations is th e6\nprimary factor inﬂuencing the lattice constant of CFO. Thes e\nresultspavethewayforthefabricationofmagnetoelectric het-\nerostructuresandnovelspintronicdevicesbasedonDLI-CV D\ngrownCFO ﬁlms.\nV. ACKNOWLEDGEMENTS\nThis work was supported by NSF ECCS Grant No.\n1102263. 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(Springer, New York,\n2006)."    },    {        "title": "2212.10284v1.Steel_Phase_Kinetics_Modeling_using_Symbolic_Regression.pdf",        "content": "Steel Phase Kinetics Modeling using\nSymbolic Regression\nDavid Piringer\nHeuristic & Evolutionary Algorithms Lab\nUniversity of Applied Sciences Upper Austria\n4232 Hagenberg, Austria\ndavid.piringer@fh-hagenberg.atBernhard Bloder\nMCL Forschungs GmbH\n8700 Leoben, Austria\nbernhard.bloder@extern.mcl.atGabriel Kronberger\nHeuristic & Evolutionary Algorithms Lab\nUniversity of Applied Sciences Upper Austria\n4232 Hagenberg, Austria\ngabriel.kronberger@fh-hagenberg.at\nAbstract —We describe an approach for empirical modeling\nof steel phase kinetics based on symbolic regression and genetic\nprogramming. The algorithm takes processed data gathered from\ndilatometer measurements and produces a system of differential\nequations that models the phase kinetics. Our initial results\ndemonstrate that the proposed approach allows to identify\ncompact differential equations that ﬁt the data. The model\npredicts ferrite, pearlite and bainite formation for a single steel\ntype. Martensite is not yet included in the model. Future work\nshall incorporate martensite and generalize to multiple steel types\nwith different chemical compositions.\nIndex Terms —steel, phase kinetics, genetic programming, sym-\nbolic regression, dynamic models\nI. I NTRODUCTION\nSteels are alloys consisting of the main elements iron and\ncarbon and often contain additional elements, which may be\ndeliberate additions for beneﬁcial effects or be present as\nlimited but unavoidable residuals from steelmaking, so called\ntramp elements. All of these elements are incorporated in\nthe crystalline solid microstructure, which directly affects the\nproperties and performance of the steel [1]. The fact that\npure iron has two different crystalline structures at different\ntemperatures allow for steelmaking with great versatility [2]:\nAustenite at high temperatures ( 912°C–1394 °C) and ferrite\nat low and room temperatures ( 912°C).\nThe crystal structures of pure iron are shown in Figure 1.\nThe shown unit cells deﬁne the repeating iron atom arrange-\nments in the crystal. For austenite the structure is face-centered\ncubic (FCC), for ferrite body-centered cubic (BCC). The FCC\nstructure of austenite is more densely packed than the BCC\nstructure of ferrite, leading to a volume change of approxi-\nmately 1–3% for the austenite to ferrite transformation [3].\nDilatometry is a technique that utilizes the change in volume\nto study phase transformations [4].\nHeat treatment can be used to achieve desired mechani-\ncal steel properties such as hardness, strength, ductility or\ntoughness, which are dependent on microstructure and grain\nsize [5]. Steel which is heated to its austenitising tempera-\nture may transform to different phases during cooling, each\nproviding a unique combination of properties. Some of these\ntransformations products are (listed in order of their formation\nwith increasing cooling velocity):\n(A)Face-centered cubic struc-\nture: Atoms are on each corner\nand the centers of each face of\nthe cube.\n(B)Body-centered cubic struc-\nture: Atoms are on each corner\nand the center of the cube.\nFig. 1: FCC and BCC structures.\n\u000fFerrite: BCC crystal structure with other elements in solid\nsolution.\n\u000fPearlite: Phase consisting of alternate lamellae of ferrite\nand cementite (iron carbide, Fe3C), formed on slow\ncooling.\n\u000fBainite: Non-lamellar mixture of ferrite and carbides,\nwhich can be classiﬁed into upper and lower bainite.\nUpper bainite consists of plates of cementite in a ma-\ntrix of ferrite. Lower bainite consists of ferrite needles\ncontaining carbide platelets\n\u000fMartensite: Metastable phase resulting from the diffusion-\nless athermal decomposition of austenite below a certain\ntemperature at sufﬁciently high cooling rates.\nA. Transformation model\nTo model the transformation behavior of austenite to its trans-\nformation products many models exist, which often rely on the\nJohnson-Mehl-Avrami-Kolmogorov model (JMAK) [6]–[10].\nThe basic equation for isotherm transformation reads as\nX= 1\u0000exp(\u0000ktn), whereXis the volume of the trans-\nformed phase, and kandnare material dependent parameters.\nJMAK models may incorporate different nucleation and\ngrowth theories and are applicable for the transformation\nof austenite to ferrite, pearlite and bainite. To describe the\ntransformation to martensite, most commonly the Koistinen-\nMarburger equation or a variation of it is used [11]: X=\n1\u0000exp(\u0000\u000b(Ms\u0000T)), where\u000bis a parameter, \u000b\u00190:011,\nMsis the martensite start temperature (highest temperaturearXiv:2212.10284v1  [cs.LG]  19 Dec 2022at which martensite can form) and Tthe temperature of the\nsample, which is cooled below Ms.\nFurther information on steels with complex microstructures\nand various modeling approaches can be found in [12].\nII. D ATA ACQUISITION\nThe data is obtained by dilatometer measurements of one\nsteel type. Several measurements with different cooling rates\nranging from 0:6K\nsto120K\nswere made. The data was\nprocessed and for the transformation rates, ﬁts for each phase,\nbased on additional information, were calculated such that the\noverall transformation is described by the transformation of\nthe individual phases. The result of the data preprocessing\nare ten data ﬁles which contains the cooling rate ( Ks), the\ntemperature T, timet, the change rates of volume fractions\n(_P1;_P2;_P3;_P4) and the retained fraction of austenite ( RA).\nBecause of the high computation time, we decided to use\nonly ﬁve data ﬁles, which are shown in Figure 2. We used the\ncombination of the data ﬁles to ﬁnd a model that describes the\nkinetics for all cooling rates. In this ﬁrst set of experiments\nwe ignore martensite ( _P4).\nIII. M ODELING APPROACH\nWe used tree-based genetic programming (GP) with a multi-\ntree representation. Each solution candidate consists of four\ntrees, whereby each tree represents one expression on the right\nhand side (RHS) of the system of differential equations (DE).\nP1=fP1(_P1;_P2;_P3;RA;Ks;T;\u0012 P1)\nP2=fP2(_P1;_P2;_P3;RA;Ks;T;\u0012 P2)\nP3=fP3(_P1;_P2;_P3;RA;Ks;T;\u0012 P3)\n_RA=fRA(_P1;_P2;_P3;RA;Ks;T;\u0012 RA)\nThe expression trees are evolved using GP. The algorithm\nautomatically selects features that are included in each ex-\npression. In the same way the functions and operators used\nin each expression are evolved automatically. We use the\nsame function and terminal set for each expression and apply\nthe same limits for the maximum number of nodes and the\nmaximum depth of each tree.\nThe evolved expressions may also include numeric param-\neters\u0012which are memetically optimized as described in [13].\nFor ﬁtness evaluation of a solution candidate we ﬁrst solve the\nDE system using the measured initial values with a numeric\nintegration scheme (RK45) and then compare to the target\nvalues for _P1;_P2;_P3and RA in the data ﬁles.\nWe used all data rows for training to check whether it is\nin principle possible to evolve a system of DE that ﬁts the\nmeasured steel phase kinetics. An analysis of the expected\nerror of the approach with separate training and test sets is\nleft for future work.\nA. Algorithm Conﬁguration\nWe used the Age-Layered Population Structure Genetic Al-\ngorithm (ALPS-GA), which was invented by Gregory S.\nHornby [14]. Similar to the original Genetic Algorithm (GA)by J. Holland [15], is uses the same genetic cycle (with selec-\ntion, crossover and mutation), but isolated for each age-layer.\nEach layer represents an age bracket with its own population.\nThe only exception for this isolation is the selection operator,\nwhich typically selects parents out of a layer-overlapping\nmating pool. The default conﬁguration for this mating pool is\nthe current layer (wherein the selection operator is currently\nworking) and the underlying layer with younger individuals.\nThe ﬁrst layer (youngest age bracket) is regularly reseeding\nwith random individuals (deﬁned by the age gap). New layers\nopen up after a predeﬁned aging scheme, which uses the\nage gap for calculation. The maximum number of layers is\nconﬁgurable. The age of an individual is deﬁned as follows\n1 +AgeoldestParent .\nFor this paper we tried several different parameter settings\nthat produced good results with ALPS-GP for similar prob-\nlems. A more detailed sensitivity analysis is left for future\nwork. The parameter values are shown in table I. We use and\nextend the framework HeuristicLab1[16] for this paper. More\ndetails about the assigned operators can be read in [17], [18].\nParameter Value\nCrossover Subtree Swapping\n(Probability = 25%)\nMutator Multi Symbolic Expression\nTree Manipulator\nMutation Probability 10%\nElites 1\nPopulation Size (per Layer) 200\nSelector Generalized Rank\n(Pressure = 5)\nMating Pool Range 1 (Current + Underlying)\nAge Gap 8\nAging Scheme Polynomial\nAge Layers 16\nmax. Generations 2000\nTerminal Set State variables and real-valued\nparameters\nFunction Set +;\u0002;\u0004; x2;exp;tanh ,\nanalytic quotient1\n1AQ(x; y) =x=p\n1 +y2\nTABLE I: Parameter conﬁguration for the ALPS-GA.\nIV. R ESULTS\nFigure 2 shows data and predictions for ﬁve different cooling\nrates (in K=s). Because we use all data rows for training,\nall predictions reﬂect the training set. Each panel in the\nﬁgure shows the change rate of the transformed amount over\ntemperature T, whereTranges from 830°Cdown to 34°C,\nfor_P1(ferrite), _P2(pearlite) and _P3(bainite) as well as their\npredictions. At the current stage, the amount of _P4(martensite)\nis ignored.\nA. Models\nThe equations 1 (phase 1), 2 (phase 2), 3 (phase 3) and 4\n(retained austenite) are parts of one solution, which represent\n1https://github.com/heal-research/HeuristicLab(A)Cooling rate 0:6K\ns\n(B)Cooling rate 2:5K\ns\n(C)Cooling rate 10K\ns\n(D)Cooling rate 40K\ns\n(E)Cooling rate 80K\ns\nFig. 2: Data and predictions (training) for ﬁve cooling rates. Each plot shows the rate of change of the volume fraction for\n_P1(ferrite), _P2(pearlite) and _P3(bainite).the best result found in terms of the normalized mean squared\nerror (NMSE). Every equation contains numeric parameters\n(tagged with cx).\nP1= (AQ(c0;tanh(c1RA)) +c2)\u0010\nc3_P1+AQ(c4;(c5Ks)2)\u0011\nc6+c7(1)\nc0= 3:591 c3= 6:4351\u000110\u00002c6= 5:1832\nc1= 2:2151 c4= 4:2717 c7= 5:0527\u000110\u00004\nc2=\u00003:031 c5= 56:255\nP2= (c0_P1(c0_P2+c1)c2+c3_P3c3_P2c4+c5)c6+c7\n(2)\nc0= 0:96475 c3= 1:0441 c6= 0:65098\nc1=\u00007:6099\u000110\u00005c4=\u00003:9235 c7= 2:0752\u000110\u00007\nc2= 0:68128 c5= 2:5571\u000110\u00005\nP3= (c0_P1+c1_P3tanh(tanh(c2_P1+c3))c4)c5+c6\n(3)\nc0= 3:1903\u000110\u00004c3= 0:75539 c6= 1:4733\u000110\u00005\nc1= 0:99678 c4=\u00009:0097\u000110\u00002\nc2=\u0000406:1 c5= 0:63386\n_RA= (AQ(c0;c1RA+c2)+\nc3_P1(AQ(c4;c5_P1) +c6) +c7)c8+c9(4)\nc0= 0:50793 c4=\u00001:5443\u000110\u00003c7=\u00000:28694\nc1= 9:0808 c5= 0:7967 c8= 6:6264\u000110\u00002\nc2=\u00003:368 c6= 0:12045 c9= 1:3127\u000110\u00002\nc3= 0:91246\nV. D ISCUSSION\nThe results of our approach are promising and indicate the\npossibility to use GP with a multi-tree representation for DE\nsystems. Despite the strong simpliﬁcation, we are optimistic\nto ﬁnd good and suitable solutions for the given problem\nformulation of steel phase kinetics modeling. The found for-\nmulas are compact and readable. Additionally, each formula\ncontains self-references or references to other formulas, which\nis preferred and necessary for a functional DE system. The\nlearned system ﬁts the targets quite well but still has some\nmargin for improvement. The high computational cost of the\ncurrent implementation has to be considered, as it makes it\ndifﬁcult to experiment with a broader spectrum of conﬁgura-\ntions. Nevertheless, there are still improvements and extensive\nexperiments planned for future work. Martensite should be\nincorporated into the model and the model must be generalized\nto multiple steel types. For this it could be useful to include\nthe grain size as an additional parameter. Furthermore, it is\nnecessary to split the data set into training and test sets to make\nsure the model is not overﬁtting. A severe limitation of the\ncurrent approach is that the model uses estimated values for the\nvolume fractions which were ﬁt using additional information.\nThese estimates might be incorrect and lead to an inaccurate\nmodel. It remains an open question how we can learn a model\nfor the kinetics solely from the dilatometer data.ACKNOWLEDGEMENTS\nThe authors gratefully acknowledge the ﬁnancial support under the\nscope of the COMET program within the K2 Center “Integrated Com-\nputational Material, Process and Product Engineering (IC-MPPE)”\n(Project No 886385). This program is supported by the Austrian Fed-\neral Ministries for Climate Action, Environment, Energy, Mobility,\nInnovation and Technology (BMK) and for Digital and Economic\nAffairs (BMDW), represented by the Austrian Research Promotion\nAgency (FFG), and the federal states of Styria, Upper Austria and\nTyrol.\nREFERENCES\n[1] G. Krauss, Steels: Processing, Structure, and Performance . ASM\nInternational, Jan. 2015.\n[2] G. Krauss, “Physical metallurgy of steels,” in Automotive Steels , pp. 95–\n111, Elsevier, 2017.\n[3] H. Bhadeshia and R. Honeycombe, Steels: Microstructure and Proper-\nties. Elsevier Science and Technology, Jan. 2017.\n[4] J.-C. Zhao, ed., Methods for Phase Diagram Determination . Elsevier\nScience, 2007.\n[5] W. F. Gale and T. C. Totemeier, eds., Smithells Metals Reference Book .\nAmsterdam Boston: Elsevier Butterworth-Heinemann, 2004.\n[6] W. A. Johnson and R. F. Mehl, “Reaction kinetics in processes of\nnucleation and growth,” Trans. Am. Inst. Min. Metall. Eng. , vol. 135,\npp. 416–442, 1939.\n[7] M. Avrami, “Kinetics of phase change. I general theory,” The Journal\nof Chemical Physics , vol. 7, pp. 1103–1112, Dec. 1939.\n[8] M. Avrami, “Kinetics of phase change. II transformation-time relations\nfor random distribution of nuclei,” The Journal of Chemical Physics ,\nvol. 8, pp. 212–224, Feb. 1940.\n[9] M. Avrami, “Granulation, phase change, and microstructure kinetics of\nphase change. III,” The Journal of Chemical Physics , vol. 9, pp. 177–\n184, Feb. 1941.\n[10] A. Kolmogorov, “A statistical theory for the recrystallisation of metals,”\nIzvestiya Akademii Nauk SSSR , vol. 3, 1937.\n[11] D. Koistinen and R. Marburger, “A general equation prescribing the\nextent of the austenite-martensite transformation in pure iron-carbon\nalloys and plain carbon steels,” Acta Metallurgica , vol. 7, pp. 59–60,\nJan. 1959.\n[12] C. Tasan, M. Diehl, D. Yan, M. Bechtold, F. Roters, L. Schemmann,\nC. Zheng, N. Peranio, D. Ponge, M. Koyama, K. Tsuzaki, and D. Raabe,\n“An overview of dual-phase steels: Advances in microstructure-oriented\nprocessing and micromechanically guided design,” Annual Review of\nMaterials Research , vol. 45, pp. 391–431, Jul. 2015.\n[13] G. Kronberger, L. Kammerer, and M. Kommenda, “Identiﬁcation of\ndynamical systems using symbolic regression,” in Computer Aided\nSystems Theory – EUROCAST 2019: 17th International Conference,\nLas Palmas de Gran Canaria, Spain, February 17–22, 2019, Revised\nSelected Papers, Part I , (Berlin, Heidelberg), p. 370–377, Springer-\nVerlag, 2019.\n[14] G. S. Hornby, “ALPS: The age-layered population structure for reducing\nthe problem of premature convergence,” in Proceedings of the 8th\nAnnual Conference on Genetic and Evolutionary Computation , GECCO\n’06, (New York, NY , USA), p. 815–822, Association for Computing\nMachinery, 2006.\n[15] J. H. Holland, Adaptation in natural and artiﬁcial systems: an intro-\nductory analysis with applications to biology, control, and artiﬁcial\nintelligence . MIT press, 1992.\n[16] S. Wagner and M. Affenzeller, “HeuristicLab: A generic and extensible\noptimization environment,” in Adaptive and Natural Computing Algo-\nrithms (B. Ribeiro, R. F. Albrecht, A. Dobnikar, D. W. Pearson, and\nN. C. Steele, eds.), (Vienna), pp. 538–541, Springer Vienna, 2005.\n[17] M. Kommenda, G. Kronberger, S. Wagner, S. Winkler, and M. Affen-\nzeller, “On the architecture and implementation of tree-based genetic\nprogramming in heuristicLab,” in Proceedings of the 14th Annual Con-\nference Companion on Genetic and Evolutionary Computation , GECCO\n’12, (New York, NY , USA), p. 101–108, Association for Computing\nMachinery, 2012.\n[18] M. Affenzeller, S. Wagner, S. Winkler, and A. Beham, Genetic al-\ngorithms and genetic programming: modern concepts and practical\napplications . Chapman and Hall/CRC, 2009."    },    {        "title": "0707.0350v1.A_new_concept_and_design_of_ferrite_based_microwave_vortex_devices.pdf",        "content": "A new concept and design of ferrite-based microwave vortex \ndevices \n \nM. Sigalov1, E. Kamenetskii2, and R. Shavit3 \n   \n                                                           \nDepartment of Electrical and Computer Engineering, Ben Gu rion University of the Negev,  Beer Sheva 84105, Israel, \n1e-mail: sigalov@ee.bgu.ac.il , tel.: +972 8 6472402, fax: +972 8 6472402, \n2e-mail: kmntsk@ee.bgu.ac.il , tel.: +972 8 6472407, fax: +972 8 6472949, \n3 e-mail: rshavit@ee.bgu.ac.il , tel.: +972 8 6471508, fax: +972 8 6472949. \n Abstract  − In microwave resonant systems with ferrite \nsamples, one becomes faced with specific phase relations for the electromagnetic fields.  Such specific phase relations may lead to appearance of nontrivial states: electromagnetic vortices. This paper provides some general ideas and design principles for microwave devices with the field vortex structures.   \n1 INTRODUCTION \nGeneral scattering and propagation characteristics \nof microwave waveguides containing ferrite samples were under investigations in numerous works during many years (see, e.g. [1]).  Specifically, in questions \nof microwave theory and techniques, an interest in such systems was devoted to development of general and rigorous formulations of the problem and an analysis of properties of the scattering matrix. At the same time, the studies of quadratic relations do not touch upon a question about specific phase relations for the fields in such microwave systems. \nIn a case of ferrite inclusions acting in the \nproximity of the ferromagnetic resonance (FMR), the phase of the wave reflected from the ferrite boundary depends on the direction of the incident wave. This fact, arising from special boundary conditions for the \ntangential components of the fields on the dielectric-ferrite interface, causes the time-reversal symmetry breaking effect in microwave resonators with inserted ferrite sample s [2]. When microwave \nresonators contain enclosed gyrotropic-medium samples, the electromagnetic-field eigenfunctions will be complex, even in the absent of dissipative losses. It means that the fields of eigen oscillations are not the fields of standing waves in spite of the fact that the eigen frequencies of a cavity with gyrotropic-medium samples are real [3]. This leads to very specific topological-phase characteristics. The power-flow lines of th e microwave-cavity field \ninteracting with a ferrite sample, in the proximity of its FMR, may form whirlpool-like electromagnetic vortices [4]. Vortices can be easily observed in water. They are \na common occurrence in plas ma science. In recent \nyears, there has been a c onsiderable interest in \nvortices of electromagnetic fields. Study of vortices with optical phase singul arities has opened up a new \nfrontier in optics [5]. In microwaves, there is a special interest in elect romagnetic vortices created \nby ferrite particles with the FMR conditions.  \nIt can be shown analytica lly that for TE polarized \nelectromagnetic waves, the singular points of the Poynting vector (the vortex cores) are directly related to the zero-electric-field topological features in a vacuum region of the cavity space [2, 4]. To analyze the vortex structure for a case when a ferrite \nsample is placed in a maximal cavity electric field and to study the fields inside a ferrite region one has to use numerical simulation methods. \n The main purpose of this paper is to develop novel \nconcepts and design principles for microwave devices with the field vorte x structures created by \nferrite samples. In our studies, we use the HFSS (the software based on FEM method produced by ANSOFT Company) CAD simulation programs for 3D numerical modeling of Maxwell equations. In our numerical experiments, both modulus and phase of the fields are determined.  \n2 MICROWAVE VORTICES IN THE   \nCAVITY-FERRITE-DISK SYSTEM \nThe most known examples of systems used for an \nanalysis of ferrite-based  microwave vortices are \nmicrowave cavities with enclosed ferrite samples [2, 4]. It was shown in [4] that in the rectangular-waveguide cavity a small ferrite disk acts as a topological defect causing induced vortices. Figure 1 gives a picture of the Poynting vector distribution in a cavity when a thin normally magnetized ferrite disk (being oriented so that its axis is perpendicular to a wide wall of a waveguide) is placed in the maximal cavity electric field. In the vicinity of a ferrite disk  2and inside it, one has strong localization of the \npower flow density due to the vortex creation. As it is shown in Figure 1, the power input is at the left-hand side of a system. If one interchanges the microwave source and receiver positions, leaving fixed a direction of the bias magnetic field, the vortex will have the same rotation direction. So the vortex rotation direction is invariant with respect to mirror reflection along a waveguide axis. When one reverses the DC magnetic field together with an interchange of the microwave source and receiver positions, the vortex changes its rotation direction. It means that the vortex rotation direction is not invariant with respect to a combined symmetry operation: mirror reflection a nd time reversal. This is \na distinctive feature regarding the known reciprocity-theorem relationships for gyrotropic media [6].   \n \nFigure 1:  A rectangular-waveguide cavity with two \nirises and an enclosed ferrite disk. \nThe observed Poyning-vector vortices are \ncharacterized by unique symmetry properties of the fields. The symmetry breaking of the field structures \nbecome evident from the pict ures shown in Figures 2 \nand 3. Figure 2 shows the RF magnetic field and Figure 3 gives the RF electric field in the vicinity of \na ferrite disk. The pictures in Figures 2 and 3 have \nthe \no90 phase shift. \n \n               \n  \n \nFigure 2:  Top and side views of the RF magnetic \nfield near a ferrite disk.        \n  \nFigure 3:  A view of the RF electric field near a \nferrite disk. \n The field localization takes place also when a \nferrite disk is placed not in a maximal cavity electric field, but in a maximal cavity magnetic field. In this case, however, a structure of the Poynting-vector \nvortices and symmetry properties of the fields are completely different from those shown in Figures 1 – 3 [7]. It is necessary to note that in all cases the \nvortex does not accumulate energy. Due to the vortex one just has redistribution of energy inside a cavity.  \nAn analysis of the Poyn ting-vector structure as \nwell as the field structures is an important tool in developing design principles for novel microwave-vortex devices.      \n3 DESIGN OF MICROWAVE-VORTEX \nDEVICES \nIn attempts to use unique properties of microwave \nvortices for design of new microwave devices, one becomes faced with serious  difficulties. The vortices \nare very sensitive to the field structure in a microwave system. When one strongly decreases the standing-wave ratio in a cavity, the vortex disappears. At the same time, creation of vortices in standard devices may completely destroy the functional ability of a system. Such an \"anti-design\" can be clearly demonstrated in a standard circulator. \nWhen one creates a vortex in a ferrite disk in a circulator, the entire syst em becomes completely \nreciprocal. \nIn this paper, we show some examples of proper \napplication of the vortex concept in the design of microwave antennas and near-field microwave sensors. \n3.1 A vortex concept in the microwave antenna  \ndesign \nWe analyze a microwave patch antenna with an \nenclosed normally magnetized ferrite disk. Excitation of an antenna is due to a microstrip line. A general structure of an antenna is shown in Figure 4.    3\n \nFigure 4: A microwave patch antenna with an \nenclosed ferrite disk.  \nOur main purpose is to realize a circularly polarized \nantenna in the best way. Figures 5 (a) and 6 (a) show the Poynting-vector pictures for two different positions of a ferrite disk. On e can clearly see that the \nvortex structure in the second case is much better than in the first case. It is very important to see that the \"vortex quality\" is strongly correlated with the circular characteristics of the antenna far fields.  \n \n   \n \n                                     (a) \n2.7 2.8 2.9 3 3.1 3.2 3.3 3.405101520\nFrequency [GHz]Axial Ratio [dB]\n2.7 2.8 2.9 3 3.1 3.2-40-30-20-100\nFrequency [GHz]S11 [dB]\n \n(b) \nFigure 5: A ferrite disk placed  at the center of a patch. \n(a) The Poynting-vector pict ure;  (b) The axial ratio \nand 11S parameters.         \n  \n                                        (a) \n2.7 2.8 2.9 3 3.1 3.2 3.3 3.405101520\nFrequency [GHz]Axial Ratio [dB]\n2.5 3 3.5-20-15-10-50\nFrequency [GHz]S11 [dB]\n \n                                          (b) \nFigure 6: A ferrite disk shifted from the center  of \na  patch. (a) The Poynting-vector picture;  (b) \nThe axial ratio and 11S parameters. \nFor a \"better vortex\", one has better antenna properties \ncharacterizing by a very small axial ratio (see Figure 6). At the same time, when a ferrite disk is placed at the center of a patch (see Figure 5) an excitation line is better matched.   \n3.2 Near-field microwave sensors \nAt present, there is a strong interest in different \nmicrowave sensors for the near-field characterization \nof material properties (see, e.g. [8]). In design of such sensors, the vortex concept could be especially useful. The fact that due to a vortex structure one has the microwave energy localization will be very important for increasing the sensor sensitivity. On the other hand, special structures of the vortex electric and \nmagnetic fields can be very useful for characterization \nof biological materials with the symmetry breaking properties. Figure 7 shows an example of a vortex-type near-field microwave sensor realized as a microstrip resonator with a ferrite cone. When the Poynting-vector vortex appears, one has strong field localization in the vicinity of a ferrite sample (see Figure 8). Figure 9 shows the Poynting-vector structure inside a ferrite cone.    4\n \n \n     Figure 7: A vortex-type near-field microwave \nsensor   \n \n     Figure 8: Field localization due to creation of the \nPoynting-vector vortex.  \n \n \n     Figure 9:  The Poynting-vector distribution inside a \nferrite cone. \nThe most attractive and useful feature of this device is \nthe rotating electric field on the tip of a cone. (see Figure 10). A direction of rotation is correlated with a direction of a bias magnetic field. \n \n \nFigure 10: Rotating electric field in a ferrite cone. \n4 CONCLUSION \nIn this paper we discuss a new concept and design \nprinciples for microwave devices with the field vortex structures. It b ecomes evident that the problem of microwave vortices of the Poynting \nvector created by ferrite samples can be very important for many modern applications, e.g., for processing of guided si gnals, for near-field \nmicrowave lenses, for field concentration in patterned microwave metamaterials, for new microwave antennas. It is supposed that these nontrivial states – the \"s wirling\" entities – can, in \nprinciple, be used to carry data and point to new communication systems. Application of such generic ideas to microwave systems is of increasing importance in numerous utilizations.  \nReferences \n[1] N. Okamoto, I. Nishioka, and Y. Nakanishi, \nIEEE Trans. Microw. Theory. Techn. MTT-19 , \n521 (1971); T. Yoshida, M. Umeno, and S. Miki, \nIEEE Trans. Microw. Theory. Techn. MTT-20 , \n739 (1972). \n[2] P. So et al, Phys. Rev. Lett. 74, 2662 (1995); M. \nVraničar et al , J. Phys. A: Math. Gen. 35, 4929 \n(2002); H. Schanze et al , Phys. Rev. E 71, \n016223 (2005). \n[3] A. Gurevich and G. Melkov, Magnetic \nOscillations and Waves  (CRC Press, New York, \n1996). \n[4] E.O. Kamenetskii, M. Sigalov, and R. Shavit, \nPhys. Rev. E 74, 036620 (2006). \n [5] M.S. Soskin et al, Phys. Rev. A 56, 4064 (1997); \nJ. Leach et al , New J. Phys. 7, 55 (2005); M. V. \nBashevoy, V.A. Fedotov, and N.I. Zheludev, Opt. \nExpress 13, 8372 (2005). \n[6] R.F. Harrington and A.T. Villeneuve, IRE Trans. \nMicrow. Theory Tech. MTT-6 , 308 (1958).  \n[7] M. Sigalov, E.O. Kamenetskii, and R. Shavit (to \nbe published). \n[8] C. Gao and X.-D. Xiang, Review Scientific \nInstruments 69, 3846 (1998).   \n "    },    {        "title": "1102.0432v2.Interstitial_Fe_Cr_alloys__Tuning_of_magnetism_by_nanoscale_structural_control_and_by_implantation_of_nonmagnetic_atoms.pdf",        "content": "arXiv:1102.0432v2  [cond-mat.mtrl-sci]  11 Nov 2011EPJ manuscript No.\n(will be inserted by the editor)\nInterstitial Fe-Cr alloys: Tuning of magnetism by nanoscal e\nstructural control and by implantation of nonmagnetic atom s\nInterstitial Fe-Cr alloys\nN. Pavlenko1, N. Shcherbovskikh2, and Z.A. Duriagina2\n1Institute for Condensed Matter Physics, National Academy o f Sciences of Ukraine, Svientsitsky str.1, 79011 Lviv, Ukra ine;\ne-mail:pavlenko@mailaps.org\n2Institute for Applied Mathematics and Fundamental Science s, Lviv Technical University, Ustyianowycha str. 10, 79013 Lviv,\nUkraine\nthe date of receipt and acceptance should be inserted later\nAbstract. Using the density functional theory, we perform a full atomi c relaxation of the bulk ferrite\nwith 12.5%-concentration of monoatomic interstitial Cr periodica lly located at the edges of the bcc Fe α\ncell. We show that structural relaxation in such artiﬁciall y engineered alloys leads to signiﬁcant atomic\ndisplacements and results in the formation of novel highly s table conﬁgurations with parallel chains of\noctahedrically arranged Fe. The enhanced magnetic polariz ation in the low-symmetry metallic state of\nthis type of alloys can be externally controlled by addition al inclusion of nonmagnetic impurities like\nnitrogen. We discuss possible applications of generated in terstitial alloys in spintronic devices and propose\nto consider them as a basis of novel durable types of stainles s steels.\n1 Introduction\nLast years demonstrate increased activities in the search\nfor novel materials exhibiting controlled modiﬁcation of\nelectronic properties by inclusion or implantation of dif-\nferent atoms or ionic groups. A prominent example of the\nimplantation-altered systems is the stainless steel. In the\nsteels, the implantation of chromium, molibdenium, ni-\ntrogen and other chemical elements substantially changes\nthe microstructure of subsurface layers and modify their\ncorrosion resistance and hardness [1].\nIn the development of novel eﬃcient multifunctional\nmaterials for technological applications in the long-term\ndevices, the properties like hardness, corrosion, heat resis-\ntance and other types of mechanical and chemical dura-\nbility are of central interest [2,3]. It frequently appears in\nscience and technology that well known materials doped\nby diﬀerent chemical elements exhibit unexpected physi-\ncal properties not revealed previously.\nAs an example of such a new unexpected behavior, in\nthe present work we consider an alloy Fe-Cr. The alloys\nof Fe and Cr, doped by C, Ni and by other elements, are\nwidely used as basic components for ferritic and marten-\nsitic steels. Substitutional alloys of Fe and Cr have at-\ntracted much attention of theory and experiment due to\ntheir rich magnetic properties characterized by local an-\ntiferromagnetism in the proximity of Cr atoms implanted\ninto ferromagnetic iron [4,5,6,7]. Due to small diﬀerences\nbetween the atomic radii of iron and chromium, the mod-\niﬁcation of the substitutional alloy properties is limitedto the local magnetic transformation due to local changes\nin the electronic orbital occupancies, without signiﬁcant\nstructural modiﬁcations. In contrast to the substitutional\nstructural conﬁgurations, the interstitial Fe-Cr alloys con-\nsidered in the present work contain Cr impurities which\nare located in the interstitial positions of the bcc lattice of\nFeα. In the recent theoretical studies of the Cr intersitials\nin Fe-Cr alloys, diﬀerent types of interstitial conﬁgura-\ntions were analyzed. Among them, a pair conﬁguration\n/angbracketleft111/angbracketrightdumbbell is considered as the most energetically\nfavourable which requires about 4.2eV for its formation\nunder irradiation [8,9].\nIn the present work, we consider a novel monoatomic\ninterstitial conﬁguration which contains single Cr atoms\npositioned in the centers of the edges of the bcc ferrite. In\ncontrast to the substitutional alloys, the signiﬁcant forces\ndue to the interstitial atoms induce substantial structural\noptimization which enhances the volume due to modiﬁed\nlattice constants and leads to the relaxation of the atomic\npositions in the unit cell. We ﬁnd that the relaxation of\nthe initial bcc unit cell results in signiﬁcant atomic dis-\ntortions and in the formation of atomic chain-like struc-\ntures. As appears in the density-functional-theory (DFT)\ncalculations of the optimized structures, the energy gain\nachieveddue to the structural relaxationof the considered\ninterstitial alloy can approach 6.17 eV which makes this\ntype of systems highly stable and durable. In the present\nwork, we propose to consider these artiﬁcially generated\nalloys as candidates for novel types of stainless steels.2 N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title Suppressed Due to Excessive Length\nThe fundamental diﬀerence between the industrial al-\nloys and the alloys studied in the present work is the\nordered and periodic character of the latters. In the in-\ndustrial steels, the amorphic character of the systems is\nrelated to the random distribution of the impurities. The\nhardeningof the steels proceeds throughthe surface treat-\nment and is accompanied by formation of granular mi-\ncrostructure with the spatially inhomogeneous impurity\nconcentration and modiﬁed subsurface properties [11]. In\nthe studies of the subsurface Cr-doped alloyed ferrite, we\nconsider the supercells containing periodically located Cr\natoms in the cubic lattice of Fe α. The interstitial Cr in-\nduces signiﬁcant atomic reconstruction with consequent\nbreak of initial cubic symmetry and stabilization of a new\nlower-symmetry state. The appearing structural transfor-\nmation has a character of a phase transition which occurs\ndue to nanoscale tailoring of cubic Fe by interstitial inclu-\nsion of Cr atoms, the eﬀect which can be experimentally\nveriﬁed by the means of modern methods like AFM spec-\ntroscopy .\nUsing the DFT-based structural optimization, we ob-\ntain the optimized atomic microstructure of a chain-like\ncharacter where the chains of octahedrically arranged Fe\natoms are formed along the (001)-axis. We ﬁnd that the\ncompeting ferromagnetic and antiferromagnetic interac-\ntions lead to spatially inhomogeneous spin polarization.\nThe magnetization of the structurally relaxed system is\nsigniﬁcantly enhanced as compared to the pure ferrite\nwithout Cr inclusions. The obtained enhancement makes\nthe generated alloys perspective candidates for spin polar-\nizersin spintronicapplications.In the generatedchain-like\nstructures,therelaxationisaccompaniedbytheformation\nof spatial channels with extremely low carrier density. We\nsuggest that these channels can be considered as paths for\nthe low-barrier-migration of light impurities like H, N, Li\nor C. As an example of a light atom in the interstitial al-\nloy, we study ofthe migrationpaths of nonmagnetic nitro-\ngen and calculate the energy barriers along the migration\npaths. We obtain a strong inﬂuence of the nonmagnetic\nN on the alloy magnetization. Our ﬁndings show that the\nstructural modiﬁcations due to possible nanoscale tuning\nof Cr impurities on the edges of bcc cubic cells of iron\ncan play a central role in the control of their electronic\nproperties.\n2 Structural relaxation of interstitial alloy\nFeα-Cr\nThe present studies of the electronic properties of the con-\nsidered interstitial Fe-Cr alloy are based on the DFT cal-\nculations of the electronic structure of the systems gener-\nated by periodic translation of specially chosen supercells.\nThe initial supercell shown in Fig. 1 contains the dou-\nbled 2×2 cubic bcc cell of ferrite (Fe α) and a single Cr\natom centered in one of the edges of the Fe αcubic unit\ncell with the lattice constant a= 3.85˚A. The obtained\nstructure is described by a chemical formula Fe 8Cr and\ndetermines an interstitial Fe-Cr alloy with the Cr con-\ncentration n= 0.125 which is typical for stainless steels.The presence of interstitial Cr leads to signiﬁcant local\nforces acting on the neighbouring Fe atoms. To minimize\nthe forces, the coordinates of all atoms have been relaxed.\nIn the present studies, the optimization of the supercell\nhas been performed by employing the DFT approach im-\nplemented withing the linearized augmented plane wave\n(LAPW) scheme in the full potential Wien2k code [13].\nTo study the role of the spin polarization in the structural\nrelaxation, two diﬀerent relaxation procedures have been\nemployed. In the ﬁrst procedure, the atomic optimal po-\nsitions are calculated in the local density approximation\n(LDA) on a 2 ×2×5 k-points grid. To explore the role\nof spin degrees of freedom in the relaxation, in the second\nprocedure the local spin density approximation (LSDA)\nhas been used in the optimization of the structure. The\nresults of both methods of the structural relaxation are\npresented in Fig. 2.\nFig. 1.Schematic view of unrelaxed 2 ×2 Feαcell which\ncontain 12.5% of edge-centered interstitial Cr.\nA central common feature which characterizes both\n(LDA- and LSDA-relaxed) structures is the clusterization\nof the sublattice of the iron atoms. In the LDA-optimized\nstructure(Fig. 2(a)), the relaxationresultsin formationof\na high-symmetry clusterized network. This network con-\nsists of the Fe 6-octahedra which form the square plaque-\nttesinthe( x,y)((a,b))planewithCratomslocatedinthe\ncenter of each plaquette. The distance from the centered\nCr to each nearest iron octaherda amounts 1 .9˚A. Despite\nthe signiﬁcant displacements of the iron atoms from their\ninitial positions, the net electric polarization of the cell\nis zero due to high structural symmetry C4/m obtained\nafter the relaxation.\nTheformationenergyoftherelaxedFe 8Cr-conﬁguration\ncan be expressed as\nEf(LDA) = Etot(Fe8Cr)−8Etot(Fe)−Etot(Cr),N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title S uppressed Due to Excessive Length 3\nwhere the last two terms identify the total energies of the\nbulk bcc Fe αand Cr, respectively. To determine Etot(Fe),\nwe have calculated the total energy value of the bulk Fe α\nin the ferromagneticstate.As the LSDA-calculationofthe\nspin-polarized conﬁgurations of the bulk Cr are converged\nto the paramagnetic state, we consider the total energy\nEtot(Cr) for the paramagnetic Cr. With these values, we\nﬁnd that Ef(LDA) = 4 .82eV. To analyze the role of the\nrelaxation, we have also calculated the energy Ef(unrel)\nofthe formationofinitial unrelaxed conﬁgurationwhich is\nequal to 5.02eV. As a consequence, the signiﬁcant energy\ngain due to the structural relaxation\n∆E(LDA) = Ef(unrel)−Ef(LDA) = 0 .196eV,\nshows a central importance of the atomic displacements\nfor the stability of the considered systems.\nThe optimization procedure based on the LSDA ap-\nproach accounts for additional corrections due to spin po-\nlarization and produces new ordered structural patterns\npresented in Fig. 2(b) and Fig. 2(c) for two diﬀerent (un-\nrelaxeda=b= 2.86˚A and relaxed a=b= 3˚A) lattice\nconstants. The volume-optimized structure (c) is signiﬁed\nby the 13%th increase of the unit cell volume due to the\ninsertionoftheinterstitialCr.TheLSDA-optimizedstruc-\ntural pattern is characterized by the chains of atomic Fe-\ngroups along the x(a)-direction, each group containing six\nFe-atoms. The nearest chains are separated by a distance\nabout 4˚A and are connected to each other by the Fe-Cr\nbonds of the length about 2 .4˚A for the structure (b) with\na= 2.86˚A, and 2 .7˚A for the structure (c) with the op-\ntimizeda= 3.0˚A. The local antiferromagnetic ordering\nin the vicinity of Cr is characterized by the magnetic mo-\nmentsµCr=−0.72µBandµ5= 2.4µBandµ6= 1.25µB\nof the neighbouring atoms Fe5 and Fe6, respectively. The\nmagnetic moments of more distant iron atoms have the\nvalues around 2.5 µB, which is close to results obtained\nfor substitutional alloys and in pure Fe α[8].\nAs compared to the tetragonal structure of the LDA-\noptimized system, the chain-like structure of the LSDA-\nrelaxed supercell is characterized by substantially lower\ncrystal symmetry and by the absence of the inversion\ncenter. In contrast to the LDA-based conﬁgurations, the\nformation energy of the LSDA-relaxed Fe 8Cr conﬁgura-\ntionEf(LSDA) = Etot(Fe8Cr)−8Etot(Fe)−Etot(Cr) =\n−1.15eV is negative which implies its high stability. We\ncan also calculate the energy gain due to the structural\nrelaxation by the LSDA approach\n∆E(LSDA) = Ef(unrel)−Ef(LSDA) = 6 .17eV,\nwhich also demonstrates the high stability of the relaxed\nspin-polarized structure and a necessity to account for a\nspin polarization in the structural optimization of the sys-\ntems with strong magnetoelastic eﬀect.\n2.1 Magnetic properties\nIn the considered systems, we have also analyzed modiﬁ-\ncation of the local magnetic properties due to the relax-\nation of the interatomic distances. To see how the atomicdisplacements inﬂuence the spin polarization of the sur-\nrounding atoms, in Fig. 3 we present the dependences of\nthe local moments of Cr and of two nearest neighboring\nFe on the Cr displacement along the bond [Fe5-Cr-Fe6]\n∆= [x(Cr)−x(Fe5)]−[x(Cr)−x(Fe5)]0,\nwhere [x(Cr)−x(Fe5)]0is the optimized [Fe5-Cr]-bond\nlength. The increase of ∆leads to the change of µCrfrom\n-0.7µBtothe valueabout-0.73 µB.In addition,the larger\n∆implies the elongation of the [Fe5-Cr] bond and lead to\nthe reduced µ5= 2.39µBdue to the tendency for a sup-\npression of antiferromagnetism in the vicinity of Fe5. The\nincrease of ∆also produces an enhancement of µ6from\n1.25µBto the values about 1 .28−1.3µB, an opposite\ntrend which occurs due to the shortening of the bond be-\ntween Cr and Fe6.\nIn Fig. 3, the ∆-dependences of the atomic magnetic\nmoments arehighly asymmetricwith respect to ∆. Conse-\nquently,theobtainedmagnetoelasticcouplingproducesan\nanisotropy of the magnetic moments and is accompanied\nby the loss of the inversion center due to the atomic dis-\nplacements, the eﬀect which can be observed in Fig. 2(b)\nand (c). In Fig. 2(c), the low-symmetry structure corre-\nsponds to the minimum of the total energy. As a con-\nclusion, the neglect of the magnetoelastic coupling in the\nelectronic structure calculations does not allow to achieve\na full optimization in this type of interstitial alloys.\n2.2 Electronic structure\nFig. 4 shows the 3 dspin-polarized electronic density con-\ntours of the LSDA-optimized structure in the ( x,z) plane.\nOne can see that the majority 3 dspin-up states of Fe are\nhighly occupied by the electrons whereas the electron con-\ncentration of Cr spin up states is substantially lower. In\ncontrast to this, the spin-down (minority) electrons are\ncharacterized by high electron occupation of Cr and lower\nelectron density on Fe. In Fig. 4, the chain-like structures\nFe-Cr in the z-direction are characterized by strong hy-\nbridization between the intra-chain 3 dspin-down orbitals\nof Fe and Cr. The last feature leads to the spatial charge\nredistribution and to higher charge densities on the bonds\nbetween spin-down Cr and Fe. In the LSDA-optimized\nsystem, the structural optimization produces areas with\nlow charge density in the y(b)-direction, where each area\ncan be identiﬁed between the chains of Fe-octahedra. As\ncan be seen in Fig. 4, these areas are almost free of the\nchargeandcanbeconsideredaschannelsforthemigration\nof light atoms like H, Li or N. Similarly to the contours\nin Fig. 4, the electron density of the majority Fe and Cr\norbitals and on the bonds between Cr and Fe calculated\nfor the LDA-optimized structure (Fig. 5) is substantially\nlower than the charge density on the spin-down contours,\nalthough the spatial charge distribution is more homoge-\nneous as compared to that in Fig. 4.\nFor the LDA-relaxed structure, the density of states is\ncharacterized by strong suppression of the majority spin-\nup DOS at the Fermi level (Fig. 6(a)), whereas the minor-\nity DOS at the Fermi level remains signiﬁcant. Similar,4 N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title Suppressed Due to Excessive Length\nFig. 2.RelaxedstructureofFewith 12.5%ofCr:(a) LDA\ncalculations, (b) spin-polarized LSDA calculations in the\nstructure with a=b= 2.86˚A and (c) spin-polarized\nLSDA calculations in the structure with a=b= 3.0˚A.\nThepath1andpath2identifypossiblepathesfordiﬀusion\nthrough the channels formed due to atomic relaxation.\nalthough much stronger, suppression of majority DOS is\ntypically observed in half-metallic systems where the elec-\ntric current is conducted by the electrons with the same\ndirection of spin [14]. In contrast to the half-metallic-\nlike features of the LDA-relaxed structure, the DOS of\ntheLSDA-optimizedsystem(Fig.6(b)) demonstratessub-\nstantial values at the Fermi level for both spin directions\nwhich implies an enhancement of the metallic state for the\nmajority electrons.\nIn transition metal oxides, the metallic state obtained\nin the LDA approach is usually strongly inﬂuenced by ad-\nditional account for the local Coulomb corrections for the\nFig. 3. Local magnetic moments (in µB) of the atoms\nin Fe5-Cr-Fe6 triad versus the displacement ∆=[Fe5-Cr]-\n[Fe5-Cr]0of Cr along the (100) axis. Here [Fe5-Cr]0is the\nequilibrium distance between Fe6 and Cr.\nd-electronicstates[15,16,17,18]. In ourwork,the Coulomb\ncorrections are incorporated within the SIC-variant of the\nLSDA+Uapproximation introduced in Ref. [15]. The re-\nsults are presented in Fig. 7 for two diﬀerent values of\nU= 2 eV and U= 4.5 eV estimated and employed in\nRef. [19,20,21] to account for the electron repulsion of 3d\nelectrons of Fe and Cr. Fig. 7 shows the ﬁnite density\nof states at the Fermi ( E= 0) level, although larger U\nleads to a signiﬁcant suppression of the majority DOS at\nEFwhich suggests a prevailing tendency towards a half-\nmetallic behavior.\nIn the LSDA-optimized structure, we ﬁnd that the\ncell magnetic moment MLSDA= 3.84µBis larger then\nthe magnetic moment MLDA= 2.88µBin the LDA-\noptimized cell. Such an enhancement of the magnetic po-\nlarization is connected with the substantial distortions\n∆Riin the range 0 .2−0.84˚A and can be considered as a\ndirectevidenceofsigniﬁcantmagnetoelasticeﬀect. Thelo-\ncal Coulomb corrections in the LSDA+ U-calculations re-\nsult in enhanced spin polarization. Speciﬁcally, we obtain\nMLSDA= 4.07µBforU= 2 eV, and MLSDA= 4.66µB\nforU= 4.5 eV. It is remarkable that in the substitutional\nalloy Fe-Cr with 12.5% of Cr, the LSDA approach gives\nthe value 3.8 µBfor the cell magnetic moment which is\nslightly lower than the magnetic moment for the consid-\nered LSDA-relaxed substitutional alloy.\nThe obtained high spin polarization of the considered\ninterstitial alloys Fe-Cr allows us to suggest these materi-\nals as possible candidates for spin polarizers in the spin-\ntronic devices. Possible technological applications of ar-\ntiﬁcially generated interstitial alloys would be related to\nthe thin ﬁlms produced for the needs of modern electronic\nindustry.Insuchartiﬁcialsystems,acentralquestionisre-\nlatedtothemethodsofimplantationandpositioningofCr\nin centersof the edges ofcubic bcc lattice ofbulk Fe. With\nthe current state of the art, such structural nanoscale ma-\nnipulation can be conﬁned to the ﬁrst subsurface layers of\nferrite ﬁlms by the using for instance the methods of opti-N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title S uppressed Due to Excessive Length 5\nFig. 4.Contours of electron density maps in the ( x,z)-\nplane(y/b= 0.25,xandzgivenin˚A)obtainedbyintegra-\ntion of electronic states in the energy window Ebetween\n−3 eV below the Fermi level and the Fermi level. The\nresults obtained by the structural optimization using the\nLSDA approximation.\ncal trapping by lasers [22] or by atomic force microscopy\n[11]. To estimate the stability of ultra-thin ﬁlms of iron-\nchromium alloys which can be also considered as a basis\nfor stainless steels, we have also extended our calculations\nto the nanoscale two-monolayers-thick iron ﬁlms contain-\ning one interstitial Cr atom per each 20-25 subsurface Fe\natoms [12]. For such type of ﬁlms, we have performed the\ncalculations of the surface formation energy and of the\nelectronicworkfunctions,whichwerealsocomparedtothe\ncorrespondingquantities in the ﬁlms of standard substitu-\ntional Fe-Cr alloys. In the interstitial Fe-Cr ﬁlms, we ﬁnd\nthat the energy of the surface formation is about 1.69eV\nand the electronic work function amounts to 1eV, whereas\nfor the substitutional Fe-Cr ﬁlms we obtain 2.56eVfor the\nsurface energy and 0.57eV for the work function. These\nresults allow us to expect high stability and durability of\nﬁlms generated on the basis of nanoscale-manipulated in-\nterstitial Fe-Cr alloys, as compared to the iron ﬁlms with\nsubstitutional Cr impurities.\nFig. 5.Contours of electron density maps in the ( x,z)-\nplane (y/b= 0.25,xandzgiven in ˚A) calculated by in-\ntegration of electron states in the energy window Ebe-\ntween−3 eV below the Fermi level and the Fermi level.\nThe LSDA results obtained in the initially LDA-relaxed\nstructure.\n3 Migration paths of N in interstitial alloys\nFeα-Cr\nA central question related to the stability of the consid-\nered interstitial alloys is how various atomic impurities\ncan modify the electronic properties. In the structurally\nrelaxed alloys, the chains of atomic Fe-groups are sepa-\nrated by 4 ˚A-wide atomic empty channels, which are ex-\npected to contain pathways for light impurity atoms like\nH, N or Li. To explore a possibility of the migration of the\nimpurities, we consider possible migration paths of a sin-\ngle nitrogen atom in the vicinity of the atomic Fe-chains\nin the interstitial Fe α-Cr.\nIn the studies of the migration paths of N impurities,\nwe employed the nudged-elastic-band (NEB) method im-\nplemented in the Quantum-Espresso(QE) Packageforthe\nDFT calculations with the use of plane-wave basis sets\nand pseudopotentials [23,24]. In these calculations, for the\natomic cores of Fe and Cr we employ the Perdew-Burke-\nErnzerhof (PBE) norm-conserving pseudopotentials [25].6 N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title Suppressed Due to Excessive Length\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s49/s56/s45/s49/s50/s45/s54/s48/s54/s49/s50/s49/s56\n/s40/s97/s41/s115/s112/s105/s110/s32/s100/s111/s119/s110/s115/s112/s105/s110/s32/s117/s112\n/s32/s32/s84/s111/s116/s97/s108/s32/s68/s79/s83\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s68/s79/s83/s32/s117/s112\n/s32/s68/s79/s83/s32/s100/s110\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48\n/s40/s98/s41/s115/s112/s105/s110/s32/s100/s111/s119/s110/s115/s112/s105/s110/s32/s117/s112\n/s32/s32/s84/s111/s116/s97/s108/s32/s68/s79/s83\n/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s68/s79/s83/s32/s117/s112\n/s32/s68/s79/s83/s32/s100/s110\nFig. 6. Total density of states (in eV−1) for structures\noptimized using (a) LDA approach and (b) spin-polarized\nLSDAapproximation.TheFermilevelcorrespondsto E=\n0.\nFor each stage of the nitrogen transport, the NEB method\ninvolvesarelaxationoftheatomicpositionsandofthedis-\ntances between the diﬀerent atoms in the supercell until\ntheforcesactingontheatomsreachtheirminima.Inthese\ncalculations, we use the plane-wave cutoﬀ 680 eV and the\nenergy cutoﬀ for charge and potential given by 1360 eV.\nIn the NEB-approach, the relaxation of the atomic po-\nsitions along the nitrogen migration path is performed\nby the minimization of the total energy of each interme-\ndiate conﬁguration (image). These images correspond to\ndiﬀerent positions of N on the migration path and they\nare produced by the optimization of a specially generated\nobject functional (action) with the consequent minimiza-\ntion of the spring forces perpendicular to the path. In\nour calculations, the convergence criteria for the norm of\nthe force orthogonal to the path is achieved at the val-\nues below 0.05eV/ ˚A. As the initial atomic conﬁguration,\nthe supercell Fe-Cr relaxed by the full-potential LSDA-\napproach [13] has been considered.\nRecent studies of the migration paths of single hydro-\ngen atoms by the pseudopotential NEB method demon-\nstrate a good agreement of the obtained transport mecha-\nnisms and energy barrierswith the experimental measure-\nments [26]. Similarly to Ref. [26], in the present studies,\neach NEB-generated conﬁguration has been modiﬁed by\nthe introduction of the N atom and the obtained in this-5 0 5\nEnergy (eV)-30-20-100102030Total DOSup, U=2eV\ndn, U=2eV\nup, U=4.5eV\ndn, U=4.5eV\nFig. 7.Total densities of states (in eV−1) for the LSDA-\noptimized structure calculated by the LSDA+ Umethod\nwith the local Coulomb corrections for the 3d-orbitals of\nFe andCr U= 2eV(black curves)and 4eV(blue curves).\nThe Fermi level corresponds to E= 0.\nway extended supercell has been fully structurally opti-\nmized.\nTo study the migration of N, we consider two diﬀerent\nmigration paths across the atomic Fe-chains in the Fe-Cr\nsupercell.Theﬁrstpath(path(a))describesthemigration\nof N from the initial position inside the cell ( z/c= 0.5)\nnear the chain (1) across the channel to the chain (2)\nschematically presented in Fig. 8(a). In distinction to the\npath (a), the second path (b) reﬂects path2 in Fig. 2(c)\nand contains additional migration step of N from the su-\npercell boundary ( z≈0) along the c-direction inside the\nsupercell, with the further relocation through the channel\nto the atomic Fe-chain (2) indicated in Fig. 8(b).\nFig. 9 shows the proﬁles of the total energy calculated\nalong the N migration paths (a) and (b). The path (a) is\ncharacterized by the high energy barriers about 0 .8 eV\nin the path coordinate range (0 ≤r/RN≤0.4) and\n(0.8≤r/RN≤1) which corresponds to the migration of\nN within the two Fe-chains (1) and (2). Here RNdenotes\nthe maximal length of the N path in the supercell which\nreaches about 1 nm for the path (a). The interchain mo-\ntion inside the channel is signiﬁed by a low energy barrier\nabout 0.2 eV (0.4≤r/RN≤0.7 in Fig. 9(a)).\nIn contrast to this, the energy proﬁle for the migra-\ntion path (b) (Fig. 9(b)) contains a plateau-like region at\n0.1≤r/RN≤0.4. This miration step indicates the in-\ntracell replacement of N near the Fe-chain (1) along the\n[001]-directiondemonstrated in Fig. 4, with a further relo-\ncationbetween the atomicchains acrossthe atomic-empty\nchannel with a low energy barrier about 0 .2 eV.\nAs a conclusion, we can note that the possible migra-\ntion paths of the light atoms in the considered intersti-\ntial Fe-Cr alloys contain a combination of the motion (i)\nwithin the atomic-empty channels and (ii) along the c-\ndirection along to the Fe-contained atomic chains.N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title S uppressed Due to Excessive Length 7\nFig. 8.Two diﬀerent migration paths of nitrogen through\nthe channel of the optimized crystal cell of Fe-Cr. The top\npicture (a) represents the interchain migration of N inside\nthe cell with the coordinate z/cnear 0.5. The bottom pic-\nture (b) corresponds to the migration of N from the cell\nboundary ( z= 0) along the zdirection with the further\nmigration between two neighbouring Fe-chains. The sym-\nbols (1) and (2) denote the diﬀerent atomic chains; (i) and\n(f) correspond to the initial and ﬁnal positions of nitrogen\nin the migration paths.\nThe question which arise due to the inclusion of N into\nthe magnetic Fe-Cr alloy is how the N impurities mod-\nify the magnetic properties of the system. In the work of\nI.Mazin[27],acomparisonofthedegreesofspin-polarization\n(DSP)calculatedforFeinthestaticlimitthroughtheden-\nsity of electronic states, via the current densities and in\nthe ballistic limit is presented. It is shown that all three\ndeﬁnitions of the DSP give very similar behavior for Fe\ndue to strong hybridization of the spanddstates at the\nFermilevel.Thusweexpect thatinthe consideredFealloy\nwith relatively low concentration of Cr it is suﬃcient to\nstudy the static spin polarization in order to capture the\nmain properties of the alloy. Fig. 10 presents the change\nof the cell and atomic magnetic moments at the migra-\ntion of N along the path (a) and path (b). Although the\nnitrogen is initially nonmagnetic in the bulk, it becomes\nweakly magnetic inside the Cr-Fe alloy with a small mag-\nnetic moment −0.04µBinduced by the magnetism of the\nsurrounding. It is noteworthy that the cell magnetic mo-\nment is increased to 4 µBas N approaches Cr and the\ndistance [N-Cr] becomes about 1.95 ˚A. Such an enhance-\nment ofMtotis explained by the strong atomic distortions0 0.2 0.4 0.6 0.8 1-0.4-0.200.20.4Total energy E-E0 (eV)\n0 0.2 0.4 0.6 0.8 1\nN migration path coordinate (r/RN)-1.5-1-0.50Total energy E-E0 (eV)(a)\n(b)\nFig. 9.Total energy proﬁles of the system along the mi-\ngration paths of N. Here E0denotes the energy of the\nsystem in the initial position of N and RNis the N coor-\ndinate in the ﬁnal position of the path.\n-2-101234\nMtot\nM(Fe5)\nM(Fe6)\nM(Cr)\n0 2 4 68 10 12\nN migration path coordinate r (Å)-3-2-101234Magnetic moments ( µB)\n(a)\n(b)\nFig. 10. Cell (Mtot) and atomic magnetic moments (in\nµB) along the migration paths of N. The red arrows iden-\ntify the maximal cell magnetic moments approachedupon\nthe minimization of the distance [N-Cr] along the N mi-\ngration paths.\nin the range between 0 .04˚A(Fe7) up to 0 .2˚A(Fe3) caused\nby the replacement of N and by the consequentmagnetoe-\nlastic eﬀect. In Fig. 10,the increase of the distance from N\nto Cr suppresses the magnetic moment of N and decreases\nthe cell magnetic polarization to the typical values about\n3.5−3.8µBobtained in LSDA-calculations for the ar-\ntiﬁcial Fe-Cr alloys. The obtained drastic change of the\nmagnetic polarization clearly demonstrates a crucial im-\nportance of the location of nonmagnetic impurities like N\nfor the electronic properties of alloy. As follows from our\nﬁndings, a control of the location of N, for example by ex-\nternal electric ﬁeld, can lead to externally tuned changes\nof the magnetic polarization, a feature which is of central\nimportance for possible spintronic devices based on the\nartiﬁcial Fe-Cr alloys.8 N. Pavlenko, N. Shcherbovskikh, and Z.A. Duriagina: Title Suppressed Due to Excessive Length\n4 Conclusion\nWe have performed the DFT studies of the bulk ferrite\nwith 12.5%-concentration of monoatomic interstitial Cr\nperiodically located at the edges of the bcc Fe αcell. We\nhave shown that the full atomic relaxation of the obtained\ninterstitial Fe-Cr stabilizes a new chain-like low-symmetry\nstructure. In this structure, the monoatomic Cr at the\nedges of ferrite bcc cells leads to the local atomic dis-\ntortions and results in the formation of parallel chains\nof Fe6-ochahedra, which are connected by the interchain\nFe-Cr bonds. The signiﬁcant energy gain caused by such a\nstructuralrelaxationapproaches6.17eVwhich makesthis\ntype of interstitial alloy highly stable and energetically fa-\nvorable with the negative formation energy approaching\n−1.15 eV. The novel electronic state of the system can be\ncharacterized as metallic, where the metallic properties is\nthe result of strong Fe-Cr hybridization of the structurally\nrelaxed alloy. In the investigations of the magnetic state\nof the generated relaxed structures, we have obtained a\nlocal antiferromagnetic order in the close proximity of Cr\natoms, whereasthe more distant Fe atomsarecoupled fer-\nromagnetically. We also ﬁnd that the nonmagnetic impu-\nrities like nitrogen can substantially modify the magnetic\nproperties of the interstitial alloy which can be considered\nas an additional manifestation of the strong magnetoelas-\ntic eﬀect in this type of alloys. We propose to consider\nthe generated interstitial alloys as perspective candidates\nfor fabrication of novel highly durable stainless steels and\nfor possible applications in spintronic and multifunctional\ndevices.\n5 Acknowledgements\nThisworkhasbeedpartiallysupportedthroughtheproject\n”Models of quantum statistical description of catalytic\nprocesses on metallic substrates” of the Ministry of Edu-\ncationandSciencesofUkraineandthegrant0108U002091\nof the National Academy of Science of Ukraine. A grant\nof computer time from the Ukrainian Academic Greed is\nacknowledged.\nReferences\n1. D. Peckner and I.M. Bernstein. Handbook on Stainless\nSteels, McGraw-Hill Book Co., New York, 1977.\n2. A. Mai, V.A.C. Haanappel, S. Uhlenbruck, F. Tietz, and\nD. St¨ over, Solid State Ionics, 176, 1341 (2005).\n3. H. Yokokawa, H. Tu, B. Iwanschitz, and A. Mai, J. Power\nSources, 182, 400 (2008).\n4. R.H. Victora and L.M. Falicov, Phys.Rev.B 31, 7335\n(1985).\n5. A.T. Paxton and M.W.Finnis, Phys.Rev.B 77, 024428\n(2008).\n6. C. Paduani and J.C. Krause, Braz. Journ. of Physics, 36,\n1262 (2006).\n7. A. Davies, J.A. Stroscio, D.T. Pierce, and R.J. Celotta,\nPhys.Rev.Lett, 764175 (1996).8. T.P.C. Klaver, P. Olsson, and M.W. Finnis, Phys.Rev.B\n76, 214110 (2007).\n9. P. Olsson, C. Domain, and J. Wallenius, Phys.Rev.B 75,\n014110 (2007).\n10. Z.A.Duriagina and M.I.Pashechko, Metal Science and\nTreatment of Metals, 4, 34 (2000).\n11. Y. Sugimoto et al., Science 322, 413 (2008).\n12. N. Pavlenko, unpublished.\n13. P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, and\nJ. Luitz, WIEN2K ,An Augmented Plane Wave + Lo-\ncal Orbitals Program for Calculating Crystal Properties ,\nISBN 3-9501031-1-2 (TU Wien, Austria, 2001).\n14. J.-H. Park et al., Nature 392, 794 (1998).\n15. V.I. Anisimov, I.V. Solovyov, M.A. Korotin,\nM.T. Czyzyk, and G.A. Sawatzky, Phys.Rev. B 48,\n16929 (1993).\n16. M.T. Czyzyk and G.A. Sawatzky, Phys.Rev. B 49, 14211\n(1994).\n17. N. Pavlenko, Phys. Rev. B 80075105 (2009).\n18. N. Pavlenko, I. Elﬁmov, T. Kopp, and G.A. Sawatzky,\nPhys. Rev. B 75, 140512(R) (2007).\n19. T. Bandyopadhyay and D.D. Sarma, Phys.Rev. B 39,\n3517 (1989).\n20. Ze Zhang and S. Satpathy,Phys.Rev. B 44, 13319 (1991).\n21. M.A. Korotin, V.I. Anisimov, D.I. Khomskii, and\nG.A. Sawatzky, Phys.Rev.Lett. 80, 4305 (1998).\n22. A. Ashkin. Optical trapping and manipulation of neutral\nparticles using lasers , World Scientiﬁc Pub., Singapore,\n2006.\n23. P. Giannozzi et al., J. Phys. Cond. Matter 21,\n395502(2009).\n24. H. Jonsson, G. Mills, and K.W. Jacobsen, Nudged Elas-\ntic Band Method for Finding Minimum Energy Paths\nof Transitions in Classical and Quantum Dynamics in\nCondensed Phase Simulations inClassical and Quan-\ntum Dynamics in Condensed Phase Simulations , ed. by\nB.J. Berne, G. Ciccoti, andD.F. Coker (Singapore: World\nScientiﬁc, 1998).\n25. J.P.Perdew,S.Burke,andM.Ernzerhof,Phys.Rev.Lett.\n77, 3865(1996).\n26. N. Pavlenko, A. Pietraszko, A. Pawlowski, M. Polomska,\nI.V.Stasyuk, and B. Hilczer, Phys. Rev. B 84, 064303\n(2011).\n27. I. Mazin, Phys. Rev. Lett. 83, 1427(1999)."    },    {        "title": "1103.5541v2.Probing_Magnetoelastic_Coupling_and_Structural_Changes_in_Magnetoelectric_Gallium_Ferrite.pdf",        "content": "Page 1 of 16  \n Probing Magnetoelastic Coupling and Structural Chan ges in Magnetoelectric \nGallium Ferrite  \n \nSomdutta Mukherjee 1, Ashish Garg 2, Rajeev Gupta 1, 3* \n1 Department of Physics  \n2 Department of Materials Science and Engineering   \n3Materials Science Programme  \nIndian Institute of Technology Kanpur, Kanpur 20810 6, India. \n \nAbstract \nTemperature dependent X-ray diffraction and Raman s pectroscopic studies were carried \nout on the flux grown single crystals of gallium fe rrite with Ga:Fe ratio of 0.9:1.1. Site \noccupancy calculations from the Rietveld refinement  of the X-ray data led to the \nestimated magnetic moment of ~0.60 µ B /f.u. which was in good agreement with the \nexperimental data. Combination of these two measure ments indicates that there is no \nstructural phase transition in the material between  18 K to 700 K. A detailed line shape \nanalysis of the Raman mode at ~375 cm -1 revealed a discontinuity in the peak position \ndata indicating the presence of spin-phonon couplin g in gallium ferrite. A correlation of \nthe peak frequency with the magnetization data led to two distinct regions across a \ntemperature ~180 K with appreciable change in the sp in-phonon coupling strength from ~ \n0.9 cm -1 (T < 180 K) to 0.12 cm -1 (180 K < T < Tc). This abrupt change in the coupling \nstrength at ~180 K strongly suggests an altered spi n dynamics across this temperature.  \n \nKeywords:  Gallium ferrite, Raman spectroscopy, Spin-phonon c oupling. \n \nPACs Numbers: 75.47.Lx, 61.05.C-, 78.30.-j, 63.20.k k \n                                                 \n* Corresponding authors, Email: guptaraj@iitk.ac.in Page 2 of 16  \n 1. Introduction \nGallium ferrite (GaFeO 3 or GFO) is a room temperature piezoelectric [1, 2] and a \nferri/paramagnet [3]. It possesses a non-centrosymm etric orthorhombic structure with \nspace group Pc2 1n [4].  Its magnetic transition temperature ( Tc) is sensitive to the \nstoichiometry of the material i.e.  Ga:Fe ratio [5]. Initial interest in GFO arose fro m a \nreport by Rado [6] showing a large magnetoelectric coupling in stoichiometric GFO ( αbc  \n≈ 1 × 10 -11  S/m at 77 K). Moreover, in recent years, multiferr oics and study of \nmagnetoelectric phenomenon has gained prominence af ter illustration of this effect in \nperovskite oxides such as BiFeO 3 [7], TbMnO 3 [8]. In GFO, recently Ogawa et al.  [9] \ndemonstrated large magnetoelectric coupling as show n by magnetization induced second \nharmonic generation and large angle Kerr rotation o f the second harmonic light in the \nferrimagnetic state. In another study, Jung et al.  [10] reported optical absorption in the \npresence of a magnetic field of 500 Oe suggesting a  large optical magnetoelectric effect \nbelow Tc. Moreover, in materials exhibiting large magnetoele ctric coupling [11, 12], one \nmay expect structural and magnetic degrees of freed om behaving in tandem. However, \nwhile GFO’s magnetoelectric characteristics are rea sonably well demonstrated, its \nmagnetoelastic behavior demands studies on high qua lity samples to elucidate structure-\nproperty correlations in GFO.  \nA recent structural study [10] on single crystal GF O using neutron and \nsynchrotron scattering has reported absence of any structural change across the magnetic \ntransition. In addition, significant distortion of the FeO 6 octahedra in the unit-cell resulted \nin the shift of Fe ions from the center of FeO 6 octahedra, leading to a spontaneous \npolarization along the crystallographic b-direction [5].  In the context of structure relate d \nstudies of GFO, we note that many such studies have  been conducted using tools which \nyield information at a rather macroscopic scale. It  would, therefore, be interesting to \nexamine the structure of GFO at nm-length scales us ing techniques such as Raman \nspectroscopy. Further, to the best our knowledge, t here is no report on structural study of \nGFO beyond room temperature. Such a study, both at macroscopic and microscopic \nlength scales, could shed light on the possible tem perature driven structural phase \ntransformation imparting higher symmetry to the sys tem.  \nRaman spectroscopy has been used extensively to pro be intricate structural details \nof various multiferroic materials such as structura l distortion [13], spin dynamics [14] \nand any kind of coupling between structure and magn etic degrees of freedom [15, 16]. \nSpin-phonon coupling has been demonstrated using Ra man spectroscopy in a variety of \nmaterials: in rare earth manganites [17] showing la rge phonon softening across magnetic \nTc, in TbMnO 3 [15] and in BiFeO 3 [18]. In rare-earth manganites, it was also shown that \nthe manganites with smaller rare earth ion show neg ligible spin lattice coupling [17].  All \nthese reports conclusively prove the utility of Ram an spectroscopy to investigate \nmultiferroic materials. In this manuscript, we repo rt a detailed x-ray diffraction and \nRaman study on flux-grown GFO single crystals to in vestigate the structural evolution \nand role of low energy excitations across different  transitions as a function of temperature \nin the temperature range 18 K to 700 K. \n \n2. Experimental Details \nSingle crystals of GFO were synthesized by flux gro wth process using β-Ga 2O3, α-Fe 2O3 \nas constituents and Bi 2O3 and B 2O3 as flux in weight ratio 1: 1: 5.4:0.6 using the Page 3 of 16  \n methodology similar to that reported in Ref. [1]. D ark brown crystals with dimensions ~ \n2 × 4 × 1 mm were extracted. Compositional analysis  of the samples was carried out \nusing energy dispersive spectroscopy (EDS) using Ox ford EDS Spectrometer in a Zeiss \nscanning electron microscope. Average EDS data indi cates that crystals are chemically \nhomogeneous with the Ga to Fe ratio of ~ 0.93: 1.11 vis-à-vis initial mixing \nstoichiometry of 0.93: 1.09 (Ga:Fe). Therefore, we specify our samples as Ga 2-xFe xO3 \nwith 1.09 ≤ x ≤ 1.11. Temperature dependent powder X-Ray Diffracti on (XRD) of the \ncrushed single crystals was carried out using a hig h resolution Philips X’Pert PRO MRD \ndiffractometer with an angular resolution of 0.0001 °.  Micro-Raman study was carried out \nin back scattering geometry using a high resolution  (~ 0.6 cm -1/pixel) system equipped \nwith Ar + 514.5 nm laser as an excitation source and a liqui d nitrogen cooled CCD \ndetector. The sample temperature was varied between  18 K - 450K using a closed cycle \ncryostat for low temperature region while a hot-col d Linkam THMS600 stage was used \nfor high temperature region. Magnetization data as a function of temperature was \nacquired using a Quantum Design SQUID magnetometer.   \n \n3. Results and Discussions \n3.1   X-ray Diffraction Analysis \nXRD data of the crushed single crystals were acquir ed from 300 to 700 K at 50 K interval \nand within the 2 θ region of 26°-120°. Fig. 1 (a) shows representativ e XRD plots of the \ndata taken at room temperature and 700 K. The spect ra were indexed to GFO ICDD data \ncard no. 76-1005 and the peak match suggests single  phase formation with no signature \nof intermediate or secondary phase(s). Same is true  for all other temperatures whose plots \nare not shown here. Further, to understand any subt le changes in the structure as well as \nto quantify the occupancy of lattice sites, we carr ied out Rietveld refinement of XRD data \nusing FULLPROF 2000 package [19]. We used orthorhom bic Pc2 1n (Pna2 1: notation \nused in international table of crystallography) sym metry of GFO for refinement at all \ntemperatures along with the site occupancies determ ined at room temperature. To avoid \nambiguities in the structural refinement, we exclud ed the 2 θ region between 30.2°-32.45° \nconsisting of a major peak of the high temperature dome material (PEEK).  The \nrefinement of the XRD data, also shown in Fig. 1(a) , did not suggest any major change in \nthe peak profiles or relative intensities, ruling o ut any anomalous structural change or \ndistortion in GFO over the temperature range of 300 -700 K . The resolution of our \nmeasurements is better than 0.003Å for all the d-values, well within the instrumental \nresolution. Further we analyzed the refined XRD dat a to understand the effect of \ntemperature on the unit-cell parameters, bond lengt hs and polyhedral distortion. \n From the refinement, we obtained the atomic positi ons of the constituent ions and \nrefinement parameters (not shown here). Further, we  estimated the room temperature \nlattice parameters of GFO from the refined data: a = 8.744 Å, b = 9.395 Å, c = 5.079 Å \nand unit-cell volume, V cell , of 417.26 Å 3. These values are in excellent agreement with \nthose reported previously [4, 5].  Similarly, at 70 0 K the unit cell parameters and volume, \ncalculated for the refined structure are a = 8.769 Å, b = 9.422 Å, c = 5.097 Å and V cell  = \n421.12 Å 3, respectively. We investigated the variation in th e lattice parameters and unit \ncell volume at temperatures from 300-700K and a plo t of these parameters versus \ntemperature is shown in Fig. 1(b). As can be seen f rom the plot, the lattice parameters \nvary linearly with temperature. From the linear fit ting of above data, the coefficients of Page 4 of 16  \n thermal expansion ( α, K -1) of GFO along three principal axes ( a, b and c) were calculated \nto be 7.2 ×10 -6, 7.5 ×10 -6 and 8.8 ×10 -6, respectively. Negligible difference among the thr ee \nvalues suggests that GFO is a thermally isotropic m aterial. From the slope of unit cell \nvolume vs. temperature plot, the thermal coefficien t for volume expansion ( γ) was \ncalculated to be 2.37 ×10 -5 K-1 which follows the relation γ = 3 α, true for an isotropic \nmaterial. Since, low temperature XRD experiments co uld not be carried out; we \nestimated the lattice parameters at low temperature  by extrapolation. The estimated lattice \nparameters at 4 K i.e .  a ~ 8.725 Å, b ~9.374 Å and c ~ 5.066 Å,  are in excellent \nagreement with those reported by Arima  et al.  [5], ( a ~ 8.71932(13) Å, b ~9.36838(15) Å \nc ~ 5.06723(8) Å). The fact that extrapolated lattice  parameters agree with the reported \nlow temperature data, it can be concluded that ther e is no structural distortion or lattice \nparameter anomaly in GFO down to 4K.   \nFurther, the refined structural data were used to c alculate the bond lengths using \nan approach used by Momma and Izumi [20]. For clari ty, a simulated crystal structure of \nGFO has been shown in Fig. 1(c) where positions of Fe and Ga ions are marked: Ga1 is \ntetrahedrally coordinated while Ga2, Fe1 and Fe2 ar e octahedrally coordinated by oxygen \natoms. Fig. 1(d) shows four different cation-oxygen  polyhedra at room temperature. It \nwas found that Ga1–O tetrahedron shares its corners  with surrounding cation octahedra \nwhich share their edges among neighboring octahedra . The estimated average bond \nlengths for Ga1-O, Ga2-O, Fe1-O and Fe2-O ions at r oom temperature are 1.860 Å, \n2.023 Å, 2.033 Å and 2.039 Å, respectively. It was observed that upon increasing the \ntemperature from 300 K to 700 K, Ga1-O, Ga2-O and F e1-O bonds stretched by 0.3 % , \n0.23 % and 1.25 %, respectively while Fe2-O bond co ntracted by 0.21 %. This translates \nto 0.54% reduction in the volume of Fe2-O octahedro n, and an increase of 0.75%, 0.45% \nand 2.9 % in Ga1-O tetrahedron, Ga2-O and Fe1-O oct ahedron volume, respectively. \nFurther, the changes in the bond lengths result in polyhedral distortion which is \nquantified  by calculating the distortion index ( ∆) as defined by Baur [21];  \n .\n1 .( ) 1n\ni avg \ni avg l l \nn l =−∆ = ∑      (1) \nwhere li = distance from the central atom to the ith  coordinating atom, n is the number of \nbonds and lavg.  = average bond length. It is observed that, at roo m temperature, Ga1-O \ntetrahedron is least distorted ( ∆ = 0.0065) among all the polyhedra and remains clos e to \nits ideal shape. While, both Fe1-O and Fe2-O octahe dra are significantly distorted with ∆ \n= 0.0650 and 0.0760, respectively and Ga2-O octahed ron is comparatively less distorted \n(∆ = 0.0182).  Upon increasing the temperature to 700  K, the distortion indices of the \nabove polyhedra change to 0.0051, 0.0806, 0.0701 an d 0.0260, respectively. This shows \nthat with increasing temperature Ga1-O tetrahedron tends to move toward the shape of a \nregular tetrahedron ( ∆ = 0.0), the shape of Fe2-O octahedron remains almo st identical to \nits RT structure and Ga2-O and Fe1-O octahedra are further distorted. Such details on the \nstructural distortion of GFO can be crucial to unde rstand the temperature dependence of \nmicroscopic polarization in GFO. Previous first pri nciple studies have shown important \ncorrelation between the structural distortion and p olarization in GFO [22]. \nStructural refinement of the XRD data also gives an  estimate of the cation \noccupancies which can prove important in explaining  the observed magnetic behavior as \ncation site disordering can lead to significant cha nges in the magnetic properties [3, 5]. \nSite disordering in GFO is expected due to the fact  that, in addition to being isovalent, the Page 5 of 16  \n ionic radii of Fe and Ga are quite close to each ot her (0.645 and 0.62 Å, respectively). \nFrom the room temperature data, we refined the cati on occupancies keeping anion \noccupancies fixed at 1.0. The occupancies of iron a t Ga1, Ga2, Fe1 and Fe2 sites are \nfound to be 0.14, 0.32, 0.83 and 0.89, respectively .  From the partial occupancies, we \ncalculated Ga to Fe ratio in GFO as ~ 0.93:1.11 whic h is in very good agreement with the \nEDS data, as mentioned in section 2. In the followi ng paragraph, we discuss the \ncorrelation between cation site occupancies and the  observed magnetic behavior of the \nGFO samples.  \n  If GFO were to behave as a perfect antiferromagnet,  the Fe spins on two \noctahedral sites would be antiparallel resulting in  a net zero magnetic moment [5, 23]. In \ncontrast, experimental studies show a large magneti c moment in GFO below Tc [5, 24].  \nIt has been suggested that this observed magnetic m oment at low temperature can be \nattributed to the cation site disorder, i.e. , presence of octahedral Fe ions predominantly, \non the octahedral Ga sites [5]. Our temperature dep endent magnetization measurement \n(not shown here) showed a ferri- to para-magnetic t ransition at ~ 290 K and also yielded \na magnetic moment per Fe site of ~ 0.67 µB at 4 K. Previous neutron diffraction study  [5] \nshows Fe at Fe2 and Ga2 sites are ferromagnetically  coupled. However, neutron \ndiffraction data did not comment about the spin con figuration of Fe at Ga1 site. \nAssuming ferromagnetic coupling of Fe at Fe1 and Ga 1 sites and high spin moment of Fe \n[3] and, using the cation  partial occupancies from  our Rietveld data, we estimated net \nmagnetic moment of GFO to be ~ 0.60 µB/ f.u. which is in excellent agreement with our \nexperimental observation. On the contrary, if we as sume Fe at Ga1 and Fe1 are \naniferromagnetically coupled, the estimated moment is ~ 1.3 µB/ f.u. which is very large \nin comparison to our experimental data. Thus, we co nclude that Fe at Ga1 site is \nferromagnetically aligned with respect to Fe at Fe1  site and antiferromagnetically aligned \nto Fe at Fe2 (and also Fe at Ga2) site. We also cal culated net magnetic moment using \npartial occupancies and magnetic moments determined  by Arima et al  [5]. These results \nare tabulated in Table 1. Based on our assumption t hat Fe at Fe1 and Ga1 sites are \nferromagnetically coupled and using the site occupa ncy data in stoichiometric GFO as \nreported by Arima et al. [5], we estimated the net magnetic moment of stoich iometric \nGaFeO 3, as ~ 0.55 µB/ f.u. which is close to their experimental observa tion (~ 0.65 µB/ \nf.u.). Here we assumed that the moment of Fe at Ga1  site is same as the moment of Fe at \nFe1 site. Finally, we calculated the net magnetic m oment of GFO (0.86 µB/ f.u.) using \npartial occupancies of our Rietveld data and the ma gnetic moments from Arima et al. [5].  \nThe observed difference between the calculated and experimental moments might be \nattributed to the difference in the magnetic moment s of Fe at cation sites (with respect to \nArima et al. [5]) due to nonstoichiometry of our sample. From th is discussion, we \nconclude that the magnetism in GFO is highly sensit ive to the cation site occupancies. \nThus it is possible to tune the magnetic behavior o f GFO by careful compositional \ntailoring. \n \n3.2   Temperature dependent Raman Spectroscopy \n \nIn a recent theoretical work  Fennie et al.  [25], predicted that combination of strain, spin-\nphonon coupling and optical modes can play an impor tant role in simultaneously \nstabilizing both ferroelectric and ferromagnetic ph ases. The veracity of Fennie et al. ’s Page 6 of 16  \n work was demonstrated in a recent report by Lee et al. [11] in which tuning of bi-axial \nstrain in EuTiO 3 thin films led to a large spin-lattice coupling re sulting in simultaneous \nferroelectricity and ferromagnetism. Further, since  Raman spectroscopy can probe lattice \nexcitations i.e.  phonons, and magnetic excitations i.e.  magnons as well as their \ninteractions, it is an ideal tool to investigate th e spin-lattice coupling in materials. Spin \ncorrelation among the nearest neighbors i.e.  <Si.Sj> can be used to relate the behavior of \nthe phonons to determine the spin-lattice coupling strength [26]. \nIn GFO, while the neutron studies made by Arima et al.  did not make any \nmention of the first-order spin-lattice coupling [5 ], a recent Raman study on \npolycrystalline bulk stoichiometric GFO [27] specul ated the existence of spin-lattice \ncoupling based on the discontinuity in the line wid th of one of the phonons at 200 K. \nHowever, poor quality of Raman spectra, possibly du e to polycrystalline nature of the \nsamples, does not exude the confidence to draw thes e conclusions. In contrast, our \ntemperature dependent Raman measurements have been carried out on high quality single \ncrystals of GFO to investigate temperature dependen t behavior of Raman modes from the \nperspective of understanding the role of phonons ac ross the phase transitions in this \nmaterial. \nGroup theoretical methods predict that there are to tal of 120 normal modes (117 \nRaman and 97 IR active) of vibration in GFO conside ring 8 f.u. in a primitive cell and \nPna2 1 (C 2v 9) space group [27]. Since above space group does not  contain an inversion \ncenter, the IR active modes are simultaneously Rama n active and these modes are non-\ndegenerate [28].  In the present study, we acquired  Raman spectra of single crystals and \nobserve 31 Raman active modes in the spectral range  of 90-900 cm -1 at room temperature. \nIt is likely that while some modes are below our de tection limit on the lower wave \nnumber side, orientation of the crystal may also re strict the observation of some other \nmodes due to selection rules. \nAs can be seen from Fig. 2(a), 31 distinct Raman pe aks occur at 99, 106, 118, 121, \n129, 138, 149, 154, 173, 199, 211,219, 240, 257, 27 0, 303,350, 359, 370, 394, 438, 465, \n521, 575, 604, 655, 674 688, 723, 743 and 759 cm -1. In contrast, the report by Sharma et \nal.  [27] on polycrystalline GFO samples showed a rathe r broad spectra with fewer \nnumber of Raman modes. The representative spectra m easured on our samples at various \ntemperatures between 18-450 K are shown in Fig. 2(a ). A first glance suggests that the \nRaman modes harden with lowering of temperature acc ompanied by narrowing of the \npeak widths. Moreover, upon cooling across the magn etic transition ( Tc ~ 290 K), the \nnumber of Raman peaks remains same suggesting absen ce of a structural change near Tc, \nconsistent with temperature dependent Neutron diffr action studies reported earlier [5]. \nFor further analysis of the Raman data to examine a ny subtle changes in the structure, we \ndivided the entire spectrum into two ranges i.e.  275-550 cm -1 and 620-820 cm -1.  The \nspectra in these two ranges were deconvoluted into a sum of nine and six Lorentzians, \nlabeled as M1, M2, …., M9 and X1, X2,….,X6 respecti vely, as shown in Fig. 2 (b) and \n(c) along with the fitted curves. The extracted lin e shape parameters characterizing a \nLorentzian function i.e.  peak positions and line widths for modes M1-M9 are  plotted as a \nfunction of temperature in  Fig. 3. The figure depi cts that the peak positions shift towards \nlower frequencies and the line widths or FWHM incre ase with increasing temperature. \nSimilar behavior also holds true for X1-X6 modes. S uch a behavior is an expected Page 7 of 16  \n outcome of lattice expansion and increase in the ph onon population as a result of \nincreasing temperature. \nThis hardening of phonons with decreasing temperatu re can be described by the \nfollowing relation [29], \n( ) (0) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) j j j qh j anh j el ph j sp ph T T T T T ω ω ω ω ω ω − − = + ∆ + ∆ + ∆ + ∆          (2) \nwhere ωj(T) and ωj(0) are the phonon frequencies of j th mode at any temperature T and \nat 0 K, (∆ωj)qh , (∆ωj)anh, (∆ωj)el-ph  and (∆ωj)sp-ph  are the changes in phonon frequencies, \nrespectively due to change in the lattice parameter s of the unit cell (quasi-harmonic \neffect); intrinsic anharmonic interactions; electro n-phonon coupling and spin-phonon \ncoupling in magnetic materials caused by the modula tion of exchange integral by lattice \nvibrations [29].  \nThe quasi-harmonic effect on the vibrational mode d ue to change in the unit cell \nvolume triggered by thermal effects can be approxim ated by Grüneisen law relating the \nchange in the frequency to the change in the lattic e volume [29] i.e. \n( )j\nj\njqh VVωγω∆  ∆ = −      (γj: Grüneisen  parameter  for the normal mode j and ∆V/V : the \nfractional unit cell volume change due to thermal e xpansion). Since, XRD analysis shows \nminute change (<1%) in the unit cell volume of GFO over the temperature range of 18 K \nto 450 K, therefore, it is expected that the contri bution of quasiharmonic effect on the \nmode frequency is negligible. \nNext, we examine the contribution due to intrinsic anharmonic interactions i.e.  \n(∆ω j)anh . Considering only the contribution from cubic and quartic anharmonic processes \nand further assuming that each phonon decays into t wo lower energy phonons of equal \nenergy i.e.  a phonon with frequency ω decays into two (three) phonons of frequency ω/2 \n(ω/3) for the cubic (quartic) anharmonic process. The refore,  the temperature dependence \nof frequency of jth  mode can be represented by the following relation (eq. (3)) [30]  \n(0) / 2 (0) / 3 (0) / 3 22 3 3 ( ) (0) 1 1 \n1 1 ( 1) j B j B j B j j j j k T k T k T T A B \ne e e ω ω ω ω ω     = − + − + +     − − −     /planckover2pi /planckover2pi /planckover2pi  (3) \nwhere, (0) jω  is 0 K frequency of the mode in harmonic approxima tion while \njAand jBare anharmonicity coefficients, giving the strength  of cubic and quartic \nanharmonic processes, respectively.  \nSimilarly, the line width of a Raman mode has follo wing temperature \ndependence: \n(0) / 2 (0) / 3 (0) / 3 22 3 3 ( ) (0) 1 1  \n1 1 ( 1) j B j B j B j j j j k T k T k T T C D \ne e e ω ω ω     Γ = Γ + + + + +     − − −     /planckover2pi /planckover2pi /planckover2pi (4) \nwhere, (0) jΓ  is intrinsic broadening of jth  mode arising from factors other than phonon \ndecay such as presence of structural defects while jC and jDare parameters for cubic \nand quartic anharmonic processes, respectively. \nEquations (3) and (4) have been used to fit the pea k frequency and line width data \nfor modes M1-M9 (shown in Fig. 3) and modes X1-X6 ( not shown here). It was observed \nthat temperature dependence of line shape parameter s of modes M1, M2, M3, M4, M6 \n(see Fig. 3), fits reasonably well to eq. (3) and ( 4). However, the frequency shifts of Page 8 of 16  \n modes M5, M7, M8 and M9 cannot be adequately descri bed by eq. (3) using one set of \nfitting parameters. Hence, the parameters (0),    A and B ω for these modes were optimized \nby fitting the experimental data above Tc. Below Tc, the most pronounced deviation from \nthe fit is found for modes M5 and M9, with M5 softe ning and M9 hardening. Fig. 3 also \nshows the plot of variation of line width (right ax is) of the above nine modes (M1-M9) as \na function of temperature along with the fits using  eq. (4). Here, the modes M2, M3, M4, \nM6 and M8 in Fig. 3 particularly fit well to eq. (4 ) throughout the entire temperature \nrange. As far as modes M1, M5, M7, M9 are concerned , the line widths of these modes \nfollow the anharmonic interaction model reasonably well below Tc. However, above Tc, \nline widths of these modes become nearly temperatur e independent resulting in \nsignificant deviation from the fit to eq. (4).  The se changes in the line widths for some of \nthe modes below Tc indicate towards a spin-phonon interaction in GFO.  The apparent \ntemperature independent behavior of line widths of a few modes can be argued as a \nconsequence of the competition between decrease in the line widths due to absence of \nmagnon-phonon interaction above Tc and increase in the line widths with increased sit e \ndisordering with increasing temperature. Since, GFO  is an insulator within the studied \ntemperature range, electron-phonon interaction is u nlikely to be temperature dependent \nand hence its contribution to change in line width with temperature can be neglected [29].  \n Finally, to quantify the spin-phonon coupling stre ngth as given by the last term in \neq. (2), we utilize the formalism proposed by Grana do et al.  [29] suggesting a mechanism \nfor phonon renormalization due to spin-phonon inter actions. The spin-phonon coupling \nstrength for a given mode can be estimated by relat ing the change in the peak position \nfrom the conventional anharmonic behavior below Tc to the nearest neighbor spin-spin \ncorrelation function as given by i j S .S spin phonon ω η → → \n− ∆ ≈ . Here, { }2\ni j ( ) S .S 2 \nsM T \nM→ → \n= ; \nM(T) is the magnetization of GFO per Fe site below Tc, M s is the saturation \nmagnetization and η represents the spin-phonon coupling strength. The factor of two on \nthe right arises due to two nearest neighbors in th e ferromagnetic ac -plane for each type \nof Fe ion in the unit-cell. Since, mode M5 at 374 c m -1 exhibits the largest deviation from \nthe conventional anharmonic behavior below Tc, this should correspond to the possibly \nlargest value of η. As per above relation, a plot between ( ) Tω vs. { }2( ) \nsM T \nM, as shown \nin Fig. 4, is used to estimate the value of η (½ of the slope, m) for GFO. The figure shows \ntwo distinct regions across a temperature ~180 K, d efined as Tf, as depicted by the sharp \nchange of slope: a region on the low temperature si de below Tf with relatively large value \nof η ~ 0.9 cm -1 and another region above Tf but below Tc with a much smaller value of η \n~ 0.12 cm -1. Such an abrupt change in η across  Tf is a strong indication of change in the \nspin dynamics in GFO. Moreover, these values of η obtained for GFO are comparable to \nearlier estimates reported on other systems using R aman scattering. For instance, on \nantiferromagnetic rutile structured MnF 2 and FeF 2, Lockwood et al.  [26] showed the spin \nphonon coupling strength for different modes in the  range from 1.3 cm -1 to 0.4 cm -1 while \nGupta et al.  [31] found a very large spin phonon coupling stren gth (~ 5.2 cm -1) for \nSr 4Ru 3O10 . Therefore, our measurements suggest that the spin -phonon coupling of GFO \nis rather weak immediately below Tc until Tf before increasing substantially below Tf. It \nwould be interesting to further examine this phenom enon across Tf by carrying out ac  Page 9 of 16  \n magnetic measurements as a function of temperature and frequency to understand the \nspin dynamics and to probe deeper into the nature o f this transition.  \n \nConclusions \nIn conclusion, temperature dependent X-ray and Rama n studies of GFO single crystals \nruled out any structural transition between 18 K an d 700 K. Rietveld refinement of the \nXRD data showed a thermally isotropic nature of the  material. Calculated magnetic \nmoment based on the cation occupancies, determined from Rietveld refinement, matched \nvery well with the experimentally measured values. The variation of the peak position for \nthe Raman mode at ~ 374 cm -1 with temperature suggested spin-phonon interaction s in \nthe material with a coupling strength of ~ 0.9 cm -1 below ~180 K. The abrupt change in \nthe slope of phonon frequency versus the square of normalized magnetization at ~180 K \nindicates the change in the nature of spin-lattice interactions across this temperature.  \n Page 10  of 16  \n Acknowledgements \nAuthors acknowledge the financial support from Coun cil of Scientific and Industrial Research \n(CSIR) and Department of Science and Technology (DS T), Govt. of India. \n \nReferences:  \n1. Remeika J P 1960 J. Appl. Phys.  31  S263. \n2. White D L 1960 Bull. Am. Phys. Soc.  5 189. \n3. Frankel R B, N A Blum, Foner S, Freeman A J and Schieber M 1965 Phys. Rev. Lett.  15  \n958. \n4. Abrahams S C, Reddy J M and Bernstein J L 1965 J. Chem. Phys.  42  3957. \n5. Arima, T et al  Phys. Rev. B  2004 70  064426. \n6. Rado G T 1964 Phys. Rev. Lett.  13  335. \n7. Wojdeł J C and Íñiguez J 2009 Phys. Rev. Lett.  103 267205. \n8. Kimura T, Goto T, Shintani H, Ishizaka K, Arima T and Tokura Y  2003 Nature 426  55. \n9. Ogawa Y, Kaneko Y, He J P, Yu X Z, Arima T and T okura Y 2004 Phys. Rev. Lett.  92  \n047401. \n10. Jung J H, Matsubara M, Arima T, He J P, Kaneko Y and Tokura Y 2004 Phys. Rev. Lett.  \n93  037403. \n11. Lee, J H et al  2010 Nature  466  954. \n12. Oh Y S et al  2011 Phys. Rev. B  83 060405. \n13. Singh M K, Ryu S and Jang H M 2005 Phys. Rev. B  72 132101. \n14. Suzuki N and Kamimura H 1973 J. Phys. Soc. Japan  35  985. \n15. Rovillain P, Cazayous M, Gallais Y, Sacuto A an d Measson M-A 2010 Phys. Rev. B  81 \n054428. \n16. Ferreira W S et al  2009 Phys. Rev. B  79  054303. \n17. Laverdière et al  2006 Phys. Rev. B  73  214301. \n18. Singh M K, Katiyar R S, and Scott J F 2008 Xiv: 0712.4040v2 [cond-mat.mtrl-sci]. \n19. Rodríguez-Carvajal J 1993 Physica B: Condensed Matter  192  55. \n20. Momma K and Izumi F 2008 J. Appl. Crystallogr.  41 653. \n21. Baur W H 1974 Acta Crystallogr.  B 30  1195. \n22. Roy A, Mukherjee S, Gupta R, Auluck S, Prasad R  and Garg A 2011 J. Phys.: Condens. \nMatter  23 325902. \n23. Han M J, Ozaki T and Yu J, 2007 Phys. Rev. B  75 060404. \n24. Nowlin C H and Jones R V 1963 J. Appl. Phys.  34 1262. \n25. Fennie C J and Rabe K M 2006 Phys. Rev. Lett.  97  267602. \n26. Lockwood D J and Cottam M G 1988 J. Appl. Phys.  64  5876. \n27. Sharma K, Reddy R, Kothari D, Gupta A, Banerjee  A and Sathe V G 2010 J. Phys.: \nCondens. Matter  22  146005. \n28. Xia H R et al  2004  J. Raman Spectroscopy 35 148. \n29. Granado E et al   1999 Phys. Rev. B  60  11879. \n30. Balkanski M, Wallis R F and Haro E 1983 Phys. Rev. B 28  1928. \n31. Gupta R, Kim M, Barath H, Cooper S L and Cao G 2006 Phys. Rev. Lett.  96  067004. \n Page 11  of 16  \n List of Tables \n \nTable 1 – Estimation of magnetic moments ( µ) in GFO using site occupancies (occ.) calculated \nfrom the XRD data and comparison with those calcula ted using Arima et al. ’s [5] data. (u.c.: \nunit-cell and f.u.: formula unit)  \n \nSite \n(Occ.*) \n Fe \nOcc.**  Fe \nOcc.† µ (Fe 3+  )  \n(µµ µµB)**  µ (Fe 3+ ) \n(µµ µµB)†  Total µ/u.c. ( µµ µµB) \n** $ † \nGa1 \n(0.86) 0.14 0.18 -5 -3.9 -2.8 -2.18 -2.81 \nGa2 \n(0.68) 0.32 0.35 5 4.7 6.4 6.2 6.58 \nFe1 \n(0.83) 0.83 0.77 -5 -3.9 -16.6 -12.95 -12.01 \nFe2 \n(0.89) 0.89 0.70 5 4.5 17.8 16.02 12.60 \n     4.8 7.09 4.36 \n     0.6 \n(/f.u.) 0.89 \n(/f.u.) 0.54 \n(/f.u.) \n \n*Cationic site occupancies obtained after Rietveld refinement  \n** Present calculation assuming high spin Fe 3+  moment of 5µ B \n† Calculated Fe site occupancies and magnetic moment  using Neutron diffraction data by \nArima et al.  [5] \n$ Calculation assuming Fe site occupancies from the  present study and Fe 3+  magnetic \nmoments by Arima et al. [5] \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Page 12  of 16  \n  \nList of Figures \n \nFig. 1. (Color online) (a) Rietveld refined XRD pat terns of crushed GFO single crystals at 300 K \nand 700 K, (b) lattice parameters and unit cell vol ume obtained from Rietveld refinement plotted \nas a function of temperature, (c) simulated crystal  structure of GFO and (d) distortion in oxygen \npolyhedra with cation-oxygen bond lengths and angle s in GFO unit cell. \n \nFig. 2. (Color online) Temperature dependent Raman spectra of GFO single crystal (a) at \nselected temperatures between 18 K to 450 K and in the frequency range (b) 275-550 cm -1 and \n(c) 620-820 cm -1. Each spectrum was fitted with sum of Lorentzian l ine shapes and fitted spectra \nare superimposed. \n \nFig. 3. (Color online) Temperature dependence of th e line shape parameters of nine modes \nbetween 250-550 cm -1. The solid and dotted lines represent the simulate d peak position and line \nwidth data according to eq. (3) and (4), respective ly. The ferri to paramagnetic transition \ntemperature (T c = 290 K) is marked for reference.   \n \nFig. 4. (Color online) Plot of ( ) Tω vs. 2( ) \nsM T \nM  \n  \n   for mode M5. The lines are linear least square \nfits to the data and Tf  (~ 180 K) marks the point at which abrupt change of slope ( m) occurs.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Page 13  of 16  \n \n \n \n \nFig. 1- Mukherjee et al.  \n \n \n \n \n \n \n \n \n \n \n \n Page 14  of 16  \n \n \n \n \nFig. 2- Mukherjee et al.  \n \n \n \n \n \n \n \n \n \n \n \n Page 15  of 16  \n  \n \n \n \nFig. 3- Mukherjee et al.  \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n Page 16  of 16  \n  \n \n \n \n \nFig. 4- Mukherjee et al.  \n  "    },    {        "title": "1010.1882v1.Temperature_dependence_of_Magnetic_properties_in_Nanocrystalline_copper_ferrite_thin_films.pdf",        "content": "    \n                  \n \nTemperature dependence of Magnetic properties in \nNanocrystalline copper ferrite thin films  \n \n \nPrasanna D. Kulkarni1, Shiva Prasad1, N. Venkataramani2 , R. Krishnan3 , Wenjie \nPang4, Ayon Guha4 , R. C. Woodward4 , R.L. Stamps4 \n1Physics Department, IIT Bombay, 2ACRE, IIT Bombay, Mumbai, 400076,  3Laboratoire de Magnetism et \nd’optique de Versailles, CNRS, 78935, France, 4School of Physics, M013, The University of Western \nAustralia, 35 Stirling Hwy, Crawley WA. \n \n \n \nAbstract  \n \nThe copper ferrite thin films have been deposited by RF sputtering at a 50W rf power. The As-deposited films \nare annealed in air at 800ºC and then slow cooled. The As-deposited (AD) as well as slow cooled (SC) films \nare studied  using a SQUID Magnetometer. The M Vs H curves have been recorded at various temperatures between 5K to 300K. The coercivities obtained from the MH curves are then plotted against temperature (T). \nThe magnetization in the films does not saturate, even at the highest field of 7T.  The high field part of the M \nVs H curves is fitted using the H\n1/2 term of  Chikazumi expression M(H)= Q*(1- a /Hn), with n=1/2. The \nvariation of coefficient ‘a’  of H1/2 term has been observed with temperature (T). An attempt has been made \nto correlate this with the coercivity (Hc) in the case of annealed films.   \n \n \nINTRODUCTION  \n \nThe copper ferrite can be stabilized in two different phases \nin thin film form, viz., a cubic and a tetragonal phase. The \ncubic phase is stabilised in the as deposited (AD) films. The slow cooling (SC) of the films after ex-situ annealing \nresults in the tetragonal phase[1]. It has also been observed \nthat the RF sputtered films are nanocrystalline in nature[1]. The magnetic measurements of the nanocrystalline film \nshows non saturation of the MH loop even at high fields. \nThis is termed as High Field Suceptibility (HFS). In the case of rf sputtered nanocrystalline ferrite thin films, the \nexpression, M(H)= Q*(1- a /H\nn) fitted the approach to \nsaturation best with n=1/2 [2]. The coefficient ‘a’ of the \nH1/2 term is attributed to point like defects present in the \nmaterial. In the present paper, the temperature (T) dependence of coercivity (H\nc) has been discussed for both \ncubic and tetragonal phases of  the Cu ferrite films. The \nvariation of H c with T is also compared with the variation of \ncoefficient ‘ a’ of H1/2 term of Chikazumi expression [3].  \n  \nEXPERIMENTAL    \n \nThe copper ferrite films were deposited using the Leybold \nZ400 rf sputtering system. The films were deposited on amorphous quartz substrates at  ambient temperature during \nsputtering. The rf power employed during the deposition is \n50W. The thickness of the films are ~2400 Å. The AD \nfilms were ex-situ annealed in air at 800 °C for 2 hours, \nfollowed by slow cooling. The MH loops were measured   \nfor AD and SC films in a field up to 7T using a\n SQUID \nmagnetometer at various temperatures between 5K and \n300K.  \n \nRESULTS  \n \nFig. 1 shows the Hc Vs T data for AD and SC samples of \n50W copper ferrite films. At all temperatures, the H c values \nfor AD film are lower than the SC film. For the AD film, \nHc varies from 35 Oe at 300K to 2210 Oe at 5K. For the SC \nfilm also coercivity increases  with decreasing temperature. \nHowever, the variation is over a smaller range from 800 Oe \nat 300K to 2300 Oe at 5K. Moreover the functional form of \nthe variation of the coercivity appears to be quite different \nfor the two cases.   \n \n          \nFig 1 : Temperature dependence of Hc for 50W, AD and \nSC Cu ferrite films.   0 50 100 150 200 250 30005001000150020002500 SC\n AD\n  Hc (Oe)\nT (K)DISCUSSIONS  \n \nThe approch to saturation was discussed by Chikazumi [3]. \nIn the Chikazumi expression, the mangetization at a given \napplied field is written in the following way,               \n \n4πM = Q * (1 – a/H1/2 – b/H – c/H2 - ….) + eH       (1) \n \nHere 4πM is the actual value of magnetization that is \nobserved at a field H, and Q, a,  b, c, and e are constants. \nThe value of Q should correspond to a value of 4 πM in \ninfinite field. The second term, a/H1/2 is attributed to the \npoint like defects or magnetic anisotropy fluctuations on atomic scale. We have fitted the magnetization data from \n0.8T field to 7T field to an expression involving just H\n1/2 \nterm in eq. (1). The value of ‘ a’ have been obtained at \nvarious temperatures between 5K and 300K.   \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n \n \n  \n \n  \nFig 2 :  The variation of H\nc Vs ‘ a’ for 50W, (a) AD and (b) \nSC, Cu ferrite films.   \nThe H\nc and ‘ a’ values at various temperatures are plotted \nagainst each other in Fig. 2. The H c values are plotted in the \nincreasing order along y-azis, which is equivalent to \ndecreasing T. For the SC film, the H c and ‘ a’ values both \nincrease with decrease in T.  While for AD film, the ‘ a’ \nvalue decreases and H c increases with decreasing T. For \nAD film, the variations of H c and ‘ a’ with temperature are \nopposite. If anisotropy is the only parameter controlling both H\nc and ‘ a’, their values are exp ected to increase with \ndecrease in temperature. This is  not observed for AD film.  The grain sizes in the case of rf sputtered AD \nfilms, have been observed to be ~5 – 10 nm [1]. In this \nrange, the grains may become superparamagnetic [4]. \nHowever, if superparamagnetis m is dominant in AD films \nat lower temperatures, the  value of ‘ a’ is expected to \nincrease, which is not observed. It is also  possible that \ndecrease in ‘ a’ with lowering T in AD films is because of \nblocking of superparamagnetic particles at lower \ntemperatures.  \n \nCONCLUSIONS \n \nThe variation of H c and ‘ a’ with temperature shows \nopposite behaviour for AD rf sputtered films. This can not \nbe completely understood on the basis of anisotropic effects or superparamagnetism.  \n \nACKNOWLEDGEMENT \n \nThe author Prasanna D. Kulk arni acknowledge the CSIR, \nIndia for financial support.  \n \nREFERENCES \n \n[1] M.Desai, S. Prasad, N. Ve nkataramani, I. Samajdar,   \n     A.K. Nigam, R. Krishnan, J. Appl. Phys. , 91 (2002)  \n     2220. [2] J. Dash, S. Prasad, N. Venkataramani, R. Krishnan, P.  \n     Kishan, N. Kumar, S.D. Kulkarni, S.K. Date, J. Appl.  \n     Phys. , 86  (1999) 3303. \n[3] S.\n Chikazumi, S.Charap, Physics of Magnetism , John  \n      Wiley & Sons, New York (1964). \n[4] C. P. Bean,  J.D.  Livingston,  J. Appl. Phys.,  30  (1959)   \n     120S. 40 42 44 46 48 5005001000150020002500\n  Hc (Oe)\na (Oe1/2) (a) AD \n24 26 28 30 328001200160020002400\n  Hc (Oe)\na (Oe1/2) (b) SC "    },    {        "title": "1305.0484v3.Chiral_properties_of_bismuth_ferrite__BiFeO3__inferred_from_resonant_x_ray_Bragg_diffraction.pdf",        "content": "Chiral Properties of Bismuth Ferrite (BiFeO3) Inferred from \nResonant x-ray Bragg Diffraction\nAngel Rodriguez-Fernandez  1*, Stephen William Lovesey 2,3, Steve Patrick Collins 3, \nGareth Nisbet 3 and Jesus Angel Blanco 1\n1 Physics Department, University of Oviedo, C/ Calvo Sotelo s/n Oviedo, Spain\n2 ISIS  Facility, STFC  Oxfordshire OX11 0QX, UK \n3 Diamond Light Source Ltd, Oxfordshire OX11 0DE, UK\nA new chiral phase of ferric ions in bismuth ferrite, the only  \nmaterial  known  to  support  multiferroic  behaviour  at  room  \ntemperature, is  inferred from extensive  sets  of data gathered by  \nresonant  x-ray  Bragg  diffraction.  Values  of  all  ferric  multipoles  \nparticipating  in  a  minimal  model  of  Fe  electronic  structure  are  \ndeduced from azimuthal-angle scans.  Extensive sets of azimuthal-\nangle data, gathered by resonant x-ray Bragg diffraction, yield \nvalues of all ferric multipoles participating in a minimal model of Fe \nelectronic structure. Paramagnetic (700 K) and magnetically ordered \n(300 K) phases of a single crystal of BiFeO3 have been studied with \nx-rays tuned near to the iron K-edge (7.1135 keV). At both \ntemperatures, intensities at a Bragg spot forbidden in the nominal \nspace-group, R3c, are consistent with a chiral motif of ferric ions in \na circular cycloid propagating along (1, 1, 0)H. Templeton and \nTempleton scattering at 700 K is attributed in part to charge-like \nquadrupoles in a cycloid. The contribution is not present in a \nstandard, simplified model of electronic states of the resonant ion \nwith trivial cylindrical symmetry. \nKEYWORDS: multiferroic state, chirality, resonant Bragg diffraction\nAn electronic state in which charge and magnetic polarizations coexist has been at the \ncentre of materials science in the past decade. Such a state exists in bismuth ferrite (BiFeO3) \nat room temperature, while in all other known cases a multiferroic state emerges upon \ncooling. This makes bismuth ferrite a unique candidate for potential application in electronic \ndevices, such as sensors or multi-state memory storage-units. 1-4) We have gathered evidence  \nfor a new\n chiral phase of BiFeO3 in the paramagnetic phase using resonant x-ray Bragg diffraction and,\n by way of a test for the chirality property, we demonstrate that our proposed electronic\n structure allows coupling to radiation with a like property, namely, x-rays with circular\n polarization (helicity).\n Moreover, our extensive sets of diffraction data enable us to infer values of ferric multipoles \n*E-mail: angelrf86@gmail.comin both the paramagnetic and magnetically ordered phases.\nBismuth ferrite forms in a rhombohedrally distorted perovskite crystal structure of R3c-\ntype (#161). Weak ferroelectricity develops below a Curie temperature Tc ≈ 1100 K, and G-\ntype antiferromagnetic order of ferric (Fe3+) dipole moments is observed below a Néel \ntemperature TN ≈ 640 K. The antiferromagnetism  coexists with a long-period modulation (≈ \n620 Å) in the hexagonal plane, as shown in figure 1.5,6) This coexistence is a curious property \nof a quite simple compound, created by the tension between two interactions that favor \nparallel and orthogonal spin arrangement, respectively. \nFig. 1.  Scheme of the crystal and magnetic structures of bismuth ferrite (BiFeO3), with \nhexagonal setting. The [0,0,1]H axis is vertical. Directions of magnetic dipoles of the \nFe ions at room temperature are indicated by arrows.\nFor studies of electronic magnetism, the experimental technique of resonant x-ray \nBragg diffraction we use has advantages over non-resonant diffraction that is difficult to \nexploit quantitatively, because uncertainty surrounds the deployment of a scattering length \nasymptotically valid in the Compton region of scattering.7-11) Resonant x-ray diffraction has \nproved its worth in many studies, particularly those that focus on one or more of the raft of \nelectronic properties driven by angular anisotropy in valence states. 12-13) Intensities collected \nat weak, space-group forbidden reflections access directly information about complex \nelectronic structure manifest in atomic multipoles, including, magnetic charge (or magnetic \nmonopole),14) electric dipole,15) anapole,16,17) quadrupole,18) octupoles,19,20) and \nhexadecapoles.21,22) In consequence, weak reflections are extremely sensitive to charge, orbital \nand spin electron degrees of freedom and BiFeO3, with a chemical structure similar to \nhaematite, is no exception.23) \nCrystal growth was performed in platinum crucibles with content of about 90 g, using \nthe accelerated rotation technique, and a platinum cover welded tightly to the crucible, \nleaving only a central hole of 0.1 mm diameter, as explained in reference.24) The size of the \nsample was about 5 x 5mm2 and a thickness of 0.5 mm, showing a polished surface in the [0, \n0, l]H direction. \n*E-mail: angelrf86@gmail.comFig. 2.  Cartesian coordinates (x, y, z) and x-ray polarization and wavevectors. The plane \nof scattering spanned by primary (q) and secondary (q') wavevectors coincides with the x-\ny plane. Polarization labelled σ and σ' is normal to the plane and parallel to the z-axis, and \npolarization labelled π and π' lies in the plane of scattering. The beam is deflected through \nan angle 2θ. Nominal setting of the crystal is indicated with ah\n antiparallel to σ-polarization, together with sense of rotation in an azimuthal-angle\nOur hexagonal crystal coordinates are ah = a(1, 0, 0), bh = a(−1/2, √3/2, 0) and ch = c(0, \n0, 1), with a = 5.58Å and c = 13.88Å.25) Basis vectors, or principal crystal axes, are ξ = (1, 0, \n0), η = (0, 1, 0) and ζ = (0, 0, 1), and they coincide with (x, y, z) in figure 2 at the nominal \nsetting of the crystal. The Bragg wavevector (0, 0, l)H is aligned with − x, as shown in figure 2. \nIntensities are measured as a function of rotation of the crystal about the Bragg wavevector \nthrough an angle ψ.\nThe (0, 0, 9)H reflection is forbidden in the nominal space group R3c. Bragg diffraction \ndue to angular anisotropy in available valence states is weak but, none the less, visible in \ndiffraction enhanced by an atomic resonance, as evident in data displayed in figure 3. \nResonant x-ray diffraction experiments were performed at the Diamond Light Source (UK), \non beamline I16. The horizontally polarized beam, σ, was tuned near the iron K-edge (7.1135 \nkeV). We observed intensity at the (0, 0, 9)H reflection in two studies with the sample held at a \ntemperature below (300 K) and above (700 K) the Néel temperature. The change in intensity \nthat we observed with cooling, between the two temperatures, confirmed the magnetic origin \nof the difference signal; relevant data are displayed in Figure 5. All data were collected in the \nrotated channel of polarization π'σ, where states of polarization labelled π' and σ are defined \nin figure 2. During the experiment we scan the surface of the sample to try to determine the \n*E-mail: angelrf86@gmail.comsize of the domains, and select the appropriate region of the sample where a likely single \ndomain could be involved in the scattering process. While it is possible that the sample \nsupports different domains, the present results were consistent with a single domain \nilluminated by the very small size of the primary beam (180 x 40 µm2). \nThe azimuthal scans presented in Figure 4 were obtained performing “θ scans” with the \ndetector around the Bragg condition for different azimuthal angles. This method was used \npreviously by Finkelstein et al.19) and by Kokubun et al.44), among others. The experimental \nvalues displayed in Figures 4 and 5 are the integrated intensity of each of the curves \nnormalized to intensity in the primary x-ray beam. Due to the small penetration depth of an allowed \nBragg spot (0, 0, 6)H, where the diffraction is mostly following a dynamical process, we have not \nconsidered to use this kind of reflection to normalize a forbidden Bragg spot (0, 0, 9)H that, due to its \nweakness has a kinematical behaviour a larger penetration depth and less affected by possible defects from \nthe surface.\nAll reasonable steps have been taken to arrive at sound data. Subtraction of the \nbackground intensity due to Renninger reflections (multi-beam peaks), observed in an \nazimuthal scan, was done using a Matlab program available at the instrument.  For the case of \nroom temperature, an azimuthal scan was done to select optimum, ﬂat positions between \npeaks and avoid the Renninger effect (therefore measured points in the azimuth dependence \nare not equidistant). Due to the fact that we have collected resonant x-ray data for a certain \nselected reflection at different parts of the single crystal, we consider that the experimental \ndata shown in figures 4 & 5 are related to the resonant event rather than to the tail of the \nRenninger effect. The high-quality crystal used for the experiment has a face perpendicular to \nthe (0, 0, 1)H direction, so the experiment was performed with a specular geometry. \nFig. 3. (Colour Online) X-ray spectrum in the vicinity of the Fe K-edge for the (009)H reflection. Diffraction \ndata reported in figures 4 and 5 were collected tuning the primary energy to E = 7.1135 keV . Inset: (Red dots) \nEnergy scan data and (blue line) approximation to a harmonic oscillator.\n*E-mail: angelrf86@gmail.comWe address, first, Templeton and Templeton (T & T) scattering reported in figure 4 \n(filled dots) measured with the sample at 700 K (TN ≈ 640 K).26) Resonant x-ray diffraction \nenhanced by an electric dipole - electric dipole (E1-E1) event is forbidden at the (0, 0, 9)H \nBragg spot of a R3c-type chemical structure. Diffraction enhanced by an electric quadrupole-\nelectric quadrupole (E2-E2) event is allowed, however, and it is produced by an electric, time-\neven hexadecapole. This diffraction is part of what we have observed, as we now explain.\nFig. 4.  (full dots) Intensity of the Bragg spot (0, 0, 9)H as a function of azimuthal angle, \nψ, with a sample temperature of 700 K , forbidden in the R3c-type structure and called \nTempleton and Templeton (T & T) scattering. Rotation of the crystal is counter clockwise \nabout the Bragg wavevector, and the origin ψ = 0 is ah antiparallel to σ-polarization, \nfigure 2. (empty dots) Intensity as a function of azimuthal angle obtained at room \ntemperature, 300K. Corrections to raw data are described in the text. Solid (dashed) line \nis a fit to our model of diffraction by a motif of charge-like quadrupoles and \nhexadecapoles, namely,Fπ'σ2 and expression (6) with v = 0 (v ≠ 0).\nOur notation for parity-even atomic multipoles is 〈TKQ›, with a complex conjugate \n〈TKQ›* = (−1)Q 〈TK−Q›, where the positive integer K is the rank and Q the projection, with − K \n≤ Q ≤ K. Angular brackets 〈...› denote the time average of the enclosed spherical tensor \noperator, i.e., multipoles are properties of the electronic ground-state, and the time-signature \nof 〈TKQ› is (− 1)K.12,13) The hexadecapole (K = 4) in question is the real part of 〈T4+3›, denoted \nby 〈T4+3›'. A triad axis of symmetry, C3z, passes through the iron sites, 6a in R3c. Diffraction \nby these ions using a Bragg wavevector (0, 0, l)H is described by an electronic structure factor,\n ΨKQ = {1 + 2 cos(2πl/3)} [〈TKQ› + (−1)l (−1)K 〈TK−Q›].(1) \nSpace-group allowed reflection are defined by diagonal elements ΨK0 with K even, and \nΨK0 ≠ 0 is allowed for l = 6n. The identity C3z 〈TKQ› = 〈TKQ› requires Q = ± 3m. As \nanticipated, E1-E1 is forbidden for l odd, because, of course, ΨK0 = 0 for a space-group \nforbidden reflection, while Q = ± 3 does not contribute to a dipole-dipole event where K does \nnot exceed 2.  Using (1) for an E2-E2 event, we find the corresponding unit-cell structure \n*E-mail: angelrf86@gmail.comfactor is a three-fold periodic function of the azimuthal angle, ψ,27)  \nFπ'σ = (3/√2) cos3θ cos(3ψ) 〈T4+3›'. (2)\nIn this expression, θ is the Bragg angle, and ah is antiparallel to σ-polarization, normal \nto the plane of scattering in figure 2, at the origin of an azimuthal-angle scan, ψ = 0. Intensity \ncorresponding to (2), Fπ'σ2 ∝ cos2(3ψ), is symmetric about ψ = 90o, which does not agree \nwith our data for T & T scattering displayed in figure 4 (filled dots). \nMissing in what has been described thus far, we propose, is T & T scattering caused by \ncharge-like quadrupoles (K = 2) in a circular cycloid, using an E1-E1 event. An electric dipole \n(E1) is expected to be appreciably stronger than an electric quadrupole (E2) event. But \ndiffraction from the quadrupoles is weak, being the responsibility of components absent in a \nstandard stick-model, in which the electronic state of the resonant ion is restricted to \ncylindrical symmetry.28) Whence, the minimal model that explains measurements in figure 4 \n(700 K, filled dots) is a sum of two forms of weak T & T scattering. Adding the \ncorresponding magnetic scattering, we achieve a model that explains data displayed in figure \n4 (300 K, empty dots). \nWe invoke a (circular) cycloid with the plane of the cycloid parallel to the plane \nspanned by ch and ah + bh. This motif is one candidate considered by Przeniosło et al; see \nModel 1 in figure 1.29) We will assume that the cycloid, composed of charge-like multipoles, is \nconstant, independent of temperature, to a good approximation. This is a sound assumption \nfor the paramagnetic phase, and not unreasonable at lower temperatures for multipoles not \ninduced by magnetic order. \nStarting with an explanation of T & T scattering in figure 4, we utilize quadrupoles for a \ncircular cycloid, 〈C2Q›, introduced by Scagnoli and Lovesey27) and recently reviewed by \nLovesey et al. 45) These quadrupoles, in common with all cycloid multipoles, are not subject to \nthe symmetry operations in the point group for sites 6a in the R3c group. In the general case, \none finds 〈C20〉 = 0 for the first harmonic of the cycloid, which is the one of interest. For a \ncircular cycloid rotating in the x-z plane,\n〈C2+1› = (1/4) [〈T2+1 + T2−1› + i〈T2+2 − T2−2›] ≡ − (1/√6)[i(yz) + (xz)], (3)\nwhere (αβ) is a standard, traceless second-rank Cartesian tensor. A representation of the \nquadrupole, 〈T2›, in terms of standard operators is available.30) \nBy way of an orientation to the result (3) we consider its value for a standard stick-\nmodel.28) In this case, all electronic properties of the resonant ion are manufactured from one \nmaterial vector. Using α and β to represent Cartesian coordinates, a general second-rank \ntensor (αβ) is to be replaced by a simple product 〈α› 〈β›, leading to (yz) = (xy) = 0 for a \n*E-mail: angelrf86@gmail.commaterial vector confined to the x-z plane with 〈y› = 0. \nGuided by R3c, we need the quadrupole (3) and the quadrupole derived from it by\nrotation about (ah + bh) by 180o. The sum of the two correctly related quadrupoles is\ntransformed to principal crystal-axes with the result,\n \nΨ2+1 = √(3/2) [(√3/4) {(ξ2 − η2) + 3(ζζ)} + (ηζ)], (4)\nwhere Cartesian quadrupoles are referred to previously defined principal crystal-axes. Note \nthat Ψ2+1 is purely real. Turning to data obtained with a sample temperature of 300 K and \ndisplayed in figure 4 (empty dots), magnetic diffraction by the cycloid is created by a time-\nodd dipole,\n〈C10› = (1/2) [〈T10› + i 〈T1+1 − T1−1›/√2]. (5)\nFor a reflection (0, 0, l)H with l odd, it actually contributes a magnetic dipole parallel \nto     (ah + bh), namely, 〈T1-1 - T1+1›/√2 calculated with principal crystal axes. At the K-edge, a \ndipole 〈T1› is simply orbital angular momentum.31)  \nFig. 5. Difference of two sets of data displayed in figure 4, I(v) − I(v = 0), together with \nexpression (7) derived from our model of electronic structure in bismuth ferrite. Multipoles \nt, u and v in (7) are set to values derived from fits to data in figure 4, namely, t = +1.19 ± \n0.07, u = − 6.20 ± 0.16 and v = 0.673 ± 0.014. Inset: Temperature dependence of the Bragg \nspot (0, 0, 9)H at an azimuthal angle ψ = 73°.\nWe use the purely real quadrupole (4), with projections Q = ± 1, as a common factor in \nthe final expression for the unit-cell structure factor. The remaining charge-like quadrupole, Q \n= ± 2, is accounted for in a ratio Ψ2+2/Ψ2+1 = −it. Calculations using an ideal cycloid show that \nt is purely real and t = + 1 (Scagnoli and Lovesey.27)). The contribution from the \nhexadecapole in (2) is captured by u = 3〈T4+3›'/(Ψ2+1√2). In the absorption profile we invoke \noverlap of the two events, E1-E1 and E2-E2, which occur at different energies with different \n*E-mail: angelrf86@gmail.comwidths. Lastly, the magnetic contribution to the structure factor is v = 3〈T1−1 − T1+1›/ (2Ψ2+1). \nNote that t, u and v are all treated as purely real quantities to be inferred from our data. Since t \nand v both relate to an E1-E1 event they are nothing more than ratios of the appropriate \nmultipoles that we have shown. On the other hand, u has to include a ratio of radial integrals \nfor E2 and E1 events, namely, (ƒ [q {R2}sd]2/[{R}sp]2) where {R}sp and {R2}sd, respectively, are \nradial integrals for E1 and E2 events at the K-edge of iron. A multiplicative factor in u, \ndenoted here by ƒ, measures the admixture of E1-E1 and E1-E2 events. While ƒ might depend \non energy it can be taken purely real without influencing conclusions, because it accompanies \nthe principal harmonic (2). \n Incorporating the two types of T & T scattering, expressions (2) and (4), and scattering \nby magnetic dipoles, we arrive at a unit-cell structure factor that gives an adequate account of \nall our data,\nFπ'σ = t cosθ sin(ψ) + u cos3θ cos(3ψ)−i sinθ cos(2ψ)−v[sinθ−i cosθsin(ψ)].   (6)\nWriting I(v) =Fπ'σ(v)2 the difference in intensity at the two temperatures is, \nI(v) − I(v = 0) =\nv[v(1 − cos2θ cos2(ψ)) − sin2θ (t sin(ψ) + u cos2θ cos(3ψ) + cos(2ψ) sin(ψ))]. (7)\nFits of I(0) to data for T & T scattering displayed in figure 4 (700 K, filled dots) yield \nvalues t = +1.19 ± 0.07, which is close enough to the ideal value to give great confidence, \nand       u = − 6.20 ± 0.16. A fit of I(v) to data gathered at 300 K, figure 4 (empty dots), yields \nv = 0.673 ± 0.014 for the magnetic dipole, with t and u set to aforementioned values. For \ncompleteness, we show in figure 5 difference data, taken from figure 4, together with the \nappropriate expression for intensity (7) evaluated with our estimates of the three multipoles. \nWe bring our Letter to a close with a survey of our observations and the interpretation \nwe construct. Above the Néel temperature, TN ≈ 640 K, our azimuthal-angle data are \nadequately explained by a model with minimal complexity. It includes Templeton and \nTempleton (T & T) scattering from charge-like quadrupoles and hexadecapoles.25) A \ncontribution by quadrupoles heralds a new chiral phase, in which quadrupoles participate in a \ncircular cycloid.  A test of\n chirality in electronic structure is to see whether or not it couples to circular polarization\n (helicity) in the x-ray beam\n An expression for intensity associated with circular polarization (helicity) in the primary \nbeam, Ic, is derived from our unit-cell structure factors 22) and we arrive at,\n*E-mail: angelrf86@gmail.comIc = Im. [(Fσ'π)*(Fσ'σ) + (Fπ'π)*(Fπ'σ)] = 2 cosθ cos(ψ) [v sinθ − cosθ sin(ψ)]\nx (t cosθ sin(ψ) + u cos3θ cos(3ψ) − v sinθ).   (8)\nIn expression (8), t and u are charge-like multipoles, which generate T & T scattering, \nand v is a magnetic dipole absent above TN. Values of the three multipoles, t, u and v, are \ninferred from data displayed in figure 4, collected above and below the Néel temperature, that \nare adequately described by Fπ'σ2 derived from (6). Note that expression (8) does not vanish \nfor v = 0, meaning resonant reflections are affected by circular polarization above TN with a \nhitherto unknown phase of the material. \nExistence of T & T scattering by quadrupoles in a cycloid implies that the actual \nchemical structure belongs to an enantiomorphic crystal class lacking a centre of symmetry. \nSpace-group R3 (#146), one of 65 members of the Sohncke sub-group of crystal structures, is \na maximal non-isomorphic subgroup of the nominal R3c-group, and thus a likely candidate \nfor a commensurate chiral motif in bismuth ferrite. In which case, a chiral motif and a single \ndomain are implied for the magnetically-ordered  state, and this does appear to be the case.33) \nA high-quality crystal, from which satellite peaks can be resolved, should show satellite \nintensity above TN. The domain pattern of propagation vectors should be reproduced on \ntemperature cycling above and below TN since they are driven by the pre-aligned quadrupoles. \nThis issue could be checked using circular polarized x-rays, due to the helicity properties of \nthis kind of x-rays.\nParallel scenarios merit a mention, e.g., the weak itinerant ferromagnet MnSi, and \nrelated materials.34-36) The compounds use a cubic group P213 (#198), and exist in both right-\nhanded and left-handed enantiomorphs. A single-valued handedness persists in the \nferromagnetic and paramagnetic phases,37) with chiral fluctuations in MnSi above the Curie \ntemperature observed by inelastic neutron scattering.38) Notably, a standard example for \nspontaneous homochirality, sodium chlorate (NaClO3), forms in the chemical structure \ndescribed by P213.39)\nMnSi has a Curie temperature Tc ≈ 29.5 K, and deep in the paramagnetic phase spin \nfluctuations are isotropic. Perhaps more relevant to the present discussion of bismuth ferrite is \nanother iron-based chiral magnet. FeGe, iso-structural with MnSi, has a high Curie \ntemperature, Tc ≈ 278.2 K, and precursor activity is well-established.36,40) Ferromagnetic \nspirals have a period ≈ 180 Å (MnSi) and ≈ 700 Å (FeGe), to be compared with a period ≈ \n620 Å in bismuth ferrite. On its own, an antisymmetric exchange-interaction  (Dzialoshinskii-\nMoriya) will promote an orthogonal arrangement of spins that can disturb an arrangement of \nparallel spins, supported by an isotropic Heisenberg exchange plus relatively weak magnetic \n*E-mail: angelrf86@gmail.comanisotropy.\nQuadrupoles (also higher-order multipoles) as a primary order-parameter is not unusual. \nHowever, again, order is achieved at low temperatures, because the underlying mechanism is \nweak. 21,41-43\nThe origin of the charge-like quadrupoles that contribute T & T scattering could be \nrelated to bismuth 6s-6p lone pairs, known to drive certain structural distortions. Apart from \nexpected direct hybridization of lone pairs, there is scope for admixture through the agency of \noxygen 2p states that contribute to angular anisotropy in valence states observed at iron sites. \nLastly, we examine the possibility that our azimuthal-angle scan at 700 K can be \nexplained by the parity-odd event E1-E2 using the R3c-group, in addition to E2-E2.12,27) We \nfind polar multipoles, 〈UKQ›, do not contribute intensity to the (0, 0, 9)H Bragg spot in \nchannels with unrotated polarization, σ'σ and π'π. The contribution from E1-E2 in the π'σ \nchannel of immediate interest comes from a purely real polar quadrupole, namely, i(3/√5) \ncos2θ 〈U20〉 that is added to the hexadecapole contribution (2). The two contributions to the \nunit-cell structure factor, E1-E2 plus E2-E2, are in phase quadrature, so there can be no \ninterference between them to lift the pure six-fold periodicity in the E2-E2 contribution to \nintensity that is lacking in figure 4 (filled dots).\nIn summary, Bragg diffraction intensities at the nominally forbidden reflection (0, 0, 9)H \nof bismuth ferrite, observed below and above the Néel temperature, are consistent with a \nchiral structure formed by a circular cycloid propagating along [1, 1, 0]H not previously \ndetected in the paramagnetic phase. The new chiral phase is responsible for some Templeton \nand Templeton (T & T) scattering at 700 K due to charge fluctuations not contained in the \nplane of the cycloid. Our extensive sets of azimuthal-angle diffraction data have been used to \ninfer good values of three atomic multipoles involved in the scattering process. A satisfactory \nminimal model of Fe electronic structure includes a quadrupole (E1-E1 event) and a \nhexadecapole (E2-E2 event) contributing T & T scattering, plus a magnetic dipole (E1-E1).\nAcknowledgments\nWe acknowledge the Diamond Light Source for the beam-time allocation on I16. We \nhave benefited from discussions and correspondence on the question of normalization of our \ndata with Dr K Finkelstein. One of us (SWL) is grateful to Dr D D Khalyavin and Dr K S \nKnight for valuable discussion about the explanation of results offered in the communication. \nWe are also grateful with G Catalan, who has provided the single crystal for performing the \nexperiment. Financial support has been received from Spanish FEDER-MiCiNN Grant No. \nMat2011-27573-C04-02.  One of us (ARF) is grateful to Gobierno del Principado de Asturias \n*E-mail: angelrf86@gmail.comfor the financial support from Plan de Ciencia, Tecnología e innovación (PTCI) de Asturias. \nWe thank Diamond Light Source for access to beamline I16 (MT7720) that contributed to the \nresults presented here.\nReferences\n1) A. M. Kadomtseva, A. K. Zvezdin, Y . F. Popov, A. P. Pyatakov, and G. P. Vorob’ev: J. \nExp. Theor. Phys. Lett. 79 (2004) 571.\n2) G. Catalan and S. F. Scott: Adv. Mater. 21 (2009) 2463.\n3) J. F. Scott: Adv. Mater. 22 (2010) 2106.\n4) D. C. Arnold, K. S. Knight, G. Catalan, S. A. T. Redfern, J. F. Scott, P. Lightfoot, and F. \nD. Morrison: Adv. Funct. Mater. 20 (2010) 2116.\n5) D. Lebeugle, D. Colson, A. Forget, M. Viret, A. M. Bataille, and A. Gukasov: Phys. \nRev. Lett. 100 (2008) 227602.\n6) S. Lee, T. Choi, W. Ratcliff, R. Erwin, S. W. Cheong, and V . Kiryukhin: Phys. Rev. B 78 \n(2008) 100101.\n7) F. de Bergevin and M. Brunel: Acta Crystallogr. 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B 75 (2007) 014409.\n17) J. Fernandez-Rodriguez,  V . Scagnoli, C. Mazzoli, F. Fabrizi, S. W. Lovesey, J. A. \nBlanco, D. S. Sivia, K. S. Knight, F. de Bergevin, and L. Paolasini: Phys. Rev. B 81 \n(2010) 085107.\n18) S.B Wilkins, R. Caciuffo, C. Detlefs, J. Rebizant, E. Colineau, F. Wastin, and G. H. \nLander: Phys. Rev. B 73 (2006) 060406.\n19)   K. Finkelstein, Q. Shen, and S. Shastri: Phys. Rev. Lett. 69 (1992) 1612.\n20) S. W. Lovesey and K. S. Knight: J. Phys.: Condens. Matter 12 (2000) 2367.\n*E-mail: angelrf86@gmail.com21) Y . Tanaka, T. Inami, S. W. Lovesey, K. S. Knight, F. Yakhou, D. Mannix, J. Kokubun, \nM. Kanazawa, K. Ishida, S. Nanao, T. Nakamura, H. Yamauchi, H. Onodera, K. \nOhoyama, and Y . Yamaguchi: Phys. Rev. B 69 (2004) 024417 .\n22) J. Fernandez-Rodriguez,  S. W. Lovesey,and J. A. Blanco: Phys. Rev. B 77 (2008) \n094441.\n23) S. W. Lovesey, A. Rodriguez-Fernandez  and J. A. Blanco: Phys. Rev. B 83 (2011) \n054427.\n24) R. Palai, R. S. Katiyar, H. Schmid, P. Tissot, S. J. Clark, J. Robertson, S. A. T. Redfern, \nG. Catalan, and J. F. Scott: Phys. Rev. B 77 (2008) 014110 .\n25) A. Palewicz, I. Sosnowska, R. Przeniosło and A. Hewat: Acta Physica Polonica A 117 \n(2010) 296.\n26) D. H. Templeton and L. K. Templeton: Acta Crystallogr. 41 (1985) 365; ibid 42 (1986) \n478.\n27) V . Scagnoli and S. W. Lovesey: Phys. Rev. B 79 (2009) 035111.\n28) J. P. Hannon, G. T. Trammell, M. Blume, and D. Gibbs: Phys. Rev. Lett. 61 (1988) \n1245; ibid 62 (1989) 2644 (E).\n29) R. Przeniosło, M. Regulski, and I. Sosnowska: J. Phys. Soc. Japan 75 (2006) 084718.\n30) S. W. Lovesey and E. Balcar: J. Phys.: Condens. Matter 9 (1997) 8679.\n31) P. Carra, B. T. Thole, M. Altarelli, and X. Wang: Phys. Rev. Lett. 70 (1993) 694.\n32) S. W. Lovesey: J. Phys.: Condens. Matter 10 (1998) 2505.\n33) R. D. Johnson, P. Barone, A. Bombardi, R. J. Bean, S. Picozzi, P. G. Radaelli, Y. S. Oh, \nS. W. Cheong, and L. C.Chapon: Phys. Rev. Lett. 110 (2013) 217206. \n34) M. Ishida, Y . Endoh, S. Mitsuda, Y . Ishikawa, and M. Tanaka: J. Phys. Soc. Japan 54 \n(1985) 2975.\n35) V . A. Dyadkin, S. V . Grigoriev, D. Menzel, D. Chernyshov, V . Dmitriev, J. Schoenes, S. \nV . Maleyev, E. V. Moskvin, and H. Eckerlebe: Phys. Rev. B 84 (2011) 014435.\n36) H. Wilhelm, M. Baenitz, M. Schmidt, C. Naylor, R. Lortz, U. K. Rössler, A. A. Leonov, \nand A. N. Bogdanov: J. Phys.: Condens. Matter 24 (2012) 294204.\n37) V . Dmitriev, D. Chernyshov, S. Grigoriev, and V. Dyadkin: J. Phys.: Condens. Matter 24 \n(2012) 366005.\n38) B. Roessli, P. Böni, W. E. Fischer, and Y. Endoh: Phys. Rev. Lett. 88 (2002) 237204.\n39) C. Viedma and P. Cintas: Chem. Commun. 47 (2011) 12786.\n40) E. Moskvin, S. Grigoriev, V . Dyadkin, H. Eckerlebe, M. Baenitz, M. Schmidt, and H. \nWilhelm: Phys. Rev. Lett. 110 (2013) 077207.\n41) P. Morin, D. Schmitt, and E. de Lacheisserie: J. Magn. Magn. Mater. 30 (1982) 257.\n42) T. Sakakibara, T. Tayama, T. Onimaru, D. Aoki, Y. Onuki, H. Sugawara, Y . Aoki, and \nH.Sato: J. Phys.: Condens. Matter 15 (2003) S2055.\n43) Y . Kuramoto, H. Kusunose, and A. Kiss: J. Phys. Soc. Japan 78 (2009) 072001.\n*E-mail: angelrf86@gmail.com44)   J. Kokubun, A. Watanabe, M. Uehara, Y . Ninomiya, H. Sawai, N. Momozawa, K.    \nIshida, and V . E. Dmitrienko: Phys. Rev. B 78 (2008) 115112.\n45)    S. W. Lovesey, V . Scagnoli, M. Garganourakis, S. M. Koohpayeh, C. Detlefs, and \nU. Staub: J. Phys.: Condens. Matter 25 (2013) 362202.\n*E-mail: angelrf86@gmail.com"    },    {        "title": "2309.13724v1.FeCo_Nanowire_Strontium_Ferrite_Powder_Composites_for_Permanent_Magnets_with_High_Energy_Products.pdf",        "content": "FeCo Nanowire −Strontium Ferrite Powder Composites for\nPermanent Magnets with High-Energy Products\nJ. C. Guzma ́n-Mínguez, S. Ruiz-Go ́mez, L. M. Vicente-Arche, C. Granados-Miralles,\nC. Ferna ́ndez-Gonza ́lez, F. Mompea ́n, M. Garc ía-Herna ́ndez, S. Erohkin, D. Berkov, D. Mishra,\nC. de Julia ́n Ferna ́ndez, J. F. Ferna ́ndez, L. Pe ́rez, and A. Quesada *\nCite This: ACS Appl. Nano Mater. 2020, 3, 9842 −9851 Read Online\nACCESS Metrics & More Article Recommendations *sıSupporting Information\nABSTRACT: Due to the issues associated with rare-earth elements, there\narises a strong need for magnets with properties between those of ferritesand rare-earth magnets that could substitute the latter in selectedapplications. Here, we produce a high remanent magnetization compositebonded magnet by mixing FeCo nanowire powders with hexaferriteparticles. In the ﬁrst step, metallic nanowires with diameters between 30\nand 100 nm and length of at least 2 μm are fabricated by electrodeposition.\nThe oriented as-synthesized nanowires show remanence ratios above 0.76and coercivities above 199 kA/m and resist core oxidation up to 300 °C due\nto the existence of a >8 nm thin oxide passivating shell. In the second step, acomposite powder is fabricated by mixing the nanowires with hexaferriteparticles. After the optimal nanowire diameter and composite composition are selected, a bonded magnet is produced. The resultingmagnet presents a 20% increase in remanence and an enhancement of the energy product of 48% with respect to a pure hexaferrite(strontium ferrite) magnet. These results put nanowire −ferrite composites at the forefront as candidate materials for alternative\nmagnets for substitution of rare earths in applications that operate with moderate magnet performance.\nKEYWORDS: composite permanent magnets, nanowires, ferrites, rare-earth-substitution, improved energy product,\nmagnetostatic interactions\n■INTRODUCTION\nThe best magnets in the world, amounting to 80% of the\nworldwide sales of the $20 billion permanent magnet market,contain rare-earth elements (REEs).\n1Their properties are far\nsuperior, which creates a large gap between the energy\nproducttheﬁgure of merit of a magnet of REE magnets\nand of the other families of magnets, such as hard ferrites andalnicos.\n2Unfortunately, as critical raw materials, REEs are\nassociated with supply risk, price volatility, and environmentallyharmful extraction and separation.\n3Al a r g en u m b e ro f\napplications, covering an important slice of the market, onlyrequire moderate magnet performance to operate; nevertheless,in the absence of a gap magnet, these technologies have noalternative but to rely on critical REE magnets. In thisframework, there is a strong demand for the development ofalternative REE-free or REE-lean gap magnets that couldeﬀectively substitute REEs in these applications.\n4,5\nA considerable amount of projects and initiatives have focused\ntheir e ﬀorts on this issue in the past few years2,6by exploring a\nrelatively wide variety of solutions that include developingcompletely new magnetic materials,\n5improving the properties\nof existing magnetic phases,7−9reducing the content of heavy\nREEswhich are the real critical issues in existing REEmagnets,10and fabricating composite materials by combining\nphases with di ﬀerent magnetic properties.11−13\nOne of the main challenges associated with the development\nof permanent magnets is that it is hard to come across a materialin nature that simultaneously hosts a large coercivity and high\nmagnetization.\n14For this reason, shape anisotropy has been\noften employed as a tool to sustain a certain resistance to\ndemagnetization in high-magnetization systems, for instance, by\nfabricating high-aspect-ratio transition-metal structures.15,16A\ngreat deal of work has been focused on Co nanowires (NWs)\nowing to the relatively large magnetocrystalline anisotropy of Co\ncompared to other magnetic transition metals such as Fe andNi.\n14−18Promising results19,20,22have been obtained in this\nsystem, including the fabrication of NW-only dense pellets with\nimproved magnetic properties.21,23Fe, FeNi, and FeCo\nnanowires have been investigated as well.24−29They present\nReceived: July 14, 2020\nAccepted: September 11, 2020\nPublished: September 11, 2020\nArticle www.acsanm.org\n© 2020 American Chemical Society\n9842https://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −9851\nThis is an open access article published under an ACS AuthorChoice License, which permits\ncopying and redistribution of the article or any adaptations for non-commercial purposes.\nDownloaded via Cecilia Granados-Miralles on September 24, 2023 at 18:56:12 (UTC).\nSee https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.\nlarger magnetization and lower coercivity than Co, which makes\nthem suitable for a wide array of application sectors that includebiomedical,30−33environmental (in this case, ferrite NWs),34\nand sensors,35among others. In addition, arrays of low-\nFigure 1. Schematic of the fabrication process of the FeCo NWs, the ferrite −FeCo NW composites, and the composite bonded magnet.\nFigure 2. (a) TEM image of a 50 nm diameter isolated NW. (b) SEM image of the 50 nm diameter NW powder. (c) HRTEM image of the side of the\nNW from (a). The yellow line delimits the core −shell frontier. Fast Fourier transform (FFT) patterns of (d) core and (e) shell. (f) XRD pattern and\ncorresponding Rietveld model of the 50 nm NWs and (g) TGA curve from room temperature (RT) to 900 °C measured in air.ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519843\ncoercivity NWs are excellent systems to study and analyze the\ncomplex hysteretic and reversal mechanisms of ferromag-\nnets.33,36−39\nIn view of their potential application as permanent magnets\nand to minimize the low coercivity problem, they have been used\nas a secondary phase in composite systems together with harder\nmagnetic materials, with reports mainly focused on MnBi as the\nhost phase.40−42Again, promising results have been achieved in\nthese systems, showing the potential of magnetic nanowires as\nbuilding blocks for permanent magnets. However, the\nproduction yield of these nanostructures is usually low and the\nassociated fabrication costs are high, which explains their\nabsence in commercial products nowadays.\nIn an e ﬀort to circumvent these issues, in this work, we focus\non the development of composite magnets based on FeCo NWs\nand strontium ferrite (SrFe12O19), as ferrites are known for their\navailability and low price ($1.5/kg). In the ﬁrst part of the work,\nthe experimental and theoretical study of the structures and\nproperties of electrodeposited NWs as a function of their\ndiameter is presented by means of electron microscopy, X-ray\ndiﬀraction (XRD), thermogravimetric analysis (TGA), and\nmagnetometry. Micromagnetic simulations were performed to\nunderstand the dimension de pendence of the magnetic\nproperties of the FeCo NWs and to determine the magnetic\nstructure. High-magnetization FeCo NWs of di ﬀerent diameters\nwere mixed and oriented with di ﬀerent concentrations of hard\nhexaferrite micropowders. After selecting an optimal diameter,\nthe most adequate mixture composition is decided upon by\ncharacterizing powder composite samples of a few milligrams. A\nbulk bonded composite magnet (0.3 g) is fabricated out of the\noptimal composition and compared to a pure ferrite reference\nmagnet fabricated and characterized under identical conditions.Figure 1 presents a schematic of the experimental fabrication\nprocess of the FeCo NWs, the composites, and the bonded\nmagnet.■RESULTS\nThe morphological and structural properties of the FeCo NW\ndry powder were characterized by transmission and scanningelectron microscopy (TEM and SEM) and X-ray di ﬀraction\n(XRD). Figure 2 a shows the TEM images of a FeCo nanowire\ngrown in the 50 nm pore template. First, the NW diameter is\nmeasured to be D\n50nm= 49 nm. The darker core of the nanowire\nis observed to be surrounded by a thin layer that appears slightlybrighter in the image. This is consistent with the formation of apassivating oxide layer on the surface of the FeCo NWs,resulting in a core −shell type of structure that protects the\nmetallic core from further oxidation.\n43The diameter of the\nmetallic core is Dcore= 35 nm. This value is in agreement with\nprevious reports,43,44as a shell with a thickness of 7 nm is\ninferred by subtracting outer and inner diameters. The SEMimage in Figure 2 b shows that all FeCo NWs obtained in the\nbatch present a diameter close to 50 nm, con ﬁrming a very\nnarrow dispersion of values. In addition, an average length of 6μm is observed. The 30 and 100 nm diameter powders were\nanalyzed by SEM as well, as shown in Figure S1 of the\nSupporting Information (SI), con ﬁrming a narrow dispersion of\ndiameters. For the 100 nm FeCo NWs, an average length of 2μm is observed.\nFigure 2 c shows a high-resolution TEM (HRTEM) image of\nthe 50 nm FeCo NW. Lattice fringes can be observed, althoughcrystallite sizes and lattice constants are not easily extractable.\nTwo fast Fourier transform (FFT) patterns are shown in Figure\n2d,e. A body-centered cubic (bcc) structure is revealed at the\ncore, as expected for metallic FeCo alloys. The pattern of theshell is signi ﬁcantly noisier and no structure is easily identi ﬁable,\nalthough it is clearly di ﬀerent from the core FFT pattern.\nThe XRD pattern of the powder in Figure 2 d only shows\ndiﬀraction maxima at the (110), (200), and (211) Bragg\npositions corresponding to a bcc lattice, with a lattice parameterof 2.856 Å. Using the equation in ref 45, from the lattice\nparameter, we infer a composition Fe\n49Co51, with no detection\nof other phases up to the resolution limit of the instrumentFigure 3. Magnetization curves measured at RT of (a) 30 nm, (b) 50 nm, and (c) 100 nm diameter NW powders oriented in an external magnetic ﬁeld\ninside a bonding glass matrix. Black and blue curves correspond to the applied ﬁeld being parallel (black) and perpendicular (blue) to the nanowire\nlong axis. (d) Evolution of the remanence-to-saturation ratio ( Mr/Ms) and coercivity ( Hc) as a function of the NW diameter.ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519844\n(approximately 5 wt %) and that the Fe and Co atoms are evenly\ndistributed among the sites of the bcc cell. The average crystallite\ndiameter obtained from the Rietveld model is approx. 12 nm.This implies that the depth of a NW comprises severalcrystallites that may explain the di ﬃculties in observing clear\nfeatures in the HRTEM image. The lack of other di ﬀraction\nmaxima, in particular, related to FeCo oxides, suggests a poorcrystallinity/amorphous structure of the oxide shell layer, asalready indicated by the di ﬀused FFT spots in Figure 2 e.\nThe TGA curve in Figure 2 e shows a very mild weight gain\nfrom RT to 300 °C, a temperature at which a more pronounced\nweight gain commences, reaching a maximum value of 25.3% at\n520°C and subsequently staying relatively constant up to 900\n°C. Previous studies allow us to correlate this TGA curve with\nthe oxidation of the FeCo bcc alloy nanoparticles to CoFe\n2O4\n(Co ferrite) under high-temperature annealing in air.43,46\nAssuming that is the case here if the FeCo NWs were 100%\nmetallic, a 37% weight gain should be observed during theoxidation to Co ferrite. From the values of the length and outer/\ncore diameter of the 50 nm NW, the volume of the metallic core\nin an average FeCo NW is calculated to be V\ncore= 24.4 ×106\nnm3, while that of the oxide shell is Vshell= 20.5 ×106nm3. Using\nthe density of FeCo of 8.1 g/cm3and that of Co ferrite of 5.23 g/\ncm3,47we calculate the mass of the oxide shell to amount to\n28.6% of the total mass of the FeCo NW powder. Thus, sinceonly 71.4% of the sample mass is metallic, the expected weightgain during oxidation of the 50 nm MWs is 26.4% (this number\nis calculated as the 37% of the metallic part of the sample, i.e.,\n71.4%, with 37% being the gain weight associated to theoxidation of FeCo alloy to CoFe\n2O4). The measured 25.3%\nweight gain is in good agreement with this value, experimentally\nsupporting that the FeCo NW powders approximately present a\n71 wt % FeCo −29 wt % Co ferrite. Moreover, we conclude from\nthe TGA results that the passivating layer protects the metalliccore from further oxidation at temperatures up to 300 °C.\nThe magnetization curves of the three samples, measured in\nparallel (black) and perpendicular (blue) to the alignment\ndirection, are presented in Figure 3 .\nThe saturation magnetization ( M\ns) values obtained for each\ndiameter are 115, 150, and 176 Am2/kg for 30, 50, and 100 nm,respectively. As a comparison, recent reports found Ms= 240−\n248 Am2/kg for Fe 65Co35NWs,32,48although it is important to\npoint out that NWs were fully metallic in those cases wherein thetemplate was not removed and the NWs were not exposed to air.The M\nsvalue for 50 nm FeCo NWs is in agreement with the fact\nthat the sample is composed of 71 wt % FeCo, which is known tohave M\ns≈210−220 Am2/kg.44,49The Co −Fe oxide shell thus\nhas a small contribution to the overall Ms. This can be explained\nby the fact that, on the one hand, Co ferrite with a 1:1 Fe/Coratio has a signi ﬁcantly smaller M\ns, around 40 −50 Am2/kg, than\nstoichiometric CoFe 2O4, around 80 Am2/kg.50On the other\nhand, the spin disorder due to low crystallinity, inferred from theTEM and XRD data, further reduces that value. To study thelong-term stability of the 50 nm NWs, we remeasured themagnetization curves 1 year after drying the powder and ﬁrst\nexposure to air. No signi ﬁcant changes were observed (see\nFigure S3 of the SI), from which it can be concluded that the\noxide shell prevents long-term oxidation (up to 1 year at least) aswell. For the 30/100 nm FeCo NWs, the M\nsvalues imply a 52/\n78 wt % of metallic phase in the nanowires, which is inagreement, following the same calculation executed above, witha passivating shell of thickness t≈6−8 nm, the same as for the 50\nnm case.\nNotable di ﬀerences can be observed between the parallel and\nperpendicular magnetization curves, particularly in theremanence ( M\nr) values, which demonstrate that the samples\nare signi ﬁcantly anisotropic remanence-to-saturation ratios\n(Mr/Ms) between 0.8 and 0.76 are observed as shown in Figure\n3dand have indeed been aligned inside the bonding glass. For\ninstance, Mr= 122 Am2/kg for the 50 nm FeCo NWs. It is\nimportant to point that, since we are measuring ensembles ofNWs and demagnetizing ﬁelds exist in the samples, the M\nrvalues\nmeasured constitute a lower limit for the real remanence of asingle FeCo NW of a given diameter.\nGiven that FeCo is a material with relatively low magneto-\ncrystalline anisotropy,\n46it is reasonable to assume that the shape\nanisotropy of the wires is responsible for this behavior,\nsupporting a large Mralong the NW long axis. This picture\nsuggests that the FeCo NWs are mainly in a single-domain\nFigure 4. Results of micromagnetic simulations: (a) hysteresis curves, (b) coercivity Hc, and (c) remanence ratio along the long axis ( zdirection) Mz/\nMsas a function of the length of a NW with D= 30 nm. Curves and dots of the same color present di ﬀerent geometrical realizations of the ﬁnite mesh\nelement structure. (d) Spin structure of the NW with D= 30 nm and L= 100 nm at remanence (0.0 T) and −200 kA/m negative ﬁeld. For a better\nvisibility, only magnetic moments for the mesh elements on the surface are shown.ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519845\nmagnetic state at remanence, with most of the spins pointing\nalong the long axis.\nMoreover, a decrease of coercivity ( Hc) with FeCo NW\ndiameter is observed, from 96 kA/m for 30 nm NW to 20 kA/m\nfor 100 nm. Recently, coercivities between 55 and 60 kA/m have\nbeen reported for 30 nm FeCo NWs (with no oxide shell).38,48\nFor 50 nm diameter, a kink in the easy-axis curve is observed in\ntheﬁrst and third quadrants. This is likely due to the presence of\na secondary phase with higher Hc. A small amount of Co ferrite\nat the shell and/or metallic Co at the core could explain the\nobservation. Coercivity is in any case considerably higher than\nthe normal values for FeCo isotropic particles Hc≈8 kA/m.44\nIt is true that, assuming a diameter-independent passivating\nlayer thickness, as will be evidenced later in the manuscript, the\noxide concentration increases as the diameter decreases, and Co\nferrite is known to present a large Hc. However, for the 1:1 Fe/\nCo ratio of our samples, the coercivity of Co ferrite is below 10\nkA/m.50Thus, it seems more likely that this behavior is\nexplained by an increase in shape anisotropy as the aspect ratio\nincreases and diameter decreases.15To understand the observed high remanence and coercivity\nvalues, micromagnetic simulations were performed. Figure 4 a−c\npresents the hysteresis loops, Hc, and remanence ratio obtained\nfrom the micromagnetic study on FeCo NWs with diameter D=\n30 nm and various lengths L. In terms of magnetic performance,\nhysteresis curves ( Figure 4 a) and corresponding coercivities\n(Figure 4 b) suggest that for L= 100 nm, Hcachieves 83% of its\nmaximum value, while the remanence ( Figure 4 c) is within a few\npercentage points of the saturation magnetization starting for L\n≥50 nm. The considerably higher remanence ratios obtained in\nthe simulations, compared to those experimentally measured in\nFigure 3 , are due to the fact that the simulations are performed\nfor single NWs and thus in the absence of interaction/\ndemagnetizing ﬁelds. For L= 200 nm, i.e., an aspect ratio of 7,\nthe maximum HcandMrare already attained.\nThe details of the spin structure within the wire at remanence\nand−200 kA/m negative ﬁeld are shown in Figure 4 d for a FeCo\nNW with D= 30 nm and L= 100 nm. The typical “ﬂower ”\nmagnetization distribution at remanence demonstrates an\nalmost single-domain state along the wire axis, with magneticmoments slightly deviating from this axis on top and bottomFigure 5. (a) Magnetization curve of nonoriented NW (50 nm) −ferrite composite powders with di ﬀerent NW wt %. (b) Coercivity ( Hc), saturation\nmagnetization ( Ms), and remanence ( Mr) as a function of the soft content in the ferrite −NW (50 nm) composites.\nFigure 6. (a) SEM image of the 70 wt % strontium ferrite −30 wt % FeCo NW powder. (b) Schematic of the bonded magnet fabricated out of the\ncomposite powder, showing NWs (in red) dispersed inside the ferrite magnet (in blue). (c) Magnetization curves at RT of the composite magnet (red)\nand the reference pure SrM magnet (black). The inset shows the second quadrant. (d) Derivative of the magnetization vs applied ﬁeld for the isolated\n50 nm NWs (blue), the 30 −70 wt % composite powder (green), and the composite bonded magnet (red); the latter two are composed of SFO and 50\nnm NWs.ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519846\nedges. The spin structure at −200 kA/m is presented by the\nlateral and top views of both NW edges. The spins on the wiresurface deviate from the long axis, forming vortices at the top andbottom edges, as observed in the top view. The totalmagnetization at these ﬁelds suggests that spins of the central\ncore part of the wire do not deviate from the long axis.\nWhile we acknowledge that it is not straightforward to fully\ntranslate this model to the FeCo NWs fabricated in this studythat have aspect ratios L/Dbetween 60 (for D= 100 nm) and\n200 ( D= 30 nm) and present an outer Co ferrite shell,\nmicromagnetic simulations clearly support the hypothesis thatthe FeCo NWs present a single-domain magnetic con ﬁguration\nthat enables the large remanence and competitive coercivityvalues. In addition, they hint at the formation of the vortex at thecore of wires during the magnetization reversal process.\nAs mentioned in the Introduction section, the two main\ndisadvantages of high-magnetization metallic nanowires asconstituents for permanent magnets are as follows: (1) theirproduction usually gives low yields at high cost and (2) theircoercivity is not as competitive as that of hard magneticmaterials with high magnetocrystalline anisotropy. To circum-vent these issues, we fabricate here composites based on NWs asthe minority phase and a majority hard ferrite phase. We selectedthe 50 nm NWs for the composites by discarding the 100 nmFeCo NWs because of their low H\ncand the 30 nm ones because\nof their lower Ms.\nFigure 5 a shows the magnetization curves of oriented\nSrFe 12O19(strontium ferrite, SFO) −NW (50 nm) composites\nwith di ﬀerent NW concentrations between 10 and 40 wt %.\nFigure 5 b summarizes the evolution of Hc,Ms, and Mras a\nfunction of the NW concentration. As expected, given that forSrFe\n12O19,Ms=7 2A m2/kg;51the Msof the composites\nincreases, while Hcdecreases with the NW content, reaching Ms\n= 100 Am2/kg and Hc= 104 kA/m for 40 wt %. The identical\nmeasurements on composites fabricated with 100 nm NWspresented in Figure S2 of the SI con ﬁrm that the NW content\nneeded for a substantial increase in magnetization leads todetrimental coercivity decay.\nWith the goal of assessing the real potential of these\ncomposites as permanent magnets, a real dense compositebonded magnet was fabricated. Based on the results from Figure\n5, the composition 70% SrFe\n12O19−30 wt % FeCo NW (50 nm)\nwas selected. A SEM image of the composite powder obtainedafter mixing is shown in Figure 6 a, where the nanowires are seen\nto be dispersed in close proximity with SrFe\n12O19particles. The\nresulting oriented bonded magnet, schematized on Figure 5 b, is\nthus composed of aligned vertical FeCo NWs embedded in amatrix of oriented SrFe\n12O19(SrM) particles. For comparison, a\npure SrM bonded magnet was fabricated in identical conditionsto the composite magnet.\nIn the magnetization curves of Figure 6 c, we observe that the\ncomposite magnet presents a larger M\nsandMrand a decreased\nHc. The magnetic parameters are shown in Table 1 . Given that\nthe theoretical saturation magnetization for SrM is 72 Am2/kg\nand the Mrfor 50 nm FeCo NW is 120 Am2/kg, the 20% increase\ninMris in agreement with a linear combination of the Mr’so f\nboth materials. However, given the existence of magnetostaticinteracting ﬁelds between all particles, this agreement may just\nbe a coincidence as the exact magnetic state of each phase isunknown. Interwire distances may be larger in the compositewith respect to the pure NW sample, which reduces interwiredemagnetizing ﬁelds, leading to a larger M\nrfrom the NW\ncontribution.36,37,48At the same time, these ﬁelds may be\ninducing reversible switching at the zero external ﬁeld in some of\nthe neighboring ferrite particles, reducing their impact on Mr.33\nThis complex nature of the reversal process of magnetic\nmaterials, particularly in multiphase systems, also makes Hc\nvalues di ﬃcult to predict. It is important to remark that the\nmixing of the two phases was performed by mild sonication, andsince no temperature or high-energy milling was employed, wedo not expect any exchange coupling to take place between bothphases. Nevertheless, based on the coercivity values of thepowder composites in Figure 5 ,t h e H\ncobserved in the\ncomposite magnet, Hc= 227 kA/m, is much higher than the\none measured for the equivalent composite powders, Hc= 131\nkA/m. In addition, the bonded composite magnet presents a\nsigniﬁcant squareness in the magnetization curve, which is not\ncommon in hard −soft composites, particularly in the absence of\ninterparticle exchange coupling.\nTo further understand this behavior, we plot in Figure 6 d the\nderivative of magnetization vs the applied ﬁeld for the FeCo NW\npowder, the composite powder, and the bonded magnet. Themaxima in the derivative reveal information about the mainreversal events in the samples.\n52The pure FeCo NW sample\npresents a relatively narrow maximum at 36 kA/m, as expectedsince the narrow distribution of NW diameters and sizes shouldentail a relatively narrow distribution of coercivities andinteraction ﬁelds.\n33The oriented composite powder presents\ntwo broader maxima, at 57 and 350 kA/m. Given that thesevalues are relatively close to the coercivities of the FeCo NW andthe ferrite, respectively, we interpret them as associated with themagnetization reversal of each separate phase. In the bondedmagnet, a single broad maximum is observed centered at 215kA/m, with a small shoulder at 92 kA/m.\nThe e ﬀective anisotropy in a pure NW system is expected to\nbe dominated by the competition between magnetostatic terms:\n(1) shape anisotropy that favors M\nrandHcand (2) interwire\ndemagnetizing ﬁelds that hinder them.48An increased packing\nfraction brings the NW closer and is expected to enhanceinterwire demagnetizing ﬁelds. In the composite magnet, the\nmagnetostatic interactions balance and thus understanding thereversal process is even more complex. On the one hand, theproximity of the platelet-shaped hexaferrite particles modi ﬁes\nthe internal ﬁelds depending on their exact shape and\ngeometrical arrangement. On the other hand, as explainedabove, the presence of ferrite particles may be reducing interwiredemagnetizing ﬁelds with respect to the pure NW sample. More\nsophisticated characterization methods are needed to fullyunravel this balance and the speci ﬁc reversal mechanisms and\nswitching events.\n33,36−39In any case, the derivatives in Figure 6 d\nsuggest that the internal ﬁelds and the particle arrangement\ncreated by the hard ferrite particles support the initial single-domain state inside the wires, which is likely responsible for thehigh squareness and H\ncof the composite magnet.Table 1. Magnetic Properties of the Composite Bonded Magnet and a Pure Ferrite Reference Magnet\noriented samples Hc(kA/m) Mr(Am2/kg) Ms(Am2/kg) BHmax(kJ/m3)\ncomposite magnet (30 wt % NW −70 wt % SrM) 227 24 28 0.46\nferrite magnet (100 wt % SrM) 242 20 24 0.31ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519847\nBoth magnets were measured to have the same density (1.7 g/\ncm3). Using this value, the energy product (BH max) was obtained\nand is presented in Table 1 . A 20% remanence increase in\ncombination with the good squareness of the magnetizationcurve leads to a BH\nmaxvalue of the FeCo NW composite that is\n48% higher than that of the pure ferrite magnet. It is important toremark that the BH\nmaxvalues are considerably lower than those\nof commercial bonded magnets due to the relatively lowdensities of the bonded magnets fabricated here. As there wasonly enough FeCo NW powder for one sample, very mildprocessing conditions were employed to ensure the survival ofthe specimen. This led to low densities. The proof of concept isin any case perfectly valid since the reference sample wasprepared identically.\nFrom what we learned from comparing the hysteresis loops of\ncomposite powders and magnets with equal compositions\n(Figure 6 d), an even greater betterment of the magnetic\nproperties, in this case, H\ncand squareness of the loop, would be\nexpected for a composite magnet with a higher density whencompared to its corresponding pure SFO reference sample. It isremarkable as well that this increase in magnetic performance isachieved without the need for exchange coupling between bothphases, and just as a consequence of the competitive propertiesof the FeCo NWs and the internal magnetostatic interactions,which simpli ﬁes composite fabrication and processing. Future\nwork will include magnetic measurements to further understandthe reversal mechanisms and interaction ﬁelds inside the\ncomposite magnet to optimize particle geometries and arrange-ments.\n38\n■CONCLUSIONS\nIn summary, we have fabricated high-remanence composites\ncomposed of FeCo nanowires and Sr-hexaferrite commercial\npowders. Soft FeCo NWs with controlled diameters 30, 50, and100 nm and at least 6 μm long exhibit high remanence under\nmagnetic orientation. Selecting 50 nm as the optimal diameter, abonded magnet composed of 70 wt % strontium ferrite −30 wt %\nFeCo NWs is produced and compared to a pure ferrite referencemagnet. The observed 20% increase in remanence together witha mild decrease in coercivity, a consequence of the magneto-dipolar interaction between ferrite particles and FeCo NWs,yield an enhancement of the energy product of 48% in thecomposite magnet compared to the pure ferrite magnet. Thisextremely promising result brings forward nanowire −ferrite\ncomposites as a clear candidate for gap magnets that could ﬁll\nthe void between ferrites and rare-earth magnets. The mainbottleneck toward industrial implementation at this stage is thelow yield and large production cost. Strategies toward solvingthe issue are being currently investigated.\n■METHODS\nFeCo NWs were prepared by template-assisted electrochemical\ndeposition using two di ﬀerent nanoporous templates. On the one\nhand, commercial polycarbonate nanoporous membranes were used,\nwith three di ﬀerent diameters (30, 50, and 100 nm), supplied by\nSterlitech. On the other hand, we have fabricated alumina templates,\nwith 50 nm diameter pores, following a one-step anodization process.31\nIn this process, high-purity Al foils (99.999%) were anodized at 40 V in\n0.3 M C 2H2O4solutions at 25 °for 7 h. The nanopore diameter was\nenlarged by etching treatment under phosphoric acid solution (5%) at\n50°C for 40 min.\nElectrodeposition was carried out in a three-electrode electro-\nchemical cell in an Ecochemie Autolab PGSTAT potentiostat using a Ptmesh as a counter electrode and an Ag/AgCl (3 M NaCl) electrode as areference electrode. Before electrodeposition, a thin Au ﬁlm was\nthermally evaporated on one side of the membrane to act as a working\nelectrode. The electrolyte was composed of CoSO\n4(0.09 M) as a Co2+\nsource, FeSO 4(0.1 M) as an Fe2+source, and H 3BO 3(0.4 M) as an\nadditive. All chemicals were of analytical grade and were used withoutfurther puri ﬁcation and mixed in deionized water. The pH was adjusted\nto 2.7 using 10% vol. H\n2SO4. FeCo NWs were grown at RT under a\nconstant voltage of −1.1 V. After growth, the Au layer was removed\nusing a 0.1 M I 2and 0.6 M KI solution. Then, the polycarbonate\nmembrane was dissolved using cycles of sonication in dichloromethane,acetone, and ethanol. Alumina templates were dissolved using a 0.4 M\nH\n3PO4and 0.2 M H2CrO4solution. The released NWs were left to dry\nat RT in ethanol to recover a dry powder. In addition, to upscale the\nproduction yield at lower costs, polycarbonate membranes were\nsubstituted by alumina templates with a larger pore density thanpolycarbonate. Alumina nanoporous templates with a pore of 50 nm\nwere fabricated following a redesigned and faster RT procedure. The\ndetails of this process will be published elsewhere. Sr-hexaferrite\nmicrosized powders were supplied by Max Baermann GmHb.\nThe morphology and composition of the NWs were measured with a\ntransmission electron microscope (TEM) JEOL JEM 2000FX. Thediameter of the wire and its core were obtained by measuring at three\ndiﬀerent regions of the wire in the TEM image and by averaging the\nvalue. Further structural characterization was performed using\nsecondary electron images from ﬁeld-emission scanning electron\nmicroscopy, ﬁeld-emission scanning electron microscopy (FE-SEM,\nHitachi S-4700), and scanning electron microscopy (TM-1000) at an\nacceleration voltage set at 15 kV. Structural analysis was performed by\nX-ray di ﬀraction (XRD) with a Bruker D8 di ﬀractometer using Cu K α\nradiation ( λ= 1.5418 Å) and a Lynxeye XE-T detector. Subsequent\nRietveld analysis of the XRD data was performed by FullProf Suite.\n53\nThermogravimetric analysis (TGA) in air was employed to\ndetermine the oxidation stage of the wires between RT and 900 °C\nusing a TA Instruments Q50 system.\nFor the theoretical study of magnetization reversal in fabricated\nFeCo NWs, we employ a micromagnetic algorithm initially developedfor the simulation of the magnetization distribution of magnetic\nnanocomposites.54,55The magnetization distribution in an isolated\ncylinder with length Land diameter Dunder the in ﬂuence of the\nexternal magnetic ﬁeld is simulated by taking into account the four\nstandard contributions to the total magnetic energy: external ﬁeld,\nmagnetic anisotropy, exchange, and dipolar interaction. The following\nmaterial parameters (typical for FeCo alloys) were used for simulating\nthe magnetic nanowires: saturation magnetization, Ms= 220 Am2/kg;\ncubic magnetocrystalline anisotropy, Kcub= 20 kJ/m3with easy axis\nalong the ⟨100⟩directions; and exchange sti ﬀness constant, Abulk=2 1\npJ/m.46NWs under study have the D= 30 nm and Lvarying in the\nrange 30 −400 nm and are discretized in the nonregular mesh with the\ntypical mesh element size of 3 nm. One of the cubic crystallographic\naxes and the direction of the external magnetic ﬁeld coincide with the\nFeCo NW rotation axis; the directions of the other two axes areidentical for all mesh elements. Open boundary conditions were applied\nin all simulations. Note that every set of parameters employs di ﬀerent\nirregular mesh realizations, resulting in slightly di ﬀerent coercive ﬁelds.\nFeCo NW −strontium ferrite mixture composites in the powder form\nwere fabricated by mixing a few milligrams of each phase at 10, 20, 30,\nand 40 wt % of FeCo NWs and sonicating in an ethanol bath for 5 min.The solution was left to dry and the composite powder was recovered\n(approximate mass around 1 −3 mg).\nTo fabricate a consolidated bonded magnet, 90 mg of NW ( D=5 0\nnm) powder was fabricated and mixed by sonication in ethanol with\n210 mg of strontium ferrite (SrFe\n12O19). The resulting composite\npowder was dried and added to a mix of epoxy resin and hardener in a1:1 ratio. The mixture was homogenized and added to a mold while\napplying an external magnetic ﬁeld of 0.4 T to orient the magnet.\nSubsequently, constant uniaxial pressure was applied for 24 h. On the\nother hand, using 300 mg of ferrite powder, a pure ferrite reference\nbonded magnet was fabricated for comparison purposes.\nMagnetic characterization of the powders was carried out at RT by\nmeans of a Lakeshore vibrating sample magnetometer (VSM), with aACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519848\nmaximum applied ﬁeld of 1.8 T, and a Quantum Design MPMS SQUID\nmagnetometer with a maximum applied ﬁeld of 5 T. Each of the\nnanowire powders, corresponding to the three di ﬀerent diameters, and\nthe resulting composites with strontium ferrite were individually\ndispersed inside a bonding glass matrix (509 Crystalbond) and left toconsolidate under an applied magnetic ﬁeld. Per our alignment\nprocedure, the mass of powder introduced in the bonding glass cannot\nbe accurately determined. For this reason, nonoriented samples were\nmeasured to calibrate the magnetization per sample mass. Besides, a\nhomemade VSM, whose maximum magnetic ﬁeld is 1.3 T, was used to\ncharacterize the magnetic properties of the bonded magnets.56\n■ASSOCIATED CONTENT\n*sıSupporting Information\nThe Supporting Information is available free of charge at\nhttps://pubs.acs.org/doi/10.1021/acsanm.0c01905 .\nSEM images of the 30 and 100 nm diameter NW powders\n(Figure S1), magnetization curves of ferrite −NW\ncomposites made with 100 nm diameter FeCo NWs(Figure S2), and magnetization curves of the 50 nm FeCoNWs measured several days and 1 year after exposure toair (PDF)\n■AUTHOR INFORMATION\nCorresponding Author\nA. Quesada −Instituto de Cera ́mica y Vidrio (CSIC), Madrid\n28049, Spain; orcid.org/0000-0002-6994-0514 ;\nEmail: a.quesada@icv.csic.es\nAuthors\nJ. C. Guzma ́n-Mínguez−Instituto de Cera ́mica y Vidrio (CSIC),\nMadrid 28049, Spain; orcid.org/0000-0003-1017-0225\nS. Ruiz-Go ́mez−Departamento de Fi ́sica de Materiales,\nUniversidad Complutense de Madrid, Madrid 28040, Spain\nL. M. Vicente-Arche −Instituto de Cera ́mica y Vidrio (CSIC),\nMadrid 28049, Spain; Unite ́Mixte de Physique, CNRS, Thales,\nUniversite ́Paris-Saclay, 91767 Palaiseau, France\nC. Granados-Miralles −Instituto de Cera ́mica y Vidrio (CSIC),\nMadrid 28049, Spain; orcid.org/0000-0002-3679-387X\nC. Ferna ́ndez-Gonza ́lez−IMDEA Nanociencia, 28049 Madrid,\nSpain\nF. Mompea ́n−Instituto de Ciencia de Materiales de Madrid\n(CSIC), Madrid 28049, Spain\nM. Garc ía-Herna ́ndez−Instituto de Ciencia de Materiales de\nMadrid (CSIC), Madrid 28049, Spain\nS. Erohkin −General Numerics Research Lab, 07745 Jena,\nGermany\nD. Berkov −General Numerics Research Lab, 07745 Jena,\nGermany\nD. Mishra −Institute of Materials for Electronics and Magnetism-\nCNR, 43124 Parma, Italy; Department of Physics, IndianInstitute of Technology Jodhpur, Jodhpur 342037, Rajasthan,India\nC. de Julia ́n Ferna ́ndez−Institute of Materials for Electronics\nand Magnetism-CNR, 43124 Parma, Italy;\norcid.org/0000-\n0002-6671-2743\nJ. F. Ferna ́ndez−Instituto de Cera ́mica y Vidrio (CSIC), Madrid\n28049, Spain; orcid.org/0000-0001-5894-9866\nL. Pe ́rez−Unite ́Mixte de Physique, CNRS, Thales, Universite ́\nParis-Saclay, 91767 Palaiseau, France; IMDEA Nanociencia,\n28049 Madrid, Spain; orcid.org/0000-0001-9470-7987\nComplete contact information is available at:\nhttps://pubs.acs.org/10.1021/acsanm.0c01905Notes\nThe authors declare no competing ﬁnancial interest.\n■ACKNOWLEDGMENTS\nWe would like to thank Dr. Vi ́ctor Fuertes for his advice on the\nprocessing of the bonded magnets. This work is supported bythe Spanish Ministerio de Economi ́a y Competitividad y\nMinisterio de Ciencia e Innovacio ́n (Project Nos. MAT2017-\n86450-C4-1-R, MAT2015-64110-C2-1-P, MAT2015-64110-C2-2-P, MAT2017-87072-C4-2-P, RTI2018-095303-A-C52,and FIS2017-82415-R) and by the European Commissionthrough Project H2020 (No. 720853; AMPHIBIAN). C.G.-M.acknowledges ﬁnancial support from MICINN through the\n“Juan de la Cierva ”Program (FJC2018-035532-I). A.Q.\nacknowledges ﬁnancial support from MICINN through the\n“Ramo ́n y Cajal ”Program (RYC-2017-23320). The work also is\nfunded by the Regional Government of Madrid (Project S2018/NMT-4321; NANOMAGCOST).\n■REFERENCES\n(1)Permanent Magnets Market Size, Share &Trends Analysis Report By\nMaterial (Ferrite, Neodymium Iron Boron, Aluminum Nickel Cobalt), By\nApplication (Industrial, Energy), By Region, And Segment Forecasts, 2020 -\n2027 ; Grand View Research, Inc.. https://doi.org/978-1-68038-058-3,\n2020.\n(2) Lewis, L. H.; Jime ́nez-Villacorta, F. Perspectives on Permanent\nMagnetic Materials for Energy Conversion and Power Generation.Metall. Mater. Trans. A 2013 ,44,2−20.\n(3) Coey, J. M. D. Permanent Magnets: Plugging the Gap. Scr. Mater.\n2012 ,67, 524−529.\n(4) Sander, D.; Valenzuela, S. O.; Makarov, D.; Marrows, C. H.;\nFullerton, E. 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A Simple Vibrating Sample Magnetometer\nfor Macroscopic Samples. Rev. Sci. Instrum. 2018 ,89, No. 034707.ACS Applied Nano Materials www.acsanm.org Article\nhttps://dx.doi.org/10.1021/acsanm.0c01905\nACS Appl. Nano Mater. 2020, 3, 9842 −98519851\n"    },    {        "title": "2309.09754v1.Computational_Exploration_of_Magnetic_Saturation_and_Anisotropy_Energy_for_Nonstoichiometric_Ferrite_Compositions.pdf",        "content": " \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 1 \nComputational Exploration of Magnetic Saturation \nand Anisotropy Energy for Nonstoichiometric \nFerrite Compositions  \n \nVenkata Rohit Punyapu1,2, Jiazhou Zhu1,3,∆, Paul Meza -Morales1,4,∆, Anish \nChaluvadi1,5,6, O. Thompson Mefford5,7, and Rachel B. Getman1,2,* \n \n1 Department of Chemical & Biomolecular Engineering, Clemson University, Clemson, SC 29634, USA  \n2 Current affiliation: William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State \nUniversity, Columbus, OH 432 10, USA  \n3 Current affiliation: Suzhou Novartis Technical Development Co., Ltd, Changshu, China  \n4 Current affiliation: Intel Corporation, Hillsboro, Oregon, 97124 , USA  \n5 Department of Materials Science and Engineering, Clemson University, Clemson, SC 29634, USA \n6 Current affiliation: Department of Materials Science and Metallurgy, University of Cambridge, Cambridge, \nCambridgeshire CB3 0FS, UK  \n7 Department of Bioengineering, Clemson University, Clemson, SC 29634, U SA \n∆ Equal authorship  \n* Corresponding  author:  getman.11@osu.edu  \n \n1.0 Introduction . The magnetite -derived ferrites \ncomposed of metallic elements in some \ncombination such as Fe, Mn, Ni, Co, Cu and Zn \nhave been widely studied for their structure and \nmagnetic properties  [1–6]. Typical ferrites have \nspinel -type (normal, inverse) crystal structures with \nO2– anions packed in a face -centered cubic (fcc ) \narrangement, such that there are two types of sites \nbetween them, i.e., tetrahedrally and octahedrally \ncoordinated sites (see Figure 1). The general \nempirical formula for the stoichiometric class of \nferrites is M xFe3-xO4, where M can be different \nsubstit uent metals (e.g., Mn, Ni, Co, Cu, Zn and \nother divalent metal cations) and 0  x  3. Non -\nstoichiometric ferrites, i.e., materials with the \ngeneral formula M1 xM2 yFe3–x–yO4, where M1 and M2 can be Mn, Ni, Co, Cu and/or Zn and  0  x  1 \nand y = 1 – x, offer  even greater compositional \ndiversity.  \nFerrite nanoparticles are widely used in the \ncores of transformers, antenna rods, \nelectromagnets, and magnets used in imaging \napplications  [7–11]. Another well -studied \napplication is magnetically mediated energy \ndelivery, which has most often been applied to \nbiomedical devices (e.g., heating a cell via \nmagnetic hyperthermia, i.e., MagMED) and more \nrecently been applied to catalysts (e.g., supplying \nthe heat needed to break and form chemical bonds \nvia magnetic induct ion heating; i.e., MIH)   [12–16]. \nSpecifically, an oscillating magnetic field is \napplied, and hysteresis in the magnetic properties of Abstract . A grand challenge in materials research is identifying the relationship between composition and \nperformance. Herein, we explore this relationship for magnetic properties, specifically magnetic saturation (M s) \nand magnetic anisotropy energy (MAE) of ferrites. Ferrites are materials derived from magnetite (which has the \nchemical formulae Fe 3O4) that comprise metallic elements in some combination such as Fe, Mn, Ni, Co, Cu and \nZn. They are used in a variety of a pplications such as electromagnetism, magnetic hyperthermia, and magnetic \nimaging. Experimentally, synthesis and characterization of magnetic materials is time consuming. In order to create \ninsight to help guide synthesis, we compute the relationship betwe en ferrite composition and magnetic properties \nusing density functional theory (DFT). Specifically, we compute M s and MAE for 571 ferrite structures with the \nformulae M1 xM2 yFe3–x–yO4, where M1 and M2 can be Mn, Ni, Co, Cu and/or Zn and 0  x  1 and y = 1 – x. By \nvarying composition, we were able to vary calculated values of M s and MAE by up to 9.6 ×105 A m–1 and 14.08×105 \nJ m–3, respectively. Ou r results suggest that composition can be used to optimize magnetic properties for \napplications in heating, imaging, and recording. This is mainly achieved by varying M s, as these applications are \nmore sensitive to variation in M s than MAE.   \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 2 \nthe nanoparticle during oscillation r esults in \nconversion of magnetic energy into thermal energy \n[8,11 –13]. The heat generated in hysteretic losses is \ndue to Nèel and Brown relaxations, which are \ndetermined by the magnetic saturation (M s), \nmagnetic anisotropy energy (MAE), and the size of \nthe nanoparticle  [17,18] . M s and MAE, in turn, are \ndetermined by the particle composition  [19–22]. \nA benefit to ferrites is that the compositions \ncan be tuned. Indeed, multiple groups have \ninvestigated the influence of ferrite composition on \nperformance for magnetic hyperthermia  [17,23] . \nSpecifically, these groups showed how varying \ncomposition results in values of M s and MAE that \nvary by up to a full order of magnitude. They further \nestimated the potential for heat generation as a \nfunction of size and composition, showing that this \nvalue could be varied by two orders of magnitude. \nComposition is important in other applications as \nwell. For example, Co, Ni, Fe and Cu ferrites \nperform well for catalysis due to large hysteresis \nlosses  [12–14,16,21,24] , while Zn, Co and Mn \nferrites are often used in magnetic resonance \nimaging (MRI) because of the resulting high Ms \nfrom their substitution   [4,11,25] . An \nunderstanding of how composition influences \nmagnetic properties is hence imperative to \nmaximizing performance for the variety of \napplications that utilize ferrites and other magnetic \nmaterials.  \nWhile M s and MAE as well as other \nmagnetic properties  can be measured \nexperimentally, the process is laborious; often requiring multiple attempts at syn thesis to achieve \nthe expected structure in addition to state -of-the-art \nmeasurement techniques to learn the magnetic \nproperties  [26–28]. Further, general rules linking \nmagnetic properties to composition do not yet \nexist  [1,17,20,25,29,30] . Filling thi s knowledge \ngap would greatly reduce the time and money \nrequired to design magnetic materials and devices \nfor a wide range of applications; however, it would \nbe impossible to accomplish this with experiments \nalone.  On the other hand, computational \napproach es can provide estimates of magnetic \nproperties relatively quickly. In such approaches, \nmagnetic properties of model structures are \ncomputed with quantum mechanics. While these \nmodel structures are simplifications of the \nstructures used in real -life applic ations – which is \nrequired for computational feasibility – they \nprovide useful estimates of the magnetic properties \nof a given composition and are vastly more efficient \nat doing so than experiments. A database of \nmagnetic material compositions and their \nassociated magnetic properties would greatly \nfacilitate design of magnetic materials for a variety \nof applications.   \nTo this end, in this work we generate a \ndatabase of ferrite compositions and their \nassociated magnetic properties. Specifically, we \ncompute values of M s and MAE for singly and \ndoubly substituted  non-stoichiometric ferrites using \ndensity functional theory (DFT). We investigate \n571 total compositions and create an open access \ndatabase that includes each composition’s specific \ncrystal structure (either normal or inverse spinel), \ncalculated M s, and ca lculated MAE. We further \nprovide insight about the influence of composition \non M s and MAE, showing  that these values can vary \nby up to 103 A m-1 and 106 J m-3, respectively.  \n \n2.0 Methodology   \n \n2.1 Ferrite Model Setup.  The ferrite model \nemployed herein is based on the calculated bulk \nunit cell of magnetite  \n(Fe 3O4). We specifically employ a unit cell with \nspace group of F d̅3m. To create models with \ndiverse compositions, we use eight repeats of the \nformula unit, giving a base stoichiometry of Fe 24O32 \n(Figure 1). This model comprises eight Fe  ions in \ntetrahedral sites (purple tetrahedra in Figure 1) and  \nFigure  1. Left: Polyhedral representation of a bulk \nferrite structure with stoichiometry Fe 24O32. Purple = \ntetrahedral sites, orange and gray = octahedral sites. \nRight: Sites where substitutions were considered in this \nwork.    \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 3 \nsixteen  Fe ions in octahedral sites (orange and gray \noctahedra in Figure 1). Up to eight Fe ions are \nsubstituted with Mn, Ni, Co, Cu, and/or Zn in the \npurple and orange sites labeled 1 through 8 in \nFigure 1. We consider both singly (e.g., Cu 1Fe23O32, \nMn 7Fe17O32) and doubly substituted (e.g., \nMn 2Co2Fe20O32, Ni 1Cu2Fe21O32) ferrites in this \nwork. Singly substituted ferrites include \nsubstitutions involving Mn and Cu and are denoted \nFeMn and FeCu for simplicity. Similarly, doubly \nsubstituted ferrites include combinat ions of Co and \nCu, Ni and Zn, Co and Ni, Mn and Ni, Cu and Ni, \nMn and Co, and Co and Zn, and are denoted \nFeCoCu, FeNiZn, FeCoNi, FeMnNi, FeCuNi, \nFeMnCo, and FeCoZn, respectively. The number of \neach general composition considered in this work is \nprovided in  Table 1. In total, 571 structures are \nconsidered, of which 99 are singly substituted and \n472 are doubly substituted. Prior literature suggests \nthat substitution into the same type of site (i.e., \neither tetrahedral or octahedral) but different \nlocation wit hin the crystal lattice (e.g., different \nnumbered tetrahedron or octahedron in Figure 1) \nhas an influence on magnetic properties  [22]. \nHence, we also consider structures that have the \nsame composition and substitution into the same \ntype of site, but with  the metal ions substituted into \ndifferent tetrahedra or octahedra (e.g., \nCu1tet,site1Fe23O32 and Cu 1tet,site4Fe23O32). In this way, \nout of 571 structures, there are 204 unique \ncompositions.  \nIn general, ferrites can crystalize in either \nthe normal (e.g., [M1 x2+M2 y2+]tet[Fe3+]oct2O4) or \ninverse spinel structure (e.g., [Fe3+]tet[M1 x2+M2 yFe3+]oct2O4) [5,6] . We consider \nboth structures for each composition. Magnetic \nproperties are reported for the structure that gives \nthe lowest electronic energy in DFT. In rare cases, \nthe electronic structure of one structure (i.e., either \nnormal or inverse spinel) did not converge. In these  \ncases, magnetic properties are reported for the \nstructure that converged. The specific models \nemployed in this work are available in the ioChem -\nBD database  [31] along with their calculated M s \nand MAE.  \n \n2.2 Magnetic property calculations . Ms is \ncomputed as  \n            \n Ms= Total  magnetic  moment  ×𝜇𝐵\nunit  cell  volume       Eq. (1) \n \nwhere the total magnetic moment is the total \nnumber of unpaired electrons in the unit cell \ncalculated in DFT, and 𝜇𝐵 is the Bohr magneton \nequal to 9.27 × 10−24 A m2. MAE is calculated as \nthe difference in energy between the hard axis and \nthe easy axis  [32], i.e.,  \n \nMAE = 𝐸hard − 𝐸easy                      Eq. (2) \n            \nwhere E is the electronic energy calculated in DFT. \nThe [0,0,1], [1,0,0] and [0,1,0] crystallographic \ndirections are evaluated as the easy and hard axes \nfor each composition. These directions were chosen \nsince test calculations on the [0,1,1], [1,0,1], [1,1,0] \nand [1,1,1] directions often resulted in electronic \nenergies significantly more positive than the General \nComposition  Total \nStructures  Ms ×105 [A m-1] MAE ×105 [J m-3] Crystal Structure  \nFeCu  24 1.2 – 9.4 1.3 – 5.5 62.5% Normal spinel  \nFeMn  75 2.0 – 5.3 0.1 – 6.4 62.5% Inverse spinel  \nFeCoCu  69 1.2 – 9.6 0.6 – 11.4 61.2% Inverse spinel  \nFeMnCo  187 3.3 – 7.4 0.07 – 8.5 66.6% Normal spinel  \nFeNiZn  34 0.3 – 8.9 *  0.05 – 3.7 93.5% Inverse spinel  \nFeMnNi  28 2.5 – 7.5 0.1 – 6.8 76.1% Inverse spinel  \nFeNiCo  47 0.6 – 4.8 0.02 – 14.1 93.5% Inverse spinel  \nFeNiCu  78 1.2 – 7.5 0.1 – 4.3 92% Inverse spinel  \nFeCoZn  29 0.4 – 8.8 *  0.06 – 11.1 64.5% Inverse spinel  Table  1. General compositions considered in this work, along with their range of M s, range of MAE, and most \nprominent crystal structure.  \n \n*Not included here is the M s of Zn 8Fe16O32 which has a M s of 0.04 A m–1.  \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 4 \n[0,0,1], [1,0,0] and [0,1,0] directions, suggesting \nthat the [0,0,1], [1,0,0] and [0,1,0] directions are \nmore reliable for large -scale DFT calculations. \nAmong t hese three directions, the direction that \ngave the lowest electronic energy was taken as the \neasy axis, and the direction that resulted in the \nhighest electronic energy was taken as the hard \naxis. Calculated easy and hard axis directions for \nsingly substit uted ferrites partially agree with prior \nresults. For instance, in Co 8Fe16O32, the easy axis \nwas found to be [100] in agreement with our \nresults  [33], while in Ni 8Fe16O32, the easy axis was \nfound to be [111]  [34], which does not match with \nour results  (which determined the easy axis to be  \n[010]). However, we find that MAEs calculated in \nthis work generally follow experimental trends in \ncases where such data is available experimentally. \nFurther details are provided in SI Section S7.  \n \n2.3 Data Visualization . The values of M s and MAE \nas functions of composition for doubly substituted \nferrites are presented as contour plots. These plots \ninterpolate between explicitly calculated points in \norder to create a continuous colormap. Each \ncolormap is base d on 18 – 31 explicitly calculated \ndata points (see SI Section S6 for a sample plot with \nonly data points). This is done using the OriginPro \nsoftware  [35] using the data boundary algorithm \nwithout smoothening. A total of 27 plots are \ngenerated for FeCoCu , FeNiZn, FeCoNi, FeMnNi, \nFeCuNi, FeMnCo FeCoZn, FeCu and FeMn, i.e., 9 \nfor crystal structure, 9 for M s and 9 for MAE.  \n \n2.4 Density Functional Theory Calculations .  \nDFT calculations are performed using the Vienna \nAb initio Simulation Package (VASP)  [36,37] . \nBoth collinear (i.e., all spins are aligned along the \n[0,0,1] direction) and non -collinear (i.e., spins are \naligned along the [1,0,0], [0,1,0] and [0,0,1] \ndirections) calculations are performed. Non -\ncollinear calculations are always started from the \nwavefunction and charge density generated from a \ncollinear calculation on the same system.  Initial \nguesses for magnetic moments of the Fe, Mn, Co, \nNi, Cu, and Zn cations are based on experimental \nfindings by de Berg et al.  [38] and reported in SI \nSection S9. The DFT+U formalism  [30,39 –41] is \nemployed to capture the strong Coulombic \nrepulsion on 3d electrons and to prevent the \ndelocalization of electrons in these semiconducting \nmaterials. We specifically employ an effective U parameter, U eff, equal to U − J, where U and J are \nthe spherically averaged screened Coulomb and \nExchange energies, respectively  [42]. Values of U \nand J used to compute crystal structure and M s are \ntaken from the Materials Project Database  [43]. \nThese v alues are provided in SI Section S9. \nCalculation of MAE requires a more stringent value \nof J in order to capture the spin orbit \ninteraction  [30] and achieve the magnetic ground \nstate energies. We hence varied this value while \nholding values of U constant  at the values taken \nfrom the Materials Project Database  [43] and \ncompared the resulting MAE with values from \nexperiment  [7,44]  (see SI Section S10). These \ncalculations were specifically done for the \nstoichiometric ferrites, i.e., Fe 24O32, Mn 8Fe16O32, \nCo8Fe16O32, Ni 8Fe16O32, Cu 8Fe16O32, and  \nZn8Fe16O32. Resulting values of J were then used for \nthe corresponding metal cations when calculating \nMAE values for the non -stoichiometric ferrites. We \nfound that a value of U eff  of 1.5 eV resulted in \nvalues of MAE  that were in good agreement with \nexperiment for the stoichiometric ferrites. Hence, J \nvalues for MAE calculations were taken as 1.5 eV \n– U. \nIn all calculations, electron exchange and \ncorrelation are treated using the Perdew -Burke -\nErnzerhof (PBE) form of t he generalized gradient \napproximation  [45], the energies of core electrons \nare simulated with projector augmented wave \n(PAW) pseudopotentials  [37,46]  up to a cut -off \nenergy of 550 eV, and spin polarization is turned \non. Gamma -centered k -point meshes of 4×4×4 are \nused to sample the first Brillouin zones. Validation \nof the cut -off energy and k -point mesh are provided \nin SI Section S10. Electronic structures are  \ncalculated self -consistently and considered to be \nconverged when the difference in energy between \nsubsequent iterations falls below 10–6 eV for MAE \ncalculations and 10–5 eV for everything else. During \ngeometry relaxations, all atom positions as well as \nthe unit cell shape and volume are allowed to relax. \nThis strategy is validated in SI Section S10. Unit \ncell relaxations are considered to be converged \nwhen the absolute values of the forces on all atoms \nare smaller than 0.02 eV/Å. Most unit cells become \nslightly non -cubic during relaxation; however, the \ndeviation from cubic is typically less than 0.3° . \nExamples of VASP input files are available in SI \nSection S9.  \n  \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 5 \n3.0 Results .  \n \n3.1 Crystal Structure.  Calculated crystal structure \npreferences are shown in Figures 2 and S1 and \ntabulated in Table 1. We find that the stoichiometric \nferrites Fe 24O32, Co 8Fe16O32, Ni 8Fe16O32, \nCu8Fe16O32, and Zn 8Fe16O32 crystalize in the inverse \nspinel structure, whereas Mn 8Fe16O32 crystallizes in \nthe normal spinel structure, in agreement with \nexperiments  [44,47 –53]. In general, we find that \nnon-stoichiometric ferrites largely crystalize in the \ninverse spinel structure (see Table 1). The \nexceptions are FeCu (Figure S1i) and FeMnCo \n(Figure 2f) which form mostly normal spinel \nstructures (FeCu at low Cu content and FeMnC o at \nhigh Mn content). Of the remaining compositions \nthat we investigated, FeNiCo forms inverse spinel \nstructures over the majority of compositional space \n(Figure 2a)  [52]. FeNiCu and FeNiZn also form \nmostly inverse spinel structures (Figure S1a and \nS1d) . The remaining compositions form a mixture \nof normal and inverse spinel structures, depending \non the composition (see Table 1 and Figures S1b, c \nand e). For example, FeMnNi forms a normal spinel \nstructure at high Mn content and inverse spinel \notherwise (F igure S1c), while FeCoZn and FeCoCu \nform inverse spinel structures at high Zn and Cu \ncontent and a mixture of inverse and normal spinel \notherwise (Figure S1b and S1e).  \n \n3.2 Saturation Magnetization . Calculated M s are \nshown in Figures 3 and S2 and tabulated in Table 1.  \nMs for the stoichiometric ferrites Fe 24O32, \nMn 8Fe16O32, Co 8Fe16O32, Ni 8Fe16O32, Cu 8Fe16O32, \nand Zn 8Fe16O32 are 4.8, 4.9, 3.6, 2.5, and 1.2, and \n0.4×10-6 × 105 A m–1, respectively, approximately \nfollowing the trends of the substituent cations \nthemselves determined experimentally [17,54] . M s \nfor most of the non -stoichiometric ferrites show \nlarge variations with composition, spanning more \nthan 5×105 A m–1. Exceptions to this are \ncompositions involving Mn, which have narrower \nranges of M s and hence cannot be as finely tuned as \nthe other compositions investigated in this work. \nComparing Figures S1 and S2, we  observe the \ncompositions that crystallize in the normal spinel \nstructure exhibit higher M s than compositions that \ncrystallize in the inverse spinel structure. For \nexample, values of M s in the largely inverse spinel \nregions of FeNiZn, FeCoCu, FeCoZn and FeMnNi \nare relatively low, whereas values of M s in the normal spin el regions of these compositions are \nhigher. In fact, the normal spinel regions of \nFeNiZn, FeCoCu and FeCoZn exhibit some of the \nhighest M s calculated in this work, with \ncompositions such as Co 4Cu1Fe19O32 and \nCo5Zn1Fe18O32 exhibiting M s of 9.6 and 8.8×105 A \nm–1, respectively. Conversely, the inverse spinel \nregions of these compositions exhibit some of the \nlowest values of M s calculated in this work, with \ncompositions such as Co 1Zn7Fe16O32 and \nCo6Zn2Fe16O32 (Figure 3b) exhibiting M s of 0.45 \nand 0.74×105 A m–1 respectively . These \nFigure 2.  Calculated crystal structures of FeNiCo (a)  and \nFeMnCo (b). Fe and substituent comp ositions span from  \n0 to 16 and 0 to 8, respectively, with the stoichiometric \nferrites represented at the vertices.  \n \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 6 \ncompositions hence show good promise for tuning \nMs through composition.  \n \n3.3 Magnetic Anisotropy Energy . Calculated \nMAE are shown in Figures 4 and S3 and tabulated \nin Table 1. MAE for the stoichiometric ferrites \nFe24O32, Mn 8Fe16O32, Co 8Fe16O32, Ni 8Fe16O32, \nCu8Fe16O32, and Zn 8Fe16O32 are 2.2, 0.1, 1.7, 1.3, \n1.3, and 0.6×105 J m-3, respectively. We observe \nthat similar to M s, MAE is also dependent on \ncomposition; however, most compositions span \nwithin the same order of magnitude (Figures 4 and \nS3). For example, FeCu and FeMn only show \nvariation in MAE when there is an almost equal \nnumber of substituent ions and Fe ions, possi bly \ndue to the resulting transformation of spinel \nstructure (Table 1 and Figures S1 and S3 parts h and \ni). Exceptions are FeNiCo, FeCoZn and FeCoCu, \npresented in Figure 4 and S3b, which span three \norders of magnitude. Comparing Figures S1 and S3, contrary to M s, we observe that compositions that \ncrystallize in inverse spinel result in higher MAE. \nThe largest MAE value of 13.6×105 J m–3 is found \nin FeNiCo (Figure 4a) in agreement with prior \nliterature  [55], and the lowest value is from \nZn8Fe16O32 of 0.06×1 05 J m–3 (Figure S3b).    \n \n4.0 Discu ssion .   Comparison of crystal structures, \nMs, and MAE calculated in this work with \nexperimental observation is provided in Tables 2 \n(specific compositions) and Section S1 (general \ncompositions). Several compositions, e.g., \nMnFe 2O4, CuFe 2O4, and CoFe 2O4 are in excellent \nagreement. Others, e.g., Co 4.8Ni3.2Fe16O32, show \nnotable differences. We have previously shown that \nthese differences can be attributed to the simplicity  \nof our models compared to real expe rimental   \nFigure 3.  Calculated M s of FeCoCu ( top) and FeCoZn \n(bottom ).  Fe and substituent compositions span from  0 \nto 16 and 0 to 8, respectively, with the stoichiometric \nferrites represented at the vertices.  \nFigure 4.  Calculated MAE (logarithmic scale ) of \nFeNiCo  (top) and FeCoCu  (bottom ). Fe and substituent \ncompositions span from  0 to 16 and 0 to 8, respectively, \nwith the stoichiometric ferrites represented at the \nvertices . \n \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 7 \nsystems  [56]. Specifically, our model systems are \nnearly pristine bulk structures, whereas  \nexperimental systems are particles with finite sizes, \nsurfaces, and defects, as well as different ligands, \ncrystallographic domains, etc.  [1,10,56,57] . \nFurther, experimental observation is an average \nvalue from a distribution of these properties, \nwhereas the calculations presented herein are for \nindividual structures. Unfortunately, it is n ot \npresently possible to model even one single \nnanoparticle with DFT, let alone a distribution for \nany one composition, and certainly not for a \ndistribution of compositions. Hence, at present, a  \nbetter use for these results is to learn how trends \ninfluenc e magnetic properties. To this end, Figure \n5 shows calculated M s and MAE for the various \ndoubly substituted non -stoichiometric ferrites.  \nFigure 5a shows that FeCoZn, FeCoCu, \nand FeNiCu can achieve relatively large ranges in \nMs from varying composition, whe reas FeNiZn, \nFeNiCo, FeMnNi, and FeMnCo tend to have more \nuniform M s. MAE for all compositions modeled in \nthis work varies by ~ 2 orders of magnitude. To \nunderstand how these compositions could be used \nin practice, Figure 5b illustrates optimal \ncombination s of M s and MAE for various \napplications. For example, our results suggest that \nFeMnCo and FeMnNi as well as some \ncompositions of FeCoZn will be optimal for \nmagnetic induction heating (evaluating these \nmaterials for toxicity for biomedical applications is \na concern that is beyond the scope of this paper). \nMRI requires materials with high M s [9,11,63]  and \ntherefore FeCoZn, FeCoCu, and FeNiCu are promising. Permanent magnets require modest M s \nand high MAE  [20,64]  and hence FeMnNi and \nFeNiCo  are promising. Ferrites with high M s and \nlow MAE could potentially replace rare earth \nmaterials in antennas  [10], and compositions such Specific Composition  Crystal structure  \nThis work / expt  Ms ×105 A m-1 \nThis work / expt  MAE ×105 J m-3 \nThis work / expt  \nFe24O32 Inverse / Inverse Ꝭ 4.8 / 4.6 Ꝭ 2.2 / 0.1 Ꝭ \nFe16Mn 8O32 Normal / Normal ꟼ 4.9 / 5.9 ꟼ 0.1 / 0.2 ꟼ  \nFe16Cu8O32 Inverse / Inverse Ꞵ 1.2 / 1.3 Ꞵ 1.3 / 1.4 Ꞵ \nFe16Co8O32  Inverse / Inverse § 3.7 / 3.5 § 1.7 / 2.2 £ \nFe16Ni8O32  Inverse / Inverse € 3.6 / 2.0 € 0.2 / 0.1 Ꝭ \nFe16Zn8O32 Inverse / Normal ¥ 0.04 / 0.09 ¥ 0.6 / N/A  \nCo1.6Cu6.4Fe2O32 Normal / N/A  1.8 / 2.3 Ω 7.4 / 3.6 Ω \nCo4.8Ni3.2Fe16O32 Inverse / N/A  1.7 / 3.2 ∑ 9.0 / N/A  \nꝬRef. [17] ; ꟼRef. [58] ; ꞴRef.  [59]; ΩRef.  [60]; £Ref.  [61]; ¥Ref.  [4]; §Ref.  [54]; €Ref.  [62]; ∑Ref.  [52]; N/A: \nnot available  \nFigure 5.  Top: Calculated Ms and MAE (log scale). \nDotted lines indicate the regions in the bottom graph. \nBottom:  Ranges of M s and MAE for various applications \nof magnetic materials.   \nTable 2. Comparison of DFT calculated crystal structure, M s, and MAE with experiment.  \n  \n____________________________________________________________________________  \n \nPunyapu , et al.   Page 8 \nas such as FeNiZn could achieve this. Transformers \nutilize magnetic induction and require minimal heat \nlosses  [65], i.e., high M s and low MAE. Based on \nFigure 5, FeCoZn and FeMnNi compositions could \nbe promising . \n \n5.0 Conclusions .  In summary, structural and \nmagnetic properties of non -stoichiometric bulk \nferrites in the formula M1 xM2 yFe3–x–yO4 where M1 \nand M2 = Mn, Ni, Co, Cu, and/or Zn and 0  x  1 \nand y = 1 – x have been investigated using DFT. \nThrough varying the composition, we found \nchanges in crystal structure (from normal to inverse \nspinel), which resulted in variations in the magnetic \nsaturation and magnetic anisotropy energy of \n9.6×105 A m–1 and 14.1×105 J m–3, respectively. We \nfound that magnetic properties are influenced by \ncomposition through their crystal structures, with \nnormal spinel compositions resulting in higher M s \nand invers e spinel compositions resulting in higher \nMAE. Our results suggest that composition can be \nused to optimize magnetic properties for \napplications in heating, imaging, and recording. \nThis is mainly achieved by varying M s, as these \napplications are more sensi tive to variation in M s \nthan MAE (Figure 5). Moving forward, developing \na strategy to achieve greater variation of MAE \nwould lead to greater technological applicability. \nOur calculations suggest that doubly substituted \nnon-stoichiometric ferrites based on Mn, Ni, Co, \nand Zn coul d achieve this. Comparison with \navailable experimental data suggests DFT \nunderpredicts M s and overpredicts MAE. As a \nmajor difference between experiments and our DFT \nsimulations is that our simulations were performed \non pristine bulk structures, whereas ex periments \nwere performed on nanoparticles comprising \ndifferent crystallographic domains and surfaces \nwith ligands and defects, these results suggest that \na way to maximize control over magnetic properties \nin practice is to minimize these effects in order t o \nhave the greatest control over MAE while using \ncomposition to control M s. This is a topic of \nongoing work.   \nAcknowledgments.  We thank Dr. Megan Hoover, \nProf. Lindsay Shuller -Nickles, Prof. Steven \nPellizzeri, Dr. Benjamin Fellows, and Dr. Zichun \n“Tony” Ya n for helpful discussions. This work was \npartly supported as part of the Center for \nProgrammable Energy Catalysis, an Energy \nFrontier Research Center funded by the U.S. \nDepartment of Energy, Office of Science, Basic \nEnergy Sciences at the University of Min nesota \nunder award #DE -SC0023464 (VRP & RBG: MAE \ncalculations, magnetic property analysis, \ncomparisons to experimental data, linking results to \npotential applications). We would also like to \nacknowledge support by Materials Assembly and \nDesign Excellence i n South Carolina (MADE in \nSC; VRP, JZ, PMM, AC: Model development, M s \nand crystal structure calculations), National \nScience Foundation award no. OIA -1655740 and \nGrants for Exploratory Academic Research \n(GEAR; OTM). We would also like to thank the \nsupport o f National Science Foundation award no. \nCBET -2146591 (OTM). This work was supported \nin part by the Center for Integrated \nNanotechnologies, an Office of Science User \nFacility operated for the U.S. Department of Energy \n(DOE) Office of Science (OTM).  \n \nKeyword s.  Structure function relationships, \nmagnetic inductive heating, magnetic imaging, \nmagnetic recording, spinel structures.  \n \nSupporting Information.  Additional figures \nillustrating preferred crystal structure, calculated \nMs, and calculated MAE; additional figures \nillustrating ranges of M s and MAE used in different \napplications; simulation input files; validations of \nthe computational model and methods.  This data \ncan be obtained by emailing the corresponding \nauthor until the data is officially published in the \npeer reviewed literature  (after which it will be \nfreely available on the publisher’s website) . \n \nReferences : \n \n[1] S. J. 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Janghorban, Soft \nMagnetic Composite Materials (SMCs) , J Mater \nProcess Technol 189, 1 (2007).  \n \n \n \n \n \n "    },    {        "title": "1205.4696v1.Resistive_and_ferritic_wall_plasma_dynamos_in_a_sphere.pdf",        "content": "arXiv:1205.4696v1  [physics.plasm-ph]  21 May 2012Resistive and ferritic-wall plasma dynamos in a sphere\nI. V. Khalzov,1,2B. P. Brown,1,2E. J. Kaplan,1,2N. Katz,1,2\nC. Paz-Soldan,1,2K. Rahbarnia,1,2E. J. Spence,3,2and C. B. Forest1,2\n1University of Wisconsin-Madison, 1150 University\nAvenue, Madison, Wisconsin 53706, USA\n2Center for Magnetic Self Organization in Laboratory and Ast rophysical Plasmas\n3Princeton Plasma Physics Laboratory,\nP.O. Box 451, Princeton, New Jersey 08543, USA\n(Dated: November 13, 2018)\nAbstract\nWe numerically study the eﬀects of varying electric conducti vity and magnetic permeability of\nthe bounding wall on a kinematic dynamo in a sphere for parame ters relevant to Madison plasma\ndynamo experiment (MPDX). The dynamo is excited by a laminar , axisymmetric ﬂow of von\nK´ arm´ an type. The ﬂow is obtained as a solution to the Navier -Stokes equation for an isothermal\nﬂuid with a velocity proﬁle speciﬁed at the sphere’s boundar y. The properties of the wall are taken\ninto account as thin-wall boundary conditions imposed on th e magnetic ﬁeld. It is found that an\nincrease in the permeability of the wall reduces the critica l magnetic Reynolds number Rmcr. An\nincrease in the conductivity of the wall leaves Rmcrunaﬀected, but reduces the dynamo growth\nrate.\n1Over the past decade, signiﬁcant eﬀort has been directed at the e xperimental demonstra-\ntion of dynamo action – self-excitation and maintenance of the magn etic ﬁeld in a ﬂowing\nelectrically conducting ﬂuid. A number of experiments with liquid metals have been con-\nstructed to test this phenomenon in various settings [1–6], and suc cessful observations of\ndynamo action have been reported in three of them [4–6]. Among oth er things, these ex-\nperiments revealed the critical importance of the magnetic proper ties of the ﬂow-driving\nimpellers. Namely, the von K´ arm´ an sodium experiment only self-sus tained a dynamo ﬁeld if\nthe impellers were ferromagnetic [6, 7]. In addition, the ﬁnite resistiv ity of the experimental\ncontainer is expected to be crucial for the dynamo instability – a situ ation similar to the re-\nsistive wall mode (RWM) in tokamaks [8]. Normally stable for the perfec tly conducting wall,\nthe RWM can become unstable if the wall has ﬁnite resistivity; in this ca se the instability\ndevelops on the wall’s resistive time scale. These facts initiated more t horough theoretical\nstudies of the eﬀect of the imposed boundary conditions on the dyn amo in experimentally\nrelevant models [9–16]. The studies established that there are no ge neral dependences of\ndynamo properties on conductivity and permeability of the boundar y; the dependences are\ndiﬀerent for diﬀerent models and ﬂows.\nThis circumstance motivates us to perform an analogous study for the Madison plasma\ndynamo experiment (MPDX, Fig. 1), currently under construction at the University of\nWisconsin-Madison. The experiment is aimed at investigations of fund amental properties\nof dynamos excited by controllable ﬂows of plasmas. Its original des ign was proposed in\nRefs. [17, 18] and conceptual features were successfully teste d in the plasma Couette ex-\nperiment (PCX) [19]. The experimental vessel is a sphere of 3 meter in diameter. An\naxisymmetric multicusp magnetic ﬁeld (created by 36 equally spaced r ings of permanent\nmagnets with alternating polarity) conﬁnes the plasma. The ﬁeld is loc alized near the vessel\nwall and a large volume of unmagnetized plasma occupies the experime nt’s core. An electric\nﬁeld applied across the multicusp ﬁeld drives the edge of the plasma az imuthally. Arbitrary\nproﬁles of azimuthal ﬂow vφ(θ) can be imposed at the spherical boundary by modulating\nthe electric ﬁeld as a function of polar angle θusing discrete electrodes.\nResults of Ref. [18] show that some ﬂows generated in such a way ca n lead to a dynamo\ninstability. However, in Ref. [18] insulating boundaries are assumed, whereas in MPDX the\nvessel is made of aluminum whose conductivity is much higher than tha t of the plasma\nunder expected experimental conditions. The goal of this paper is to generalize the results\n2S\nN\nS      NSN\n+ -B\nE\nV(b)\nR0Wall PlasmaVacuum \n(air)\nThin insulating layerσ\nμσw\nμwσv=0 \nμv=1 \nd(c)\n(a) Electrodes\nHelmholtz \ncoilsRings of permanent \nmagnets\nFIG. 1: Madison plasma dynamo experiment (MPDX): (a) sketch of the experiment; (b) electrode\nconﬁguration near the wall for driving plasma velocity vφ(θ); (c) model for deriving thin-wall\nboundaryconditions, with σandµdenotingconductivity andrelativepermeabilityoftheres pective\nmedia.\nof Ref. [18] by considering eﬀects of varying conductivity and perm eability of the vessel on\nthe dynamo in a model relevant to MPDX.\nTo describe the plasma we use dimensionless numbers:\nM=V0/radicalbiggρ0\nP0, Re=R0V0\nν, Rm=R0V0\nη, Pm=ν\nη\n– Mach, ﬂuid Reynolds, magnetic Reynolds and magnetic Prandtl, res pectively. Here V0\nis the peak driving velocity, ρ0andP0are the average plasma mass density and pressure,\nR0is the radius of the sphere (a unit of length throughout the paper) ,νandηare the\nplasma kinematic viscosity and magnetic diﬀusivity (assumed to be con stant and uniform).\nFor given plasma parameters these numbers can be estimated from the Braginskii equations\n[20] (see corresponding formulas in Refs. [18, 21]). Their expected values for MPDX are\nlisted in Table I. By varying temperature, density and ion species of t he plasma one can\nchange its magnetic Prandtl number by several orders of magnitu de. Such ﬂexibility makes\nit possible to demonstrate a dynamo in a laminar ﬂow by choosing a regim e withPm∼1\nandRm∼Re∼102. This is an advantage over the liquid metal dynamo experiments,\nwherePm∼10−5and the ﬂows are always turbulent.\nOur ﬁrst step is to ﬁnd an equilibrium velocity ﬁeld capable of dynamo ac tion. For\nsimplicity, we do not focus on the details of plasma driving near the wall. We neglect\n3TABLE I: Expected parameters of MPDX\nQuantity Symbol Value Unit\nRadius of sphere R0 1.5 m\nWall thickness d 0.05 m\nPeak driving velocity V0 0−20 km/s\nAverage number density n01017−1019m−3\nElectron temperature Te2−10 eV\nIon temperature Ti0.5−4 eV\nIon species H, He, Ne, Ar\nIon mass µi1, 4, 20, 40 amu\nMach M 0−8\nFluid Reynolds Re 0−105\nMagnetic Reynolds Rm0−2×103\nMagnetic Prandtl Pm10−3−5×103\nthe multicusp magnetic ﬁeld and applied electric ﬁeld in our consideratio n and assume\nthat the velocity proﬁle is speciﬁed at the boundary. As shown in Ref . [21] for the model\nrelevant to PCX (cylindrical prototype of MPDX), the velocity stru cture obtained under\nsuch assumption is the same as the velocity structure obtained with a more realistic E×B\nforcing, except in a thin boundary layer.\nThe velocity ﬁeld is found using the hydrodynamic part of the extend ed MHD code\nNIMROD [22] with an isothermal ﬂuid model, which in non-dimensional fo rm is\n∂n\n∂τ=−∇·(nv), (1)\nn∂v\n∂τ=−n(v·∇)v−∇n\nM2+1\nRe/parenleftbigg\n∇2v+1\n3∇(∇·v)/parenrightbigg\n, (2)\nwhereτ,nandvstand for normalized time, density and velocity, respectively: τ=tV0/R0,\nn=ρ/ρ0,v=V/V0. The diﬀerential plasma driving near the wall of MPDX is represented\nby the velocity boundary condition:\nv/vextendsingle/vextendsingle\nr=1=vφ(θ)eφ,0≤θ≤π, (3)\nwherevφ(θ) is a function of polar angle θwith physical restriction vφ(0) =vφ(π) = 0. In\ngeneral, this function may be expressed as vφ(θ) =/summationtextaksinkθ, whereakare real coeﬃcients.\n40.51\n0\n-0.5\n-1 vφ\nθ0o       30o       60o      90o     120o     150o    180o(a) Velocity boundary condition\n-1        -0.5         0          0.5         1 0.98    0.99     1      1.01   1.02   1.03\n(b) Velocity structure\nnv polvφ\n(c) Number density\nn\nFIG. 2: Axisymmetric equilibrium ﬂow of von K´ arm´ an type fo r Mach number M= 1 and ﬂuid\nReynolds number Re= 300 used in kinematic dynamo study: (a) velocity boundary c ondition\nvφ(θ) adopted from Ref. [18]; (b) structure of normalized veloci ty; (c) contour plot of normalized\ndensity (dashed lines denote n <1). Left half of (b) shows stream lines of poloidal ﬂux nvpol\nsuperimposed on its absolute values depicted in colors, rig ht half of (b) shows contour plot of\nazimuthal velocity vφ(dashed lines denote vφ<0). Vertical lines in (b) and (c) represent the axis\nof symmetry.\nWe use a velocity boundary condition of the von K´ arm´ an type from Ref. [18], shown to\nresult in a dynamo in an incompressible ﬂow with Re= 300 and Rm>∼237. It is given by\na2=−0.4853,a4=−0.5235,a6=−0.0467,a8= 0.1516 (Fig. 2a). In present study we take\nthe Mach number M= 1, the ﬂuid Reynolds number Re= 300 and the magnetic Reynolds\nnumbers up to Rm= 400. These parameters can be achieved in MPDX by creating, for\nexample, an argon plasma with V0= 5 km/s, n0= 1018m−3,Te= 10 eV and Ti= 1 eV.\nIn the NIMROD simulation, we used a meshing of the poloidal plane with 4 608 quadri-\nlateral ﬁnite elements of polynomial degree 2, and 6 Fourier harmon ics in the φ-direction\n(the azimuthal mode numbers are 0 ≤m≤5). This resolution was suﬃcient for the laminar\nﬂow under consideration. We took a non-moving ﬂuid ( v= 0) with uniform density ( n= 1)\n5as the initial state and evolved Eqs. (1), (2) with the boundary con dition given by Eq. (3)\nuntil a steady state was reached. The resulting ﬂow for Re= 300 and M= 1 is shown in\nFig. 2. The velocity ﬁeld v(r,θ) is axisymmetric and hydrodynamically stable with respect\nto perturbations with m >0.\nThe main results of the paper are obtained by solving the kinematic dy namo problem\nwith this velocity ﬁeld,\nγB=Rm∇×(v×B)+∇2B,∇·B= 0, (4)\nforunknown magneticﬁeld Bandnormalizeddynamo growthrate γ= ΓR2\n0/η. Werepresent\nthe divergence-free ﬁeld as an expansion in a spherical harmonic ba sis [23]:\nBr=L/summationdisplay\nl=ml(l+1)SlYm\nl\nr2eimφ, (5)\nBθ=L/summationdisplay\nl=m/bracketleftbigg1\nr∂Sl\n∂r∂Ym\nl\n∂θ+imTlYm\nl\nrsinθ/bracketrightbigg\neimφ, (6)\nBφ=L/summationdisplay\nl=m/bracketleftbiggimYm\nl\nrsinθ∂Sl\n∂r−Tl\nr∂Ym\nl\n∂θ/bracketrightbigg\neimφ, (7)\nwhereSl(r) andTl(r) are functions of ronly and Ym\nl(θ) are spherical harmonics related\nto the associated Legendre polynomials by Ym\nl(θ) =Pm\nl(cosθ). Since the velocity is ax-\nisymmetric, we consider each azimuthal mode mseparately. The summation in Eqs. (5)-(7)\nis truncated at some spherical harmonic L(L= 20 provides a satisfactory convergence in\nthese studies). Substituting Eqs. (5)-(7) into Eq. (4) and using t he orthogonal properties of\nspherical harmonics, one obtains for m≤l≤L:\nγSl=∂2Sl\n∂r2−l(l+1)Sl\nr2+RmAm\nlL/summationdisplay\nj=m/bracketleftbigg\nI(1)\nljSj\n−I(2)\nlj∂Sj\n∂r+I(3)\nljTj/bracketrightbigg\n, (8)\nγTl=∂2Tl\n∂r2−l(l+1)Tl\nr2−RmAm\nlL/summationdisplay\nj=m/bracketleftbigg\nI(1)\njlTj\n+∂\n∂r/parenleftbigg\nI(2)\nljTj+I(3)\nlj∂Sj\n∂r+I(4)\nljSj/parenrightbigg\n+I(4)\njl∂Sj\n∂r/bracketrightbigg\n. (9)\nHere the bar above a symbol denotes its complex conjugate, Am\nlis a numerical factor\nAm\nl=(2l+1)(l−m)!\n2l(l+1)(l+m)!,\n6andI(1−4)\nlj(r) are functions of rgiven by the integrals:\nI(1)\nlj=j(j+1)\nrπ/integraldisplay\n0Ym\nj/bracketleftbigg\nvθ∂Ym\nl\n∂θsinθ−imvφYm\nl/bracketrightbigg\ndθ, (10)\nI(2)\nlj=π/integraldisplay\n0vr/bracketleftbigg∂Ym\nl\n∂θ∂Ym\nj\n∂θsinθ+m2Ym\nlYm\nj\nsinθ/bracketrightbigg\ndθ, (11)\nI(3)\nlj=imπ/integraldisplay\n0∂vr\n∂θYm\nlYm\njdθ, (12)\nI(4)\nlj=j(j+1)\nrπ/integraldisplay\n0Ym\nj/bracketleftbigg\nvφ∂Ym\nl\n∂θsinθ+imvθYm\nl/bracketrightbigg\ndθ. (13)\nNote that Eqs. (8)-(13) are valid for any axisymmetric velocity ﬁeld . To calculate the\nintegrals in Eqs. (10)-(13), we interpolate the velocity ﬁeld on a unif orm polar grid (typically\nwithNr= 50 radial and Nθ= 1000 angle grid points) and use the trapezoidal rule of\nintegration.\nEqs. (8), (9) should be supplemented with boundary conditions for functions Sl(r) and\nTl(r). The absence of a singularity in the ﬁeld at the center of the spher e requires\nSl/vextendsingle/vextendsingle\nr=0= 0, Tl/vextendsingle/vextendsingle\nr=0= 0. (14)\nThe outer boundary conditions depend on the properties of the sh ell. To avoid undesired\ndiversion of ﬂow-driving current into the shell, the inner surface in M PDX is covered with\nan insulating coating (Fig. 1c). Thus, the normal component of cur rent is zero at r= 1, i.e.,\nTl/vextendsingle/vextendsingle\nr=1= 0. (15)\nTo derive the condition for Slatr= 1, we consider the model shown in Fig. 1c and use the\ngeneral boundary conditions for normal and tangential compone nts of the magnetic ﬁeld at\nthe interface between two media with diﬀerent relative magnetic per meabilities µ1andµ2:\nB1n=B2n,B1t\nµ1=B2t\nµ2.\nWe also assume that the insulating coating is thin enough that it has no impact on proﬁle\n7ofSl. Then the resulting equations are (omitting “ l” inSl)\nr= 1 :S=Sw,1\nµ∂S\n∂r=1\nµw∂Sw\n∂r, (16)\n1< r <1+d\nR0:η\nηwγSw=∂2Sw\n∂r2−l(l+1)Sw\nr2, (17)\nr= 1+d\nR0:Sw=Sv,1\nµw∂Sw\n∂r=∂Sv\n∂r, (18)\nr >1+d\nR0:Sv∝r−l, (19)\nwhere Eq. (17) is derived for a stationary wall with thickness d, symbols with subscripts\nrefer to wall (“ w”) and vacuum (“ v”), and symbols without subscript refer to plasma. We\nassume that dis small, so that the variations of Slin the wall are small too. This is the thin-\nwall approximation [14], it applies if d≪R0andd≪ |ηw/Γ|1/2. Under these assumptions,\nEqs. (16)-(19) are reduced to\n/parenleftbigg∂Sl\n∂r(1+lcµ)+S(lµ+γcσ)/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=1= 0, (20)\nwhere we have used the relation η=c2/(4πσµ) between magnetic diﬀusivity ηand electric\nconductivity σof a medium ( cis the speed of light), and introduced the wall conductivity\nparameter cσand the wall permeability parameter cµ,\ncσ=σwd\nσR0, cµ=µwd\nR0. (21)\nEq. (20) is obtained for a stationary wall without requiring the no-s lip boundary condition\nfor plasma velocity, in contrast to analogous equation (14a) from R ef. [14].\nEqs. (8), (9), (14), (15), (20)constituteaneigenvalueproblem forthedynamogrowthrate\nγand unknown eigen-functions SlandTl. In order to solve it, we apply the ﬁnite diﬀerence\nmethod to Eqs. (8), (9) and discretize SlandTlfor each harmonic l(m≤l≤L) on a\nuniform grid at 0 ≤r≤1 withNrequal intervals. Eqs. (14) and (15) are straightforward to\nimplement in the ﬁnite diﬀerence scheme, while Eq. (20) is taken into ac count by using an\nextra (ghost) grid point to approximate the derivative. The result ing system of ( L−m+\n1)(2Nr−1) linear algebraic equations for ( L−m+ 1)(2Nr−1) unknowns is cast in the\nform of a matrix eigenvalue equation, which is solved in MATLAB. The de veloped scheme\nhas been successfully benchmarked against the results of the kine matic dynamo study from\nRef. [24].\n8-12-8 -4 048\nγ\n  \ncσ=0 \n0 50 100 150 200 250 300 350 400-12-8 -4 048\nRm γ\n  cσ=1 \ncσ=10\ncσ=0 \ncσ=1 \ncσ=10(a)  cµ=0 \n(b)  cµ=10\nRm cr ≈161Rm cr ≈244\nFIG. 3: Dependence of dynamo growth rate γon magnetic Reynolds number Rmfor diﬀerent\nvalues of the wall parameters cσandcµ.\n10  -410  -310  -210  -11 10 10  2 10  3 10  4 140160180200220240260\ncµRm cr \nFIG.4: Dependenceofcritical magneticReynoldsnumber Rmcronthewallpermeabilityparameter\ncµ.\nHere we report the results of solving the kinematic dynamo eigenvalu e problem [Eqs. (8),\n(9), (14), (15), (20)] with the velocity shown in Fig. 2, and for the relative permeability of\nthe plasma µ= 1, magnetic Reynolds numbers Rm= 0−400 and varying wall parameters\ncσandcµ. We consider only the most unstable (or least decaying) m= 1 azimuthal mode.\nThe convergence of the numerical scheme is checked by comparing simulations at diﬀerent\n9resolutions. Results reported here are obtained by using a maximum number of spherical\nharmonics L= 20 and number of radial grid points Nr= 50.\nThe results are summarized in Figs. 3 and 4. In the present case γis always real, so\nthe dynamo threshold Rmcrcorresponds to the condition γ= 0. As seen in Fig. 3, Rmcr\nrequired for the onset of the dynamo does not depend on the wall c onductivity parameter\ncσ. This is because cσdrops out of the problem when γ= 0, as follows from Eq. (20).\nHowever, cσaﬀects the dynamo growth rate: larger values of cσ(larger wall conductivity)\nlead to lower |γ|. Therefore, in the limit of a perfectly conducting shell no growing ﬁe ld is\npossible, since γ→0.\nThe wall permeability parameter cµhas a strong inﬂuence on both dynamo threshold\nRmcrand growth rate γ. A ferritic wall facilitates dynamo action. As shown in Fig. 4, the\ncritical magnetic Reynolds number Rmcrdecreases with increase of cµ: fromRmcr≈244\nwhencµ= 0 toRmcr≈154 when cµ→ ∞. These results are consistent with previous\ntheoretical dynamo studies in other geometries [9, 10, 12, 13], whic h indicated reduction of\nRmcrfor the ferritic-wall boundary conditions.\nEstimates for typical parameters of MPDX show that its wall is very conducting and\nnon-ferritic with cσ≈30 andcµ≈0. Under these conditions, dynamo action is achievable\nfor the considered ﬂow if Rm>∼244, the respective dynamo growth rate at Rm= 400 is\nΓ≈3.6 s−1.\nIn summary, we have studied the inﬂuence of ﬁnite conductivity and permeability of\nthe wall on a plasma dynamo in a sphere. Our results show that the dy namo threshold is\naﬀected only by the wall permeability, while the dynamo growth rate d epends on both wall\nproperties.\nThe authors wish to thank C. Sovinec for valuable help and discussion s related to\nNIMROD.\n[1] N. L. Peﬄey, A. B. Cawthorne, and D. P. Lathrop, Phys. Rev. E61, 5287 (2000).\n[2] C. B. Forest, R. A. Bayliss, R. D. Kendrick, M. D. Nornberg , R. O’Connell, and E. J. Spence,\nMagnetohydrodynamics 38, 107 (2002).\n[3] M. Bourgoin, L. Mari´ e, F. P´ etr´ elis, C. Gasquet, A. Gui gon, J.-B. Luciani, M. Moulin, F.\n10Namer, J. Burguete, A. Chiﬀaudel, F. Daviaud, S. Fauve, Ph. Od ier, and J.-F. Pinton, Phys.\nFluids14, 3046 (2002).\n[4] A. Gailitis, O. Lielausis, S. Dement’ev, E. Platacis, A. Cifersons, G. Gerbeth, T. Gundrum,\nF. Stefani, M. Christen, H. H¨ anel, and G. Will, Phys. Rev. Le tt.84, 4365 (2000).\n[5] R. Stieglitz and U. M¨ uller, Phys. Fluids 13, 561 (2001).\n[6] R. Monchaux, M. Berhanu, M. Bourgoin, M. Moulin, Ph. Odie r, J.-F. Pinton, R. Volk,\nS. Fauve, N. Mordant, F. P´ etr´ elis, A. Chiﬀaudel, F. Daviaud , B. Dubrulle, C. Gasquet,\nL. Mari´ e, and F. Ravelet, Phys. Rev. Lett. 98, 044502 (2007).\n[7] G. Verhille, N. Plihon, M. Bourgoin, P. Odier, and J.-F. P inton, New J. Phys. 12, 033006\n(2010).\n[8] M.S.ChuandM.Okabayashi, PlasmaPhys.Control.Fusion 52, 123001 (2010), andreferences\ntherein.\n[9] R. Avalos-Zuniga, F. Plunian, and A. Gailitis, Phys. Rev . E68, 066307 (2003).\n[10] R. Avalos-Zuniga and F. Plunian, Eur. Phys. J. B 47, 127 (2005).\n[11] R. Laguerre, C. Nore, J L´ eorat, and J.-L. Guermond, C. R . Mecanique 334, 593 (2006).\n[12] C. Gissinger, A. Iskakov, S. Fauve, and E. Dormy, Europh ys. Lett. 82, 29001 (2008).\n[13] C. Gissinger, Europhys. Lett. 87, 39002 (2009).\n[14] P. H. Roberts, G. A. Glatzmaier, and T. L. Clune, Geophys . Astrophys. Fluid Dyn. 104, 207\n(2010).\n[15] C. Guervilly and P. Cardin, Geophys. Astrophys. Fluid D yn.104, 221 (2010).\n[16] A. Giesecke, C. Nore, F. Stefani, G. Gerbeth, J. L´ eorat , F. Luddens, and J.-L. Guermond,\nGeophys. Astrophys. Fluid Dyn. 104, 505 (2010).\n[17] C. B. Forest, R. A. Bayliss, D. D. Schnack, E. J. Spence, a nd K. Reuter, Bull. Am. Phys. Soc.\n53, 222 (2008).\n[18] E. J. Spence, K. Reuter, and C. B. Forest, Astrophys. J. 700, 470 (2009).\n[19] C. Collins, N. Katz, J. Wallace, J. Jara-Almonte, I. Ree se, E. Zweibel, and C. B. Forest, Phys.\nRev. Lett. 108, 115001 (2012).\n[20] S. I. Braginskii, Reviews of Plasma Physics (Consultants Bureau, New York, 1965), Vol. 1, p.\n205.\n[21] I. V. Khalzov, B. P. Brown, F. Ebrahimi, D. D. Schnack, an d C. B. Forest, Phys. Plasmas 18,\n032110 (2011).\n11[22] C. R. Sovinec, A. H. Glasser, T. A. Gianakon, D. C. Barnes , R. A. Nebel, S. E. Kruger, D.\nD. Schnack, S. J. Plimpton, A. Tarditi, M. S. Chu, and the NIMR OD Team, J. Comp. Phys.\n195, 355 (2004).\n[23] E. C. Bullard, and H. Gellman, Phil. Trans. R. Soc. Lond. A247, 213 (1954).\n[24] M. L. Dudley, and R. W. James, Proc. R. Soc. Lond. A 425, 407 (1989).\n12"    },    {        "title": "1312.2428v2.Giant_Photogalvanic_Effect_in_Metamaterials_Containing_Non_Centrosymmetric_Plasmonic_Nanoparticles.pdf",        "content": "arXiv:1312.2428v2  [physics.optics]  11 Apr 2014GiantPhotogalvanicEffect inMetamaterials\nContaining Non-Centrosymmetric PlasmonicNanoparticles\nSergei V. Zhukovsky,1,2,∗Viktoriia E. Babicheva,2,1,3Andrey B. Evlyukhin,4\nIgor E. Protsenko,5,6Andrei V. Lavrinenko,1and Alexander V. Uskov5,6\n1DTU Fotonik – Department of Photonics Engineering,\nTechnical University of Denmark, Ørsteds Plads 343, DK-280 0 Kgs. Lyngby, Denmark\n2National Research University of Information Technology,\nMechanics and Optics, Kronverksky pr. 49, St. Petersburg, 1 97101, Russia\n3Birck Nanotechnology Center, Purdue University, 1205 West State Street, West Lafayette, IN, 47907-2057 USA\n4Laser Zentrum Hannover e.V., Hollerithallee 8, D-30419 Han nover, Germany\n5P. N. Lebedev Physical Institute, Russian Academy of Scienc es, Leninskiy Pr. 53, 119333 Moscow, Russia\n6Advanced Energy Technologies Ltd, Skolkovo, Novaya Ul. 100 , 143025 Moscow Region, Russia\nPhotoelectric properties of metamaterials containing non -centrosymmetric, similarly oriented metallic\nnanoparticles embedded in a homogeneous semiconductor mat rix are theoretically studied. Due to the asym-\nmetric shape of the nanoparticle boundary, photoelectron e mission acquires a preferred direction, resulting ina\nphotocurrentﬂowinthatdirectionwhennanoparticlesareu niformlyilluminatedbyahomogeneous planewave.\nThis effect is a direct analogy of the photogalvanic (or bulk photovoltaic) effect known to exist in media with\nnon-centrosymmetric crystal structure, suchas doped lith iumniobate orbismuth ferrite,butis several orders of\nmagnitude stronger. Termedthe giant plasmonic photogalvanic effect ,the reportedphenomenon is valuable for\ncharacterizing photoemission and photoconductive proper ties of plasmonic nanostructures, and can ﬁnd many\nuses for photodetection andphotovoltaic applications.\nI. INTRODUCTION\nThe recent decade in modernphysicshasfeaturedthe con-\nceptofopticalmetamaterials. Thecentralideaofthisconc ept\nis to bestow the role of known, ordinary constituents of mat-\nter(atoms,ions,ormolecules)uponartiﬁcial“meta-atoms ”—\nnanosized objects purposelydesigned to have the desired op -\nticalproperties[1]. Theassemblyofsuchmeta-atoms—anar -\ntiﬁcial composite metamaterial—exhibits the desired prop er-\nties macroscopically, provided that the meta-atoms are muc h\nsmallerthanthewavelengthoflightinteractingwiththem.\nGreat as the variety of naturally occurring atoms and\nmolecules (and, in turn, of natural materials) may be, the in -\nherent total freedom in choosing the shape and composition\nofartiﬁcialmeta-atomsisbelievedtobeevengreater—near ly\narbitrary. Thus, a prominentsuccess of optical metamateri als\nis the design of materials with optical properties that eith er\ndonotexistoraremuchweakerinnaturallyoccurringmedia.\nNotable examples include metamaterials with a negative re-\nfractive index, near-zero, or very large permittivity; met ama-\nterials with magnetic permeability at optical frequencies ; ex-\ntremelyanisotropichyperbolicmetamaterialsthatbehave like\nmetals in some directions and like dielectrics in others; ch i-\nral metamaterials with giant magnetooptical properties, a nd\nmanyothers[2–4].\nMost meta-atom designs proposed to date are based on\nmetallicnanoparticles,nanoantennas,orresonatorsofva rious\nshapes[5]. Insuchmetallicstructures,thesizeprerequis itefor\nmeta-atom design is fulﬁlled by subwavelength conﬁnement\nofelectromagneticﬁelddueto localizedsurfaceplasmonre s-\nonance excitation. At the same time, localized plasmons are\n∗sezh@fotonik.dtu.dkknown to cause strong local ﬁeld enhancement, which can\nenhance the functionality of metamaterials in the context o f\nbiological and chemical sensing, as well as give rise to new\nconceptsofopticalmetamaterialsbasedonstronglyenhanc ed\nnonlinear, photorefractive, and photoconductive effects . In\nparticular,plasmonics-enhancedphotoconductivity–the emis-\nsion of photoelectrons from nanoparticles due to action of\nstronglocal ﬁelds in the localizedplasmonicresonances–w as\nrecentlyshowntobepromisingforphotodetectionandphoto -\nvoltaicapplications[6–8].\nTranscending the purely electromagnetic approach tradi-\ntionallyadoptedinthestudyofplasmonicnanostructuresa nd\naccounting for processes when light can cause electrons to\nleave the nanoparticles has far-reaching implications, pu tting\nforth a new concept of photoconductive metamaterials. The\nenhanced photoelectric effect from plasmonic nanoantenna s\nwith generation of “hot” electrons [9, 10] can be used to im-\nprovethecharacteristicsoflight-harvestingdevices(e. g.,pho-\ntoconductive plasmonic metamaterials, photodetectors, s olar\nandphotochemicalcells)[9–19],aswell asmoregenerallyi n\noptoelectronics, photochemistry, and photo electrochemi stry\n[20–22].\nIn this paper ,we predict and numerically demonstrate an\neffect related to new functionality of photoconductive met a-\nmaterials: the giant plasmonic photogalvanic effect . Named\nafter photogalvanic(or bulk photovoltaic)effect in bulk n on-\ncentrosymmetric media [23], plasmonic photogalvanic effe ct\nis shown to exist in a metamaterial containing similarly ori -\nented non-centrosymmetricmetallic nanoparticles embedd ed\nin a homogeneoussemiconductor matrix. The low degree of\nsymmetryof the nanoparticleshape causes the net ﬂux of the\n“hot” electrons emitted from the nanoparticles via the reso -\nnant plasmonic excitation to be directional. This directio nal-\nityleadstoaphoto-electromotiveforceasaresultofhomog e-\nneousexternallightillumination(thephotogalvaniceffe ct).2\nFigure 1. Schematics of (a) “hot” photoelectron emission th rough\nthe Schottky barrier at a metal/semiconductor interface; ( b) several\nnanoparticle arrays studiedinthepresent paper (showing t hreechar-\nacteristiccasesofcylinders,truncatedcones,andcones) ;(c)enlarged\nview of one nanoparticle showing itsgeometrical parameter s.\nWe reportthat the resultingphotocurrentdensity generate d\nin a layer of nanoparticles emerges and grows as the parti-\ncle shape changes from cylindrical to conical, i.e., with th e\nincrease of the particle asymmetry. We calculate the com-\nponents of the effective third-rank tensor relating the cur rent\ndensity to the incident electric ﬁeld, and show that this eff ec-\ntive tensor for the nanoparticlearray exceedsthat for the n at-\nurally occurring ferroelectrics that exhibit bulk photovo ltaic\neffect. Hence, the reported plasmonic effect can be regarde d\nasa“giant”versionoftheconventional(non-plasmonic)ph o-\ntogalvaniceffectoccurringinnaturalmaterials.\nThe paper is organized as follows. In Section II we re-\nviewthetheoreticalbackgroundfor“hot”electronphotoem is-\nsion at metal-semiconductor interfaces containing Schott ky\nbarriers. In Section III we numerically investigate arrays of\nnanoparticleswhose shape varies from cylindrical to conic al.\nWe discuss the observed increase of photocurrent direction -\nality and induced electromotive force as a result of increas ed\nspatialasymmetryofthenanoparticles. InSectionIVwedra w\nparallels between the predicted plasmonic photogalvanic e f-\nfect and the known photogalvanic effect in certain naturall y\noccurring media. Finally, in Section V we summarize and\noutlinepossibleapplicationsfortheproposedeffect.\nII. THEORETICALBACKGROUND\nWeconsiderametallicnanoparticleembeddedinauniform\nsemiconductormatrix(Fig. 1) in presenceof a normallyinci -\ndentlightwaveoffrequency ω, suchthat\nWb<¯hω<Eg. (1)\nIt is assumed that the photon energy ¯hωis insufﬁcient\nto excite electron-hole pairs in the semiconductor matrixwith gap energy Eg, but exceeds the work function for the\nmetal/semiconductor interface Wb(see Fig. 1a), so that the\nphoton energy transferred to an electron in metal can cause\nit toleavethenanoparticle.\nTwo mechanismsof energytransfer fromthe photonto the\nelectron can be identiﬁed [24–26]. One is absorption of a\nphoton by an electron in the bulk of the nanoparticle with\nsubsequent transport of the “hot” electron to the surface an d\nits emission by overcoming the Schottky barrier (the volume\nphotoelectriceffect[27]). Theotherisabsorptionofapho ton\nby an electron as it collides with the nanoparticle boundary ,\ncausing emission of that particular electron from the metal\n(the surface photoelectric effect). In both of these mecha-\nnisms, the spatial and momentum distribution of the emitted\nelectronsisstronglyinﬂuencedby(i)thespatialconﬁgura tion\nofthenanoparticlesurfaceand(ii)resonantﬁeldenhancem ent\nnearthat surfaceduetothe plasmonicresonance.\nWhile the question on which of the two mechanisms is\nstrongerinnanostructuresisyettobeanswered,itwasshow n\nthat electron collisions after photon absorption leading t o\nrapid “hot” electron cool-down make surface-driven effect s\nprevail over bulk effects in many cases [24, 28]. So, follow-\ning the previously developed Tamm theory of photoemission\nfrom plasmonic nanoparticles[8, 17, 18], we can express the\nphotocurrentfroma nanoparticleas\nINP(λ)=Cem(λ)˛\nparticle|En(r)|2d2r. (2)\nThe integrationis performedover the surface of the nanopar -\nticle. The expression in Eq. (2) only takes into account the\nincreased number of photoelectrons without account for the\ndirection of their momentum as they leave the nanoparticle.\nAsa quantityintegratedforall electronsin all directions ,it is\napplicable in the photoconductivity scenario when the emit -\nted electrons are subsequently directed using an externall y\napplied or a built-in potential, and their momentum directi on\nuponleavingthenanoparticlecanthusbetotallydisregard ed.\nIn the absence of such potential, the initial velocity of the\nemitted photoelectrons starts to play a role in the deﬁnitio n\nof the photocurrent from a nanoparticle, which then has to\nassumea modiﬁedform,\nINP(λ)=−Cem(λ)˛\nparticle|En(r)|2nd2r,(3)\nwherenis the unit normal vector at the nanoparticle surface\natpointr. InbothEqs.(2)and(3),thecoefﬁcient Cemequals\nCem=ηoe\n¯hωYm=ηoe\n¯hω·ε0cnm\n2, (4)\nwherenmis the refractive index of the matrix and ηois\nthe external quantum efﬁciency of the electron photoemis-\nsion through the potential barrier at the metal/matrix inte r-\nface [8, 29]; it strongly depends on the photon energy and\nvaries from zero at ¯hω=Wb(see Fig. 1a) to non-zero val-\nues for higher photon energies [28]. The admittance (invers e\nimpedance) Ymofthematrixmediumrelatestheintensity Sof\naplanewavewith theelectricﬁeldstrength EasS=Ym|E|2.3\nWhen a plasmonic resonance is excited in the nanoparti-\ncle, the amplitude of the resonant ﬁelds inside it and near it s\nsurface usually greatly exceeds the amplitude of the incide nt\nﬁeld. Therefore,wecanseefromEqs.(2)–(3)thatinnanopar -\nticles with centrosymmetric shapes (such as nanospheres,\nnanocubes,ornanodisks)thespatialsymmetryoftheresona nt\nmodes will lead to the cancellation of the directed photocur -\nrent,sothat INP=0eventhough INP/negationslash=0. Inotherwords,even\nthoughthe incidentwave can cause additionalphotoelectro ns\nto be emitted over the Schottky barrier, the collective moti on\noftheseelectronswill notinduceanelectromotiveforce.\nThe situation changes drastically when the shape of the\nnanoparticleand/or the ﬁeld distributionof the resonantp las-\nmonicmodelacksthecenterofsymmetry. Eq.(3)thenshows\nthat the resulting photocurrent acquires directionality, so that\nphotoemissionfromsuchanon-centrosymmetricnanopartic le\nresults in net photocurrent ( INP/negationslash=0) and induces electromo-\ntive force. Theratio ρ=|INP|/INP, whichcan vary from0 to\n1, can be regarded as a measure of directionality for photoe-\nmissionwithrespectto onenanoparticle.\nIn a metamaterial comprising an oriented arrangement of\nsuchnon-centrosymmetricnanoparticlesthatisnottooden se,\nthe satisfactory approximation is that the neighboring par ti-\ncles do not modify the plasmonic resonance of each other in\na signiﬁcant way. In this case, the individual photocurrent s\nfrom each particle sum up, with the resulting current den-\nsity from a square nanoparticle lattice with period awritten\nasj=INP/a2. Since the nanoparticles are axially symmet-\nric with respect to the z-axis (see Fig. 1c), and the lattice has\n4-fold rotational symmetry, we expect that for a normally in -\ncidentwave jx=jy=0andjz=|j|,sowe canwrite\njz=|INP|/a2=ρINP/a2=ρCem|E2\n0|ξ (5)\nwhereE0istheﬁeld incidentonthelattice, and\nξ=1\n|E2\n0|a2˛\nparticle|En(r)|2d2r (6)\nrelates the incident ﬁeld E0with the local ﬁeld E(r)and has\nthemeaningofaﬁeldenhancementfactorduetothelocalized\nplasmonresonanceinthenanoparticles[8,29]. Iftheincid ent\nﬁeldis, e.g., x-polarized,Eq.(5)canberewrittenas\njz=ρξCemE0xE∗\n0x=˜αzxxE0xE∗\n0x, (7)\nwhich has the form equivalent to the photocurrentinduced in\nsome media with non-centrosymmetric crystal structure due\ntothe photogalvanic(orbulkphotovoltaic)effect[30–32] ,\nji=αijkEjE∗\nk, (8)\nwhereEjare again the components of the incident ﬁeld,\nand coefﬁcients αijk, related to the components of the third-\nrank piezoelectric tensor [30], are non-zero only for nonce n-\ntrosymmetric media. Examples include piezoelectrics or fe r-\nroelectrics such as LiNbO 3:Fe, quartz with F-centers, or p-\nGaAs [30]. We see that the photocurrent jin Eq. (8) is non-\nlinear(quadratic)withrespecttothe ﬁeldstrength E.We canalso rewriteEq.(8) ina modiﬁedform[33],\nji=/parenleftbig\nβijk/2/parenrightbig\n(eje∗\nk+e∗\njek)S0, (9)\nwhereej=Ej/|E|is thejthcomponent of the incident light\npolarizationvector,and S0isitsintensity. InSIunits,thecom-\nponents of the tensor βhave the dimension of inverse volts;\nnormalized by the absorption coefﬁcients, βijkare related to\nthe Glass coefﬁcients for the high-voltage bulk photovolta ic\neffectinnonlinearcrystals[33,34]. Onecansimilarlyrew rite\nEq.(7), introducingthe plasmonicequivalentofEq.(9)as\njz=˜βzxxS0|ex|2,˜βzxx=ρξηoe/(¯hω).(10)\nWe see that Eq.(7) is formallyequivalentto Eq.(8), standar d\nfor describing the photogalvanic effect in bulk media. How-\never,oneimportantdifferencehastobenoted. Inbulkmedia ,\ndirected photoelectrons are generated throughout the volu me\nof the material. They have a ﬁnite lifetime since their initi al\nvelocity decays as they move. Hence the coefﬁcients αijkin\nEq. (8) are proportional to that lifetime [23]. In contrast, the\ngeometry in Fig. 1b considered here deals with the injection\nof directed photoelectrons from a nanoparticle array into t he\nsurrounding medium. Hence there is no lifetime in Eqs. (5)\nand (7), which is emphasized by using a tilde over the coef-\nﬁcients˜αijkin that formula. The behavior of photoelectrons\naftertheyhavebeenemittedisasubjectforfurtherdiscuss ion.\nIt is expectedthat whenmanynanoparticlelayersare stacke d\ntogether,thephotoelectronscanberecapturedandre-emit ted,\nand in the case of bulk metamaterial the effective lifetime f or\nthedirectionallyemittedelectronscouldbereintroduced .\nIII. DIRECTIONALPHOTOEMISSION\nFROMCONICALNANOPARTICLES\nTo conﬁrm the predicted photogalvanic effect in a plas-\nmonic metamaterial, we consider an array of gold nanoparti-\ncles whose shape is gradually varied from cylindrical to con -\nical (Fig. 1b–c). The choice of conical nanoparticles is wel l-\nmotivated from a fabricational standpoint, since nanodisk s\nfabricated using lithographic means often acquire asymme-\ntry and resemble cone-likeshapes [35, 36]; other asymmetri c\nshapessuchashemispheresandnanopyramidsarealsoreadil y\nfabricableusingvarioustechniques[37–39].\nLet the nanocones under study have height h; the larger\n(bottom)facethasradius r,andtheradiusofthesmaller(top)\nfacet is given by r(1−ζ)so thatζcan be understood as\nthe “asymmetry” or “conicity” parameter. The case ζ=0\ncorresponds to the centrosymmetric, disk-shaped particle s,\nwhereas the opposite case ζ=1 corresponds to maximally\nasymmetric nanocones. For the computational example, we\nuser=25nmand h=18nm.\nPermittivityofgoldwasdescribedbytheDrudemodelwith\nplasma frequency 2 .18×1015s−1and collision frequency\n6.47×1012s−1[40]. The particles are embedded in a homo-\ngeneousGaAsmatrix( nm=3.6,Wb=0.8eV,Eg=1.43eV),\nwhich results in the operating range between 870 and 1550\nnmaccordingtoEq.(1). ForAu/GaAsinterface,thevaluesof4\nFigure 2. (a) Absorption spectra for the nanoparticle latti ce with the\nconicity parameter ζranging from 0 (cylinders) to 1 (cones). The\nlightisnormallyincidentontothelatticeplaneandlinear lypolarized\nalong the x-axis. For easier readability, each line is offset by0.1. (b )\nImagesofthenormalcomponentoftheelectricﬁeldatthepla smonic\nresonance(thefrequencyoftheabsorptionmaximum)for ζ=0,0.5,\nand1,viewedfromtwodifferentangles. (c)Extinctioncros s-section\nfor a single nanoparticle withnear-cylindrical ( ζ=0.02) andhighly\nconical (ζ=0.87) shape, calculated with the DDA method com-\nparing the full cross-section (dots) with the electric dipo le moment\ncontribution(blueline);theinsetsshowtheimagesofthed iscretized\nnanoparticles.\nηorange from zero at ¯hω=Wbto about 0.0025 for ¯hω/lessorsimilarEg\n[28]. The particles are arranged in a 2D square lattice with\nperioda=100 nm. According to the earlier results [8, 18],\nsuch a dense lattice prevents the appearance of higher-orde r\ndiffraction and ensures that the lattice effects are outsid e of\nthe operating range, so the resonant mode of a particle in a\nlattice largely coincides with that of an isolated particle [8].\nAnother beneﬁt of making the lattice dense is the increase of\nthe nanoparticles concentration, resulting in the increas e of\nthetotal inducedphotocurrentasperEq.(5).\nSimulationswerecarriedoutinthefrequencydomainusing\nCST Microwave Studio. The results show that all the struc-\ntures in question feature a rather broad absorption resonan ce\ncorrespondingtotheexcitationofalocalizedsurfaceplas mon\n(Fig. 2a). As expected, small amakes the lattice-related res-\nonances occur outside of the operating frequency range, so\nthe resonance is dominated by the response of a single plas-\nmonicnanoparticle,broadenedduetothepresenceofmanyof\nthem [8]. Note that to reduce the inﬂuence of the discretiza-\ntion(“staircasing”)artifactsatsharpedges(whichwould any-way be unphysical from the fabricational point of view), all\nedges of the nanoparticles were smoothed with the curvature\nradiusof δr=1 nm. Togetherwith the adaptivemesh reﬁne-\nment,thisprovedsuccessfulineliminatingnumericalarti facts\nfrommeshing-related“hotspots”.\nWe further see that the resonance strongly depends on the\nnanoparticleshape[18]. Astheshapechangesfromacylinde r\nto a cone, the resonance undergoesa very slight blue shift up\ntoζ=0.4;furtherincreaseof ζleadstoastrongredshiftac-\ncompaniedbyarathersigniﬁcantnarrowingoftheresonance .\nPlotting the normal componentof the resonant mode ﬁeld on\nthenanoparticlesurface(Fig.2b),wenoticethatalthough the\nﬁeld at the smaller base(or tip)of the conebecomesprogres-\nsively weaker as ζincreases, the mode maintains the charac-\nteristicoutsideﬁeldpatternofafundamentaldipolereson ance\nforallnanoparticles. Toreconﬁrmthis,wehavecalculated the\nextinctioncross-sectionofconicalnanoparticlesusingt hedis-\ncretedipoleapproximation(DDA)method[41,42]. Figure2c\nshows that extinctionpropertiesof the nanoparticlesare f ully\nreproduced if only the electric dipole moment is kept in the\ncasesofbothlowandhighasymmetry.\nThe DDA results reproduce the red shift of the resonance\nasζincreases, whichis qualitativelysimilar to what happens\nin a metal spheroid as it becomes more oblate [43]. More-\nover, the results in Fig. 2c show the decrease of the effectiv e\ndipole moment of a nanocone compared to a nanodisk. This\ndecrease diminishes the coupling between the individualpa r-\nticlesinthelattice,weakeningtheabsorptionpeakbroade ning\nforlarger ζ, whichis thoughto be the primarymechanismof\ntheresonancenarrowingforthenanocones.\nFrom the distribution of the local ﬁeld at the nanoparti-\ncle surface, which was determined numerically at every fre-\nquency, we then calculate the photoemission current from\neach nanoparticle using Eqs. (2) and (3). The results are\nshowninFigs.3a–b,respectively. Itcanbeseenthatthecyl in-\ndrical nanoparticles have no preferred direction of the emi t-\ntedphotoelectrons. However,themorethenanoparticlesha pe\nevolvestowards conical with z-axis symmetry, the greater di-\nrectionalityalongthe z-axisthephotoelectronsacquire.\nThe sign of jzindicates that electronstend to be emitted in\nthedirectionofthebaseofthecone. Thiscanbeexplainedby\nFigure 3. (a) Total photocurrent INPand (b)z-component of the di-\nrectionalphotocurrent INPfordifferentvaluesoftheconicityparam-\neterζ. The linesare offset by 25for easier readability.5\nthe ﬁelddistributionsin Fig. 2b. Ina nanodisk,the two face ts\nhaveequalﬁelddistributionandthereforephotocurrentre sult-\ningfromemissionfromthetwofacetsbalanceseachother;th e\nsamehappenswiththesidewalls,resultinginoverall INP=0.\nIn a cone,the ﬁeld distribution(andhence,photoemission) at\nits base is similar to the facet of the nanodisk, but the ﬁeld a t\ntheremainingsurfaceoftheconeissigniﬁcantlyweakercom -\npared to the base because the center of mass of a nanocone\ngetsshiftedtowardsitsbaseasitsasymmetryincreases,ag ain\nqualitatively similar to an oblate metal spheroid [43]. The re-\nfore,asζincreases,thesmallerfacetoftheconehasa gradu-\nallydecliningcontributiontothetotalphotoemissionpro cess,\nthusincreasingthephotocurrentdirectionality.\nSincetheoperatingfrequencyrangeonlycontainsoneplas-\nmonic rsonance, the increase in ρis almost uniform across\nthe spectrum. We can thus introducean overallspectrally av -\neraged¯ρfor the structure with a given ζ; the resulting de-\npendence ¯ρ(ζ)is shown in Fig. 4. We can see that highly\nasymmetric nanoparticles( ζ>0.5) display signiﬁcant ρthat\nexceeds0.5,whileforfullyconicalnanoparticles ρ>0.8.\nMoreover, we see that the ﬁeld enhancement ξalso be-\ncomesgreater whenparticles becomemoreasymmetric, with\nmaximum ξchanging from about 40 for cylinders to almost\n100 for cones. Using Eq. (10) and the expression for ηofor\nAu/GaAs interface calculated in [28] at the maximum of the\ndependence ξ(¯hω)(see Fig. 3a), we can ﬁnally derivethe ef-\nfective tensor component ˜βzxx, also shown in Fig. 4. We see\nthatthemaximumvalueofaround0.07isreachedforconicity\nparametervalues ζ≃0.6...0.7 (Fig. 4). For more asymmet-\nricshapes,theincreasein ξandρiscompensatedbytheshift\nof plasmonic resonance towards ¯hω=Wb, whereηorapidly\napproacheszero, leadingto a slight decrease in ˜βzxxto values\naround0.05. Changingthe nanoparticle’saspect ratio in su ch\na way that its plasmonic resonance gets shifted to higher fre -\nquencies(exceeding1eV)isexpectedtocounteractthiseff ect\nandfurtherboostthedirectionalphotocurrent.\nThe resulting values of ˜βzxxin Fig. 4 are seen to greatly\nexceed the typical values for ferroelectric crystals such a s\nFigure4. Photocurrent directionalityratio ¯ρaveraged forthe photon\nenergiesbetween1.0and1.1eV(squares)andcalculatedval ueofthe\neffective ˜βzxxfor the value of ηofor the photon energy of maximum\nphotocurrent (the peak in Fig. 3a) calculated according to R ef. [28]\n(circles),depending onthe conicity parameter ζ.the experimentally determined βxxx=3.1×10−12V−1in\nLa3Ga5SiO14:Fe [33]. Theobtained˜βzxxisalso foundtoex-\nceed the anomalously high values for bismuth ferrite known\nto outperformtypical ferroelectric materials by about ﬁve or-\nders of magnitude in the thin-ﬁlm conﬁguration ( βijjaround\n2...3×10−4V−1accordingto the recent measurements[44]\nandﬁrst-principletheoreticalcalculations[32]). Furth ermore,\nit exceeds the values βijj∼10−4...10−3V−1calculated for\nthe photogalvanic effect based on ratchet photocurrent fro m\ninteraction of free electrons with asymmetric nanoscatter ers\nwithouttakinganyplasmoniceffectsintoconsideration[3 1].\nHence, the plasmonic photogalvanic effect in metamateri-\nalsbasedonasymmetricallyshapednanoparticlescanbecon -\nﬁrmed to constitute a “giant” versionof bulk photovoltaice f-\nfectpresentinnon-centrosymmetriccrystals.\nIV. PLASMONICVERSUSCONVENTIONAL\nPHOTOGALVANICEFFECT\nHaving established formal equivalence between the con-\nventional bulk photovoltaic effect in non-centrosymmetri c\ncrystalline materials [30] and the reported plasmonic effe ct\nin nanoparticles through the parallels between Eqs. (8) and\n(7), or similarly between Eqs. (9) and (10), we would like to\ndiscussthesimilaritiesintheunderlyingphysicsbehindt hese\ntwoeffectsin moredetail.\nConventional photogalvaniceffect [45]orbulkphotovoltaic\neffect[30] (sometimes called high-voltage bulk photovoltaic\neffect [46]) refers to the generation of intrinsic photocur rents\noccurringinsingle-phasematerialswithoutinversionsym me-\ntry[47]. Microscopically,itisassociatedwithviolation ofthe\nprincipleofdetailedbalanceforphotoexcitednon-equili brium\ncarriersinnoncentrosymmetriccrystals: theprobability ofthe\nelectron transition between the states with momentum kand\nk′,W(k,k′), does not, in general, equal the probability of the\nreverse transition: W(k,k′)/negationslash=W(k,−k′)[48]. This gives rise\nto a ﬂux of photoexcited carriers, which manifests itself as\nphotocurrentwithacertaindirectioneventhoughthemediu m\nishomogeneousanduniformlyilluminated.\nThe principle of detailed balance may be violated due to\na variety of mechanisms, e.g., inelastic scattering of carr iers\non non-centrosymmetric centers, excitation of impurity ce n-\nterswithanasymmetricpotential,orhoppingmechanismtha t\nacts between asymmetrically distributed centers [30]. Oth er\neffects that have been pointed out are excitation of non-\nthermalized electrons having asymmetric momentum distri-\nbution due to crystal asymmetry, delocalized optical trans i-\ntions in lattice excitation of pyroelectrics [46], spin-or bital\nsplitting of the valence band in gyrotropic media [30], and\nsecond-order nonlinear optical interaction known as “shif t\ncurrents”, shown to be the dominating photocurrent cause in\nferroelectrics[32, 47].\nThelattereffectisstrikinglysimilartotheplasmoniceff ect\nreported here because the expression for the “shift current ”,\nJq=σrsqErEs, is essentially coincident with Eqs. (7)–(8); in\nboth cases, the photocurrent is quadratic with respect to th e\nﬁeldstrengthofincidentlight[17].6\nAnother striking similarity arises when one compares the\nconsidered photoemission from nanoparticles with photoio n-\nization from atoms. Indeed, one can regard nanoparticles as\natoms whose electrons are placed into a highly asymmetric\npotential well, which is formed by the boundary of metal\nwith thesurroundingmedium. Then,it isknownfromatomic\nphysicsthatthepatternofphotoeffectfromsuchatomswoul d\ndepend on the shape of the atomic potential well. From\nthis point of view, it is clear that changing the shape of the\nnanoparticles can efﬁciently deform the pattern of electro n\nphotoemission from such nanoparticles in much the same\nway as what happens in non-centrosymmetriccrystals where\nasymmetryisinherentinthecrystallattice structure.\nHence, we have solid grounds to regard the reported plas-\nmonic effect in nanoparticle arrangements as the plasmonic\nanalogy (or “metamaterial counterpart”) to the convention al\nphotogalvanic or bulk photovoltaic effect, if we think of\nnanoparticles as “meta-atoms” and equate the absence of the\ncenter of symmetry in them to a similar geometric feature of\ncrystal lattice in bulk media. In some ways, it resembles the\nmesoscopicphotovoltaiceffect thatwasreportedearliertooc-\ncur in ensembles of semiconductor microjunctions of larger\ndimensions(about1 µm)in themicrowaverange[49,50].\nChoosingbetweentheterms photogalvanic andbulkphoto-\nvoltaictonamethereportedeffectisworthanotherdiscussion.\nIn homogeneous media, these two terms are essentially syn-\nonymousandhavebeenusedinterchangeably. Indeed,histor -\nically the words “galvanic” and “voltaic”, attributed to Lu igi\nGalvani and Alessandro Volta respectively (both of whom\nwere behindthe inventionof a battery), have the same mean-\ning. Inlateruse,though,theterm photovoltaiceffect gaineda\nmuch wider recognition and at the same time, became much\nmore generic; it came to mean anyeffect of electric energy\ngeneration as a result of light illumination (perhaps with t he\nexception of photoelectron emission into vacuum, for which\nthe termphotoelectriceffect remainsmorepopular). Still, the\npredominantusage realm of photovoltaic effects became tha t\noftheprocessesinthemodern-daysemiconductorsolarcell s,\ni.e.,theeffectsrelatedtogenerationandsubsequentsepa ration\nof electrons and holes in semiconductor structures. To dis-\ntinguish these heterostructure effects from photocurrent gen-\nerationin the bulk of a non-centrosymmetric homogeneous\nmedium, the latter adopted the name “ bulkphotovoltaic ef-\nfect”; its much less popular synonym “photogalvanic effect ”\nhasnotneededthisaddition.\nForthisreason,inourattempttoclassifythepredictedpla s-\nmoniceffect,adoptingthename“plasmonicbulkphotovolta ic\neffect” can be confusing because one is tempted to forget\nthat a plasmonic metamaterial is only “effectively bulk” in\nthe sense that a macroscopic excitation such as an incident\nplane wave will not discriminate the individual nanopartic les\nand will interact with the metamaterial as if it were homoge-\nneous. Microscopically, though, it is not homogeneous; the\nvery existence of localized surfaceplasmon excitations im-\nply that there are surfaces that give rise to them. The word-\ning“bulkplasmonicphotovoltaiceffect”wouldbeevenmore\ndangerousbecauseit may mislead oneinto thinkingthat bulk\nplasmons,ratherthansurfaceplasmons,areat work,whichi snot the case. Therefore, to avoid such confusions, we argue\nthatplasmonicphotogalvaniceffect isthepropernameforthe\nreportedphenomenon.\nInthe broaderpicture,it hasattributesofbotha bulkeffec t\nandasurfaceeffectdependingonthelengthscale, andinthi s\nway it bridges the gap between the inner photoelectric effec t\n(deﬁnedaselectricchargecarriergenerationduetophoton ab-\nsorption in a bulk material) and the outer photoelectric eff ect\n(deﬁnedas electric chargecarrieremission fromonemedium\ninto another across an interface) [51]; this is not to be con-\nfused with internal vs. external photoelectric effect, bot h of\nwhicharesubclassesoftheouterphotoelectriceffectdepe nd-\ningonwhetherthesecondmediumissolidornot[52].\nOne does need to keep in mind, however, that the present\nnumerical demonstration of plasmonic photogalvanic effec t,\nbased on a 2D arrangement of nanoparticles, has only suc-\nceeded in demonstrating the equivalence for a “thin slab”\nof metamaterial, analogous to a thin ﬁlm of a bulk non-\ncentrosymmetric crystalline medium. Further comparison o f\nplasmonic and conventional photogalvanic effect involvin g a\n3D arrangement of nanoparticles should therefore be forth-\ncoming. It is rather straightforward, the key challenge be-\ning the means to provide uniform illumination in a medium\nwithsufﬁcientlymanylossymetallicinclusionsunderthec on-\ndition of localized plasmonic resonance (and hence, highly\ninhomogeneous and strongly enhanced local ﬁelds). Over-\ncoming this challenge may result in a cap on the maximum\nnanoparticle density and therefore, in a limit on how strong\nplasmonicphotogalvaniceffectcanbe.\nThatsaid,theresultspresentedinthenumericaldemonstra -\ntion in Section III point out that in terms of the relevant ten -\nsor components, plasmonic photogalvanic effect can be sev-\neral orders of magnitude stronger than the conventional one .\nHence,callingtheeffect giantplasmonicphotogalvaniceffect\niswarranted,onparwithgiantmagnetoopticaleffectspres ent\nin chiral metamaterials and similarly surpassing the natur ally\noccurringchiralityinbulkmediabyordersofmagnitude[53 ].\nV. CONCLUSIONSAND OUTLOOK\nTo summarize, we have theoretically predicted new\nfunctionality in photoconductive metamaterials: the gi-\nant plasmonic photogalvanic effect , analogous to photogal-\nvanic (or bulk photovoltaic) effect in homogeneous non-\ncentrosymmetric media [23]. The reported effect is numer-\nically demonstrated in a metamaterial containing similarl y\noriented non-centrosymmetricmetallic nanoparticles emb ed-\nded in a homogeneous semiconductor matrix, when illumi-\nnated by a wave with photon energies insufﬁcient for the in-\nternal photoelectric excitation in the semiconductor. Due to\nthe lower degree of symmetry in the nanoparticles (the ab-\nsence of mirror symmetry in the x-yplane), the ﬂux of “hot”\nelectronsemittedfromthenanoparticleswiththeassistan ceof\na resonant plasmonic excitation aquires directionality. A ver-\naged over the volume of the metamaterial, this directionali ty\nis manifest as an electromotiveforce resulting from homoge -\nneousexternallightillumination(thephotogalvaniceffe ct).7\nWe have also found that the resulting current density gen-\nerated in a layer of nanoparticles grows as the particle shap e\nchanges from cylindrical to conical. Furthermore, we have\ncalculated the component ˜βzxxof the effective third-rank ten-\nsorthatrelatestheinducedcurrentdensitytotheincident elec-\ntric ﬁeld intensity [see Eq. (10)]. We have shown that the ef-\nfective˜βzxxforthenanoparticlearrayexceedsthecomponents\nβijkfor the naturally occurring ferroelectrics with bulk pho-\ntovoltaic effect [32, 33, 44] by orders of magnitude. Hence,\nthereportedplasmoniceffectcanberegardedasa“giant”ve r-\nsionofthephotogalvaniceffectoccurringinnaturalmater ials,\nadding to the assortment of effects that are much stronger in\nartiﬁcialmetamaterialsthaninnaturalmedia.\nOnafundamentallevel,theproposedeffectisimportantfor\nour understanding of plasmon-assisted electron photoemis -\nsion processes, and constitutes a new way of exploringlight -\nmatterinteractionatthemesoscopicscale. Onamoreapplie d\nlevel, our results can be used in a variety of ways, from a\nnewwayofcharacterizingplasmonicstructures(distinctf rom\npurelyopticalorelectron-microscopyapproaches),tonew de-\nsignsofphotodetectorsoperatingoutsideofthespectralr ange\nfor band-to-band transitions for semiconductors. It can al so\nbe used to increase the performanceof photovoltaicelement s\nby making use of longer-wavelengthphotons, which are nor-\nmally lost in traditionalcells based on the inner photoelec tric\neffect [13, 18]. The result that photoemission predominant ly\noccurs at the base of the nanocones makes them particularly\nappealingforphotovoltaicdeviceswherenanoparticlesar ede-\npositedona semiconductorsubstrate.\nMoreover, we can regard photoemission in non-\ncentrosymmetric plasmonic nanoantennas as a “ratchet”(Brownian-motor) mechanism [54, 55] that works as an\noptical rectiﬁer or “rectenna” [56]. The fundamentalconce pt\nof optical ratchet devices and optical rectennas attracts m uch\nattentioninrecentdevelopmentsinnanotechnology[57–60 ].\nFinally, we note that we have only considered the plas-\nmonic analogue of the linear bulk photovoltaic effect, sinc e\nthenanoparticleshapewaschosentobeachiral. Itisexpect ed\nthat chiral (or planar chiral) nanoparticles would provide the\nplasmonic analogue to the circular bulk photovoltaic effec t\n[23, 33],describedonanalogywith Eq.(9)as\njC\ni=iβC\nij[e×e∗]jS0. (11)\nThus, designing plasmonic nanostructures with anomalousl y\nhigh effective ˜βC\nijis expected to result in new ways to char-\nacterize both chiral plasmonic nanostructures and chirali ty-\nrelatedpropertiesoflight.\nACKNOWLEDGMENTS\nThe authors would like to thank Jesper Mørk for valuable\ncomments. 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The \naverage value of in plane spin component in this 2- D model reaches zero at one particular \ntemperature. This particular temperature obtained u sing our theoretical model agrees with \nthe experimental value of the temperature of rf spu ttered polycrystalline strontium ferrite \nthin films deposited on polycrystalline Al 2O3 substrates (500 0C). This spin reorientation \ntemperature solely depends on the values of energy parameters used in our modified \nHeisenberg Hamiltonian equation. \n  \n1. Introduction: \n         Ferrite thin films are prime candidates in  the application of magnetic memory \ndevices and monolithic microwave integrated thin fi lms (MMIC). Both strontium and \nbarium ferrites belong to M-type hexagonal category . Due to its hard magnetic and \nuniaxial properties, hexagonal ferrites are unique among other ferrite materials. Strontium \nferrite thin films have been synthesized on Al 2O3 polycrystalline substrates using rf \nsputtering [1, 2], magnetron sputtering [3] and pul sed laser deposition [4]. The \norientation of magnetic easy axis of ferrites vastl y depends on the deposition or annealing \ntemperature, orientation of the substrate and gas p ressure inside deposition chamber.  \n       Spin was assumed to be in the plane of y-z, and only two spin components (S y and \nSz) were taken into account in this 2-D model. The av erage value of in plane (S y) \ncomponent was determined as a function of the tempe rature. By plotting S y versus \ntemperature, the temperature at which S y approaches was investigated. Below this \nparticular temperature (T s), the magnetic easy axis orients in the plane of t he film.  2Different values of spin exchange interaction, long  range magnetic dipole interaction, \nsecond order magnetic anisotropy, fourth order anis otropy and stress induced anisotropy \nwere plugged in the equation of our modified Heisen berg Hamiltonian, in order to \ndetermine the variation of T s. The out of plane easy axis orientation of barium ferrite thin \nfilms belonging to hexagonal ferrite has been previ ously explained by us using this model \n[5]. In addition, the easy axis orientations of sof t spinel ferrite [6] and ferromagnetic [7] \nthin films have been explained previously. In all t hese cases, S y component was plotted \nagainst the temperature in order to investigate the  orientation of magnetic easy axis. \nFurthermore, the total magnetic energy of Nickel fe rrite and ferromagnetic films has been \nexplained using unperturbed [11], the second order [8, 12, 16] and third order perturbed \nHeisenberg Hamiltonian [9, 14, 15]. According to ou r previous studies, stress induced \nanisotropy effects on coercivity [13]. However, the  unperturbed Heisenberg Hamiltonian \nhas been employed in this report.     \n \n2. Model:  \nThe total energy of a magnetic thin film is given b y following modified Heisenberg \nHamiltonian [8, 9].  \nH= -∑ ∑ ∑∑ − − − +\n≠ n m m mz\nmz\nm\nn m mn n mn mn m\nmn n m\nn m S D S D\nrSr rS\nrSSSSJ\nm m\n,4 ) 4 ( 2 ) 2 (\n5 3) ( ) ( )).)( .( 3 .( .λ λ ωrrrrrrrr \n             ∑∑− −\nm mm s m Sin K SH θ2.. rr\n                                                               (1)                                                                   \nHere J, ω, θ,  ,, ,, ,) 4 ( ) 2 (\ns out in m m K HH D D  m and n represent spin exchange interaction, \nstrength of long range dipole interaction, azimutha l angle of spin, second and fourth order \nanisotropy constants, in plane and out of plane int ernal magnetic fields, stress induced \nanisotropy constant and spin plane indices, respect ively. When the stress applies normal \nto the film plane, the angle between m th  spin and the stress is θm.  \n  \nThe long range magnetic dipole interaction of hexag onal ferrite calculated in one of our \nprevious research articles has been used to find th e total energy per unit spin given in \nfollowing equation [5].   \nE( θ) = 3NJ+5(N-1)J+ ω[N(88.3197 sin 2θ+11.3541 sin θcos θ -127.9435 cos 2 θ)   3            +(N-1) (93.0605 sin 2θ +25.3002 sin θ cos θ - 15.423 cos 2θ)] \n) sin cos sin (3 cos cos 2 ) 4 (\n1 14 ) 2 ( 2θ θ θ θ θs out in mN\nmN\nmm K H HN D D + + + − −∑ ∑\n= =     (2) \nHere ∑\n=N\nmmD\n1) 2 (and ∑\n=N\nmmD\n1) 4 ( represent the total second and fourth order anisot ropy \nconstants in the whole film.  \n \nBecause only the Fe +3  ions contribute to the net magnetic moment of hexa gonal ferrites, \nthe equation derived for barium ferrite can be appl ied for strontium ferrite too. \n \n3. Results and Discussion: \nThe average value of in plane spin component is giv en by \nθθθ\nππ\nded e\nS\nkT EkT E\ny\n∫∫\n−−\n=\n00sin \n                     (3) \nHere E, k and T indicate the total magnetic energy given in equati on (2), Boltzmann’s \nconstant and absolute temperature. Thickness of the  strontium ferrite films incorporated \nfor these simulations were approximately 2.5 µm thick. So value of N employed for these \ninvestigations was 998.  \nFigure 1 indicates the variation of ySwith temperature. When J = 10 -33  Joules, ω=10 -30 \nJoules, ∑\n=N\nmmD\n1) 2 (=10 -29  Joules, ∑\n=N\nmmD\n1) 4 (=10 -42  Joules, K s= 10 -30  Joules, H in =10 -32  Am -1 \nand H out =10 -39 Am -1, yS reaches zero at 773 K. This implies that in plane orientation of \neasy axis vanishes above 500 oC. Therefore, our experimental results of polycryst alline \nstrontium ferrite thin film can be explained using this theoretical model [1]. The spin \nreorientation temperature (T s) vastly depends on the energy parameters. When J =  10 -33  \nJoules, ω=10 -29  Joules, ∑\n=N\nmmD\n1) 2 (=10 -29  Joules, ∑\n=N\nmmD\n1) 4 (=10 -42  Joules, K s= 10 -30  Joules,  4Hin =10 -32  Am -1 and H out =10 -39  Am -1, yS approaches zero at 160 K as shown in figure 2. \nThis means that T s can be reduced by increasing ω.    \n \n100 200 300 400 500 600 700 800 900 1000 00.1 0.2 0.3 0.4 0.5 0.6 0.7 \ntempurature(K) Sy \n \nFig 1: yS versus temperature for the first set of values of energy parameters. \n \n \n \n \n \n \n \n \n \n \n  5100 200 300 400 500 600 700 800 900 1000 00.2 0.4 0.6 0.8 11.2 1.4 1.6 x 10 -4 \ntempurature(K) Sy \n \n \nFig 2: yS versus temperature for the second set of values of  energy parameters. \n \nThe variation of T s with J is given in figure 3. Other values of energ y parameters were \nkept at ω=10 -30  Joules, ∑\n=N\nmmD\n1) 2 (=10 -29  Joules, ∑\n=N\nmmD\n1) 4 (=10 -42  Joules, K s= 10 -30  Joules, \nHin =10 -32  Am -1 and H out =10 -39  Am -1 for this simulation. So T s gradually increases with J. \nAbove J=10 -30  Joules, a rapid variation of T s can be observed. Below J=10 -31 Joules, T s \ndoesn’t vary with J. Between J=10 -30  and 10 -31 Joules, T s slightly varies with J. Spin \nexchange interaction is related to the coupling bet ween spins. So spins are restricted to \nrotate freely in a particular direction at higher v alues of J. As a result, T s increases with J. \n   6\n \nFig 3: Variation of T s with J. \n \nFigure 4 shows the variation of T s with K s. Below K s=10 -29  Joules, T s slightly varies with \nKs. Above K s=10 -29  Joules, T s drastically decreases with K s. Other energy parameters \nwere set to J = 10 -33  Joules, ω=10 -30  Joules, ∑\n=N\nmmD\n1) 2 (=10 -29  Joules, ∑\n=N\nmmD\n1) 4 (=10 -42  \nJoules, H in =10 -32  Am -1 and H out =10 -39  Am -1 in this simulation. Due to in plane stress, \nspins prefer to align in the in plane direction [10 ]. As a matter of fact, T s decreases with \nKs. As shown in figure 5, T s varies with H in . Other parameters were kept at J = 10 -33  \nJoules, ω=10 -30  Joules, ∑\n=N\nmmD\n1) 2 (=10 -29 Joules, ∑\n=N\nmmD\n1) 4 (=10 -42  Joules, K s= 10 -30  Joules \nand H out =10 -39  Am -1 for this simulation. Below H in =10 -29 Am -1, Ts doesn’t vary with H in .  \nAbove H in =10 -29  Am -1, T s rapidly decreases with H in . At larger values of internal in plane \nmagnetic field, spins can easily rotate in the in p lane direction. Then T s decreases with \nHin .  \n \n \n \n \n  7 \n \nFig 4: Plot of T s versus K s. \n \n \nFig 5: Dependence of T s on H in . \n \n4. Conclusion: \n        The easy axis orientation of polycrystallin e strontium ferrite thin films sputtered on \npolycrystalline Al 2O3 substrates could be explained using our modified H eisenberg \nHamiltonian model. The total energy of oriented hex aferrite thin films derived from this \nmodel was employed in this case, rather than consid ering 2nd  or 3 rd  order perturbation.  8Variation of average value of in plane spin compone nt with temperature was investigated. \nThe spin reorientation temperature solely depends o n ω,  J, K s and H in . However, T s is \nslightly sensitive to other energy parameters too. Below 500 oC, the easy axis of \nstrontium ferrite is oriented in the plane of the f ilm. This particular spin reorientation \ntemperature could be obtained at J = 10 -33  Joules, ω=10 -30  Joules, ∑\n=N\nmmD\n1) 2 (=10 -29  Joules, \n∑\n=N\nmmD\n1) 4 (=10 -42  Joules, K s= 10 -30  Joules, H in =10 -32  Am -1 and H out =10 -39  Am -1. However, \nthe spin reorientation temperature could be varied in a wide range by changing the values \nof J, ω,∑\n=N\nmmD\n1) 2 (, ∑\n=N\nmmD\n1) 4 (, K s, H in  and H out . Because the exact experimental values of J, \nω,∑\n=N\nmmD\n1) 2 (, ∑\n=N\nmmD\n1) 4 (, K s, H in  and H out  of strontium thin films can’t be found, a \nreasonable set of values has been employed for thes e explanations.  \n   References \n1.  H. Hegde, P. Samarasekara and F.J. Cadieu, 1994. No nepitaxial sputter synthesis of \naligned strontium hexaferrites, SrO.6(Fe 2O3), films. J. Appl. Phys. 75(10), 6640-\n6642. \n2.  Antony Ajan, B. Ramamurthy Acharya, Shiva Prasad, S . N. Shringi and N.   \nVenkataramani, 1998. Conversion electron Mössbauer studies on strontium ferrite \nfilms with in-plane and perpendicular anisotropies.  J. Appl. Phys. 83, 6879. \n3.  A. Kaewrawang, A.Nagano Ghasemi, Xiaoxi Liu and  A.  Morisako, 2009. Properties \nof Sr Ferrite Thin Films on Al-Si Underlayer. IEEE trans. on Mag. 45(6), 2587-2589.   \n4. M.E. Koleva, S. Zotova, P.A. Atanasov, R.I. Tomo v, C. Ristoscub, \n    V. Nelea, C. Chiritescu, E. Gyorgy, C. Ghica an d I.N. Mihailescu, 2000. Sr-  \n     ferrite thin films grown on sapphire by pulsed  laser deposition. Applied Surface   \n     Science 168, 108-113. \n5. P. Samarasekara and Udara Saparamadu, 2013. Easy  axis orientation of Barium hexa-  \n      ferrite films as explained by spin reorientat ion. Georgian electronic scientific   \n      journals: Physics 1(9), 10-15.  96. P. Samarasekara and Udara Saparamadu, 2012. Inve stigation of Spin Reorientation \n     in Nickel Ferrite Films. Georgian electronic s cientific journals: Physics 1(7), 15-20.  \n7. P. Samarasekara and N.H.P.M. Gunawardhane, 2011.  Explanation of easy axis   \n    Orientation of ferromagnetic films using Heisen berg Hamiltonian. Georgian electronic    \n    scientific journals: Physics 2(6), 62-69. \n8. P. Samarasekara, 2010. Determination of Energy o f thick spinel ferrite films using   \n    Heisenberg Hamiltonian with second order pertur bation. Georgian electronic scientific   \n    journals: Physics 1(3), 46-52. \n9. P. Samarasekara, 2011. Investigation of Third Or der Perturbed Heisenberg  \n    Hamiltonian of Thick Spinel Ferrite Films. Inve nti Rapid: Algorithm Journal 2, 1-3.  \n10. S. Chikazumi, Physics of magnetism, 1964, John Wiley & Sons., pages182-184.  \n11. P. Samarasekara, 2007. Classical Heisenberg Ham iltonian solution of oriented \nspinel ferrimagnetic thin films. Electronic Journal  of Theoretical Physics    \n4(15), 187-200.  \n12. P. Samarasekara and S.N.P. De Silva, 2007. Heis enberg Hamiltonian solution of   \n       thick ferromagnetic films with second order   perturbation. Chinese Journal of  \n       Physics 45(2-I),  142-150. \n13. P. Samarasekara, 2003. A pulsed rf sputtering m ethod for obtaining higher deposition   \n      rates. Chinese Journal of Physics 41(1), 70-7 4. \n14. P. Samarasekara and William A. Mendoza, 2011. T hird order perturbed Heisenberg  \n      Hamiltonian of spinel ferrite ultra-thin film s. Georgian electronic scientific   \n      journals: Physics 1(5), 15-18. \n15. P. Samarasekara and William A. Mendoza, 2010. E ffect of third order perturbation on  \n      Heisenberg Hamiltonian for non-oriented ultra -thin ferromagnetic films. Electronic   \n      Journal of Theoretical Physics 7(24), 197-210 .  \n16. P. Samarasekara, M.K. Abeyratne and S. Dehipawa lage, 2009. Heisenberg   \n      Hamiltonian with Second Order Perturbation fo r Spinel Ferrite Thin Films. Electronic   \n      Journal of Theoretical Physics 6(20), 345-356 . \n \n \n \n "    },    {        "title": "2301.01728v1.Preparation_and_Characterization_of_NixMn0_25_xMg0_75Fe2O4_Nano_ferrite_as_NO2_Gas_Sensing_Material.pdf",        "content": "Preparation and Characterization of Ni xMn 0.25-xMg 0.75Fe2O4 Nano -ferrite  \nas NO 2 Gas Sensing Material   \n \nHussein I. Mahdi 1, Nabeel A. Bakr 2, Tagreed M. Al -Saadi 3 \n1,2 Department of Physics, College of Science, University  of Diyala, Diyala, IRAQ  \n3 College of Education for Pure Science, Ibn Al Haitham, University of Bagdad, Bagdad, IRAQ  \n*Corresponding author: sciphydr2110@uodi yala.edu.iq  \n \nAbstract   \n       NixMn 0.25-xMg 0.75Fe2O4 nano -ferrites (where x = 0.00, 0.05, 0.10, 0.15 and 0.20)  were \nproduced via sol-gel auto-combustion technique. Investigations were done into how the \nincorporation of Ni ions affects  the Mn 0.25Mg 0.75Fe2O4 ferrite's structure, morphological, magnetic, \nand NO 2 gas sensing features. All the samples are single -phase,  based on the structural study \nutilizing the X -ray diffraction (XRD) pattern. In terms of the structure of the cubic spinel, \naccording to the XRD study, the crystallite sizes range from 24.30 to 28.32 nm, indicating nano -\ncrystallinity. The synthesis of sph erical nanoparticles with a small modification in particle size \ndistribution was verified via FE -SEM images. The study  found that  the size of particles is tiny \nenough to act superparamagnetically . The area of hysteresis loop is almost non -exist ing, thus \nreflecting typical soft magnetic materials according to magnetic measurements by VSM carried \nout at room temperature. Furthermore, the conductance responses of the Ni xMn 0.25-xMg 0.75Fe2O4 \nnano -ferrite were measured by exposing the ferrite to oxidiz ing (NO 2) gas at different operating \ntemperatures. The results show that the sensor boasts shorter response and recovery times, as well \nas a higher sensitivity 707.22% of the sample (x=0.20) for nano -ferrite.  \n \nKeyword:  Mn-Mg ferrite, Ni ions  substitution , sol- gel auto -combustion technique , XRD, VSM, \nNO 2 gas sensor.  \n \n1. Introduction  \n       Because chemical sensors may control emissions and identify dangerous contaminants , their \ndemand has risen dramatically.  The most promising chemical sensors are metal oxide \nsemiconductor ones since they offer several benefits like low cost, compact size, low power \nconsumption, and online operation. They have received extensive research for a long time because \nthey are very suitable with microelectronic processes [1]. Utilizat ion of nanocrystalline materials \nfor gas sensing have recently sparked a great deal of curiosity [2]. Ferrites have proven to be \neffective materials for gas semiconductor detectors [ 3]. Whenever a semiconductor gas sensor is \nexposed to various gas environments, it acquires the ability to modify the conductivity of the \ndetecting material.         \n       The surface -controlled technique of gas sensing depends on the interaction among both gas \nmolecules to be identified and adsor bed oxygen. The operating  temperature, the type of gas being used, and the type of detector all affect how the detector responds to gas [ 4]. The oxides having a \nstructural formula of AB 2O4 are significant for gas detection purposes and were studied for the  \nidentification of both oxidizing and reducing gases. These oxides are preferred above all spinel -\ntype metal oxide semiconductor detector, due to the magnetic materials used in high frequency \napplications as micro -electronic/magnetic devices [5]. The most exciting features of spinel ferrites \nfor gas detecting are their chemical makeup and structure , in which transition or post -transition \ncations occupy two different cation positions [6]. The spinel ferrites, including MgFe 2O4, ZnFe 2O4, \nMnFe 2O4, NiFe 2O4, and  CoFe 2O4, have shown  excellent sensitivity for  a wide range  of gases due \nto their stability in thermal and chemical atmospheres, quick reaction and recovery times, \ninexpensive, and straightforward electronic structures [ 7,8]. Magnesium ferrite is specifically \namong the most significant ferrites due to its low magnetic and dielectric losses , high resistivity, \nand other properties that make it an essential component in catalytic reactions, detectors, and \nadsorption  [9]. Depend ing on the preferred energies for divalent and trivalent ions in the spinel \nstructure, it possesses an inverse spinel structure with Mg2+ ions in octahedral sites and Fe3+ ions \nequally divided over tetrahedral and octahedral sites [ 10].  \n       The sol-gel, molten -salt approach, hydrothermal , co-precipitation, and microemulsion \ntechniques were all employed to obtain nano -sized spinel ferrite powder [ 11,12]. Among the \nnumerous techniques, the sol-gel technique is a convenient, environmentally friendly, and l ow-\ncost technique for synthesizing ferrites at relatively low temperatures in a short period of time [ 13]. \n       Doping is a significant and successful method for fine -tuning the required properties of \nsemiconductors [ 14,15]. The dopant might improve the gas-sensing characteristics of metal -oxide \nsemiconductors by modifying the energy -band structure, improving the morphology and surface -\nto-volume ratio, and developing extra active centers at the grain boundaries [ 16].  \n       In the present work, we report the synthesis of Ni xMn 0.25-xMg 0.75Fe2O4 nano -ferrite by using  a \nsimple sol-gel auto-combustion technique  and its application  as NO 2 gas senso r has been \nsystematically investigated , where  the results  are presented and discussed.  \n2. Experimental Part  \n2.1. Materials and method       \n       The general formula of the spinel ferrite of NixMn 0.25-xMg 0.75Fe2O4 (where x = 0.00, 0.05, \n0.10, 0.15 and 0.20) has been produced via sol-gel auto-combustion  technique. Analytical -grade \nmaterials of f erric nitrate nonahydrate Fe(NO 3)3.9H 2O, m agnesium nitrate  hexahydrate  \nMg(NO 3)2.6H 2O, manganese nitrate  mono hydrate  Mn(NO 3)2.H 2O, and nickel nitrate  hexahydrate  \nNi(NO 3)2.6H2O are used as precursors  of iron and  other metals , whereas  citric acid (C 6H8O7) is \nused as a complexant/fuel agent for the auto -combustion process. The required masses of the raw \nmaterials required to prepare the ferrite are  shown in Table 1. Th ese values are  obtained using the \nfollowing equation:  \nWt (g)= Mw (g/mol) × M (mol/L) × V (L)          ……….………. (1) \nWhere, Wt is the  mass of the raw material , Mw is the  molecular weight  of the raw material , M is \nthe number of moles required for the material in one liter of solvent , and V is the  volume of solvent . \nMetal nitrates were entirely dissolved  in small quantities of distilled water after being weighed. \nThis solution was then mixed with citric acid to achieve a molar ratio of these nitrates and citric acid of 1:1 in the final sample. After that, ammonia is added to the mixture  in droplets to balance \nthe (pH) to (~7) while mixing it . Combustion  reaction  occurs  among nearby metal nitrates and \ncitrate molecules, resulting in a polymer network with  colloidal dimensions recognized as sol  [17-\n19]. While continuous ly mixing and heating the solution for one hour at 90 °C, the solution \nis evaporated, and  then it held at this temperature until it solidified in a gel form. The gel  then is \ncooked to 120 ◦C in order to trigger auto -combustion  where t he dried gel  is burnt until it is totally \nconsumed to produce loose powder. Finally, to get the required ferrite , the resultant powder is \ncrushed in an agate mortar . The freshly as-prepared  ferrite powder is then heated  for two hours at \n600 ◦C. \n \nTable 1 . The masses of raw materials required to obtain NixMn 0.25-xMg 0.75Fe2O4 ferrite . \n \n2.2. Fabrication of gas sensors   \n       For each sample, 1.75 g  of powder is collected  and a pressure of 200 bar  is applied by manual \npress for 120  seconds  to produce a disc  with a diameter of 1 cm and a thickness of 3.5 mm . The \ndisc is then placed  in furn ace at a temperature of 900 ◦C for a period of two hours . Thin copper \nwires are used as connecting leads, and silver paste is used to construct the electrodes on one side \nof the sample , while electrodes are placed on all specimen surfaces to obtain Ohmic contacts  [20]. \nThe electrodes are fabricated for the five nano -ferrite samples, then the sensitivity of each sample \nto NO 2 gas at a constant concentration (65 ppm) is tested by a gas sensitivity test system.  \n \n2.3. Characterization  \n       By using  powder  X-ray diffractometer (Philips PW1730 ), the ferrites'  XRD (X-ray diffraction) \npattern is obtained via  Cu-Kα (Wavelength -1.5406  Å) radiation , scan range: 20o – 80o, and scan \nspeed: 6 deg./min.  The ferrites' surface morphology was investigated  utilizing (MTRA 3 LMU) \nfield emission scanning electron microscope (FE -SEM)  combined with Energy Dispersive X -ray \nAnalyzer (EDX) . A vibrating sample magnetometer  (EZ VSM model 10) was used to measure the \nmagnetism of some specimens . In order to detect (NO 2) gas at various temperatures, the gas \nresponse characteristics of sintered discs  (900°C) were investigated. The resistance  of gas sensor \nsamples  is measured  by using Impedance Analyzer (UNI -TUT81B)  equipped with a computerized  \ntesting tool.  \n x Composition  Ferric  \nnitrate (g)  Magnesium \nnitrate (g)  Manganese \nnitrate (g)  Nickel \nnitrate (g)  Citric \nacid (g)  \n0.00 Mn 0.25Mg 0.75Fe2O4 32.32  7.6923  1.8900  0.00 23.0556  \n0.05 Ni0.05Mn 0.20Mg 0.75Fe2O4 32.32  7.6923  1.5120  0.5816  23.0556  \n0.10 Ni0.10Mn 0.15Mg 0.75Fe2O4 32.32  7.6923  1.1340  1.1632  23.0556  \n0.15 Ni0.15Mn 0.10Mg 0.75Fe2O4 32.32  7.6923  0.7560  1.7448  23.0556  \n0.20 Ni0.20Mn 0.05Mg 0.75Fe2O4 32.32  7.6923  0.3780  2.3264  23.0556  3. Results and Discussion  \n \n3.1. X -Ray Diffraction  \n       X-ray diffraction (XDR) analysis was carried out to determine the phase formation of the \nNixMn 0.25-xMg 0.75Fe2O4 nano -ferrite in the 2θ range 10o ≤ 2θ ≤ 80o. Figure  1 shows the indexed x -\nray diffraction patterns of the NixMn 0.25-xMg 0.75Fe2O4 ferrite annealed at 600 ◦C. The presence of \n(220), (311), (400), (422), (511), (440), and (533) planes confirms  the formation of cubic spinel \nstructure.  The diffraction peaks agree  with the JCPDS card number  89-3084  [21]. Additionally, \nthe size of the crystallites gradually decreased  as the amount of Ni doping increased. This was \nshown in the XRD pattern, where the  NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite peaks get shifted to \nhigher angles , as the angle value increased, as listed in Table 2.   \nBy using the Scherrer’s equation, the crystallite size D of the NixMn 0.25-xMg 0.75Fe2O4 specimens \nwas determined from the broadening of the (311) peak in the XRD patterns.  \n            𝐷=K λ\n𝛽cos θ                                         ……….………. ( 2) \nWhere, K is constant assumed to be 0.9, λ is X -ray wavelength equal to 1.5406 (Å), β is the full \nwidth at half maximum (FWHM) of the highest intensity diffraction peak expressed in radians, \nwhile θ is the Bragg's angle of the diffraction peak [22,23].  \n       By using the following equation, the cubic unit cell lattice parameter (a)  for all compounds \nwas computed via diffraction planes:  \n            a = d hkl √ℎ2+𝑘2+𝐼2                            ……….………. (3) \n       Where, d is the interplanar  spacing and  (h, l and k)  are the Miller indices of the crystal planes \n[24]. The X -ray density ( 𝜌𝑥) can be computed via the following equation:  \n           𝜌𝑥= 8 Mw\nNA a3                                     ……….………. ( 4)  \n       Where, M W represents the molecular weight  and NA is the Avogadro's number [25].  \nThe lattice parameter  (a), XRD density (ρx), and crystallite size (D) for all samples are  given in \nTable 3.  10 20 30 40 50 60 70 80(533) x=0.20\n x=0.15\n x=0.10\n x=0.05\n x=0.00\n(440)(511)(422)(400)(311)(220)Intensity (arb.u)\n2q (degree) \nFigure 1. X-ray diffraction patterns of NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite prepared by auto -\ncombustion method.  \n \n       Increasing the concentration of Ni2+ leads to increase the lattice constant  of ferrite compounds \nas listed in Table 3 . Smaller Fe3+ ions have been observed to migrate from tetrahedral to octahedral \npositions in response to Ni2+ addition [ 26,27], therefore tetrahedral sites are enlarged as a result of \nincreasing the lattice constant [ 28,29]. Moreover, this caused the lattice to grow an d the density to \ndrop,  indicating that the lattice constant  has changed as a result of the dopant ions being absorbed \ninto the lattice could have taken an interstitial position s among  the hosting ions [ 20]. \n \nTable 2. Structure properties of  the NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite.  \nh k l 2θ (deg)  \n(JCPDS)  2θ (deg)  \n(x=0.00)  2θ (deg)  \n(x=0.05)  2θ (deg) \n(x=0.10)  2θ (deg) \n(x=0.15)  2θ (deg) \n(x=0.20)  \n220 30.115  30.1365  30.4563  30.3111  30.3932  30.3938  \n311 35.466  35.4950  35.8238  35.7308  35.8876  35.7541  \n400 43.123  43.2299  43.5441  43.4461  43.4725  43.3345  \n422 53.478  53.5835  53.9189  53.7877  53.8403  53.6563  \n511 57.000  57.1528  57.4708  57.3573  57.4057  57.2337  \n440 62.594  62.7239  62.8946  62.9067  62.9564  62.8185  \n533 74.049  74.2529  74.3735  74.2861  74.3755  74.2936  \n Table 3. Unit cell constant (a), density (ρ x) and crystallite size (D) of NixMn 0.25-xMg 0.75Fe2O4 \nnano -ferrite prepared by  auto-combustion method.  \nx  Composition  a (Å)  ρx (g/cm3) D (nm)  \n0.00 Mn 0.25Mg 0.75Fe2O4 8.36743  5.250  28.31  \n0.05 Ni0.05Mn 0.20Mg 0.75Fe2O4 8.37691  5.232  24.34  \n0.10 Ni0.10Mn 0.15Mg 0.75Fe2O4 8.38131  5.224  24.34  \n0.15 Ni0.15Mn 0.10Mg 0.75Fe2O4 8.38245  5.222  28.32  \n0.20 Ni0.20Mn 0.05Mg 0.75Fe2O4 8.38717  5.213  24.30  \n  \n3.2. FE -SEM and EDX Analysis  \n       To assess the morphology of the fabricated samples, (FE -SEM ) was used. Figure 2 illustrates \nthe NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite  micro images at a 200 nm scale  after annealing at 600 °C . \nThe observed FE-SEM  images  made it extremely apparent that the magnetic ferrite particles were \ncreated through some aggregation at the nanoscale . The FE -SEM images show porous, sponge -\nlike shape particles of the samples (x  = 0.00, and 0.05). Most likely, the gases released during the \ngel's combustion process are what caused the pores to form [ 30]. In addition, the images show \nparticles that are spherical or semi -spherical and nonhomogeneous in form of the samples (x=0.10, \nand 0.15 ), as well as the images show homogeneous distributi on and  spherical nanoparticles of the \nsample (x  = 0.20). The FE -SEM images  also show the formation of tiny  agglomerated grains with \nsurface spaces or  voids  and no distinct shape.  The a gglomerates are where the porosity is located . \nSince gas detecting is a surface phenomenon and porosity is essential, the reported porous \nmicrostructure is beneficial for sensing purposes [ 31]. It is obviously shown in the micrographs \nthat the particles  structures  of the NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite  are very coarse, whi ch \nfacilitate adsorption of oxygen species on the detecting surface. Adsorption of oxygen species is \nresponsible for gas detecting [ 32]. \n \n \n \n \n \n \n \n \n \n \n \n   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2. FE -SEM images of NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite . \n \n \n \nx = 0.05  \nx = 0.10  \n x = 0.15  \nx = 0.20  \nx = 0.00         The EDX spectra  of the NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite  (where x = 0.00, 0.05, 0.10, 0.15 \nand 0.20)  are illustrated in Figure 3, referring that  the spectral lines related to  (Ni, Mn, Mg, Fe and \nO), verify that the synthesized compound NixMn 0.25-xMg 0.75Fe2O4 was achieved.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \nFigure 3. EDX spectra of  NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite.  \n \nx = 0.00  \n x = 0.05  \nx = 0.10  \n x = 0.15  \nx = 0.20  3.4. Magnetic Characteristics  \n       Hysteresis loop is measured utilizing a (VSM)  system , and magnetic characteristics of samples \nwere examined at room temperature (300 K).  Figure 4 shows the hysteresis loop curves of \nNixMn 0.25-xMg 0.75Fe2O4 (x = 0.00, and 0.20).  (S) shaped curves indicate that  standard soft magnetic \nmaterial and magnetic coercivity can be  ignored . In addition, the particles are so small that they \nbehave like superparamagnetic material. Due to the small crystallite size, as i s evidenced by the \nXRD analysis in Table 3 , nanoparticles have superparamagnetic behavior, in which their magnetic \nmoments attempt to align with one another in a specific way [3 3,34]. \n       According to Neel, the distribution of cations among the octahedral and tetrahedral locations \nin spinel ferrite determines the overall magnetic moment  [35]. Saturation  magnetization (M s), \nremnant magnetization (M r), and magnetic coercivity (H c) values were computed from the M -H \ncurves depending on (Ms) measured values .      \n       M-H curves  have demonstrated how chemical compound affe cts magnetic properties. Table 4  \nillustrates the variation in saturation magnetization (M s) values for specimens captured from \nhysteresis loop curves. As 0.20 of the Ni2+ ions were swapped out for Mn2+ ions, the M s value \ndropped from 28.980 (emu/g) for x = 0.00 to 23.400 (emu/g). According to experimental \nobservations, as nickel content rises, the ratio of ferric, manganese, or magnesium ions on the A -\nlocation  decreases , while at the same time, the of  Fe3+ ions grows by the same amount on the \nlocation B. As a result, the A -B interaction is reduced. As a consequence of the ionic moments on \nthe B -sites no longer being maintained parallel to each other, the angles among them st art to form, \nwhich lowers the moment of the B sub lattice itself. Most likely, nickel ions have been replaced \nby cations in the B -sites [34]. Figure 4 shows how the observed values of the remnant \nmagnetization (M r) and coercive field (H c) are so small , demonstrating that the grain size does not \npass the critical diameter of single -domain grain  [34]. The cation distribution has a significant \nimpact on the net magnetic moments and magnetocrystalline anisotropy. Table 4 lists the magnetic \nfactors.  \n \n \n \n \n \n \n \n \n \n \n \nFigure 4. Magnetization (M) versus applied magnetic field (O e) of NixMn 0.25-xMg 0.75Fe2O4   \n(x = 0.00, and 0.20) nanoparticles at 300K.  \n-10000 -8000 -6000 -4000 -2000 0 2000 4000 6000 8000 10000\n-40-30-20-10010203040\n X= 0.00\n X= 0.20\nMagntization(emu/g)\nApplied Magntic Field (Oe)Table 4. Variation of magnetic factors for NixMn 0.25-xMg 0.75Fe2O4 (x =0.00, and 0.20) \nnanoparticles.  \nx Compound  Ms (emu/g)  Mr (eum/g)  Hc (Oe) \n0.00 Mn 0.25Mg 0.75Fe2O4 28.98  10.95 61.50 \n0.20 Ni0.20Mn 0.05Mg 0.75Fe2O4 23.40  7.54 94.00 \n \n3.3. Gas Sensing Features  \n       The gas concentration, material composition, type of conductivity, operating temperature, and \ndifferent controlling parameters are considered as important factors which  affect the gas sensitivity \nor gas response of the metal oxide semiconductor sensor [36]. Depending on the  compound and \noperating temperature , the gas sensitivity of the NixMn 0.25-xMg 0.75Fe2O4 (where x= 0.00, 0.05, \n0.10, 0.15, and 0.20) nano -ferrite against NO 2 gas is studied and computed using  following \nequation:   \n            S = │𝑅ɡ−𝑅𝑎\n𝑅𝑎│× 100 %        [Oxidizing gas]           ……….……….  (5) \nWhere R g and R a represent the electrical resistances in the NO 2 gas and air, respectively [37, 38]. \n       Figure 5 shows the sensing characteristics and variation for each sample against nitrogen \ndioxide NO 2 gas when exposed and removed the examined gasses of the NixMn 0.25-xMg 0.75Fe2O4 \nnano -ferrite . As can be seen from the figure, the resistance value increases when the discs are \nexposed to NO 2 gas (Gas ON), and subsequently decreases when  the gas is closed (Gas OFF) for \nall samples.  At concentration of 65 ppm of  NO 2, the sensor's sensitivity was examined at various \noperating temperatures (200  ◦C, 250 ◦C, and 300 ◦C). In the existence of an oxidizing gas, the \noperating temperature is required to change the material's oxidation state and the conductivity  of \nNixMn 0.25-xMg 0.75Fe2O4 nano -ferrite.  The response time  is defined as the amount of time needed \nto reach  90% of the equilibrium response of the gas , while the recovery time,  is defined as the  \namount of time needed to reach  10% of the baseline resistance [ 39]. From Table  5, it can be seen \nthat samples demonstrate a high sensitivity to nitrogen dioxide gas at 250 ◦C while it is around 300 \n◦C for sample x=0.00. As shown in the FE -SEM images, the sensitivity of the doped samples \nincreases because it has the highest  roughness , and t his is agreement with the findings of \nresearchers  [20,32]. Additionally, the figure also demonstrates that the Ni 0.20Mn 0.05Mg 0.75Fe2O4 \nferrite compound has its highest gas response 707.22% of the sample (x=0.20) at 250 ◦C.  Since \nthe sensitivity process in metal oxides occurs through the adsorption of oxy gen ions on the surface, \ndoping of Mn by Ni generally often enhances the sensitivity because a lack of oxygen causes the \nformation of oxygen voids;  (When the oxygen concentration in the Ni xMn 0.25-xMg 0.75Fe2O4 lattice \nincreases, more oxygen ions (O-2 and -O) adsorb to the sensor's surface due of the gaps or voids ) \n[20]. In contrast to the pre -adsorbed oxygen and other test gases, NO 2 gas has a greater electron \naffinity and is a very reactive and oxidizing gas [40]. After the covalent bond between nitrogen \nand oxygen is formed, NO 2 has an unpaired electron, and remains as one of the atoms with a single \nunpaired electron. Because the nano -ferrite has a short response time (1.2 -11.4) s at 200 ◦C and a \nshort recovery time (1.5 -4.4) s at 250 ◦C , it is possible to  conclude that the sensor has excellent \nsensing characteristics. This fast response of the sensor could be a result of the small particle size, \nwhich causes the particle  boundaries to enlarge. The values of sensitivity, response time, and \nrecovery time are  tabulated in Table 5.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 5. The variation in resistance with time of NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite at different \noperating temperatures.  \nx=0.05  \n x=0.0 0 \nx=0.15 \n x=0.10 \nx=0.20 Table 5. NO 2 gas sensitivity,  response time and  recovery time values of NixMn 0.25-xMg 0.75Fe2O4 \nnano -ferrite  at different operating temperatures . \n \n4. Conclusions  \nUtilizing a simple sol -gel auto -combustion process, Ni xMn 0.25-xMg 0.75Fe2O4 nano -ferrite was \nsynthesized  using metal nitrates as a source of cations and citric acid (C6H8O7) as a \ncomplexant/fuel agent for the auto -combustion process . The NixMn 0.25-xMg 0.75Fe2O4 nano -ferrite \nwith the spinel structure peaks in the XRD patterns corresponding to the investigated systems, and \nno unidentified peaks are observed. The FE -SEM images show microstructures with open pores \nand nanoscale grains with agglomeration, which is nea rly comparable to the crystalline  size \ndetermined by XRD. These findings reveal that, due to the particles being small, the prepared  \nsamples  at-room -temperature hysteresis loop curves exhibit superparamagnetic behavior. \nFurthermore, the results of the NO 2 gas sensing showed that the gas sensor had a good performance \nin terms of its response to the gas. The sensitivity increases with the increasing concentration of \nNi in composition , as well as it also boasts shorter response and recovery times. For gas sensing \napplications, in Mn 0.25Mg 0.75Fe2O4 it is concluded that it is desirable to substitu te manganese ions \nby nickel ions.  \n \nReferences  \n[1] E. Rossinyol , J. 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Using the symmetry analysis, we show that unlike most antiferromagnetic rare-\nearth transition -metal perovskites , a larger structural distortion leads to  a high er TN in hexagonal ferrites \nand manganites , because the K3 structural distortion induces the three -dimensional magnetic ordering, \nwhich is forbidden in the undistorted structure  by symmetry . We also revealed  a near-linear relation \nbetween TN and the tolerance factor and a power -law relation between TN and the K 3 distortion amplitude . \nFollo wing the analysis, a record -high TN (185 K) among hexagonal ferrites was predicted in hexagonal \nScFeO 3 and experimentally  verified in ep itaxially stabilized films . These results add to the paradigm of \nspin-lattice coupling in antiferromagnetic oxides  and suggest s further tunability of hexagonal ferrites if \nmore  lattice distortion can be achieved.  \n  2 \n Spin-lattice couplings  have a significant impact on magnetic properties. In antiferromagnetic (AFM) \northorhombic RTMO3 (o-RTMO3) for example, where R stands for  rare earth, Y, or Sc, and TM stands for \ntransition metal, a larger orthorhombic distortion from the cubic perovskite  structure  correlates with  a lower  \nNeel temperature  (TN) [see supplementary information] , which  may be understood  as the reduction of the \nAFM super -exchange interactions caused by the smaller  TM-O-TM bond angles due to the orthorhombic \ndistortions  [1,2] .  \nThe effect of spin-lattice coupling s may be employed to tune the magnetic properties . Here  we focus on \nincrea sing the TN of hexagonal RFeO 3 (h-RFeO 3), a family of multiferroics material s that are promising  \ncandidates for applications  because of  their spontaneous electric and magnetic polarizations , and  potential \nmagnetoelectric effects  due to the coupling between the ferroelectric and the magnetic  orders  [3,4] . For \nwidespread applications , it is important  to increase the TN of h-RFeO 3 [5], by, e.g.  atomic -scale structural \nengineering based on  the spin-lattice coupling s. \nOn the other hand , in h-RFeO 3, TN increases with the lattice distortion, which is a puzzling trend opposite \nto that in the AFM o -RTMO3[see supplementary information ]. Previously, Disseler  et al. discovered a \ncorrelation between  TN and lattice constants  in h-RMnO 3 and h -RFeO 3 [6]. The higher TN for smaller  R has \nbeen attributed to closer Fe -Fe (or Mn -Mn) distances  [6,7] . This understanding is worth revisiting , since it \ncannot explain that in  AFM  o-RTMO3, the smaller lattice constants  do bring the TM atoms closer, but the \nreduced TM-O-TM bond angle s actually decreases  the AFM exchange interaction s and TN. Hence , there \nshould be a distinct mechanism  of magnetic ordering  and spin -lattice coupling  in h-RFeO 3. Elucidating this \nmechanism will not only provide guidance in increasing TN of h-RFeO 3, but also add to the  paradigm s of \nspin-lattic e coupling in AFM  materials . \nIn this work, we examine the role of the structural distortion in the magnetic ordering in h-RMnO 3 and h -\nRFeO 3. A symmetry analysis  shows  that the three -dimensional magnetic ordering is forbidden in the \nundistorted structure  by symmetry , but can be induced by the K 3 distortion  with a power -law relation \nbetween  TN and K 3 magnitude . Based on these revelation s, we have predicted a record -high TN in h-RFeO 3 \nwhen R=Sc and experimentally  confirmed it in epitaxially stabilized films . \nHexagonal ScFeO 3 (001) and YbFeO 3 (001) films (5  × 5 mm2 and 10  × 10 mm2 surface area, 70 -200 nm \nthick) have been grown on Al 2O3 (001) and yttrium stabilized zirconia (YSZ) (111) respectively using \npulsed laser (248 nm) deposition in a 5 m Torr oxygen environment, at 750 ℃ with a laser fluence of about \n1.5 J/cm2 and a repetitio n rate of 2 Hz  [8]. The film growth was mon itored using the reflection high -energy \nelectron diffraction (RHEED). The structural and magnetic properties have been studied using x -ray \ndiffraction and spectroscopy, magnetometry and neutron diffraction. X -ray diffraction  experiments , \nincluding θ/2θ sca n, φ scan, and reciprocal space mapping were carried out using a Rigaku D/Max -B \ndiffractometer with Co -Kα radiation (1.793 Å  wave length ) and a Rigaku SmartLab diffractometer with \nCu-Kα radiation (1.5406 Å). X -ray absorption spectroscopy (including x -ray linear dichroism) with a 20  \nincident angle was studied at beamline 4 -ID-C at the Advanced Photon Source at Argonne National \nLaboratory . Neutron diffraction  experiment s were carried out at beamline C ORELLI  at the Spallation \nNeutron Source (SNS) and HB3A four-circle diffractometer (FCD) at  the High Flux Reactor (HFIR) with \na thermal neutron wavelength of 1.546 Å , in the  Oak Ridge National Lab oratory . Temperature and \nmagnetic -field dependence of the magnetization was measured using a superconducting qu antum \ninterference device (SQUID)  magnetometer  with the field along the film normal direction . \nThe crystal structure of isomorphic hexagonal RMnO 3 and RFeO 3 (h-RMnO 3 and h -RFeO 3) has a P6 3cm \nsymmetry, consisting of alternating FeO (or MnO) and RO2 layers [Fig. 1(a)]. AFM orders occur in h -\nRMnO 3 and h -RFeO 3 below about 70 -140 K with spins in the FeO (or MnO) layers forming 120 -degree 3 \n structures  [6,9–12]. Below about 1000 K, ferroelectricity in h -RMnO 3 and h -RFeO 3 is induced by a lattice \ndistortion (K 3) [Fig. 1(a)] that tilts the FeO 5 (or MnO 5) local environment, shifts the R atoms along the c \naxis, and trimerizes the unit cell, with a sizable electric polarization (P ~ 10 µC/cm2) [13–16]. In addition, \nhexagonal RFeO 3 exhibits a weak ferromagnetism  [10,12,14,15,17,18]  [Fig. 1(a)] due to the canting Fe \nspins . \nMagnetic ordering relies on the underlying  exchange interaction s. In h-RFeO 3 and h-RMnO3, although the \nexchange interaction s within the FeO (or Mn O) layers are strong , the inter -layer exchange interaction s are \nweakened by the topology of layered structure and hexagonal  stacking . Using h-RFeO 3 as an example, F ig. \n1(b) shows the arrangement of the Fe atoms and their spins  in two neighboring FeO layers . The Fe atoms \nare on the hexagonal A and C sites  in the two layers respectively . One Fe atom (Fe 0) in the z = c/2 layer is \nhighlighted by its  tilted  FeO 5 trigonal bipyramid. T he interlayer nearest -neighbor exchange energy  for Fe 0 \nis  𝐸𝑖𝑛𝑡𝑒𝑟=∑𝐽0𝑖𝑆⃗0⋅𝑆⃗𝑖3\n𝑖=1 , where 𝑆⃗𝑖 is the spin on Fe i, and 𝐽0𝑖 is the exchange interaction coefficient \nbetween Fe 0 and Fe i. When there is no lattice distortion, the local symmetry  of Fe 0 is C3v, leading to  \nJ01=J02=J03 and Einter=0 because  ∑𝑆⃗𝑖3\n𝑖=1=0. In other words, the interlayer exchange are canceled ; the spin \nalignment between the two layers is lost.  Therefore, the three -dimensional magnetic ordering is forbidden \nin the undistorted  P63/mmc  structure  by symmetry . \nOn the other hand , the K3 lattice distortion  [Fig. 1(a)]  reduce s the symmetry to CS, making  J01=J02J03. \nConsequently, nonzero  lattice distortion  leads to  the three -dimensional magnetic ordering  because \nEinter=(J01-J03)S(S+1)0  [19]. Since the inter -layer exchange interaction is the bottleneck of the three -\ndimensional magnetic ordering, one has TN  Einter= (J01-J03)S(S+1). The dependence of TN on the K 3 \ndistortion then hinges on the relation between J01-J03 and the magnitude of K 3 (QK3). Previously, Das et al . \nanalyzed the relation between J01-J03 and QK3 [3]. Expanding J01 and J03 with respec t to QK3 around QK3=0, \nthe odd terms are expected to be zero due to the symmetry at QK3=0, leaving  J01-J03  a2Q2K3 + a 4Q4K3, \nwhere a 2 and a 4 are coefficients . In Fig. 1(c), we plot the log{ TN/[S(S+1)]} as a function of log( QK3) of h-\nRMnO3 measured using the neutron diffraction from the literature  [20,21]  [see supplementary information ], \nwhere  spin S is 2 for Mn . The data appear to fall on a straight line, indicating a power -law relation \nTN/[S(S+1)] QnK3; a fit shows n = 2.7±0.05. Given that the tilt of FeO 5 and MnO 5 caused by the K 3 \ndistortion is on the order of 5 degrees  [9,20,21]  which is not so small, both the a2Q2K3 and the a 4Q4K3 terms  \ncould  play a role, resulting 2<n<4. \nIt is challenging to predict the TN using the direct dependence of TN on the K 3 distortion  though , because \nthe K 3 distortion, which involves the displacement of oxygen  atoms , is difficult to measure precisely.  \nTolerance factor t=(rR+rO)/(rTM+rO)2 where rR, rTM, and rO are atomic radius of R, TM and oxygen, is a \ngood measure of lattice distortion from the cubic perovskite structure in o -RTM O3. It could also be used to \ngauge the structur al distortion in h-RFeO 3 and h-RMnO3, because a smaller R atom is expected to reduce \nthe in -plane lattice constant, which needs to be accommodated by a larger K 3 distortion  to reduce the \ndistances between Fe (or Mn) atoms within the FeO (or MnO) layers.  In other words, smaller t is expected \nto lead to larger TN, which is consistent with  the data from the literature  [Fig. 1(c) inset]   [11,20,21] , where \na linear correlation  between TN/[S(S+1) and t in h-RFeO 3 and h-RMnO3 is observed  (S = 2 and 2.5 for Mn \nand Fe respectively ). \nFollowing the trend in Fig. 1(c), a small er R, correspond ing to a smaller t, will lead to a higher TN in h-\nRFeO 3 and h-RMnO3. Since Sc has much smaller atomic radius than the rare earth and Y  [22], TN in h-\nScFeO 3 is expected to be higher than that of other h-RFeO 3. To verify  the prediction, we studied the \nmagnetic ordering temperature in h-ScFeO 3. ScFeO 3 naturally crystallizes in bixbyite structure in bulk; high \npressure growth of ScFeO 3 results in a corundum structure  [23]. Previous studies show that partially 4 \n substituting  Lu with Sc in LuFeO 3 may stabilize the hexagonal structure  [6,7,24] . However, t he stabilization \nof pure ScFeO 3 in the P63cm structure  has never been reported . In this study, w e have successfully  grown \nh-ScFeO 3 epitaxial films on Al 2O3 (001) substrates . The crystal structure and epitaxial relations of the h -\nScFeO 3 films w ere characterized using x -ray diffraction. As shown in Fig. 2(a), the θ/2θ scan shows a \ntypical pattern of the P63cm structure with the epitaxi al relation : h-ScFeO 3 (001) || Al 2O3 (001). The φ scan \n[see supplementary information] demonstrates the six -fold rotation symmetry and the in -plane epitaxial \nrelation : h-ScFeO 3 (100) || Al 2O3 (100). The RHEED patterns [Fig. 2(b) and (c)], which are signatu res of \nthe h -RFeO 3 structure, indicate a flat surface. The FeO 5 trigonal bipyramid configuration is confirmed by \nthe similarity between the x -ray linear dichroism spectroscopy of h -ScFeO 3 [Fig. 2(d) ] and those of h-\nLuFeO 3 and h -YbFeO 3 observed previously  [8,25 –27]. From  the x-ray reciprocal space mapping [see \nsupplementary information] , the lattice constants of  h-ScFeO 3 were determined: a = 5.742 Å and c = 11.690 \nÅ, smaller than the values of other h -RFeO 3 [28–30], suggesting a larger lattice distortion  [31]. \nTN of h-ScFeO 3 was measured  by the neutron diffraction  experiments at CORELLI  in addition to that of h -\nYbFeO 3. Using a wide wavelength -band  neutron beam and a two -dimensional detector at CORELLI , a \nthree -dimensional portion of the reciprocal space [using the Miller indices  (H, K, L) as the coordinates ] can \nbe measured  without rotating the sample [See supplementary materials] . The (101) and (114) diffraction \npeaks , were mapped out  in the three -dimensional reciprocal space. As shown in Fig. 2(e), the two magnetic \nBragg diffraction peaks  (101) and (1 -11), which are equivalent because of  the six-fold rotational symmetry \nalong the c axis, were  observed. The (101) Bragg peak is forbidden for the nuclear diffraction  due to the \ncrystal structure symmetry of h -RFeO 3 (space group P6 3cm), but it  is allowed  for magnetic diffraction  [9]. \nThe observation of the (101) peak  confirms the magnetic ordering in h -RFeO 3, as previously shown in h-\nLuFeO 3 and h-RMnO 3 [6,9,12] . The temperature dependence of the (101) peak intensity s uggests a \ntransition at about 200 K, which is corroborated by the measurement s at HB3A  [Fig. 2(f)]. In contrast, the \nintensity of the (114) peak , which mainly comes from the nucl ear scattering,  show s an insignificant \ntemperature dependence.  As shown in Fig. 2(g) , a similar transition temperature is observed in the \ntemperature dependence of the magnetization measured using a SQUID  magnetomet er on warming , after \ncooling the sample in a 10 kOe magnetic field (field cool or FC) and after cooling in a zero magnetic field \n(zero -field cool or ZFC). The FC and ZFC curves diverge at around 185 K, giving a more precise \ndetermination of  TN.  \nAs predicted , h-ScFeO 3 shows a high TN among all h-RMnO 3 and h -RFeO 3, as shown in Fig. 3(a) , where \nour measurement on h -YbFeO 3 and data in the literature are also included  [See supplementary \ninformation]  [6,9,11 –15,17,21] ; the measured TN of h-ScFeO 3 is slightly lower than the value predicted by \nextrapolating the linear relation between TN and t, which is also true for h -ScMnO 3 [9]. The reduction of \nmagnetization at low temperature in Fig. 2(g) hints  a possible spin reorientation at about 100 K in h-ScFeO 3. \nHowever, the temperature dependence of the (101) peak intensity in Fig. 2(f) indicates t hat spin \nreorientation in h -ScFeO 3 may not be significant  enough  to change the spin structure from A 2 to A 1 [6].  \nFinally , we discuss the effect of lattice distortion  on the canting of Fe moments, which is responsible for \nthe net magnetization MFe along the c axis. The MFe in h-ScFeO 3 can be inferred from the magnetometry \ndata. As shown in Fig. 2(h), the M-H curve shows a soft and a hard component , corresponding to two steps \nat H ≈ 0 and H ≈ 30 kOe  respectively . This two -component feature has been observed in both h -LuFeO 3 \nand h -YbFeO 3 films  [18,25] . The jump of magnetization at the higher field corresponds to the intrinsic \ncoercivity of the h -RFeO 3, while the jump at low field corresponds to the unavoidable structural boundaries \nin film samples of h-RFeO 3 that create uncompensated spins.  From the 30 -kOe jump, we found that MFe = \n0.015+/ -0.002 µB/Fe in h-ScFeO 3, which is smaller than that of h -LuFeO 3 and h-YbFeO 3 [18,25] , as shown \nin Fig. 3(b). This result is  counter -intuitive , because  a large  K3 distortion , correspond ing to a larger tilt 5 \n angle of the FeO 5 would  seemingly  generate  a larger canting angle  of the Fe moments  (θcant). However, θcant \nresults  from a competition between the exchange interaction and the Dzyaloshinskii -Moriya (DM) \ninteraction  [32–34]. Relation between the canting angle  (θcant) of the Fe moments , tilt angle of  the FeO 5 \n(γtilt), lattice constant in the basal plane ( a), and the intralayer  exchange interaction coefficient J can be \nderived as  θcanta2γtilt/J [See supplementary  material s]. Although h-ScFeO 3 is expected to have a larger γtilt \nand smaller  J, a is also smaller . Hence,  the size of θcant cannot be simply linked to the amplitude of  γtilt. If \nthe effect of a dominates, MFe would decrease for smaller R, which is what we found in  our previous first -\nprinciple calculations  [31]. \nIn conclusion , using symmetry analysis, we showed that the three -dimensional magnetic ordering in h-\nRMnO 3 and h -RFeO 3 are forbidden in undistorted structures  by symmetry , but can be induced by the \nstructural distortions. We also showed that dependence of  TN on structural distortions  manifest s as a near -\nlinear relation with the tolerance factor and a possible power law with QK3, suggesting  a higher TN in h-\nScFeO 3 with respect to other hexagonal  ferrite s studied so far, which was realized in this work  in epitaxially \nstabilized films . In addition to indicating  that the multiferroic ordering in h -RFeO 3 and h -RMnO 3 may be \nfurther enhanced with larger lattice distortion s, these results also establish a paradigm of structural origin \nof magnetic ordering  and spin -lattice coupling  in AFM oxides . \nAcknowledgement  \nThis work  was primarily supported by the National Science Foundation (NSF), Division of Materials \nResearch (DMR) under Award No. DMR -1454618. X.W. was supported by National Science Foundation \nthrough Award No. DMR -1552287 . X.M.C. acknowledges partial support from N SF DMR -1708790 and \nDMR -1053854 . 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The arrows from the atoms indicate the atomic displacement s of the K 3 lattice \ndistortion. The table indi cates the hexagonal stacking. (b) The geometric arrangement of Fe atoms in the \nz=0 and z=c/2 layers. The arrows through the Fe atoms indicate the spin directions. The atom Fe 0 is \nhighlighted by its FeO 5 trigonal bipyramid to depict the local symmetry . (c) log{TN/[S(S+1)] } as a function \nof log(Q K3). Inset:  TN/[S(S+1) ] as a function of the tolerance factor.  The dashed line is a linear fit to the \ndata. The data are from the literature (see text).  \n \n \n  \n9 \n  \nFigure 2 . Structural and magnetic characterizations of h -ScFeO 3(001)/Al 2O3 films. (a) θ/2θ x-ray \ndiffraction measured using an x -ray wavelength 1.789 Å. (b) and (c) are the RHEED diffraction patterns \nmeasured when the electron beam are along the (1 -10) and (100) direction s. (d) X -ray absorption spectra \nmeasured at the Fe L edges using s (in plane) and p (out of plane) linearly polarized x -ray beams. (e) A \nslice of the reciprocal space of h -ScFeO 3 at L=1 measured using neutron diffraction  at CORELLI . (f) \nTemperature dependence of the neutron diffraction intensities of the (101) and (114) peaks measured at \nCORELLI and H B3A . (g) Temperature dependence of the magnetization  per formula unit (f.u.)  measured \nduring warming after field -cool (10 kOe) and zer o-field-cool using 100 Oe and 500 Oe . (h) Magnetization -\nfield hysteresis loop measured at 100 K . The field is along the film normal direction.  \n \n  \n10 \n  \nFigure 3 . (a) The dependence of TN and TN/S(S+1) on the tolerance factor. The dashed line is a guide to the \neye. (b) The magnetization from the canting of Fe spins ( MFe) as a function of the in-plane lattice constant.  \nExcept for h -YbFeO 3 and h -ScFeO 3 in (a) and h -ScFeO 3 in (b), the data are from the literature (see text).  \n \n \n \n1 \n Supplementary material  \nS1. Neel temperature s of antiferromagnetic RTM O3 \nIn Fig. S1, we surveyed the Neel temperature of antiferromagnetic RTM O3 with orthorhombic and \nhexagonal structure s, where R stands for rare earth, Y, or Sc, TM stands for transition metal. The \nresults are plotted as a function of the tolerance factor  [1,2] , defined as t=(rR+rO)/(rTM+rO)2, \nwhere rR, rTM, and rO are atomic radius of R, TM and oxygen. For orthorhombic RTM O3 (o-RTM O3), \nwhen t decreases from 1, corresponding to orthorhombic distortions from the cubic perovskite \nstructure, TN decreases. In contrast, for hexagonal RTM O3 (h-RTM O3), when t decreases, TN \nincreases.  \nRTiO 3 crystalize in orthorhombic structure.  When the size of R is smaller  than that of Sm, the \nmaterials become ferromagnetic  [3]. \nRVO 3 and RCrO 3 both crystalize in orthorhombic structure; all members are antiferromagnet ic [4–\n9].  \nRMnO 3 crystalize in orthorhombic structure when the size of R is larger  than that of Dy, with \nantiferromagnetic order  [10–14].  Otherwise, RMnO 3 crystalize in hexagonal structure , with 120 -\ndegree antiferromagnetic order  [15]. \nRFeO 3 crystalize in orthorhombic structure; all members exhibit antiferromagnetism  [15]. The \nhexagonal structures can be stabilized using thin film epitax y or doping , resulting the 120 -degree \nantiferromagnetic order  [16–21] similar to that in h -RMnO 3. The plot includes the new data from \nthis work.  \nRNiO 3, except for LaNiO 3 which crystalizes in rhombohedra structure with a paramagnetism, have \northorhombic  structures  with antiferromagnetism  [22–24]. \n  \n0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92100200700\n  Neel Temperature\nTolerance FactorOthorhmobic\n Ti  V\n Cr  Mn\n Fe  Ni\nHexagonal \n Mn  Fe \nFigure S1 . The Neel temperature of orthorhombic antiferromagnetic RTM O3. \n 2 \n S2. K3 lattice distortion in  hexagonal  RFeO3 and RMnO 3 \nIn Fig. S 2, we surveyed the K3 lattice distortion  of hexagonal  RFeO 3 and RMnO 3. The experimental \ndistortion amplitude (QK3) was calculated from the crystal structure measured using neutron \ndiffraction, since the oxygen positions are critical  [25,26] . The theoretical  distortion amplitudes  \ncalculated based on the first principles  [27,28] , are systematically larger than the experimental  \nvalues but follow the similar trend.  \n  \n0.840 0.845 0.850 0.8550.91.01.1\nY0.5Lu0.5\nHoLuTm\nEr\nY\n  \n h-RMnO3 Theory\n h-RFeO3 Theory\n h-RMnO3 Experiment, 300 KQK3 (Å)\nTolerance Factor \nFigure S 2. The K 3 lattice distortion in hexagonal RFeO3 and RMnO3. \n 3 \n S3. X-ray diffraction characterization of the h -ScFeO 3 films  \nFigure S 3 shows the x -ray diffraction characterization of h -ScFeO 3 films. Hexagonal coordinates \nare used for both the h -ScFeO 3 film and the Al 2O3 substrat e. As shown in Fig. S 3(a), the three -\nfold and the six -fold rotational symmetries of Al 2O3 and of h -ScFeO 3 respectively are revealed by \nthe φ scan, consistent with their rhombohedral and hexagonal structures respectively. From the \nalignment of the peaks, the epitaxial relation in the basal plane h -ScFeO 3(100) || Al 2O3(100) is \nrevealed. In Fig. S 3(b), the recip rocal space mapping around the h -ScFeO 3 (308) peak is plotted. \nFrom the peak position, one can calculate the lattice constants: a = 5.742 Å, and c = 11.690 Å. \nFigure S 3(c) shows the rocking curve of the h -ScFeO 3 (004) peak. Using the Scherrer \nequation  [30], one can estimate the in -plane size of the crystallites: 0.9𝜆\nΔ𝜃cos 𝜃, where 𝜆 is the x -ray \nwavelength, Δ𝜃 is the full width half maximum (FWHM) in Fig. S 3(c); the result shows the in -\nplane crystallite size is approximately 25 nm.  \n   \nFigure S 3. X-ray diffraction measurement on h -ScFeO 3(001)/Al 2O3(001) films: (a) φ scan, (b) reciprocal space \nmapping (RSM) of the h -ScFeO 3 (038) peak, and (c) rocking curve on the h -ScFeO 3 (004) peak. The φ scan and \nRSM were measured on a Rigaku Smart Lab with 1.54 06 Å x -ray wavelength while the rocking curve is measured \nusing a Rigaku  D/Max -B with 1.789 Å x -ray wavelength.  \n \n4 \n S4. Neutron  diffraction measurements  of the h -YbFeO 3 films  \n \nFigure S 4. (a) A slice of the reciprocal space of h -YbFeO 3 at L=1 measured using neutron \ndiffraction at CORELLI  (see text). (b) Temperature dependence of the integrated neutron \ndiffraction intensities of the (101) and (114) peaks measured at CORELLI and HB3A.  \nTo study the magnetic ordering of h -YbFeO 3, we carried out neutron diffraction measurements. \nFigure S 4(a) shows a slice of the reciprocal space at L=1. Three Bragg peaks (101), (011), and ( -\n111), which are equivalent because of the six -fold rotational symmetry along the c axis, are \nobserved . The (101) Bragg peak is forbidden for the nuclear diffraction due to t he crystal structure \nsymmetry of h -RFeO 3 (space group P6 3cm), but it is allowed for magnetic diffraction  [29]. Figure \nS4(b) shows the temperature dependence of the integrated peak intensities for selected peaks. A \nclear change of the slope of the (101) peak intensity is observed at 125 ± 5 K, suggesting a magnetic \ntransition [Fig. S4(b)]. In contrast, the intensity of the (11 4) peak, which mainly comes from the \nnuclear scattering, shows an insignificant temperature dependence. The strong temperature \ndependence of the (101) peak is confirmed by single -crystal neutron diffraction at HB3A, as also \nshown in Fig. S4(b) and Fig. S 5. The magnetic transition temperature of 125 K is consistent with \nthe reported TN for h -YbFeO 3 from magnetometry  [20]. \n \nFigure S 5. Neutron diffraction measurements on h -YbFeO 3 films at HB3A. (a) Diffraction \nintensity as a function of temperature for the (101) peak. The insets are the detector images \nshowing the (101) peak at low temperature which vanishes at high temperature. (b) The calculated \nv.s. observed magnetic structural  factor s which lead to a magnetic moment of 2.0 µ B per Fe.  \n \n \n  \n5 \n S5. Neutron diffraction data  measured at the  Spallation neutron  source  (SNS)  \nElastic diffractions with unpolarized pulsed neutron beams were carried out on h -RFeO 3 films at \nthe beamline CORELLI  of the Spallation neutron source (SNS) with a pulsed neutron source. With \nthe two-dimensional detector, and the pulsed neutron beam which contain s neutrons with various \nwavelength s, a three -dimensional portion of the reciprocal space can be measured with out rotating \nthe sample . In this work, the incident angle of the neutron beam is set so that both the magnetic \nBragg diffraction peak (101) and the nuclear Bragg diffraction peak (114) are included in this \nportion of the reciprocal space and measured at the same time.  \nFigure S 6 and S 7 show the raw data of the neutron diffraction on h -YbFeO 3 and h -ScFeO 3 \nrespectively.  Slices at L=4 and L=1 reveals the diffraction of (114) and (101) [Fig . S6 and S 7 (a) \nand (b)] peaks and their equivalences respectively. The white space in the slices are  parts of the  Figure S 6. Neutron diffraction data of \nh-YbFeO 3. (a) and ( c) are slice s of \nthree -dimensional reciprocal space at \nL=4 and L=1 respectively. (b) and (d) \nare the temperature dependence of the \nL scan near the Bragg diffraction \npoints (1 -24) and (011) respectively.  \n \n \n Figure S 7. Neutron diffraction data of \nh-ScFeO 3. (a) and ( c) are slice of three -\ndimensional reciprocal space at L=4 \nand L=1 respectively. (b) and (d) are \nthe temperature dependence of the L \nscan near the Bragg diffraction points \n(1-24) and (101) respectively.  \n \n \n6 \n reciprocal space that are  not covered by the detector.  From the temperature dependence of the L \nscan, one can observ e the disappearance of the  magnetic diffractions peak (101) but not the nuclear \ndiffraction peak (114)  within the temperature range of the measurements. It is obvious that the \ntransition temperature of h -ScFeO 3 is significantly higher than that of h -YbFeO 3. \n \n  7 \n S6. Dzyaloshinskii -Moriya (DM) interaction between Fe sites  and the canting of the Fe spins  in h-\nRFeO 3 \n• DM interactions  \nHere we discuss the DM interaction  [31,32]  using the symmetry analysis and the Keffer’s \nestimation  [33]. As shown in Fig. S 8, the Fe sites in h -RFeO 3 trimerizes due to the collective tilt \nof the FeO 5. We can assume the positions of the three Fe atoms (Fe 1 to Fe 3) and the oxygen in the \ncenter (O C) as \n𝑟⃗𝐹𝑒1=𝑎\n3(−1\n2,√3\n2,0) \n𝑟⃗𝐹𝑒2=𝑎\n3(−1\n2,−√3\n2,0) \n𝑟⃗𝐹𝑒3=𝑎\n3(1,0,0) \n𝑟⃗𝑂𝐶=(0,0,−𝛿), \nwhere a is the lattice constant in the basal plane of h -RFeO 3. \nWe analyze the DM interaction between Fe 1 and Fe 2 as an example. Accord ing to Moriya  [31], the \nDM interaction between two magnetic ions results from the on -site spin -orbit coupling and the \nhopping between the electronic states of the magnetic ions through a diamagnetic atom (or atoms). \nThe symmetry of the geometric arrangement of the two magnetic ions and the diamagnetic atom(s) \nis critical. For the DM interaction between Fe 1 and Fe 2, the symmetry of the Fe 1-OC-Fe2 \nconnectivity determines the direction of the vector coefficient 𝐷⃗⃗⃗12 in the D M interaction 𝐷⃗⃗⃗12⋅\n(𝑆⃗1×𝑆⃗2). Two symmetry rules  [31] can be applied here: 1) if there is a mirror plane including Fe 1 \nand Fe 2, 𝐷⃗⃗⃗12⊥ the mirror plane; 2) if there is a mirror plane perpendicular to 𝑟⃗𝐹𝑒1−𝑟⃗𝐹𝑒2 that \npasses the p oint 𝑟⃗𝐹𝑒1+𝑟⃗𝐹𝑒2\n2, 𝐷⃗⃗⃗12∥ the mirror plane. Therefore, 𝐷⃗⃗⃗12  is perpendicular to the plane \nincluding the Fe 1-OC-Fe2 atoms.   \nFigure S 8. Schematics of the FeO 5 trimer. (a) Side view of the trimer highlighting the downward displacement \nof the atom O C and the DM interaction vector between Fe 1 and Fe 2. (b) Top view of the trimer.  \n \n8 \n According to Keffer  [33], if the DM interaction is mediated mainly by one diamagnetic atom, one \ncan estimate the magnitude and the orientation of the vector interaction coefficient as 𝐷⃗⃗⃗𝑖,𝑗∝\n𝑟⃗𝑖3×𝑟⃗𝑗3, where 𝑟⃗𝑖3 and 𝑟⃗𝑗3 are the vectors from the two magnetic ions to the diamagne tic (3rd) atom \nrespectively. In the case of Fe 1-OC-Fe2, \n𝐷⃗⃗⃗12∝𝑟⃗𝐹𝑒1𝑂𝐶×𝑟⃗𝐹𝑒2𝑂𝐶=(𝑎𝛿\n√3,0,√3𝑎2\n18), \nwhere 𝑟⃗𝐹𝑒1𝑂𝐶=𝑟⃗𝐹𝑒1−𝑟⃗𝑂𝐶 and 𝑟⃗𝐹𝑒2𝑂𝐶=𝑟⃗𝐹𝑒2−𝑟⃗𝑂𝐶. \nAssuming 𝐷⃗⃗⃗12=−𝐷(sin𝜙,0,cos𝜙), one finds  \ntan𝜙=6𝛿\n𝑎. \nNote that the tilt angle of the FeO 5 local environment can be defined using tan𝛾=𝛿\n𝑎/3 =3𝛿\n𝑎. Since \nall the angles are small (in h -LuFeO 3, 𝛾 ≈9.5 degree), one  finds approximate ly  \n𝛾≈3𝛿\n𝑎,𝜙≈6𝛿\n𝑎. \nIn addition, since 𝛿≪𝑎, the magnitude of DM interaction strength follows  \n𝐷∝𝑎2. \n• Canting of the Fe spin and the DM interaction  \nThe canting of Fe spins mainly comes from the DM interaction and the tilt of the FeO 5 local \nenvironment which tilts the vector coefficient 𝐷⃗⃗⃗𝑖,𝑗 (by angle 𝜙) and causes the Fe spins to reorie nt \nto minimize the DM interaction energy 𝐷⃗⃗⃗𝑖,𝑗⋅(𝑆⃗𝑖×𝑆⃗𝑗). At the same time, canting of the Fe spins \nreduces the Fe -Fe spin angle and increases the antiferromagnetic exchange interaction. Therefore, \nthe canting angle 𝜃 is a result of the competition  between the exchange interaction and the DM \ninteraction. Below, we derive the relation between 𝜃 and 𝜙 (the tile angle of 𝐷⃗⃗⃗𝑖,𝑗). \nAs shown in Fig . S8 we can assume the spin vectors as the following   \n𝑆⃗1=(1\n2cos𝜃,−√3\n2cos𝜃,−sin𝜃)𝑆 \n𝑆⃗2=(1\n2cos𝜃,√3\n2cos𝜃,−sin𝜃)𝑆 \n𝑆⃗3=(−cos𝜃,0,−sin𝜃)𝑆, \nwhere S the magnitude of the Fe spin.  \nWe consider the exchange  interaction between the spins 𝑆⃗1 and 𝑆⃗2 as an example , which shows  \n𝐸12(𝑒𝑥)=𝐽𝑆⃗1⋅𝑆⃗2=𝐽(−1\n2+3\n2sin2𝜃)𝑆2, \nwhere 𝐽>0 is the exchange interaction coefficient.  9 \n For the DM interaction, if we assume 𝐷⃗⃗⃗12=−𝐷(sin𝜙,0,cos𝜙), where 𝐷>0 is the magnitude \nof the interaction (see above  discussion ), the interaction energy is  \n𝐸12(𝐷𝑀)=𝐷⃗⃗⃗12⋅(𝑆⃗1×𝑆⃗2)=−𝐷(√3sin𝜃cos𝜃sin𝜙+√3\n2cos2𝜃cos𝜙). \nThe canting angle 𝜃 will be determined by the minimization of the total energy 𝐸12(𝑒𝑥)+𝐸12(𝐷𝑀). By \nnoting that both 𝜃 and 𝜙 are small, one can make the approximation that  \n𝐸12(𝑒𝑥)+𝐸12(𝐷𝑀)≈𝐽(−1\n2+3\n2𝜃2)−𝐷(√3𝜃𝜙+√3\n2). \nTherefore, the minimizatio n results in  \n𝜃≈𝐷\n√3𝐽𝜙. \nSince it is expected that the DM interaction is much weaker than the exchange interaction (or 𝐷≪\n𝐽), one expects 𝜃≪𝜙, which agrees with the experimental observation ( typically  𝜙 is a few  \ndegree s while  𝜃 is a fraction of a  degree). \nAccording to the discussion above , 𝐷∝𝑎2, where a is the lattice constant of the basal plane. \nHence the canting angle of the Fe spin  follows  \n𝜃∝2𝑎2𝜙\n√3𝐽=4𝑎2𝛾\n√3𝐽. \nIn the case of hexagonal ferrites, the smaller R leads to larger  𝜙, which decreases J, but also \ndecreases a. Thus , it is possible that the combined effect may lead to a reduction of the canting \nangle 𝜃 under the compressive strain.  \n  10 \n [1] A. S. Bhalla, R. Guo, and R. Roy, Mater. Res. Innov. 4, 3 (2000).  \n[2] R. D. Shannon, Acta Cryst. A32, 751 (1976).  \n[3] H. D. Zhou and J. B. 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Janolin, J. M. Kiat, and \nA. Fuwa, Phys. Rev. B 92, 054101 (2015).  \n[18] S.-J. Ahn, J. -H. Lee, H. M.  Jang, and Y. K. Jeong, J. Mater. Chem. C 2, 4521 (2014).  \n[19] J. A. Moyer, R. Misra, J. A. Mundy, C. M. Brooks, J. T. Heron, D. A. Muller, D. G. \nSchlom, and P. Schiffer, APL Mater. 2, 012106 (2014).  \n[20] Y. K. Jeong, J. Lee, S. Ahn, S. -W. Song, H. M. Jang , H. Choi, and J. F. Scott, J. Am. \nChem. Soc. 134, 1450 (2012).  \n[21] S. M. Disseler, X. Luo, B. Gao, Y. S. Oh, R. Hu, Y. Wang, D. Quintana, A. Zhang, Q. \nHuang, J. Lau, R. Paul, J. W. Lynn, S. W. Cheong, and W. Ratcliff, Phys. Rev. B 92, \n054435 (2015).  11 \n [22] M. L. Medarde, J. Phys. Condens. Matter 9, 1679 (1997).  \n[23] J. A. Alonso, M. J. Martínez -Lope, M. T. Casais, J. L. Martínez, G. Demazeau, A. \nLargeteau, J. L. García -Muñoz, A. Muñoz, and M. T. Fernández -Díaz, Chem. Mater. 11, \n2463 (1999).  \n[24] M. T. Fernández -Díaz, J. A. Alonso, M. J. Martínez -Lope, M. T. Casais, an d J. L. García -\nMuñoz, Phys. Rev. B 64, 1444171 (2001).  \n[25] A. Munoz, J. Alonso, M. Martinez -Lopez, M. Cesais, J. Martinez, and M. Fernandez -\nDiaz, Chem. … 13, 1497 (2001).  \n[26] S. Lee, A. Pirogov, M. Kang, K. H. Jang, M. Yonemura, T. Kamiyama, S. W. Cheong , F. \nGozzo, N. Shin, H. Kimura, Y. Noda, and J. G. Park, Nature 451, 805 (2008).  \n[27] H. Tan, C. Xu, M. Li, S. Wang, B. L. Gu, and W. Duan, J. Phys. Condens. Matter 28, \n126002 (2016).  \n[28] C. Xu, Y. Yang, S. Wang, W. Duan, B. Gu, and L. Bellaiche, Phys. Re v. B 89, 205122 \n(2014).  \n[29] A. Munoz, J. A. Alonso, M. J. Martinez -Lope, M. T. Casais, J. L. Martinez, and M. T. \nFernandez -Diaz, Phys. Rev. B 62, 9498 (2000).  \n[30] B. D. Cullity, Elements of X -Ray Diffraction.  (Addison -Wesley Pub. Co., Reading, Mass., \n1956). \n[31] T. Moriya, Phys. Rev. 120, 91 (1960).  \n[32] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).  \n[33] F. Keffer, Phys. Rev. 126, 896 (1962).  \n "    },    {        "title": "0811.3491v1.Frequency_dependent_reflection_of_spin_waves_from_a_magnetic_inhomogeneity_induced_by_a_surface_DC_current.pdf",        "content": "arXiv:0811.3491v1  [cond-mat.other]  21 Nov 2008Frequency-dependent reﬂection of spin waves\nfrom a magnetic inhomogeneity induced by a surface DC-curre nt\nT. Neumann,∗A. A. Serga, and B. Hillebrands\nFachbereich Physik and Forschungszentrum OPTIMAS\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nM. P. Kostylev†\nSchool of Physics\nUniversity of Western Australia, Crawley,\nWestern Australia 6009, Australia\n(Dated: October 30, 2018)\nThe reﬂectivity of a highly localized magnetic inhomogenei ty is experimentally studied. The\ninhomogeneity is created by a dc-current carrying wire plac ed on the surface of a ferrite ﬁlm. The\nreﬂection of propagating dipole-dominated spin-wave puls es is found to be strongly dependent on\nthe spin-wave frequency if the current locally increases th e magnetic ﬁeld. In the opposite case the\nfrequency dependence is negligible.\nPACS numbers: 75.30.Ds, 85.70.Ge\nThe interaction of spin-wave packets propagating in\na ferromagnetic ﬁlm with localized artiﬁcial inhomo-\ngeneities is of interest for numerous applications. For\nexample, the tunneling [1], the guidance [2], the ﬁlter-\ning through band gaps [3], and shaping [4] of spin-wave\npulses has been realized in this way.\nThere are diﬀerent possibilities to create inhomo-\ngeneities, such as by mechanical structuring of the mag-\nnetic material [3, 5, 6] or by variation of the satura-\ntion magnetization by ion irradiation [7]. However, these\nmodiﬁcations of the magnetic properties are irreversible\nand do not allow any adjustment or real-time control.\nA more promising approach is to change the magnetic\ninhomogeneitydynamicallyascanbedonee.g.withpara-\nmetric pumping by fast variation of the bias magnetic\nﬁeld [4, 8]. In this process the increase of the precess-\ning transversal magnetization component leads to a de-\ncrease of the static saturation magnetization. However,\none should keep in mind the high complexity of the pro-\ncess and the relatively long time scale in the microsecond\nrange to reach a quasi-equilibrium regime [9, 10].\nInstead, many recent experiments [11, 12, 13] have re-\nlied on the technique to locally modify the bias magnetic\nﬁeldbytheOerstedﬁeldofacurrent-carryingwireplaced\non the ﬁlm surface. With this setup, XOR and NAND\ngates, milestones in the development of spin-wave logic,\nhave been realized [14] and resonant spin-wave tunneling\nhas been discovered [15]. Moreover, periodic structures,\nso called magnonic crystals, of such design [16] have the\nadvantage of being controllable on a time scale shorter\nthan the spin-wave relaxation time.\nIn the current work, we directly measure the reﬂected\n∗Electronic address: neumannt@physik.uni-kl.de\n†On leave from FET Department, St. Petersburg Electrotechni cal\nUniversity, St.Petersburg 197376, Russiaand transmitted intensity of spin-wave pulses propagat-\ning in a thin yttrium-iron-garnet (YIG) ﬁlm through a\ncurrentinduced magneticinhomogeneityforawide range\nof spin-wave carrier frequencies and applied dc-currents.\nThe results provide evidence for the potential of this\nstructure as an eﬀective frequency ﬁlter and adjustable\nenergy divider.\nAll experiments were performed in the pulse regime.\nThis approach was chosen to check the applicability of\nthe method for microwavesignal processing used in mod-\nerndigitaltechnique andexcludeanyinﬂuence ofheating\neﬀects caused by the dc-current applied to the wire.\nFigure 1(a) shows a sketch of the experimental setup.\nA triggered microwave switch transforms the cw-signal\nof a microwave generator with a carrier frequency fbe-\ntween 7.010 GHz and 7 .115 GHz into 280 ns long pulses\nwith a 1 ms repetition rate. These pulses are sent to a\n50µm wide strip-line transducer placed on the surface\nFIG. 1: (Color online) (a) Sketch of the experimental mi-\ncrowave setup and of the section layout. (b) Schematic rep-\nresentation of the dc-current inﬂuence on the dispersion re la-\ntion.2\nof a 5µm thick, longitudinally magnetized single crystal\nYIG-stripe of 15 mm width.\nThe microwave signal excites packets of backward vol-\nume magnetostatic waves (BVMSW) whose wave vector\nis aligned in the direction of the applied bias magnetic\nﬁeldH= 1800 Oe with a typical value between 10 cm−1\nand 200 cm−1. The spin waves propagate through the\nﬁlm and are picked up by a second, identical antenna\nsituated at a distance of 8 mm. By amplifying and de-\ntecting the obtained microwavesignal we can observethe\ntransmitted pulse in real time.\nWhile the spin-wave pulse is propagating in the ﬁlm\na 180 ns long and between −2 A and +2 A strong dc-\ncurrent pulse with a rise time of less than 20 ns is ap-\nplied to a 50 µm thick wire placed on the ﬁlm surface\nhalfway between the antennae (Fig. 1(a)). It creates a\nhighly localized magnetic inhomogeneity acrossthe YIG-\nwaveguideofupto ±200Oeandwidth comparabletothe\nspin-wavewavelength. Thelength ofthe dc-currentpulse\nis chosen in order to, on the one hand, avoid heating the\nsample and, on the other hand, reach a stationary state\nof the spin-wave propagation. For such a structure, two\nprincipally diﬀerent operation regimes exist (Fig. 1(b)).\nIf the dc-current locally decreases the bias magnetic ﬁeld\nwe operate in the tunneling regime [1]. We will refer to\nthe opposite case as diﬀraction regime .\nSpinwavesreﬂectedfromtheinhomogeneityarepicked\nup by the input antenna. A Y-circulator in the input\nchannel allows to detect and observe this signal on the\noscilloscope simultaneously to the transmitted one.\nIn Fig. 2(a) the transmission and reﬂection character-\nistics for exemplary currents −0.5 A and −1.0 A repre-\nsentative for the tunneling regime as well as +0 .5 A and\n+1.0Afor the diﬀractionregimeareshowntogetherwith\nthe characteristics obtained when no current is applied.\nIf no current is applied the measuredintensities are de-\ntermined by the excitation and detection characteristics\nof the used microstrip antennae and the spin-wave spec-\ntrum. Because of the ﬁnite size of the antennae, the exci-\ntation and detection is limited to spin waves with a long\nenoughwavelength. Thisexplainsthe vanishing intensity\nfor frequencies below 7 .02 GHz. At the same time, the\nfrequency of ferromagnetic resonance fFMR≈7.10 GHz\nconstitutes an upper frequency limit for the BVMSW in\nthe dipole approximation. Hence, the intensity of the\ndetected signal quickly drops above it. Note, that in the\nabsence of a current, only a relatively small part of the\nspin wave is reﬂected by the metal wire placed on the\nsample surface [17], so that most of the spin-wave signal\nis observed in transmission. The dips observable in the\nreﬂected and transmitted signalarecaused byresonances\nbetween the input antenna and the central wire.\nWhen a dc-current is applied, the ratio between reﬂec-\ntion and transmission changes signiﬁcantly depending on\nthe direction and magnitude of the applied dc-current.\nIn the tunneling regime, i.e. when the bias magnetic\nﬁeld is locally decreased by the current, the reﬂected sig-\nnal monotonically increases over the whole investigatedFIG. 2: (Color online) (a) Reﬂection andtransmission chara c-\nteristics in the tunneling and diﬀraction regime. The verti cal\ndashed line indicates f= 7.09 GHz. (b) Current dependent\nreﬂection and transmission for f= 7.09 GHz.\nfrequency range with increasing current modulus. In\nthe diﬀraction regime the behavior is non-monotonous.\nWhen the current is increased two reﬂection resonances\nare clearly observed. They are most pronounced for\nf= 7.09 GHz, which is slightly below fFMR(see Fig. 2).\nFor 0.5 A the reﬂected signalintensity increasesdrasti-\ncallycomparedtothecasewithoutcurrent. Thereﬂected\nsignal rises within 30 ns after the application of the dc-\npulse by a factor of 10. The intensity then stays constant\nfor the remainder of the microwave signal pulse.\nFurther increase of the current leads to a decrease of\nthe reﬂected signal intensity. At a current of 1 .0 A the\nreﬂected signal is eﬀectively suppressed and the trans-\nmitted pulse shape is almost undisturbed. This behavior\nis repeated when the current is further increased. The\nreﬂected spin-wave intensity rises till it reaches a second\nmaximum at 1 .8 A and then drops again. Fig. 2(b) dis-\nplays the current dependence of the reﬂected and trans-\nmitted signal intensity for f= 7.09 GHz.\nBy adding the reﬂected and transmitted signal inten-\nsities, it can be veriﬁed that their sum stays constant.\nTo explain the two diﬀerent regimes, consider Fig. 1(b)\nagain. In the tunneling regime, for large enough cur-\nrent modulus a barrier is formed which reﬂects the sig-\nnal. A larger absolute current leads to an increase of\nthe zone where the spin-wave propagation is prohibited\nand through which, consequently, the spin waves can-3\nFIG. 3: (Color online) Frequency-dependent reﬂectivity.\nnot propagate and need to tunnel [1]. In the diﬀraction\nregime, constructive interference of spin waves reﬂected\nfrom the region of inhomogeneous magnetic ﬁeld occurs.\nThe observed data on spin-wave reﬂection supports very\nwell the theoretical prediction reported in [11] by the au-\nthors.\nWe determine the frequency-dependent reﬂectivity R\nof the investigated structure via\nR=Ireﬂ\nItrans+Ireﬂ\nwhereIreﬂandIreﬂdenote the reﬂected respectively\ntransmitted intensity (Fig. 3). In the tunneling regime,\nfor large enough currents the reﬂectivity is constant over\nthe whole range of accessible frequencies. The picture\nis completely diﬀerent in the diﬀraction regime. Here,\nthe reﬂectivity is clearly frequency-dependent. For low\nfrequencies between 7 .01 GHz and 7 .04 GHz it is con-\nstant and restricted to R <0.2. For higher frequencies\nwhich arecloserto fFMR, a strongdc-currentdependence\nis observed. In particular, we note that the reﬂectivityforf= 7.09 GHz and I=−0.5 A is only 0 .78 while it\nreaches almost 1 for the opposite dc-current polarity.\nIn conclusion, we have investigated the reﬂection\nof dipole-dominated spin waves propagating through a\nhighly localized dc-current induced magnetic inhomo-\ngeneity. The reﬂection on the inhomogeneity created by\na surface dc-current proves to be dependent on the fre-\nquency of the propagating spin wave as well as on the\nmagnitude and polarity of the dc-current. While the lat-\nter two dependencies are monotonous for a current de-\ncreasing the local magnetic ﬁeld ( tunneling regime ) they\nexhibit a resonant structure in the opposite case ( diﬀrac-\ntion regime ) which is well pronounced only in a small fre-\nquency range below fFMR. For a given current of 0 .5 A\nthe ratio of transmitted signals in tunneling and diﬀrac-\ntion operation regime is more than 25.\nThese results have to be considered for the future de-\nsign of current controlled spin-wave devices, e.g. spin-\nwave logic gates and dynamic magnonic crystals. While\nthe diﬀraction regime has the advantage of a high re-\nﬂectivity for relatively low and easily reachable currents\n(which allow for long current pulses without disturb-\ning heating eﬀects), the tunneling regime allows the de-\nsign of frequency-independent structure. This may be\nespecially interesting for ultra-short pulses with a wide\nFourier spectrum. In addition, the frequency-dependent\nreﬂection in the diﬀraction regime can be used to create\ntunable frequency selective devices. By using multiple\nwires instead of just a single one, the characteristics of\nthe structure can be further improved.\nFinancial support by the MATCOR Graduate School\nof Excellence, the DFG SE 1771/1-1, the Australian Re-\nsearch Council, and the University of Western Australia\nis gratefully acknowledged.\n[1] S.O.Demokritov, A. A.Serga, A.Andr´ e, V.E. Demidov,\nM. P. Kostylev, and B. Hillebrands, Phys. Rev. Lett. 93,\n047201 (2004).\n[2] J. Topp, J. Podbielski, D. Heitmann, and D. Grundler,\nPhys. Rev. B 78, 024431 (2008).\n[3] A. V. Chumak, A. A.Serga, B. Hillebrands, and M. P.\nKostylev, Appl. Phys. Lett. 93, 022508 (2008).\n[4] A. A. Serga, T. Schneider, B. Hillebrands, M. P.\nKostylev, and A. N. Slavin, Appl. Phys. Lett. 90, 022502\n(2007).\n[5] A. Maeda, and M. Susaki, IEEE Trans. Magn. 42, 3096\n(2006).\n[6] M. P. Kostylev, P. Schrader, R. L. Stamps, G. Gubbiotti,\nG. Carlotti, A. O. Adeyeye, S. Goolaup, and N. Singh,\nAppl. Phys. Lett. 92, 132504 (2008).\n[7] R. L. Carter, J. M. Owens, C.V. Smith, and K.W. Reed,\nJ. Appl. Phys. 53, 2655 (1982).\n[8] G. A. Melkov, A. A. Serga, A. N. Slavin, V. S. Tiberke-\nvich, A. N. Oleinik, and A. V. Bagada, JETP 89, 1189\n(1999).\n[9] V. E. Demidov, O. Dzyapko, S. O. Demokritov, G. A.Melkov, and A. N. Slavin, Phys. Rev. Lett. 99, 037205\n(2007).\n[10] T. Neumann,A.A.Serga, andB.Hillebrands, submitted,\narXiv:0810.4033v1 [cond-mat.other] (preprint).\n[11] M. P. Kostylev, A. A. Serga, T. Schneider, T. Neumann,\nB. Leven, B. Hillebrands, and R. L. Stamps, Phys. Rev.\nB76, 184419 (2007).\n[12] V. E. Demidov, U. H. Hansen, and S. O. Demokritov,\nPhys. Rev. B 78, 054410 (2008).\n[13] K. R. Smith, M. J. Kabatek, P. Krivosik, and M. Wu, J.\nAppl. Phys. 104, 043911 (2008).\n[14] T. Schneider, A.A.Serga, B. Leven, B. Hillebrands, R.L .\nStamps, andM. P.Kostylev, Appl.Phys.Lett. 92, 022505\n(2008).\n[15] U. Hansen, M. Gatzen, V. E. Demidov, and S. O.\nDemokritov, Phys. Rev. Lett. 99, 127204 (2007).\n[16] Y. K. Fetisov, J. Comm. Tech. Electron. 10, 1171 (2004).\n[17] I. V. Krutsenko, G. A. Melkov, and S. A. Ukhanov, Ra-\ndiotexn. i Elektron. 32, 1976 (1987)."    },    {        "title": "2401.11267v1.Charge_transfer_transitions_and_circular_magnetooptics_in_ferrites.pdf",        "content": "arXiv:2401.11267v1  [cond-mat.mtrl-sci]  20 Jan 2024Charge transfer transitions and circular magnetooptics in ferrites\nA.S. Moskvin\nUral Federal University, 620083 Ekaterinburg, Russia and\nM.N. Mikheev lnstitute of Metal Physics of Ural Branch of Rus sian Academy of Sciences, 620108 Ekaterinburg, Russia\nThe concept of charge-transfer (CT) transitions in ferrite s is based on the cluster approach and\ntakes into account the relevant interactions as the low-sym metry crystal ﬁeld, spin-orbital, Zeeman,\nexchange and exchange-relativistic interactions. For all its simplicity, this concept yield a reliable\nqualitative and quantitative microscopic explanation of s pectral, concentration, temperature, and\nﬁeld dependences of optic and magneto-optic properties ran ging from the isotropic absorption as\nwell as the optic anisotropy to the circular magneto-optics . In this review paper, starting with a\ncritical analysis of the fundamental shortcomings of the \"ﬁ rst-principles\" DFT-based band theory\nwe present the main ideas and techniques of the cluster theor y of the CT transitions to be main\ncontributors to circular magneto-optics of ferrites.\nI. INTRODUCTION\nOver the past 175 years since Michael Faraday’s discov-\nery of the relation between light and electromagnetism,\nmagneto-optics has become a broad ﬁeld of fundamen-\ntal and applied research. On the one hand, magneto-\noptics is aimed at the experimental study of the electronic\nand magnetic structure, magnetic anisotropy, magnetic\nphase transitions, spin-orbital, exchange and exchange-\nrelativistic eﬀects, and on the other hand, at the search\nfor new materials with high magneto-optical character-\nistics, improvement and development of new magneto-\noptical applications. Various ferrites and, especially,\nbismuth-substituted iron garnets R 3Fe5O12(R = Y, or\nrare-earth ion) occupy a special place among magneto-\noptical (MO) materials, being one of the main objects\nof fundamental research and basic materials for creating\nvarious devices of applied magneto-optics from magneto-\noptical sensors and visualizers, the terahertz isolators,\ncirculators, magneto-optical modulators, optical magne-\ntoelectric sensors, nonreciprocal elements of the inte-\ngrated optics, to promising applications in high density\nMO data-storage and low-power consumption spintronic\nnanodevices.\nRare-earth orthoferrites RFeO 3, which have been stud-\nied since the 60s of the last century, have attracted\nand continue to attract the particular attention of re-\nsearchers for several decades owing to their weak fer-\nromagnetism, remarkable magneto-optical properties,\nspin-reorientation transitions between antiferromagnet ic\nphases, high velocity of domain walls, and many other\nproperties. Their physical properties remain a focus\nof considerable research due to promising applications\nin innovative spintronic devices, furthermore, they con-\ntribute to an emerging class of materials, multiferroics\nwith strong magnetoelectric coupling.\nThe problem of describing the optical and magneto-\noptical properties of ferrites is one of the most challeng-\ning tasks in the theory of strongly correlated 3d com-\npounds. Despite many years of experimental and the-\noretical research, the nature of their optical and, espe-\ncially, magneto-optical response remains a subject of de-bate. This concerns both the identiﬁcation of electronic\ntransitions responsible for the formation of the main op-\ntical and magneto-optical properties and the compre-\nhensive calculation of their contribution to the optical\nand magneto-optical response functions. The solution of\nthis problem largely depends on the choice of the opti-\nmal strategy for taking into account the eﬀects of charge\ntransfer and strong local correlations, which can be for-\nmulated as the choice of a compromise between the one-\nelectron band and atomic-molecular description of elec-\ntronic states.\nThe nature of the low-energy optical electron-hole ex-\ncitations in the insulating transition metal 3d oxides rep-\nresents one of the most important challenging issues for\nthese strongly correlated systems. All these excitations\nare especially interesting because they could play a cen-\ntral role in multiband Hubbard models used to describe\nboth the insulating state and the unconventional states\ndeveloped under electron or hole doping. Because of the\nmatrix element eﬀect the optical response does provide\nonly an indirect information about the density of states.\nNevertheless it remains one of the most eﬃcient tech-\nnique to inspect the electronic structure and energy spec-\ntrum.\nIn this review paper, we present a critical analysis of\nband approaches to describing the optical and magneto-\noptical response of 3d ferrite-type compounds based on\nthe use of density functional theory (DFT) and ar-\ngue that the traditional physically transparent atomic-\nmolecular cluster approach (see, e.g., [ 1–3] and references\ntherein) based on local symmetry, strong covalence and\ncharge transfer (CT) eﬀects with strong local correla-\ntions, provides a consistent description and explanation\nof the optical and magneto-optical response of various\nferrites in a wide spectral range. The review was stim-\nulated by the lack of detailed and reliable studies of\nelectron-hole excitations and of a proper understanding\nof the relative role of diﬀerent transitions to optical and\nmagnetooptical response for ferrites.\nThe rest of the paper is organized as follows. In Sec.\n2 we present a critical overview of the DFT based ap-\nproaches for description of the optical and magnetoopti-\ncal properties of strongly correlated 3d compounds and2\npoint to the cluster model as a comprehensive physically\nclear alternative to the DFT approach. In Sec. 3 we ad-\ndress the charge transfer (CT) states and CT transitions\nin octahedral [FeO 6]9−and tetrahedral [FeO 4]5−clusters\nas basic elements of crystalline and electronic structure\nfor most ferrites. Here we also show that the CT transi-\ntions provide an adequate description of the optical spec-\ntra for a wide range of ferrites and other 3d oxides. In\nSec. 4 we discuss diﬀerent interactions for the CT states\nwith a speciﬁc focus on so-called exchange-relativistic in -\nteractions, in particular, novel \"spin-other-orbit\" inte r-\naction. In Sec. 5 we analyze the polarisability tensor for\nthe octahedral [FeO 6]9−cluster and argue that its con-\ntribution to the optical and magnetooptical anisotropy is\ndetermined by diﬀerent interactions in excited states. In\nSec. 6 we overview diﬀerent points of a microscopic theory\nof circular magnetooptics for ferrite-garnets and weak fer -\nromagnets, including Bi-substituted garnets, speciﬁc rol e\nof the \"spin-other-orbit\" coupling in weak ferromagnetic\nferrites, the temperature dependence of circular magne-\ntooptics, and the role of the 4f-5d transitions in rare-eart h\nions. A brief summary is given in Sec. 7.\nII. DENSITY FUNCTIONAL THEORY OR\nCLUSTER MODEL?\nA. So-called \"ab initio\" DFT based approaches\nThe electronic states in strongly correlated 3 doxides\nmanifest both signiﬁcant localization and dispersional\nfeatures. One strategy to deal with this dilemma is to\nrestrict oneself to small many-electron clusters embedded\nto a whole crystal, then creating model eﬀective lattice\nHamiltonians whose spectra may reasonably well repre-\nsent the energy and dispersion of the important excita-\ntions of the full problem. Despite some shortcomings the\nmethod did provide a clear physical picture of the com-\nplex electronic structure and the energy spectrum, as well\nas the possibility of a quantitative modeling.\nHowever, last decades the condensed matter commu-\nnity faced an expanding ﬂurry of papers with the so called\nab initio calculations of electronic structure and physical\nproperties for strongly correlated systems such as 3d com-\npounds based on density functional theory [ 4,5]. Only in\nrecent years has a series of papers been published on ab\ninitio calculations of the electronic structure, optical and\nmagneto-optical spectra of iron garnets (see e.g., Refs. [ 6–\n8])\nHowever, DFT still remains, in some sense, ill-deﬁned:\nmany of DFT statements were ill-posed or not rigorously\nproved. All eﬀorts to account for the correlations beyond\nLDA (local density approximation) encounter an insolu-\nble problem of double counting (DC) of interaction terms\nwhich had just included into Kohn-Sham single-particle\npotential.\nMost widely used DFT computational schemes start\nwith a \"metallic-like\" approaches making use of approx-imate energy functionals, ﬁrstly LDA scheme, which are\nconstructed as expansions around the homogeneous elec-\ntron gas limit and fail quite dramatically in capturing the\nproperties of strongly correlated systems. The LDA+U\nand LDA+DMFT (DMFT, dynamical mean-ﬁeld the-\nory)[9] methods are believed to correct the inaccuracies\nof approximate DFT exchange correlation functionals.\nThe main idea of these computational approaches con-\nsists in a selective description of the strongly correlated\nelectronic states, typically, localized dorforbitals, using\nthe Hubbard model, while all the other states continue\nto be treated at the level of standard approximate DFT\nfunctionals. At present the LDA+U and LDA+DMFT\nmethods are addressed to be most powerful tools for the\ninvestigation of strongly correlated electronic systems,\nhowever, these preserve many shortcomings of the DFT-\nLDA approach.\nUsually the values of eﬀective on-site Coulomb pa-\nrametersUeff=U−J, whereUrepresent the ad hoc\nHubbard on-site Coulomb repulsion parameter and J\nthe intra-atomic Hund’s exchange integral, are ordinar-\nily determined by seeking a good agreement of the cal-\nculated properties with the experimental results such as\nband gaps or oxidation energies. The values of Ueffaf-\nfect strongly the calculated material properties even in\nthe ground state, so that it is desirable to ﬁnd its opti-\nmal values, which also depend on the chosen exchange-\ncorrelation functional. Recent studies have attempted to\ncalculate these parameters directly based on ﬁrst princi-\nples approaches. Nevertheless, the calculated values dif-\nfer widely, even for the same ionic state in a given mate-\nrial, due to a number of factors such as the choice of the\nDFT scheme or the underlying basis set. Although it has\nbecome a common practice that a certain Ueffvalue is\nchosen a priori during the setup of a ﬁrst principles-based\ncalculation, it is also well known that a certain Ueffvalue\nmay not work deﬁnitively for all calculation methods and\nDFT schemes. By independently constraining the ﬁeld\non the Fe atoms at the octahedral and tetrahedral sites\nin YIG, the authors [ 8] have obtained two diﬀerent values\nofUeff, i.e., 9.8 eV for octa-Fe and 9.1 eV for tetra-Fe.\nThese values are considerably diﬀerent from those used\nfor iron garnets in previous works, e.g. U= 3.5 eV and\nJ= 0.8 eV[ 10] using the orthonormalized linear combina-\ntion of atomic orbitals basis set within constrained LDA\napproach and Ueff= 5.7 eV[ 11],U= 4 eV[ 7,12]. The\nHubbard and Hund’s UandJparameters were chosen\nasUeff= 2.7 eV for YIG, 4.7 eV for LuIG, and 5 eV for\nBi-substituted garnet Bi xLu3−xFe5O12[13].\nDespite many examples of a seemingly good agree-\nment with experimental data (photoemission and inverse-\nphotoemission spectra, magnetic moments,...) claimed\nby the DFT community, both the questionable starting\npoint and many unsolved and unsoluble problems give\nrise to serious doubts in quantitative and even qualita-\ntive predictions made within the DFT based techniques.\nStrictly speaking, the DFT is designed for description\nof ground rather than excited states with no good scheme3\nfor excitations. Because an excited-state density does not\nuniquely determine the potential, there is no general ana-\nlog of the Hohenberg-Kohn functional for excited states.\nThe standard functionals are inaccurate both for on-site\ncrystal ﬁeld and for charge transfer excitations [ 14]. The\nDFT based approaches cannot provide the correct atomic\nlimit and the term and multiplet structure [ 15,16], which\nis crucial for description of the optical response for 3d\ncompounds. Although there are eﬀorts to obtain correct\nresults for spectroscopic properties depending on spin\nand orbital density this problem remains as an open one\nin DFT research. Clearly, all these diﬃculties stem from\nunsolved foundational problems in DFT and are related\nto fractional charges and to fractional spins. Thus, these\nbasic unsolved issues in the DFT point toward the need\nfor a basic understanding of foundational issues.\nIn other words, given these background problems, the\nDFT based models should be addressed as semi-empirical\napproximate ones rather than ab initio theories. M. Levy\nintroduced in 2010 the term DFA to deﬁne density func-\ntional approximation instead of DFT, which is believed\nto quite appropriately describe contemporary DFT [ 17].\nBasic drawback of the spin-polarized approaches to de-\nscription of electronic structure for spin-magnetic sys-\ntems, especially in a simple LSDA scheme [ 6], is that\nthese start with a local density functional in the form\nv(r) =v0[n(r)]+∆v[n(r),m(r)](ˆσ·m(r)\n|m(r)|),\nwheren(r),m(r)are the electron and spin magnetic den-\nsity, respectively, ˆσis the Pauli matrix, that is these\nimply presence of a large ﬁctious local one-electron spin-\nmagnetic ﬁeld ∝(v↑−v↓), wherev↑,↓are the on-site\nLSDA spin-up and spin-down potentials. Magnitude of\nthe ﬁeld is considered to be governed by the intra-atomic\nHund exchange, while its orientation does by the eﬀec-\ntive molecular, or inter-atomic exchange ﬁelds. Despite\nthe supposedly spin nature of the ﬁeld it produces an\nunphysically giant spin-dependent rearrangement of the\ncharge density that cannot be reproduced within any con-\nventional technique operating with spin Hamiltonians.\nFurthermore, a direct link with the orientation of the\nﬁeld makes the eﬀect of the spin conﬁguration onto the\ncharge distribution to be unphysically large. However,\nmagnetic long-range order has no signiﬁcant inﬂuence on\nthe redistribution of the charge density. In such a case\nthe straightforward application of the LSDA scheme can\nlead to an unphysical overestimation of the eﬀects or even\nto qualitatively incorrect results due to an unphysical ef-\nfect of a breaking of spatial symmetry induced by a spin\nconﬁguration. The DFT-LSDA community needed many\nyears to understand such a physically clear point.\nOverall, the LSDA approach seems to be more or less\njustiﬁed for a semi-quantitative description of exchange\ncoupling eﬀects for materials with a classical Néel-like\ncollinear magnetic order. However, it can lead to er-\nroneous results for systems and high-order perturbation\neﬀects where the symmetry breaking and quantum ﬂuc-tuations are of a principal importance such as: i) non-\ncollinear spin conﬁgurations, in particular, in quantum\ns = 1/2 magnets; ii) relativistic eﬀects, such as the sym-\nmetric spin anisotropy, antisymmetric DM coupling; iii)\nspin-dependent electric polarization; iv) circular magne -\ntooptical eﬀects.\nIn general, the LSDA method to handle a spin de-\ngree of freedom is absolutely incompatible with a con-\nventional approach based on the spin Hamiltonian con-\ncept. There are some intractable problems with a match\nmaking between the conventional formalism of a spin\nHamiltonian and LSDA approach to the exchange and\nexchange-relativistic eﬀects. Visibly plausible numeric al\nresults for diﬀerent exchange and exchange-relativistic\nparameters reported in many LSDA investigations (see,\ne.g., Refs. [ 18]) evidence only a potential capacity of the\nLSDA based models for semiquantitative estimations,\nrather than for reliable quantitative data.\nIt is rather surprising how little attention has been\npaid to the DFT based calculations of the optical prop-\nerties for the transition metal oxides (TMO). Lets turn\nto a recent paper by Roedl and Bechstedt[ 19] on NiO\nand other TMOs, whose approach is typical for DFT\ncommunity. The authors calculated the dielectric func-\ntionǫ(ω)for NiO within the DFT-GGA+U+ ∆tech-\nnique and claim:\"The experimental data agree very well\nwith the calculated curves\" (!?). However, this seem-\ning agreement is a result of a simple ﬁtting when the two\nmodel parameters Uand∆are determined such (U= 3.0,\n∆= 2.0 eV) that the best possible agreement concerning\nthe positions and intensities of the characteristic peaks\nin the experimental spectra is obtained. In addition, the\nauthors arrive at absolutely unphysical conclusion: \"The\noptical absorption of NiO is dominated by intra-atomic\nt2g→egtransitions\" (!?).\nThere are still a lot of people who think the Hohenberg-\nKohn-Sham DFT within the LDA has provided a very\nsuccessful ab initio framework to successfully tackle the\nproblem of the electronic structure of materials. How-\never, both the starting point and realizations of the DFT\napproach have raised serious questions. The HK \"the-\norem\" of the existence of a mythical universal density\nfunctional that can resolve everything looks like a way\ninto Neverland, the DFT heaven is probably unattain-\nable. Various DFAs, density functional approximations,\nlocal or nonlocal, will never be exact. Users are will-\ning to pay this price for simplicity, eﬃcacy, and speed,\ncombined with useful (but not yet chemical or physical)\naccuracy [ 14,20].\nThe most popular DFA fail for the most interesting sys-\ntems, such as strongly correlated oxides, in particular fer -\nrites. The standard DFT approximations over-delocalize\nthe 3d-electrons, leading to highly incorrect description s.\nSome practical schemes, in particular, DMFT can correct\nsome of these diﬃculties, but none has yet become a uni-\nversal tool of known performance for such systems [ 14].4\nB. Cluster model approach\nAt variance with the DFT theory the cluster model\napproach does generalize and advance crystal-ﬁeld and\nligand-ﬁeld theory. The method provides a clear phys-\nical picture of the complex electronic structure and the\nenergy spectrum, as well as the possibility of a quan-\ntitative modeling. In a certain sense the cluster calcu-\nlations might provide a better description of the over-\nall electronic structure of insulating 3d oxides than the\nband structure calculations [ 21,22], mainly due to a bet-\nter account for correlation eﬀects, electron-lattice cou-\npling, and relatively weak interactions such as spin-\norbital and exchange coupling. Moreover, the cluster\nmodel has virtually no competitors in the description of\nimpurity or dilute systems. Cluster models do widely use\nthe symmetry for atomic orbitals, point group symme-\ntry, and advanced technique such as Racah algebra and\nits modiﬁcations for point group symmetry [ 23]. From\nthe other hand the cluster model is an actual proving-\nground for various calculation technique from simple\nquantum chemical MO-LCAO (molecular orbital-linear-\ncombination-of-atomic-orbitals) method to a more elab-\norate LDA+ MLFT (MLFT, multiplet ligand-ﬁeld the-\nory)[24] approach. The LDA+ MLFT technique implies\na sort of generalization of conventional ligand-ﬁeld model\nwith the DFT-based calculations. Haverkort et al.[24]\nstart by performing a DFT calculation for the proper,\ninﬁnite crystal using a modern DFT code which employs\nan accurate density functional and basis set [e.g., lin-\near augmented plane waves (LAPWs)]. From the (self-\nconsistent) DFT crystal potential they then calculate a\nset of Wannier functions suitable as the single-particle\nbasis for the cluster calculation. The authors compared\nthe theory with experimental spectra (XAS, nonresonant\nIXS, photoemission spectroscopy) for diﬀerent 3d oxides\nand found overall satisfactory agreement, indicating that\ntheir ligand-ﬁeld parameters are correct to better than\n10%. However, the authors have been forced to treat\non-site correlation parameter Uddand orbitally averaged\n(spherical) ∆pdparameter as adjustable ones. Despite\nthe involvement of powerful calculation techniques the\nnumerical results of the LDA+ MLFT approach seem to\nbe more like semiquantitative ones. Nevertheless, any\ncomprehensive physically valid description of the electro n\nand optical spectra for strongly correlated systems, as we\nsuggest, should combine simple physically clear cluster\nligand-ﬁeld analysis with a numerical calculation tech-\nnique such as LDA+MLFT [ 24], and a regular appeal to\nexperimental data.\nIt is now believed that the most intensive low-energy\nelectron-hole excitations in insulating 3d oxides corre-\nspond to the charge transfer (CT) transitions while diﬀer-\nent phonon-assisted crystal ﬁeld transitions are generall y\nmuch weaker. Namely the CT transitions are considered\nas a likely source of the optical and magneto-optical re-\nsponse of the 3d metal-based oxide compounds in a wide\nspectral range of 1-10 eV, in particular, of the fundamen-tal absorption edge. The low-energy dipole-forbidden d-d\norbital excitations, or crystal ﬁeld transitions, are char -\nacterized by the oscillator strengths which are smaller by\na factor 102−103than those for the dipole-allowed p-d\nCT transitions and usually correspond to contributions\nto the dielectric function ε′′of the order of 0.001-0.01.\nDespite CT transitions are well established concept in\nthe solid state physics, their theoretical treatment re-\nmains rather naive and did hardly progress during last\ndecades. Usually it is based on the one-electron approach\nwith some 2 p-3dor, at best, 2 p→3dt2g, 2p→3degCT\ntransitions in 3 doxides. In terms of the Hubbard model,\nthis is a CT transition from the nonbonding oxygen band\nto the upper Hubbard band. But such a simpliﬁed ap-\nproach to CT states and transitions in many cases ap-\npears to be absolutely insuﬃcient and misleading even\nfor qualitative explanation of the observed optical and\nmagneto-optical properties. First, one should general-\nize the concept of CT transitions taking into account\nthe conventional transition between the lower and up-\nper Hubbard bands which corresponds to an inter-site\nd-dCT transition, or intersite transition across the Mott\ngap.\nSeveral important problems are hardly addressed in\nthe current analysis of optical spectra, including the rel-\native role of diﬀerent initial and ﬁnal orbital states and\nrespective CT channels, strong intra-atomic correlations ,\neﬀects of strong electron and lattice relaxation for CT\nstates, the transition matrix elements, or transition prob -\nabilities, probable change in crystal ﬁelds and correlatio n\nparameters accompanying the charge transfer.\nOne of the central issues in the analysis of electron-\nhole excitations is whether low-lying states are comprised\nof free charge carriers or excitons. A conventional ap-\nproach implies that if the Coulomb interaction is eﬀec-\ntively screened and weak, then the electrons and holes are\nonly weakly bound and move essentially independently as\nfree charge-carriers. However, if the Coulomb interaction\nbetween electrons and holes is strong, excitons are be-\nlieved to form, i.e. bound particle-hole pairs with strong\ncorrelation of their mutual motion.\nDespite all the shortcomings the cluster models have\nproven themselves to be reliable working models for\nstrongly correlated systems such as 3d compounds.\nThese have a long and distinguished history of appli-\ncation in electron, optical and magnetooptical spec-\ntroscopy, magnetism, and magnetic resonance. The au-\nthor with colleagues has successfully demonstrated great\npotential of the cluster model for description of the p-d\nandd-dcharge transfer transitions and their contribu-\ntion to optical and magneto-optical response in various\n3d oxides such as ferrites [ 25–34], cuprates [ 35–39], man-\nganites [ 34,40,41], and nickelates [ 42,43].5\n3d\n2peg\nt2g\nt□□( )2gp\ne□( )gs\na□□( )1gst□□( )1ust□□( )1gp\nt□□( )2up\nt□□( )1up\nDistorted□MeO octahedron6□=10Dq\nNondistorted□MeO octahedron6\nFIG. 1. The diagram of Me 3d-O2p molecular orbitals for the\nMeO6octahedral center. The O 2p - Me 3d charge transfer\ntransitions are shown by arrows: strong dipole-allowed σ−σ\nandπ−πby thick solid arrows; weak dipole-allowed π−σand\nσ−πby thin solid arrows; weak dipole-forbidden low-energy\ntransitions by thin dashed arrows, respectively.\nIII. CLUSTER MODEL: THE CT\nCONFIGURATIONS AND CT TRANSITIONS IN\nFERRITES\nA. Electronic structure of octahedral [FeO 6]9−\nclusters in ferrites\nThe slightly distorted octahedral [FeO 6]9−clusters are\nmain optical and magneto-optical centers in weak fer-\nromagnetic orthoferrrites RFeO 3, hematite α-Fe2O3,\nborate FeBO 3, cubic antiferromagnetic garnets like\nCa3Fe2Ge3O12, and, together with tetrahedral [FeO 4]5−\ncomplexes in other ferrites as well.\nFive Me 3d and eighteen oxygen O 2p atomic orbitals\nin octahedral MeO 6complex with the point symmetry\ngroupOhform both hybrid Me 3d-O 2p bonding and anti-\nbondingegandt2gmolecular orbitals, and purely oxygen\nnonbonding a1g(σ),t1g(π),t1u(σ),t1u(π),t2u(π)orbitals\n(see, e.g., Refs.[ 3,23,40]). Nonbonding t1u(σ)andt1u(π)\norbitals with the same symmetry are hybridized due to\nthe oxygen-oxygen O 2p π- O 2pπtransfer. The relative\nenergy position of diﬀerent nonbonding oxygen orbitals\nis of primary importance for the spectroscopy of the\noxygen–3d–metal charge transfer. This is ﬁrstly deter-\nmined by the bare energy separation ∆ǫ2pπσ=ǫ2pπ−ǫ2pσ\nbetween O 2p πand O 2pσelectrons.\nSince the O 2p σorbital points towards the two neigh-\nboring positive 3d ions, an electron in this orbital has itsenergy lowered by the Madelung potential as compared\nwith the O 2p πorbitals, which are oriented perpendic-\nular to the respective 3d–O–3d axes. Thus, Coulomb\narguments favor the positive sign of the π−σseparation\nǫpπ−ǫpσwhich numerical value can be easily estimated\nin frames of the well-known point charge model, and ap-\npears to be of the order of 1.0eV. In a ﬁrst approxi-\nmation, all the γ(π)statest1g(π),t1u(π),t2u(π)have the\nsame energy. However, the O 2p π-O 2pπtransfer yields\nthe energy correction to bare energies with the largest\nvalue and positive sign for the t1g(π)state. The energy\nof thet1u(π)state drops due to a hybridization with the\ncation 4 pt1u(π)state. In other words, the t1g(π)state is\nbelieved to be the highest in energy non-bonding oxygen\nstate. For illustration, in Figure 1we show the energy\nspectrum of the 3d-2p manifold in the octahedral com-\nplexes MeO 6with the relative energy position of the lev-\nels according to the quantum chemical calculations [ 44]\nfor the [FeO 6]9−octahedral complex in a lattice environ-\nment typical for perovskites such as LaFeO 3. It should be\nemphasized one more that the top of the oxygen electron\nband is composed of O 2p πnonbonding orbitals that pre-\ndetermines the role of the oxygen states in many physical\nproperties of 3d perovskites.\nTheconventional ground state electronic structure of\noctahedral Fe3+O6clusters is associated with the conﬁg-\nuration of the completely ﬁlled O 2p shells and half-ﬁlled\nFe 3d shell. The typical high-spin ground state conﬁg-\nuration and crystalline term for Fe3+in the octahedral\ncrystal ﬁeld or for the octahedral [FeO 6]9−center ist3\n2ge2\ng\nand6A1g, respectively.\nThe excited CT conﬁguration γ1\n2p3dn+1arises from\nthe spin-conserving transition of an electron from the pre-\ndominantly anionic molecular orbitals γ2pinto an empty\n3d type MO ( t2goreg). The transition between the\nground and the excited conﬁguration can be presented\nas the intra-center p-d CT transition γ2p→3d(t2g,eg)\n.\nThe p-d CT conﬁguration consists of two partly\nﬁlledmolecular-orbital subshells, localized predominan tly\non 3d cation and ligands, respectively. The excited cation\nconﬁguration (3d6) nominally corresponds to the Fe2+\nion. Strictly speaking, the many-electron p-d CT conﬁg-\nuration should be written as tn1\n2gen2gγ2pwithn1+n2= 6,\nor((tn1\n2gen2g)2S′+1Γ′\ng;γ2p)2S+1Γ(S=S′±1\n2,Γ∈Γ′\ng×γ2p,\n2S+1Γis a crystal term of the CT conﬁguration), if we\nmake use of the spin and orbital quasimomentum addi-\ntion technique [ 23].\nB. Intra-center electric-dipole p-d CT transitions\nThe conventional classiﬁcation scheme of the intra-\ncenter electric-dipole p-d CT transitions in the octahe-\ndral [FeO 6]9−clusters ﬁrst of all includes the electric-\ndipole allowed transitions from the odd-parity oxygen\nγu=t1u(π),t2u(π),t1u(σ)orbitals to the even-parity iron6\n3dt2gand 3degorbitals, respectively. These one-electron\ntransitions generate the many-electron ones6A1g→\n6T1u, which diﬀer by the crystalline term of the respec-\ntive 3d6conﬁguration:\n(t3\n2g4A2g;e2\ng)6A1g→((t4\n2g;e2\ng)5T2g;γu)6T1u,(1)\n(t3\n2g4A2g;e2\ng)6A1g→((t3\n2g;e3\ng)5Eg;γu)6T1u,(2)\nforγu→3dt2gandγu→3degtransitions, respec-\ntively. We see that in contrast to the manganese centers\nMn3+O9−\n6[40] each one-electron γu→3dt2gtransition\ngenerates one many-electron CT transition.\nMeO6octahedral center can be written with the aid of\nWigner-Eckart theorem [ 23] as follows (see Ref. [ 40] for\ndetails)\n∝angbracketleftγuµ|ˆdq|γgµ′∝angbracketright= (−1)j(γu)−µ/angbracketleftbiggγut1uγg\n−µ q µ′/angbracketrightbigg∗\n∝angbracketleftγu∝bardblˆd∝bardblγg∝angbracketright,\n(3)\nwhere/angbracketleftbigg\n· · ·\n· · ·/angbracketrightbigg\nis the Wigner coeﬃcient for the cubic\npoint group O h[23],j(Γ)is the so-called quasimomen-\ntum number, ∝angbracketleftγu∝bardblˆd∝bardblγg∝angbracketrightis the one-electron dipole mo-\nment submatrix element. The 3d-2p hybrid structure of\nthe even-parity molecular orbital γgµ=Nγg(3dγgµ+\nλγg2pγgµ)and a more simple form of purely oxygen odd-\nparity molecular orbital γuµ≡2pγuµboth with a sym-\nmetry superposition of the ligand O 2p orbitals point to\na complex form of the submatrix element in ( 3) to be a\nsum oflocalandnonlocal terms composed of the one-site\nand two-site ( d-pandp-p) integrals, respectively. In the\nframework of a simple \"local\" approximation that implies\nthe full neglect of all many-center integrals\n∝angbracketleftt2u(π)∝bardblˆd∝bardbleg∝angbracketright= 0;∝angbracketleftt2u(π)∝bardblˆd∝bardblt2g∝angbracketright=−i/radicalbigg\n3\n2λπd;\n∝angbracketleftt1u(σ)∝bardblˆd∝bardblt2g∝angbracketright= 0;∝angbracketleftt1u(σ)∝bardblˆd∝bardbleg∝angbracketright=−2√\n3λσd;\n∝angbracketleftt1u(π)∝bardblˆd∝bardbleg∝angbracketright= 0;∝angbracketleftt1u(π)∝bardblˆd∝bardblt2g∝angbracketright=/radicalbigg\n3\n2λπd. (4)\nHere,λσ∼tpdσ/∆pd,λπ∼tpdπ/∆pdareeffective\ncovalency parameters for eg,t2gelectrons, respectively,\nd=eR0is an elementary dipole moment for the cation-\nanion bond length R0. We see, that the \"local\" approx-\nimation results in an additional selection rule: it for-\nbids theσ→π, andπ→σtransitions, t1u(σ)→t2g,\nandt1,2u(π)→eg, respectively, though these are dipole-\nallowed. In other words, in frames of this approximation\nonlyσ-type (t1u(σ)→eg) orπ-type (t1,2u(π)→t2g) CT\ntransitions are allowed. Hereafter, we make use of the\nterminology of \"strong\" and \"weak\" transitions for the\ndipole-allowed CT transitions going on the σ−σ,π−π,TABLE I. Parameters (energies, oscillator strength, line\nwidth) of the dipole allowed intra-center CT transitions in oc-\ntahedral (6A1g→6T1u, No.= 1-6) and tetrahedral (6A1g→\n6T2, No.= 7-13) clusters in Y 3Fe5O12[26,31]. Ecompand E fit\nare the computed and ﬁtted CT transition energies, respec-\ntively.\nNo.Transition Ecomp(eV)Efit(eV)f (×10−3)Γ(eV)\n1t2u→t2g 3.1 2.8 4 0.2\n2t1u(π)→t2g 3.9 3.6 30 0.3\n3t2u→eg 4.4 4.3 60 0.3\n4t1u(σ)→t2g 5.1 4.8 40 0.3\n5t1u(π)→eg 5.3 5.2 200 0.3\n6t1u(σ)→eg 6.4 6.1 200 0.3\n71t1→2e 3.4 3.4 30 0.4\n86t2→2e 4.3 4.6 20 0.3\n91t1→7t2 4.5 4.7 40 0.3\n105t2→2e 5.0 4.9 30 0.3\n116t2→7t2 5.4 5.1 20 0.3\n121e→7t2 5.6 5.6 10 0.3\n135t2→7t2 6.0 6.0 20 0.3\nandπ−σ,σ−πchannels, respectively. It should be\nemphasized that the \"local\" approximation, if non-zero,\nis believed to provide a leading contribution to transi-\ntion matrix elements with corrections being of the ﬁrst\norder in the cation-anion overlap integral. Moreover,\nthe nonlocal terms are neglected in standard Hubbard-\nlike approaches. Given typical cation-anion separations\nRMeO≈4a.u. we arrive at values less than 0.1 a.u.\neven for the largest two-site integral, however, their ne-\nglect should be made carefully. Exps.( 3),(4) point to\nlikely extremely large dipole matrix elements and oscil-\nlator strengths for strong p-dCT transitions, mounting\ntodij∼eÅ andf∼0.1, respectively.\nHence, starting with three nonbonding purely oxy-\ngen orbitals t1u(π),t1u(σ),t2u(π)as initial states for\none-electron CT, we arrive at six many-electron dipole-\nallowed CT transitions6A1g→6T1u. There are two tran-\nsitionst1u(π),t2u(π)→t2g(π−πchannel), two transi-\ntionst1u(π),t2u(π)→eg(π−σchannel), one transi-\ntiont1u(σ)→t2g(σ−πchannel), and one transition\nt1u(σ)→eg(σ−σchannel).\nIt should be noted that the dipole-forbidden t1g(π)→\nt2gtransition seemingly determines the onset energy of\nall the p-d CT bands.\nFor our analysis to be more quantitative we make two\nrather obvious model approximations. First of all, we as-\nsume that as usually for cation-anion octahedra in 3d ox-\nides [3,44,45] the non-bonding t1g(π)oxygen orbital has\nthe highest energy and forms the ﬁrst electron removal\noxygen state. Furthermore, to be deﬁnite we assume that\nthe energy spectrum of the non-bonding oxygen states\nfor [Fe3+O6]9−centers coincides with that calculated in\nRef. [44] for [Fe3+O6]9−in orthoferrite LaFeO 3, in other7\nwords, we have (in eV):\n∆(t1g(π)−t2u(π))≈0.8; ∆(t1g(π)−t1u(π))≈1.8;\n∆(t1g(π)−t1u(σ))≈3.0.\nSecondly, we choose for the Racah parameters B=\n0.09eV andC= 0.32eV, the numerical values typical\nfor the Fe3+ion [3].\nThe energies of the intra-center CT transitions for oc-\ntahedral FeO 6and tetrahedral FeO 4clusters in Y 3Fe5O12\nwere calculated using the spin-polarized X αdiscrete vari-\national (SP-X αDV) method [ 26,31]. These results are\npresented in Table Itogether with the results of ﬁtting\nthe experimental optical data [ 3,46], taking into account\nonly the contribution of the intra-center CT transitions\nwith a Lorentzian line shape.\nIn addition to several dipole-allowed CT transitions,\nthe CT band will also include various forbidden transi-\ntions. First of all, these are dipole-forbidden p-d transi-\ntions between states with the same parity of the 2p t1g-\n3dt2gtype, as well as satellites of allowed transitions\nhaving the same electronic conﬁguration, but diﬀerent\nterms of the ﬁnal states. For instance in the FeO 6-\noctahedron, these are the6A1g→6Γutransitions\n(Γ =A1, A2, E, T 1) forbidden by the quasimoment\nselection rule, and the6A1g→4Γuspin forbidden\ntransitions (if Γ∝negationslash=T1u, then quasimoment forbidden,\ntoo). The forbiddenness of these transitions is lifted\neither by the electron-lattice interaction, low-symmetry\ncrystal ﬁeld, spin-orbital interaction, or the exchange in -\nteraction with neighbouring clusters. A detailed analysis\nof the energy spectrum of the CT band requires taking\ninto account the d-d, p-d, and p-p correlation eﬀects.\nC. Inter-center d-dCT transitions\nStrictly speaking, reliable identiﬁcation of the intra-\ncenter p-d CT transitions is possible only in highly\ndilute or impurity systems such as YAlO 3:Fe or\nCa3FexGa2−xGe3O12, while in concentrated systems\n(YFeO 3, Ca3Fe2Ge3O12, Y3Fe5O12, ..) these transi-\ntions compete with inter-center d-d CT transitions [ 35–\n37,39,41,42].\nThe inter-center d-dCT transitions between two MeO n\nclusters centered at neighboring sites 1 and 2 deﬁne inter-\ncenter d-dCT excitons in 3d oxides [ 35–37,39,41,42].\nThese excitons may be addressed as quanta of the dis-\nproportionation reaction\nMe1Ov\nn+Me2Ov\nn→Me1Ov−1\nn+Me2Ov+1\nn,(5)\nwith the creation of electron MeOv−1\nnand hole MeOv+1\nn\ncenters. Depending on the initial and ﬁnal single particle\nstates all the inter-center d-dCT transitions may be clas-\nsiﬁed to the eg−eg,eg−t2g,t2g−eg, andt2g−t2gones.\nFor the 3 doxides with cations obeying the Hund rulethese can be divided to so-called high-spin (HS) transi-\ntionsS1S2S→S1±1\n2S2∓1\n2Sand low-spin (LS) transi-\ntionsS1S2S→S1−1\n2S2−1\n2S, respectively.\nAn inter-center d-dCT transition in iron oxides with\nFe3+O6octahedra\n[FeO6]9−+[FeO6]9−→[FeO6]10−+[FeO6]8−(6)\nimplies the creation of electron [FeO 6]10−and hole\n[FeO6]8−centers with electron conﬁgurations formally\nrelated to Fe2+and Fe4+ions, respectively. The low-\nenergy inter-center d-dCT transitions from the initial\nFe3+O6(t3\n2ge2\ng) :6A1gstates can be directly assigned to\neg→eg,eg→t2g,t2g→eg, andt2g→t2gchannels with ﬁnal\nconﬁgurations and terms\neg→eg:t3\n2ge1\ng;5Eg−t3\n2ge3\ng;5Eg,\neg→t2g:t3\n2ge1\ng;5Eg−t4\n2ge2\ng;5T2g,\nt2g→eg:t2\n2ge2\ng;5T2g−t3\n2ge3\ng;5Eg,\nt2g→t2g:t2\n2ge2\ng;5T2g−t4\n2ge2\ng;5T2g. (7)\nIn the framework of high-spin conﬁgurations the eg→t2g\nCT transition has the lowest energy ∆ = ∆ eg−t2g,\nwhile theeg→eg,t2g→t2g, andt2g→egtransitions have\nthe energies ∆ + 10Dq(3d6),∆ + 10Dq(3d4), and∆ +\n10Dq(3d6) + 10Dq(3d4), respectively. The transfer en-\nergy in the Fe3+-based ferrites for the eg→t2gCT tran-\nsition\n∆Fe−Fe\negt2g=A+28B−10Dq\ncan be compared with a similar quantity for the eg→eg\nCT transition in Mn3+-based manganite LaMnO 3\n∆Mn−Mn\negeg=A−8B+∆JT,\nwhere∆JTis the Jahn-Teller splitting of the eglevels\nin manganite. Given B≈0.1eV,Dq≈0.1eV,∆JT≈\n0.7eV,∆Fe−Fe\negeg≈2.0eV (see, e.g., Ref. [ 48]) we getA≈\n2.0eV,∆Fe−Fe\negt2g≈4.0eV. In other words, the onset of\nthed-dCT transitions in Fe3+-based ferrites is strongly\n(∼2eV) blue-shifted as compared to the Mn3+-based\nmanganite LaMnO 3.\nAnother important diﬀerence between ferrites and\nmanganites lies in the opposite orbital character of initia l\nand ﬁnal states for the d-dCT transitions. Indeed, the\nlow-energy d4d4→d3d5CT transition in manganites\nimplies an orbitally degenerate Jahn-Teller initial state\n5Eg5Eg[49] and an orbitally nondegenerate ﬁnal state\n4A2g6A1gwhile the low-energy d5d5→d4d6CT transi-\ntions in ferrites imply an orbitally nondegenerate initial\nstate6A1g6A1gand an orbitally degenerate Jahn-Teller\nﬁnal states such as5Eg5Egforeg→egor5Eg5T2gfor\neg→t2gCT transitions. An unconventional ﬁnal state\nwith an orbital degeneracy on both sites, or Jahn-Teller\nexcited states may be responsible for the complex multi-\npeak lineshape of the inter-center d-dCT band in ferrites.8\nD. Interplay of the CT transitions in ferrites\nThe most complete and detailed analysis of the opti-\ncal spectra for a wide range of ferrites has been carried\nout in relatively recent papers [ 33,34]. The authors an-\nalyze optical ellipsometry data in the spectral range of\n0.6-5.8 eV for two groups of the iron oxides with more\nor less distorted FeO 6octahedral and FeO 4tetrahedral\nclusters. One of the two groups of materials includes or-\nthoferrites RFeO 3, bismuthate BiFeO 3, Y.95Bi.05FeO3,\nhematiteα−Fe2O3, Fe2−xGaxO3, and borate Fe 3BO6\nin which iron Fe3+ions occupy only octahedral centro-\nor noncentrosymmetric positions and distortions range\nfrom 1 to 20 %. The second group includes lithium fer-\nrite LiFe 5O8, barium hexaferrite BaFe 12O19, iron garnets\nR3Fe5O12, and calcium ferrite Ca 2Fe2O5in which Fe3+\nions occupy both octahedral and tetrahedral positions\nwith a rising tetra/ortho ratio. Experimental data were\ndiscussed within the cluster model which implies an inter-\nplay of intra- ( p-d) and inter-center ( d-d) CT transitions.\nSome previously reported optical data on ferrites were\nin most cases obtained with the use of conventional reﬂec-\ntion and absorption methods. The technique of optical\nellipsometry provides signiﬁcant advantages over conven-\ntional reﬂection and transmittance methods in that it\nis self-normalizing and does not require reference mea-\nsurements. The optical complex dielectric function ε=\nε′−iε′′is obtained directly without a Kramers-Krönig\ntransformation. The dielectric function εwas obtained\nin the range from 0.6 to 5.8 eV at room temperature. The\ncomparative analysis of the spectral behavior of ε′and\nε′′is believed to provide a more reliable assignement of\nspectral features. The spectra were analyzed using the\nset of the Lorentz functions\nTo begin our discussion of the CT transitions in\nferrites we refer to the spectroscopic data for garnets\nY3FexGa5−xO12(x=5, 3.9, 0.29, 0.09)[ 51]. They demon-\nstrate that the optical response in the spectral range up\nto 30 000 cm−1(∼3.7 eV) is governed by the intra-center\ntransitions for both octahedral and tetrahedral Fe3+cen-\nters. It means that the onset energy for diﬀerent d-dCT\ntransitions in ferrites is expected to be >3.7 eV in agree-\nment with our model estimates discussed in Sec.3.3.\nTo uncover the role played by the octahedral Fe3+cen-\nters we turn to the optical response of the orthoferrites\nRFeO3.\nThese compounds contain the only type of centrosym-\nmetric, slightly ( ∼1%) distorted, FeO 6octahedra. De-\nspite the long story of optical and magneto-optical stud-\nies (see, e.g. Refs. [ 3,52]) the microscopic origin of\nthe main spectral features in orthoferrites remains ques-\ntionable and the transition assignments made earlier in\nRef. [3] need a comprehensive revisit. The ε′,ε′′spectra\nof ErFeO 3for three main polarizations shown in Fig. 2\nare typical for orthoferrites RFeO 3[3,52,53]. The low-\nenergy intense band around 3 eV may be assigned to a\nstrong dipole allowed intra-center t2u(π)→t2gCT tran-\nsition as was proposed in Ref. [ 3]. This is a characteristicPhoton energy, eV1 2 3 4 5Dielectric function i,/c101 /c101 /c101= -’ ’’ErFeO3\n/c101b/c101a\n/c101c/c101’\n/c101’\n/c101’/c101’’\n/c101’’\n/c101’’\n2468\n02468\n02468\n0\nFIG. 2. (Color online) The dielectric function spectra\nin ErFeO 3orthoferrite for three main polarizations. The\nLorentzian ﬁtting is marked by dotted curves and ﬁlling. In-\nsets show indices of absorption and refraction.\nfeature of the octahedral Fe3+centers in oxides. How-\never, such an assignment also implies the existence of a\nweak band due to a low-energy dipole-forbidden intra-\ncentert1g(π)→t2gCT transition, red-shifted by about\n0.8 eV as expected from estimates [ 44]. Indeed, a band\naround 2.5 eV is found in the optical and magneto-optical\nspectra of diﬀerent orthoferrites [ 3]. This band is clearly\nvisible in hematite α-Fe2O3near 2.4 eV[ 33,34] where the\nt1g(π)→t2gtransition becomes allowed due to a break-\ning of the centro-symmetry for Fe3+centers.\nThe nearest high-energy neighborhood of the 3 eV\nband is expected to be composed of t1u(π)→t2gCT\ntransitions with a comparable intensity and estimated\nenergy about 4 eV. All the dipole-allowed intra-center\np-dCT transitions to the egstate are blue-shifted by\n10Dq(3d5)as compared to their γ→t2gcounterparts\nwith the onset energy of the order of 4 eV. Interestingly,\nfor the dipole-allowed γu→t2gtransitions the maximum\nintensity is expected for the low-energy t2u(π)→t2g\ntransition while for γu→egtransitions the maximum\nintensity is expected for the high-energy ( ∼6−7eV)\nt1u(σ)→egtransition. The analysis of the experimen-\ntal spectra for orthoferrites demonstrates the failure of9\nthe intra-center p-dCT transitions to explain the broad\nintensive band centered near 4.5 eV together with a nar-\nrow low-energy satellite peaked near 3.9 eV. Both fea-\ntures are typical for orthoferrites [ 3,52] and may be as-\nsigned to a eg→t2glow-energy inter-center CT tran-\nsition6A1g6A1g→5Eg5T2gto an unconventional ﬁnal\nstate with an orbital degeneracy on both sites. These\nJahn-Teller excited states are responsible for the complex\nlineshape of the eg→t2gCT band which is composed of\na narrow exciton-like feature and a broad intense band\nseparated by ∼0.5 eV, which is believed to be a measure\nof the Jahn-Teller splitting in the excited state. Thus\nwe see that all the spectral features observed in the opti-\ncal spectra of orthoferrites for energies below 5 eV can be\ndirectly assigned to the low-energy intra-center p-dand\ninter-center d-dCT transitions.\nIt is worth noting that the dielectric function in ortho-\nferrites is nearly isotropic due to very weak ( ∼1%) rhom-\nbic distortions of FeO 6octahedra and nearly equivalent\ndiﬀerent Fe-O-Fe bonds. Nevertheless a ﬁne structure of\nthe main CT bands is clearly revealed in magneto-optical\nspectra of orthoferrites, which was earlier assigned to the\ndipole-forbidden d-dcrystal ﬁeld transitions [ 3,52]. In\nour opinion, their relation to the low-symmetry distor-\ntions in the p-dCT band seems to be more reasonable.\nThe eﬀect of a strong change in bulk crystalline sym-\nmetry and local trigonal noncentrosymmetric distortions\nof FeO 6octahedra is well illustrated by the optical re-\nsponse of hematite α-Fe2O3[33,34]. First of all there is\na noticeable rise of intensity and a splitting for dipole-\nforbiddent1g(π)→t2gtransition at 2.4 eV, which is\nclearly visible in the spectra of the gallium-substituted\nsample. Second, one should note a clear splitting on the\norder of 0.3-0.4 eV of the 3 eV band due to a sizeable trig-\nonal distortion of the FeO 6octahedra. In both cases theband splitting eﬀect reﬂects the singlet-doublet splittin g\nof the initial orbital triplets, t1g(π)andt2u(π), respec-\ntively, due to the low-symmetry trigonal crystal ﬁeld.\nInterestingly, the integral intensity of the t2u(π)→t2g\nband at 3 eV is visibly enhanced in hematite as compared\nto similar bands in orthoferrites that may result from the\nmore covalent Fe-O bonding in hematite.\nIV. EFFECTIVE HAMILTONIAN FOR\nFE-CLUSTERS IN FERRITES\nAs the principal interactions determining the CT tran-\nsitions contribution to the optics and magneto-optics of\nferrites, we note the symmetrysymmetry crystal ﬁeld\n(LSCF), Zeeman interaction VZ, spin-orbit interac-\ntionVSO, exchange interaction Vex, and the exchange-\nrelativistic interactions Vex\nso. The CT conﬁgurations\nhave two unﬁlled shells – the 3d6(t4\n2ge2\ngort3\n2ge3\ng)-\nshell and γ2p- shell ( ˜γ1\n2p-hole), which distinguishes\nthem considerably from the ground state conﬁguration\nhaving only one unﬁlled shell 3d5(t3\n2ge3\ng)and leads\nto the speciﬁcity of the manifestation of various interac-\ntions, especially anisotropic ones. Below, we consider the\naforenamed interactions in the cluster approach.\nA. Low-symmetry crystal ﬁeld\nUsing the the cubic group irreducible tensor operator\ntechnique, in particular, the Wigner-Eckart theorem [ 23]\nwe can write the matrix of the eﬀective Hamiltonian of\nthe low-symmetry crystal ﬁeld, ˆHLSCF,in general as\nfollows\n∝angbracketleftκSMsΓM|ˆHLSCF|κ′S′M′\nsΓ′M′∝angbracketright=/summationdisplay\nγν/summationdisplay\nΓΓ′Bγ⋆\nν(κΓκ′Γ′)(−1)Γ−M/angbracketleftigg\nΓγΓ′\n−M ν M′/angbracketrightigg\nδSS′δMsM′s, (8)\nwhereγ=E,T2,∝angbracketleft:::∝angbracketrightis the3Γsymbol [ 23],κ,κ′are\ncertain CT conﬁgurations, Bγ\nν(ΓΓ′)are crystal ﬁeld pa-\nrameters.\nFor a certain T1(T2)term the ˆHLSCF can be written\nas an eﬀective operator\nVLSCF=/summationdisplay\nijBCF\nij/bracketleftbigg\n/tildewideLiLj−1\n3L(L+ 1)δij/bracketrightbigg\n.(9)\nHere,BCF\nijis the symmetric traceless matrix of the\nLSCF parameters; /tildewideLiLj= (LiLj+LjLi)/2,Lis\nthe eﬀective orbital moment of the T1- ,T2- term(L=\n1). However, in general, the LSCF can lead to the\nmixing of diﬀerent cubic terms2S+1Γ,2S+1Γ′(E,T2∈\nΓ×Γ′) of identical or diﬀerent CT conﬁgurations withthe same spin multiplicity. All these eﬀects may be of\nimportance, since HLSCF reaches the magnitude up to\n∼0.1eVunder the low-symmetry distortions of the\n[FeO6]9−complex of order 10−2.\nB. Conventional spin-orbital interaction\nThe conventional \"intra-center\" spin-orbital interac-\ntionVSO=/summationtext\nia(ri)li·sifor a certain T1(T2)term\ncan be written as follows\nVso=λL·S, (10)\nwhereλis the eﬀective spin-orbit coupling constant,\ntabulated for the CT states of the [FeO 6]9−, [FeO 4]5−10\nclusters in Refs.[ 26,31]. The contributions to λare due\nboth to the ligand (oxygen) 2p-subsystem and the iron\nsubsystem, the latter contribution being dominant. VSO\nleads to the terms splitting and mixing, the latter being\nespecially signiﬁcant in case of identical conﬁgurations\nor those diﬀering from each other in the state of the 3d-\nshell, only. However, in general, the VSOcan lead to the\nmixing of diﬀerent cubic terms2S+1Γ,2S′+1Γ′(|S−S′| ≤\n1≤S+S′;T1∈Γ×Γ′).\nC. Zeeman interaction\nThe Zeeman interaction VZ=/summationtext\niµB(li+ 2si)·H\ncan be written for a certain T1(T2)term as an eﬀective\noperator\nVZ=µB(gLL+gSS)·H, (11)\nwheregSandgLare respectively the spin g-factor\n(gS≈2) and the eﬀective orbital g-factor whose values\nare listed in Refs. [ 26,31]. Note, that gLcan disagree\nwith the classical orbital value gL= 1not only in magni-\ntude, but even in sign. In particular, the CT state of the\nt5\n2u(t4\n2ge2\ng5T2)conﬁguration dominating the magneto-\noptics of ferrites at the long wavelength tail, has the value\ngL=−3\n4. It is worth noting that at variance with the\nspin-orbital coupling the contributions to gLdue to the\noxygen ˜γ2p-hole and the 3 d-electrons have comparable\nvalues.\nD. Exchange interaction\nThe Heisenberg exchange interaction of the [FeO 6]9−\nm-cluster in the CT state with the neighbouring n-cluster\nin the ground6A1gstate can be written in a simpliﬁed\nform as follows\nVex=−2/summationdisplay\nm>nJmn(Sm·Sn), (12)\nwhereJmnis the exchange integral, although in general\nit should be replaced by the orbital operator, e.g. for acertain6T1uterm for the m-cluster\nˆJmn=J0\nmn+/summationdisplay\ni=αβJαβ\nmn(/tildewiderˆLαˆLβ−2\n3δαβ).(13)\nIn general, the cluster spin momentum operators in ( 12)\nshould be replaced by the ﬁrst rank spin operators, which\ncan change the spin multiplicity. The Vexgives rise to the\norbital and spin splitting and mixing of the CTconﬁg-\nuration terms. The exchange parameters in Vexare\ndetermined not only by the ordinary cation-anion-cation\nsuperexchange Fe3+–O2−–Fe3+, but also by the\nconsiderably stronger direct cation-anion Fe3+–O2−\nexchange reaching the magnitude on the order ∼0.1eV.\nStrictly speaking, at variance with the antiferromagnetic\nexchange interaction between the ground states the ex-\nchange in the CT state can lead to both antiferro- and\nferromagnetic spin coupling. Interestingly, the matrix of\nthe orbital operator ˆJmnin (13) has a structure similar\ntoVLSCF (8) with the main orbitally isotropic γ=A1g\nterm included. In other words, nontrivial orbital part of\nˆVexcan be considered as a spin-dependent contribution\nto the low-symmetry crystal ﬁeld.\nIt should be noted that, in addition to the spin-\ndependent part, the exchange interaction also contains\na spin-independent contribution, which has a similar or-\nbital structure.\nE. Exchange-relativistic interactions\nCombined eﬀect of a conventional intra-center spin-\norbital coupling and orbitally nondiagonal exchange cou-\npling for an excited orbitally degenerated state of the Fe-\ncluster within the second-order perturbation theory can\ngive rise to a novel type of exchange-relativistic interac-\ntion, modiﬁed spin-orbital coupling ˆVex\nSO, which can be\nwritten as a sum of isotropic, anisotropic antisymmetric,\nand anisotropic symmetric intra-center and inter-center\nterms, respectively [ 25,26,31,54]\nˆVex\nSO=/summationdisplay\nm,nλ(0)\nmn(Lm·Sn)+/summationdisplay\nm,n(λλλmn·[Lm×Sn])+/summationdisplay\nm,n(Lm↔\nλλλmnSn). (14)\nIt is worth noting that λλλmnhas the symmetry of the\nDzyaloshinskii vector [ 55–58], while the last term has\nthe symmetry of the two-ion quasidipole spin anisotropy.\nGenerally speaking, all the three terms can be of a com-\nparable magnitude.\nThe contribution to the intra-center ( m=n) bilinear\ninteraction is determined by the spin-independent purelyorbital exchange, while the inter-center ( m∝negationslash=n) term, or\n\"spin-other-orbit\" coupling ˆVSoO, is determined by the\nspin-dependent exchange interaction. However, the spin-\ndependent exchange leads to the occurrence of additional\nnonlinear spin-quadratic terms, the contribution of which\ncan be taken into account by the formal replacement of\nthe linear spin operator Snin (14) for the nonlinear op-11\neratorSmn\nˆSq(mn) =ˆSq(n)+γ/bracketleftig\nˆV2/parenleftig\nS(m)/parenrightig\n×S1(n)/bracketrightig1\nq=ˆSq(n)+γ/summationdisplay\nq1,q2/bracketleftigg\n2 1 1\nq1q2q/bracketrightigg\nˆV2\nq1/parenleftig\nS(m)/parenrightig\nSq2(n), (15)\nwhere [:::] is the Clebsch-Gordan coeﬃcient[ 59],V2\nq(S)\nis the rank 2 spin irreducible tensor operator. In partic-ular,\nˆV2\n0(S) = 2/bracketleftbigg(2S−2)!\n(2S+ 3)!/bracketrightbigg1/2/parenleftig\n3ˆS2\nz−S(S+ 1)/parenrightig\n.(16)\nThe coeﬃcient γin (15) can be calculated for speciﬁc\nterms. The isotropic part of VSoOcan be presented, in\nthe general case, as follows\nViso\nSoO=/summationdisplay\nmnλ(mn)(L(m)·S(n))+/summationdisplay\nm/negationslash=nλ′(mn)/parenleftig\nL(m)·S(m)/parenrightig/parenleftig\nS(m)·S(n)/parenrightig\n. (17)\nSimilarly to the Dzyaloshinskii vector, to estimate the\nparameters of the spin-other-orbit coupling, we can use\nthe simple relation [ 60]\nλ(m)≈λ(mn)≈λ′J′\n∆ESΓ, (18)\nwhereλ′andJ′are the spin-orbital constant for the\nT1-,T2-states and the nondiagonal exchange parameter,\nrespectively, ∆ESΓis a certain excitation energy. Pa-\nrameters like λ(m), λ(mn)can be considerably larger\nthan typical values of the Dzyaloshinskii vector [ 56–58],\ndue both to smaller values of ∆ESΓand to the direct\n2p–3d-exchange which, as stated above, is stronger than\nthe 3d–2p–3d superexchange determining d(mn). Eﬀec-\ntive orbital magnetic ﬁelds acting on the T1andT2orbital\nstates, e.g., for Fe3+ions in ferrites due to Vex\nSOcan reach\nthe magnitude larger than 10 T ( λ′≥102cm−1, J′≥\n102cm−1,∆ESΓ∼104cm−1).\nThe approach presented here can be immediately ex-\ntended to tetrahedral clusters [FeO 4]5−.\nV. ANISOTROPIC POLARIZABILITY OF THE\nOCTAHEDRAL [FEO 6]9−-CLUSTER\nAlmost all ferrites are low anisotropic optical media in\na wide spectral range : ∆ǫ/ǫ0≤10−2,ǫ0and∆ǫ\nbeing respectively the isotropic and anisotropic parts of\nthe permittivity tensor ˆǫ. The latter can be written as\nthe sum of the symmetric and antisymmetric parts:\n∆ǫ= ∆ǫs\nij+ ∆ǫa\nij, (19)characterizing the linear birefringence/dichroism and th e\ncircular birefringence/dichroism, respectively. The lat ter\ncan be described by axial gyration vector g[61] which is\ndual to∆ǫa\nij:\ngi=1\n2eijk∆ǫa\njk, (20)\nwhereeijkis the Levi-Civita tensor.\nWithin a linear approximation the Fe-cluster contribu-\ntion to anisotropic permitivity tensor can be expressed in\nterms of the cluster anisotropic polarizability tensor ˆα:\nas follows\n∆ˆǫ= 4πNLˆα, (21)\nwhereNis the number of clusters per unit volume;\nL=n2\n0+2\n9is the Lorentz-Lorenz factor. Hence, for the\ngyration vector we have\ng= 4πNLα, (22)\nαbeing the \"microgyration vector\", related to the anti-\nsymmetric part of the cluster polarizability tensor by an\nexpression analogous to ( 20).\nA. Simple microscopic theory\nThe microscopic analysis of the optical anisotropy is\nusually being carried out on the basis of the Kramers-\nHeisenberg formula [ 62] for the electronic polarizability;\nin case of the microgyration vector it takes on following12\nform :\nα=1\n/planckover2pi1/summationdisplay\nijρi[dij×dji]·F1(ω, ωij).(23)\nFor the symmetric part of ˆα, the Kramers-Heisenberg\nformula reduces to\nαsym\nkl=1\n/planckover2pi1/summationdisplay\nijρi∝angbracketlefti|dk|j∝angbracketright∝angbracketleftj|dl|i∝angbracketright·F2(ω, ωij).(24)\nIn these formulae, dijis the matrix element of the\nelectric dipole moment d(dk,lbeing its Cartesian pro-\njections) between the initial state |i∝angbracketrightand the ﬁnal state\n|j∝angbracketrightfor the CT transition; ρiis the statistical weight\nof the|i∝angbracketrightstate.Fk(k= 1 2) is the Lorentz dispersion\nfactor\nFk(ω, ωij) =(ω+iΓij)[1−(−1)k] +ωij[1 + (−1)k]\n(ω+iΓij)2−ω2\nij.\n(25)\nHere,ωijdenotes the CT transition frequency, Γijis\nthe line width.\nInstead of the Cartesian tensor, one can introduce theirreducible polarizability tensor [ 26,31] :\nαk\nq=1\n/planckover2pi1/summationdisplay\nij/summationdisplay\nq1q2ρi/bracketleftigg\n1 1k\nq1q2q/bracketrightigg\n∝angbracketlefti|dq1|j∝angbracketright∝angbracketleftj|dq2|i∝angbracketright·Fk(ω, ωij),\n(26)\nwhere [:::] is the Clebsch-Gordan coeﬃcient[ 59],dqis\nthe irreducible tensor component of the dipole moment\nd(d±1=∓1√\n2(dx±idy), d0=dz).\nAn important advantage of the irreducible tensor form\nis the natural separation of isotropic and anisotropic\ncontributions: α0\n0describes the isotropic refrac-\ntion/absorption; α1\nqandα2\nqdescribe the circular and\nlinear birefringence/dichroism, respectively.\nFor octahedral [FeO 6]9−(tetrahedral [FeO 4]5−) clus-\nters with an orbitally nondegenerate ground state6A1g\nin ferrites, the contribution of the CT transitions6A1g→\n6T1u(6A1g→6T2) to the anisotropic polarizability will\nbe associated only with certain \"perturbations\" in ex-\ncited6T1u- (6T2-) CT states.\nIn the linear approximation, we single out two main\ncontributions αk\nq(split)andαk\nq(mix), associated with\nthe orbital splitting of excited6T-states and mix-\ning/interaction of diﬀerent6T-states, respectively, un-\nder the action of various perturbations, VLSCF,VZ,VSO,\nVex\nSO[26,31].\nαk\nq(split) =1\n/planckover2pi12/summationdisplay\ni=6A1g/summationdisplay\nj=6T1u/summationdisplay\nµµ′/summationdisplay\nq1q2ρi/bracketleftigg\n1 1k\nq1q2q/bracketrightigg\n×∝angbracketlefti|dq1|jµ∝angbracketright∝angbracketleftjµ|ˆV|jµ′∝angbracketright∝angbracketleftjµ′|dq2|i∝angbracketright·∂Fk(ω, ω0\nij)\n∂ω(0)\nij(27)\nαk\nq(mix) =1\n/planckover2pi1/summationdisplay\ni=6A1g/summationdisplay\n(j,j′=6T1u\nEj>Ej′)/summationdisplay\nq1q2ρi/bracketleftigg\n1 1k\nq1q2q/bracketrightigg\n×∝angbracketlefti|dq1|j∝angbracketright·∝angbracketleftj|V|j′∝angbracketright\nEj−Ej′·∝angbracketleftj′|dq2|i∝angbracketright·Fk(ω, ωij) (28)\nA simple illustration of the nature of circular and linear\nbirefringence due to a splitting mechanism is presented\nin Figure 3.\nNote that in ferrites with an orbitally nondegenerate\nground6A1gstate of Fe-clusters, both linear and cir-\ncular birefringence will be associated with orbital split-\nting/mixing in excited states. Obviously, the Fe-cluster\ncontribution to the linear birefringeance/dichroism will\nbe related with low-symmetry crystal ﬁeld VLSCF in ex-\ncited6T1ustates, while the contribution to circular bire-\nfringence/dichroism will be determined by the orbital\nZeeman interaction or complex spin-orbital interaction\nsuch asVSOandVex\nSO. Large exchange spin ﬁelds up to\n103T and large spin Zeeman splittings do not make a\ndirect contribution to circular magnetooptics in ferrites .\nDue to a competition of the splitting and mixing mech-\nanisms the spectral dependence of the polarizability can-\nnot be considered to be a sum of separate individual6A1g→6TCT transitions.\nB. Symmetry considerations\nAccounting for local point symmetry, crystal and mag-\nnetic symmetry in many cases provides important qual-\nitative and even quantitative information about various\nanisotropic eﬀects, in particular, the role of certain mi-\ncroscopic mechanisms.\n1. Linear birefringeance in orthoferrites\nSimple symmetry considerations within the framework\nof the so-called \"deformation\" model made it possible to\nexplain the dependence of linear birefringence on the type\nof R-ion in orthoferrites RFeO 3[63].13\n6T1u\n6A1gM□=□+1\nM□= 1 /c45M□=□06T1u\n6A1g|y/c62|z/c62\n|x/c62\nH=0 H||z V =0rhomb V /c1850rhomba b\nFIG. 3. An illustration of the nature of circular and linear\nbirefringence due to a splitting mechanism: ( a) schematic for\nthe dipole allowed CT transitions6A1g→6T1ufor the light\nwith right and left circular polarization under external ma g-\nnetic ﬁeld and orbital Zeeman splitting; ( b) schematic for the\nCT transitions6A1g→6T1ufor the light with a linear polar-\nization in a low-symmetry (rhombic) crystal ﬁeld and Stark\nsplitting for excited6T1ustate. Note that we are dealing with\nﬁnite current (a) and currentless (b) states, respectively .\nThe real FeO 6cluster in orthoferrites can be repre-\nsented as a homogeneously deformed ideal octahedron.\nTo ﬁnd the degree of distortion, we introduce a symmet-\nric strain tensor εijaccording to the standard rules. In\nthe local system of cubic axes of the octahedron\nεij=1\n4l26/summationdisplay\nn=1(Ri(n)uj(n)+Rj(n)ui(n)), (29)\nwhereR(n)is the radius-vector of the Fe-O nbond,u(n)\nis the O n-ligand displacement vector, or\nˆε=\n1−l1\nl1\n2(π\n2−θ12)1\n2(π\n2−θ13)\n1\n2(π\n2−θ21) 1−l2\nl1\n2(π\n2−θ23)\n1\n2(π\n2−θ31)1\n2(π\n2−θ32) 1−l3\nl\n,(30)\nwherelis the Fe-O separation in an ideal octahedron, li\nare the Fe-O iinteratomic distances1\n3(l1+l2+l3) =l,\nandθijare the bond angles O i-Fe-Ojin a real complex.\nLocalx,y,z axes in octahedron are deﬁned as follows:\nthez-axis is directed along the Fe-O I, thex-axis is along\nFe-OIIwith the shortest Fe-O bond length. In general,\nthe deformations of octahedra in orthoferrites are small\nand do not exceed 0.02.\nDiagonal components of the traceless strain tensor ( 30)\n(tensile/compressive deformations) can be termed as E-\ntype deformations since εzzand1√\n3(εxx−εyy)trans-\nform according to the irreducible representation (irrep)\nEof the cubic group O h, while oﬀ-diagonal components(shear deformations) can be termed as T2-type deforma-\ntions since εyz,εxz, andεxytransform according to the\nirrepT2of the cubic group O h.\nIn the linear approximation, the symmetric anisotropic\npolarizability of the octahedron FeO 6can be related to\nits deformation by the following relation\nαij=/braceleftigg\npEεij, i=j\npT2εij, i∝negationslash=j,(31)\nwhereεijis the FeO 6-octahedron deformation tensor\n(Trˆε=0);pE,T2are the photoelastic constants, re-\nlating the polarizability to E,T2-deformations, respec-\ntively. The relation ( 31) is valid in the local coordinate\nsystem of the FeO 6-octahedron. In the abc-axes system,\nit can be rewritten as\nαij=pEεE\nij+pT2εT2\nij, (32)\nwhereεE\nijandεT2\nijare the components of the tensor of\ntheE- andT2-deformations of the octahedron in the\nabc- system, respectively.\nProceeding to the permittivity tensor ˆǫand summing\nover all Fe-ions sites, we arrive at nonzero diagonal com-\nponents of ˆǫ:\nǫii=PEεE\nii+PT2εT2\nii, (33)\nwherePE,T2= 4πN/parenleftig\nn2\n0+2\n3/parenrightig2\npE,T2;Nis the number\nof Fe3+ions per 1 cm3. Components of ˆεE,ˆεT2ten-\nsors serve as the structure factors and may be calculated\ntaking into account the known components of the tensor\nof FeO 6octahedron local deformations and the Eulerian\nangles relating the local axes to the abcones.\nThus, we have a two-parameter formula ( 33) for the\nbirefringence of orthoferrites as a function of rhombic\ndistortions of their crystal structure. The photoelastic\nconstantsPE, PT2can be found from the comparison of\nexperimental data [ 64,65] with the theoretical structure\ndependence of the ab-plane birefringence :\n∆nab=na−nb=1\n2n0/bracketleftbig\nPE(εE\nxx−εE\nyy) +PT2(εT2xx−εT2yy)/bracketrightbig\n(34)\ntreated as a dependence on the type of the orthoferrite.\nThe Figure 4shows both experimental and calculated\n∆nabgivenPE= 6.2n0, PT2= 4.0n0(values ob-\ntained from the least-squares ﬁtting). A very nice agree-\nment of the two-parameter formula ( 34) with experiment\ntestiﬁes to the validity of the deformation model of the\nbirefringence.\nUsing the found parameter PE,T2values, we are able\nto describe all the peculiarities of the orthoferrite bire-\nfringence. In particular, Figure 4shows the theoreti-\ncal predictions for the orientation angles ±θof opti-\ncal axes, measured from the c-axis for the ac- andbc-\nplanes and from the a-axis for the ab-plane, together with\nscarce experimental data on Eu, Tb, Dy, Y, Yb orthofer-\nrites [65,66]. Quite good agreement with the available14\n+\n+\n++\nRFeO3LaPrNdSmEuGdTb Dy YHoErTmYbLu1030507090\n+\n+- ab-plane\n- ac-plane\n- bc-plane/c113/c111\n2\n/c68n/c18010\nab5\n4\n3\n2\n1\n0\n/c451\n/c452\n/c453\n/c454\n/c455EuLaPr\nNdSm Gd\nTbDy\nYHoErTm Lu\nYb\nRFeO3\nFIG. 4. Left panel: Linear birefringeance ∆nabfor orthoferrites\nRFeO3inab-plane, solid circles are predictions of the deformation\nmodel, hollow circles are experimental data ( λ= 0.633µm) [64].\nRight panel: The orientation angles ( ±θ) of optical axes in respec-\ntive planes of orthoferrites predicted by the deformation m odel.\nThe solid black circles are scarce experimental data for bc-plane\n(λ= 0.68µm) [65,66].\nexperimental data is another conﬁrmation of the validity\nof the deformation model of birefringence of orthoferrites .\nIn general, for all its simplicity, the deformation model\nreﬂects quite correctly the main peculiarities of the nat-\nural birefringence of orthoferrites.\n2. Circular birefringeance/dichroism in ferrites\nThe gyration vector and the magnetic moment (or the\nferromagnetic vector m) have the same transformation\nproperties.\nForferrimagnetic iron garnets\ng=ˆAama+ˆAdmd+ˆCH, (35)\nwheremaandmdare magnetic moments, or ferromag-\nnetic vectors, of octahedral and tetrahedral sublattices,\nrespectively.\nIn weak ferromagnets like RFeO 3and in a number of\nother magnetic compounds with non-equivalent magnetic\nsublattices, certain components of the ferromagnetic vec-\ntormand the antiferromagnetic vector lin a two-\nsublattice model transform identically, what enables one\nto write gin the linear approximation through m,l,\nand the external magnetic ﬁeld Has\ng=ˆAm+ˆBl+ˆCH,(m2+l2= 1) (36)\n(the ferromagnetic (FM), antiferromagnetic (AFM),\nand ﬁeld contributions, respectively)\nThe form of each of ˆA,ˆB,ˆCtensors is determined\nby the crystal symmetry. For example, in orthorhombic\nweak ferromagnetic orthoferrites RFeO 3\nˆA=\naxx0 0\n0ayy0\n0 0azz\n,ˆB=ˆBs+ˆBa=\n0 0bxz\n0 0 0\nbzx0 0\n,\naxx∝negationslash=ayy∝negationslash=azz, bzx∝negationslash=bxz.In rhombohedral weak ferromagnets (α-\nFe2O3, FeBO 3, FeF3,etc.)\nˆA=\na⊥0 0\n0a⊥0\n0 0a/bardbl\n,ˆB=ˆBa=\n0bxy0\nbyx0 0\n0 0 0\n,\ni.e.,byx=−bxy, and the ˆBtensor, in contrast with\northoferrites, is antisymmetric. The symmetry properties\nof theˆAandˆCtensors are identic.\nThe special role of the antiferromagnetic contribution\nto the gyration vector for weak ferromagnets is due to\nthe fact that for them, as a rule, m≪l, for example,\nm/l≈0.01in YFeO 3andm/l≈0.001inα-Fe2O3, re-\nspectively [ 57,58,67]. However, the components of the\ngyration vector ginα-Fe2O3and YFeO 3are compa-\nrable in magnitude with those for the yttrium iron gar-\nnet, Y 3Fe5O12[3,68] although the magnetization of the\nlatter is approximately by two orders larger than in the\nhematite and by one order larger than in orthoferrites. It\nseems impossible to explain this phenomenon other than\nin terms of the AFM contribution. Hence, it appears\nthat there must be microscopic mechanisms causing the\nantisymmetric relations of the gyration vector to spins :\ng=/summationdisplay\nmn[B(mn)× ∝angbracketleftS(n)∝angbracketright], (37)\nwhere the vector B(mn)is determined by the antisym-\nmetric part of ˆB.\nVI. CHARGE TRANSFER TRANSITIONS AND\nMAGNETO-OPTICAL EFFECTS (MOE) IN\nFERRITES\nA. Working microscopic models for circular MOE\nThe main contribution to the microgyration vector for\n[FeO6]9−and [FeO 4]5−clusters and the circular MOE for\nferrites is determined by the splitting and mixing mecha-\nnisms [ 26]. To the ﬁrst order of the perturbation theory,\nonly the interactions VSO, VZ, Vex\nSOplay part, as these\nare odd in the orbital moment and enable the orbital\nsplitting and mixing of excited CT states of the6T1u\ntype. Note that the spin part of VZjust as the isotropic\nHeisenberg spin exchange of the [FeO 6]9−cluster with\nits magnetic surroundings, characterized by the spin ex-\nchange ﬁeld Hex, do not contribute in the linear ap-\nproximation to the circular MOE. VSOand the orbital\npart ofVZyield the FM and ﬁeld contributions to the\ngyration vector; their combined action for the \"octahe-\ndral\" CT transitions due to the splitting of the excited\n6T1ustates is given by\ngsplit\na= 2/summationdisplay\nj=6T1uπe2LN\n/planckover2pi1meω0j/parenleftig\nλj∝angbracketleftS∝angbracketright+µBgj\nLH/parenrightig\nfj∂F1(ω, ω0j)\n∂ω0j.\n(38)15\nwhere∝angbracketleftS∝angbracketrightis the thermodynamic spin average, fjis the\noscillator strength for6A1g−6T1uCT transition, λjand\ngj\nLare eﬀective spin-orbital constant and orbital g-factor\nfor a certain6T1uterm (see tables 1 and 2 in Ref. [ 26]).The contribution of the mixing mechanism, that is of\nthe interaction of diﬀerent6T1uCT terms of the oc-\ntahedral [FeO 6]9−(6T2CT terms of the tetrahedral\n[FeO4]5−) cluster can be written as follows [ 26])\ngmix\na=/summationdisplay\n(j,k=6T1u\nE0j>E0k)4πe2LN\nme/parenleftig\nλjk∝angbracketleftS∝angbracketright+µBgjk\nLH/parenrightig/parenleftbiggfjfk\nω0jω0k/parenrightbigg1/2\n∝angbracketleft6A1g∝bardbld∝bardblj∝angbracketright∝angbracketleft6A1g∝bardbld∝bardblk∝angbracketrightF1(ω, ω0j)−F1(ω, ω0k)\nE0j−E0k,(39)\nwhere∝angbracketleft6A1g∝bardbld∝bardblj∝angbracketrightis the dipole moment submatrix ele-\nment. The parameters of the type of eﬀective orbital\ng-factorsgjk\nLand spin-orbit coupling constants λjk\ngjk\nL=∝angbracketleftκj6T1u∝bardbl/summationtext\nnln∝bardblκk6T1u∝angbracketright\n∝angbracketleft1∝bardblˆl∝bardbl1∝angbracketright;gL≡gjj\nL≡gj\nL;(40)\nλjk=∝angbracketleftκj6T1u∝bardblˆQ11∝bardblκk6T1u∝angbracketright\n∝angbracketleft1∝bardblˆl∝bardbl1∝angbracketright∝angbracketleft5\n2∝bardblˆs∝bardbl5\n2∝angbracketright;λ≡λjj≡λj,(41)\nare determined by the submatrix elements of the sum/summationtext\nnlnof one-particle orbital moment operators acting\non all atomic orbitals in the molecular orbitals, and by\nthe submatrix element of the double irreducible spin-\norbit tensor operator ˆQ11[69]. Numerical values of gjk\nL\nandλjkfor theCTstates of the [FeO 6]9−and [FeO 4]5−\nclusters are given in Tables 1 and 2 [ 26]. In ( 40), (41),\nboth the splitting (j=k)and mixing (j∝negationslash=k)are\ntaken into account. κjis the set of intermediate quan-\ntum numbers, necessary for distinguishing diﬀerent6T1u\nterms.fjis the oscillator strength of the6A1g→κj6T1u\nCT transition, E0jis its energy.\nThus,VSOandVZto the 1st order of the perturba-\ntion theory, give rise to isotropic ˆA,ˆCtensors ( 36). The\nfrequency dependences of the real and imaginary parts of\nthe splitting contribution to gfor aCTtransition have\nrespectively the \"dissipative\" and \"dispersive\" form.\nThe splitting contribution of the exchange-relativistic\ninteraction Vex\nSO(14) for isolated6T1uterm to the gyra-\ntion vector can be represented as follows [ 25,26,31,54]:\ng=2πLe2fAT\nm/planckover2pi1ω0/parenleftigg\n↔\nλλλ∝angbracketleftˆS∝angbracketright+/summationdisplay\nn↔\nλλλn∝angbracketleftˆSn∝angbracketright/parenrightigg\n∂F(ω,ω0)\n∂ω0,\n(42)\nwhere ﬁrst and second terms in brackets correspond\nto intra-center and inter-center, or spin-other-orbit\nexchange-relativistic contributions, respectively,↔\nλλλand\n↔\nλλλnare the eﬀective tensors of the respective interactions.\nIn other words, these terms correspond to contributions\nwithm=nandm∝negationslash=ninVex\nSO(14). The summa-\ntion overnin (42) extends to the nearest neighbors of\nthe considered center, fATis the oscillator strength of\nthe6A1g−6T1utransition. In general, in accordancewith ( 14) the tensors↔\nλλλand↔\nλλλnof the intra- and inter-\ncenter exchange-relativistic contributions in ( 42) con-\ntain isotropic, antisymmetric, and symmetric anisotropic\ncomponents.\nIn addition to the \"gyroelectric\" contribution to the\ngyration vector that we have considered, we should note\nthe existence of a small \"gyromagnetic\" contribution re-\nlated with the magnetic susceptibility, which determines\nthe frequency-independent contribution to the Faraday\nrotation [ 70]\n∆ΘF=2πn0\ncγm, (43)\nwhereγis gyromagnetic ratio, mis magnetic moment.\nIt is interesting that yttrium iron garnet in the wave-\nlength range λ>5µmis a gyromagnetic medium, since\nthe gyromagnetic contribution to the Faraday rotation is\npredominant ( ΘF≈60 deg/cm at T=300 K), although in\nthe wavelength range λ <4µmit can be considered as\nan ordinary gyroelectric medium due to a sharp increase\nin the gyroelectric contribution in ΘF[70].\nB. Fe3+diluted nonmagnetic compounds\nThe most suitable objects for the application and jus-\ntiﬁcation of the cluster theory for ferrites are the Fe3+\ndiluted nonmagnetic compounds such as YAlO 3and\nCa3Ga2Ge3O12with the crystal structure close to or-\nthoferrite YFeO 3and iron garnet Ca 3Fe2Ge3O12, respec-\ntively. In such dilute systems, band models are inapplica-\nble for describing Fe 3d states, so that the cluster model\nhas virtually no competitors in describing the optical and\nmagneto-optical response of dilute systems in the O 2p-\nFe 3d charge transfer range, especially since it becomes\npossible to restrict ourselves to taking into account only\nintra-center p-d transfer.\nThe Faraday eﬀect was measured in single-crystalline\nsamples of diluted garnet Ca 3Ga2−xFexGe3O12\n(x= 0.15)[ 25], where the Fe3+ions occupy only\nthe octahedral positions, and the [FeO 6]9−octahedrons\nare assumed to be essentially noninteracting. Making\nuse of the splitting ( 38) and mixing ( 39) contributions\nto the gyration vector with the data for eﬀective orbital\ng-factors and spin-orbital parameters from Table 1 in16\nh/c119/c44eVImgz\n-0.040.02\n0.00\n-0.02\n3 4 5RegzYIG\nFIG. 5. Spectral dependence of the real and imaginary parts of\nthez-component of the gyration vector in YIG: experimental data\nare shown by dotted curves, model ﬁtting is shown by solid cur ves.\nRef. [26] and assuming that energies of all \"octahedral\"\nCT transitions in this garnet are blue-shifted by 1.4 eV in\ncomparison with corresponding energies in \"orthoferrite\"\ncomplexes (see Table I), the authors calculated both the\nferromagnetic and ﬁeld contributions to the Faraday\nrotation\nΘF=ω\n2n0cg=AFm+CFH, (44)\nover the entire CT band. As a result, good agreement\nwas obtained with the experimental values of the ferro-\nmagnetic and ﬁeld contributions to ΘF, measured in the\nspectral range 1.4-3.1 eV (see Figure 2 in Ref. [ 25]).\nUnfortunately, there are few examples in the literature\nof a systematic study of the concentration dependence of\noptical and magneto-optical eﬀects in diluted systems.\nC. The yttrium iron garnet\nThe absence of the magneto-optically active rare-earth\nsublattice in yttrium iron garnet Y 3Fe5O12permits the\nevaluation of the \"undistorted\" iron sublattices contri-\nbution. In addition, experimental studies of YIG\nmagneto-optics are abundant [ 3,50,51,71]. The au-\nthors [ 26] have undertaken a theoretical model computa-\ntion of the FM and ﬁeld contributions ( 36) to the gy-\nration vector of the YIG, taking into account the CT\ntransitions both in octahedral [FeO 6]9−and tetrahedral\n[FeO4]5−clusters.\nFigure 5shows the results of the theoretical simula-\ntion of the spectral dependence of the real and imaginary\nparts of the gyration vector z-component, Regzand\nImgz, in YIG (solid lines), with dipole allowed and a\nnumber of dipole forbidden CT transitions (marked by\nlong and short line segments at the bottom of the Fig-\nure5, taken into account. The parameters of the main\nCT transitions used in the model simulation are pre-\nsented in Table I. Besides a satisfactory agreement with\nthe experimental data in a wide spectral range, 2.5 –5.5 eV, the computed Regzvalue on the long wave-\nlength tail of the CTtransitions band (λ=0.63µm)\nyields the Faraday rotation in YIG ΘF= 860deg/cm ,\npractically coinciding with the experimental value 830\ndeg/cm [72,73]. The computed values of the partial\nFaraday rotation contributions due to octahedral CT\ntransitions (6500 deg/cm ) and tetrahedral ones (- 5640\ndeg/cm ) satisfactorily agree with the experimental val-\nues 8670 and −7840deg/cm , respectively [ 72,73]. As\nexpected for a longitudinal ferrimagnet, we see the eﬀect\nof signiﬁcant mutual compensation for the contributions\nof the octa- and tetra-sublattices.\nBoth in the octahedral CT transitions contribution to\nRe gz, and in that of the tetrahedral transitions, the\nmain role belongs to the mixing mechanism, in agreement\nwith the predominance of paramagnetic-shaped lines in\nmagneto-optical spectra of YIG noted in Ref. [ 74].\nThe authors [ 26] have also computed the ﬁeld contribu-\ntion (36) to the YIG gyration vector g, with theoretical\nvalues of the orbital Landé factors gjk\nL(see Tables 1 and 2\nin Ref. [ 26]), taking into account the main electric-dipole-\nallowed CT transitions, only. Rough as it is, the approx-\nimation of allowed CT transitions gives nevertheless the\nΘF/Hvalues of −10◦·cm−1·T−1(λ= 0.7µm)and\n−2.4◦·cm−1·T−1(λ= 1.1µm)– near to corre-\nsponding experimental data ( −12.4◦·cm−1·T−1[75]\nand−2.5◦·cm−1·T−1[76], respectively). The lack\nof experimental data precluded a comparison at shorter\nwavelengths.\nThe electronic structure, magnetic, optical and\nmagneto-optical properties of yttrium iron garnet were\ninvestigated recently [ 8] by using \"ﬁrst principles\"\nGGA+U calculations with Hubbard energy correction for\nthe treatment of the strong electron correlation. The au-\nthors boldly make a too strong statement that \"the cal-\nculated Kerr spectrum which included on-site Coulomb\ninteraction of Fe 3d electrons described well the experi-\nmental results\", which clearly does not follow from the\ndata presented in Figure 6 from their article, especially\nsince the calculated dielectric function shows a dramatic\ndiscrepancy with experiment.\nD. Bi-substituted iron garnets\nAlthough pure yttrium iron garnet has several advan-\ntages in terms of magneto-optical response, it has not\nbeen widely applied in integrated devices due to its lim-\nited Faraday rotation. However, decompensation of the\ncontributions of the octa- and tetra-sublattices, in par-\nticular, due to the replacement of R-ions in R 3Fe5O12\ngarnets by Bi3+or Pb3+ions, makes it possible to in-\ncrease the Faraday rotation of iron garnets by one or two\norders of magnitude in the visible and near-infrared re-\ngion (see, e.g., Ref. [ 46]).\nWittekoek et al. [ 46] proposed in a purely qualitative\nmanner that the origin of the large Faraday rotation\nin Bi,Pb-substituted iron garnets is the hybridization of17\nBi,Pb 6p orbitals, which possess anomalously large spin-\norbit coupling ( ζ6p≈2 eV), with the O 2p and Fe 3d or-\nbitals. Later this idea was supported and developed\nwithin cluster molecular orbital theory [ 30,32,77]. The\nenhancement of spin-orbit coupling in Fe 3d orbitals was\nassumed to be much smaller than that in O 2p orbitals,\nbecause Fe sites are located more distant than O sites\nfrom Bi sitess.\nTaking account of the overlap of 2p(O2−)and\n6p(Bi3+)electronic shells as well as the virtual tran-\nsition of the oxygen 2p- electron to the bismuth empty\n6p- shell, the wave function of the outer 2p- electrons of\nthe neighboring oxygen ion acquires thereby an admix-\nture of Bi 6p-states [ 30,32]:\nϕ2pm−→ψ2pm=ϕ2pm−/summationdisplay\nm′∝angbracketleft6pm′|2pm∝angbracketright∗ϕ6pm′,\n(45)\nwhereϕ2pmandϕ6pmare atomic wave functions.\nThe Bi 6p-O 2p hybridization results in the modiﬁca-\ntion of the spin-orbit interaction on the oxygen ion :\nVSO=VSO(2p) + ∆Viso\nSO(2p) + ∆Van\nSO(2p),(46)\nwhereVSO(2p) =ζ2p(l·s)is conventional spin-orbital\ninteraction with ζ2p≈0.02 eV,∆Viso\nSO(2p)and∆Van\nSO(2p)\nare eﬀective isotropic and anisotropic terms due to the\nBi 6p-O 2p hybridization:\n∆Viso\nSO(2p) = ∆ζ2p(l·s), (47)\nwhere eﬀective spin-orbital parameter is estimated in\nRef. [32] to be∆ζ2p≤0.1 eV per one Bi3+-ion, that is\nseveral times larger than conventional parameter ζ2p:\n∆Van\nSO(2p) =λijˆliˆsj, (48)\nwhere the eﬀective spin-orbit interaction tensor λijde-\npends on the geometry of the Bi-O bond [ 30,32]\nλij∝ζ6p/parenleftbigg\nRiRj−1\n3δij/parenrightbigg\n, (49)\nwhereRis a unit vector along the Bi-O bond direction.\nThus, the eﬀect of the bismuth ions on the circular\nMOE in iron garnets is essentially related to the oxygen\nO 2p-states in [FeO 6]9−and [FeO 4]5−clusters. The Bi3+\nions, leading to an increase in the eﬀective spin-orbital\ncoupling constant for oxygen ions, have a signiﬁcant eﬀect\non the circular magneto-optics of iron garnets, through a\nchange in the eﬀective spin-orbital coupling parameters\nλ=λ(3d) +λ(2p)\nfor the excited6T-states with the p-d charge transfer.\nThe simple theory we are considering allows us to make\na number of predictions. First, the eﬀect of the Bi 6p-\nO 2p hybridization may be particularly signiﬁcant for\nthe CT transitions, whose ﬁnal state spin-orbit coupling\nconstantλcontains the ligand contribution λ(2p)only, e.g., the transitions t2u−egandt1u(π)−egin the\n[FeO6]9−clusters (predicted energies 4.4 and 5.3 eV, re-\nspectively). Since ζ2p≪ζ3d≈0.1 eV the contribution\nof such transitions to the FM part of the gyration\nvector ( 36) in unsubstituted garnets is practically van-\nishing. The Bi substitution makes these transitions ob-\nservable. On the contrary, the CT transitions whose ﬁnal\nstate V SOconstantλincludes only the 3d-contribution,\ne.g., transition t1u(σ)−egin the [FeO 6]9−clusters (pre-\ndicted energy 6.4 eV) are not appreciably inﬂuenced by\nthe Bi3+-ions. Thus, the spectral dependence of the gy-\nration vector in YIG and Bi-substituted compounds can\ndiﬀer greatly. Second, the Bi 6p-O 2p hybridization in-\nduces the anisotropy of the ˆAtensor in the FM contri-\nbution to the gyration vector ( 36), which diﬀers for the\nocta- and tetra-positions of the Fe clusters. Third, in\nour model, bismuth ions do not directly aﬀect the value\nof the ﬁeld contribution ˆCH(36) to the gyration vector.\nAt variance with the cluster model, the \"ﬁrst-\nprinciples\" band calculations indicate a slightly diﬀer-\nent, albeit contradictory, picture of Bi 6p-O 2p-Fe 3d hy-\nbridization. Thus, analyzing the electronic structure\nof Bi3Fe5O12(BIG) calculated by the fully relativistic\nﬁrst-principles method based on the full-potential linear -\ncombination-of-atomic-orbitals (LCAO) approach within\nthe local-spin-density-approximation (LSDA), Oikawa et\nal. [6] found that the enhancement of the spin-orbit cou-\npling due to the hybridization of Bi 6p is considerably\nlarger in the Fe 3d conduction bands than in the O 2p\nand Fe 3d valence bands. The origin of this enhancement\nis that the Fe 3d conduction bands energetically overlap\nwith Bi 6p bands. Their results indicate the signiﬁcance\nof spin-orbit coupling in Fe 3d conduction bands in rela-\ntion to the large magneto-optical eﬀect observed in BIG.\nHowever, the results of recent GGA+U calculation by\nLi et al. [ 12] show that quite the contrary, Bi 6p orbitals\nin BIG hybridize signiﬁcantly with Fe 3d orbitals in the\nlower conduction bands, leading to large V SO-induced\nband splitting in the bands. Consequently, the transi-\ntions between the upper valence bands and lower con-\nduction bands are greatly enhanced when Y is replaced\nby Bi. Such contradictions turn out to be typical for\nvarious \"ab-initio\" DFT based calculations.\nE. Exchange-relativistic interaction and\nunconventional magnetooptics of weak\nferromagnetic orthoferrites\nInterestingly that circular magnetooptic eﬀects in weak\nferromagnets are anomalously large and are comparable\nwith the eﬀects in ferrite garnets despite two-three orders\nof magnitude smaller magnetization [ 3,65,68,78–81].\nIn 1989 the anomaly has been assigned to a novel type\nof magnetooptical mechanisms related with exchange-\nrelativistic interactions, in particular, with so-called spin-\nother-orbit coupling [ 54].\nWe have shown that an antisymmetric exchange-18\nrelativistic spin-other-orbit coupling gives rise to an un -\nconventional \"antiferromagnetic\" contribution to the cir -\ncular magnetooptics for weak ferromagnets which can\nsurpass conventional \"ferromagnetic\" term [ 25–29,31,54]\n(see, also Ref. [ 79]).\nThe gyration vector in weak ferromagnets is a sum\nof so-called ferromagnetic and antiferromagnetic terms\nwith identical transformation properties, see Exp. ( 36).\nIt should be noted that within the two-sublattice model\nfor orthoferrites we neglect weak antiferromagnetic A-\nand C-modes (see, e.g., Refs. [ 56–58,63]).\nFor the ﬁrst time the antiferromagnetic contribution\nto circular MOE was experimentally identiﬁed and eval-\nuated in orthoferrite YFeO 3[54]. An analysis of the ﬁeld\ndependence of the Faraday rotation ΘF(Hext)made it\npossible to determine all the contributions to the gyra-\ntion vector ( λ= 0.6328µm):\nAzzmz= (0.95±0.55)·10−3;Bzx|lx|= (3.15±0.55))·10−3;\nAxxmx= (0.2±0.7)·10−3;Bxz|lz|= (−2.1±1.0))·10−3;\nCzz≈Cxx= (−1.1±2.8)·10−6kOe−1, (50)\nwhere|lx| ≈ |lz| ≈1. Interestingly, rather large measure-\nment errors allow for certain to determine only the fact\nof a large if not a dominant antisymmetric antiferromag-\nnetic contribution related with antisymmetric spin-other -\norbit coupling. Strictly speaking, the mutual orientation\nof the ferro- ( m) and antiferromagnetic ( l) vectors de-\npends on the sign of the Dzyaloshinskii vector [ 56–58].\nInterestingly, a rather arbitrarily chosen relative orien ta-\ntion of these vectors in Ref. [ 54] with positive sign of mz\nandlxexactly matches the theoretical predictions about\nthe sign of the Dzyaloshinskii vector [ 56–58].\nExistence of spontaneous spin-reorientational phase\ntransitions Γ4(FzGx)→Γ2(FxGz)in several rare-earth\northoferrites does provide large opportunities to study\nanisotropy of circular magnetooptics [ 3,27–29,31,65,\n78]. Gan’shina et al.[28] measured the equatorial Kerr\neﬀect in EuFeO 3, TmFeO 3, and HoFeO 3and have found\nthe the gyration vector anisotropy in a wide spectral\nrange 1.5-4.5 eV. The magnetooptical spectra, both real\nand imaginary parts of the gyration vector, were nicely\nﬁtted within a microscopic model theory based on the\ndominating contribution of the O2 p–Fe3dcharge trans-\nfer transitions and spin-other-orbit coupling in [FeO 6]9−\noctahedra. An example of modeling the spectrum of the\nreal part of the gyration vector in orthoferrite EuFeO 3\nis shown in Figure 6. Let us again pay attention to the\ncomparable values of circular MOEs in orthoferrites and\nferrite garnets at more than an order of magnitude lower\nmagnetic moment in weak ferromagnets of the YFeO 3\ntype and longitudinal ferrimagnets of the YIG type. The\nauthors [ 28] have demonstrated a leading contribution\nof the antisymmetric spin-other-orbit coupling and es-\ntimated eﬀective orbital magnetic ﬁelds in excited6T1u\nstates of the [FeO 6]9−octahedra, HL∼100T. TheseRegz\n0.02\n-0.022 3 4 50.04\n0EuFeO3\nh/c119/c44eV\nFIG. 6. Spectral dependence of the real part of the z-component\nof the gyration vector ib EuFeO 3: experimental data are shown by\ndotted curve, model ﬁtting is shown by solid curve.\nanomalously large ﬁelds can be naturally explained to\nbe a result of strong exchange interactions of the charge\ntransfer6T1ustates with nearby octahedra that are de-\ntermined by a direct p-dexchange.\nWhereas the existence of the antiferromagnetic contri-\nbution to the gyration vector is typical of a large number\nof multisublattice magnetic materials, the antisymmetry\nof the tensor↔\nBis a speciﬁc feature of weak ferromagnets\nalone. In the case of rhombohedral weak ferromagnets\nsuch as FeBO 3, FeF3, orα-Fe2O3, the tensor↔\nB, govern-\ning the antiferromagnetic contribution to the Faraday ef-\nfect is entirely due to the antisymmetric contribution, in\nview of the requirements imposed by the crystal symme-\ntry. In crystals of this kind the appearance of the antifer-\nromagnetic contribution to the gyration vector is entirely\ndue to allowance for the antisymmetric spin-other-orbit\ncoupling.\nHowever, the data on the anisotropy of the Faraday\neﬀect in TmFeO 3[78] and the values of the Faraday ef-\nfect in SmFeO 3(m∝bardbla-axis) and a number of other\northoferrites with m∝bardblc-axis [ 65] bear evidence of the\nexistence of an appreciable symmetric AFM ˆBslcon-\ntribution to the gyration vector of orthoferrites. Indeed,\nthe Faraday eﬀect in the Γ4phase(m∝bardblc)and in the\nΓ2phase(m∝bardbla)is determined, respectively, by the z-\nandx- component of g:\ngz=Amz+Bzxlx;gx=Amx+Bxzlz(51)\n(under the justiﬁed assumption that ˆAbe isotropic).\nSincem⊥landmx≈mz=m, lettinglx= 1\nwith the view of the deﬁnitude, we obtain :\ngz=Am+Ba\nzx+Bs\nzx;gx=Am+Ba\nzx−Bs\nzx,\n(52)\nso that the experimentally found ratio [ 65,78]\nRegz/Regx≈2.5−3 (atλ≈1−2µm) indicates un-\nambiguously the existence of an appreciable symmetric\nAFM termBs\nzx:\nBs\nzx\nAm+Bazx∼0.5.19\n0.0 0.2 0.4 0.6 0.8 1.0 1.2<S >z <S >z2<S >z1.0\n0.5\n0.0/c1160.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\nFIG. 7. Temperature dependence of the normalized thermody-\nnamic quantities determining the temperature dependence o f the\ncircular MOE. The inset shows an example of ﬁtting the experi -\nmental data on the temperature dependences of the equatoria l Kerr\neﬀect in hematite α-Fe2O3(see Figure 6 in Ref. [ 68]) using the two-\nparameter formula ( 56), dotted curve is the /angbracketleftSz/angbracketrightdependence.\nF. The temperature dependence of the circular\nmagneto-optics of ferrites\nThe analysis of the temperature dependences of MOE\ncan yield an important information about the role of var-ious mechanisms of the circular MOE . Experimental\nstudies of the Faraday and Kerr eﬀects in weak ferro-\nmagnetsα-Fe2O3[79],FeBO 3[80,81],YFeO 3[68] have\nshown that their circular MOE and the magnetic mo-\nment, both total and that of each sublattice, have diﬀer-\nenttemperature dependences. In Refs. [ 68,79,81], an\nattempt was made to connect this phenomenon with the\nso-calledpairtransitions.\nHowever, we show here that all peculiarities of the tem-\nperature dependence of the Faraday and Kerr eﬀects for\nweak ferromagnets can be naturally and consistently ex-\nplained by taking into account the AFMˆBlcontribution\nto the gyration vector due to the exchange-relativistic in-\nteractions. Whereas the FMˆAmcontribution to g\n(36), theAFMˆBsymlcontribution due to LSCF\nin6T1uCT states ( 36), and the contributions due\nto intra-center Vex\nsohave the temperature dependence\ndetermined by the ordinary thermodynamic average of\nthe spin ∝angbracketleftS(m)∝angbracketright, theAFM contribution owing to the\n\"spin-other orbit\" interaction is related to the average\nvalue of a complicated spin operator ˜S(mn)(15). In the\nmolecular ﬁeld approximation the thermodynamic aver-\nage of the nonlinear operator Smnin (15) can be written\nas follows [ 82]\n∝angbracketleftˆSq(mn)∝angbracketright=∝angbracketleftˆSz(n)∝angbracketrightC1\nq(S(n)) +γ∝angbracketleftˆV2\n0(S(m))∝angbracketrightT∝angbracketleftˆSz(n)∝angbracketright/summationdisplay\nq1,q2/bracketleftigg\n2 1 1\nq1q2q/bracketrightigg\nC2\nq1(S(m))C1\nq2(S(n)), (53)\nwhereC2\nq1(S(m)),C1\nq2(S(n))are spherical tensorial har-\nmonics (Ck\nq=/radicalig\n4π\n2k+1Ykq) as the functions of classical\nspin direction;\n∝angbracketleftSz∝angbracketright=SBS(x),\nwhereBS(x)is the Brillouin function\nBS(x) =2S+1\n2Scoth2S+1\n2Sx−1\n2Scoth1\n2Sx;x=3S\nS+1σ\nτ\n(σ=Sz/Sandτbeing the reduced magnetic moment\nand temperature, respectively);\n∝angbracketleftˆV2\n0(S)∝angbracketrightT= 2/bracketleftbigg(2S−2)!\n(2S+ 3)!/bracketrightbigg1/2/parenleftig\n3∝angbracketleftˆS2\nz∝angbracketright −S(S+ 1)/parenrightig\n,(54)\nwhere\n/parenleftig\n3∝angbracketleftˆS2\nz∝angbracketright −S(S+ 1)/parenrightig\n=\n/parenleftig\n2S(S+ 1)−3Scothx\n2S·BS(x)/parenrightig\n, (55)Thus, the temperature dependence of the gyration vector\nin the molecular ﬁeld approximation is determined by the\nfollowing two-parameter formula :\ng(T) =a∝angbracketleftSz∝angbracketright+a′∝angbracketleftSz∝angbracketright∝angbracketleftS2\nz∝angbracketright ≈Am+A′m3, ,(56)\nwith the frequency dependent coeﬃcients a, b. Tem-\nperature dependence of the thermodynamic factors ∝angbracketleftSz∝angbracketright\nand∝angbracketleftSz∝angbracketright∝angbracketleftS2\nz∝angbracketrightare presented in Figure 7, where the inset\nshows an example of ﬁtting experimental data on the\ntemperature dependences of the equatorial Kerr eﬀect\nin hematite α-Fe2O3(see Figure 6 in Ref. [ 68]) using the\ntwo-parameter formula ( 56).\nIn other words, the MOE in weak ferromagnets will\nbe characterized by a clear nonlinear dependence on the\nmagnetic moment of sublattices, the presence of which\nis a direct indication of the contribution of exchange-\nrelativistic interactions of the spin-other-orbit type. A s\nexpected, the nonlinear contribution, both in magni-\ntude and in sign, will depend substantially on the fre-\nquency [ 68,79–81].\nIt is worth noting that Exp. ( 53) provides a dependence\nof the exchange-relativistic contribution to the gyration\nvector on the mutual orientation of neighboring spins.20\nG. The high-energy optics and magneto-optics of\nferrites\nThe availability of modern high-intensity synchrotron\nradiation has facilitated the reﬁnement of conventional\nspectroscopy. This is especially true in the ﬁeld of\nMOE , where the synchrotron radiation is a convenient\ntool of obtaining the spectra at high energies.\nKučeraetal.[84] have obtained the reﬂectivity spectra\nof a number of iron and non-iron garnets and yttrium or-\nthoferrite in the vacuum ultraviolet 5 to 30 eV range us-\ning synchrotron radiation as the light source. Contrary\nto the visible and near UV regions, all the spectra ob-\ntained are strikingly similar in this spectral range. Two\nbroad bands sited at about 10 and 17 eV have been found\nin both garnet and orthoferrite reﬂectivity and optic ab-\nsorption spectra. The 10 eV band was assigned to the\nCT transition from the oxygen 2p valence band to the\nyttrium 4d or 5s conduction states. The band centered\nnear 17 eV was attributed to the \"orbital-promotion\"\ninter-conﬁgurational Fe 3d →Fe 4p transition. Despite\nthe large peak values, the contribution of these transi-\ntions to the MOE of ferrites in the visible region, being\nstructureless, is signiﬁcantly inferior to the contributi on\nof O 2p-Fe 3d CT transitions.\nH. Rare-earth ions in ferrites\nThe simplest expression for the contribution of the\ndipole-allowed 4f–5d transition to the rare-earth ion po-\nlarizability tensor can be obtained by neglecting the split -\nting of the 4fn−15d- conﬁguration [ 85]\nαk\nq= (−1)1+k3√\n2k+11\n/planckover2pi1/braceleftigg\n3 3k\n1 1 2/bracerightigg\ne2r2\nfdFk(ω, ωfd)∝angbracketleftˆUk\nq(J)∝angbracketright\n(57)\nwhere {:::} is the 6j-symbol [ 59],rfd=∝angbracketleft4f|r|5d∝angbracketrightis the\nradial integral, ∝angbracketleftˆUk\nq(J)∝angbracketrightis the thermodynamical average\nof the irreducible tensor ˆUk\nq(J)with submatrix element\nU(k)\nSLJ;SL′J′[59].\nThe components of the tensor α1\nq, which determines\nthe contribution of the rare-earth ion to the circular\nmagneto-optics, can be written as follows\nα=−1\n7√\n2e2r2\nfdF1(ω,ωfd)2−gJ\ngJµBmR,(58)\nwheremRis the magnetic moment of the R-ion, gJis\nthe Lande-factor. The symmetric anisotropic part of the\npolarizability tensor determines the eﬀects of linear bire -\nfringence and dichroism. In Cartesian form, we get[ 85]\nαij=√\n3\n14e2r2\nfdF2(ω,ωfd)α∝angbracketleft3/tildewidestˆJiˆJj−J(J+1)∝angbracketright,(59)\nwhere/tildewidestJiJj=1\n2(ˆJiˆJj+ˆJjˆJi),αis the Stevens parame-\nter [86].A detailed analysis of the role of the eﬀects of a\nstrong crystal ﬁeld for the 5d electron was carried out\nin Refs. [ 85,87].\nVII. CONCLUSIONS\nThe paper presents the theory of the optical and\nmagneto-optical properties of strongly correlated iron ox -\nides, primarily ferrite garnets and orthoferrites, based o n\nthe cluster model with the leading contribution of the\ncharge transfer transitions.\nAt variance with the \"ﬁrst-principles\" DFT based band\nmodels the cluster model is physically clear, it allows one\nto describe both impurity and dilute and concentrated\nsystems, provides a self-consistent description of the op-\ntical, magnetic, and magneto-optical characteristics of F e\ncenters with a detailed account of local symmetry, low-\nsymmetry crystal ﬁeld eﬀects, spin-orbit and Zeeman in-\nteractions, and also relatively new exchange-relativisti c\ninteraction, which plays a fundamental role for the circu-\nlar magneto-optics of weak ferromagnets.\nThe cluster approach provides a regular procedure for\nclassifying and estimating the probability of allowed and\nforbidden electric-dipole CT transitions and their contri -\nbution to optical and magneto-optical anisotropy.\nThe cluster model makes it possible to describe all the\nspeciﬁc features of the inﬂuence of Bi ions on the circular\nmagneto-optics of ferrites by the Bi 6p-O 2p hybridization\nand partial Bi-O \"transfer\" of the large Bi 6p spin-orbit\ninteraction. The cluster model predicts the \"selective\"\nnature of the inﬂuence of Bi only on certain CT transi-\ntions, the appearance of an anisotropy of the ferromag-\nnetic contribution, and the absence of any inﬂuence on\nthe ﬁeld contribution to the gyration vector.\nThe contribution of the exchange-relativistic interac-\ntion for the excited6T1uterms in [FeO 6]9−clusters leads\nnot only to the appearance of an \"antiferromagnetic\"\ncontribution to the gyration vector of weak ferromag-\nnets such as orthoferrite RFeO 3and hematite α-Fe2O3\nbut also to the deviation of the temperature dependence\nof circular MOE from the simple proportionality to the\nmagnetization m. The appearance of a nonlinear m-\ndependence is an indication of the contribution of the\nunusual \"spin-other-orbit\" interaction in excited6T1u\nstates.\nUndoubtedly, the considered version of the cluster the-\nory requires more detailed development both in terms of\nimproving the used MO-LCAO scheme and in terms of\nthe possible application of the \"hybrid\" LDA+ MLFT\nscheme [ 24]. In any case, development the cluster model\nof magneto-optical eﬀects in ferrites needs data from sys-\ntematic experimental studies of the concentration, spec-\ntral, and temperature dependences of various optical and\nmagneto-optical eﬀects for Fe centers in oxides.21\nACKNOWLEDGMENTS\nThis study was supported by the Ministry of Science\nand Higher Education of the Russian Federation, project\nFEUZ-2023-0017REFERENCES\n[1] Moskvin, A. S., Cluster theory of charge-transfer exci-\ntations in strongly correlated oxides. Optics and Spec-\ntroscopy, 111, 403-410 (2011).\n[2] Moskvin, A. 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It is observed that the 1.15:2 \nprecursor ratio gives better phase control but poor coercivity. On the other hand, 1.05:2 \nprecursor ratio results in subs tantially better coercivity values (274kA/m; almost 300% the \nvalue reported for co -precipitated cobalt ferrite by Praveena et al.), but moderate BH product \nmaximum (2.25 kJ/m3; ~ comparable to many reports on cobalt ferrite nanoparticles so far). \nThe nanoparticles with best coercivity are annealed at 873K for varying durations (2, 4, 6, 12 \nhrs). It is observed that the coercivity drops drastically (almost by 80%) after annealin g for 2 \nhours. However thereafter coercivity and saturation magnetization improves marginally with \nincreasing duration of annealing. These studies, along with thermogravimetric analysis, and \ninfrared spectroscopic results indicate that a hydroxide nanophas e based flux pinning \nmechanism at the grain boundary plays an important role in explaining the observed magnetic \nproperty trends. It is believed that this result will be generically helpful in developing soft \nchemically derived ferrites with higher coerciv ity and moderate (BH) max. However to develop 2 \n plausible applications using the reported ferrites that use nanophase flux pinning, soft materials \nand device processing methods will need to be explored.  \nKeywords:  Cobalt ferrite nanoparticles, Co -precipitation , Soft chemistry, P inning \nnanohydroxide phases.  \n1. Introduction  \nCobalt ferrite (CoFe 2O4) is a ferrimagnetic material with tunable coercivity (1.67x103 -\n4.65x105 A/m) [1], moderate magnetization (around 80Am2/kg) and good thermal and \nchemical stability [2]. It  is being used in several technological applications such as computer \nmemory core elements and switches, magnetic recording media [3], core of coils in microwave \nfrequency devices, contrast agents for magnetic resonance imaging (MRI) [4], probes for \nmagnet ic force microscopy, hyperthermia based tumour treatment [5], drug delivery [6], and \nmagnetically recoverable catalyst [7 -9]. Crystallographically, the ferrite spinel structure \n(AB 2O4) is a closed -packed oxygen lattice, in which tetrahedral (A sites) and o ctahedral (B \nsites) interstices are occupied by the cations. In the inverse spinel structure, all the Co2+ ions \noccupy the octahedral sites of the lattice. Half of the Fe3+ ions occupy the octahedral sites and \nthe rest of the Fe3+ ions are present in tetrahedral sites [10].  \nLarge -scale industrial applications of nanoscale ferrites have motivated the development of \nwidely used chemical methods, including solution combustion [11], co -precipitation [12], sol -\ngel [13] and precipitation  [2] for the synthesis of stoichiometric and chemically pure spinel \nferrite nanoparticles. Magnetization can be varied by introducing off -stoichiometry (like \noxygen vacancies and excess metal ions) in the lattice structure. Controlling the stoichiometry \nin turn can be achieved by varying the synthesis conditions [12]. Co -precipitation is a simple \nand green synthesis method for producing nanoparticles. The advantage of this method is that \nit offers better size control; the polydispersity is largely controlle d through the handle on the 3 \n relative rates of nucleation and growth during the synthesis process [14]. Size control is \nachieved by the addition of a precipitating agent (eg. NaOH) to the precursor solution; vigorous \nstirring is usually employed in this pro cess [2].  \nIt may be noted that the coercivity of a magnetic material depends on the particle dimensions \n(in addition to composition). Hence particle size control and appropriate grain boundary \nprocessing is critical for obtaining optimum coercivity. In thi s work, we explore (i) precursor \nratio, and (ii) annealing temperature and time in obtaining optimal coercivity. Results indicate \nthat nanophases undetectable via x -ray diffraction seem to be responsible for the observed \ncoercivity in the as prepared prist ine samples (with Co:Fe=1.05:2). Any heat treatment affects \nthis coercivity, indicating the interfacial origins of pinning and hence coercivity enhancement \nin this sample.  \n2. Experimental section  \nTo study the effect of off -stoichiometry on the magnetic proper ties, solutions with 0.21, 0.22, \n0.23, 0.24 M cobalt nitrate (50ml) and 0.4M (200ml) ferric nitrate are prepared separately using \nde-ionised water. The suitably chosen cobalt nitrate solution is mixed with the iron containing \none (50ml:50ml). A particular pH range (approx. 11.7) is maintained by the addition of 2M \nsodium hydroxide. Higher pH is chosen to ensure fast growth kinetics, which in turn results in \nlarger crystallite sizes [12]. This precipitate is collected, washed several times to remove \nsoluble impurities and pelletized. Products made using different precursor concentration are \nlabelled accordingly (ref: Table 1).  \nSample with best coercivity value (274.7KA/m) is annealed at 873K for varied durations (2, 4, \n6, and 12 hours). The sintering tempera ture is chosen based on the observations made by \nPraveena et.al. [12], in order to obtain cobalt ferrite nanoparticles with moderate Fano factor \n(σ2/d, where σ -variance and d= mean of Average crystallite size). Fano factor is a statistical 4 \n measure of size dispersion in nanoparticles. As the calcination temperature increases, the \ncrystallite size increases while the Fano factor decreases [12]. Therefore to obtain ferrite with \ngood magnetization values as well as relative monodispersity, a moderate annealing \ntemperature (873K) is chosen.  \nTo characterize pristine and sintered samples, PANalyticalX’pert powder X -ray diffractometer \n(XRD) is used with Cu K α (1.54A) radiation. The morphology and composition are examined \nusing Energy Dispersive X -ray Spectroscopy (E DS) and Scanning Electron Microscope \n(Inspect F SEM). Room temperature measurements are made using Vibrating Sample \nMagnetometer (Lakeshore VSM 7410). Weight loss measurements are done using \nThermogravimetric analyser (SDT Q600 V20.9 Build 20). FTIR analys is is done using Perkin \nElmer Spectrum1 FT -IR instrument. Reitveld analysis is carried out using MAUD (Materials \nAnalysis Using Diffraction) software. .  \nTable 1:  Sample label s used in this work  \nPrecursor ratio  \n(Fe : Co)  Sample name  \n(Product)  \n2 : 1.05  C1.05 \n2 : 1.1  C1.1 \n2 : 1.15  C1.15 \n2 : 1.2  C1.2 \n \n \n \n \n 5 \n 3. Results and discussion  \nThe X -ray diffractograms (ref: Fig. 1) of the samples Co 1.05, Co 1.1, Co 1.15 and Co 1.2 (ref: Table \nI), show the peaks for CoFe 2O4 (reference code: 98 -077-9266; ICSD code: 98553). The \ncharacteristic peak of ferrites (311) is seen in every sample processed [15]. This indicates the \nformation of cobalt ferrite nanoparticles and apparent absence of any secondary phases \n(discernible from XRD). The crystallite size (approximately 24 nm) of as prepared pristine \nsamples is calculated using Scherrer formula at [311] peak. Rietveld refinement based analysis \nalso suggests the absence of secondary phases as the refined calculated parameters as cl osely \nmatching with ASTM data. For example, the refined calculated lattice parameter for co -\nprecipitated C 1.05 is 0.84nm, which is almost equal to reported ASTM data (0.84nm) [16].  \n \nFig.1:  XRD pattern of the samples Co 1.05, Co 1.1, Co 1.15, and C o1.2 confirming the formation of \ncobalt ferrite nanoparticles  (reference code: 98 -077-9266; ICSD code: 98553).  [311] is \nobserved throughout indicating cobalt ferrite phase in all samples . \nTable 2 indicates that C 1.05 shows coercivity values as high as 274.7 kA/m (ref: Fig 2) ; this is \nalmost thrice the value reported by Praveena et.al. (87.9 kA/m) and is reproducibly obtained. \nHigh coercivity is observed in cobalt ferrite nanoparticle prepared by water -in-oil emulsions \n6 \n when precursor ratio is changed [17]. This high coercivity value is likely related to an unusual \natom ratio in the sample (1:3.5 = Co:Fe; obtained from EDX/SEM). In fact this suggests the \nformation of a small volume fraction nanophase, which is likely segregating at the grain \nboundaries. Furthermor e it may be expected that this large coercivity can be a result of flux \npinning because of the presence of these pinning secondary nanophase (which is undetectable \nby SEM and XRD). This is a reasonable expectation considering prior reports [18].  \n \nFig.2:  M-H hysteresis curves of CoFe 2O4 with different precursor ratio (C 1.05, C1.1, C1.15, and \nC1.2). C 1.05 shows the best H c (274.7 kA/m) and M r (14.3A.m2/kg) values. The coercivity of this \nsample is almost thrice the value previously reported by Praveena et.al.  The (BH) max is almost \n12MGOe suggesting the hard magnet properties of the cobalt ferrite nanoparticles (C 1.05). \nTable 2: Composition of precursor, H c (coercive field), M s (saturation magnetization), M r \n(remnant magnetization), and Fe/Co (ratio of cations  in the products) of co -precipitated \nsamples. A substantially high H c is observed in C 1.05, indicating the fundamental difference in \nthe sample compared to the rest.  \n7 \n Composition  Hc (kA/m)  Ms (A.m2/kg) Mr (A.m2/kg) Fe/Co (determined via \nenergy dispersive X-rays)  \nC1.05 274.75  23.8 14.3 3.5 \nC1.1 12.3 17.7 2.1 2.8 \nC1.15 53.1 41.1 12.5 2 \nC1.2 15.8 8.6 1.3 1.85 \n \nIn order to confirm this “pinning by nanophase” hypothesis, annealing for several durations \nhas been carried out. The values of coercivity  dropped drastically with annealing  (ref: Fig.3)  \nas shown in Table 3. This drop can be due to the loss of pinning sites (plausibly due to \nhydroxide phases, considering the starting precursors). Interestingly this property is not seen \nin other co-precipitated samples (C 1.1, C 1.15, and C 1.2), indicating relevance of starting \nprecursors (and hence the role of nucleation and kinetics of co -precipitation on obtaining high \nHc). C 1.1, C 1.15, and C 1.2 are primarily cobalt ferrite ultrafine nanoparticle s (average particle \nsize: 17nm, 20nm, and 17nm respectively; ref: Fig. 4) which are superparamagnetic in nature \n(ref: Fig. 2). Hence the overall coercivity values of these samples are not high. Size distribution \nanalysis of samples show log -normal distribu tion based on histogram obtained from SEM \nimages. This is consistent with what Praveena et al. observed [12]. The average particle size of \nthe samples C 1.1, C 1.15, and C 1.2 (around 18± 2nm) is close, but different with C 1.05 (around \n25nm) suggesting the di fference in growth kinetics. Also particles in samples (C 1.1, C1.15, and \nC1.2) seem to be clustered or segregated, very likely because of small particle sizes in these \nsamples [19].  \n 8 \n  \nFig.3:  M vs H of C 1.05 sintered for 2 , 4, 6, and  12 hrs at 873K. Coercivity values maintain an \nincreasing trend with annealing duration (indicating improved magnetocrystalline anisotropy, \nwith grain coarsening , as expected) . Pinning nanophases likely reduce in volume fraction or \ndisappear during annealing, resulting in initial coercivity loss, post annealing in C 1.05. \nTable 3: Change in H c (coercive field), M s (saturation magnetization), and M r (remnant \nmagnetization) with annealing durations. ( Sample  studied : C1.05) \nSintering Duration (Hrs)  Hc (kA/m)  Ms (A.m2/kg) Mr (A.m2/kg) \n0 (pristine sample)  274.7  23.8 14.3 \n2 60.6 21.3 7.6 \n4 62.3 21.4 7.5 \n6 61.8 28.6 10.8 \n12 86.36  45.6 16.9 \n \n9 \n  \nFig. 4:  SEM images of (a) C1.05 sample. Average particle size is around 25nm which suggests \nthat the sample is still in single domain stage. Hence upon annealing the coercivity increases \nwith annealing duration as grain size increases. ( b, c, d)  C1.1,C1.15, andC 1.2 respectively. \nAverage  particle size s of these samples are approximately 17 , 20, and 17nm respectively \n(measured using ImageJ).  \nTo test this hypothesis, FTIR analysis is performed (ref: Table. 4). The spectra indicate the \npresence of hydroxide phases in the C 1.05 (ref: Fig. 5). This is not surprising because hydroxide \nphases have been reported in other co -precipitated ferrites as well. These hydroxide phases are \nremnants of unreacted phases, which are observed and often times reported in co -precipitation \nof cobalt  ferrite [20]. The peak observed around 1600 cm-1 is likely to be due to stretching of \nOH-1 or molecular H 2O, similar to the modes observed in Zn -Ni ferrite nanoparticles \n10 \n synthesized by co -precipitation [21]. Presence of residual nitrates (i.e. NO 3-1moieti es) is \nconﬁrmed by the 1400cm-1 IR peak. The FTIR spectra obtained is quite broad for all the \nsamples, as is commonly observed in inverse spinel ferrites [12]. This broadening increases \nwith reduction of particle size, as the cation disorder is known in in crease in nanoferrites [22]. \nThis reduction in particle size [23] is responsible for increment in inverse nature of spinel \nferrites (such as increment in coercivity with particle size) [24]. The main broad peaks seen \naround 600 cm-1 and 3400 cm-1 correspon ds to stretching vibrations associated with metal -\noxygen bonds, O -H, and N -H respectively. The peaks at about 2925, 1050 cm-1 are due to C –\nH stretching and O –H bending vibrations, respectively. These indicate the presence of residual \nmolecules adsorbed to the particle surface [25]. From Fig. 5, it can be realized that the \ndisappearance of hydroxides due to annealing is likely responsible for the loss of coercivity. \nThis explanation is substantiated by the broad peaks at 1600 cm-1, 3400 cm-1 and absence of \nintensity at 1050 cm-1 in annealed sample. Hence it can be concluded that the presence of \nnanohydroxide phases in the particles, most likely at the interfaces, are responsible for the huge \ncoercivity via pinning mechanism.  \n \n11 \n Fig. 5: FTIR spectrum of cobalt  ferrite nanoparticles sintered for 2, 4, 6 , and 12  hrs durations \nat 873K. The peaks at 3400 cm-1 and 1600 cm-1 indicates the presence of hydroxides. The \nintensity of these peaks reduces with annealing duration explaining the loss in high coercivity \nwith a nnealing.  \nTable 4: List of peak IR positions in C 1.05  \nPeak position (cm-1) Assignments  \n600 metal -oxygen bond vibration  \n1050  O-H bending  \n1400  NO 3-1 \n1600  OH-1 or molecular H 2O \n2922  C-H stretching  \n3395  O-H, N -H stretching  \n \nIt may be noted that from the FTIR alone we may deduce that there are hydroxide \ndecomposition steps occurring in these samples during annealing. Literature suggests that there \nwould be at least two decomposition steps in the as prepared material. The first decomposition \nstep (label led: step I) would be for cobalt hydroxide (Co(OH) 2) and cobalt oxyhydroxide \n(CoO(OH)); these occur  at around 130o and 252oC respectively [26] forming cobalt oxide \n(Co 3O4)  and H 2O [27]. The second decomposition step (labelled: step II), i.e. decomposition \nof cobalt oxide (Co 3O4) formed after first decomposition step occurs at 790o and 800oC \nrespectively [28] forming CoO and oxygen. To examine these, thermogravimetric analysis \n(TGA) is performed. From the TGA analysis of C 1.05 (ref: Fig. 6), it  is evident that there is \nsubstantial mass loss seen over a broad range (100 -200oC); indicating step -I [29]. Breadth of \nthe TGA peak is reasonable considering compositional heterogeneities often reported in nano -\nferrites [30] (and their grain boundaries) [ 31]. TGA shows substantial weight loss around 12 \n 650oC, readily attributable to the decomposition of cobalt oxides. The small weight loss \nobserved is due to the small volume fraction of the oxide phase at the grain boundary. Other \nweight losses indicated by c hanges in slope (ref: region -i and region -ii in Fig. 5), observed \naround 300 and 400oC, are likely due to decomposition of Fe -rich nitrates [29] and hydroxides \n[32] respectively. Since the annealing temperature is around 600oC, it can be safely assumed \nthat the hydroxide nanophases are absent in the annealed samples. The increasing trend in \ncoercivity of annealed samples is due to increment in particle size up to a particular value (54 \n± 2nm) seen in inverse spinel ferrites as mentioned by Praveena et.al as  well. As the particle \nsize observed in the sample prepared is around 25nm (ref: Fig. 5), the coercivity increases due \nto increased role of magnetocrystalline anisotropy [33].  \n \nFig.6:  TGA analysis of C 1.05 sample indicating the weight loss step involving decomposition \nof cobalt hydroxides and oxides. Readily observed Steps I and II indicate  weight loss due to \ndecomposition of (I) cobalt hydroxides and H 2O, and (II) cobalt oxide decomposition. \n13 \n Discernible changes in slope at regions – i and ii are attributed to decomposition of ferric nitrate  \nand hydroxides.  \nThe relevance of the results presented here may be noted in the context of the current quest for \nreplacements for champion rare earth permanent magnets (REPMs) for novel magnetic storage \ndevices is an important and emerging areas. Hence for novel magnetic nanomaterials, there is \nvalue in looking at the (BH) max parameter as well. For competing with REPMs, the (BH) max \nvalue must be as high as possible [34]. Here interestingly C1.05 shows high coercivity along \nwith moderate magnetic saturation. However, only moderate (BH) max [35] value (almost \n0.25MGOe or 2.25kJ/m3) is observed for this sample ( highest (BH) max observed in cobalt \nferrite nanoparticles is 2.1MGOe [36 -37]). This moderate (BH) max is because of the moderate \nmagnetic saturation values. This result is in concurrence with previously reported high \ncoercivity cobalt ferrite nanoparticles with (BH) max values ranging from 0.2 -1.1MGOe [38 -39]. \nThus we obtain cobalt ferrite nanoparticles with  both high H c as well as moderate (BH) max \nvalues. (BH) max  calculations are done in SI unit system using the formula B =  μ0 (H + M) \n[40]. Hence we suggest that C 1.05 can potentially be adapted and deployed for application in \nhigh magnetic storage applicat ions. This result is interesting because of the green chemical \nnature of the co -precipitation. Hence it is believed that this result will be generically helpful in \ndeveloping soft chemically/ chimie douce  (i.e low temperature chemistry) derived ferrites wit h \nhigher coercivity and desirable (BH) max. However since the coercivity drops significantly with \nannealing from 2 hours at 873K, to develop plausible applications using the reported ferrites \nthat use nanophase flux pinning, soft materials and device proces sing methods will need to be \nexplored.  \n \n \n 14 \n 4. Conclusion  \nHigh coercivity  (~274 kA/m, almost 300% the previously reported value for co -precipitated \ncobalt ferrite nanoparticles) materials are produced via choice of a suitable precursor ratio (Co \nrich reaction mix; Co: Fe = 1.05:2). The product obtained is a Co rich, and apparen tly single \nphase. High H c (274kA/m) in the sample is evidently composition dependent, and concomitant \nwith the presence of interfacial nanohydroxide phases, suggesting an interfacial pinning \nmechanism. This is corroborated by annealing experiments where co ercivity dropped \ndrastically (almost by 80%), while saturation magnetization increased consistently. The \n(BH) max product is moderate indicating its potential for improvement through further work and \nuse in magnetic memory applications. Annealing past the i nitial coercivity drop however \nmarginally improves both (i) coercivity and (ii) saturation magnetization indicating grain \ngrowth and associated magnetocrystalline anisotropy.  \nWe believe that the result presented here will be generically helpful in develop ing soft \nchemically/ chimie douce  (i.e low temperature chemistry) derived ferrites with higher \ncoercivity and sufficient (BH) max. However since the coercivity drops significantly with \nannealing, to develop plausible applications using the reported ferrites that use nanophase flux \npinning, soft materials and device processing methods are perhaps necessary.   \nAcnowledgements  \nWe thank the Department of Metallurgical and Materials Engineering, Indian Institute of \nTechnology Madras. 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"    },    {        "title": "1503.08976v1.Ultrafast_near_infrared_photoinduced_absorption_in_a_multiferroic_single_crystal_of_bismuth_ferrite.pdf",        "content": "arXiv:1503.08976v1  [cond-mat.str-el]  31 Mar 2015Ultrafast near infrared photoinduced absorption in a multi ferroic single crystal of\nbismuth ferrite\nEiichi Matsubara1,2, Takeshi Mochizuki2, Masaya Nagai2, Toshimitsu Ito3, and Masaaki Ashida2\n1Department of Physics, Osaka Dental University, 8-1,\nKuzuha-Hanazono, Hirakata, Osaka, 573-1121,Japan,\n2Graduate School of Engineering Science, Osaka University,\n1-3, Machikaneyama, Toyonaka, Osaka, 560-8531, Japan, and\n3Electronics and Photonics Research Institute, National In stitute of Advanced Industrial Science and Technology (AIS T),\n1-1-1, Higashi, Tsukuba, Ibaraki, 305-8562, Japan\nWe studied the ultrafast third-order optical nonlinearity in a single crystal of multiferroic bismuth\nferrite (BiFeO 3) in the near-infrared range of 0.5–1.0 eV, where the materia l is fundamentally trans-\nparent, at room temperature. With pump pulses at 1.55 eV, whi ch is oﬀ-resonant to the strong\ninter-band charge transfer (CT) transition, we observed in stantaneous transient absorption with\npencil-like temporal proﬁle originating from the two-phot on CT transition from the oxygen 2 pto\nthe iron 3 plevels. In contrast, under pumping with 3.10-eV photons, th e pencil-like absorption\nchange was not observed but decay proﬁles showed longer time constants. Although the two-photon\nabsorption coeﬃcient is estimated to be 1.5 cm/GW, which is t en (hundred) times smaller than that\nof two(one)-dimensional cuprates, it is larger than those o f common semiconductors such as ZnSe\nand GaAs at the optical communication wavelength.\nPACS numbers: 42.70.Nq, 75.50.Ee, 78.20.Ci, 79.20.Ws, 78. 47.jb\nI. INTRODUCTION\nA lot of eﬀorts have been devoted for seeking materials\nfor all optical switching, where large nonlinearity, quick\nresponse time, and operability at room temperature are\ndemanded. Transition metal oxides of Mott Hubbard in-\nsulators are known to show ultrafast photoinduced phe-\nnomena reﬂecting broadband optical absorption. Espe-\ncially in an one dimensional cuprate of Sr 2CuO3, gigan-\ntic photoinduced absorption with an ultrafast decay con-\nstant of 1 ps occurs owing to the two-photon absorption\nprocess [1–3]. Since the wavelengthrange of probe pulses\nfor the process includes the optical communication wave-\nlength, such kind of materials are hopeful candidates for\nultrafast all optical switches. So far, one and two dimen-\nsional cuprates have been well studied, however, there\nare not so many reports on other transition metal oxides\nespecially three dimensional ones.\nBismuth ferrite (BiFeO 3) is one of the multiferroics\nwhich possesses both ferroelectricity (FE) and antiferro-\nmagnetism ( TC∼1100 K, TN∼640 K), this material\nis studied from viewpoints of both fundamental physics\nand applications [4, 5]. Optical pump-probe measure-\nments under excitation with above bandgap (2.6–2.8 eV)\nphotons have been performed in ﬁlms and single crys-\ntals of BiFeO 3, where the authors discussed various pho-\ntoinduced phenomena such as coherent oscillations due\nto magnon and phonon [6–8], and spectral modulations\nin connection with the transport property [9, 10]. As\na nonlinear medium, BiFeO 3has been studied in terms\nof the second-order nonlinearity in studies such as tera-\nhertz emission through optical rectiﬁcation [11] and ul-\ntrafast modulation of spontaneous polarization [12], and\nsecond harmonic generation spectroscopy [13, 14]. How-\never, there has been almost no study on the third-ordernonlinearityespeciallyaroundtheopticalcommunication\nwavelength in the near infrared region.\nBiFeO 3is also one of the Mott insulators with ultra-\nbroadband optical response continuously ranging from\nthe far-infrared (terahertz) to the ultraviolet regions, re-\nﬂecting strong interactions not only between electrons\nbut also between electrons and lattice, spin, orbital de-\ngrees of freedoms, so that the material makes us expect\nstrong and ultrafast nonlinear optical responses in the\nnear-infrared below-gap region as ever reported in low-\ndimensional cuprates [1, 2]. Furthermore, BiFeO 3has\nthe aspect of a ligand system, in which d-dtransitions\nplay important role in the optical responses in the visible\nand the near infrared regions, which makes the physics\nof the material more interesting.\nAs for the sample preparation, the fabrication of a sin-\ngle crystal of BiFeO 3has long been diﬃcult. Recently,\nsingle crystals grown by the ﬂux method have been uti-\nlized in many studies [10, 11, 15–21], however, the ob-\ntained size of the crystals has been limited. In 2011, Ito\net al.realized the fabrication of large single crystals of\nBiFeO 3by using a modiﬁed ﬂoating zone method with\nlaserdiodes for heating [22]. Besides the largesize (4 mm\nin diameter), the leakage current under high bias voltage\nis extremely low. Hence, we think this crystal will enable\nustoextractonlytheintrinsicopticalpropertyofthema-\nterial by making full use of the interaction length. Thus,\nin the present study, we optically pump the large single\ncrystal of BiFeO 3with femtosecond pulses and probe the\ntransient absorption change mainly in the near infrared\nregion, to explore unknown ultrafast nonlinear response\nof BiFeO 3.2\nII. EXPERIMENTAL\nA large single crystal of BiFeO 3with a diameter of 4\nmm was grown with the ﬂoating zone method utilizing\nlaser diodes as heat sources [22]. By controlling the oxy-\ngen pressure during growth, oxygen deﬁciency is reduced\na lot so that the leakage current ﬂowing in the crystal\nunder high bias-voltage is extremely low. Also the crys-\ntal does not show any magnetic hysteresis in the M-H\ncurve which is usually observed due to the weak parasitic\nferromagnetismin in crystals grown by the ﬂux method.\nThe unit cell of BiFeO 3has the rhombohedral ( R3c)\nstructure, which is essentially a cubic elongated along\nthe [111]direction. The ferroelectricpolarizationis along\nthis axis. BiFeO 3is a G-type antiferromagnet in which\nall the nearest neighbor spin couplings, including both\ninter- and intra-plane ones, are antiferromagnetic. Due\ntothemagneticorder,BiFeO 3formsthespiralspinstruc-\nture with a period of 62 nm along the three equiva-\nlent wavevectors of [10-1], [01-1], and [1-10] directions.\nBiFeO 3is also a charge transfer (CT) type insulator with\na bandgap energy of 2.6 −2.8 eV, which corresponds to\nthe lowest inter-band transition from the oxygen 2 pto\nthe iron 3 dstate. This broad CT band can be further\ndecomposed into several peak structures centered at 2.5,\n2.9, 3.2, 4.0, 4.5, and 6.1 eV [21]. This CT band with\nstrongoscillatorstrengthsbringsaboutstrongabsorption\nwith coeﬃcients of the order of 105cm−1[19]. BiFeO 3\nalso shows relatively weak ( ∼100 cm−1)d-dabsorptions\nwith broad peaks centered at 1.4 and 1.9 eV, which are\nalso split into two by taking the spin degrees of freedom\ninto account [14].\nPump-probe measurements were carried out with a ti-\ntanium sapphire regenerative ampliﬁcation system (800\nnm, 35 fs, 1 kHz) as a light source. A (100)-oriented\nspecimen of BiFeO 3with a thickness of 170 µm was\npumped with the fundamental (800 nm, 1.55 eV) or the\nsecond harmonic pulses (400 nm, 3.10 eV), and the tran-\nsient transmission change was probed with signal or idler\npulses from an optical parametric ampliﬁer. The dura-\ntion of the pump pulse was measeued to be 60 fs, and\nthe cross correlation width of the pump and probe pulses\nwas 100–110 fs, both at the sample. The penetration\ndepths given by the inverse of the absorption coeﬃcients\n[19] are 33 µm at 1.55 eV, and 0.49 µm at 3.10 eV, re-\nspectively. The near-infrared transmission was measured\nwith a combination of a monochromator and a cooled\nHgCdTe detector. The 1 /e2beam sizes of the pump\nand probe pulses were 1.35 mm and 130 µm in diam-\neter, respectively. First we pumped the specimen with\n3.1-eV pulses and measured the transient optical con-\nductivity with air-plasma based terahertz time domain\nspectroscopy. As a result, we observed no Drude-like re-\nsponse but observedan oscillation of transmission resem-\nbling a monocyclic terahertz pulse proﬁle (not shown).\nThis result is consistent with the high resistivity con-\nﬁrmed with the dc measurements [22] and the oscillation\nproﬁle seems to be caused by the generation of terahertzpulses due to the modulation of polarization by ultravi-\nolet pulses [12]. Since the electrons transport property\nis determined by oxygen deﬁciencies, the absence of the\ntransient Drude responseprovesthe suppression of them.\nSuch high quality of the crystal will enable us to extract\nintrinsic property without artifacts.\nIII. RESULTS\nFigure1showsthe timeevolutionsofthephotoinduced\nabsorption change at each probe photon energy (0.5–1.0\neV). The pump photon energy is ﬁxed at 1.55 eV. At\nall probe photon energies, we can see steep increase of\nabsorption and its ultrafast recovery. The amount of ab-\nsorption change increases with the probe photon energy.\nThe inset shows the expanded proﬁles around the time\norigin. Here the temporal proﬁles, especially for probe-\nphoton energies greater than 0.8 eV, characteristically\nhave pencil-like structures with a duration of 100–150 fs.\nThis is close to the cross correlation width of the pump\nand probe pulses (100–110 fs). The decay proﬁles were\nwell ﬁtted by the expression assuming two exponential\nfunctions and a constant as,\n∆αL=A0+A1exp(−t/τ1)+A2exp(−t/τ2).(1)\nThe constant component ( A0) certainly has a decay pe-\nriod beyond 1 ns, however, it did not exceed 1 ms, which\nis the separation period of the laser source. The values\nofτ2andτ1are ﬁtted to be 50–60 fs and 1–3 ps, re-\nspectively. In the inset of Fig. 2 (a), we present how the\ntransient absorption proﬁles probed at 0.9 eV are decom-\nposed into three components. Importantly, the dotted\ncurve showing the cross-correlation proﬁles of the pump\nand the probe pulses coincides with the dominant part of\nthe transient absorption change. They appear to be es-\nsentially independent of the probe photon energy (wave-\nlength) within the accuracyofthe measurements. In Fig.\n3 (a) we present the dependence of the magnitudes of\nthe three components on the pump photon density. All\nthe amplitudes of the three components increased with\nthe probe-photon energy. More closely, A1andA2were\nproportional to the excitation density, while A0showed\nthe quadratic dependence. We also examined the polar-\nization dependence and found the temporal proﬁle did\nnot depend on the polarization of the pump and probe\npulses, although the magnitude became smallerwhen the\npolarizations of two pulses were crossed.\nTo elucidate the physical mechanism, we changed the\nexcitation photon energy to 3.10 eV and measured the\ntransient absorption signal at the probe photon energy\nof 0.9 eV. The proﬁle is shown in Fig. 2 (b). The\ndecay curve was also decomposed into three parts with\ntimeconstantsof140–170fs, 1.9–2.6ps, andnanoseconds\n(which can be regarded as constant). Hereafter, we de-\nnotetheﬁttingresultsas A0(fun) orA0(sh)todistinguish\nthose pumped by the fundamental (1.55 eV) and second\nharmonic (3.10 eV) pulses. The inset shows the proﬁles3\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00/s68/s97L \n86420 -2\nTime [ps]Probe at\n 0.5 eV\n 0.6 eV\n 0.7 eV\n 0.8 eV\n 0.9 eV\n 1.0 eV\n 0.25\n0.20\n0.15\n0.10\n0.05\n0.00 /s68/s97L \n-1.0-0.50.00.51.0\nTime [ps]\nFIG. 1: Probe photon energy dependence of the transient\nabsorption proﬁles ∆ αLin a single crystal of BiFeO 3. The\npump photon energy is ﬁxed at 1.55 eV, while the probe pho-\nton energy is varied from 0.5 to 1.0 eV. The excitation densit y\nis 40 GW/cm2.\n0.15\n0.10\n0.05\n0.00 /s68/s97L\n43210-1Time [ps]A0(fun)\nA1(fun)\nA2(fun)0.15\n0.10\n0.05\n0.00/s68/s97L\n2.0 1.5 1.0 0.5 0.0 -0.5 -1.0\nTime [ps]0.10\n0.08\n0.06\n0.04\n0.02\n0.000.15\n0.10\n0.05\n0.00 /s68/s97L\n43210-1Time [ps]A0(sh)\nA1(sh)\nA2(sh)(a)\n(b)Pump@1.55 eV\nPump@3.10 eV\nFIG.2: Pumpphoton-energydependenceofthephotoinduced\nabsorption ∆ αLat the probe-photon energy of 0.9 eV. (a)\nTemporal proﬁle pumped at 1.55 eV, and (b) that pumped at\n3.10 eV. Insets show temporal proﬁles of each ( A0,A1,A2)\ncomponent. The excitation density is 56 GW/cm2. Dotted\ncurves show the cross-correlation proﬁles of the pump and th e\nprobe pulses at each pump photon energy. Here, the proﬁle\nat the pump photon energy of 1.55 eV was directly measured,\nwhile that at 3.10 eV was estimated as the possible maximum\none taking into account the group delay dispersion in a BBO\ncrystal and optical ﬁters employed for the measurement.of each decomposed decay component. Apparently, the\ndecay time of the fastest component for pumping with\n3.10-eV pulses ( τ2(sh)) is longer than that with 1.55-eV\npulses (τ2(fun)), where the magnitude of A2(sh) is rel-\natively smaller than A2(fun). The proﬁle under pump-\ning with 3.10-eV pulses is sharper at the top and does\nnot have the pencil-like structure, which appeared under\npumping with 1.55-eV photons, although the duration\nof 3.10-eV pulses is expected to be longer than that of\nthe 1.55-eV ones due to the material dispersion in the\nnonlinear crystal. We also examined the dependence on\nthe excitation density as presented with closed markers\nin Fig. 3 (b). A2(sh) was quadratically proportional to\nthe excitation density up to 60 GW/cm2, approximately,\nwhileA0(sh) and A1(sh) showed linear dependence and\nthe slopes become gentle around 40 and 55 GW/cm2, re-\nspectively. From these dependences and the absence of\nthe pencil-like structure, the origin of photoinduced ab-\nsorption under pumping with the second harmonic (3.10\neV) pulses is diﬀerent from that with fundamental (1.55\neV) pulses.\n8060402000.12\n0.10\n0.08\n0.06\n0.04\n0.02\n0.00\n/s68/s97L (A2(fun, sh))/s32   A0(sh) \n A1(sh) \n A2(sh) 25x10-3\n20\n15\n10\n5\n0/s68/s97L (A0,1(fun, sh))\n806040200/s32  A0(fun) \nA1(fun) \nA2(fun) (a) (b)\nExcitation density [GW/cm2]\nFIG. 3: Excitation density dependence of the magnitude of\neach decay component under pumping with (a) fundamental\n(1.55 eV)pulses , and (b) second harmonic pulses (3.10 eV) at\nthe probe photon-energy of 0.9 eV. For clarity, vertical err or\nbars are not indicated, however, the values of error are smal l\nso as not to eﬀect the dependence.\nIV. DISCUSSIONS\nNow let us elucidate the microscopic origin of the\npresent photoinduced phenomena. First, we focus on the\npencil-like top structure in the transient absorption pro-\nﬁleand the originofthe fastest decaycomponent τ1(fun).\nOgasawara et al.observed a similar structure in the\nphotoinducedtransmissionchangeproﬁlesofaquasi-one-\ndimensional cuprate of Sr 2CuO3, which they attributed\nto the two-photon excitation process from the oxygen 2 p\nto the even-parity band of Cu which lies at a higher fre-\nquency than the one-photon CT band [1–3]. We plot the\namplitude of the fastest decay component as a function\nof the probe photon energy as shown in Fig. 4. By over-\nlaying the absorption spectrum in such a manner that\nthe horizontal axis (top) is shifted by the pump photon\nenergy of 1.55 eV towards higher energy, the magnitude4\n0.30\n0.25\n0.20\n0.15\n0.10\n0.05\n0.00/s68/s97 L\n1.2 1.0 0.8 0.6 0.4\nProbe photon energy [eV]1.0\n0.8\n0.6\n0.4\n0.2\n0.02.8 2.6 2.4 2.2 2.0Photon Energy [eV]\n/s97 [105\n cm-1]/s32A0(fun)\nA1(fun)  x3\nA2(fun)  x3\n Absorption\nFIG. 4: Relationship between the magnitude of decay com-\nponents and the linear absorption spectrum of BiFeO 3.A0,\nA1, andA2are represented as closed markers as a function\nof probe photon energy (left and bottom axis). Solid curve\nshows the absorption spectrum (right and top axis) adopted\nfrom the data shown in Ref. [21].\nof the fastest component A2(fun) signiﬁcantly rises up at\n0.7 eV, where the linear absorption with the correspond-\ning photon energy (2.2–2.3 eV) also starts to increase.\nFrom this and the pencil-like structure, we conclude that\nthe two-photonabsorptionprocessfromthe O 2 pand the\nFe 3dlevelgives the fastest decay component asschemat-\nically shown in Fig. 5 (a). Note the unit cell of BiFeO 3\nlacks inversion symmetry at temperatures below 1100 K,\nso that the two-photon transition to the one-photon al-\nlowed level is allowed.\nLetus furtherexaminetheprobephotonenergy(wave-\nlength) dependence. While A0(fun) signiﬁcantly in-\ncreases with the probe photon energy, the increase of\nA1(fun) is minor. According to the result of the pump-\nprobe measurements in the ultraviolet range, the to-\ntal lifetime of the CT excited states is in the order of\nnanoseconds [10]. In the present case, one pump photon\ncannot create the CT excited state, however, the two-\nphoton process can realize it. Hence, we consider the\nA0(fun) component comes from the lifetime of the two-\nphoton excited CT states. We can attribute the gradual\nincrease of A0(fun) with the probe photon-energy to the\ndependence of the oscillator strength of the CT transi-\ntion on the photon energy. The pump photon energy\nof 1.55 eV is also close to the broadband d-dtransition\nfrom6A1gto4T1g, so that the A1(fun) component with\nthe decay constant of 1–3 ps probably corresponds to the\nlifetime of the d-dexcited states, and the transition to\nthe higher d-dlevels or upper Hubbard band induces the\ntransient absorption as schematically shown in Fig. 5\n(b). The relatively small amplitude A1(fun) is due to the\nsmaller oscillator strength compared with that of the CT\ntransition by nearly one hundred times. As for the chan-\nnel of the decay, we can think of some process involving\nthe spin system, because the d-dtransitions are knownto be accompanied by magnon excitation in the form of\nsidebands [14].\nNext let us discuss the photoinduced phenomena un-\nder pumping with 3.10-eV photons. The time evolution\nbasically resembles the ones reported by Sheu et al.[10],\nwho pumped a single crystal of BiFeO 3with 3.1-eV pho-\ntons and measured the transient reﬂection change also\nat 3.1 eV. From the quadratic dependence of A2(sh)\non the excitation density, some cooperative interaction\nis certainly involved in the decay process. Similarly to\nthe case in the previous report, the relaxation from the\nCT excited states to the bottom of the band through\nelectron-electron or electron-phonon scatterings is highly\npossible as the origin. As already written, τ2(sh) slightly\ndecreaseswiththeexcitationdensity, whichseemstosup-\nport this speculation. Furthermore, while A0(fun) shows\nthe quadratic dependence up to the excitation density of\n56 GW/cm2,A0(sh) shows the linear dependence only\nup to 40 GW/cm2and then the slope becomes gentle. In\nthe case of excitation with fundamental (1.55 eV) pulses,\nsome cooperative interaction seemingly works between\nthe two-photon excited CT states. This is the origin of\nparabolic dependence of A0on the excitation density.\nIn contrast, A0(sh) reﬂects the one-photon excited CT\nstates, it can easily be saturated at the relatively weak\nexcitation density as we observed in Fig. 3(b).\nNow let us discuss the two-photon absorption coeﬃ-\ncient ofβin BiFeO 3. Using the transient transmission\nchange of –0.11 at the probe photon energy of 1.0 eV\n(1.24µm),βis estimated to be 1.5 cm/GW. The βin\nthe one-dimensional cuprate of Sr 2CuO3is 160 cm/GW\nat the peak of the two-photon band[1], while that of\nthe two-dimensional Sr 2CuO2Cl2is 12 cm/GW. This di-\nmensionality dependence of the optical nonlinearity in\nmaterials relating to high TCcuprate superconductors\nwas closely examined by Ashida et al., in which the β\nin the two-dimensional cuprates was typically ten times\nsmallerthanthatoftheone-dimensionalones. Themech-\nanism was explained using the cluster model[3]. Intu-\nitively, to limit the coherent vibrational oscillations of\nelectrons with large amplitudes in one direction can en-\nhancetheopticalnonlinearity. Whileinthecaseoftwoor\nthree dimensional materials, motion of electrons spreads\ninto multiple directions even if electrons and lattice are\nvibrated in one direction by linearly polarized optical\npulses. Thus the low-dimensional cuprates may be more\nbeneﬁcial in terms of application to optical device. How-\never, the βvalue is still larger than those of common\nsemiconductors such as ZnSe and GaAs at the optical\ncommunication wavelength [23]. BiFeO 3also has the\nstrong aspect as a multiferroic, multifunctional responses\nrelating to the magneto-optic eﬀect can be expected in\nlow temperature or polarization sensitive measurements.\nTo explore such phenomena is our future work.\nFinally, let us give a brief comment on the relationship\nbetween the present results and the optical anisotropy.\nBiFeO 3shows uniaxial birefringence with an extraordi-\nnaryaxisalongthe [111]direction. However,weobserved5\n4T1g 4T2g \n6A1g 1.4 eV 1.9 eV (b) \nFe 3d \n(Upper Hubbard) \nVirtual \nintermediate \nstate \nO 2p ~2.2 eV \nPump Probe (a)\nPump Probe \nFIG.5: Schematicdrawings oftheelectronic statesofbismu th\nferrite to show the origin of the ultrafast decay components\nunder pumpingwith fundamental (1.55 eV) pulses. (a) Upper\nHubbard band of Fe 3 dand O 2plevel, in connection with the\nfastest (A2(fun)) and the slowest A0(fun) decay components.\n(b) Relationship between the d-dlevels and the second fastest\n(A1(fun)) component.\nno signiﬁcant polarization dependence in the results of\nthe pump probe spectroscopy. We can understand the\nreason considering the three dimensional chemical bond-\ning structure in BiFeO 3; in the present phenomenon, the\nCT excitation and d-dtransition play important roles,\nwhile absorption due to phonons and magnons shows\nsigniﬁcant polarization dependence. At low tempera-\ntures, the coherence of elementary excitation will grow\nup enough to defeat dephasing factors and we will ob-\nserve the transient anisotropic optical response originate\nfrom coherent phonon and magnon.\nV. CONCLUSIONS\nIn conclusion we demonstrated the ultrafast pho-\ntoinduced absorption spectroscopy in a single crystalof BiFeO 3using sub-100-fs optical pulses in the near-\ninfrared and ultraviolet ranges. With pump pulses at\n1.55 eV, we observed sharp decrease in transmission\nwhich remained constant around the minimum for ap-\nproximately 100 fs, it decayed with a time constant as\nshort as 50–60 fs. Examining the dependences of the\nsignal intensity on pump ﬂuence and probe photon en-\nergy,weconcludethedominanttransientabsorptionorig-\ninates from the two-photon absorption process from the\noxygen 2 pto the iron 3 dlevel. The decay proﬁle also\nhas two other components with constants of 1–3 ps, and\nnanoseconds which can be ﬁtted as a constant in the\npresent observation range. We attribute them as some\nrelaxation process involving spins, and the total lifetime\nof the CT excited electrons via the two-photon process.\nThe two-photon absorption coeﬃcient is estimated to be\n1.5 cm/GW, which is ten (hundred) times smaller than\nthat oftwo(one)-dimensionalcuprates. This result shows\nsuch ultrafast response is common in strongly correlated\nsystems and may reﬂect the dimension dependence of the\nnonlinearity. Thepresent βvalue isstilllargerthan those\nofcommonsemiconductorssuchasZnSeandGaAsatthe\nopticalcommunicationwavelength. With pump pulsesat\n3.10eV,thepencilliketwo-photonabsorptiondidnotap-\npear, butthe proﬁleshadlongertime componentsof140–\n170 fs, 1.9–2.6 ps, and nanoseconds. We attribute them\nto the collective electron-electron and electron-phonon\nscattering in the CT band, some decay process involving\nspin or band to band relaxation, and the total lifetime of\nthe CT excited levels, respectively.\n[1] T. Ogasawara, M. Ashida, N. Motoyama, H. 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B\n11, 2065-2074 (1992)."    },    {        "title": "1902.09315v1.Elemental_substitution_tuned_magneto_elastoviscous_behavior_of_nanoscale_ferrite_MFe2O4_M___Mn__Fe__Co__Ni_based_complex_fluids.pdf",        "content": "1 \n Elemental substitution tuned magneto -elastoviscous \nbehavior  of nanoscale ferrite MFe2O4 (M = Mn, Fe, \nCo, Ni) based complex fluids  \n \nAnkur Chattopadhyay a,†, Subha jyoti  Samantab, Rajendra  Srivastavab\n, Rajib  \nMondalc  and Purbarun Dhar a,*\n \na Department of Mechanical Engineering, Indian Institute of Technology Ropar,  \nRupnagar –140001, India  \nb Department of Chemistry, Indian Institute of Technology Ropar,  \nRupnagar –140001, India  \nc Department of Condensed Matter Physics and Material Science,  \nTata Institute of Fundamental Research, Mumbai –400005 , India  \n \n* Corresponding author :  \nE–mail: purbarun@iitrpr.ac.in  \nPhone: +91 –1881 –24–2173  \n† E–mail: ankur.chattopadhyay@iitrpr.ac.in  \n \n \n 2 \n Abstract  \n \nThe present article reports  the governing influence of substituting the M2+ site in  nanoscale  \nMFe 2O4 spinel ferrite s by different magnetic metals (Fe/Mn/Co/Ni) on magnetorheological and \nmagneto -elastoviscous behaviors of the corresponding magnetorheological fluids (MRFs) . \nDifferent doped MFe 2O4 nanoparticles have been synthesized using the polyol -assisted \nhydrothermal method. Detailed steady and oscillatory shear rheology have been performed on \nthe MRFs to det ermine the magneto -viscoelastic responses  The MRFs  exhibit shear thinning \nbehavior and augmented yield characteristics  under influence of magnetic field . The steady state  \nmagneto viscous  behaviors are scaled against the governing Mason number and self -similar \nresponse from all the MRFs have been noted. The MRFs conform to an extended  Bingham \nplastic model under  field effect.  Transient magnetoviscous  responses show distinct hysteresis \nbehaviors when the MRFs are exposed to time varying  magnetic fields. Oscillatory shear studies \nusing frequency and strain amplitude sweeps exhibit predominant solid like behavior s under field \nenvironment . However, the relaxation behavior s and strain amplitude sweep tests of the MRFs \nreveal that while the fluids show solid -like behavior s under field effect, they cannot be termed as \ntypical elastic fluids. Comparison s show that the MnFe 2O4 MRFs  have superior yield \nperformance among all. Howe ver, in case of dynamic and oscillatory systems, CoFe 2O4 MRFs \nshow the best performance . The viscoelastic responses of the MRFs are noted to correspond to a \nthree element viscoelastic model.  The study may find importance in desi gn and development \nstrategies  of nano -MRFs for different applications.  \n \nKeywords: Viscoelasticity, magnetorheology , yield stress, nanoparticles, f errite s, smart fluids , \ncolloids  \n \n \n 3 \n 1. Introduction  \n \nMagnetorheological fluids (MR Fs), which are colloidal dispersions of magnetic nanoparticles, \ncan be dubbed  as ‘smart fluids ’ due to their magnetic field dependent tunable visco sity. \nConventional  MR fluids  consist of magneti c submicron particles dispersed stably in non -\nmagnetic liquids  and these  fluids show altered rheological behavior in presence of external \nstimulus (magnetic field) [1].  Consequently, MR fluids have received considerable attention  in \ndifferent technological applications , such as shock absorbers, brakes, clutches, seismic vibra tion \ndampers, control valves , torque  transducers, polishing fluids , and have also been studied for \nbiomedical  applications like drug delivery, cell separation, diagnostics sensor, etc. [2 -5]. \nTherefore, in -depth understanding of the fundamental  rheological  behavior of MR Fs is essential  \nfor design and development of such field-responsive systems.  \n \nWhen the MRFs are exposed to a magnetic field, the dispersed magnetic nanoparticles \n(MNP s) coalesce to form a fibrillar  micro structure . Under the action of shear , these fibrils or \nchains  resist  hydrodynamic  deformation forces , leading to improved  rheological  parameters [6].  \nIt has been theorized that attachment  and detachment of  the sidechains  from  dense  clusters of \nparticles is responsible for the dynamic viscoelastic behaviors of MR Fs [6]. Some  studies have \nshown that the  dipole interactions among MNPs are primarily responsible for the difference in \naggregation behavior [7, 8] , which in turn governs rheological features . Measurement techniques \nlike neutron  scattering [9, 10] and X-ray scattering [11, 12] have been employed by researchers \nto reveal information  regarding the field induced micro structure s of the MRFs .  Reports have \nalso shown  [13] that the stress relaxation process in a ferrofluid can be attributed  to both linear \nchains and the dense  bulk aggregates. Theoretical models have also been put forward to  \ndetermine  the viscoelastic responses of MRFs [14, 15]. These models propose that for highly \nconcentrat ed MRFs, complex ly shaped structures , larger than single chains  observed in dilute \nMRFs , are majorly responsible for the augmented viscoelastic responses  under field effect .  \n \n Zubarev  et al. have suggested a few analytical models to explain the rheological behavior \nof ferrofluids [16]. To examine the structural configurations of MR suspensions under steady and 4 \n dynamic flow, molecular dynamics [17, 18] simulation s have been performed an d the results \nindicate that the responses of chainlike  structures are strongly dependent on the orientation \nrelative to the direction of applied shear flow under field stimul us. This  ultimately lead s to \npronounced anisotropic viscous behavior. Brownian dyn amics simulations have also been used \nto understand magnetoviscous  responses when the MRF is exposed to weak and strong dipole \ninteractions with varying magnetic fields [19].  \n \nSince the morphology  and structure of MNPs influence the magnetoviscous  responses  of \nthe MRFs, selection of proper magnetic nanocolloids  have strong implications  on the overall \nrheological behavior. Binary transition  metal  oxides (BTMOs) (denoted by AB 2O4, where  A, B = \nCo, Ni, Fe, Mn) have attracted  considerable  attention  in this field due to their strong  size and \nshape dependent magnetic properties [20, 21]. Ferrite nanoparticles (FNPs), which fall into the \ncategory of BTMOs , possess typical spinel structure s. Here  A and B are metallic cations \npositioned at two different crystallographic sites, the tetrahedral (A site) and the octahedral (B \nsite) [22]. The common examples for ferrites are MFe 2O4 (where M = Mn, Fe, Co, Ni etc.) and \nmany  of these nanoparticles demonst rate superparamagnetic (SPM) properties . The \nsuperparamagnetism is manifested below  a critical size of the nanoparticles of around 30 nm in \ndiameter [22]. By regulating the M2+ appropriately, the magnetic configurations of MFe 2O4 can \nbe tuned to obtain  high magnetic permeability and electrical resistivity . These nanoscale ferrites \nhave been considered as the potential candidate s in high-performance electromagnetic [23,  24] \nand spintronic devices [25, 26].  \n \nIf the  M2+ occupies only the tetrahedral sites , the spinel is termed as direct, whereas when \nit occupies the octahedral sites only, the spinel is called inverse [27]. Thus , the preferred location \nof M2+ is critical to tune the magnetic behavior. Although significant progress has been made in \npreparing  typical FNPs, there are few gener ic process es for producing MFe 2O4 nanoparticles \nwith the desired size and acceptable size distribution [28]. Many studies report the various \nsynthesis  processes to obtain MFe 2O4 FNPs and  their magnetic  properties [28 -31]. However , \ndespite their excellent magnetic behavior even at the nanoscale,  the employability  of such FNPs \nin nanocolloidal  MRFs have  not been explored . In recent years, a few studies have investigated \nthe magnetorheological characterist ics of CoFe 2O4 [32-34], ZnFe 2O4 [35], MgFe 2O4 [36] based 5 \n suspensions . However, these  studies primarily concentrate upon steady state rheological \ncharacteristics and the more on the methods of preparing the MRFs  than their fluid dynamics . \n \nThe present study  explore s the magnetoviscous  and magneto -elastoviscous responses of \nMFe 2O4 ferrite based nanocolloid al MRFs .  The study also focuses on the examin ation  of the role \nof the dopant metal  ion M2+on the overall magnetorheological response of the fluids.  The \nobjective  of the study is also to understand magneto -viscoelastic behaviors of the colloids by \ntuning the M2+ (Fe/Mn/Co/Ni)  crystalline location . The ferrites have been synthesized by \nchemical routes and have been characterized in detail for their physical structure and properties. \nSteady and oscillatory rheological measurements, at small and medium shear rates, have been \nstudied and the response s are analyzed to deduce the viscoelastic nature of the colloids. Standard \nand extended viscoelastic and rheological models have also been used to determine the elastic \nand viscous response s of the colloids under magnetic field. The present article may find \nimpo rtance in design and development of nano -MRFs . \n \n \n2. Materials and method ologie s \n2.1. Synthesis of nanomaterials  \nThe base chemical ingredients, such as FeCl 3.6H 2O, MnCl 2.2H 2O, CoCl 2.4H 2O, NiCl 2.6H 2O, \nand PVP (polyvinyl pyrrolidone)  were procured and used as  is (Loba Chime Pvt. Ltd. , India).  \nEthylene glycol and poly-ethylene  glycol (MW 400) were procured from Spectrochem , India , \nand NH 4F from Sigma -Aldrich , India . MFe 2O4 nanoparicles were synthesized using the polyol  \nassisted  hydrothermal method [37]. In a typical synthesis, 20 mmol of FeCl 3.6H 2O and 10 mmol \nMCl 2.2H 2O (M = Mn, Co, Ni) were taken in a beaker containing 200 mL ethylene glycol and \nstirred for 30 mins. Then 3.2 gm of polyethylene glycol (M.W 400) was added to this  solution \nunder stirring condition and further stirred for 30 minutes. After complete dissolution , 80 mmol \nof NH 4F was added slowly to th e solution  and allowed to settle for an additional 2 h to allow  a \nviscous solution to form. Finally, the solution was transferred to a Teflon lined autoclave and \nkept at 180 °C for 24 h. After the hydrothermal treatment, the autoclave was cooled to room 6 \n temperature, and the batch was filtered, washed several times with deionized water and dried \novernight (10 -12 h) at 80 °C  which  result s in the formation of a black powder  of MFe 2O4. \n \n2.2. Nanomaterial c haracterization  \nX-ray diffraction (XRD) patterns for the nanomaterials were recorded in the 2θ range of 5 °-80° \nwith a scan rate of 2°/min (PAN analytical X’PERT PRO diffractometer, the Netherland s) using \nCu Kα radiation (λ=0.1542 nm, 40 KV, 45 mA) (Fig. SF1, Refer Supplementary). Nitrogen \nadsorption -desorption measurements were performed at -200 C (Quantachrome Instruments, \nAutosorb -IQ volum etric adsorption analyzer, USA ). The specific surface area of the material was \ncalculated from the adsorption data points obtained at P/P 0 between 0.05 -0.3 from the Brunauer -\nEmmett -Teller (BET) equation. Field enhanced  scanning electron microscopy (FESEM) \nmeasurements (ZEISS Supra ) were done to determine  the morphology of the materials  (figure 1 \n(a)-(h)). \nThe crystallinity, phase purity, and successful formation of all the ferrite samples were \nconfirmed by Powder X -ray (P-XRD) diffraction in the range of 2θ (5° -80°). All the samples \nexhibit reflections at 2θ = 30.27°, 35.46°, 43.25°, 53.62°, 57.26°, and 62.97° , which indicates \nFCC framework structure . Except  NiFe 2O4, none of the sample s exhibit additional reflection \nother than the FCC framework , and this eliminate s the possibility of presence of any other phase s \n(such as CoO, MnO 2, Fe 2O3, etc. in COFe 2O4, and MnFe 2O4). In the case of NiFe 2O4,  low \nintensity  peaks at 2θ = 48.09° and 51.56° (in addition to the standard FCC of NiFe 2O4)  are \npresent due to the formation of impurity phase of NiO/Fe 2O3 [38]. The XRD reflection patterns \nof spine  slightly  vary from one another due to the difference in their crystal field stabilization \nenergy in their respective coordination ge ometry.  \n \n \n 7 \n  \nFigure 1 . FESEM images of (a -b) NiFe 2O4, (c-d) Fe 3O4, (e-f) CoFe 2O4, and (g -h) MnFe 2O4 \nnanoparticles.  \n \nThe detailed information regarding crystallite size, FWHM, and other parameters are \nenlisted in table ST1 (Refer Supplementary). The average crystallite size obtained from P -XRD \n8 \n analysis for MnFe 2O4, CoFe 2O4, NiFe 2O4and Fe 3O4 are 10.7, 7.7, 10.4, and 9.2 nm, \nrespectively. The presence and amount of all the elements in a representative material  CoFe 2O4 \nis confirmed from EDAX analysi s (Fig. SF2, Refer Supplementary). The surface area obtained \nfrom BET measurements and  textural properties are summarized in ST 2 (Refer Supplementary). \nBET analysis further depicts that CoFe 2O4 exhibits highest surface area among the different \nmaterials synthesized. Fig. 2 illustrates the  room temperature (300K) magnetization curves (M -\nH) of different MFe 2O4 nanoparticles, measured by Vibrating Sample Magnetometer (VSM). \nNone of the MNPs possess  magnetic hysteresis , which confirms superparamagnetic behavior. \nThe Mn based ferrite exhibits  the highest saturation magnetization MS ~ 74 emu /g (achieved \nwithin ~ 0.8 T). The saturation magnetization of the ferrites are ~ 62, 44 and 37 emu/g for Fe 3O4, \nCoFe 2O4, and NiFe 2O4, respectively.  \n \n \n \nFigure 2 . Magnetization curves of various MFe 2O4 ferrites at 300K . The absence of \nmagnetization hysteresis confirms the superparamagnetic phase.  \n \n \n9 \n 2. 3. Instrumentation  \nA non-polar liquid, silicone oil (SO) (procured from Avra Synthesis Ltd., India) was used to \nprepare the MRFs . The SO has a Newtonian viscosity of 350  cSt at 25 °C. Anhydrous FNPs  \nwere  dispersed in the SO as per concentration requirements  (40 wt. % in the present stud ies) and \nstirred mechanically and ultrasonicated to obtain a homogeneous colloid. T o prevent moisture  \nadsorption , the MRFs were stored in a desiccator. The MR properties of the colloids were  \nmeasured using a rotational rheometer (MCR 102, Anton Paar, Germany) with parallel plate \nconfiguration at constant gap of 1 mm. The rheometer is connected to a Magnetorheological \nmodule, capable of generating magnetic fields up to 1 T for sample thicknesses of 1 mm  and \nlower . The MRFs are tested at  four different m agnetic field intensit ies (0, 0.35, 0.7, and 1 T). \nDuring the experiments, the sample temperature has been maintained constant at 300K using a \nPeltier controller.  The rheological behavior is obtained by measuring the shear stress and \nviscosity as a function of shear rate from 0.01 to 100 s−1. The viscoelastic responses have been \nstudied in terms of storage modulus (G'), loss modulus (G\"), dissipation  (loss)  factor (tan δ), and \nthe complex viscosity (η*). To probe the dynamic rheological response,  oscillatory tests with \nstrain amplitude s of 0.01 to 1 % and frequenc ies of  1 to 100 Hz  have been performed . \nMeasurements of stress relaxation behaviors and magnetoviscous  hysteresis have also been \nperformed . The typical uncertainty involved in the measure ments was within  ±5%.  \n \n3. Results and discussion  \n3.1. Magnetorheology  \n \n Figures SF4-SF7 (Refer supplementary) illustrates  the role of magnetic field  in modifying the  \nstatic rheological behaviors of the colloids . The increase in viscosity is due to  the formation of \nchain -like structures by the nanoparticles within  the colloids under the action of  magnetic field. \nAt lower magnetic field  strengths , the chains or fibrils  formed  within the MRF are typically  \nlinear. With incre ase of  field strength , the n umber densities of chains improve  with increasing \naspect ratio of the fibril structures [39]. Further increase of field strength can lead to lateral \ncoalescence of the chains , resulting in thick columns [ 40, 41], and ultimately decrease in the 10 \n magnetic field induced viscosity . All the samples  (MRFs)  exhibit shear thinning behaviors for \nthe entire range of shear rates  and magnetic field strengths . Reports have  shown that the chains \nunder confinement can show different  responses  compared to unconfined  chains [42], which is \npossibly the reason why ferrites with higher saturation magnetization show  low viscous response \nto magnetic fields compared to the lower magnetization ferrites . The higher magnetization leads \nto agglomeration of the chains at lower field strengths, and the viscous response of the colloid \ndoes not improve at higher fields.  \n \n \nFigure 3. (a) Viscosity behavior against shear rate for various FNPs based MRFs at 1T (b)  Shear \nstress response to shear rates for various FNPs based MRFs at 1T.  \n \nThe Mn-based  ferrite colloids show the highest viscous response, followed by the Fe 3O4, \nwhich is followed by the Co, Cu and Ni-based  ferrites. This can be explained using the coupling  \nconstant ( λ). For a constant magnetic field and temperature,  \n2\n           (1) \nwhere,  χ is the magnetic susceptibility of the material [4 3].  Higher the value of λ more is the \nlikelihood of formation of chains that are aligned along the direction of the magnetic field. This \ndirectly results in  enhanced magneto -static particle interaction, leading to enhanced viscosity \n[42]. The coupling constant is known to have a q uadratic relationship with the magnetic \nsusceptibility. Thus, the slope of the M -H curve has a strong effect on the viscosity of the \n11 \n associated MRFs. The magnitudes of χ for the FNPs follow the order as Mn>Fe>Co>Ni (fig. 2) \nand consequently, the magnetovis cous effect of their colloids obey the same order (fig. 3 (a)).  \n \n \nThe yield stress es of the MRFs also reveal similar magnetorheological behaviors  (fig. 3 \n(b)). The yield stress provides an estimate of the force required to continuously deform the \nparticle aggregates and chains which tend to reform in the presence of the magneto -static forces \ndue to the applied magnetic field. Enhancement of yield values has been observed when the \nMRFs are exposed to increased magnetic field intensities (Figs. SF4 -SF7). The yield stress is \nknown to increase  in a quadratic  manner  with increasing magnetic field strengths at low field  \nregimes as  \n2\n0()yc H    \n          (2) \nwhere \n0 is the permeability of the vaccum, \nc is the relative permeability of the carrier fluid. In \neqn. 2, \n is the contrast factor, defined as  \n2pc\npc\n           (3) \nwhere \np is the relative permeability of the particle.  For instance the \n  of Fe 3O4 based MRF \ncomes around 2.7 [44]. The respective values of \n  of other MRFs vary from  2 to 3, based on \nthe choice of nanomaterials.   \n \nThe augmentation of yield stress is pronounced  in the linear regime of magnetization [45 \n- 47]. At higher  magnetic fields, around the magnetic saturation limit, the yield stress becomes \nfield independent  \n2\n0 ys M\n           (4) \nwhere Ms represents the saturation magnetization. It is important to understand the comparative \nyield  stresses  of the MRFs (fig. 4 (a)) caused due to  substituti on of Fe by other metals (Mn, Co, \nNi etc.)  in the ferrites . Doping with Mn in MFe 2O4 results in superior yield stress  compared  to \nFe. However, doping with Co and Ni leads to reduction  in yield stresses than Fe3O4 MRFs . These \nobservations signify that there is a strong dependence of the yield stress on the magnetic moment 12 \n of the M2+ (nμ B, where n = 5, 4, 3 and 2 , for Mn, Fe, Co, an d Ni, respectively) [29, 48, 49]. \nHence, the static magnetorheological behavior in ferrite based MRFs is a strong function of the \nmagnetic properties of the dopant atom.  \n \nFig. 4 (a) illustrates the yield stress values for three values  of applied magnetic fields. \nTypically, the rate of enhancement of yield stress values is high for low magnetic field  strengths , \nwhereas , at high field strength s, the yield stress values reach saturation . An extended  Bingham \nmodel  has been used to predict  the yield stress at both low  and high regimes of magnetic field \nstrength. It is  expressed as [50]  \n2\n02( )( )2H\nH\ny y y yee\n   \n\n   \n        (5)  \nwhere, τy is the yield stress at magnetic  field H. In eqn. 4, \n0y represents  the yield stress at  zero-\nfield,  and\ny represents the corresponding  yield stress once it has attained saturation. The \nparameter α is a fit variable in the extended Bingham model, which have been shown in table 1 . \nThe Mn-based  MRF has the highest saturation yield stress  and hence for a particular imposed \nfield, Mn-based  MRFs can resist higher deformation.  \n \n \n13 \n Figure 4.  (a) Comparison of experimental yield stress values and the model predictions (eqn. 4) \n[symbols refer to experimental observations and line s represent  the model], (b) comparison of \nviscous  behavior s of different ferrite (Fe/Mn/Co) MRFs as function of Mason number  \n \nTable 1. Fit parameters of yield stress model for different MRFs  \nParameters  Mn Fe Co Ni \ny\n(Pa) 1050  820 635 50 \n0y\n(Pa) 10 150 6.5 10 \nα 7 4.3 3.6 4.2 \n \nAs observable in fig.4, the present MR Fs conform to the Bingham plastic model, \nexpressible as  \ny pl   \n          (6) \nwhere,  the shear stress  (\n) is related to the yield stress (\ny ), the plastic viscosity (\npl ) and the \nimposed shear rate  (\n\n). The apparent viscosity \napp for the MRF is defined as  \napp\n           (7) \nIntroducing the non -linear nature of the plastic viscosity, the apparent viscosity can be remodeled  \nin terms of the infinite shear viscosity as \napp y pl  \n     \n          (8) \nIt is shown by reports that \npl would ultimately be equal to\n , especially at high shear limits \n[51, 52]. The non -dimensionalized form of the eqn. 6 can be expressed as   \n1app\nMn\n\n          (9)  14 \n where, \n is a constant which  is determined from fitting the experimental observations  to the \neqn. 9. The Mason number  (\nMn ) is the ratio of the hydrodynamic shear forces to magnetic  \n(ferrodynamic ) forces in MRFs and is expressed as \n2\n02 ( )c\ncMnH\n  \n          (10) \nFig. 4 (b) illustrates the viscosity ratio as a function of Mason  number  by adjusting the set of \n\nand\n to obtain the best possible fit. In case  of the present MR Fs, \nis deduced to range between \n4 × 10-4 and 2 × 10-3 and \n  ranges in ~ 7-12 Pas for  the entire range of magnetic field strengths . \n \nUnderstanding the transient responses of MRFs is  important to assess their performance s \nin dynamic environments. Figure 5 (a)  illustrates  the magnetic field sweep  test results  for Fe3O4 \nMRFs  at different shear rates (0.01 s-1 to 1 s-1). The viscosity enhances with increase in magnetic \nfield and saturate s at high fields , irrespective of the magnitude of  the shear rates. The magnetic \nfield leads to formation of the chained fibrils by the aligned nanoparticles within the MRFs. Such \nstructure induces localized elasticity to the fluid phase and leads to enhanced shear resistance,  or \nviscosity. The number density of the fibril s reach a maximum at a particular field, and beyond \nthat, no additional  fibril  formation takes place. Thereby, the magnetoviscous  behavior reaches a \nplateau . The MRFs  show  hysteresis in the magnetoviscous  effect (fig. 5 (a)) upon application and \nwithdrawal  of magnetic field. It can be seen in  fig. 5 (a) that the curve s for the decreasing field \ncase lies above the increasing field case . This suggests that the field -induced chains or fibrils \npossess  a structural hysteresis  [53], even though the constituent na noparticles are \nsuperparamagnetic. With the withdrawal of the field, the magnetization of the nanoparticles relax \nimmediately (due to zero magnetic hysteresis), however, the interparticle magneto -static \ninteractions require a finite relaxation period. This  causes the elasticity of the microstructure to \nrelax in a lagging manner to the external field, leading to the positive magnetoviscous  hysteresis \nat lower fields.  The comparative performances of all the ferrite MRFs  for a shear rate of 1 s-1 \nhave been illustrated in fig. 5 (b). It is noteworthy that all the doped ferrite MRFs show reduced \nmagnetoviscous  hysteresis compared to the Fe 3O4 MRFs.  15 \n  \nFigure 5.  (a) Comparison of magnetic hysteresis of Fe 3O4 MRFs for different shear rates (0.01 , \n0.1, 1 s-1) (‘L’ represents loading – increasing magnetic field – solid symbol , and ‘UL’ represents \nunloading - decreasing magnetic field - open symbol ), (b) comparison of magnetic hysteresis of \nMFe 2O4 (M = Mn, Fe, Co, Ni )  MRFs at shear rate of 1 s-1, (c) comparison of magnetoviscous \nhysteresis area between experimental data and theoretical model of Fe/Mn/Co /Ni ferrite based  \nMRFs [symbols represent  experimental observations and line s represent eqn. 13]  \n \n16 \n Upon application or withdrawal of the magnetic field, the magnetic moment of a particle \ncan relax by two mechanisms. In case of the Brownian relaxation, the magnetic moment remains \nfixed within the particle,  and it reorients  its position  as a whole. The ro tation of magnetic \nmoment instead of the actual particle conformation is  called the Neel relaxation. It is common  \nfor smaller particles (size of the order ~ nm) to magnetically relax by Neel mechanism, while \nlarger particles tend to follow the Brownian mechanism. The Brownian relaxation time (\nB ) and \nNeel relaxation time (\nN ) scales can be expressed as per eqns. 11 and 1 2 respectively.  \n3c\nB\nBV\nkT\n           (11) \n01expN\nBKV\nf k T\n\n         (12)  \nwhere, f0 is the frequency of domain flipping, whose value has been assumed to be 109 Hz for \nmagnetic particles in non -polar media  [54], K is the magnetic anisotropy constant, kB is the \nBoltzmann constant, T represents the absolute temperature and V is the volume of the FNP. The \ntypical values of \nB  and \nN  of the FNPs in the present study are ~  10−6 s and ~  10−9 s, \nrespectively. The process with the smaller relaxation time governs the overall relaxation of the \nMRFs [54], and in the present case, the Neel relaxation is the dominant mechanism.  \n \nIt can be observed from fig. 5 (a) that  magnet oviscous remanence  is a function of the \nimposed  shear rates. The area enclosed by the loading (increasing field) and unloading \n(decreasing field) curves, referred to as the magnetoviscous hysteresis area , can be used to model \nsuch MRFs.  When the hysteresis areas are plotted as function of shear rate, relative decrease are \nobservable with increasing shear values  (fig. 5 (c)). This behavior can be predicted  using power \nlaw model as per  [42] \nvAu\n           (13) \nwhere, A is the predicted magnetoviscous hysteresis area, u is a scaling factor, and v is an \nexponent of the shear rate (\n\n ). The respective values of u and v of different MR Fs have been 17 \n listed in Table 2. The reduction of hysteresis areas at increased shear rates indicates that the \nmagnetic moment remanence due to interparticle interactions within the fibrils are overcome by \nthe higher shear, which leads to loss of structural integrity of the fibrils. This leads to quicker \nlowering of the viscosity  on w ithdrawal of the field , leading to lower viscous hysteresis. A point \nof interest may be noted from table 2. For the doped ferrite based MRFs, the values of u and v \nare very similar, which suggests that these fluids exhibit self -similar magnetoviscous hysteresis. \nThis point may find useful implications in dopant based magnetic nanoparticle synthesis for \ncontrol of magnetoviscous  hysteresis in the corresponding MRFs.  \n \nTable 2. Fit parameters of eqn. 13 of different ferrite MRFs (goodness of fit >0.95)  \nParameters  Mn Fe Co Ni \nu 13 50 15 15 \nv -1.38 -1.1 -1.36 -1.34 \n \n \n3.2. Magneto -viscoelasticity  \nOscillatory rheological responses of the MRFs have been examined to understand their dynamic \nbehavior s. Fig. 6 (a) illustrates the magneto -viscoelastic behavior of the MRFs . The storage ( G') \nand loss ( G\") moduli for the different MRFs are obtained from frequency sweep tests at different \nmagnetic field  strengths at constant  strain amplitude s. In absence of magnetic field, for the entire \nrange of frequenc ies, the G\" is higher than the G', signifying predominantly liquid behavior. \nUnder the influence of magnetic field, the  improvemen t of G' component indicates  \nmicrostructur e elasticity within the MRF. The particles within an MRF experience rotational \nresistance when in the presence of a magnetic field, due to competition between magnetic and \nhydrodynamic torques [55, 56]. The hydrodynamic torque tends to distort the alignment of \nparticles induced by the magnetic field. At higher fields, the hydrodynamic torque on the fibrils \nis overcome by the magnetic t orque, thus leading to a microstructure with elastic integrity. This 18 \n leads to a behavior which mimics polymeric or elastic fluids (typically linear viscoelasticity \nbehavior at high fields (fig. 6 (a)). The loss factors also exhibit improvement under field s timuli \n(fig. 6 (b)) , which indicates enhanced dissipative behavior in addition to improved elasticity and \nis a preferred characteristics for dynamic applications . With increase in frequency, the adjacent \nchains are repeatedly sheared in each other’s vicini ty. Consequently, the hydrodynamic torque \nexceeds the magnetic torque,  leading to viscous behavior  (loss factor approaches unity)  at high  \nfrequencies .  \n \n \nFigure 6.  Frequency sweep responses at  different magnetic field  strengths  (0, 0.35, 0.7, 1 T) (a) \nstorage and loss moduli of Fe 3O4  MRF at 1% strain  amplitude  [open symbol – storage modulus, \nclosed symbols – loss modulus], (b) complex viscosity and loss factors as function of oscillatory \nfrequency 1% strain amplitude [symbol – complex viscosity, line – loss factor], (c) storage and \nloss moduli at 10% strain  amplitude  [open symbol – storage modulus, closed symbols – loss \n19 \n modulus], (d) Complex viscosity and loss factor  at 10% strain amplitude [symbol – complex \nviscosity, line – loss factor]  \n \nThe comparisons between frequency sweep responses at strain amplitudes of 1% and \n10% for Fe3O4 MRFs  have  been illustrated in figs. 6 (c) and 6 (d). The behaviors of 10 % are \nsimilar in nature to the 1 % case, which signifies good phase stability at higher osci llatory \nfrequency values. S crutiny of fig. 6 (d) reveals that while the 10 % case shows higher lossy \nbehavior than the 1 % case at 0 T, the loss factors are similar at 1 T. This signifies that the MRFs \npossess higher microstructural  elasticity at higher frequencies and high fields. This behavior \ncould be of potential interest for usage in utilities.  Fig. 7 illustrates the role of elemental \nsubstitution on the frequency sweep viscoelastic behavior of the different MRFs.  It is \nnoteworthy that the doping by  the Mn does not lead to improvement in the magneto -viscoelastic \nresponse of the MRFs compared to the Fe 3O4 based MRFs. While in the steady shear rheology \nthe role of magnetization moment of the doping element was prominent, the dynamic or \noscillatory rheo logy case is not so.  \n \nFigure 7.  Frequency sweep responses of various MFe 2O4 MRFs  at 1 T at strain amplitude 1% (a) \nstorage and loss moduli [open symbol – storage modulus, solid symbols – loss modulus], (b) \ncomplex viscosity and loss factor  as function s of oscillatory frequency [ symbol – complex \nviscosity, line – loss factor]  \n20 \n  \nTo understand the role of imposed oscillatory strain on the viscoelastic behavior of the \nMRFs, amplitude sweep experiments ( from strains of 0.01% to 1%) are performed for different \noscillatory frequencies. Figs . 8 (a) and 8 (b) illustrate the amplitude sweep viscoelastic response s \nof Fe3O4 MRFs  at oscillatory  frequencies of 1 and 10 Hz. The compar ison of the behavior s of the \ndoped ferrite MRFs have been illustrated in figs.  8 (c) and 8 (d).  The G\" is higher than the G' at \nzero field case (figs. 8 (a) and (b))  with no distinct region of linear viscoelasticity.  With magnetic \nfield, the G' is higher than the G\" at low  values of  strain  amplitude . As the strain amplitude \nincreases, a distinct crossover is noted and the viscous behavior overshoots the elastic behavior.  \n \n \n \nFigure 8.  Amplitude sweep responses (a) storage and loss moduli of Fe 3O4 MRFs at different  \nmagnetic fields (0 , 0.35, 0.7, 1 T) at 1 Hz oscillatory frequency [open symbol – storage modulus, \n21 \n closed symbols – loss modulus], (b) storage and loss moduli at 10 Hz [open symbol – storage \nmodulus, closed symbols – loss modulus], (c) comparison of various MR Fs at 1 Hz and 1T [open \nsymb ol – storage modulus, closed symbols – loss modulus], (d) comparison of various MR Fs at \n10 Hz and  1T [open symbol – storage modulus, closed symbols – loss modulus].  \n \nUnder the effect of magnetic field also no linear regime is observed. The crossover \namplitudes have been illustrated in fig. 9 (a). Increase in the field strength increases the \ncrossover amplitude, thereby postponing the initiation of viscous deformation. Additionally, \nhigher oscillatory frequencies lead to reduction in the crossover amplitude due to lesser \nrelaxation time available to the fibrils to align with respect to the magnetic field. This leads to \nloss of microstructure elasticity locally, and leads  to reduction in crossover amplitude. It is \nnoteworthy that the MRFs of ferrites of higher magnetic moments (such as Mn based ferrite and \nFe3O4) exhibit reduction in the crossover amplitude at higher magnetic field strengths. This can \nbe explained based on  the inter -chain or inter -fibril magnetic interactions in such MRFs. Due to \nthe high magnetic moment of the constituent particles in the chains, the magneto -static repulsion \nbetween neighboring chains will be high. This leads to decrease of the effective e lasticity of the \nmicrostructure, and even more so at higher fields. Consequently, the crossover amplitude of the \nMRFs of such ferrites deteriorates at high fields.  22 \n  \nFigure 9.  Crossover amplitude (a) comparison of various MR Fs at 1 Hz for various magnetic \nfield strengths  (0.35 , 0.7, 1 T), (b) comparison for different MRFs at  1 Hz and 10 Hz . \n \nThe relaxation behaviors of the MRFs have also been characterized to determine their \nviscoelastic nature. Figs. 10 (a) and (b) illustrate the stress relaxation behaviors of the Fe 3O4 and  \nMnFe 2O4 MRFs  respectively for a given magnitude of 10% strain . The relaxation modulus \nimproves with increase in field strength. At higher strains, the nature of the transient evolution of \nthe relaxation curve is similar  for MnFe 2O4 MRFs , however in case of Fe 3O4 MRFs , the transient \nevolution changes with field strength. The presence of the field induces fibrillation within the \nMRF. The fibrils lead to local elasticity within the fluid. The viscoelastic nature leads to \nincreased re laxation modulus as well as weaker temporal relaxation (fig. 10 (b)), implying that \nthe microstructure withstands the applied strain without plastic deformation to a greater extent. \nTo quantify the magnitude of stress relaxation, a relaxation ratio has bee n defined and the values \nhave been illustrated (figs. 10 (c) and 10 (d)). It is defined as the ratio of the initial relaxation \n23 \n modulus to the relaxation modulus (at 100s in the present case). It is observable from fig. 10 (c) \nthat Mn based MRFs exhibit the  highest stress relaxation caliber under field constraints.  \n  \nFigure 10.  (a) relaxation modulus of Fe 3O4 MRFs at 1 0% strain, (b) relaxation modulus of \nMnFe 2O4 MRFs at 10% strain, (c) relaxation ratios of various MRFs at 10 % strain, (d) \ncomparison of relaxation ratios of different MRFs at 1% and 10% strains.  \n \nThe viscoelastic response s of the MRFs  have been compared against theoretical model s \nto determine the type of elastic fluids that the present fluids conform to . Typically,  visco elastic \nmaterials can be modeled  as different combinations of elastic  springs (signifies conservation of \nenergy) and viscous dashpots  (represents dissipation of energy), which ultimately leads to \ndifferent constitutive stress –strain relations  [57]. A simple viscoelastic sys tem can be modeled as \ntwo element classical models , like Maxwell model (fluid like behavior) or Kelvin -Voigt model \n(solid like behavior) [ 58, 59 ]. The present MRFs, especially under magnetic field, do not \nconform to the two element models . Accordingly,  3 element elastic and viscous models have \n24 \n been implemented to model the experimental observations.  It has been found that a three element \nelastic model also does not hold good for the MRFs, and hence the three element viscous model \nhas been employed. The model consists of a viscous dashpot , which is  connected  in series with a \nclassical Kelvin -Voigt element (a spring and dashpot connected in parallel) . The schematic of the \nmodel element has been shown in fig. SF8 (Refer Supplementary ). For a Newtonian fluid  based \ndamping element , the stress within the dashpot  element is expressed as \nd\ndt\n           (14) \nwhere,  σ is the stress, ε is the strain and η is the viscosity of the fluid.  \nHowever, s uch a Newtonian three element fluid  model fails to rep licate  the experimental  \nviscoelastic moduli. This is atypical for the present case, as prominent linear viscoelastic regimes \nwere not identifi able from the amplitude and frequency sweep studies. A  fractional order time \nderivative q is introduced in the constitutive equation  of the dashpot  to provide a realistic \nrepresentation of  a non-linear, non-Newtonian fluid system. For a non-linear, non-Newtonian \ndashpot , the stress is expressed as  \n1( ')( / )q q q\nq q q qd d dt k kdt d t dt      \n       (15) \nwhere, β = kτ-q is a variable equivalent to viscosity  and governs the viscous relaxation behavior,  τ \nis the characteristic time  and t’ is the dimensionless time  [60-62]. The net stress within the Voigt \nelement of the 3 element systems  is [63] \n2\n22 ( ')q\nqdtEdt  \n          (16) \nwhere, E is the elastic modulus of the associated spring component within the Voigt element.  \nThe stress within the dashpot which is placed in series with the  Voigt  element  is as (the \nindex p governs the fraction response of the series dashpot)  \n1\n1 ( ')p\npdtdt\n          (17) 25 \n The total strain ε within the 3  element system is determined via  Laplace transform  as \n12( ) ( )()pqssss E s\n         (18) \nThe final form of the  stress -strain relationship in the frequency domain is [63]  \n12\n21( ) ( ( ) )( ) ( )( ) ( )pq\nqpi E i\nE i i         \n        (19) \nExpanding the eqn. 19 in terms of the storage and loss moduli yields [63] \n2 2 2 2 2 2 2\n1 2 1 1 1 2 1 2\n22\n2 1 2 12()( ) ( )p q p p p q p q\nq p q pE fg E E f g fGE g f e d                         \n   (20) \n2 2 2 2 2\n1 2 1 1 2 1 2\n22\n2 1 2 12()( ) ( )p q p p q p q\nq p q pE dg E d e dGE g f e d                      \n    (21) \nwhere, \nsin( / 2)pd , \nsin( / 2)qe , \ncos( / 2)pf and \ncos( / 2)qg .  Value sets of  E, β1, \nβ2, p and q are estimated to predict the frequency sweep observations for the MRFs.  The  Mn \nbased MRF has been used as a representative system and eqns. 20 and 21 have been fit to the \nviscoelastic moduli ( fig. 1 1). The set values of the different parameters that fit the viscoelastic \nresponses have been given in tables ST3 and ST4 (Refer Supplementary ). The values of  p and q \nlie between 0 and 1 for entire range of magnetic field intensities. This justifies the use of \nfractional viscoelasticity model for the present MRFs. With increase in field strength, the values \nof E, β1 and β2 are found to enhance .  \n 26 \n  \nFigure 11 . Comparison  of experimental observation s (at 1% strain)  for Mn based MRFs with the \n3 element fractional fluid  model (a) storage modulus and  (b) loss modulus. [The symbols \nrepresent the experimental  observations and the theoretical predictions are indicated by solid \nlines]  \n \nAdditionally,  the relaxation modulus (G(t))  also can be  deduced  from eqn. 16  and can be \nexpressed as  [64] \n( / )( ) 1(1 )q\ncttG t Eq\n         (22) \nWhere, \nct  is the time constant  and follows the relation ship \n2 q\nctE\n            (23) \nSubstituti on of  eqn. 23 in the eqn. 22 leads to the simplified form of the relaxation modulus  as \n2()(1 )qtG t Eq\n\n           (24) \n Fig. 12 illustrates the comparison  of the estimated G(t) from the eqn. 24 and the experimental \ndata. The  responses of MnFe 2O4 and Fe 3O4 based MRFs can be distinguished from the  respective \n27 \n choices of E, \n2and q; and have been tabulated in tables ST5 and ST6 (Refer Supplementary). \nThis indicates that the viscous dashpot and the elastoviscous Voigt element are play similarly \nimportant roles on the viscoelastic responses  and temporal relaxation behaviors  under magnetic \nfield.  \n \nFigure 12. Comparison of experimental relaxation moduli (at 10% strain) with the estimations \nfrom eqn. 24 (a) Fe 3O4 based MRFs, (b) MnFe 2O4 based MRFs [The symbols represent the \nexperimental observations and the theoretical predictions are indicated by sol id lines]  \n \n4. Conclusions  \nThe present article discusses the magnetoviscous and magneto -elastoviscous behaviors of \nnanoscale MFe 2O4 ferrites based MRFs. The article is aimed at understanding the role of cation \nsubstitution of magnetic ferrites on the magnetic behavior of their MRFs.  MFe 2O4nanoparticles  \nhave been synthesized using polyol -assisted hydrothermal method. The detailed characterization \nconfirm s the presence of Mn, Co , Ni etc. in MFe 2O4 nanoparticles  as the doped atom. The \nnanoparticles have b een dispersed in silicone oil ( 40 wt . % concentration ) and ultrasonicated as \nrequired to obtain the corresponding MRFs . The steady shear rheometry reveal s that MnFe 2O4 \nbased MR Fs show the  yield stress among all  the other fluids . The order of yield stress \nimprovement under magnetic field is noted to a trend similar to the magnetic moments of the \nferrite nanoparticles. The MR Fs behave like  typical Bingham plastic under magnetic field. \n28 \n Comparison with an extended Bingham plastic model shows good agreem ent with the \nexperimental observations for all magnetic field  strengths . The magnetorheological  behaviors of \nall the MRFs are self -similar, and all the apparent viscosities under field influence conform to a \nmaster curve against the governing Mason numbers . The transient responses of the MRFs show \ndistinct magnetoviscous remanence  patterns. The hysteresis areas of  the MRFs comply to a \nproposed simple power law correlation .  \nImproved viscoelastic responses , in terms of loss factors , have been observed for the \ndifferent MRFs during magneto -viscoelasticity tests. While in the absence of magnetic field , the \nMRFs  are predominantly fluidic, presence of magnetic field induces elastic nature to the fluids . \nIn the viscoelastic studies also the role of the doped magnetic cation is prominent, and conforms \nto the nature of magnetization of the cation. Oscillatory s train tests reveal that the MRFs do not \npossess any distinct linear viscoelastic response, and hence the fluids are not typically elastic \nfluids  by definition despite possessing G’ higher than G”.  But the MRFs show  distinct crossover \nlocations  and these  have been identified for each fluid . The Co based MRFs have the highest \nresistance to viscous transition among the different MRFs. The magnetic stimul us aids the \nretaining of the elastic nature up to higher strains . Further,  the Mn based MRFs also possess the \nhighest stress relaxation caliber,  which is further enhance d in the presence of field and higher \nstrain s. The study shows that the  Mn ferrite based MRFs are the most suitable when the utility \nrequires  higher yield stress. However, when applications require the micro -structural integrity of \nthe MRF to be retained in dynamic conditions,  Co ferrite based MR Fs are more robust , due to \ntheir i mproved  magneto -viscoelastic behavior. It is thus shown that  introduction of a suitable \ndoping atom in the M2+ location of MFe 2O4 structure can definitely improve the \nmagnetorheological and magneto -elastoviscous performances of the corresponding MRFs \ncompared to simple Fe3O4 based MRFs . The findings may have important implications in \nefficient design and development of nano -MRFs for variant applications and utilities.  \n \n \n 29 \n Supplementary M aterial   \nThe supplementary material document contains additional information, data, tables and plots of \nthe complex fluid characterization, rheological behavior and additional data on the magneto -\nelastoviscous response of the fluids.  \nConflict of Interest  \nThe authors declare having no conflicts of interest with any individual or agency with respect to \nthis article.  \nAcknowledgements  \nPD thanks the Department of Mechanical Engineering, IIT Ropar for financial support towards \nthe present work. 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Samarasekara \nDepartment of Physics, University of Peradeniya, Pe radeniya, Sri Lanka. \nAbstract \n             The third order perturbed Heisenberg H amiltonian was employed to investigate \nthe spinel thick nickel ferrite films. The variatio n of energy up to N=10000 was studied. \nAt N=75, the energy required to rotate from easy to  hard direction is very small. For film \nwith N=10000, the first major maximum and minimum c an be observed at 202 0 and 317 0, \nrespectively. This curve shows some abrupt changes after introducing third order \nperturbation. The maximum energy of this curve is h igher than that of spinel thick ferrite \nfilms with second order perturbed Heisenberg Hamilt onian. At some values of stress \ninduced anisotropy, the maximum energy is less than  that of spinel thick ferrite films \nwith second order perturbed Heisenberg Hamiltonian derived by us previously.    \n1. Introduction: \n                           The effect of third orde r perturbation on the Heisenberg Hamiltonian of \nspinel ferrite thick films will be described in det ail in this report. Previously, the structure \nof spinel ferrites with the position of octahedral and tetrahedral sites is given in detail 1-5. \nOnly the occupied octahedral and tetrahedral sited were used for the calculation in this \nreport although there are many filled and vacant oc tahedral and tetrahedral sites in cubic \nspinel cell 1. Few previous reports could be found on the theore tical works of ferrites 6-8. \nThe solution of Heisenberg ferrites only with spin exchange interaction term has been \nfound earlier by means of the retarded Green functi on equations 6.     \n           All the relevant energy terms such as sp in exchange energy, dipole energy, second \nand fourth order anisotropy terms, interaction with  magnetic field and stress induced \nanisotropy in Heisenberg Hamiltonian were taken int o consideration. These equations \nderived here can be applied for spinel ferrites wit h unit cell AFe 2O4 such as Fe 3O4, \nNiFe 2O4 and ZnFe 2O4 only. The spin exchange interaction energy and dip ole interaction \nhave been calculated only between two nearest spin layers and within same spin plane. \nAlso the angle within one cubic cell is assumed to be constant. The change of angle at the \ninterface of cubic cell will be considered. Earlier  the second order perturbed Heisenberg \nHamiltonian of spinel ferrite thick films 9 and third order perturbed Heisenberg  2Hamiltonian of spinel ferrite thin films 10 have been studied. Heisenberg Hamiltonian was \nemployed to find the energy of spinel ferrite films  12, 17, 18 and energy of ferromagnetic \nfilms 13, 17, 11 .  According to our previous experimental studies, the stress induce \nanisotropy plays a major role in sputtered ferromag netic and ferrite thin films 14, 15, 16, 19 .     \n \n2. Model: \n \n           The Heisenberg Hamiltonian of a thin fil m can be written as following. \nH= -∑ ∑ ∑∑ − − − +\n≠ n m m mz\nmz\nm\nn m mn n mn mn m\nmn n m\nn m S D S D\nrSr rS\nrSSSSJ\nm m\n,4 ) 4 ( 2 ) 2 (\n5 3) ( ) ( )).)( .( 3 .( .λ λ ωrrrrrrrr\n  \n        ∑∑− −\nm mm s m Sin K SH θ2 .. rr\n              (1)  \n            Here J, ω, θ,  ,, ,, ,) 4 ( ) 2 (\ns out in m m K HH D D  m, n and N are  spin exchange \ninteraction, strength of long range dipole interact ion, azimuthal angle of spin, second and \nfourth order anisotropy constants, in plane and out  of plane applied magnetic fields, stress \ninduced anisotropy constant, spin plane indices and  total number of layers in film, \nrespectively. When the stress applies normal to the  film plane, the angle between m th  spin \nand the stress is θm.  \n            The cubic cell has been divided into 8 spin layers with alternative A and Fe spins \nlayers. The spins of A and Fe will be taken as 1 an d p, respectively. While the spins in \none layer point in one direction, spins in adjacent  layers point in opposite directions. A \nthin film with (001) spinel cubic cell orientation will be considered. The length of one \nside of unit cell will be taken as “a”. Within the cell the spins orient in one direction due \nto the super exchange interaction between spins (or  magnetic moments). Therefore the \nresults proven for oriented case in one of our earl y report will be used for following \nequations 13 . But the angle θ will vary from θm to θm+1  at the interface between two cells.   \nFor a thin film with thickness Na, \nSpin exchange interaction energy=E exchange = N(-10J+72Jp-22Jp 2)+8Jp ∑−\n=+−1\n11 ) cos( N\nmm m θ θ  \nDipole interaction energy=E dipole   3∑ ∑\n=−\n=+ + + + − + + −=N\nmN\nmm m m m m dipole p E\n11\n11 1 )] cos( 3) [cos( 41 . 20 )2cos 31 ( 415 . 48 θ θ θ θ ω θ ω  \nHere the first and second term in each above equati on represent the variation of energy \nwithin the cell and the interface of the cell, resp ectively. Then total energy is given by \nE= N(-10J+72Jp-22Jp 2)+8Jp ∑−\n=+−1\n11 ) cos( N\nmm m θ θ  \n      ∑ ∑\n=−\n=+ + + + − + + −N\nmN\nmm m m m m p\n11\n11 1 )] cos( 3) [cos( 41 . 20 )2cos 31 ( 415 . 48 θ θ θ θ ω θ ω  \n       ∑\n=+ −N\nmm m m m D D\n14 ) 4 ( 2 ) 2 (] cos cos [ θ θ  \n     ∑\n=+ + − −N\nmm s m out m in K H H p\n1]2sin cos sin [) 1 ( 4 θ θ θ                          (2)  \nHere the anisotropy energy term and the last term h ave been explained in our previous \nreport for oriented spinel ferrite. If the angle is  given by θm=θ+εm with perturbation εm, \nafter taking the terms up to third order perturbati on of ε, \nThe total energy can be given as E( θ)=E 0+E( ε)+E( ε2)+E( ε3) \nHere \nE0= -10JN+72pNJ-22Jp 2N+8Jp(N-1)-48. 415ωΝ -145. 245ωΝ cos(2 θ) \n      +20.41 ωp[(N-1)+3(N-1)cos(2 θ) ] \n     )2sin cos sin () 1 ( 4 cos cos ) 4 (\n1 14 ) 2 ( 2θ θ θ θ θs out in mN\nmN\nmm K H HNp D D + + − − − −∑ ∑\n= =  (3)   \n∑ ∑−\n= =+ − =1\n1 1) ()2sin( 23 . 61 )2sin( 5 .290 ) (N\nmn mN\nmm p E ε ε θ ω ε θ ω ε  \n         ∑ ∑\n= =+ +N\nmN\nmm m m m D D\n1 1) 4 ( 2 ) 2 (2sin cos 2 2sin ε θ θ ε θ  \n         ∑ ∑ ∑\n= = =− + − − +N\nmN\nmN\nmm s m out m in K H Hp\n1 1 1] 2cos 2 sin cos )[ 1 ( 4 εθ εθ εθ                          (4)  \n \n \n  4∑ ∑ ∑−\n= =−\n=− − + − −=1\n1 121\n12 2 2) ( 2 . 10 )2cos( 5 .290 ) ( 4 )(N\nmN\nmmN\nmn m m n p Jp E ε ε ω ε θ ω ε ε ε  \n           ∑−\n=+ −1\n12) ()2cos( 6 . 30 N\nmm n p ε ε θ ω      \n           ∑ ∑\n= =− + − −N\nmN\nmm m m m D D\n1 12 ) 4 ( 2 2 2 2 ) 2 ( 2 2) sin 3 (cos cos 2 ) cos (sin ε θ θ θ ε θ θ  \n          ∑ ∑\n= =+ − +N\nmN\nmmout \nmin H Hp\n1 12 2cos 2sin 2)[ 1 ( 4 εθ εθ ] 2sin 2\n12∑\n=+N\nmm sK εθ                      (5)  \n∑ ∑ ∑\n= = =− − + =N\nn mN\nmm mN\nmm n m D p E\n1 , 13 ) 2 (\n13 3 3sin cos 342sin 66 .193 ) ( 2sin 2 . 10 )( ε θ θ εθ ω ε ε θ ω ε  \n                ∑\n=− −N\nmm mD\n13 ) 4 ( 2 2) sin cos 35( sin cos 4 ε θ θ θ θ  \n                 ∑\n=− +N\nmmin Hp\n13cos 6)[ 1 ( 4 εθ ] 2cos 34sin 6 1 13 3∑ ∑\n= =+ −N\nmN\nmms\nmout K Hεθ εθ  \n \nThe sin and cosine terms in equation number 2 have been expanded to obtain above \nequations. Here n=m+1. \nUnder the constraint ∑\n==N\nmm\n10 ε , first and last three terms of equation 4 are zero .  \nTherefore, E( ε)= εαrr.  \nHere θ θ εα 2sin ) ( ) (Brr=  are the terms of matrices with \nθ ω θλ λ λ2 ) 4 ( ) 2 (cos 2 46 .122 ) ( D Dp B + + −=                                        (6)  \nAlso εε εrr. .21)(2C E = , and matrix C is assumed to be symmetric (C mn =C nm ). \nHere the elements of matrix C can be given as follo wing, \nCm, m+1 =8Jp+20.4 ωp-61.2p ωcos(2 θ) \nFor m=1 and N,  \nCmm = -8Jp-20.4 ωp-61.2p ωcos(2 θ)+581 ωcos(2 θ) ) cos (sin 22 2θ θ− −) 2 (\nmD  \n      ) sin 3 (cos cos 42 2 2θ θ θ − +) 4 (\nmD )] 2sin( 4 cos sin )[ 1 ( 4 θ θ θs out in K H Hp + + − +   (7)  \nFor m=2, 3, ----, N-1  5Cmm = -16Jp-40.8 ωp-122.4p ωcos(2 θ)+581 ωcos(2 θ) ) cos (sin 22 2θ θ− −) 2 (\nmD  \n      ) sin 3 (cos cos 42 2 2θ θ θ − +) 4 (\nmD )] 2sin( 4 cos sin )[ 1 ( 4 θ θ θs out in K H Hp + + − +    \nOtherwise, C mn =0 \nAlso ε βε εr. )(2 3= E  \nHere matrix elements of matrix β can be given as following. \nWhen m=1 and N, \n) 2 (sin cos 342sin 2 . 10 2sin 66 .193 m mm D p θ θ θ ω θ ω β − + −=                               \n               \nθ θ θ θ θ θ sin 6cos 6)[ 1 ( 4 ) sin cos 35( sin cos 4) 4 ( 2 2 out in \nmH Hp D − − + − − ]2cos 34θsK+  \n                                                                            \nWhen m=2, 3, ------, N-1 \n) 2 (sin cos 342sin 4 . 20 2sin 66 .193 m mm D p θ θ θ ω θ ω β − + −=  \n            \nθ θ θ θ θ θ sin 6cos 6)[ 1 ( 4 ) sin cos 35( sin cos 4) 4 ( 2 2 out in \nmH Hp D − − + − − ]2cos 34θsK+  \nθ ω β 2sin 6 . 30 1 , pm m=+                                        (8) \n \nOtherwise βnm =0. Also βnm =βmn  and matrix β is symmetric. \n                                          \nTherefore, the total magnetic energy given in equat ion 2 can be deduced to  \nE( θ)=E 0+εαrr. + ε βεεεrrr. . .212+C                       (9)    \nBecause the derivation of a final equation for ε with the third order of ε in above equation \nis tedious, only the second order of ε will be considered for following derivation. \nThen E( θ)=E 0+εαrr. + εεrr. .21C \nUsing a suitable constraint in above equation, it i s possible to show that α εrr.+−=C   6Here C+ is the pseudo-inverse given by \nNECC −=+1 . .                       (10)  \nE is the matrix with all elements given by E mn =1.  \nAfter using ε in equation 9, E( θ)=E 0 α αrr..21+− C - ) () (2α βα+ +C Cr\n                  (11)  \n3. Results and discussion: \n               When N is very large (Ex: N=10000), CC +=1, and C + is the standard inverse \nmatrix of C. When the difference between m and n is  one, C m, m+1 =8Jp+20.4 ωp-\n61.2p ωcos(2 θ). If H in , H out  and K s are very large, then C 11 >>C 12 . If this C m, m+1 =0, then \nthe matrix C becomes diagonal, and the elements of inverse matrix C + is given by \nmm mm CC1=+. Therefore all the derivation will be done under a bove assumption for the \nconvenience. \nThen )2sin( ] cos 2 46 .122 [2 ) 4 ( ) 2 (\n1 θ θ ω α αλ λ D Dpn + + −= =−− − − =  \n22 2\n12\n111 ) 2 ( 2 ..+ + +− + + = C N C C α α α α =\n22 2\n1\n11 2\n1 ) 2 ( 2\nCN\nC−+α α \nFor Nickel ferrite with p=2.5, \nE0= 52.5JN-20J-48. 415ωΝ -145. 245ωΝ cos(2 θ)+51.025 ω(N-1)[1+3cos(2 θ) ] \n     )] 2sin cos sin ( 6 cos [cos ) 4 ( 4 ) 2 ( 2θ θ θ θ θs out in m m K H H D D N + + − + −    \nC11 =C NN = -20J-51 ω+428 ωcos(2 θ) θ2cos 2+) 2 (\nmD ) sin 3 (cos cos 42 2 2θ θ θ − +) 4 (\nmD  \n      )] 2sin( 4 cos sin [ 6 θ θ θs out in K H H + + −    \nC22 =C 33 =------=C N-1,N-1 = -40J-102 ω+275ω cos(2 θ) ) 2(cos 2 θ +) 2 (\nmD   \n             ) sin 3 (cos cos 42 2 2θ θ θ − +) 4 (\nmD )] 2sin( 4 cos sin [ 6 θ θ θs out in K H H + + −     \n)2sin( ] cos 2 15 .306 [2 ) 4 ( ) 2 (\n1 θ θ ω αλ λ D D + + −=  \n) 2 (\n11 sin cos 342sin 16 .168 m NN Dθ θ θ ω β β − −= =                               \n               θ θ θ θ θ θ sin 6cos 6[ 6 ) sin cos 35( sin cos 4) 4 ( 2 2 out in \nmH HD − − − − ]2cos 34θsK+   7) 2 (\n22 sin cos 342sin 66 .142 mDθ θ θ ω β − − =  \n            θ θ θ θ θ θ sin 6cos 6[ 6 ) sin cos 35( sin cos 4) 4 ( 2 2 out in \nmH HD − − − − ]2cos 34θsK+  \nθ ω β 2sin 5 . 76 1 ,=+m m     \n(C +α)2β(C +α)= (C 11 +α1)2(β11 C11 +α1+β12 C22 +α2+-------+ β1N CNN +αN)  \n                          +(C 22 +α2)2(β21C11+α1+β22C22 +α2+-------+ β2NCNN +αN) \n                          +(C 33 +α3)2(β31C11 +α1+β32C22 +α2+-------+ β3NCNN +αN)+--------      \n                             ------+(C NN +αN)2(βN1C11 +α1+βN2C22 +α2+-------+ βNNCNN +αN) \n)] 2 (4) (2) (2[ ) () (22 12 3\n22 22 12 22 \n11 12 \n2\n22 22 12 \n11 11 \n2\n11 3 2β ββ β β β βα α βα +−+++ + + =+ +\nCN\nC C C C C CC C  \n                                           The tota l energy can be found from equation 11. When \n5 , 10 ) 4 ( ) 2 (\n= = = = = =ω ω ω ω ω ωm s out in m Dand K H H DJ, the 3-D plot of ωθ) (E versus θ and N \nis given in figure 1. Although the equation is vali d for large values of N only, the graph \nhas been drawn for even small values of N too in or der to study the variation of energy at \nsmall values of N too. The maximum energy of this t hick film is almost same as that of \nspinel thick ferrite films with second order pertur bed Heisenberg Hamiltonian 10 . Energy \nvariation of these two is also similar. Near N=75, the separation between maximum and \nminimum energies are very small implying that the e nergy required to rotate from easy to \nhard direction is small at this N value. The maximu m energy of this film is higher than \nthat of ferromagnetic thick films with third order perturbation 12 . Energy variation is \ndifferent from that of ferromagnetic thick films wi th third order perturbation.       \n \n \n  8\n \nFigure 1: 3-D plot of ωθ) (E versus θ and N for Nickel ferrite \n \n                       When N=10000, the graph betw een ωθ) (E and θ is given in figure 2. The \nmaximum energy of this curve is higher than that of  spinel thick ferrite films with second \norder perturbed Heisenberg Hamiltonian 10 . Due to sudden changes, this energy curve is \nless smoother compared with that of spinel thick fe rrite films with second order perturbed \nHeisenberg Hamiltonian. The first major maximum and  minimum can be observed at \n202 0 and 317 0, respectively. Angle between easy and hard directi ons is not 90 0 in this \ncase. Some sudden changes could be observed in ferr omagnetic thick ferrite films with \nthird order perturbation 12 . The maximum energy is almost same as that of ferr omagnetic \nthick ferrite films with third order perturbation. Positions of easy and hard directions and \nseparation between easy and hard directions are dif ferent from those of ferromagnetic \nthick ferrite films with third order perturbation a nd spinel thick ferrite films with second \norder perturbation.    9\n \nFigure 2: Graph between ωθ) (E and θ for N=10000 \n \n                      When N=10000 and ωsK is a variable, the 3-D plot of ωθ) (E versus θ and \nωsK is given in figure 3. The maximum energy of this c urve is less than that of spinel \nthick ferrite films with second order perturbed Hei senberg Hamiltonian 10 . The variation \nof energy is also different from that of spinel thi ck ferrite films with second order \nperturbed Heisenberg Hamiltonian.      10 \n \nFigure 3: 3-D plot of ωθ) (E versus θ and ωsK for N=10000  \n \n4. Conclusion: \n            The variation of energy with angle, thi ckness and stress was studied up to \nN=10000. Near N=75, the energy required to rotate f rom easy to hard direction is very \nsmall implying that the anisotropy energy is small at this value N. For film with \nN=10000, the major maximum and minimum can be obser ved at 202 0 and 317 0, \nrespectively. Introducing third order perturbation destroys the smoothness of energy \ncurve. The maximum energy of this curve is higher t han that of spinel thick ferrite films \nwith second order perturbed Heisenberg Hamiltonian.  In 3-D plot of ωθ) (E versus θ and \nωsK, some energy minimums can be observed indicating t hat the film can be oriented in \nsome particular directions by applying certain stre sses. \n  11 References: \n1. Kurt E. Sickafus, John M. Wills and Norman W. Gr imes, 1999. Structure of spinel.  \n    Journal of the American Ceramic Society 82(12),  3279-3292. \n2. I.S. Ahmed Farag, M.A. Ahmed, S.M. Hammad and A. M. Moustafa, 2001. Study of  \n    cation distribution in Cu 0.7 (Zn 0.3-xMg x)Fe 1.7  Al 0.3 O4 by X-ray diffraction using Rietveld  \n    method. Egyptian Journal of Solids 24(2), 215-2 26. \n3. V. Kahlenberg, C.S.J. Shaw and J.B. Parise, 2001 . Crystal structure analysis of   \n    synthesis Ca 4Fe 1.5 Al 17.67 O32 : A high pressure, spinel related phase. American   \n    Mineralogist 86, 1477-1482. \n4. I.S. Ahmed Farag, M.A. Ahmed, S.M. Hammad and A. M. Moustafa, 2001.    \n    Application of Rietveld method to the structura l characteristics of substituted copper   \n    ferrite compounds. Crystal Research and Technol ogy 36(1), 85-92. \n5. Z. John Zhang, Zhong L. Wang, Bryan C. Chakoumak os and Jin S. Yin, 1998.  \n   Temperature dependence of cation distribution an d oxidation state in magnetic Mn-Fe  \n   ferrite nanocrystals. Journal of the American Ch emical society 120(8), 1800-1804. \n6. Ze-Nong Ding, D.L. Lin and Libin Lin, 1993. Surf ace magnetism in a thin film of   \n    Heisenberg ferrimagnets. Chinese Journal of Phy sics 31(3), 431-440. \n7. D. H. Hung, I. Harada and O. Nagai, 1975. Theory  of surface spin-waves in a semi- \n     infinite ferrimagnet. Physics Letters A  53(2), 157-158. \n8. H. Zheng and D.L. Lin, 1988. Surface spin wave o f semi-infinite two-sublattice   \n    ferrimagnets. Physical Review B37, 9615-9624. \n9. P. Samarasekara, 2010. Determination of Energy o f thick spinel ferrite films using    \n    Heisenberg Hamiltonian with second order pertur bation. Georgian electronic  \n    scientific journals: Physics 1(3), 46-49. \n10. P. Samarasekara and William A. Mendoza, 2011. T hird order perturbed Heisenberg  \n      Hamiltonian of spinel ferrite ultra-thin film s. Georgian electronic scientific   \n      journals: Physics 1(5), 15-18. \n 11. P. Samarasekara, 2008. Influence of third orde r perturbation on Heisenberg    \n       Hamiltonian of thick ferromagnetic films. El ectronic Journal of Theoretical Physics  \n       5(17), 231-240. \n  12 12. P. Samarasekara, 2007. Classical Heisenberg Ham iltonian solution of oriented spinel   \n      ferrimagnetic thin films. Electronic Journal of Theoretical Physics 4(15), 187-200. \n13. P. Samarasekara and S.N.P. De Silva, 2007. Heis enberg Hamiltonian solution of thick    \n      ferromagnetic films with second order   pertu rbation. Chinese Journal of Physics  \n      45(2-I), 142-150. \n14. P. Samarasekara and F.J. Cadieu, 2001. Magnetic  and Structural Properties of RF  \n      Sputtered Polycrystalline Lithium Mixed Ferri magnetic Films. Chinese Journal of   \n      Physics 39(6), 635-640. \n15. H. Hegde, P. Samarasekara and F.J. Cadieu, 1994 . Nonepitaxial sputter synthesis of   \n      aligned strontium hexaferrites, SrO.6(Fe 2O3), films. Journal of Applied Physics    \n      75(10), 6640-6642. \n16. P. Samarasekara, 2003. A pulsed rf sputtering m ethod for obtaining higher deposition  \n      rates. Chinese Journal of Physics 41(1), 70-7 4. \n17. P. Samarasekara and William A. Mendoza, 2010. E ffect of third order perturbation on   \n      Heisenberg Hamiltonian for non-oriented ultra -thin ferromagnetic films. Electronic  \n      Journal of Theoretical Physics 7(24), 197-210 .  \n18. P. Samarasekara, M.K. Abeyratne and S. Dehipawa lage, 2009. Heisenberg   \n      Hamiltonian with Second Order Perturbation fo r Spinel Ferrite Thin Films. Electronic   \n      Journal of Theoretical Physics 6(20), 345-356 . \n19. P. Samarasekara and Udara Saparamadu, 2013. Eas y axis orientation of Barium hexa-  \n      ferrite films as explained by spin reorientat ion. Georgian electronic scientific   \n      journals: Physics 1(9), 10-15. \n \n "    },    {        "title": "1202.1040v1.High_temperature_spin_wave_propagation_in_BiFeO3__relation_to_the_Polomska_transition.pdf",        "content": "1 \n  \nHigh-temperature spin-wave propagation in BiFeO 3: relation to the \nPolomska transition \n \nAshok Kumar 1, J. F. Scott 1,2, R. S. Katiyar 1 \n \n1Department of Physics and Institute for Functional Nanomaterials, University of Puerto \nRico,San Juan, Puerto Rico, USA, PR-00936-8377  \n2Department of Physics, Cavendish Laboratory, Cambri dge University, Cambridge CB3 \nOHE, United Kingdom \nAbstract:  \nIn bismuth ferrite thin films the cycloidal spiral spin structure is suppressed, and as a \nresult the spin-wave magnon branches of long wavele ngth are reduced from a dozen to \none, at ω = 19.2 cm -1 (T=4K).  This spin wave has not been measured prev iously above \nroom temperature, but in the present work we show v ia Raman spectroscopy that it is an \nunderdamped propagating wave until 455 K.  This has  important room-temperature \ndevice implications.  The data show that ω(T) follows an S=5/2 Brillouin function and \nhence its Fe +3  ions are in the high-spin 5/2 state and not the lo w-spin S=1/2 state.  The \nspin wave cannot be measured as a propagating wave above 455 K. We also suggest that \nsince this temperature is coincident with the myste rious “Polomska transition” (M. \nPolomska et al., Phys. Stat. Sol. A  23, 567, (1974)) at 458+/-5 K, that this may be du e to \noverdamping. \nCorresponding Author:  E-mail: *J F Scott jfs32@hermes.cam.ac.uk , R S katiyar, \nrkatiyar@uprrp.edu  2 \n The production of high-quality thin films of bismut h ferrite has opened up the \npossibility of making room-temperature devices that  rely upon spin wave propagation.  In \n2008 the study of spin waves via magnon Raman spect roscopy began [1-5], and more \nrecently a review was published on its device appli cations [6].  Foremost among the \npotential device applications are THz emitters [7,8 ,9], and for the majority of devices \n(including electric-field tunable magnetic devices)  the existence of room-temperature \nspin waves (magnons)  that are propagating modes (a s opposed to overdamped or \ndiffusive modes) is important [10,11,12].  Here we report propagating spin waves in \nBiFeO 3 up to 450 K. Although in bulk there are numerous m agnon branches at long \nwavelength that scatter light in the Raman effect ( due to the cycloidal spin structure), in \nthin films this cycloid is suppressed and there is only a single magnon branch in the \nRaman spectra. Loudon and Fleury [13,14] provided t he basic theory of magnon \nspectroscopy and damping. In general, systems with Fe or Co ions can exhibit \nunquenched orbital angular momentum; but despite th e fact that they therefore are not \npure spin systems, their temperature dependence and  damping are not very different from \nthose in pure spin systems such as Mn compounds [15 ]. Low-energy spin waves are \nobserved in several ferro/antiferromagnetic systems  and well explained with a \nphenomenological theory [16,17,18].    \nHere we report the temperature dependent softening of spin waves, their \nsuppression in thin-film form, correlation with a m agnetic transition temperature, and \ndielectric loss. Softening of spin waves follows a modified Brillouin function with S=5/2 \n(which is mean field with critical susceptibility e xponent β=1/2).  3 \n  Pre-patterned platinum interdigital electrodes wer e procured from NASA Glenn \nResearch Center’s electronics division having dimen sion 1900 µm (length), 15 µm \n(interdigital spacing) and 150 +/- 25 nm (height)  with  45 parallel capacitors in series on \nsapphire substrates.   Fig. 1 (a, b, c) shows the c artoon of spin waves, their suppression to \na spin arrangement similar to that in conventional two-sublattice antiferromagnetic \nsystems, and the in-plane view of the BFO thin film s on the pre-patterned interdigital \nelectrodes. The details of crystal phase formations , surface morphology, impurities and \nelectrical testing equipments and parameters were p resented in previous report [1,5]. \nBFO thin films were grown utilizing an excimer lase r (KrF, 248 nm) with a laser energy \ndensity of 1.5 J/cm 2, laser rep-rate of 10 Hz, substrate temperature 65 0°C and oxygen \npressure at 80 mTorr. Micro-Raman spectra were reco rded in the backscattering \ngeometry using 514.5 nm monochromatic radiations ov er wide range of temperature \nutilizing low temperature cryostage from Linkam.  \n Figure 2 demonstrates the variation of low-energy spin waves at various \ntemperatures. Only one spin wave was observed in th e low-frequency Raman spectra \nwhich is sharp and can be clearly seen beyond the b aseline noise background. These low-\nenergy spin waves are underdamped until 450 K with a sharp magnon peak. Spin waves \nshift towards lower frequencies with increase in te mperature; our due to stray light and \nspectrometer characteristics, our experimental limi tation was to observe the Raman \nspectra only above 10 cm -1.  \n Fig. 3 shows the scaling behaviour magnon frequenc ies ω(T) versus temperature in one \nof our films.  Note that the frequency data ω(T) satisfy a mean-field S=5/2 high-spin \nBrillouin function better than a low-spin S=1/2 Bri llouin function.  It is already known 4 \n from a series of elegant papers by Gavriliuk et al.  in Moscow [19-22] that the Fe +3  ions in \nthe insulating state of bismuth ferrite have S=5/2,  whereas the metallic state has S=1/2, so \nthis only confirms previously established conclusio ns. However, this is the first time such \nconclusions can be inferred purely from Raman magno n spectroscopy in any material, so \nthat it provides a nice pedagogical example. \nBFO thin films show only single spin wave compare t o the electromagnons (several spin \nexcitations) in single crystal, compel us to explai n the magnon frequency to a single \nquantum excitation, a molecular field theory can pr ovide an interpretation based on the \nmacroscopic parameters, H A and H E =| λM1|=| λM2|, where H A and H E indicates anisotropy \nand exchange field[16]. \nThe solution of Bloch equations provide the low lyi ng magnon frequency \n  ...............................................(1) \n \n \nWhere γ is the gyromagnetic ratio of Fe 3+  ions, m is electronic mass, g-factor, c is speed \nof light, e is the electronic charge. \nIn the framework of molecular field theory the net magnetization is the superposition of \ntwo interpenetrating magnetization M 1 and M 2 preferentially parallel and antiparallel to \nHA. It is assumed that the H A and H E is the proportional to the saturation values of M 1and \nM2 given by; \n .................................................. ........................(2) \nwhere µB is the Bohr magneton, a is the lattice constant, kB is Boltzmann constant, and \nBS(y) the Brillouin function. βγ ω ) 2 (2\nE AAm HH H + =\n2 , 2 / 5 ,2 / = = = g Smc ge γ\n)()/ 2 (3yBaSg Ms B s µ =5 \n  ( )− + +=Syctnh S Sy Sctnh SSBJ21\n21 2\n21 2          \nfor y= 1<<< TkBBµ 45 31coth 3yy\nyy −+=  \n \n .................................................. .................(3) \n                 \n                                                           ........................................... .........................(4) \n \nSolving equation eq. 2 by assuming that ωM is proportional to the M s and equal to 19.2 \ncm-1 at T =0 K and T N~ 631.5 K (power law fitting of experimental data).  We have \nutilized the equation 3 and 4 to solve the equation  2 and Brillouin function.  This \ntheoretical model has been used to see the softenin g of low lying magnon for most of the \nantiferomagnetic systems. Our experimental data fit ted well with the theory very near to \nspin S=5/2. Darby has obtained the numerical values  and the equations for the \nspontaneous magnetization and their softening behav iour of for magnetic system with \ndifferent spin behaviour [17]. \nNear the Neel temperature ) 1 ( 1 / << → y TTN  Brillouin function and scaled \nmagnetization will follow the equation  \n.... 45 21) 1 2 (\n3 )2 (1) 1 2 ()(3\n44\n22\n+− +−− +=y\nSS y\nSSyBS ................................................... .............(5) \n... 1\n) 1 () 1 (\n310 \n2 222\n0+\n\n\n−\n+ ++=\n\n\n\nNTT\nS SS\nMM................................................... .......................(6) S\nBB\nE A\nBBMTSgH HTSgy ) ( ) ( µλκµ\nκµ+ ≡ + =\n) )( 1 ( )3 / 2 (2 2 3µλ µ + + = SS ga TkB NB6 \n whereas near the absolute 0 K temperature ) 1 ( 0 / >> → y TTN  Brillouin function and \nscaled magnetization can be obtained from the equat ion 7 and 8. \n.... exp 11)( +\n\n− −=Sy\nSyBS ................................................... ........................................(7) \n..... 13exp 11\n0+\n\n\n+− −=\n\n\n\nTT\nS S MMN................................................... ............................(8) \nEquation 6 and 8 together offer the different value s of magnetization and Brillouin \nfunction near low temperature (absolute zero Kelvin ) and Neel point respectively. \nUtilizing both equation one can sketch the complete  range of magnetization verses \ntemperature scaling behaviour. \nTheoretical modelling of magnetization for differen t spin behaviours can be obtained \nfrom equation 6 and 8. \n It is worth mentioning that magnon frequency is di rectly proportional to magnetization \nwith a factor                          that also in volves the spin factor.  \n  \n2 , 2 / 5= = g S for high spin state \nBut for the low spin the value of 1.5<g <2, we cons ider g=1.5 in our fitting parameters  \n) 1 (2) 1 ( ) 1 (\n23\n++ −++=JJLL SSgJ  \nWe have utilized the low temperature spin-wave theo ry since experimental data only \nreached up to the 70% of the Neel temperature. The scaling behaviour well follow the \nS=5/2 for both low temperature spin wave theory and  modified molecular field theory \nnear the Neel temperature. S=1/2 is very unlikely t o fit the experimental data in both βγ ω Mm= mc ge 2 / =γmc ge 2 / =γ7 \n cases. It is worth mentioning that our spin wave so ftening matches very well with the \nrecent observation of the magnon softening probed b y terahertz spectroscopic studies [9].  \nThe magnon damping is of special interest (Fig.3):  As noted above, it is modest \nuntil ca. 150 K above ambient. We find that it is n o longer measurable as a propagating \nwave above 450+/-10 K.  It is not unusual for magno ns to become overdamped at \ntemperatures of order 70-80% T(Neel), but in the ca se of bismuth ferrite, this might not \nbe entirely coincidental, since a yet enigmatic pha se transition has been reported [23,24] \nin some specimens at 458+/-5 K.  If this is indeed the temperatures of a subtle structural \nphase transformation (surface [25] or bulk), then o ne might expect anomalous damping at \nnearby temperatures. In order to test this coincide nce, we present in the final section a \ncomplementary study of in-plane dielectric loss, wh ich confirms the strong connection to \nthe Polomska transition. \nTemperature dependent line widths of the magnons ar e presented in the Fig. 4. These \nshow an anomalous decrease in the spin linewidth ne ar the so-called spin reorientation \ntransition (210 K) but with non-monotonic behaviour , increasing at higher temperatures. \nThe detailed phenomena near the presumed magnetric transition at ca. 210K are \ndiscussed in earlier reports [1-5]. It is interesti ng that the in-plane loss tangent of BFO is \nvery small up to 450 K (only 1-2% dielectric loss w as observed as can be seen from \nFig.5).  A reproducible broad loss peak (“hump”) wa s observed near the 210K magnetic \ntransition for wide range of frequency. The detaile d in-plane dielectric properties of BFO \nwill be presented elsewhere. It is not a coincidenc e that the dielectric loss also starts \nincreasing above 450 K where magnon disappears as a  propagating mode. 8 \n In summary, we report for the first time high tempe rature spin wave propagation and \ntemperature dependence in BFO thin films. The spin waves remain underdamped until \n450 K with a sharp magnon peak. Propagating spin wa ves were not observed at elevated \ntemperatures.  The magnon frequency follows an S=5/ 2 Brillouin function and hence its \nFe +3  ions are in the high-spin 5/2 state. The linewidth  demonstrates anomalous decrease \nnear the magnetic transition temperature but increa ses strongly at higher temperatures. A \ndielectric peak loss broad in temperature was obser ved near the onset of the 210 K \nmagnetic transition over wide temperature range, an d not coincidentally, it starts \nincreasing rapidly above the 450 K Polomska transit ion, at which the spin waves no \nlonger are observed as propagating waves.  \n \n \nAcknowledgement:  \nThis work was supported in parts by DOE-DE-FG02-08E R46526 and IFN-NSF-\nRII 07-01-25grants. \n \n \n \n \n \n \n \n 9 \n  \n References: \n[1] A. Kumar, N.M. Murari, and R.S. Katiyar, Applie d. Physics, Letters, 92, 152907, \n(2008). \n[2] M. Cazayous, Y. Gallais, A. Sacuto, T. De Sousa , D. Lebeugle, and D. Colson, Phys. \nRev. Lett. 101, 037601Y (2008).  \n[3] P. Rovillain, M. Cazayous, Y. Gallais, A. Sacut o,  R. P. S. M. Lobo, D. Lebeugle, D. \nColson, Phys. Rev. B 79, 180411 (R) (2009). \n[4] M. K. Singh, R. S. Katiyar, J. F. Scott, J. Phy s. Cond. Mat. 20, 252203 (2008).  \n[5] A. Kumar, J. F. Scott, R. S. Katiyar, Applied P hysics letters, 99, 062504 (2011) \n[6] G. Catalan and J. F. Scott, Adv. Mater. 21, 246 3-2485 (2009). \n[7] M. Tonouchi, 35th International Conference on I nfrared, Millimeter and Terahertz \nWaves (IEEE, Rome, 2010). \n[8] D. S. Rana, L. Kawayama, K. Takahashi, M. Tonou chi, EPL 84, 67016 (2008). \n[9] D. Talbayev, S. A. Trugman, Seongsu Lee, Hee Ta ek Yi, S.-W. Cheong, A. J. Taylor, \nPhysical. Rev. B, 83, 094403 (2011). \n[10] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, B. Hillebrands, Appl. Phys \n.Lett. 87, 153501, (2005). \n[11] V. V. Kruglyak, S. O. Demokritov &  D. Grundle r, J. Phys. D, 43, 264001 (2010). \n[12] A. Khitun, M.  Bao, K. L. Wang, J. Phys. D 43,  264005, (2010). \n[13] R. Loudon, Adv. Phys. 17, 243 (1968). \n[14] P. A. Fleury and R. Loudon, Phys. Rev. 166, 51 4 (1968). \n[15] J. Nouet, D. J. Toms, J. F. Scott, Phys. Rev. B 7, 4874-4883 (1973). 10 \n  [16] S. Venugopalan, A. Petrou, R. R. Galazka, A. K. Ramdas, S Rodriguez, Physical \nreview B, 25, 2681 (1982). \n[17] M. I. DARBY, Br. J. Appl. Phys. 18 1415, (1967 ). \n[18] T. P. Devereaux, R. Hackl, Reviews of Modern P hysics, 79, 175, ( 2007) \n[19] A. G. Gavriliuk, V. V. Struzhkin, L. S. Lyubut in et al., JETP Lett. 86, 197-201 \n(2007). \n[20] A. G. Gavriliuk, L. S. Lyubutin, and V. V. Str uzhkin, JETP Lett. 86, 532-536 \n(2007).  \n[21] L. S. Lyubutin, A. G.  Gavriliuk, and V. V.  S truzhkin, JETP Lett. 88 , 524-530 \n(2008). \n [22] A. G. Gavriliuk, V. Struzhkin, I. S.  Lyubuti n, I. A. Trojan, M. Y. Hu, P. Chow,  \nPhase transitions in multiferroic BiFeO 3, “Materials Research at High Pressure,” [Mat. \nRes. Soc. Symp. Proc. 987, 147-152 (2007); eds. M. R.   Manaa, A. F.  Goncharov, R. J.  \nHemley, R. Bini; (MRS: Boston, MA)]. \n[23] M. Polomska, W.  Kaczmare, Z.  Pajak,  Phys. S tat. Sol. A 23, 567-574 (1974). \n[24] P. Fischer, M.  Polomska, I.  Sosnowska et al. , J. Phys. C: Sol. St. Phys. 13, 1931-\n1940 (1980). \n[25] X. Marti, P. Ferrer, J. Herrero-Albillos; G. C atalan, Phys. Rev. Lett. 106, 236101 \n(2011). \n \n \n \n \n 11 \n  \nFigure captions: \n \nFig.1 Schematic diagram and cartoon of the spin wav es, spin wave generation by anti \nferromagnetic materials, and the in-plane view of t he BFO thin films. (a) Spin wave \npropagation in the BFO single crystal, (b) suppress ion of spin waves with several \nfrequencies to a single frequency and the behavior is similar to the magnon propagation \nin antiferromagnetic materials,  an incident photon interact with an electron having  spin s \nand force it to hop with the neighboring site leavi ng a hole and creating a double \noccupancy in the intermediate state with energy (E) . Particle of the double state with \nopposite spin hops back to the hole site liberating  a photon with different energy leaving \nbehind a locally disturbed antiferromagnet. (c) sch ematic diagram of the in plane view of \nthe BFO thin films grown on the pre patterned inter digited electrode. \n \n(2) Low lying spin waves probed by Raman spectrosco py at various temperatures (80 K \nto 450 K), Magnon frequency shifted towards lower f requency side with increase in \ntemperature. \n \n(3)Magnon frequency versus temperature in bismuth f errite thin films. Experimental \nvalues of spin wave frequency are presented by soli d dots with the experimental error \nbars line. The curves are least squares fits to S=1 /2 (low-spin) (top solid line) and S=5/2 \n(high-spin) for the magnon frequency for the Fe +3  ions utilizing low temperature spin \ntheory (equation 1 & 8). The theoretical fitting of  the complete scaling behavior of the 12 \n experimental data is shown by the last curve (green  solid line), that fitted well utilizing \nmodified mean molecular field model. Note that magn on frequency theoretically differs \nsomewhat from magnetization M(T). \n(4) Temperature dependent linewidths of magnon freq uency showing decrease in \nlinewidth near the 210K magnetic transition tempera ture and higher temperature increase. \n (5) In-plane loss tangent (at 10 kHz) of BFO thin films showing broad maximum near \nthe onset of magnetic phase transition temperature and sharp increase above the \npolomska transition near 458K.  \n \n \n 13 \n  \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \nFig.1 \n \n \n \n \n \n (a) \n(c) (b) 14 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.2 \n \n  \n \n \n \n \n 10 20 30 40 423 K \n373 K \n323 K \n273 K \n223 K \n173 K \n123 K \n80 K \nRaman Shift (cm -1 )Intensity (arb. units.) 15 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.3 \n \n \n \n \n \n 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 \n  \n exp. data \n S=1/2, \n S=5/2 \n S=5/2 and β β β β =1/2 ωωωω/ωωωωmax \nT/T N16 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.4 \n \n \n 100 200 300 400 500 600 04812 16 FWHM(cm -1 )\nTemperature (K) 17 \n  \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.5 \n \n \n \n 100 200 300 400 500 600 700 0.01 0.1  10 KHz Tangent loss( δδδδ)\nTemperature(K) "    },    {        "title": "1204.0177v1.Topological_phase_effects_and_path_dependent_interference_in_microwave_structures_with_magnetic_dipolar_mode_ferrite_particles.pdf",        "content": "Topological -phase effects and path -dependent interference in microwave \nstructures with magnetic -dipolar -mode ferrite particles  \n \n \nM. Berezin, E.O. Kamenetskii,  and R. Shavit  \n \nMicrowave Magnetic Laboratory  \nDepartment of Electrical and Computer Engineering,  \nBen Gurion University of the Negev, Beer Sheva, Israel  \n \nMarch 26, 2012  \n                                                                          \nAbstract  \n \nDifferent ways exist in optics to realize photons carrying nonzero orbital  angular momentum . \nSuch photons with rotating wave fronts  are called twisted photons.  In microwaves, twisted fields \ncan be produced based on small ferrite particles with magnetic -dipolar -mode (MDM) \noscillations. Recent studies showed  strong localization of  the electric and magnetic energies of \nmicrowave fields by MDM  ferrite  disks. For electromagnetic waves  irradiating MDM disks, \nthese small ferrite samples appear as singular subwavelength regions with time and space \nsymmetry breakings. The fields scattered  by a MDM disk are characterized by topologically \ndistinctive power -flow vortices  and helicity structures. In this paper we analyze twisted states of \nmicrowave fields scattered by MDM ferrite disks. We show that in a structure of the fields \nscattered by MD M particles, one can clearly distinguish rotating topological -phase dislocations. \nSpecific long -distance topological properties of the fields are exhibited clearly in the effects of \npath-dependent interference with two coupled MDM particles. Such double -twisted scattering is \ncharacterized by topologically originated split -resonance states. Our studies of t opological -phase \neffects and path -dependent interference in microwave structures with MDM ferrite  particles  are \nbased on numerical analysis and recently d eveloped analytical models. We present preliminary \nexperimental results aimed to support basic statements of our studies.  \n \nPACS number(s): 41.20.Jb; 76.50.+g ; 84.40. -x; 81.05.Xj  \n \nI. INTRODUCTION  \n \nPresent interest in electromagnetic fields  with rotating w ave fronts – twisted photons – stimulates \ninvestigations of such field structures in different frequency domains , from optical [1, 2] to \nradiowave frequencies [3 , 4]. Recently, it has been shown that in microwaves,  twisted fields can \nbe produced based on sm all ferrite particles with magnetic -dipolar -mode (MDM) oscillations.  \nThe spatial distribution of the fields originated from a MDM particle is nonhomogeneous in the \nsense that the equal phase fronts are helices. In creation of these  rotating -front microwave  fields , \na main role plays topological -phase effects  [5 – 7].  \n    The phase, known as the Berry phase, or the geometric phase, has found applications in a wide \nrange of disciplines throughout physics . Geometric phases may show up whenever the system  \nunder  study depends on several parameters and is transported  around a closed path in parameters  \nspace [ 8]. While the derivation of the Berry phase makes use of the transport of states in Hilbert \nspace, there are many classical derivations for geometric phases. Without having to appeal to \nquantum mechanics, one can consider the topological phases in such classical effects as, for \nexample, noninertial motion due to parallel transport of coordinates [9] and polarization rotation \nin helicoidal fibers [10 – 13]. Due to the Berry phase one can observe a spin -orbit interaction in  2 optics. It was shown that a spin -orbit interaction of photons results in fine splitting of levels in a \nring dielectric resonator, similarly to that of electron levels in an atom [14]. Since t he Berry \nphase  is purely geometric in origin, it does not depend on dynamic considerations. It is also \nindependent on the properties of the circuit in ordinary space.   \n    Very specific properties of geometrical phases become evident in a case of rotational  effects in \nmagnetic samples. In Ref. [15] it was shown that s ample rotation induce s frequency splittings in \nnuclear -quadrupole -resonance spectra. The observed  splittings can be interpreted in two ways: \n(a) as a manifestation of Berry's phase, associated w ith an adiabatically changing Hamiltonian, \nand (b) as a result of a fictitious magnetic field, associated with a rotating -frame transformation . \nThe main standpoint is that the dynamical phase evolution in magnetic structure is unaffected by \nmagnetic rotati onal motion and any rotational effects must arise from Berry's phase. Related \neffects were  predicted in other magnetic  resonance experiments that involve sample rotation.  \nAmong these experiments, it is worth noting ferromagnetic -resonance (FMR) rotating ma gnetic \nsamples. Frequencies of the FMR are typically in the gigahertz range, which is far above \nachievable angular velocities of mechanical rotation of macroscopic magnets. In Refs. [16, 17] it \nwas discussed that a rotating ferromagnetic nanoparticle (havi ng a very small moment of inertia ) \ncan be a resonant receiver  of the electromagnetic waves at th e frequency of the FMR . If such a  \nreceiver is rotating mechanically  at an angular velocity \n , one has splittings of the FMR \nfrequency.  The frequency of the  wave perceived by the receiver equals  \n \n                                                              \n()n\nFMR  ,                                                             (1)  \n \nwhere \nFMR  is frequency of the ferromagnetic resonance, \nn\nI\n , I is the moment of inertia of a \nnanoparticle, and \n0, 1, 2, 3,...n     The experimen tal results for the rotational motion of solid \nferromagnetic nano particles confine d inside polym eric cavities give evidence for the \nquantization  [16].  \n    There exists, however, an evidence for the FMR splittings in macroscopic magnets without \nmechanical rotation of samples. The multiresonance splittings in the FMR spectra were observed \nexperimental ly in non -rotating macroscopic quasi -2D ferrite disks with magnetostatic -wave \n(MS-wave) [or magnetic -dipolar -mode (MDM)] oscillations [18 – 20]. In these microwave \nexperiments, the MDM ferrite disk behaves as a multi resonant receiver  for the electromagneti c \nwaves  irradiating this sample. Analytical and numerical studies of this effect gave a deep insight \ninto an explanation of the experimental absorption spectra [5 – 7, 21 – 25]. It was shown that \nvery effective multiresonance interactions of a ferrite disk  with external electromagnetic fields \nare the result of energy eigenstates and specific topological properties of MDM oscillations. For \nelectromagnetic waves  irradiating a MDM ferrite disk, this small sample appears as a topological \nsingularity characteriz ed by strong 3D locali zation of electromagnetic energy with the power -\nflow-density vortices. Evidently, in such ferrite particles one has the Berry -phase generation not \nby the sample rotation, but due to topological characteristics of the fields of MDM osc illations.  \n    When we discuss unique properties of the MDM oscillations in quasi -2D ferrite disks, an \nimportant question also arises:  Could the observed effect of microwave energy localization due \nto small MDM ferrite particles (or, in other words, parti cles with the MS -magnon oscillations) be \nconsidered, somehow, as a dual effect of the energy localization in optics due to metallic \nparticles with the electrostatic -plasmon oscillations? In a case of subwavelength metallic -sphere \nparticles with plasmon osc illations, to answer the question: how can a particle absorb more than \nthe light incident on it [26], one uses the classical Mie theory [27]. A small plasmon -oscillation \nparticle localizes the electric energy of the electromagnetic wave. The scattered fiel ds correspond \nto the fields scattered by a small electric dipole and the problem is one of solving the Laplace  3 equation with the proper boundary conditions at a sphere surface [28]. An array of such particles \ncan be treated as a chain of point electric dip oles [29, 30]. For small subwavelength ferrite -disk \nparticles with MS -magnon oscillations, situation is completely different. In this case, one has \nstrong localization of both the electric and magnetic energies in a subwavelength region of the \nelectromagne tic wave. Moreover, in this subwavelength region, there is mutual coupling between \nthe electric and magnetic fields resulting in appearance of the power -flow-density vortices [5, 7]. \nIf one supposes that he has created a particle with a local (subwavelengt h-region) coupling \nbetween the electric and magnetic energies – a magnetoelectric (ME) particle – one, certainly, \nshould demonstrate a special topological structure of the fields near this particle. Would the \nfields scattered by ME particles correspond to the fields scattered by a local combination of a \nsmall electric dipole and a small magnetic dipole? A positive answer to this question  means that \nusing a gedankenexperiment with two quasistatic, electric and magnet ic, point probes for the \nnear-field charac terization, one should observe not only the electrostatic -potential distribution \n(because of the electric polarization) and not only the magnetostatic -potential distribution \n(because of the magnetic polarization), one also should observe a special cross -potential term \n(because of the cross -polarization effect). This fact contradicts to classical electrodynamics. \nWhat will be an expression for the Lorentz force acting between these particles? This expression \nshould contain an \"electric term\", a \"magnetic ter m\", and a special \"magnetoelectric  term\". Such \nan expression is unknown in classical electrodynamics. One cannot consider (in classical \nelectrodynamics) a system of two coupled electric and magnetic dipoles as local sources of the \nfields and there are no t wo coupled Laplace equations (for the magnetostatic and electrostatic \npotentials) in the near -field region  [28].  In a presupposition that a particle with an effect  of local \ncoupling between the electric and magnetic energies  is really created, one has to s how that inside \nthis partic le there are internal dynamical motion processes with special symmetry properties.   \n     Theoretically, it was shown that dynamical processes of magnetic dipole -dipole interactions \nin quasi -2D ferrite disks are characterized by s ymmetry breaking effects. These effects are well \nmodeled analytically with use of a so -called magnetostatic -potential (MS -potential) wave \nfunction \n . The boundary -value -problem solutions for a scalar wave function \n  in a quasi -2D \nferrite disk demonstrate multiresonance MDM spectra with unique topological structures of the \nmode fields [5 – 7, 21 – 25]. Based on the \n -function solutions , it was also shown that the near \nfields origin ated from ferrite -disk particles with MDM oscillations are characterized by \ntopologically distinctive power -flow vortices, non -zero helicity, and a torsion degree of freedom \n[31, 32]. The existence of the power -flow-density vortices of the MDM oscillations  presumes an \nangular momentum of the fields. The electric and magnetic fields of MDMs have both the \"spin\" \nand \"orbital\" angular momentums. One can find that in a source -free subwavelength region of \nmicrowave fields there exists the field structures with l ocal coupling between the time -varying \nelectric and magnetic fields differing from the electric -magnetic coupling in regular -propagating \nfree-space electromagnetic (EM) waves. To distinguish such field structures, the term \nmagnetoelectric (ME) fields has b een used [31]. Small ferrite -disk particles with MDM spectra, \nbeing sources of microwave ME near fields, are termed as ME particles [31, 32]. The coupled \nstates of electromagnetic fields with MDM vortices can be specified as the MDM -vortex \npolaritons [7].  \n    The properties of the MDM oscillations inside a ferrite disk are well observed numerically due \nto the ANSOFT HFSS program. In a case of ferrite inclusions acting in the proximity of the \nFMR, the phase of the wave reflected from the ferrite boundary de pends on the direction of the \nincident wave. For a given direction of a bias magnetic field , one can distinguish the right -hand \nand left -hand rays of electromagnetic waves. This fact, arising from peculiar boundary \nconditions of the fields on the dielectri c-ferrite interface s, leads to the time -reversal symmetry \nbreaking effect in microwave resonators with inserted ferrite samples  [33 – 36]. A nonreciprocal \nphase behavior on a surface of a ferrite disk results in path -dependent numerical integration of  4 the problem. For the spectral -problem solutions, the HFSS program, in fact, composes the fields \nfrom interferences of multiple plane EM waves inside and outside a ferrite particle. One has a \nvery good correspondence of the HFSS results with the analytical and experimental results. From \nthe HFSS studies one can see the same eigenresonances and the same field topology of the \nresonance eigenstates in different microwave structures {when, for example, a ferrite disk is \nplaced in a rectangular waveguide and when it is placed in a microstrip structure [32]}.  \n    It becomes evident that the helicity properties, observed in the near -field (subwavelength) \nregion of the MDM ferrite disk [5 – 7, 31, 32], can be  transferred  and transformed  to the far -field \n(on scales compar ed with the incident EM wave) region. As we will show in this paper, a tiny \nferrite disk changes the field topology in entire waveguide structure, having sizes much, much \nbigger than sizes of a ferrite particle. Based on numerical (the HFSS - program) resul ts, we are \nable to observe topological structures of the fields scattered by the MDM ferrite particle and the \nlong-distance helicity properties of the fields in a waveguide. We can see that on certain planes \nin vacuum, t he fundamental excitations appear in  loop-like forms. We show that in the field \ntopology, there can be path-dependent loop quantization  of the fields.  We show also that on \nmetallic waveguide walls, there exist electric and magnetic topological charges. We analyze the \npath-dependent interfere nce in microwave waveguides due to two MDM particles placed at \ndistances comparable with the disk diameters and at distances essentially exceeding sizes of \nthese particles. The results show that because of the path -dependent integration, an entire volume \nof a waveguide is strongly involved in the process of an interaction between small -scale MDM \nparticle oscillations and large -scale waveguide -mode fields. Our numerical studies are supported \nby recently developed analytical models for MDM oscillations in fer rite-disk particles. We \npresent preliminary experimental results aimed to support some basic statements of our studies.  \n  \nII. MDM FERRITE DISKS AS AXION -LIKE PARTICLES  \n \nIt has been shown [31, 32], that in a vacuum region, which is in close vicinity to a quasi -2D \nferrite disk with the MDM oscillations, there exists a specific parameter of the field structure:  the \nhelicity density of the fields. For a real electric field, this parameter is written as  \n \n                                                              \n1\n8F E E \n .                                                           (2)  \n \nFor time -harmonic fields, the parameter is calculated as  \n \n                                                           \n* 1Im16F E E  \n | .                                                (3) \n \nOne can also calculate a space angle between the vectors \nE\n  and \nE\n  as \n \n                                                            \n*Im\ncosEE\nEE \n\n\n .                                                (4) \n \nThis is the helicity density normalized to the field amplitudes.  The presence of non -zero helicity \nparameter for the fields near small quasi -2D ferrite disks with the MDM oscillation spectra \nmakes it evident the fact of existence of unique field structures – microwave ME fields. Such \nfields, originated from intrinsic dynamical process of magnetization inside a quasi -2D ferrite  5 disk, have a topological -phase structure [5 – 7, 31]. In the vicinity of a ferrite disk,  the fields are \nsplitted for the regions with positive and  negative helicity parameters [31 ]. \n    The analytically derived topological characteristics of the MDM fields are well illustrated by \nthe HFSS numerical results. In a case of numerical studies, one  uses an entire k-space of \nelectromagnetic waves. Ferrite inclusions are characterized by non -hard-wall boundary \nconditions with the interplay of reflection and transmission at the interfaces. These properties are \ncombined with the time -reversal symmetry b reaking effects. At the surface of a quasi -2D ferrite \ndisk, the fields inside and outside a particle are related by the electromagnetic boundary \nconditions. At resonance frequencies, one has an enhanced effect of the field localization due to \nmultiple back  and forth bouncing of the fields inside a disk.  It is very important that such a \ncomplicated process of numerical integration (when one has interferences of multiple plane EM \nwaves inside and outside a ferrite particle) is well modeled analytically. In th e analytical studies, \nhowever, an interaction of a small ferrite particle with MDM spectra is described by a quantum -\nlike model, non -trivial from a classical point of view.  \n    From the analytical consideration, observation of the helicity density in the f ield structure tells \nus about the presence of specific axion -like fields of MDM which interact with the external EM \nfields. The MDM ferrite disks appear as pseudo -scalar particles. The helicity density F, defined \nby Eq. (2), transforms as a pseudo -scalar u nder space reflection  and it is odd under time  \nreversal  . With the observed helicity properties of the fields in a vacuum region, abutting to a \nquasi -2D ferrite disk with the MDM oscillations, one becomes faced with fundamental aspects \nakin to the probl ems of axion electrodynamics. Maxwell's equations are invariant when the \nelectric and magnetic fields mix via rotation by an arbitrary angle \n . For a real angle \n , such a \nduality symmetry leaves invariant  quadratic forms for the fields. This symmetry can be extended \nto Maxwell's equations in the presence of sources, provided that additional magnetic charges and \ncurrents are introduced [28]. Whenever a pseudo -scalar axion -like field \n  is introduced in the \ntheory, the dual symmetry is spontaneously and explicitly broken. An axion -electrodynamics \nterm, added to the ordinary Maxwell Lagrangian [37]:  \n \n                                                                \nEB\nL ,                                                                (5)                     \n \nwhere \n  is a coupling constant, results in modified electrodynamics equations with the electric \ncharge and current densities replaced b y [37, 38]  \n \n                                \n( ) ( )eeB       \n      and    \n( ) ( )eej j B Et    \n .                (6)  \n  \nIntegrating Eq. (5) over a closed space -time with periodic boundary conditions, we obtain the \nquantization  \n \n                                                                   \n4S d x n L ,                                                    (7)  \n \nwhere n is an integer. It is evident that \nS  is a topological term. While \nS  generically breaks the \nparity and time -reversal symmetry, both symmetries are intact at \n0  and \n . The field \n  \nitself is gauge dependent.  \n    Axion -like fields and their i nteraction with the electromagnetic fields have been intensively \nstudied and they have recently received attention due to their possible role played in building \ntopological insulators. A topological isolator is characterized by additional parameter \n  that \ncouples the pseudo -scalar product of the electric and magnetic fields in the effective topological  6 action. Evidently  in common insulators  \n0 . Interesting possibility assuming -invariance \nexists when  \n , and this defines  the new class of insulators – topological insulators [39 – 43]. \nThe form of the effective action implies that an electric field can induce a magnetic polarization, \nwhereas a magnetic field can induce an electric pol arization. This effect is known as the \ntopological ME effect and \n  is related to the ME polarization. The topological angle \n  is a \nstatic parameter for a topological insulator. Promoting \n  to a dynamical field entails that the \nintegrand in action ( 7) would change the field equation . In Ref. [42] it was pointed out that, when \nthere is an antiferromagnetic order in an isolator, the magnetic fluctuations can couple to the \nelectromagn etic fields, playing the role of the dynamical axion fields. In some materials, \nclassified as topological magnetic isolators, the axion field couples linearly to light, resulting in \nthe axionic polariton. One has a condensed -matter realization of the axion  electrodynamics [42, \n43]. \n    In this paper we show that MDM ferrite disks can be considered as axion -like particles. The \ncoupling of such pseudo -scalar particles with microwave photons plays a key role in our studies. \nA mechanism of interaction of electr omagnetic fields with a small MDM ferrite disk is non -\ntrivial. This is purely a topological effect . In topological isolators, the main concept of \ntopological order appears due to the surface (or edge) states [39 – 41]. With a certain similarity \nto such sur face states, we can see that in MDM particles, the topological effects are exhibited due \nto chiral edge modes on a lateral surface of a ferrite disk [23]. This becomes evident from the \nspectral properties of MDMs in such a particle. To make the MDM spectra l problem analytically \nintegrable, two approaches were suggested. These approaches, distinguishing by differential \noperators and boundary conditions used for solving the spectral problem, give two types of the \nMDM oscillation spectra in a quasi -2D ferrite disk. We, conventionally, name these two \napproaches as the G- and L-modes in the magnetic -dipolar spectra [6, 25 ]. The MS -potential \nwave function \n  manifests itself in different manners for every of these types of spectra. For L \nmodes, the MS -potential wave functions \n  are odd under both  parity  and time-reversal , and \nare  -invariant [7]. The L-mode solutions show existence of four helical MS waves. Specific \ninteractions of these waves give evidence f or the double -helix resonances. In a case of the L \nmodes, the MS -potential wave function \n  is considered as a generating function for the vector \nharmonics of the magnetic fields [5 – 7, 24, 25 ]. The eigen electric and magnetic fie lds of the \nMDM oscillations one derives from the L-mode solutions. T he helicity density for the fields, \nshown in Eqs. (3), (4), is obtained based on the L-mode spectrum [6, 7, 31]. In a case of the G-\nmode spectrum, where the physically observable quantitie s are energy eigenstates and eigen \nelectric moments of a MDM ferrite disk , the MS -potential wave function \n  appears as a Hilbert -\nspace scalar wave function . There is a univocal correspondence  between the G-mode and L-\nmode spectrum  solutions. As it was shown in Ref. [7], there exists a certain operator \nˆC  which \ntransforms the L- mode spectrum to the energy eigenstate spectrum of MS -mode oscillations – \nthe G-modes. For G-modes, we have magnetically gapped chi ral edge states. These edge states \nconcern the concept of topological order. Topological phases of the edge states are accumulated \nby the double -valued border functions on a lateral surface of a ferrite disk and the topological \neffects become apparent thro ugh the integral fluxes of the pseudo -electric fields . There are \npositive and negative fluxes corresponding to the counterclockwise and clockwise edge -function \nchiral rotations.  \n    To have the standard action as a Lorentz scalar  [28], one should multiply the helicity \nparameter of MDM by a certain scalar function, which is odd under both   and . Such a scalar \nfunction appears due to topological -phase edge states on a lateral surface of a ferrite disk. \nCoupling of MDM disks to external EM fields arise s exclusively due to interaction of these \nfields with the eigen electric fluxes and eigen electric (anapole) moments originated from the  7 MDM topological -phase edge states [23]. The following below studies give evidence for such \nME coupling of the axion -like MDM oscillations to microwave EM fields.  \n \nIII. SCATTERING OF ELECTROMAGNETIC WAVES BY SMALL MDM PARTICLES  \n \nWe consider scattering of electromagnetic fields by MDM ferrite disks in a structure a \nrectangular waveguide with an enclosed ferrite particle. For bett er illustration of our studies of \nscattering effects by small ferrite particles, we use S -band rectangular waveguide which has the \nsizes of a waveguide cross section (4 cm \n  10 cm) much bigger than sizes of a ferrite disk (the \nthickness of 0.05 mm and the diameter of 3 mm).  \n    A normally magnetized ferrite disk is placed inside a waveguide (operating at the \n10TE  mode) \nsymmetrically with respect to waveguide walls so that the disk axis is perpendicular to a wide \nwall of a waveguide. Fig. 1 shows geometry of the structure.  For a numerical analysis in the \npresent paper, we use t he yttrium iron  garnet (YIG)  disk normally magnetized by a  bias magnetic \nfield \n02,930 H Oe; the saturation magnetization of the ferrite is \n 1880  4sM G. For better \nunderstanding the field structures we use a ferrite disk with very small  losses: the linewidth of a \nferrite is \n0.1 OeH . The numerically calculated reflection coefficient, represented in  Fig. 2, \nclearly verifies the known spectral properties of MDMs [5, 7]. For the first two modes, the peak \npositions of the numerically obtained spectrum in Fig. 2 are in good correlation with the MDM \nspectrum calculated based on an analytical model [24, 25 ]. However, because of a very big \ndifference between sizes of a waveguide and a ferrite resonator, one cannot observe such a good \ncorrelation for higher -order MDMs.    \n    Our analysis in the present paper we restrict with consideration of topological phas e effects \nonly for the 1st MDM. The unique helicity properties of the fields in the vicinity of a MDM \nferrite disk become evident from Fig.  3. In this figure one can see the normalized  helicity density \ncalculated numerically based on Eq. (4) at the resonan ce frequency of the 1st MDM ( f = \n3.0237GHz ). The pictures are obtained in two mutually perpendicular cross -section planes \npassing through the disk axis. The main point of the pictures shown in Fig. 3 is the fact that due \nto a MDM ferrite disk, a symmetrica l Maxwellian -field structure in vacuum becomes splitted \ninto two non -symmetrical (with non -zero helicities) field structures. These splittings are \nextended on extremely large area. One can see that the region of non -zero helicity parameters \nmay cover almos t an entire waveguide cross section. Along a waveguide axis, an entire region of \nnon-zero helicity stretches about a quarter of the waveguide wavelength. On metal walls of a \nwaveguide, the helicity is zero. We can say that below and above a ferrite disk th e fields are \ntwisted in different azimuth directions, being \"pinned\" at metal. This gives evidence for the \ntorsion degree of freedom of the fields coupled to a MDM ferrite disk [31, 32].  \n    The topological -phase effects in an entire microwave -waveguide st ructure with an enclosed \nMDM ferrite disk are well illustrated by the pictures of the time and space transformations of the \nwave fronts of the electric field. For the 1st resonance frequency of 3.0237GHz, Fig. 4 shows \ntransformations of the wave front of t he electric field in a waveguide on a vacuum plane 0.1 mm \nabove a ferrite disk at different time -phases. The inserts show enlarged pictures of the electric -\nfield distributions immediately above a ferrite disk. One can see that as the time -phase changes, \nthe wave fronts change their geometrical forms. It is very important to note that a geometrical \nstructure of the shown fronts restores twice during a time period of the microwave radiation. For \ntwo next -coming fronts of incident waveguide -mode fields with th e time -phase difference of \n , \none can see the same configuration of the wave fronts distorted due to scattering from a MDM \nparticle. T he wave front becomes dist orted, when passing through a peculiar topological  relief . \nThis relief , appearing due to non -zero field helicity, one has only at frequencies of the MDM \nresonances, when the helicity parameter is non -zero. The pictures of the \nE\n -field structures at the  8 non-resonance frequency and at the MDM -resonance   frequency, shown in Fig. 5, allows \ncompare configurations of the undistorted wave front and the wave front distorted due to a \ntopological  relief  with non -zero helicity.  \n    Together with strong localization of the electric field at the MDM resonances, on e observes \nalso strong localization of the magnetic field at the resonance frequencies. Fig. 6 shows the \nmagnetic -field distributions on a vacuum plane 0.1 mm above a ferrite disk at different time \nphases. The frequency is the 1st resonance frequency of 3. 0237GHz. The top row shows enlarged \npictures of the magnetic -field distributions immediately above a ferrite disk. In the bottom row \none can see structures of the wave fronts of the magnetic field. A character of the magnetic field \nnear a MDM ferrite disk looks like the field of a rotating magnetic dipole. Evidently, for \nmagnetic fields, there are no transformations of geometrical forms of wave fronts, as we can \nobserve for electric fields. Strong localization of both the electric and magnetic fields near M DM \nparticles results in appearance of power -flow vortices at resonance frequencies. For the 1st \nresonance frequency of 3.0237GHz, such a vortex on a vacuum plane 0.1 mm above a ferrite \ndisk is shown in Fig. 7. Such topological structures of the power -flow density of the near fields \nat the MDM resonances were studied in details in Refs. [6, 7, 24].  \n    The field distributions near a small ferrite disk, shown in Figs. 3 – 7, give evidence for the fact \nthat the observed scattering phenomena are very different  from the known classical effects of \nscattering of electromagnetic fields by small particles [27, 44]. The field in vacuum, surrounding \na ferrite -disk particle, is a superposition of the incident (the waveguide -mode) and scattered (the \nMDM) fields:  \n \n                                                          \nWG MDM H H H\n ,                                                             (9)    \n  \n                                                          \nWG MDM E E E\n .                                                              (10)                               \n \nThe incident waveguide -mode EM field excites MDM egenstates in a small quasi -2D ferrite \ndisk. There is a very strong difference between wavenumbers of the waveguide -mode EM wave \nand the MDM oscillations. We have \nMDM EMkk . Based on the field structures shown in Figs. 3 \n– 7 (and also based on our previous studies [6, 7]), we can state that interaction of \nelectromagnetic fields with a small MDM ferrite particle can not be considered as scattering by \nsmall electric or/and magnetic dipoles. In particular, one cannot find any similarity between \nscattering by a MDM ferrite particle and scattering by a small electric -dipolar particle with \nplasmon oscillations [26 – 30].  \n    How do incident EM fields interact with a MDM ferrite particle?  From the pictures in Fig. 6 \none can see that in the region close to a ferrite disk, the magnetic field \nMDMH\n  appears as the field \nof a magnetic dipole rotating in th e xy plane. A small ferrite disk is placed in a waveguide \nsymmetrically with respect o waveguide walls. On this place, the field \nWGH\n  is linearly polarized \nwith respect to the x axis (see Fig. 1). With represent ation of  this field as  two circular polarized \nfields , rotating in the xy plane, one can clearly see that magnetic -dipolar field \nMDMH\n  is \nperpendicular to both circular polarized magnetic fields of a waveguide mode. So no magnetic \ninteraction of a MDM parti cle with an incident magnetic field occurs. Does the particle interact \nwith the electric component of an incident EM field and, if does, what is the nature of this \ninteraction? One of the main features of the shown above field distributions is the fact tha t in the \nnear-field vacuum region immediately above a ferrite disk there are only in -plane components of \nthe field\nE\n . The field \nMDME\n   is perpendicular to the field \nWGE\n  and, evidently, a normal \ncomponent of the field \nE\n is zero at every time phase \nt during a time period of microwave  9 oscillations. It means that in the near -field vacuum region immediately above a ferrite disk, the \nscattered electric field completely annihilates the incident electric field. Since the incident field \nWGE\n is oriented perpendicularly to the disk plane, this is an evidence for  the property of a MDM \nferrite disk  to create an eigen electric -field flux in opposition to  the flux of  the externally applied \nRF electric field. S pectral properties of the MDMs show that on a lateral surface of a  quasi -2D \nferrite disk there exist the double -valued -function l oop magnetic currents which produce eigen \nelectric fields and hence eigen electric fluxes through the loop. In this case one can introduce a \nnotion of an electric self inductance as the ratio of the electric flux to the magnetic current. In a \nregion far from a MDM disk, it can be described as a small particle with eigen electric moment \ncounter -directed to the field \nWGE\n . Such an eigen electric moment – the anapole moment – was \ndescribed in Refs. [23, 31, 45, 46]. The anapole moment of a ferrite disk arises not from the curl \nelect ric fields of MDM oscillations. An analysis of symmetry properties of the anapole moment \nallows understanding the mechanism of interaction of MDM particle with an external electric \nfield. In Refs. [20, 45, 46], it was  show n experimentally that the MDM ferrite disks can \neffectively interact with the external RF electric field oriented normally to the disk plane.  \n    The observed interaction of an external EM field with a MDM ferrite disk becomes \nunderstandable when we consider  more in detail the  topological  properties of MDMs.  \nAnalytically, the helicity density of the MDM near fields is derived based on the properties of \nthe L-mode spectra [6, 31]. At the same time, the energy content (weight) of each MDM is \ndetermined by the G-modes. The egenstates of the o perator \nˆC  (which transforms the L- mode \nspectrum to the G-mode energy eigenstate spectrum) are chiral edge states on a lateral surface of \na ferrite disk. The edge geometrical -phase loops are penetrated by the fluxes of the pseudo -\nelectric field [7, 23].  The edge geometrical -phase loops are created by two helices. This arises \nfrom a geometrical structure of L modes, which are helical MS modes [6]. In the near -field \nregion, the electric -field distribution can be well illustrated base d on an analysis of the helical \ntopological loops composed with branches of two (left -handed and right -handed) helices. Fig. 8 \nshows transformations of the electric fields in a region very close to a ferrite disk. The electric -\nfield vectors are shown in co nnection with geometry of the MS helical -wave loops [6]. During \nthe dynamical -phase variation of \n , one has a complete circulation along a double -helix \ntopological loop, but with the electric -field vectors directed oppositely.  \n    In the region far from a ferrite disk, one has a more complicated situation. A more detailed \nanalysis of the electric -field structure clearly shows a topological character of the scattered \nfields. It becomes evident that the topologically originated scat tered electric fields have \ncomponents in the xy plane and are oppositely directed with respect to z axis. For a given time \nphase \nt , Fig. 9 ( a), (b) show the electric field distributions in vacuum above and below a ferrite \ndisk, respectively,  at the distances  0.1mm from the ferrite -disk planes and at a bias magnetic \nfield directed along z axis. During the dynamical -phase variation, the topological loops are \ntransformed both in the form and in orientation. Moreover, the topological l oops can be partially \nclosed by metallic walls of a waveguide. This is illustrated in Fig. 9 ( c), (d). There are the \nelectric field distributions  at the same vacuum planes as in Fig. 9 ( a), (b), but at another time \nphase \nt . Config urations of the geometrical -phase loops are recreated for every \n  phase \ndifference of the field vectors. The regions of non -zero helicity are the regions of pseudo -\nelectric -field fluxes. When a bias magnetic field is directed oppo site to the z-axis direction, one \nhas different orientations of both a wave front and electric -field vectors. This is illustrated in Fig. \n9 (e), where the time phase and the vacuum plane position (0.1 mm above a ferrite disk plane) \nare the same as in Fig 9  (d). In an entire space inside a waveguide, we have the geometrical -\nphase shells for the electric -field vectors. The topological -phase loops in Fig. 9 are observed on \nthe xy cross -sectional planes of these shells. Fig. 10 shows the topological -phase loops  for other \ncross -sectional planes: the xz and yz planes inside a waveguide. The pictures in Fig. 10 give  01 evidence for transformations of the electric fields in accordance with a model of the MS helical -\nwave loops (see Fig. 8). It is worth noting that the t opological -phase profiles are correlated with \nprofiles of the helicity parameter. In Fig. 11, we show the electric -field-front configuration on a \nvacuum plane 7 mm above a ferrite -disk plane combined with the cross -sectional outline of the \nhelicity density  on the same plane.  \n    The structure of the field scattered by a small ferrite particle with MDM oscillations has \nproperties resembling the field structure of axion -electrodynamics. We have non -zero helicity \ndensity F, transform ed as a pseudo -scalar under  space reflection  with odd  time reversal   [31, \n32], and a pseudo -scalar axion -like field originated from the topological -order chiral edge modes \non a lateral surface of a ferrite disk [23, 31]. If the axion fields  exist , they will couple the \nradiation EM field to the MDM field with a coupling proportional to the helicity density . The \nfact that the MDM oscillations in a ferrite disk, coupled to the electromagnetic fields, can play \nthe role of the dynamical axion fields is also well illustrated by distribu tions of the currents and \ncharges induced on metal walls of a waveguide. Fig. 12 shows modifications of the surface \nelectric charge and current densities due to the MDM -oscillation particle. P resumably, these \nmodification s can be described by the axion -electrodynamics terms  in Eq. (6). In Fig. 12, we can \nsee that surface electric charges, appearing also as topological charges, originate surface electric \ncurrents in the forms of convergent or divergent spirals. Because of such a topological effect, an \nentire  structure of the current distributions on the waveguide walls is strongly distorted. As \nunique feature of the observed topological effect, we can point out on the presence of topological \nmagnetic charges on  the waveguide walls. This is illustrated in Fig.  13 for the upper wide wall of \na waveguide. Because of symmetry of the magnetic fields in a waveguide along z-axis, one has \nthe same pictures of topological magnetic charges on  the lower waveguide wall. It is worth \nnoting also that the regions of topologic al electric charges completely coincide with the regions \nof topological magnetic charges. The superimposed positions of these topological charges are \nshown in Fig. 14.  \n    For better understanding of the shown numerical results (and, especially, for furthe r analysis \nof interactions between two MDM ferrite disks) we have to dwell more in details on the answer \nto the above question: How do the incident single -valued EM fields interact with double -valued \nfields of MDM ferrite particles? The G-mode solutions fo r magnetic -dipolar oscillations are \ncharacterized by the single -valued MS -potential membrane functions. However, to satisfy the \nboundary conditions for magnetic flux density on a lateral surface of a disk, one has to impose a \ngeometrical phase factor . This  phase factor appears due to double -valued edge wave function s \n[23, 31]. For a G-mode membrane MS -wave function  \n~, the boundary condition on a lateral \nsurface of a ferrite disk is the following:  \n \n                                                          \n0\nrrrr\n          \n ,                                            (11)  \n \nwhere \n  is a radius of the disk. O n a lateral border of a ferrite disk , the correspondence b etween \na double -valued membrane wave functi on \n\n and a singlevalued function  \n~ is expressed as: \nrr   \n, where \niqfe\n  is a double -valued edge wave function  on contour \n2L\n. The azimuth number \nq  is equal to \n1\n2l , where l is an odd quantity ( l = 1, 3, 5, …). \nFor amplitudes we have \n f f  and \nf  = 1. Function \n  changes its sign when \n  is rotated \nby \n2  so that \n21iqe . As a result, o ne has the e nergy -eigenstate spectrum of MS -mode  \noscillations with topological phases accumulated by the edge  wave function \n . On a lateral  00 surface of a quasi -2D ferrite disk, one can distinguish two different function s \n. A line integral \naround a singular contour \nL : \n2\n**\n01( )  ( )\nri d i d\n  \n\n                \nLL  is an observable \nquantity. It follows from the fact that because  of such a quantity one can restore \nsinglevaluedness (and, therefore, Hermicity) of the G-mode spectral problem. Because of a \ngeometrical  phase factor  on a lateral boundary of a ferrite disk, G-modes are characterized by a \npseudo -electric field (the gauge field)  [23, 31]. We denote this pseudo -electric field  by the letter \n€\n. The geometrical phase factor in the G-mode solution is not single -valued under continuation \naround a contour \nL  and can be correlated with a certain vector potential \n()m\n€\n [23, 31]:   \n \n                                        \n2\n* ( )\n0[( )( ) ] 2m\nr€ i d K d q\n           \nLL .                        (12)  \n \nwhere \n1\nre\n\n\n  and \ne\n is a unit vector along an azimuth coordinate, and K is a \nnormalization coefficient [31]. In Eq. (12) we inserted a connection which is an analogue of  the \nBerry phase . In our case, the Berry's phase is generated from the broken dynamical symmetry. \nThe confinement effect for magnetic -dipolar oscillations requires proper phase relationships to \nguarantee single -valuedness of the wave functions. To compensate for sign ambiguities and thus \nto make wave functions single valued we added a vector -potential -type term  \n()m\n€\n  (the Berry \nconnection) to the MS -potential Hamiltonian. On a singular contour \n2L , the vector \npotential \n()m\n€\n  is related to double -valued functions. It can be observable only via the circulation \nintegral ove r contour \nL , not poin twise.  The field \n€\n  is the Berry curvature. In contrast to the \nBerry connection \n()m\n€\n , which is physical only after integrating around a closed path, the Berry \ncurvature \n€\n  is a gauge -invariant local manifestation of the geometric properties of the MS -\npotential wavefunctions. The corresponding flux of the gauge field \n€\n  through a circle of radius \n\n is obtained as: \n \n                                      \n( ) ( )2me\n€\nSK € dS K d K q         \nLL ,                          (13) \n \nwhere \n()e\n  are quantized fluxes of pseudo -electric field s. There are the positive and negative \neigen fluxes. These differe nt-sign fluxes should be inequivalent to avoid the cancellation . It is \nevident that while integration of the Berry curvature over the regular -coordinate angle \n  is \nquantized in units of \n2 , integration ove r the spin -coordinate angle  \n \n1\n2  is quantized \nin units of  \n. The physical meaning of coefficient K in Eqs. (11), (13) concerns the property of \na flux of a pseudo -electric field. I t is related to the notion of a magnetic current in the G-mode \nanalysis. In Refs. [23, 45, 46], the coefficient K was conventionally taken as equal to unit. In our \nrecent study [31], we related quantity K to an elementary flux of the pseudo -electric field.  In the  02 next section we show that quantity K can be found in an analysis of two interacting MDM ferrite \ndisks.  \n    The velocities of two border flows \n\n  results in two edge magnetic currents on a lateral \nsurface of a ferrite disk [6, 23, 31]. For the observer placed in a laboratory frame, there will be \ntwo double -valued -function magnetic currents rotating at the same direction of the azimuth \ncoordinate. In the vicinity of a ferrite disk, the linear -polarized \nE\n -field of the incident wave can \nbe decomposed in two (clockwise and counterclockwise) circularly rotating fields. On the other \nhand, for a given direction of a DC magnetic field \n0H\n , one has two azimuth modes of pseudo -\nelectric fields originated from double -valued -function edge magnetic currents and rotating at a \ncertain direction on a lateral surface of a ferrite disk. These azimuth modes of pseudo -electric \nfields are \n -shifted. Fig. 15 illustrates the mechanism of interaction between the flux of the \nincident single -valued electric field and the flux of the double -valued pseudo -electric fields of \nMDM ferrite particles. From these pictures, one can see  that (with averaging at the azimuth \ncoordinates) the electric flux of the incident EM field  is completely annihilated by the electric \nflux of the MDM field.    \n \nIV. PATH -DEPENDENCE INTERFERENCE WITH TWO COUPLED MDM  \n      PARTICLES  \n \nWe consider now interactions between two MDM ferrite disks placed in a microw ave \nwaveguide. When the spin -coordinate and orbital  degrees of freedom of MDM oscillations are \nmixed in a particular way, the momenta of these oscillations in a coupled -particle system feel \nimportant effects of this \n  Berry’s phas e. The shown above topological -effect distributions of \nthe currents and charges induced on metal walls of a waveguide by a single MDM ferrite disk \npresume very specific conditions on coupling between these particles. When the MDM of one \nferrite disk is rot ating, it swirls its surrounding spacetime around.  Rotating spacetime can impart \nto the field of a waveguide mode an intrinsic form of orbital angular momentum . This intrinsic \nform of orbital angular momentum  can be detected by another ferrite disk . Intera ction between \ntwo MDM ferrite disks can be non -reciprocal.  \n    In Ref. [47], two interacting MDM ferrite disks were studied analytically based on the \ncoupled -mode theory for G-mode spectra and with taking into account the pseudo -electric fluxes. \nIt was sho wn that for such a structure, there can be two states of MS interaction with the \nsymmetrical and antisymmetrical field distributions. It was also shown that the MDM interaction \ndue to pseudo -electric fluxes appears with certain quantization rules for these  fluxes. In both \ntypes of interactions (MS interaction and pseudo -electric -flux interaction), the coupled -state \nresonances should be distinguished by energy. There are two energetically stable states. To a \ncertain extent, we have a model which rather more resembles coupled semiconductor quantum \ndots [48 – 52], than coupled plasmon -oscillation particles [53 – 55]. In this paper we present the \nresults of numerical studies of two interacting MDM ferrite disks. These numerical results \nillustrate, sufficiently w ell, the theoretical model in Ref. [47]. In our numerical analysis, we \ndistinguish two cases. In the first case (the short -distance interaction, when a gap between ferrite \ndisks is much less than or comparable with diameters of disks) both the MS and pseud o-electric -\nflux interactions should be taken into account. In the second case (the long -distance interaction, \nwhen a gap is much bigger than diameters of disks) we have the coupling mainly due to \npseudoelectric fluxes.  \n    The MS interaction between two M DM disks can be analyzed based on the coupled -mode \ntheory [47]. Interaction by the pseudo -electric fluxes for two laterally coupled MDM disks  appears \ndue to formation of closed topological loops between the disks. In an assumption about ability of \nthe edge  function in one location to produce phase accumulation in the edge function in another  03 location, the electric interaction presumes existence of four pseudo -electric fluxes [47]. For a given \nMDM p, we will designate a pseudo -electric flux which is connecte d with an edge function of a \nferrite disk a and penetrating the border loop of disk a, as \naa\npe , and a pseudo -electric flux which \nis connected with an edge function of a ferrite disk b and penetrating the border loop of disk b, as \nbb\npe\n. At the same time, we will designate a pseudo -electric flux connected with an edge function \nof a ferrite disk b and penetrating the border loop of disk a, as \nab\npe , and a pseudo -electric flux \nconnected with an edge functi on of a ferrite disk a and penetrating the border loop of disk b, as \nba\npe\n. To preserve the single -valuedness of the membrane wave function  \n~ of disks a and b we \nhave [47]  \n \n                                                   \n12aa ab ae e e\np p pqK                                                     (14) \n \nand \n \n                                                   \n12bb ba be e e\np p pqK       .                                            (15)  \n \nBoth \nq  and \nq  take values of  \n1 3 5, , ,...222   \n    Because of linearity, the quantities \nab\npe  and \nba\npe  can be represented as  \n \n                                                           \n12ab be ab e ab\nppppk k qK                                              (16) \n \nand \n \n                                                           \n12ba ae ba e ba\nppppk k qK     .                                        (17)  \n \nFor mode p, ME -interaction coefficients \nab\npk  and \nba\npk  determine, respectively, a fraction of a total \npseudo -electric flux of disk a perceiving the border ring of disk b and a fraction of a total pseudo -\nelectric flux of disk b perceiving the border ring of disk a. Evidently, \n01ab\npk  and \n01ba\npk . \nBased on Eqs. (14) – (17), we obtain  \n \n                                                                   \n121aae ab\nppqkK                                               (18) \n \nand \n   \n                                                                    \n121bbe ba\nppqkK    .                                         (19)  \n  04 Because of the path -dependence interaction, coupling between the disks is non -reciprocal. It \nmeans that \nab ba\nppkk . So \naa bbee\npp    and \nab baee\npp   . \n    From Eqs. (14), (15), we have  \n  \n                                                            \n12abee\nppqqK       .                                     (20)  \n \nThe minimal difference between quantities \nq  and \nq  is \n1 . This allows representing the \ncoefficient  K as \n \n                                                             \n\nmin2\nabee\nppK\n   .                                                    (21) \n \nThe fluxes   \nae\np  and \nbe\np  are normalized quantities with respect to the disk sizes and \namplitudes of the MDM oscillations [31, 47]. Since a minimal difference of the pseudo -electric \nfluxes , \n\nminabee\npp   , is a constant quantity, for a given geometry of MDM disks  one should \nobserve the quantum -like effect of the topological coupling between the disks. One can presume \nthat because of two energetically stable states in the couple d-disk structure, the frequency \ndifference between the resonance peak positions should not strongly depend on the distance \nbetween the disks. However, the intensity of the fields in the disks should depend on this distance. \nMoreover, since the coupling bet ween the disks is non -reciprocal, there should be observable \ndifference between the field intensities in the disks. Our numerical studies well illustrate these \nstatements.  \n    A structure of two coupled MDM disks embedded in a rectangular waveguide is sho wn in Fig. \n16. For  the short -distance interaction (the distance between centers of the disks is 3.6 mm), one \nobserves splitting of the resonance peak. Fig. 17 shows this splitting on the reflection -coefficient \ncharacteristic of the 1st MDM. The frequency d ifference between the splitted peaks is very small. \nIt is equal to 3.9 MHz. Such frequency difference is about 0.1% of the frequency of the 1st \nMDM . For the short distances, the main coupling due to MS interaction is accompanied with the \nauxiliary  effect o f the pseudo -electric -flux interaction. In Fig. 18 one can see the electric field  \ndistributions  on the upper surface of a ferrite disks  at a certain time phase.  One has the \nsymmetrical and antisymmetrical coupled modes.  For both modes, there is an observab le \ndifference between the field intensities in the disks. In a case of the symmetrical mode, the field \nintensity for the disk a is higher than for the disk b, while for the antisymmetrical mode, one \nobserves opposite situation with the field intensity. We can conclude that for the symmetrical \nmode, the flow of the pseudo -electric field piercing the disk a exceeds the flow piercing the disk \nb. For the antisymmetrical mode, one has the opposite situation. Fig. 19 shows positions of the \nwave fronts (localized regions where the electric field of a waveguide mode changes its sign) on \na vacuum plane passing 0.1 mm above ferrite disks. While in a case of the symmetrical mode \ntwo disks interact with the incident EM field like one disk (compare with Fig. 4), for the \nantisymmetrical mode two disks interact with the incident EM field with a specific manner. In \nthe last case, one has a very complicated structure of topological loops. It is worth noting that for \ntwo (symmetrical and antisymmetrical) types of modes, both d isks are involved in the process of \ninteraction with the incident EM field. Moreover, similarly to the situation with a single ferrite \ndisk, a normal component of the field \nE\n  above every disk is zero at any time phase \nt .  05     Let us consider now a structure when two disks are separated at the distance (between centers \nof the disks) of 6 mm. In this case, the splitting on the reflection -coefficient characteristic of the \n1st MDM shown in Fig. 20 corresponds t o the frequency of 3.8 MHz.  For such a distance \nbetween the disks, the main coupling is due to the pseudo -electric -flux interaction. In Fig. 21 one \ncan see the electric field  distributions  on the upper surface of a ferrite disks , corresponding to the \nsymme trical and antisymmetrical coupled modes.  For these modes, one observes a strong \ndifference between the field intensities in the disks. For the coupled -mode resonances, Fig. 22 \nshows positions of the wave fronts on a vacuum plane passing 0.1 mm above ferri te disks. In \ncomparison with the case of the short -distance interaction, now only one disk is mainly involved \nin the process of interaction with the incident EM field. There is disk a for the symmetrical -mode \nresonance and disk b for the antisymmetrical -mode resonance. The helicity -density distributions \nfor the coupled -mode resonances, shown in Fig. 23, give more evidence for the non -reciprocal \ninteraction between the disks. One can see that for identical disks, the behaviors of coupling \nbetween the disks \n  ab  and \n  ab  are completely different. These topological effects take \nplace only at the coupled -mode resonance. Fig. 24 shows the wave fronts on a vacuum plane 0.1 \nmm above ferrite disk s for the frequency 3.0333GHz , which is between the coupled -mode \nresonances. There are no topological distortions of the wave fronts. It is also possible to show \nthat at this frequency, the helicity density is zero.  \n    As a very attractive feature of the topological -effect coupl ing, there is evidence for long -\ndistance interaction between two MDM disks. In Fig. 25, we present the split -resonance \ncharacteristic of the 1st MDM at the distance between centers of the disks of 12 mm. The \nfrequency split is about 1 MHz. Our studies conf irm the fact that there is possibility to observe \nthe topological -effect coupling between the disks at extremely long distances between the disks. \nFig 26 shows a structure of two MDW ferrite disks placed inside a waveguide at the distance of \n30 cm. The spl it-resonance characteristic of the 1st MDM at this distance is shown in Fig. 27. \nThe frequency split is about 1.6 MHz. At the resonance frequencies, one has a strong difference \nbetween the field intensities in the disks. This is illustrated in Fig. 28, whe re one can see the \nhelicity -density distributions for the coupled -mode resonances.  \n \nV. EXPERIMENTAL SUDIES ON INTERACTION BETWEEN TWO MDM DISKS  \n \nAs a final stage of our studies on the path -dependent interference, we present some preliminary \nexperimental r esults on interaction between two MDM disks. There are supplementary results \naimed only to support some of our basic conclusions of this paper. More detailed experimental \nstudies are the purpose of our future publications. For better sensitivity, in our ex periment we use \na microstrip structure with embedded ferrite disks. As we discussed in Introduction, for the same \nparameters of ferrite disks, the resonance frequencies of MDM oscillations should be the same in \ndifferent microwave structures. Fig. 29 shows  a microstrip structure with one ferrite disk and a \nnumerical spectrum for this structure. Comparison of the numerical spectra in Figs. 2 ( a) and 29 \n(b) shows that for different types of the external EM fields (fields of a waveguide and fields of a \nmicrost rip structure) one observes the same peak positions for the main eigenmodes of MDM \noscillations.  \n    For experimental studies, we used a microstrip structure with two disks. This structure is \nshown in Fig. 30 ( a). The disks have diameter  of 3 mm. They are  made  of the yttrium iron garnet  \n(YIG) film on the gadolinium gallium garnet (GGG) substrate .  The YIG film thickness is 50 \nmkm.  The saturation magnetization  of YIG  is \nG 1880  40M  and the  linewidth is \nOe 8.0H\n. It is necessary t o note here that instead of a bias magnetic field in numerical \nstudies (\n02,930 H Oe), in the experiments we applied lower quantity of a bias magnetic field: \n02,607 H\nOe. Use of such a lower quantity (giving us the same posit ions of the resonance  06 peaks in the numerical study and in the experiment) is necessary because of non -homogeneity of \nan internal DC magnetic field in a real ferrite disk. A more detailed discussion on a role of non -\nhomogeneity of an internal DC magnetic fi eld in the MDM spectral characteristics can be found \nin Refs. [19, 22].  \n    Fig. 30 ( b) shows the numerical and experimental normalized spectra for a microstrip structure \nwith two coupled disks. The distance between the centers of the disks is 9 mm. One ca n observe \nthe split -resonance characteristics for the 1st MDM. It is worth noting that in a microstrip \nstructure, the frequency splits at the coupled -state responses are different compared to such \nresponses in a rectangular waveguide. These splits (7.2 MHz  for the numerical characteristic and \n8.9 MHz for the experimental result) are more than 0.2% of the frequency of an incident EM \nfield. As it follows from Fig. 30 ( b), there is a sufficiently good correspondence between the \nnumerical and experimental resul ts for a microstrip structure with two coupled disks. This gives \na preliminary confirmation of our numerical studies and analytical models. It is necessary to \nnote, however, that the experimental peaks in Fig. 30 ( b) are not so sharp and not identical in \ncomparison with the peaks obtained from numerical results. The reasons for this are the \nfollowing. We have different linewidth s for ferrite disks in the numerical (\n0.1 OeH ) and \nexperimental (\nOe 8.0H ) studies. Also, in experim ents, we were unable to use absolutely \nidentical disks: there are some differences with the disk geometries and material parameters.    \n \nVI. DISCUSSIONS    \n \nA non -integrable (path -dependent) electromagnetic problem of ferrite disks placed inside a \nrectangu lar waveguide, following from closed -loop nonreciprocal phase behaviors on lateral \nsurfaces of the disks, can be solved numerically based on the HFSS program. One of the main \nquestions for discussions concerns credibility  of the results, obtained based on the classical -\nelectrodynamics HFSS program, for description of the shown non -trivial topological effects \noriginated from the MDM ferrite disks. Also, a more serious question arises: Whether these \nnumerical results are applicable for confirmation of such an alytically -derived notions as pseudo -\nelectric -field fluxes, anapole moments, energy eigenstates, etc., which hardly can be classified as \npure classical notions?        \n    First of all, we have to note strong regularity of the results obtained based on the  HFSS -\nprogram solutions. From numerical studies of different microwave structures with embedded \nthin-film ferrite disks, one can see consistent pattern of both the spectral -peak positions and \nspecific topological structure of the fields of the oscillating modes. Moreover, these numerical \nresults are in very good correlations with the corresponding results obtained from analytical \nstudies and microwave experiments. The HFSS program, in fact, composes the field structures \nfrom interferences of multiple plane EM waves inside and outside a ferrite particle. In such a \nnumerical analysis one obtains the pictures of the field structures (resulting in the real -space \nintegration) based on integration in the k-space of the EM fields. In the k-space integration, the \nfields are expanded from very low wavenumbers (free -space EM waves) to very high \nwavenumbers (the region of MS oscillations). As a result of numerical integration, one has \nconvergent solutions at a given frequency. It is worth noting, however, that inside a small ferrite \nparticle (and in the near -field region outside this particle), such a very complicated EM -wave \nprocess in a numerical representation is well modeled analytically with use of the MS -potential \nwave function \n . As we di scussed before, to make the path -dependent spectral problem of \nMDM oscillations analytically integrable, two approaches – the G- and L-mode magnetic -dipolar \nspectra – have to be considered.  Evidently, inside (and nearly outside) a ferrite disk, the HFSS \nnumerical results can be well applicable for confirmation of the analytically -derived quantum -\nlike models based on scalar wave function \n . There exist many examples in physics when non -\nclassical effects are well modeled by a combine d effect of numerous classical sub -elements.   07         As a very interesting effect of the topological -effect coupling, there is observation of the \nlong-distance interaction between two MDM disks. In a waveguide structure with two interacting \ndisks, the fre quency split  in the resonance characteristics is extremely small. Its width is about \n0.1% of the frequency of an incident waveguide field. As we can see from our numerical \nsimulation, the wave front of the fields scattered by the MDM ferrite disks can be o riented at a \nsmall angle (sometimes parallel) to the waveguide axis. It means the presence of the extremely -\nlong-wavelength field structures in a waveguide. Such field structure may presume the quasi -\nstatic interaction between the particles. But how this c an be realized at distances, which are very \nlong in comparison with sizes of interacting particles? These distances are not small enough that \nretardation  effects can be ignored.  It can be supposed that there should be a certain mechanism of \ntransportation of energy. As we showed above, there exist topological sources which appear on \nwaveguide walls near the places of the location of ferrite disks. Could these topological sources \ncause a flow of topological -vortex (geometrical -phase) fluctuations against the  background of a \nlaminar EM -wave flow? The classical -type model cannot comprehensively explain the observed \ntopological effects. Together with the above question, the open questions are: Why  the frequency \nsplit in the resonance characteristics is almost in dependent on the distance between disks? Why \nthe intensities of the fields in the disks a re different for different frequencies of the split -\nresonance state. Based on our analytical semi -classical model, some answers to these questions \ncan be suggested. Th e far -field interaction, can presume the closed -loop topology of pseudo -\nelectric fluxes for the coupled -disk structure, which can be observed at very long distances inside \na waveguide. In this case, the eigen and mutual pseudoelectric fluxes can be or summ ed or \nsubtracted when they pierce a separate ferrite disk. When the fluxes are summed, the intensity of \nthe fields near a disk will be high. Contrarily, when the fluxes are subtracted, the intensity of the \nfields near a disk will be low. We may follow the above mechanism of an interaction with the \nexternal EM field. Near every (of the coupled) particle the interaction of an external EM field is \nwith an integrated flux of the pseudoelectric field. For two different interactions, one has two \nseparate eigen fr equencies. This is a quantized effect, which does not depend on distances \nbetween the disks. The eigen electric fluxes are quantized. The incident EM field cannot \"see\" \nthis quantization for a single particle, but can recognize this quantization for couple d particles at \nthe long -distance interaction. Further (and more detailed) studies on these unique topological \neffects are needed.  \n \nVII. CONCLUSION  \n \nInteraction of MDM particles with microwave radiation has the topological -effect nature. The \npresence of he licity density of the fields originated from MDM ferrite particles gives evidence \nfor axion -like properties of these fields. Because of these properties, topology of the twisted \nstates of microwave fields scattered by MDM ferrite disks strongly distinguish  from a structure \nof an optical twisted photon.    \n    In this paper,  we have  investigated the topological phase effects in a microwave waveguide \nwith embedded MDM ferrite disks. We have shown that at MDM resonances, structures of the \nfields engineered by ferrite -disk particles (both in the near - and far -field regions) are \nsubstantially different from the properties of electromagnetic fields originated from usual \nmicrowave sources. Topological structures of the fields give evidence for the regions of phase \nsingularities. One can see spontaneous generation of curved and closed -loop phase dislocations . \nThe observed continuous variations of topological -contour pictures give evidence for a certain \neffect of the topological -wave propagation.  In a system of couple d disks one has topologically \noriginated split -resonance states. 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(4), at the resonance frequency \nof the 1st MDM ( f = 3.0237GHz ). (a) Cross -section plane yz (orthogonal to a front of an incident \nEM wave), passing through the disk axis; ( b) cross -section plane xy (parallel to a front of an \nincident EM wave ), passing through the disk axis . \n \nFig. 4. Transformations of the wave front of the electric field in a waveguide (at the 1st resonance \nfrequency of 3.0237GHz) on a vacuum pla ne 0.1 mm above a ferrite disk at different time -\nphases. The inserts show enlarged pictures of the electric -field distributions immediately above a \nferrite disk.  \n \nFig. 5. The pictures of the \nE\n -field structures on a vacuum plane 0. 1 mm above a ferrite disk. ( a) \nUndistorted wave front at frequency 3.1363GHz , which is between first and  second MDM \nresonances; ( b) distorted wave front at the 1st resonance frequency of 3.0237GHz.  \n    \nFig. 6. Magnetic -field distributions on a vacuum plane  0.1 mm above a ferrite disk at different \ntime phases. The frequency is the 1st resonance frequency of 3.0237GHz. The top row: enlarged  21 pictures of the magnetic -field distributions immediately above a ferrite disk. The bottom row: \nstructures of the wave fr onts of the magnetic field.   \n \nFig. 7. Power -flow vortex on a vacuum plane 0.1 mm above a ferrite disk. The frequency is the \n1st resonance frequency of 3.0237GHz.  \n \nFig. 8. Transformations of the electric fields in a region very close to a ferrite disk. The  electric -\nfield vectors are shown in connection with geometry of the MS helical -wave loops [20].  \n \nFig. 9. The topologically originated scattered electric fields with the xy-plane components. ( a), \n(b). The electric field distributions in vacuum above and be low a ferrite disk , respectively,  at the \ndistances  0.1mm from the ferrite -disk planes and at a bias magnetic field directed along z axis. \nThe time phase is\n6t\n . (c), (d). The electric field distributions in vacuum above and below a \nferrite disk , respectively,  at the distances  0.1mm from the ferrite -disk planes and at a bias \nmagnetic field directed along z axis. The time phase is\n15t\n . (e) The same as in Fig. 9(d), but \nat a bias magnetic field directed opposite to  the z-axis direction.  \n \nFig. 10. The electric -field distrubution on the cross sectional planes at the 1st resonance \nfrequency. ( a) The xz plane in a waveguide ; (b) The yz in a waveguide.  \n \nFig. 11. The electric -field-front configuration on a vacuum plane 7  mm above a ferrite -disk plane \ncombined with the cross -sectional outline of the helicity density on the same plane.  \n \nFig. 12. Modification of the electric charge and current densities on waveguide walls at the 1st \nresonance frequency . Surface electric char ges, appearing also as topological charges, originate \nsurface electric currents in the forms of convergent or divergent spirals. ( a) The electric charge \nand current densities on upper waveguide wall; ( b) enlarged pictures of distributions on upper \nwaveguid e wall. ( c) The electric charge and current densities on lower waveguide wall; ( d) an \nenlarged picture of distributions on lower waveguide wall.  \n \nFig. 13. Modification  of the magnetic field on upper waveguide wall at the 1st resonance \nfrequency . There is e vidence for topological magnetic charges on  the waveguide walls. Because \nof symmetry of the magnetic fields in a waveguide along z-axis, one has the same pictures of \ntopological magnetic charges on  the lower waveguide wall.  (a) The magnetic field on upper \nwaveguide wall; ( b) enlarged pictures of distributions on upper waveguide wall.  \n \nFig. 14. A combined picture of t he magnetic -field (arrows) and surface current (lines) \ndistributions on upper waveguide wall  at the 1st resonance frequency . (a) Enlarge d area (with the \nimage of a ferrite disk) for the phase \n21t\n ; (b) the same,  for the phase \n30t\n . \n \nFig. 15. The mechanism of interaction between the flux of the incident single -valued electric \nfield and the flux of the double -valued pseudo -electric fields of MDM ferrite particles. W ith \naveraging at the azimuth coordinates, the electric flux of the incident EM field  is completely \nannihilated by the electric flux of the MDM field.  (a), (b) Two (clockwise and \ncounterclockwi se) circularly rotating \nE\n -fields of the incident wave. ( c), (d) Two azimuth modes \nof pseudo -electric field s originated from double -valued -function edge magnetic currents and \nrotating at a certain direction on a lateral surface of a ferrite disk.  \n \nFig. 16. A structure of two coupled MDM disks embedded in a rectangular waveguide.   20  \nFig. 17. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 3.6 mm.  \n \nFig. 1 8. The electric field  distribution  on the upper surface of ferrite disks . The distance between \ncenters of the coupled disks is 3.6 mm.  (a) At frequency ( 3.0253GHz ) of the first peak of the \nsplitted resonance; ( b) at frequency ( 3.0292GHz ) of the second peak  of the splitted resonance.  \n \nFig. 19. Positions of the wave fronts (localized regions where the electric field of a waveguide \nmode changes its sign) for coupled disks on a vacuum plane passing 0.1 mm above ferrite disks. \nThe distance between centers of the  disks is 3.6 mm.  (a) At frequency (3.0253GHz ) of the first \npeak of the splitted resonance; (b) enlarged pictures of the same distributions.  (c) At frequency \n(3.0292GHz ) of the second peak of the splitted resonance; (b) enlarged pictures of the same \ndistri butions . \n \nFig. 20. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 6 mm.  \n \nFig. 21. The electric field  distribution  on the upper surface of ferrite disks . The distance between  \ncenters of the coupled disks is 6 mm.  (a) At frequency ( 3.031GHz ) of the first peak of the \nsplitted resonance; ( b) at frequency ( 3.0348GHz) of the second peak of the splitted resonance.  \n \nFig. 22. Positions of the wave fronts (localized regions where the e lectric field of a waveguide \nmode changes its sign) for coupled disks on a vacuum plane passing 0.1 mm above ferrite disks. \nThe distance between centers of the disks is 6 mm.  (a) At frequency (3.031GHz ) of the first peak \nof the splitted resonance; (b) enla rged pictures of the same distributions.  (c) At frequency \n(3.0348GHz ) of the second peak of the splitted resonance; (b) enlarged pictures of the same \ndistributions . \n \nFig. 23. T he normalized  helicity density for coupled disks  at cross -section plane yz (orth ogonal  \nto a front of an incident EM wave), passing through the disk axes. The distance between centers \nof the disks is 6 mm. (a) At frequency (3.031GHz ) of the first peak of the splitted resonance; (b) \nat frequency (3.0348GHz ) of the second peak of the spl itted resonance.  \n \nFig. 24. The wave front of the electric field for two coupled disks  at frequency 3.0 333GHz  \nbetween two MDM resonances. Vacuum plane is 0.1 mm above ferrite disk s. The distance \nbetween centers of the disks is 6 mm.  \n \nFig. 25. Reflection coe fficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 12 mm.  \n \nFig. 26. A structure of two far -distant MDM disks embedded in a rectangular waveguide.  The \ndistance between the disks is 30cm.  \n \nFig. 27. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between the disks is 30cm.  \n \nFig. 28. T he normalized  helicity density for coupled disks  at cross -section plane yz (orthogonal  \nto a front of an incident EM  wave), passing through the disk axes. The distance between the \ndisks is 30cm. The upper picture: frequency (3.0353GHz ) of the first peak of the splitted  22 resonance. The lower picture: frequency (3.0369GHz ) of the second peak of the splitted \nresonance.  \n \nFig. 29. A microstrip structure with an embedded ferrite disk. ( a) Geometry of a structure (a bias \nmagnetic field is directed along z axis); ( b) a numerical spectrum for the structure (a transmission \ncoefficient between ports 1 and 2).  \n \nFig. 30. A microstrip structure with two coupled ferrite disks. ( a) Geometry of a structure (a bias \nmagnetic field is directed along z axis, the distance between disks is 9mm); ( b) numerical and \nexperimental split -resonance characteristics for the 1st MDM (a normalized transmis sion \ncoefficient between ports 1 and 2).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  23  \n \n \nFig. 1 . Geometry of a structure:  a normally magnetized ferrite disk inside a rectangular \nwaveguide.  For better illustration of our studies of scattering effects by small ferrite par ticles, \nwe use S -band rectangular waveguide which has the sizes of a waveguide cross section (4 cm \n  \n10 cm) much bigger than sizes of a ferrite disk (the thickness of 0.05 mm and the diameter of 3 \nmm).  \n \n \n          (a) \n \n(b) \n 0 2 3  24 Fig. 2 . MDM resonances of a quasi -2D ferrite disk. ( a) Numerically obtained reflection \ncoefficient. (b) Analytically derived spectral peak positions for the first three MDM resonances.  \n \n \n \n \n(a) \n \n \n \n                                                                           (b) \n \nFig. 3. The normalized  helicity density , calculated based on Eq. (4),  at the resonance frequency \nof the 1st MDM ( f = 3.0237GHz ). (a) Cross -section plane yz (orthogonal  to a front of an incident \nEM wave), passing through the disk axis;  (b) cross -section plane xy (parallel  to a front of an \nincident EM wave ), passing through the disk axis . \n \n  25 \n \nFig. 4. Transformations of the wave front of the electric field in a waveguide (at the 1st resonance \nfrequency of 3.0237GHz) on a vacuum plane 0.1 mm above a  ferrite disk at different time -\nphases . The inserts show  enlarged pictures of the electric -field distributions immediately above a \nferrite disk.    \n                                                                         \n                                                  (a)                                                        ( b) \n \nFig. 5. The pictures of the \nE\n -field structures on a vacuum plane 0.1 mm above a ferrite disk. ( a) \nUndistorted wave front at frequency 3.1363 GHz , which is between first and  second MDM \nresonances;  (b) distorted wave front at the 1st resonance frequency of 3.0237GHz.  \n \n  26 \n \n \n \nFig. 6. Magnetic -field distributions on a vacuum plane 0.1 mm above a ferrite disk at different \ntime phases. The frequency is  the 1st resonance frequency of 3.0237GHz. The top row: enlarged \npictures of the magnetic -field distributions immediately above a ferrite disk.  The bottom row: \nstructures of the wave fronts of the magnetic field.     \n \n \nFig. 7.  Power -flow vortex on a vacuum  plane 0.1 mm above a ferrite disk. The frequency is the \n1st resonance frequency of 3.0237GHz.  \n  27 \n \nFig. 8. Transformations of the electric fields in a region very close to a ferrite disk. The electric -\nfield vectors are shown in connection with geometry of t he MS helical -wave loops [20].  \n                \n \nFig. 9. The topologically originated scattered electric fields with the xy-plane components.  (a), \n(b). The electric field distributions in vacuum above and below a ferrite disk , respectively,  at the \ndistance s 0.1mm from the ferrite -disk planes and at a bias magnetic field directed along z axis. \nThe time phase is\n6t\n . (c), (d). The electric field distributions in vacuum above and below a  28 ferrite disk , respectively,  at the distances  0.1mm  from the ferrite -disk planes and at a bias \nmagnetic field directed along z axis. The time phase is\n15t\n . (e) The same as in Fig. (d), but at \na bias magnetic field directed opposite to the z-axis direction.  \n \n \n         \n \n \nFig. 10. The electric -field distrubution on the cross sectional planes at the 1st resonance \nfrequency. ( a) The xz plane in a waveguide ; (b) The yz in a waveguide.  \n \n \n \n \nFig. 11. The electric -field-front configuration on a vacuum plane 7 mm above a ferrite -disk plane \ncombined with the cross -sectional outline of the helicity density on the same plane.  \n \n  29 \n \n \nFig. 1 2. Modification of the electric charge and current densities on waveguide walls  at the 1st \nresonance frequency . Surface electric charges, appearing also as topolo gical charges, originate \nsurface electric currents in the forms of convergent or divergent spirals. (a) The electric charge \nand current densities on upper waveguide wall ; (b) enlarged picture s of distributions on upper \nwaveguide wall. ( c) The electric char ge and current densities on lower waveguide wall; ( d) an \nenlarged picture of distributions on lower waveguide wall.  \n \n \nFig. 13. Modification  of the magnetic field on upper waveguide wall at the 1st resonance \nfrequency . There is evidence for topological magnetic charges on  the waveguide walls. Because \nof symmetry of the magnetic fields in a waveguide along z-axis, one ha s the same pictures of  31 topological magnetic charges on  the lower waveguide wall.  (a) The magnetic field on upper \nwaveguide wall; ( b) enlarged pictures of distributions on upper waveguide wall.  \n \nFig. 14. A combined picture of t he magnetic -field (arrows)  and surface current (lines) \ndistributions on upper waveguide wall  at the 1st resonance frequency . (a) Enlarge d area (with the \nimage of a ferrite disk) for the phase \n21t\n ; (b) the same,  for the phase \n30t\n . \n \n \n \n  30 Fig. 15. The mechanism of interaction between the flux of the incident single -valued electric \nfield and the flux of the double -valued pseudo -electric fields of MDM ferrite particles. W ith \naveraging at the azimuth coordinates, the electric flux of the inc ident EM field  is completely \nannihilated by the electric flux of the MDM field.  (a), (b) Two (clockwise and \ncounterclockwise) circularly rotating \nE\n -fields of the incident wave. ( c), (d) Two azimuth modes \nof pseudo -electric field s originated from double -valued -function edge magnetic currents and \nrotating at a certain direction on a lateral surface of a ferrite disk.  \n \nFig. 16. A structure of two coupled MDM disks embedded in a rectangular waveguide.  \n     \n \n \n \nFig. 17. Reflection coe fficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 3.6 mm.  \n \n  32 \n \n                                             (a)                                                                  ( b) \n \nFig. 18. The electric field  distribution  on the upper surface of ferrite disks . The distance between \ncenters of the coupled disks is 3.6 mm.  (a) At frequency ( 3.0253GHz ) of the first peak of the \nsplitted resonance; ( b) at frequency ( 3.0292GHz ) of the second peak of  the splitted resonance.  \n \n \n \n \n \nFig. 19. Positions of the wave fronts (localized regions where the electric field of a waveguide \nmode changes its sign) for coupled disks on a vacuum plane passing 0.1 mm above ferrite disks. \nThe distance between centers of t he disks is 3.6 mm.  (a) At frequency (3.0253GHz ) of the first \npeak of the splitted resonance; (b) enlarged pictures of the same distributions.  (c) At frequency \n(3.0292GHz ) of the second peak of the splitted resonance; (b) enlarged pictures of the same \ndistributions . \n \n  33 \n \nFig. 20. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 6 mm.  \n \n \n \n \n                                  \n                                            (a)                                                                 (b) \n \nFig. 21. The electric field  distribution  on the upper surface of ferrite disks . The distance between \ncenters of the coupled disks is 6 mm.  (a) At frequency ( 3.031GHz ) of the first peak of the \nsplitted resonance; ( b) at frequency ( 3.0348GHz) of the second peak of the splitted resonance.  \n \n  34 \n \n \nFig. 22. Positions of the wave fronts (localized regions where the electric field of a waveguide \nmode changes its sign) for coupled disks on a vacuum plane pass ing 0.1 mm above ferrite disks. \nThe distance between centers of the disks is 6 mm.  (a) At frequency (3.031GHz ) of the first peak \nof the splitted resonance; (b) enlarged pictures of the same distributions.  (c) At frequency \n(3.0348GHz ) of the second peak of the splitted resonance; (b) enlarged pictures of the same \ndistributions . \n \n \n \n                                                       (a)                                                     ( b) \n \nFig. 23. T he normalized  helicity density for coupled disks  at cross-section plane yz (orthogonal  \nto a front of an incident EM wave), passing through the disk axes. The distance between centers \nof the disks is 6 mm. (a) At frequency (3.031GHz ) of the first peak of the splitted resonance; (b) \nat frequency (3.0348GHz ) of the second peak of the splitted resonance.  \n \n \n  35 \n \n \nFig. 24. The wave front of the electric field for two coupled disks  at frequency 3.0 333GHz  \nbetween two MDM resonances. Vacuum plane is 0.1 mm above ferrite disk s. The distance \nbetween centers of the disks i s 6 mm.  \n \n \n \nFig. 25. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between centers of the disks is 12 mm.  \n \n \n \n \nFig. 26. A structure of two far -distant MDM disks embedded in a rectangular waveguide.  The \ndistance between the disks is 30cm.  \n  36   \n \n \nFig. 27. Reflection coefficient for two coupled disks at the frequency region of the 1st MDM. The \ndistance between the disks is 30cm.  \n \n \n \n \n \nFig. 28. T he normalized  helicity density for coupled disks  at cross -section  plane yz (orthogonal  \nto a front of an incident EM wave), passing through the disk axes. The distance between the \ndisks is 30cm. The upper picture: frequency (3.0353GHz ) of the first peak of the splitted \nresonance. The lower picture: frequency (3.0369GHz ) of the second peak of the splitted \nresonance.  \n \n  37    \n    \n  \n(a)                 (b) \n \nFig. 29. A microstrip structure with an embedded ferrite disk. ( a) Geometry of a structure (a bias \nmagnetic field is directed along z axis); ( b) a numerical spectrum for the structure (a transmission \ncoefficient between ports 1 and 2).  \n \n \n            \n  \n \n(a)                 (b) \n \nFig. 30. A microstrip structure with two coupled ferrite disks. ( a) Geometry of a structure (a bias \nmagnetic field is directed along z axis, the distan ce between disks is 9mm); ( b) numerical and \nexperimental split -resonance characteristics for the 1st MDM (a normalized transmission \ncoefficient between ports 1 and 2).  \n \n "    },    {        "title": "1301.1945v1.High_coercivity_induced_by_mechanical_milling_in_cobalt_ferrite_powders.pdf",        "content": " \n High coercivity induced by mechanical milling in \ncobalt ferrite powders  \nPonce , A. S. 1, Chagas, E. F.1, Prado, R. J.1, Fernandes, C. H. M. 2, Terezo, A. J.2, and Baggio -\nSaitovitch, E.3 \n1Instituto de Física , Universidade Federal de Mato Grosso, 78060 -900, Cuiabá -MT, Brazil \n2Departamento de Química, Universidade Federal do Mato Grosso, 78060 -900, Cuiabá -MT, \nBrazil \n3Centro Brasileiro de Pesquisas Físicas, Rua Xavier Sigaud 150 Urca. Rio de Janeiro, Brazil.  \nPhone number: 55 65 3615 8747  \nFax: 55 65 3615 8730  \nEmail address: efchagas@fisica.ufmt.br  \n \n \nAbstract  \nIn this work we report a study of the magnetic behavior of ferrimagnetic oxide CoFe 2O4 \ntreated by mechanical milling with different grinding balls. The cobalt fe rrite nanoparticles \nwere prepared using a simple hydrothermal method and annealed at 500oC. The non -milled \nsample presented coercivity of about 1.9 kOe, saturation magnetization of 69.5 emu/g, and a \nremanence ratio of 0.42. After milling, two samples attai ned coercivity of 4.2 and 4.1 kOe, \nand saturation magnetization of 67.0 and 71.4 emu/g respectively. The remanence ratio \nMR/MS for these samples increase to 0.49 and 0.51, respectively. To investigate the influence \nof the microstructure on the magnetic beh avior of these samples, we used X -ray powder \ndiffraction (XPD), transmission electron microscopy (TEM), and vibrating sample \nmagnetometry (VSM). The XPD analysis by the Williamson -Hall plot was used to estimate \nthe average crystallite size and strain induc ed by mechanical milling in the samples.  \nKeywords: Cobalt ferrite, (BH) max product, Strain, Coercivity, Milling.  \nIntroduction  \n \nCobalt ferrite (CoFe 2O4) is a promising compound with potential biomedical applications  \n [1,2], as well as use in high density mag netic storage [3], permanent magnets [4], and \nelectronic devices. This is due to several properties, such as electrical insulation, chemical \nstability, high magnetic -elastic effect [ 5], moderate saturation magnetization, and high \ncoercivity (H C). \nParticula rly in permanent magnet applications, high HC is a fundamental characteristic mainly \nbecause  HC is an important parameter to the maximum energy product (BH)max, the figure \nof merit for permanent magnets. Many authors report tuning the HC through thermal \nannealing [6], capping [7] and mechanical milling treatment [8] of the grains . Limaye  et al.  \n[7] obtained cobalt ferrite nanoparticles with very high coercivity (9.5 kOe) by capping it \nwith oleic acid; however, the saturation magnetization decreased to 7.1 e mu/g – that is, about \n10 times less than that obtained from uncapped nanoparticles. This small value of saturation \nmagnetization (M S) is an undesirable property in hard magnetic applications. In another \ninteresting article, Liu et al.  [9] increased the H C of cobalt ferrite, from 1.23 to 5.1 kOe, with \na relatively small decrease in MS. They used a brief (1.5 h) mechanical milling process on \nrelatively large particles (average grain size of 240 nm). However, for nanoparticles with an \naverage grain size of 12 nm, no improvement in magnetic parameters was obtained.  \nApplication of mechanical milling to nanoparticles (diameter < 100 nm) seeking to increase  \nHC has not yet been successful. However, few important phenomena happen only at the \nnanoscale. Exchange spri ng (exchange coupled) is an example. High HC cobalt ferrite make \npossible an expressive increase in the (BH) max in exchange coupled nanocomposite \nCoFe 2O4/CoFe 2 [10,11 ]. The importance of studying the magnetic behavior of this \nnanocomposite, aiming to impro ve its magnetic properties, is what led to this work.  \nIn order to increase the HC of cobalt ferrite nanoparticles, we processed them with thermal \nannealing at moderate temperature (500 °C) and brief mechanical milling. Using a shake \nmiller and several diff erent milling parameters, we increased the HC of all treated samples. \nThe result of this processing was an increase of about 150% in coercivity. Also, the M R/MS \nratio improved from 0.42 to 0.49 and 0.51 for the two best samples, and the (BH) max was \namplifi ed almost 100%. This substantial enhancement in magnetic parameters justifies the \npublication of this study. Samples were analyzed by X -ray powder diffraction (XPD), \ntransmission electron microscopy (TEM), and magnetic hysteresis curves, all at room \ntemper ature.  \n  \n Experimental procedure  \nThe cobalt ferrite nanoparticles were synthesized using a simple hydrothermal method \ndescribed in a previous work [10]. The morphology and particles size distribution of the \nsamples were examined by direct observation via tra nsmission electron microscopy (TEM) \nusing JEOL -2100 apparatus installed at LNNano / LNLS – Campinas – Brazil woking at 200 \nkV . \nThe pristine sample (without thermal neither mechanical milling treatments)  was annealed at \n500 °C for 6 hours and separated in 5  amounts called CF0, CF1, CF2, CF3 and CF4. The \nsample CF0 was not milled and the other samples were milled during 1.5 hours using \ndifferent milling parameters. The milling process used was the high energy mechanical ball \nmilling in a Spex 8000 miller. Det ails of the different milling processes to each sample are \ndescribed in table I.  \nTable I – Details of mechanical milling process.  \nSample  Ball material \n(mass density)  Ball diameter  Proportion \nBall/sample  Milling time (h)  \nCF1 Stainless steel  \n(7.8 g/cm3) 6.5 mm 1:4 1.5 \nCF2 Zirconia  \n(6.1 g/cm3) 6.0 and 1.2 -1.4 \nmm*  1:7:3  1.5 \nCF3 Zirconia  \n(6.1 g/cm3) 6.0 mm  1:7 1.5 \nCF4 Tungsten carbide \n(15 g/cm3) 5.0 mm  1:9 1.5 \n* The sample CF2 was milled by balls of both indicated sizes.  \n \nThe crystalline phases of the cobal t ferrite nanoparticles were identified by X -ray powder \ndiffraction (XPD) patterns, obtained on a Shimadzu XRD -6000 diffractometer installed at the \n“Laboratório Multiusuário de Técnicas Analíticas ” (LAMUTA / UFMT – Cuiabá -MT - \nBrazil), equipped with graphi te monochromator and conventional Cu tube (0.154178 nm) \nworking at 1.2 kW (40 kV , 30 mA), using the Bragg -Brentano geometry.  \nMagnetic measurements were carried out using a vibrating sample magnetometer (VSM) \nmodel VersaLab Quantum Design installed at CBPF,  Rio de Janeiro -RJ – Brazil. Experiments  \n were done at room temperature and using magnetic field up to 2.5 T.  \n \nResults and discussion  \nThe XPD patterns obtained from samples CF0, CF1, CF2, CF3 and CF4 (see figure 1) \nconfirming that all samples are CoFe 2O4 with the expected inverse spinel structure. These \nmeasurements indicate the absence of any other phases or contamination before and after \nmilling processing.  \nTEM images of the pristine cobalt ferrite nanoparticles revealed an extremely polydisperse \nsystem wi th several forms (see figure 2 a). The particle size distribution indicates that cobalt \nferrite particles have a mean diameter of 22.6 nm; however TEM images also revealed the \npresence of particles larger than 100 nm. TEM measurements also indicate that nan oparticles \nincreased after thermal treatment, changing the mean diameter to 44.7 nm. As occurred for \nthe pristine sample, some big particles were observed to sample CF0 (figure 2 b). These \nresults indicate that, despite the relatively low annealing temperat ure, there was coalescence \nof nanoparticles causing the increase of mean size.  \nFor samples CF3 and CF4 the TEM images revealed similar particles with shear bands (see \nthe figure 3 a and 3 b). We observed both regular and irregular shear bands in good agreeme nt \nwith the microstructural evolution described by Liu et al. [9]. These observations suggest that \nthe shear bands are moiré fringes (see figure 3 c). \nTo evaluate the influence of mechanical milling on the magnetic behavior of our samples, we \nmeasured the m agnetic hysteresis loop. These measurements are shown in figure 4. Three \ninteresting characteristics can be noted from the hysteresis loops obtained from milled \nsamples: the increase in H C, the changes in MS, and the increase of MR to all milled samples. \nThese characteristics are discussed in detail below.  \nSamples CF3 and CF4 presented the highest HC values found in this work, 4.1 and 4.2 kOe, \nrespectively. This enhancement in HC, when compared with the value obtained for sample \nCF0 (no mechanical milling),  is very expressive and indicates an increase of about 120% \nafter milling, for both samples. However, the maximum energy product for sample CF4 (0.88 \nMGOe) was smaller than that observed in sample CF3 (1.1 MGOe) due the different MR for \nboth samples – 36.8 emu/g for sample CF3 and 33.0 emu/g for CF4. The main magnetic \ncharacteristics obtained for all the samples studied in this work are shown in Table II.  \n  \n Table II – Magnetic characteristics of milled cobalt ferrite at room temperature.  \nSample  Coercivity (k Oe) MR(emu/g)  Mr/Ms (BH) max (MGOe)  MS(emu/g)  \nCF0 1.9 29.2 0.42 0.54 69.5 \nCF1 2.5 29.4 0.44 0.61 66.6 \nCF2 3.7 29.3 0.45 0.66 64.6 \nCF3 4.1 36.8 0.51 1.1 71.4 \nCF4 4.2 33.0 0.49 0.88 67.0 \n \nTo investigate the influence of the annealing and milling process es on the strain and average \ncrystallite size of the samples, we used the Williamson -Hall plot [ 12-14] obtained by the \nXPD measurements,  as in reference [13]. The analysis shows that broadening of the \ndiffraction peaks after milling evidences both grain si ze reduction and increase in \nmicrostructural strain. The instrumental broadening was corrected using the Y2O3.diffraction \npattern as the standard. The results of the Williamson -Hall plot analysis are shown in figure \n5. \nTo facilitate the analysis of the inf luence of structural parameters on the magnetic behavior of \nthe samples, we included in Table III results obtained from the Williamson -Hall analysis  \ntogether with coercivity and saturation magnetization . \n \nTable III – Results obtained from Williamson -Hall a nalysis, coercivity and saturation \nmagnetization.  \n \nSample  Grain size (nm)  Strain (%)  HC  (kOe)  MS (emu/g)  \nCF0 64(5)  0.32(3)  1.9 69.5 \nCF1 34(2)  0.34(3)  2.5 66.6 \nCF2 17(2)  0.8(1)  3.7 64.6 \nCF3 25(2)  0.8(1)  4.1 71.4 \nCF4 21(1)  1.0(1)  4.2 67.0 \n \nAs one can see, milled samples presented a smaller average crystallite size, and \nmicrostructural strain -induced by milling was observed in samples CF2, CF3 and CF4. The \nstrain from samples CF0 and CF1 were similar, within the margin of experimental errors of  \n the meth od. The strain observed in sample CF4 was very close to the value obtained by Liu \nand Ding [8] for their sample with highest coercivity. The main difference between our \nsample CF4 and that of Liu and Ding's [8] work is the average size of the particles. In  Liu \nand Ding's work the average particle size was about 110 nm, while in our study it was 21 nm. \nBesides the mean diameter of the nanoparticles from sample CF4 being smaller than the \ncritical size expected for the magnetic multi -domain regime [15], many b ig particles with a \ndiameter larger than this critical size were observed in TEM images. Thus we conclude that \nparticles in the multi -domain regime play an important role in magnetic behavior. Pinning \neffects observed by Liu et al.  [9] must also be importa nt to our samples. Structural defects \nthat produce the pinning of wall domains were observed with high magnification TEM \nmeasurements (see figure 6). We conclude that a combination of strain and pinning effect is \nresponsible for the increase of H C in sampl e CF4.  \nThe increase of HC observed in the other samples (CF1, CF2, and CF3) is clearly associated \nwith the strain induced by mechanical milling. The samples presenting the two highest H C \nvalues, CF3 (H C = 4.1 kOe) and CF4 (H C = 4.2 kOe), also have the larg est strains: 0.8 and \n1.0, respectively. The sample with lowest H C, CF1 (H C = 2.5 kOe), presents the smaller strain \n(0.34). To sample CF3, as to sample CF4, shear bands were observed in TEM images (not \nshowed) suggesting that pinning effect play an importan t role in the magnitude of H C also to \nthis sample.  \nThe second important characteristic observed in the hysteresis loops was the change in \nsaturation magnetization ( MS) in the milled samples. The M S values were obtained by \nextrapolating 1/H to zero in the p lot magnetization versus 1/H. Except for the sample CF3, \nthe MS decreased in the other milled samples. MS increased from 69.5 emu/g (sample CF0) to \n71.4 emu/g in the sample CF3, but for sample CF2 there was a decrease of about 7.5%. \nNormally, when M S decre ases together with crystallite size, the decrease in MS is attributed to \nthe surface effect, sometimes called the “dead” surface [16]. The dead surface is associated \nwith disorder of surface spins. The number of surface spins in a sample increases when the  \ncrystallite size decreases, thus it is expected that the M S will also decrease. However, this \nexplanation is not consistent with all our results. If compared with the non -milled sample \nCF0, the saturation magnetization of the samples CF1, CF2 and CF4 decr eases with the \ncrystallite size, but the MS for sample CF3 was higher than that obtained for sample CF0 (see \nTable III). Thus we attributed the decrease in saturation magnetization to a combination of  \n two factors: the surface effect [17] and redistribution  of magnetic cations [18].  \nTo understand the cation redistribution it is necessary to know the cobalt ferrite structure and \nthe distribution of magnetic ions in this structure. The cobalt ferrite has an inverse spinel \nstructure with the two magnetic ions (Co2+ and Fe3+) occupying  two different sites called A \nand B. The A site is a tetrahedral site and the B site is an octahedral site. In an ideal inverse \nspinel cobalt ferrite, half of the Fe3+ cations (magnetic moment equal to 5μ B) occupy the A -\nsites and t he other half the B -sites, together with Co2+ cations. (There are two B sites for \nformula unit.) Since the magnetic moments of the ions in the A and B sites are aligned in an \nanti-parallel way, the magnetic contribution of the Fe3+ cations are mutually com pensated. \nTherefore, the net magnetic moment of the ideal inverse spinel cobalt ferrite  is due only to \nthe Co2+ cations (magnetic moment equal to 3μB). However, the cobalt ferrite normally  \npresents a mixed spinel structure, the sites A and B have a fraction of Co2+ and Fe3+ cations. \nTherefore, in a general way, the structure of cobalt ferri te can be described by reference [18]:  \n         (1) \nwhere i (the degree of inversion) describes the fraction of the tetrahedral sites occupied by \nFe3+ cations. The ideal inverse spinel structure has i = 1, i = 0 to the normal spinel and to the  \nmixed spinel structure the value of i is between 0 and 1.  \nIf Co2+ ions move from A -sites to B -sites and consequently Fe3+ cations move from B -sites to \nA-sites, there will be a change in the degree of inversion i (see equation 1), a decrease of the \nnet mag netic moment of the material, and, consequently, a decrease in M S. On the other hand, \nif the opposite process happens the saturation magnetization will increase. In a recent article, \nSato Turtelli et al.  [18] showed that samples prepared by mechanical mill ing present an i \nfactor smaller than that obtained for samples synthesized by sol gel method. Another \nindication that there is a cationic redistribution in CoFe 2O4, due mechanical milling is the \nwork of Stewart et al. [19]. They observed cation swap betwee n magnetic site A and B in \nnormal spinel ferrite ZnFe 2O4 due the mechanical milling. In addition, the cationic \nredistribution described before is consistent with the enhancement of the H C due to changes \nin the magneto -crystalline anisotropy. According to S ato Turtelli et al.  [18] more Co2+ in B-\nsite (smaller value of i factor) increases the magneto -crystalline anisotropy and, consequently, \nincreases the coercivity.  \nThe third interesting result observed in magnetic measurements is the increase of remanence \n(MR) and, consequently, the increase of the remanence ratio (M R/MS) obtained for all of the  \n milled samples. Sample CF3 presented the most expressive increase of M R and, consequently, \nhighest M R/MS ratio (0.51). There are two possible causes of this effect  (desirable for hard \nmagnetic applications): first are the changes of magnetic domain regime between sample CF0 \nand samples CF1, CF2, CF3 and CF4. The sample CF0 has an average size of about 65 nm, \nindicating that this sample is in magnetic multi -domain reg ime [15]. Thus the magnetization \nprocess is caused mainly by the movement of domain walls. However, when the grain is \nsmaller than 40 nm the cobalt ferrite is in single magnetic domain regime and the \nmagnetization curve is governed by rotation of the magne tic moments. The second possible \nexplanation is the enhancement of cubic magneto -crystalline anisotropy, indicated by the \napproximation of the theoretical value expected to non -interacting particles with positive \ncubic anisotropy (0.83 at 0K) [20]. Similar  effect occurs when the temperature decreases \n[10,21] . \n \nConclusion  \nIn agreement with other authors, our results suggest that strain plays an important role in the \nHC value. Structural defects were observed in big particles (around of 100 nm), these pinnin g \ncenters must be important to high HC behavior.  \nThe increase of M S observed in sample CF3 suggest that high energy ball milling can \npromote a cationic swap between Co2+ and Fe3+ from A and B sites respectively. However, \nthis assumption deserves more inves tigation.  \nWe achieved a significant increase of the (BH) max for sample CF3 compared with non -milled \nsamples due the  HC and the M R enhancement. The increase of the (BH) max in sample CF4 was \nsmaller, indicating that grinding ball mass density plays an import ant role in the magnetic \nbehavior of milled samples.  \n \nAcknowledgments  \nThis work was supported by CAPES Brazilian funding agency (PROCAD -NF project \n#2233/2008). The authors would like to thank the LNNano/LNLS for technical support \nduring electron microscopy  work  \n \nReferences  \n \n[1]  M. Colombo, S. Carregal -Romero, M. F. Casula, L. Gutiérrez, M. P. Morales, I. B.  \n Bohm, J. T. Heverhagen, D. Prosperi, W. J. Parak, Biological applications of magnetic \nnanoparticles, Chem Soc Rev. 41 (2012) 4306 -4334.  \n \n[2] T.E. Torr es, A.G. Roca, M.P. Morales, A. Ibarra, C. Marquina, M.R. Ibarra, G.F. Goya, \nMagnetic properties and energy absorption of CoFe2O4 nanoparticles for magnetic \nhyperthermia. J. Phys.: Conf. Ser.  200 (2010) 072101.  \n \n[3] V . Pillai, D.O. Shah, Synthesis of high -coercivity cobalt ferrite particles using water -in-oil \nmicroemulsions,  J. Magn. Magn. Mater.  163 (1996) 243 -248. \n \n[4] Y . C. Wang, J. Ding, J. B. Yi, B. H. Liu, T. Yu, Z. X. Shen, High -coercivity Co -ferrite thin \nfilms on (100) -SiO2 substrate, Appl. Phys. L ett. 84 (2004) 2596 -2598.  \n \n[5] I. C. Nlebedim, J. E. Snyder, A. J. Moses, D. C. Jiles,  Dependence of the magnetic and \nmagnetoelastic properties of cobalt ferrite on processing parameters, J. Magn. Magn. Mater. \n322 (2010) 3938 -3942.  \n \n[6] W.S. Chiua, S. Rad iman, R. Abd -Shukor, M.H. Abdullah, P.S. Khiew, Tunable coercivity \nof CoFe2O4 nanoparticles via thermal annealing treatment, J Alloy Compd.  459 (2008) 291 –\n297. \n \n[7] M. V . Limaye, S. B. Singh, S. K. Date, D. Kothari, V . R. Reddy, A. Gupta, V . Sathe, R. J. \nChoudhary, and S. K. Kulkarni, High Coercivity of Oleic Acid Capped CoFe2O4 \nNanoparticles at Room Temperature, J. Phys. Chem. B  113  (2009) 9070 –9076.  \n \n[8] B. H. Liu, J. Ding, Strain -induced high coercivity in CoFe2O4 powders, Appl. Phys. Lett. \n88 (2006) 04 2506.  \n \n[9] B. H. Liu, J. Ding, Z. L. Dong, C. B. Boothroyd, J. H. Yin, and J. B. Yi, Microstructural \nevolution and its influence on the magnetic properties of CoFe2O4 powders during \nmechanical milling, Phys. Rev. B 74 (2006) 184427.  \n \n[10] G.C.P. Leite, E.F . Chagas, R. Pereira, R.J. Prado, A.J. Terezo, M. Alzamora, E. Baggio -\nSaitovitch, Exchange coupling behavior in bimagnetic CoFe2O4/CoFe2 nanocomposite, J. \nMagn. Magn. Mater. 324 (2012) 2711 –2716.  \n \n[11] J. M. Soares, F. A. O. Cabral, J. H. de Araújo, and F.  L. A. Machado, Exchange -spring \nbehavior in nanopowders of CoFe2O4 –CoFe2 Appl. Phys. Lett. 98 (2011) 072502.  \n \n[12] G. K.Williamson, W. H. Hall, X ray line broadening from filed aluminium and wolfram \nActa Metall. 1, (1953) 22 -31. \n \n[13] V . D. Mote, Y . Purush otham, B. N. Dole, Williamson -Hall analysis in estimation of \nlattice  \nstrain in nanometer -sized ZnO particles J. Theor. Appl. Phys. 6:6 (2012).  \n \n[14] A. Khorsand Zak, W.H. Abd. Majid, M.E. Abrishami, Ramin Yousefi, X -ray analysis of \nZnO nanoparticles by Wi lliamson -Hall and size -strain plot methods Solid State Sci. 13 (2011)  \n 251-256. \n \n [15] Debabrata Pal, Madhuri Mandal,  Arka Chaudhuri, Bipul Das, Debasish Sarkar, Kalyan \nMandal, Micelles induced high coercivity in single domain cobalt -ferrite nanoparticles J . \nAppl. Phys. 108 (2010) 124317.  \n \n[16]  A. Franco, F. C. e Silva, Vivien S. Zapf,  High temperature magnetic properties of Co 1-\nxMg xFe2O4 nanoparticles prepared by forced hydrolysis method J. Appl. Phys. 111 (2012) \n07B530 . \n \n[17] M. Rajendran, R.C. Pullar, A. K. Bhattacharya, D. Das, S.N. Chintalapudi, C.K. \nMajumdar , Magnetic properties of nanocrystalline CoFe2O4 powders prepared at room \ntemperature: variation with crystallite size J. Magn. Magn. Mater. 232 (2001) 71 –83. \n \n[18] R.  Sato Turtelli, M.  Atif, N.  Mehmooda, F.  Kubel, K.  Biernacka, W.  Linert, R. \nGrössingera, Cz.  Kapusta, M.  Sikora, Interplay between the cation distribution and \nproduction methods in cobalt ferrite Mat. Chem. Phys. 132 (2012) 832 –838. \n \n[19] S. J. Stewart,  S. J. A. Figueroa, J. M. Ramallo López, S. G. Marchetti, J. F. Bengoa, R. J. \nPrado, F. G. Requejo, Cationic exchange in nanosized ZnFe2O4 spinel revealed by \nexperimental and simulated near -edge absorption structure  Phys. Rev. B 75 (2007) 073408.  \n \n[20] M. P. Pileni Size and morphol ogy control of nanoparticles growth in organized \nsurfactant assemblies, in Janos H. Fendler, Imre Dékány (Eds.) Nanoparticles and \nNanostructured Films , Kluwer Academic Publishers, Dordrecht, 1996, pp 80 . \n \n[21] E. Veena Gopalan, I. A. Al -Omari, D. Sakthi Ku mar, Yasuhiko Yoshida, P.A. Joy, M.R. \nAnantharaman, Inverse magnetocaloric effect in sol –gel derived nanosized cobalt ferrite \nAppl. Phys. A 99 (2010) 497 -503. \n \nFigures   \n \n \nFigure 1 – XPD patterns of Co ferrite.  \n \n \nFigure 2 – TEM images of a) the pristine sam ple and b) sample CF0.   \n \n \nFigure 3 – TEM images showing shear bands to a) CF4 and b) CF3 samples TEM image,  c) \nregular, and d) irregular squematic pictures of moire fringes. The insert of a) shows details of \nthe marked area.  \n  \n \n \nFigure 4 - Hysteresis loops for CoFe 2O4 nanoparticles at room temperature (300 K) for \nmaximum applied field of 25 kOe.  \n  \n \nFigure 5 – Willianson -Hall plot to milled cobalt ferrite.   \n  \n \nFigure 6 - Structural defects in crystalline plans to milled cobalt ferr ite (sample CF4). The \nmarked area indicate a structural defect.  \n "    },    {        "title": "1012.5743v1.Transmission_Electron_Microscopy_Studies_on_RF_Sputtered_Copper_Ferrite_Thin_Films.pdf",        "content": "TRANSMISSION ELECTRON MICROSCOPY STUDIES ON RF SPUTTERED COPPER \nFERRITE THIN FILMS \n \n         Prasanna D. Kulkarni and Shiva Prasad    Indradev Samajdar           \n         Department of Physics                               Metallurgical Engineering & Materials Science \n         IIT Bombay, Mumbai 400076                   IIT Bombay, Mumbai 400076 \n \n         N. Venkataramani                                      R. Krishnan \n         ACRE IIT Bombay                                   Laboratoire de Magnetisme et d’optique de Versailles,                \n         Powai, Mumbai 400076                            CNRS, 78935 Versailles, France  \n \n \n        ABSTRACT \nCopper ferrite thin films were rf sputtered at  a power of 50W. The as  deposited films were \nannealed in air at 800°C and slow cooled. The transmission electron microscope (TEM) studies \nwere carried out on as deposited as well as on sl ow cooled film. Significantly larger defect \nconcentration, including stacking faults, was obser ved in 50W as deposited films than the films \ndeposited at a higher rf power of 200W. The film annealed at 800 °C and then slow cooled \nshowed an unusual grain growth upto 180nm for a film thickness of ~240nm. These grains  \nshowed Kikuchi pattern.  \n \nINTRODUCTION \nIt has been observed that changing power signifi cantly alters the microstructure and magnetic \nproperties of the rf sputtered films. In the case of hexagonal strontium ferr ite films, the films are \noriented when deposited at a low rf power1. Recently, our group has carried out extensive studies \non copper ferrite films2. The copper ferrite films have been deposited at different rf powers and \nsubjected to ex-situ thermal treatment. Copper ferrite exhibits two different phases. A high \ntemperature cubic phase and a room temperatur e tetragonal phase. The high temperature phase \ncan be stabilised at room temperature after the films are annealed and quenched. The detailed structural and magnetic properties of the 200W deposited copper ferrite films have been reported earlier\n2.  The as deposited films are cubic. For the slow  cooled films, a phase transforation  from \ncubic to tetragonal has been observed on heating the film. When copper ferrite thin films are \ndeposited at a lower rf power of 50W, a strong (lll ) orientation is observed for the quenched \nfilms. These orientation features and magnetic  properties of the quenched 50W copper ferrite \nfilms are reported earlier3. In this paper, the detailed TEM st udies carried out to investigate the \nmicrostructure and magnetic properties of the 50W as deposited and slow cooled films are \nreported.   \n \nEXPERIMENTAL \nThe copper ferrite films are deposited using the Leybold Z400 rf sputtering system. The films are \ndeposited on amorphous quartz as well as silicon (111) substrates.  No heating or cooling was carried out during sputtering. The rf power em ployed during the deposition is 50W. The film \nthickness is ~240nm. The as deposited film is ex-situ annealed, at 800° C, followed by slow \ncooling. The structure of the as deposited and slow cooled films are studied using X-ray diffraction (Phillips X-pert system) and the micr ostructure by Transmission Electron Microscope \n(Phillips CM200). The samples for TEM are prepar ed by chemical etching of silicon in a 3:1 solution of HNO 3 and HF. Magnetization studies are ca rried out using Vibrating Sample \nMagnetometer (VSM), up to a field of 0.7T.  \n \nRESULTS AND DISCUSSION \nFig.1(a) shows the XRD pattern for the as depos ited copper ferrite film deposited at 50W rf \npower. A broad hump at a 2 θ value of 35 degree is observed in this case. A 100% intensity peak \nof copper ferrite is centred around this 2 θ value.  This has been reported as a (311) peak in the \nJCPDS data for bulk copper ferrite.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1. XRD patterns of 50W copper ferrite films, (a) As deposited (b) 800 °C Slow cooled. 40 60 80\n  Intensity (a.u.)\n2θ (degree)(a)\n20 40 60 80(311)\n(444)(333)(222) (111)\n  Intensity (a.u.)\n2θ (degree)(b)\n \nThe as deposited film has been annealed in air at 800 °C and slow cooled. The XRD pattern of the \nslow cooled film is shown in the Fig. 1(b). The unit cell of the tetragonal phase of copper ferrite \nhas been transposed to a pseudo cubic unit cell2. Miller indices of the planes indexed to the XRD \npattern are with respect to the transposed unit ce ll. This is done to facilitate the comparison with \nthe cubic phase of copper ferrite. All the peaks obs erved in the XRD pattern could be indexed to \nthe copper ferrite spinel structure. The (111), (222), (333), (444) peaks are present along with a \n(311) peak. The (222) peak which is about 8% in tense peak in the bulk copper ferrite is a 100% \nintensity peak in the slow cooled  film. This shows that the slow cooled film has a strong (111) \norientation, which is the plane c ontaining the highest packing density for the spinel structure. \nThese orientation features are not observed in the 200W deposited films2, but are seen in the 50W \nquenched film. The usual splitting of (333) and (511)  peaks, which is a signature of the presence \nof tetragonal phase is not seen in the XRD pattern  of slow cooled 50W f ilm. This may be due to \nthe strong orientation present in the film. Hence, the phase of the slow cooled film in the low \npower case is not conclusive from XRD pattern alone.   \n \nThe Transmission Electron Microscope (TEM) studies are carried out on the 50W as deposited \nand the slow cooled films. Fig.2(a) shows th e TEM bright field image and the corresponding \ndiffraction pattern for the as deposited copper ferrite film. In the bright fiel d image, large defect \nconcentration is observed, along with a presence (a t higher magnification) of stacking faults. The \ngrains with similar orientations are clusterd, as  confirmed by conical dark field imaging, with the \ncluster sizes in the range of 10-25nm.   \n \n \n \n \n \n \n \n \n \n \n \n 500nm \na \n 500nm\nb \n 500nm\nc\n \nFig. 2. TEM images for 50W copper ferrite films, (a) As deposited, (b) 800°C Slow cooled \n(bright field image), (c) 800°C Slow cooled (conical dark field image).   \n \nThe bright field image for the slow cooled film and the corresponding diffraction pattern is \nshown in Fig.2(b). The crystallites within 5 degree misorientation or falling within near Bragg, are considered as a grain. The grain sizes are identified by conical dark field imaging in TEM. \nFig. 2(c) shows the dark field image of the slow cooled film. The grain sizes lie in the range 10-\n180nm. Clear signature of abnormal grain growth is observed in the film. The large defect concentration and clustering of the grains of similar orientation is not observed in the as \ndeposited films of 200W\n2. Additionally the grain growth in 200W deposited and then annealed at \n800°C are restricted at ~40nm. This has been attributed to pinning by the low angle grain \nboundaries2. For the same growth mechanism, a trans ition from normal to abnormal grain growth \ncan be caused by pinning (either through sec ond phase or through higher low angle boundary \nconcentration) or relative differences in the concentration of defects4. In view of the fact that \ndefet concentration in as deposited 50W film is larger than 200W film, the growth seems to be \nbecause of relative difference in concentration of defects. Both 50W and 200W films go through \na phase transformation while annealing at 800°C. The 200W film, however, does not show the extent of defect elimination, as observed in the case of 50W film through Kikuchi. The Kikuchi \npattern observed for a 50W slow cooled film is  shown in the inset of Fig.2(c). To the best \nknowledge of authors, this is the first observati on of Kikuchi in ferrite thin films. Kikuchi \nformation needs a subtle balance of  elastic and inelastic scattering\n5. The later, should be stronger \nin nano crystalline ferrite films due to high defect  concentration and should not allow presence of \nKikuchi. In the 50W annealed material, however , even from convergent spots corresponding to \nlower grain sizes (~20nm) Kikuchis are clearly visible in TEM. This is an indirect indication of the elimination of defects in the 50W slow cooled films.  \n \nFig 2(a) shows the diffraction pattern for the as deposited film. The diffraction pattern of the as \ndeposited film could be indexed to the cubic phase  of copper ferrite. Inset of Fig.2(b) shows the \ndiffraction pattern for the slow cooled film. The splitting2 of the rings corresponding to the same \nh2+k2+l2 of a plane is not observed in the diffrac tion pattern. Such a splitting of the rings is \nclearly observed in 200W slow cooled film2. This may be due to the orientation present in the \nfilm. However, TEM results when taken with the magnetisation data (to be discussed next) leads \nto the inference that the slow cooled film is in tetragonal phase.  -8000 -4000 0 4000 8000-150001500\n  M (Gauss)\nH (Oe)(b)\n-4000 -2000 0 2000 4000-1000-50005001000 \n M (Gauss)\nH (Oe)(a) \n \n \n \n \n \n \n \n \n \n \n \n \nFig.3 Parallel MH loops for 50W copper ferrite films, (a) As deposited, (b) 800°C Slow cooled.  \n \nThe magnetic measurements have been done on the as deposited as well as slow cooled 50W \ncopper ferrite films. Fig.3(a) shows the paralle l MH loop for the as deposited 50W film. The \nsaturation magnetization calculated from the para llel MH loops is ~750 Gauss. This value is \nmuch lower than the saturation magnetization of ~1100 Gauss in the 200W as deposited films2. \nThe magnetic properties of the slow cooled 50W f ilm are similar to the slow cooled 200W film. \nFig. 3(b) shows the parallel MH loop for the 50W slow cooled film. The saturation magnetization \ncalculated from the parallel MH loop is ~1500 Gauss. The saturation magnetization in case of 200W deposited slow cooled flims is ~1600 Gauss.  The smaller value of saturation magnetization \nin 50W slow cooled film than  in case of 200W slow cooled film, is not well understood. The \nmangetization realised in the 50W and 200W slow cooled films may not be solely due to grain \nsizes.  \nIn any case, the saturation mangetization value of ~1500 Gauss for 50W slow cooled film is close \nto the bulk value (~1700 Gauss) for tetragonal copper ferrite. This confirms that the slow cooled \nfilms are in tetragonal phase of copper ferrite.  \n \nCONCLUSIONS \nThe present study shows the rf power employed during sputtering alters the microstructure of the copper ferrite thin films significantly. At a lower rf power the defect concentration increases along with orientation along the [111] direction. Due to the crystalline orientation, the specific \npeaks identifying the tetragonal phase are not observed in XRD and TEM data. The \nmagnetization value confirms the presence of the tetragonal phase. The slow cooled films show \nthe abnormal grain growth and Kikuchi pattern.  \n \n \nACKNOWLEDGEMENT \nThe author Prasanna D. Kulkarni thanks the CSIR , India for financial support. Two of the authors \nShiva Prasad and N. Venkataramani thank the IIT  Bombay heritage fund for the financial support \nto attend the conference.  \n \n \n REFERENCES \n        1B. Acharya, S.N. Piramanayagam, A. Ajan, S. N. Shringi, S. Prasad, N. Venkataramani, R. \nKrishnan, S.D. Kulkarni, S.K. Date, “Oriented st rontium ferrite films sputtered onto Si(111),” \nJournal of Magnetism and Magnetic Materials  140-144 (1995) 723-724.  \n     2M.Desai, S. Prasad, N. Venkataramani, I. Sa majdar, A.K. Nigam, R. Krishnan, “Annealing \ninduced structural change in sputter deposited c opper ferrite thin films and its impact on magnetic \nproperties,” Journal of Applied Physics , 91[4] 2220-27 (2002). \n        3P. Kulkarni, M. Desai, N. Venkataramani, S.  Prasad, R. Krishnan, “Low power RF sputter \ndeposition of oriented copper ferrite films,” Journal of Magnetism and Magnetic Materials  272-\n276 (2004) e793-e794.  \n       4F.J.Humphreys and M. Hatherly, “Abnormal grain growth”; pp. 314-325 in Recrystallization \nand Related Annealing Phenomena , 1st Edition, Pergamon, New York, 1995. \n       5P.J. Goodhew, J. Humphreys, R. Bean land, “Kikuchi line patterns”; pp. 58-61 in Electron \nMicroscopy and Analysis, 3rd ed., Taylor & Francis, London, 2000. "    },    {        "title": "2112.12456v1.Control_of_site_occupancy_by_variation_of_the_Zn_and_Al_content_in_NiZnAl_ferrite_epitaxial_films_with_low_magnetic_damping.pdf",        "content": "arXiv:2112.12456v1  [cond-mat.mtrl-sci]  23 Dec 2021Control of site occupancy by variation of the Zn and Al conten t in NiZnAl ferrite\nepitaxial ﬁlms with low magnetic damping\nJulia Lumetzberger,1,∗Verena Ney,1Anna Zhakarova,2Daniel Primetzhofer,3Kilian Lenz,4and Andreas Ney1\n1Institute for Semiconductor and Solid State Physics,\nJohannes Kepler University Linz, Altenberger Straße 69, 40 40 Linz, Austria\n2Swiss Light Source (SLS), Paul Scherrer Institut, 5232 Vill igen PSI, Switzerland\n3Department of Physics and Astronomy, /RingAngstr¨ om Laboratory,\nUppsala University, Box 516, SE-751 20 Uppsala, Sweden\n4Helmholtz-Zentrum Dresden – Rossendorf, Institute of Ion Be am Physics and Materials Research,\nBautzner Landstraße 400, 01328 Dresden, Germany\n(Dated: December 24, 2021)\nThe structural and magnetic properties of Zn/Al doped nicke l ferrite thin ﬁlms can be adjusted\nby changing the Zn and Al content. The ﬁlms are epitaxially gr own by reactive magnetron sputter-\ning using a triple cluster system to sputter simultaneously from three diﬀerent targets. Upon the\nvariation of the Zn content the ﬁlms remain fully strained wi th similar structural properties, while\nthe magnetic properties are strongly aﬀected. The saturati on magnetization and coercivity as well\nas resonance position and linewidth from ferromagnetic res onance (FMR) measurements are altered\ndepending on the Zn content in the material. The reason for th ese changes can be elucidated by\ninvestigation of the x-ray magnetic circular dichroism spe ctra to gain site and valence speciﬁc infor-\nmation with elemental speciﬁcity. Additionally, from a det ailed investigation by broadband FMR a\nminimum in g-factor and linewidth could be found as a functio n of ﬁlm thickness. Furthermore, the\nresults from a variation of the Al content using the same set o f measurement techniques is given.\nOther than for Zn, the variation of Al aﬀects the strain and ev en more pronounced changes to the\nmagnetic properties are apparent.\nI. Introduction\nZn/Al doped nickel ferrites (NiZAF) came into the fo-\ncus of recent research because of their promising charac-\nteristics for application in spintronics, e.g. spin pumping\n[1, 2]. The material is insulating and ferromagnetic at\nroom temperature with low magnetic damping. There-\nfore,itcanbeconsideredasagoodcandidate[3–5]among\nfew other materials, e.g. MAFO [6, 7], all-perovskite ox-\nides [8] and Fe alloyed with V and Al [9] as replacement,\nfor the well known magnetic insulator yttrium iron gar-\nnet (YIG) [10, 11]. YIG exhibits comparable magnetic\nproperties as NiZAF, however, it has the complex gar-\nnet structure and thin ﬁlm growth requires gadolinium\ngallium garnet (GGG) as substrates thus limiting a wide\napplicability. Since the number of ferromagnetic insula-\ntors with low magnetic damping is sparse [12], the search\nfor appropriate replacements for YIG such as NiZAF is\nstill ongoing and of high interest in current spintronic re-\nsearch.\nThe ideal nominal stoichiometry of NiZAF was reported\nto be Ni 0.65Zn0.35Al0.8Fe1.2O4. It is based on theoreti-\ncal values and experiments on the bulk material [13, 14].\nNiZAF grows in the cubic spinel structure and is derived\nfrom NiFe 2O4, which grows in the inverse spinel crystal\nstructure [Fe 1]A[Ni1Fe1]BO4, where A denotes the tetra-\nhedral (Td) and B denotes the octahedral (Oh) site. In\nthe ideal case Ni only occupies B sites and Fe sits in\n∗Electronicaddress: julia.lumetzberger@jku.at; Phone: + 43-732-2468-9651; FAX: -9696a 1:1 ratio on the A and B sites. However as previ-\nously reported [3, 4], due to the growth on a normal\nspinel substrate MgAl 2O4(001) and by introducing the\ntwo additional elements Zn and Al a mixed spinel is the\nresult. Therefore, Ni can be considered to also occupy\ntetrahedral sites and additionally Fe2+\nOhwas found to be\nintroduced into the system [3, 4]. In addition to this\ncomplex interplay of several elements with diﬀerent va-\nlencies, charge neutrality has to be considered as well,\nand small deviations can be compensated to some extent\nby the formation of oxygen vacancies.\nAccording to theory [13], a variation of the Zn content\nwill procure the biggest changes to the magnetic proper-\nties since it is incorporatedin the crystalstructure with a\nvalence of 2+, preferably on tetrahedral sites. Therefore,\nit can be supposed to mainly aﬀect Ni2+and Fe2+. In\nparticular, since Ni2+\nTdand Fe2+\nOhhave the biggest nega-\ntive impact on the magnetic properties, i.e., damping by\nunquenched orbital moment and hopping [Fe2+→Fe3+\n+ e-], respectively. A systematic variation of Zn is there-\nfore expected to show how the occupancy and/or valence\nof the cations change and aﬀect the structural proper-\nties, and more importantly the magnetic properties. In\ncontrast, Al doping is mainly introduced to reduce the\nstrain of epitaxial thin ﬁlms. Even though the Al con-\ntent was already varied in bulk, showing a dependence\nof the saturation magnetization [14], a systematic study\non thin ﬁlms, which may have additional eﬀects, is still\nlacking.\nThe earliest reports on NiZAF thin ﬁlms grown with\npulsed laser deposition (PLD) revealed a highly strained\nsoft magnetic material with a low magnetic damping [3].2\nHowever, due to the high preparation temperatures and\nthe volatilityof Zn, the thin ﬁlms showed a Zn deﬁciency.\nIn addition, the exact amount of Al in these PLD grown\nﬁlms was not reported. The Zn deﬁciency could be com-\npletely removed and even turned into an excess of Zn in\nNiZAF thin ﬁlms grown by reactive magnetron sputter-\ning (RMS) [4], resulting in an even higher strained mate-\nrialwithslightlyincreasedmagneticdamping. Therefore,\na systematic variation of the Zn content, and in a second\nstep, the Al content should be performed in order to op-\ntimise the composition with regard to the desired soft\nmagnetic properties of RMS grown thin ﬁlms of NiZAF.\nIn this work the composition of NiZAF was systemati-\ncally changed using RMS in a triple cluster system, devi-\nating from the nominal reported best stoichiometry [3].\nTheZncontentwasadaptedbyvaryingthesputterpower\nof the ZnO target. The resulting samples were analyzed\nfor their structural properties, actual chemical composi-\ntion and static and dynamic magnetic properties to de-\ntermine the optimum Zn content. The valence and site\noccupation of the Ni and Fe cations was studied by x-ray\nmagnetic circular dichroism (XMCD) spectra. In combi-\nnation with multiplet ligandﬁeld simulations[15], acom-\nparison to previous NiZAF thin ﬁlms was drawn [4]. In a\nnext step, the thickness of ﬁlms with optimized Zn con-\ntent was varied and the resulting series was analyzed by\nbroadband ferromagnetic resonance (FMR) to disentan-\ngle the various contributions to the linewidth broadening\nand to calculate the g-factor and Gilbert damping. In a\nlast step, the variation of the Al content was investigated\nfor optimized Zn content using the same set of methods\nas mentioned above.\nII. Experimental Details\nIn this work NiZAF epitaxial thin ﬁlms were fabri-\ncated by RMS in a triple cluster system using three\ndiﬀerent targets. The base material NiFe 2O4was co-\ndeposited using ZnO and Al as additional targets. The\nresulting composition of NiZAF was achieved by setting\ndiﬀerent sputter powers for each target. The epitaxial\nthin ﬁlms were grown in an ultra high vacuum (UHV)\nchamber with a base pressure of 4 ×10−8mbar. A dou-\nbleside polished single crystalline spinel [MgAl 2O4(001)]\nwas chosen as a substrate. The optimized growth con-\nditions are a sample temperature of 650 °C, an Ar:O 2\nratio of (10:0 .5)standard cubic centimeter and a work-\ning pressure of 4 ×10−3mbar. The sputter power of the\nZnO target was varied between 3W to 6W in steps of\n1W,while the Al target was initially kept constant at\n16W and the NiFe 2O4target at 70W. The parameter\nrange was determined by recalculation from the reported\nbest stoichiometry using the deposition rates measured\nby a quartz crystal micro balance. Note that, it was not\nfeasible to create a stable plasma with a sputter power\nof 2W at the ZnO target, which limits the lowest achiev-\nable Zn concentration. This sample serieswith a nominalthickness of 40nm is referred to as the Zn series in the\nfollowing. Additionally, on optimized preparation condi-\ntions from the Zn series the thickness (10nm to 40nm)\nand Al content (15W to 17W) were varied, which are\nreferred to as the thickness and the Al series, respec-\ntively. Note that, the sputter time for samples thinner\nthan 16nm ranges from 3min to 5min. Therefore, a\nthermal equilibrium after opening of the shutter and ex-\nposing the heated substrate to the plasma may not be\nreached during sputtering\nThe structural characterization of the samples was done\nby X-raydiﬀraction (XRD) measurementswith a Panan-\nalytical X’Pert MRD recording ω−2θscansand symmet-\nric as well as asymmetric reciprocal space maps (RSM).\nThe composition of the NiZAF thin ﬁlms was deter-\nmined by ion beam analysis, i.e. Rutherford backscatter-\ning spectrometry (RBS) using 2MeV He+and 10MeV\n12C3+primary beams at the Tandem Laboratory at Up-\npsala University. To disentangle the elemental contri-\nbutions, the spectra were analyzed using the SIMNRA\nsoftware [16]. Details of the experimental set-up are de-\nscribed elsewhere[17]. To exclude a contamination ofthe\nmaterialbylightelementslikehydrogenandcarbon,elec-\ntron recoil detection analysis (ERDA) using an 36MeV\niodine primary beam was performed. The details of the\nused set-up can be found elsewhere [18].\nThe static magnetic properties were measured with\nintegral superconducting quantum interference device\n(SQUID) magnetometry by a Quantum Design MPMS-\nXL5system applying the magnetic ﬁeld in the ﬁlm plane\n(IP) as well as out-of-plane (OOP). The magnetic be-\nhavior in a ﬁeld range of ±5T from 300K down to 2K\nand the temperature dependence of the magnetization\nup to 395K were measured. The data were background\ncorrected for the contribution of the diamagnetic sub-\nstrate and known artifacts were carefully avoided [19].\nIn particular, the superconducting magnet was resetted\nto record reliable hysteresis loops at 300K restricting the\nmagnetic ﬁeld to below 10mT [20].\nThe dynamic magnetic properties were measured by\nFMR using a conventional X-band set-up (9 .5GHz) at\n300K. Additionally, the samples from the thickness se-\nries were analyzed with a broadband vector network ana-\nlyzer ferromagnetic resonance (VNA-FMR) set-up to de-\ntermine the frequency dependence of the resonance posi-\ntion and FMR linewidth up to 50GHz and to verify the\ng-factorand disentanglethe diﬀerentcontributionstothe\nmagnetic damping. Furthermore, for selected samples\nalso the polar and azimuthal angular dependence were\nrecorded.\nFinally,X-rayabsorptionspectra(XAS)attheNi, Feand\nZn L3,2edges, as well as the O and Al K edges were mea-\nsured at the X-Treme beamline at the Swiss Light Source\n(SLS) [21]. The spectra were obtained in total electron\nyield (TEY) with 20 °grazing incidence at 300K and a\nﬁeld of 5T using circular polarized light σ. To obtain\nthe XMCD the diﬀerence between the normalized XAS\nrecorded with σ+andσ−light was taken. In Fig. 1(a)3\n(a)\n (c)\n(b) (d)(e)\n(f)\nFIG. 1: Absorption spectra recorded at 300K and 20 °grazing from the sample with optimized growth conditions of the Zn\nseries. (a) shows the σ+andσ−spectra and the resulting XMCD at the Fe L 3,2edges. In (b-f) the XAS spectra at the Fe ,\nNi and Zn L 3,2edges and Al and O K edges are shown, respectively.\ntheσ+andσ−spectra of the best sample from the Zn\nseries for the Fe L 3,2edges and the resulting XMCD are\nshown exemplarily. The XAS is obtained by the aver-\nage of the σ+andσ−spectra and shown in Fig. 1(b).\nThe multiplet splitting at the L 3,2edges is typical for\nan oxide. The XAS of the other constituents were also\nrecorded and Fig. 1summarizes them for the Ni (c) and\nZn (d) L 3,2edges as well as the Al (e) and O (f) K-edges.\nAll spectra were recorded with circularly polarized light,\nhowever, besides Ni no signiﬁcant XMCD could be de-\ntected. For the K-edges, where the XMCD is knowingly\nrather small, this is understandable, in particular for the\nrather noisy Al K-edge. Thus, no conclusive statements\nabout an eventual magnetic polarization of Al can be\nmade. On the other hand, the absence of a signiﬁcant\nXMCD at the Zn L-edges (not shown) indicates no mag-\nnetic polarization of the Zn in the sample. The XMCD\nspectra at the Fe and Ni L 3,2edges will be discussed in\ndetail further below.\nIII. Structural Properties: Zn series\nThe samples from the Zn series were routinely ana-\nlyzed by XRD and the results of the symmetric ω−2θ\nscans are shown in Fig. 2(a). The (004) reﬂection of\nthe NiZAF peak has pronounced Laue oscillations, in-\ndicating a smooth surface and excellent crystal qualityfor each sample. The second reﬂection stems from the\nMgAl2O4(004) substrate. The NiZAF(004) reﬂections\nforthe entireseriesarelocated atanangleof42 .5°with a\nfull width half maximum (FWHM) of 0 .25 for the 40nm-\nthick ﬁlms. This corresponds to a perpendicular lattice\nparameter of a⊥= (8.50±0.01)/RingA. Only the sample\ngrown with 6W Zn shifts to slightly lower angles, but\nno indication for a reduced crystal quality is obvious.\nIn Figs.2(b) and (c) the symmetric and asymmetric re-\nciprocal space map (RSM) of the 3W Zn sample from\nthe Zn series are shown, respectively. Both space maps\nhighlight the good crystal quality of the grown samples,\nsinceLaueoscillationsarevisiblyin bothcases. Fromthe\nsymmetric RSM along the (004) the peak position and\ncalculated out-of-plane lattice parameter for NiZAF are\nconﬁrmed. From the asymmetric RSM along the ( ¯1¯15)\nplane it is apparent that the substrate and ﬁlm reﬂec-\ntion are perfectly aligned in the Qyaxis, indicating a\nfully strained material in the in-plane direction. From\nthis an in-plane lattice parameter of a/bardbl= (8.08±0.01)/RingA\nidentical to the lattice parameter of the cubic MgAl 2O4\nsubstrate is obtained. This results in a tetragonal distor-\ntion ofc/a= 1.052±0.003, which is signiﬁcantly higher\nthan reported for epitaxial NiZAF ﬁlms before [3–5].\nTo conﬁrm the systematic variation of the Zn content by\nsputter power, RBS measurements were performed us-\ning a 2MeV He+ion beam (not shown) and a 10MeV\n12C3+ion beam (see Fig. 2(d)). The advantage of us-4\n(a)\n (b)\n(c)\n (d)\n(1,-1,5)\n3W Zn(0,0,4)\n3W Zn\nFIG. 2: Structural analysis on the Zn series. In (a) the symme tricω−2θscans. The symmetric and asymmetric RSM of the\n3W sample is shown exemplary for the Zn series in (b) and (c), r espectively. In (d) the RBS spectra using a 10MeV12C3+\nprimary ion beam are shown for each sample.\ning heavier ions is the better separation of Fe, Ni and\nZn, which have a similar atomic mass, even revealing the\nsplitting into the isotopes. The peaks of the energies cor-\nresponding to the diﬀerent elements in the NiZAF ﬁlm\nare indicated with arrows. From a direct comparison,\ndiﬀerences are only evident for the Fe peak, where the\n6W sample shows a reduced intensity. Any changes in\nNi and Zn concentration cannot be determined by direct\ncomparison of the spectra. The lighter element Al only\nshows as a tiny step at lower energies and assumptions\nabout the composition can only be made by using SIM-\nNRA [16] simulations to determine the percentages. The\nmain focus is on the small variation of the Zn amount.\nTo resolve the Zn amount an exact knowledge about the\npercentages of the biggest contributor oxygen would be\nnecessary, which is not feasible using a C beam. In order\nto eliminate trivial dependencies such as the charge andsolid-angle product of the experiment or the inelastic en-\nergy loss in the substrate material, which is needed for\nnormalization, we are primarily discussing the respective\nratios of two of the elements, which is inﬂuenced almost\nexclusivelybythestatisticsoftheexperiment. Note, that\nother light species were only found at insigniﬁcant con-\ncentrations according to ERDA (not shown).\nBy increasing the sputter power of the ZnO target from\n3W to 6W, the Zn:Fe ratio increased from 0 .23 to 0.35.\nThe same behavior can be observed for the Zn:Ni and\nthe Zn:Al ratio, which areincreasingas well from0 .29to\n0.61 and from 0 .37 to 0.52, respectively. Interestingly, an\nincrease in Zn seems to reduce not only the Ni, but also\naﬀects the Al and Fe concentration. According to theory\n[3, 13, 14] Zn should only substitute for Ni. The Ni:Fe\nratio on the other hand stays constant around 0 .58. Ad-\nditionally, the Ni:Al and Fe:Al ratios both decrease by5\nincreasing the sputter power (not shown). Table Isum-\nmarizes the results obtained for the chemical composi-\ntion and the XRD analysis in the ﬁrst two columns. For\ncomparison also the ﬁndings of previous publications are\nprovided [3, 4]. A comparison of the relative composi-\ntions shows that the results of the present sample series\nis in between previous results regardingthe ratio of Zn to\nthe othermetals. The Zn-deﬁcientsample grownby PLD\n[3] has similar ratios of Zn:Fe of 0 .22 and Zn:Ni of 0 .4\nas the 3W sample. Unfortunately, the Al concentration\nis not reported in [3]. Ratios of Zn:Fe of 0 .42, Zn:Ni of\n0.88 and Zn:Al of 0 .55 for the previous RMS sample [4]\nare even higher than what was found for the 6W sample\nfrom the Zn series. From the Ni:Fe ratio the low value\nof 0.48 in [4], further highlights the abundance of Zn in\nthe reported sample.\nSummarizing this part, as expected, the relative com-\nposition of the elements showed that the Zn content in-\ncreases with increasing sputter power. In comparison to\nthe two previous publications on NiZAF, the ratios of Zn\nto the other metals of the entire Zn series are between\ntheZn-deﬁcientNiZAFgrownbyPLD[3]andtheZn-rich\nNiZAF grown by RMS[4]. The structural analysis of the\nentire Zn series results in NiZAF ﬁlms, which are even\nmore strained compared to previous publications [3–5].\nNevertheless, all samples maintain their high crystalline\nquality.\nIV. Magnetic Properties: Zn series\nHaving established a good crystal quality and the sys-\ntematic increase in Zn content of the Zn series, static\nmagnetic measurements were performed. In Fig. 3(a)\ntheM(H) curves at 300K are shown in the range of\na few mT to better visualize the hysteresis opening. A\nchange in coercivity depending on the Zn content is ap-\nparent, where the lowest concentration present in the\n3W sample leads to the smallest coercivity of lower than\nµ0Hc= 0.2mT. For a Zn amount of the 6W sample\nthe coercivity is at least twice as high. The saturation\nmagnetization at 5T (not shown) is highest for the 3W\nsample with Ms= (159±16)kA/m and decreases to\nMs= (122±12)kA/m for the 6W sample. The re-\nduction of Mswith increasing Zn is reasonable, since a\npartial substitution of Fe and Ni with the nonmagnetic\nZn leads to a reduced magnetization. The M(T) curves\nmeasured at 10mT shown in Fig. 3(b) exhibit a similar\ntrend for the Zn series. All Curie temperatures TCare\nwell above 400K, however the TCof the 6W sample ap-\npears to be the lowest, because of the sharpest curvature\nof theM(T) curve close to 400K. Measurements up to\nhigher temperatures were not feasible since the magne-\ntometer is limited to 400K. The estimated M(T) be-\nhavior at higher temperatures, is illustrated exemplarily\nby the dotted line. From that a high TC= 450K com-\nparedto previousﬁndings forsputtered NiZAF thin ﬁlms\n[4] was obtained for the 3W sample. The shape of the\n(a)\n(b)\n(c)\nFIG. 3: Magnetic analysis of the Zn series. In (a) the M(H)\ncurves at 300K, in (b) the M(T) curves at 10mT with an\nextrapolation of the Curie temperature and in (c) the X-band\nFMRat300Kand9 .5GHz, for eachsample from theZnseries\nare shown.\nM(T) curve is very similar for the entire Zn series, while\nthe magnetization systematically decreased with increas-\ning Zn content, which is consistent with the decrease of\nthe saturation magnetization mentioned above.\nFurthermore, FMR measurements were performed to\nobtain a ﬁrst indication about the size of the magnetic\ndamping with the FMR line. A comparison of the res-6\nTABLE I: Summary of the structural and magnetic properties o f the samples from the Zn series including [3, 4]. The respect ive\nmeasuring errors are given in the ﬁrst row.\nGrowth RBS XRD SQUID FMR\nZn (W) Zn:Fe Zn:Ni Zn:Al Ni:Fe (004) reﬂex ( °)Ms(kA/m)µ0Hc(mT)µ0Hres(mT)µ0∆Hpp(mT)\n±0.5 (W) ±0.02 (°)±10(%) ±0.2(mT) ±0.2(mT) ±0.2(mT)\n3.00.23 0.39 0.37 0.59 42.52 159 0.2 31.3 3.9\n4.00.28 0.48 0.44 0.59 42.50 156 0.2 34.1 4.9\n5.00.31 0.53 0.48 0.58 42.50 142 0.2 39.0 4.0\n6.00.35 0.61 0.52 0.58 42.43 122 0.4 47.3 6.8\nRef. [3] 0.22 0.40 - 0.55 43.25 120 0.2 84.8 0.8\nRef. [4] 0.42 0.88 0.55 0.48 42.56 195 0.2 57.8 4.4\nonances of the samples of the Zn series measured at\n9.5GHz and 300K using a conventional X-band set-up\nis shown in Fig. 3(c). The measured resonances are\nﬁtted with a Lorentzian to obtain the resonance po-\nsition (µ0Hres) and peak-to-peak linewidth ( µ0∆Hpp).\nA clear increase of the resonance position with sputter\npower is evident, indicating reduced magnetic anisotropy\nwith increasing Zn content. The lowest resonance posi-\ntion was obtained for the 3W sample, which is around\nµ0Hres= (31.3±0.2)mT. This sample also has the\nsmallest linewidth of µ0∆Hpp= (3.9±0.2)mT of the\nZn series. The linewidth also increases with the Zn con-\ntent reaching µ0∆Hpp= (6.8±0.2)mT for the 6W\nsample. The results of the magnetic characterization of\nthe Zn series together with [3, 4] are shown in Tab. I\nyielding the following comparison: The Zn-deﬁcient sam-\nple from [3], has a saturation magnetization of Ms=\n(120±12)kA/m and a small coercivity. The resonance\nposition was found at µ0Hres= (84.8±0.2)mT with a\nvery small linewidth of µ0∆Hpp= (0.8±0.2)mT. The\nZn-rich sample from [4] has a higher Msof 195kA /m\nwith an equally small coercivity and reduced resonance\nposition of µ0Hres= (57.8±0.2)mT with a linewidth of\nµ0∆Hpp= (4.4±0.2)mT. Interestingly, the 3W sam-\nple with the lowest amount of Zn showed the best re-\nsults from the preliminary analysis. The thin ﬁlm has\na saturation magnetization between the Zn-deﬁcient [3]\nand Zn-rich [4] NiZAF with a comparably small coer-\ncivity. However, the resonance position is drastically\nshifted to µ0Hres= (31.3±0.2)mT. This can be cor-\nrelated with the higher strain in this sample favoring\nhigher magnetic anisotropy and thus a lower resonance\nﬁeld. Interestingly, the Curietemperatureof450Kfound\nfor the 3W sample is so far the highest reported value\nfor sputtered NiZAF thin ﬁlms and matches the result\nfrom PLD grown samples in [3]. Although, the sample is\neven more strained than the reported Zn-deﬁcient [3] and\nZn-rich [4] samples, the linewidth remained narrow with\nµ0∆Hpp= (3.9±0.2)mT. Obviously, the 3W sample\nhas the most promising Zn concentration with regard to\nthe desired soft magnetic properties. In a next step, the\ncation distribution of the magnetic cations in NiZAF will\nbe analyzed for the entire Zn series.V. Cation distribution: Zn series\nThe analysis of the cation distribution and valence is\nrestrictedtoNi and Fe sincethoseelements showa signif-\nicant XMCD signal and thus contribute to the magnetic\nproperties in the NiZAF thin ﬁlms. In Fig. 4the XMCD\nspectra of the Zn series at the Fe and Ni L 3,2edges are\nshown.\nAt a ﬁrst glance, the XMCDs at the Fe L 3,2edges in\nFig.4(a) exhibit a similar shape for the entire series with\nthe exceptionofthe 6W sample. Since the L 2edgeshows\nno signiﬁcantdiﬀerences between the samples besides the\nintensity, only the L 3edge is discussed in detail. The\ntwo negative peaks stem from octahedrally coordinated\nFe. The peak at lower energies contains contributions\nfrom both Fe2+and Fe3+. The one at higher energies\nsolely stems from Fe3+. The positive peak is attributed\nto tetrahedral Fe3+, which is thus coupled antiparallel to\nthe octahedral Fe, as known for nickel ferrite in general.\nThese peak assignments are based on multiplet ligand\nﬁeld theory [15] and were reported in several publica-\ntions before [4, 22].\nThe strongest deviations of the Fe L 3edge in the entire\nseries is evident for the 6W sample, since an additional\nfeature between the ﬁrst two peaks arises and its over-\nall shape and relative intensities diﬀer. The diﬀerences\nbetween the other three sample are more subtle and are\nmainly seen in the positive tetrahedral Fe3+peak, where\nthe intensity is increased for the 5W sample. This is in\na sense remarkable, because Zn is expected to predomi-\nnantlyoccupytetrahedralsites,however,hereanincrease\nin Zn increased Fe on the tetrahedral site. Furthermore,\nthe ﬁrst negative peak ofthe XMCD is lowestfor the 3W\nsamplesuggestingthe lowestcontribution ofFe2+\nOh(disre-\ngarding the 6W sample). Also here the increasing Zn2+\ncontent counterintuitively leads to an increase of the un-\nwanted Fe2+. Even though the signal for Fe2+\nOhseems\nto be lower in the 6W sample, the relative peak height\ncompared to the octahedral Fe3+is higher. This cor-\nrelates well with the results from the magnetic analysis,\nsince the presence of Fe2+\nOhincreases damping by hopping\nmechanisms and was found to be at a minimum for the\n3W sample. In turn, these ﬁndings indicate that already\nat the lowest possible Zn concentration the beneﬁcial as-\npects of the Zn incorporation are reached and a further7\n(a)\n(b)\n(c)\nFIG. 4: XMCD spectra of the samples from the Zn series.\nIn (a) the XMCD spectra at 300K in 20 °grazing at the Fe\nL3,2edges and in (b) at Ni L 3,2edges are shown. (c) depicts\na comparison of the multiplet ligand ﬁeld simulation using\nCTM4XAS [15] with the experimental data for the 3W sam-\nple.\nincrease in Zn content counteracts both the suppression\nof tetrahedral Fe3+as well as Fe2+.\nA comparison of the XMCD spectra at the Ni L 3,2edges\nis shown in Fig. 4(b). Analogous to the Fe spectra the\noverall shape is rather similar for each sample, especially\nthe L2edge only shows small variations. Focusing on the\nL3edge the 6W sample is again conspicuously diﬀerent\nexhibiting an additional positive peak at the low energyside of the negative peak, which is absent in the other\nsamples. According to multiplet ligand ﬁeld theory [13],\nthis peak is assigned to Ni2+\nTd. Since the presence of Ni2+\nTd\nis known to increase damping by unquenched orbital mo-\nmenta, this will further contribute to the increased FMR\nlinewidth in the 6W sample. A direct comparison of the\nother three samples with respect to Ni2+\nTdis not possible,\nsince the feature is absent. Solely the magnetic contri-\nbution from Ni2+\nOh, which is responsible for the overall\nshape of the magnetic dichroism spectrum can be dis-\ncussed. The octahedral contributions seem to be lesser\nin the 5W sample in comparison to the two samples with\nlower Zn concentration. Also here, the role of increased\nZn content is surprising, since the magnetic signal of Ni\non octahedral sites is reduced.\nAn analysis of the cation distribution at the Fe and Ni\nL3,2edges shows that increasing the sputter power of\nthe ZnO target up to 5W leads to small but notice-\nable changes in the site occupancy at both edges. In\ncase of Fe an increase in Fe3+at tetrahedral sites and in\ncase of Ni a reduction of the Ni2+\nOhsignal are apparent.\nSince the stoichiometry was not changed drastically by\nthe Zn variation, this is a ﬁrst indication that Zn also oc-\ncupies octahedral sites. Increasing the Zn amount even\nmore,makesthechangestotheXMCDspectraevenmore\npronounced. Moreover, a sign for Ni2+\nTdappears, which\nsuggest that Ni moved to tetrahedral sites. These ﬁnd-\nings arehighly unexpected, since accordingto theory [13]\nand previous studies on NiZAF [3, 4] the addition of Zn\nshould prevent Ni from occupying tetrahedral sites and\nnot promoting it. Obviously, a critical Zn concentration\nexists in this quaternaryoxide, above which the inﬂuence\nof Zn is detrimental to the desired magnetic properties.\nSo far only direct comparison between the samples was\ngiven using multiplet ligand ﬁeld theory to correlate va-\nlence and occupancy with the peaks observed in the\nXMCD spectra. By applying multiplet ligand ﬁeld simu-\nlation using CTM4XAS [15], a suggestion for the respec-\ntive percentages of the elemental contributions using a\nweighted linear combination can be calculated. This has\nalreadybeendoneinapreviouspublication[4], wherethe\ndetails of the used parameters are given in detail, please\nalso refer to [22–25]. In Fig. 4(c) a comparison of the ex-\nperimental XMCD spectra and the respective simulation\nof the 3W sample at the Fe L 3,2edges is shown. The ex-\nperimentaldata is in goodagreementwith the simulation\nat the L 3edge, which will also be the focus of discussion.\nThe simulation yields the following percentages: 33% of\nFe2+\nOh26% of octahedral and 41% of tetrahedral Fe3+,\nrespectively. Compared to percentages of 21% obtained\nin Ref.[4] this is an increase of Fe2+\nOhby approximately\n5%. Since the amount ofoctahedralFe3+is equal within\nerror bars, these 5% must have shifted from tetrahedral\nFe3+to Fe2+\nOh.\nSummarizing our ﬁndings, the Zn-rich NiZAF [4] has\na decreased Curie temperature below 400K, while the\nsaturation magnetization is increased. As a compari-\nson of the XMCD spectra showed this is accompanied8\nby a reduction of Fe2+\nOh. However, by adapting the Zn\ncontent a sample with a slightly higher out of plane lat-\ntice parameter of a⊥= (8.50±0.01)/RingA in comparison to\na⊥= (8.49±0.01)/RingA as well as a decrease in coerciv-\nity lower than µ0Hc∼0.2mT at 300K was fabricated.\nInterestingly, a comparison of the linewidth at 9 .5GHz\nwith [4] suggests a lower FMR linewidth in case of the\nsample from the Zn series, even though the XMCD spec-\ntrum indicates a highersignal ofFe2+\nOh. Howevera change\nin FMR linewidth at one ﬁxed frequency does not allow a\nsolid conclusionabout the magnetic damping. Therefore,\nand to identify the diﬀerent contributions to linewidth\nbroadening,athoroughinvestigationbybroadbandFMR\nis in order, which will be discussed in the next section.\nVI. Broadband FMR: thickness series\nThe preliminary analysis with X-band FMR indicates\nthat the lowest damping occurs in the 40nm-thick 3W\nsample. However, the reported ideal sample thickness for\nthe Zn-deﬁcient NiZAF was 16nm [3] and the Zn-rich\nsample was 25nm thick [4]. Therefore, a thickness se-\nries has been grown while keeping all other preparation\nparameters constant to check if the magnetic damping\ncould be lowered further.\nThe results of the XRD measurements of this thickness\nseries are shown in Fig. 5(a). It can be seen, that the\nstructural quality of the samples is retained for a thick-\nness reduction down to 10nm. Additionally, the diﬀrac-\ntograms nicely show the correlation between the sample\nthickness and the Laue oscillations. The position of the\n(004) NiZAF peak is identical for the 25nm and 40nm\nthick samples. However, reducing the thickness further\nshifts the peak to lower angles around 42 .43°indicating\nan increased a⊥and thus increased strain. From SQUID\nmagnetometry a small coercivity around µ0Hc∼0.2mT\nat 300K and a Curie temperature well above 400K could\nbe determined for all measured thicknesses (not shown).\nThesaturationmagnetizationdoesnotshowasystematic\nvariation with thickness within experimental uncertain-\nties. The values range from a Ms= (185±19)kA/m\nfor the 10nm sample to a Ms= (147±19)kA/m for\nthe 16nm sample. A comparison of the X-band FMR re-\nsults at 300K is provided in Fig. 5(b) demonstrating that\nthe 40nm sample has the lowest linewidth. However, all\nother thicknesses show only slightly diﬀerent results and\nno conclusive trend with thickness is visible. Therefore,\na detailed analysis of the magnetic properties by broad-\nband FMR up to 50GHz is required to verify and dis-\nentangle the various possible contributions to linewidth\nbroadening.\nA 2D gray-scale plot of the broadband FMR measure-\nments of the 40nm sample up to 50GHz is shown in\nFig.5(c). From 30GHz on the plot becomes visibly\nblurry. This broadening is caused by a weak second reso-\nnance mode overlapping with the main mode. The inset\nof Fig.5(c) shows the ﬁtted resonance line at 30GHz, inwhich at least a second line is evident. A possible ex-\nplanation for the appearance of a second or even more\nresonance lines could be inhomogeneities, i.e. areas with\nslightly diﬀerent magnetic properties leading to a diﬀer-\nent resonance ﬁeld. A consequence of the appearance\nof the second line is the strong scatter in the data of\nthe extracted linewidth at higher frequencies. From the\nfrequency-dependent resonance ﬁelds of the main peak\na g-factor of g= 2.17±0.09 was determined by using\nthe Kittel ﬁt [26]. In Fig. 5(d) the linewidth is plotted\noverthe same frequency range with the respective broad-\nening contributions. In addition to the Gilbert damp-\ning, which is linear in frequency, another contribution\nhas to be introduced that considers eﬀects caused by\nmosaicity (due to sample inhomogeneity). Furthermore,\na small frequency-independent inhomogeneous damping\nhas to be added to satisfyingly ﬁt the experimental data.\nNote that, no contribution from two magnon scattering\ncould be observed, which occurred in NiZAF thin ﬁlms\nas reported in Ref.[4]. Nonetheless, the obtained Gilbert\ndamping parameter of α= 9×10−3is higher than for\nthe sputtered Zn-rich NiZAF in Ref.[4]. This was un-\nexpected, since the X-band FMR at 9 .5GHz indicated\na lower linewidth than previously determined. How-\never, this highlights the necessity of measuring broad-\nband FMR above 30GHz to be able to accurately deter-\nmine the Gilbert damping parameter.\nThe same measurements were performed on the 25nm\nthick samples, which showed no indication for a second\nresonance line as can be seen in Fig. 6(a). This sam-\nple has a g-factor of g= 2.15±0.09 and the frequency\ndependence of the linewidth including the diﬀerent con-\ntributions obtained by ﬁtting are shown in Fig. 6(b).\nThe experimental data can only be ﬁtted with the lin-\near Gilbert term plus a mosaicity and a small inhomo-\ngeneous oﬀset. A contribution of two magnon scattering\nis again absent. The obtained Gilbert damping param-\neter ofα= 7.8×10−3is lower in comparison to the\n40nm-thick sample and also mosaicity and inhomoge-\nneous broadening are reduced. Since 16nm is the pro-\nposed best thickness with the lowest damping in case of\nNiZAF grown with PLD [3], a 16nm sample was ana-\nlyzed as well. The results of the 2D gray-scale plot and\nthe ﬁtted linewidth contributions are shown in Figs. 6(c)\nand (d), respectively. The quantitative analysis yields\ng= 2.18±0.09. The frequency-dependent linewidth\ncould be ﬁtted without mosaicity, however the linear\nGilbert term is increased to α= 11.4×10−3compared\nto the thicker samples; a small inhomogeneous oﬀset also\nremains.\nThe broadband FMR showed that by reducing the thick-\nnessnot onlythe structuralbut alsothe magnetic quality\nof the samples could be retained conﬁrming the repro-\nduceability of the sputter process with the found opti-\nmal preparation conditions. The unsystematic change of\nthe linewidth with thickness is uncharacteristic to previ-\nous reports on NiZAF [3], where a clear reduction with\nthickness was found. However, in particular for very thin9\n(a) (b)\n(c) (d)\nFIG. 5: In (a) the symmetric ω−2θscans and in (b) the X-band FMR at 300K and 9 .5GHz, for each sample from the\nthickness series are shown. (c) and (d) show a 2D gray-scale p lot of the resonance position and the linewidth with broaden ing\ncontributions, respectively over a frequency range of up to 50GHz for a 40nm-thick sample. The inset in (c) depicts the\nresonance line at 30GHz.\nsamples, even a low amount of defects already has a big\nimpact on site occupancy. Additionally, for 16nm and\n10nm the higher strain inferred from XRD could be an-\nother cause for the increased damping. Nonetheless, the\nthickness series showed that a minimum in the Gilbert\ndamping is found for a thickness of 25nm. In agree-\nmentwith Ref.[4], whereRMSgrownNiZAF withasimi-\nlar thickness exhibited a comparable Gilbert damping of\nα= 6.8×10−3. Nevertheless, the cationic concentra-\ntions were diﬀerent, see Tab. I. Additionally, it has to\nbe considered that the samples here are strained even\nmore and have a clearly higher Curie temperature up to\n450K, while retaining a very low damping. Furthermore,\nthe even lower g-factor of g= 2.15±0.09 compared to\ng= 2.29±0.09 [3] would suggest a reduced amount of\nNi2+\nTd, sincetheycontributemostlybyunquenchedorbital\nmomenta.\nOn the 25nm sample with lowest damping additionalmeasurements were performed to check the anisotropy\nin-plane along the azimuthal angle in a range from 0 °\nto 360 °and out-of plane along the polar angle in range\nfrom−20°to 90°at a frequency of 15GHz (not shown).\nTo ﬁt the angular dependences and determine the diﬀer-\nent contributions to anisotropy, the resonance equation\nanalogous to [4, 27] was used. The azimuthal angular\ndependence indicates a four fold symmetry as expected\nfrom a cubic system resulting in a cubic anisotropy of\n2K4/bardbl/Ms= 2.2mT, with an additional small uniaxial\ncontribution of 2 K/bardbl/Ms= 0.8mT. These ﬁndings are\ncontraryto Refs.[3] and [4], which reported higher values\nof 10mT and a slight shift towards a tetragonal crystal\nsymmetry, respectively. The polar angular dependence\nrevealed an out-of-plane anisotropy of 3 .5T, which is\nconsiderable higher than reported before and can be ex-\nplained by the increased c/aratio. It is also in quantita-\ntive agreement with the SQUID measurements at 300K10\nin in-plane and out-of-plane orientation (not shown).\nHaving now optimized the Zn content and thickness, the\nremaining parameter is the strain of the NiZAF ﬁlm,\nwhich can be controlled by the Al content as discussed\nbefore [3] and in bulk samples [14]. Therefore, the Al\nconcentrationis varied in a next step using the optimized\nparameters from the Zn series.\nVII. Dependence of Al content\nThe symmetric ω−2θscans measured by XRD of the\nfour samples from the Al series are shown in Fig. 7(a).\nThree samples have a Zn amount of 3W with a varied Al\namount between 15W to 17W and a promising sample\nwhere both doping parameters were changed is analyzed\nas well. The strongest changes in structure are appar-\nent in the sample sputtered with 17W, where the crys-\ntal quality seems most disordered. Even though, Laue\noscillations are visible, indicating a smooth surface, the\ncrystal quality is greatly reduced. The other three sam-\nplesshowadependenceofthelatticeparameterontheAl\ncontent. By increasingthe Al content the (004) ﬁlm peak\nmoves to higher angles, decreasing the lattice parameter\nand therefore reducing strain. The sample in which both\ndopants were adapted (green line) has an increased peak\nposition of 42 .65°, corresponding to a⊥= (8.47±0.01)/RingA.\nTheM(H) curves at 300K of the samples from the Al\nseries are shown in Fig. 7(b). They show that the best\nsample from the Zn series still has the lowest µ0Hc=\n0.2mT as well as the highest saturation magnetization of\nMs= (159±16)kA/m at 300K. The 15W sample shows\nthe biggest deviation, where the coercivity considerably\nincreased up to µ0Hc= 15.0mT as denoted in the ﬁgure.\nIn comparison the most similar result to the best sample\nof the Zn series (red line) is obtained from the sample\nwhere both the Zn and Al content were adapted (green\nline). In the Al series the M(T) behavior is comparable\nto the Zn series and the Curie temperature remains well\nabove 400K (not shown). However, the sample where\nboth doping parameters were changed (green line) re-\ntains a good structural quality as well as coercivity, but\nhas a decreased Curie temperature of TC= (385±2)K\nsimilar to [4]. Additionally, the saturation magnetization\nis signiﬁcantly lowered to Ms= (110±16)kA/m.\nIn a next step, changes in the cationic occupation and/or\nvalence are to be researched. XMCD spectra were\nrecorded at the Fe and Ni L 3,2edges under the same\nconditions as for the Zn series. The results of the Fe L 3,2\nedges are shown in Fig. 7(c) and the discussion is again\nfocused on the L 3edge. The elemental contributions of\nFe are very similar in the best sample from the Zn series\nas well as the sample in which both doping parameters\nwere changed (red and green line). In the other two sam-\nples the amount of octahedral Fe is heavily changed, in\nparticular for Fe3+. XMCD spectra at the Ni L 3,2edges\nshown in Fig. 7(d) also reveal signiﬁcant changes upon\nvariation of the Al content. The two samples with higherstructural perfection also have similar XMCD spectra.\nIn comparison the 15W and 17W sample exhibit an in-\ncreased Ni2+\nOhpeak. Additionally, the 17W sample has\na positive peak at lower energies as well, which indicates\nthe presence of a signiﬁcant amount of Ni2+\nTd. The sim-\nilarity of the XMCD spectra at both edges of the best\nsample from the Zn series and the sample where both\ndopants were changed (red and green line) coincides with\nthe structural and magnetic properties, which were sig-\nniﬁcantly better compared to the remaining samples of\nthe Al series. Furthermore, the deviating structural or\nmagnetic properties apparent in the 17W and 15W sam-\nple, respectively can be explained as well. The XMCD\nspectra at the Fe L 3edge have an increased amount of\noctahedral Fe, in particular the 15W sample. As a result\nthecoercivitywasdrasticallyincreased. Additionally, the\n17W sample shows the presence of Ni2+\nTd, which leads to\nstrong changes in the XRD and in combination with the\nhigher amount of Fe2+\nOhto an increased coercivity. The\ninset in Fig. 7(d), shows a comparison of the FMR lines\nof the best samples from the Al series. The sample where\nboth doping parameters were changed achieves a similar\nsmall linewidth of µ0∆Hpp= (4.6±0.2)mT as obtained\nfor the best samples from the Zn series. However, due\nto the reduced strain by the higher amount of Al the\nanisotropy decreased and the resonance moves to higher\nﬁelds of 44 .8mT. The 15W and 17W samples are not\ndiscussed further, since the ﬁrst shows an increased coer-\ncivity while the second has a drastically reduced crystal\nquality in addition to the presence of Ni2+\nTd.\nA summary of the parameters obtained from the analy-\nsis of the Al series can be seen in Tab. II. In the table\nthe elemental ratios determined by RBS using a 10MeV\n12C3+ion beam are depicted as well. However, a distinct\ndependence of the ratios on the Al content as seen for the\nZn series is not as apparent. The Al:Fe ratio increases\nwith increasing sputter power of the Al target from 0 .58\nto 0.68. Also the Al:Ni ratio shows similar results with\nan increase from 1 .06 to 1.20. No dependence for the\nAl:Zn ratio could be determined. The Ni:Fe ratio is\nrather constant around 0 .56, similar to the Zn series.\nThe analysis of the Al series experimentally conﬁrmed,\nthat the strain in the material can be controlled by ad-\njusting the Al amount, as predicted from theory [13].\nHowever, a deviation of the Al content from the opti-\nmized parameters of the Zn series resulted either in the\nlossofstructural qualityorincreasedcoercivityas well as\nthe presence of Ni2+\nTd. More importantly, a comparison of\nthe best sample from the Zn series with a sample where\nboth Al and Zn were varied showed a decrease in satu-\nration magnetization and Curie temperature. However,\nthe release in strain lead to a decreased anisotropy while\nretaining a similar small linewidth. The comparison sug-\ngeststhat due to the complexinterplay ofthe elements in\nNiZAF several favorable combinations in the parameter\nspace of the growth conditions can be found to achieve\nferromagnetic insulators with low magnetic damping for\nspintronic applications.11\n(a)\n (b)\n(c) (d)\nFIG. 6: In (a) a 2D gray-scale plot of the resonance position a nd in (b) the linewidth with broadening contributions over a\nfrequency range of up to 50GHz for a 25nm thick sample is shown . (c) and (d) depict the results of a 16nm thick sample,\nrespectively.\nTABLE II: Summary of the structural and magnetic properties of the samples from the Al series. The respective measuring\nerrors are given in the ﬁrst row.\nGrowth RBS XRD SQUID FMR\nZn (W) Al:Fe Al:Zn Al:Ni Ni:Fe (004) reﬂex ( °)Ms(kA/m)µ0Hc(mT)µ0Hres(mT)µ0∆Hpp(mT)\n±0.5 (W) ±0.02 (°)±10(%) ±0.2(mT) ±0.2(mT) ±0.2(mT)\n15.0 0.58 2.28 1.06 0.55 42.42 190 0.8\n16.0 0.62 2.69 1.06 0.59 42.50 159 0.2 31.3 3.9\n17.0 0.63 2.34 1.12 0.56 42.41 142 15.0\n18.0 (5W Zn) 0.68 2.13 1.20 0.56 42.65 110 0.2 44.8 4.6\nVI. Conclusion\nIn this work the systematic variation of the Zn and\nAl concentrations in Zn/Al doped nickel ferrite lead to\nnew insights regarding the interplay of cation distribu-\ntion, site occupancy, and magnetic damping. Three dif-ferent sample series were investigated to identify the ef-\nfects of Zn and Al content as well as thickness, due to its\nreported correlation to magnetic damping [3].\nFrom the Zn series NiZAF thin ﬁlms were found to main-\ntain the structural quality overa change in sputter power\nfrom 3W to 6W of the ZnO target. The change in Zn12\n(a)\n(c)\n(b)\n(d)\nFIG. 7: Static and magnetic analysis of the Al series. In (a) t he symmetric ω−2θscans and in (b) the M(H) curves at 300K\nand 5T, for each sample from the Al series are shown. (c) and (d ) depict the XMCD spectra at 300K in 20 °grazing at the Fe\nL3,2edges Ni L 3,2edges, respectively for the each sample from the Al series. T he inset in (c) shows a comparison of the FRM\nlines of two samples from the Al series.\ncontent was conﬁrmed by measuring the elemental ratios\nwith RBS. The magnetic analysis with SQUID magne-\ntometry showed a reduction of the saturation magnetiza-\ntion with increasing Zn, while maintaining a small coer-\ncivity. Interestingly, the Curie temperature was found\nto be very robust towards changes in the Zn content\nand above 400K throughout the whole series. By in-\ncreasing Zn the anisotropy decreased, evidenced by the\nmeasured higher resonance position and a broadening of\nthe linewidth occurred in FMR. An investigation of the\nXMCD spectra at the Fe and Ni L 3,2edges showed an\nincreasein the tetrahedralFe3+and the presenceofNi2+\nTd\nat higher Zn concentrations. Since the stoichiometry did\nnot change drastically, this suggests the presence of oc-\ntahedral Zn2+, which is contrary to theoretical studies\nreporting that Zn solely favors tetrahedral coordination\n[13].To conﬁrm these suspicions XANES on the Zn K\nedge in combination with simulations would be required,since measurements on the Zn L 3,2edges did not allow\nto draw any meaningful and direct conclusions. Further-\nmore, a variation of the thickness under the optimized\npreparation conditions from the Zn series, highlighted\nthe reproduceability of the thin ﬁlms. The samples from\nthe thickness series had equally promising structural and\nmagnetic properties suitable for a ferromagnetic insula-\ntor with a low magnetic damping.\nA comparisonof the results from the structural and mag-\nneticanalysisoftheAlseriesshowedthatthebestsample\nof the Zn series could not be further improved by varying\nthe Al concentration. Only a small variation of the Al\namountalreadyleadto drasticchangestoeither thecrys-\ntal quality or the magnetic properties. From the XMCD\nspectra it could be concluded, that only changing both\nthe Zn and Al amount keeps the site occupancy of the\nFe and Ni cations relatively unaﬀected. However, if only\none of the parameters is changed the occupancy of Ni13\nand Fe is changed signiﬁcantly manifesting in either an\nincreased coercivity or loss in structural quality.\nThe best insulating ferromagnet with a low magnetic\ndamping was obtained for a thickness of 25nm at lowest\npossibleZncontent. AGilbertdampingof α= 7.8×10−3\nand a g-factor of g= 2.15±0.09, measured over a fre-\nquency range of up to 50GHz, was achieved.\nThe investigation of the new series in comparison to pre-\nvious publications [3, 4] augments previous ﬁndings of\nthe site occupancy being the main driving force in de-\ntermining the magnetic damping in NiZAF thin ﬁlms.\nFurthermore, the comparison showed that, there exists\nmorethanonestoichiometryforthefabricationofaferro-\nmagnetic insulatorwith a lowmagnetic damping suitable\nfor spintronic applications. From this publication a Zn-\ndeﬁcient NiZAF with the so far highest reported strain\nyielded a Gilbert damping as low as α= 7.8×10−3. De-\nspitetheheavystraintheCurietemperaturewasfoundto\nbe as high as 450K, unprecedented in sputtered NiZAFthin ﬁlms.\nAcknowledgement\nThe authors gratefully acknowledge funding by FWF\nproject ORD-49. The X-ray absorption measurements\nwere performed on the EPFL/PSI X-Treme beamline at\nthe Swiss Light Source, Paul Scherrer Institut, Villigen,\nSwitzerland. In addition, support by VR-RFI (Contract\nNo. 2017-00646 9 & 2019-00191) and the Swedish Foun-\ndation forStrategicResearch(SSF, ContractNo. RIF14-\n0053) supporting accelerator operation at Uppsala Uni-\nversity is gratefully acknowledged. The research lead-\ning to this result has been supported by the RADIATE\nproject under the Grant Agreement 824096 from the EU\nResearch and Innovation programme HORIZON 2020.\n[1]S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, S.\nVon Molnar, M. Roukes, A. Y. Chtchelkanova, and D.\nTreger, Science 2945546, 1488 (2001).\n[2]Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B6622, 224403 (2002).\n[3]S.Emori, B. A.Gray, H.-M.Jeon, J.Peoples, M. Schmitt,\nK. Mahalingam, M. Hill, M. E. McConney, M. T. Gray,\nU. S. Alaan et al, Adv. Mater. 29, 1701130 (2017).\n[4]J. Lumetzberger, M. 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Cieluch 3 \n \n1 Institute of Molecular Physics, Polish Academy of Science, Smoluchowskiego 17,  \nPL-60179 Pozna ń, Poland \n2 Adam Mickiewicz University, Faculty of Chemistry, Grunwaldzka 6, PL-60780 Pozna ń \nPoland \n3 The Research Centre of Quarantine, Invasive and Genetically Modified Organisms,  \nInstitute of Plant Protection – National Research Institute, W ęgorka 20, PL-60318 \nPozna ń, Poland \n* Corresponding author: and@ifmpan.poznan.pl   \n \nIn this report we present results of field-cooled and zero field -cooled \nmagnetization measurements and investigation of aging and memory effect in \nbismuth ferrite multiferroic micro-cubes obtained by means of simpl e microwave \nsynthesis procedure. It is found that difference between field-c ooled and zero \nfield-cooled magnetizations appears at the temperature of freez ing of \nferromagnetic domain walls. The decay of the magnetic moment vs . time \ndescribed by power-law relation and the absence of memory effect  are caused by \ndomain growth mechanism rather than by the spin-glass phase. The negli gible \nvalue of remnant magnetic moment indicates that bismuth ferrit e compound \nexhibits low concentration of ferromagnetic domains and can be cl ose to \nferromagnetic to spin-glass transition.  \nKeywords: multiferroics; bismuth ferrite; spin-glass; ferromagne tic domains; \naging effect; memory effect  \n2 1.  Introduction \nBismuth ferrite BiFeO 3 (BFO) compound belongs to an interesting class of \nmagnetoelectric (ME) multiferroic materials that exhibit a coexistence of mutually \ncoupled ferroelectric and magnetic ordering. These ME materials have recently attracted \nattention of researchers all over the world because of their per spective applications in \nnanophysics, sensors, future electronic and multi-state memory devices [1-5].  \nBFO multiferroic compound shows a rhomboedrally distorted perovskite cel l \nwith space group R3c  at room temperature. Ferroelectricity in BFO arises below the \nCurie temperature TC=1143 K and is due to the ordering of lone pairs of Bi 3+  ions \nlocated at the A sites of the perovskite unit cell [6]. The ferr oelectric ordering causes \nirreversible polarisation vs. electric field P(E) with a value of remnant polarisation Pr \n≈35 µC⋅cm -2 at room temperature [4], close to the estimations from first- principles \ncalculations [7, 8]. Magnetic properties of BFO are associate d with Fe 3+  spins \noccupying the B sites in the perovskite unit cell. From the point of view of magnetic \nordering, BFO is an antiferromagnet (AFM) below the Néel tem perature TN=643 K.  \nThe magnetic order in BFO is subjected to other transitions mani fested near the \ntemperatures 140 K and 200 K. The transition at 140 K is interpreted in t erms of spin-\nglass phase appearance [9, 10] whereas the anomaly about 200 K is as cribed to spin \nreorientations assisted by magnon softening [11].  \nAFM ordering in BFO is a rather complex G-type structure, in w hich each spin \nof iron is surrounded by six antiparallel coordinated spins, with sup erimposed long-\nrange incommensurate cycloidal modulation [12]. The spin cycloid can pr opagate along \nthree equivalent vectors q1=[1,-1,0], q2=[1,0,-1] and q3=[0,-1,1] (in pseudocubic \nnotation) with the period of λ=62 nm. In the cycloid, the Fe 3+  spins rotate in the plane \ndetermined by one of the three propagation vectors q and the vector of spontaneous  \n3 electric polarization Ps in the direction [111]. The magnetoelectric interaction similar in \nthe form to that of Dzyaloshinski-Moriya type causes small canti ng of spins out of the \nrotation plane, which produces a weak ferromagnetic moment (FM): m∼Ps×L where L \nis the antiferromagnetic vector. Cycloidal modulation brings about an analogous space-\nmodulation of m vector, spin density wave (SDW) and the absence of spontaneous \nmagnetization [13]. However, the homogeneous G-type ordering wit h no cycloidal \nmodulation can be established in BFO by anisotropic strains or a str ong magnetic field \nand weak macroscopic FM with a theoretical magnitude of 0.02 ÷0.1 µB/Fe can appear. \nTherefore, in BFO multiferroic probably the FM ordered and spin-gla ss phases can \nsimultaneously exist.  \nThe systems exhibiting both ferromagnetic and spin-glass propertie s like, for \nexample chromium thiospinels, have been reported by Vincent et al. [14 ]. Depending on \nchromium content the thiospinels could be ferromagnet, spin-glass or fe rromagnet \nfollowed by re-entrant spin-glass phase at lower temperature. It  has been demonstrated \nthat many properties of disordered FM phases and spin-glasses l ike aging, rejuvenation, \nmagnetization dependence on history and response to AC magnetic fie ld are indeed \nsimilar.  \nThe aim of this work is to study the possible mutual interactions of the ordered \nFM and spin-glass phase in BFO and to answer the question which phenomenon  brings \na dominant effect on the magnetic properties: spin-glass or FM domain growth and \npinning of the domain walls.  \n2.  Experimental \nA simple microwave synthesis procedure was used to obtain bismuth ferrite multiferroic \nmicro-cubes. The micro-cubes were formed by means of microwa ve assisted  \n4 hydrothermal Pechini process. The substrates for the synthesis, B i(NO 3)3·5H 2O, \nFe(NO 3)3·9H 2O, Na 2CO 3 were mixed and added to a KOH water solution (11 M).  \nAll chemicals were of analytical grade and were used wit hout further purification. Then \nthe mixture was poured into a few teflon reactors and processed in CEM Mars-5 \nmicrowave oven. The microwave reaction was carried out at 200 °C f or 30 min.  \nThe as-obtained brown in coloration, fine powders were collected by de canting, rinsed \nby water and dried at 70 °C in an conventional oven. Some powders were al so air \ncalcined at 500 °C for 1h, however this treatment had no distinct effect  on their \ncrystallographic structure and physical properties.  \nThe samples were characterized by powder X-ray diffraction ( XRD) performed \non an ISO DEBEYE FLEX 3000 instrument, using a Co lamp generating 0.17928 nm , \nwavelength. The morphology of the samples was examined by S3000N Hi tachi \nScanning Electron Microscope (SEM).  \nThe magnetic measurements were performed by means of the Phy sical Property \nMeasurement System (PPMS Quantum Design Ltd.). The VSM (Vibra ting Sample \nMagnetometer) probe was fitted to PPMS and used for DC magn etic moment \nmeasurements. The powder samples were sealed in gelatine capsules.  \n3.  Results and Discussion \nFig. 1 represents the X-ray diffraction pattern of as-obtained mi crowave-synthesized \nproduct. The vertical lines terminated by squares correspond to the si mulated spectrum \nof BiFeO 3 (BFO) for X-ray radiation with the wavelength 0.17928 nm as used in t he \nexperiment. Comparing these two plots one can clearly see that the  as-obtained \nmicrowave-synthesized product is in good crystalline BFO rhombohedral  phase with \nthe R3c  space group and contains very low amount of Bi 25 FeO 40  parasitic phase.   \n5 SEM microscopy revealed that the powders were composed of large r \nagglomerates formed by BFO almost regular micro-cubes (see Fig. 2). The slightly \nrounded corners of the cubes were due to the effect of high KOH co ncentration in \nsolution, actually equal to 11 M. The mean size of the BFO micro-cubes  was about  \n1 µm.  \nThe zero-field cooling (ZFC) and field cooling (FC) temperature  dependences of \nmagnetization M(T), are shown in Fig. 3. For the sake of simplicity, this figure pres ents \nZFC and FC data for three values of the applied magnetic fields onl y. The ZFC \nmeasurements were performed on warming the sample, after pri or cooling it to a low \ntemperature in the absence of a magnetic field, followed by appl ying of the field of a \ngiven strength. The FC data were also recorded on warming, after  previous cooling of \nthe sample in a magnetic field. At high temperatures the ZFC and FC magnetizations \nmatch each other and gradually increase as temperature decrea ses, probably due to local \nclustering of the spins [15] or to ferromagnetic domain growth [14]. The  ZFC and FC \nmagnetizations start to differ below a certain field depended tem perature Tf(H), which is \na phenomenon which can be interpreted in terms of spin-glass transition or freezing of \ndomain walls motion [16]. The relation between Tf and applied magnetic field is shown \nin the semilogarithmic plot in Fig. 2. It is evident that Almeida -Thouless (AT) line in \nthe form of: H=HAT [1-Tf(H)/ Tf(0)] 3/2  where HAT  is a coefficient, does not fit the \nexperimental data correctly in whole explored field range i.e. f rom 3 ·10 -3 T to 3 T.  \nThe best fit to the AT line can be obtained for µ0HAT =3.1 T, Tf(0)=71 K and it matches \nthe experimental data in the range of high magnetic fields, only. T he data in the  \nlog( H)-T semilogaritmic representation, however, can be well fitted by a linea r function, \nwhich means that log( H)∝Tf(H). Therefore, the following relation is fulfilled between \nthe value of applied magnetic field and Tf temperature:   \n6   (1) \nwhere H0 and b are parameters of the fit equal to: µ0H0=4,7 T, b=-0.053 K -1.  \nThe relation described by eq. (1) can be explained within the model of domain-\nwall pinning proposed by Kersten [17] if one notices that the energy of  a domain \ncharacterized by magnetic moment m and interacting with a magnetic field is E∝mH . \nConsequently eq. (1) can be expressed in terms of energy E=E0exp[-b Tf(H)] which has \nidentical form as the wall energy density σw=a ⋅exp(-b T) [18]. In general, the pinning of \ndomain-walls proceeds along a complex scenario in which the walls a re bowed between \nadjacent pinning sites and can move to the next site when the applied f ield increases \nabove a certain critical value. However, in the most simple approx imation the domain-\nwall pinning energy Ep is proportional to the wall energy density Ep∝σw [17] and the  \neq. (1) should be obeyed. Thus, in this approach the bifurcation between FC and  ZFC \nmagnetizations can be ascribed to the freezing of domain-wall m otion below the field \ndependent temperature Tf(H) and an increase in the strength of domain wall-pinning as \nthe temperature further decreases. Above Tf the energy due to thermal activation of the \ndomain-walls exceeds the pinning energy and the walls can move betwe en the pinning \nsites.  \nAn effective test for glassy behaviour exhibited by many syst ems is the aging \neffect manifested as a magnetization dependence on time. The ag ing effect for our BFO \nsample is shown in Fig. 5, which presents the time dependence of thermor emanent \nmagnetization after earlier cooling in the applied field µ0H=0.1 T to a selected \ntemperature followed by a rapid switching off this field. To bette r visualize the details \nof the magnetic moment relaxation vs. time, all curves m(t)/ m(t0) in Fig. 5 are \nnormalized with respect to the first recorded data m(t0) where t0≈15 s is the time \nnecessary to switch off the magnetic field and to complete the first measurement.  [ ])(b - exp f 0 HT H H= \n7 When the data are plotted in log( m(t)/ m(t0)) vs. log( t/t0) form they co-ordinate linearly, \nexcept for the long term measurements for T=80 K. This means that the aging proceeds \naccording to the following relation:  \n  (2) \nwhere A is a coefficient. \nThe best fit to the experimental data in Fig. 5 wit hin the model (2) is obtained for m∞=0 \nand A=0.033, 0.097 and 0.262 at 10, 40 and 80 K, re spectively. \nThe power-law decay of the remnant magnetization m(t) or aging as in eq. (2) \ncan be an indication of FM domain development after  rapid switch-off the field in the \ninitially single domain FM state [19]. The decompos ition of the homogeneous FM state \ninto a polydomain one is probably controlled by the  thermal fluctuations in a similar \nmanner to the random-field controlled systems [20].  Indeed, identical dependences as \neq. (2) have been theoretically obtained using Mont e Carlo simulations in an assembly \nof single ferromagnetic domain nanoparticles with d ipolar interparticle interactions [21] \nand found experimentally in superferromagnetic gran ular Co 80 Fe 20 /Al 2O3 multilayers \n[19]. The negligible value of finite residual magne tic moment m∞ means that an \nintermediate FM domain concentration exists below s ome threshold above which the \nnon-vanishing remanence is observed. A relaxation o f m(t) faster than predicted by the \npower-law, recorded at T=80 K for long-term measurements is probably caused  by low \npinning energy of domain walls, which at a temperat ure above Tf(H) should be \nnegligible (see Fig. 4) \nContrary to spin-glass state, the ferromagnetic or superferromagnetic systems \ncontrolled mainly by the processes of domain growth  and domain walls pinning exhibit \npartial or negligible memory effect [14]. The resul t of test for memory effect for BFO is ( ) ( )A\ntttm mtm−\n∞ \n\n\n+ =\n00 \n8 presented in Fig. 6. The measurements were performe d in the following way: at first the \nreference curve (solid squares in Fig. 6) was deter mined on field cooling from room \ntemperature down to 10 K at the cooling rate of 1 K ⋅min -1 and at the applied field \nµ0H=0.1 T. Then, temperature was raised back and the p rocedure was repeated with \ncooling interrupted for t=104 s at Tw=50 K and the relaxation of the magnetic moment \nwas measured (open squares in Fig. 6). After resumi ng the cooling and reaching 40 K, \nanother measurement of magnetic moment m was performed on heating the sample \nfrom 40 K to 60 K (open circles). The curve obtaine d on heating coincides with the \nreference line and does not exhibit any peculiarity  at Tw. This result is in agreement \nwith FM processes controlled by domain growth mecha nisms which can erase the \nmemory associated with domain wall pattern [14]. In  contrast, for the spin-glass phase \nor domain wall reconformations (without domain grow th) the memory effect should be \npresent.  \n4.  Conclusions \nThe results of our experiments, like the ZFC-FC bif urcation at the temperature of \nfreezing of FM domain wall motion, power-law aging of magnetic moment and the \nabsence of memory effect, seem to be well explained  by the domain growth, domain \nwall pinning and reconformation processes in the BF O multiferroic. This possible \nexplanation has been already mentioned in the repor t of Singh et al. [10] as a alternative \nto spin-glass, because many other systems can exhib it properties like AT-line \noccurrence (superparamagnets), ageing and rejuvenat ion (ferroic systems) or frequency \ndependent susceptibility (relaxors).  \nHowever, these two points of view do not exclude ea ch other because the glassy \nbehaviour can also appear due to reconformations of  domain walls, which posses spin- \n9 glass like hierarchy of metastable states in real s pace [14]. Moreover, BFO is very weak \nFM with a rather low concentration of FM domains as  evidenced by a negligible value \nof remnant magnetization m∞. Such systems are very close to transition between  spin-\nglass and FM phase and can exhibit complex properti es. \nAcknowledgements  \nThis project has been granted by National Science Centre (pr oject No. N N507 229040) and \npartially by COST Action MP0904.  \n  \n10  References \n[1]  G. Smolenskii, V. Yudin, E. Sher, and Y.E. Sto lypin, Antiferromagnetic \nproperties of some perovskites , Sov. Phys. JETP 16 (1963), pp. 622-624.  \n[2]  M. Fiebig, Revival of magnetoelectric effect , J. Phys. D: Appl. Phys. 38 (2005), \npp. R123-R152.  \n[3]  A.M. Kadomtseva, Yu.F. Popov, A.P. Pyatakov, G .P. Vorobev, A.K. Zvezdin,  \nand D. Viehland, Phase transitions in multiferroic BiFeO 3 crystals, thin-layers, \nand ceramics: enduring potential for a single phase , room-temperature \nmagnetoelectric “holy grail” , Phase Transit. 79 (2006), pp. 1019-1042.  \n[4]  D. Lebeugle, D. Colson, A. Forget, M. Viret, P . Boville, J.F. Marucco, and S. \nFusil, Room-temperature coexistence of large electric pola rization and magnetic \norder in BiFeO 3 single crystals , Phys. Rev. B 76 (2007), pp. 024116-1-8.  \n[5]  T. Choi, S. Lee, Y.J. Choi, V, Kiryukhin, and S.-W. Cheong, Switchable \nferroelectric diode and photovoltaic effect in BiFe O 3, Science 324 (2009), pp. \n63-66.  \n[6]  D. Khomskii, Classifying multiferroics: Mechanisms and effects , Physics 2 \n(2009), pp. 20- 28.  \n[7]  A. Lubk, S. Gemming, and N.A. Spalin, First-principles study of ferromagnetic \ndomain walls in multiferroic bismuth ferrite , Phys. Rev. B 80 (2009), pp. \n104110-1-9.  \n[8]  C. Ederer and N.A. Spaldin, Influence of strain and oxygen vacancies on the \nmagnetoelectric properties of multiferroic bismuth ferrite , Phys. Rev. B 71 \n(2005), pp. 224103-1-11.  \n[9]  M.K. Singh, W. Prellier, M.P. Singh, R.S. Kati yar, and J.F. Scott, Spin-glass \ntransition in single-crystal BiFeO 3, Phys. Rev. B 77 (2008), pp. 144403-1-5.  \n[10]  M.K. Singh, R.S. Katiyar, W. Prellier, and J. F. Scott, The Almeida-Thouless line \nin BiFeO 3: is bismuth ferrite a mean field spin glass? , J. Phys.: Condens. Matter. \n21 (2009), pp. 042202-1-5.   \n11  [11]  M.K. Singh, S. Dussan, W. Prellier, and R.S. Katiyar, One-magnon light \nscattering and spin-reorientation transition in epi taxial BiFeO 3 thin films , J. \nMagn. Magn. Matter. 321 (2009), pp. 1706-1709.  \n[12]  I. Sosnowska, T. Peterlin-Neumaier, and E. St eichele, Spiral magnetic-ordering \nin bismuth ferrite , J. Phys. C: Solid State Phys. 15 (1982), pp. 4835 -4846.  \n[13]  M. Ramazanoglu, M. Laver, W. Ratcliff II, S.M . Watson, W.C. Chen, A. \nJackson, K. Kothapalli, S. Lee, S.-W. Cheong, and V . Kiryukhin, Local weak \nferromagnetism in single-crystalline ferroelectric BiFeO 3, Phys. Rev. Lett. 107 \n(2011), pp. 207206-1-5.  \n[14]  E. Vincent, V. Dupuis, M. Alba, J. Hammann, a nd J.-P. Bouchaud, Aging \nphenomena in spin-glass and ferromagnetic phases: D omain growth and wall \ndynamics , Europhys. Lett. 50 (2000), pp. 674-680.  \n[15]  S. Nakamura, S. Soeya, N. Ikeda, and M. Tanak a, Spin-glass behavior in \namorphous BiFeO 3, J. Appl. Phys. 74 (1993), pp. 5652-5657.  \n[16]  H. Chang, Y.-q. Guo, J.-k. Liang, and G.-h. R ao, Magnetic ordering and \nirreversible magnetization between ZFC and FC state s in RCo 5Ga 7 compounds , \nJ. Magn. Magn. Matter. 278 (2004), pp. 306-310.  \n[17]  M. Kersten, Über die Bedeutung der Versetzungsdichte für die Th eorie der \nKoerzitivkraft rekristallisierter. Werkstoffe , Z. Angew. Phys. 8 (1956), pp. 496-\n502.  \n[18]  G. Vertesy, and I. Tomas, Temperature dependence of the exchange parameter \nand domain-wall properties , J. Appl. Phys. 93 (2003), pp. 4040-4044.  \n[19]  X. Chen, W. Kleemann, O. Petracic, O. Sichels chmidt, S. Cardoso, and P.P. \nFreitas, Relaxation and aging of a superferromagnetic domain  state , Phys. Rev. \nB. 68 (2003), pp. 054433-1-5.  \n[20]  Y. Imry, and S.K. Ma, Random-field instability of ordered state of contin uous \nsymmetry , Phys. Rev. Lett. 35 (1975), pp. 1399-1401.  \n[21]  M. Urlich, J.G.-Otero, J. Rivas, and A. Bunde , Slow relaxation in ferromagnetic \nnanoparticles: indication of spin-glass behavior , Phys. Rev. B. 67 (2003), pp. \n024416-1-4.  \n12  Figure Captions \nFigure 1 XRD-pattern the BiFeO 3 bismuth ferrite micro-cubes. The asterisk * indica tes \ntraces of Bi 25 FeO 40  parasitic phase.  \nFigure 2 SEM micrograph of the BiFeO 3 bismuth ferrite micro-cubes synthesized by \nmeans of microwave activation of the Pechini proces s.  \nFigure 3. ZFC and FC magnetization temperature depe ndences recorded for three \ndifferent magnetic fields: 0.01 T; 0.1 T and 1 T. T he arrows indicate the temperature Tf \nat which the difference between ZFC and FC magnetiz ation appears.  \nFigure 4. Relation between the temperature Tf at which the difference between FC and \nZFC magnetizations appears vs. the applied magnetic  field. Solid red line corresponds \nto the domain walls pinning model equation (1), the  broken blue line represents the \nAlmeida-Thouless transition.  \nFigure 5. Normalized value of magnetic moment relax ation vs. normalized time (aging) \nafter field cooling the sample in the applied magne tic field µ0H=0.1 T to 10 K; 40 K and \n80 K  \nFigure 6. The absence of memory effect in BFO. Soli d squares represent the reference \nmagnetization measured on field-cooling in µ0H=0.1 T. Open squares represent FC \nmagnetization measurement on cooling with intermitt ent stop at 50 K for tw=10 4 s. \nOpen circles correspond to FC magnetization measure d when warming from 40 K to  \n60 K. \n \n  \n13  Figures \n \nFigure 1 \n \nFigure 2 \n  \n14  \n \nFigure 3 \n \nFigure 4  \n15  \n \nFigure 5 \n \nFigure 6 "    },    {        "title": "1608.07429v1.Optimization_of_nanocomposite_materials_for_permanent_magnets_by_micromagnetic_simulations__effect_of_the_intergrain_exchange_and_the_hard_grains_shape.pdf",        "content": "arXiv:1608.07429v1  [cond-mat.mtrl-sci]  26 Aug 2016Optimization of nanocomposite materials for permanent mag nets by\nmicromagnetic simulations: eﬀect of the intergrain exchan ge and the hard\ngrains shape\nSergey Erokhin and Dmitry Berkov\nGeneral Numerics Research Lab, Moritz-von-Rohr-Strasse 1 A, D-07749 Jena, Germany\nIn this paper we perform the detailed numerical analysis of r emagnetization processes in nanocom-\nposite magnetic materials consisting of magnetically hard grains (i.e. grains made of a material with\na high magnetocrystalline anisotropy) embedded into a magn etically soft phase. Such materials are\nwidely used for the production of permanent magnets, becaus e they combine the high remanence\nwith the large coercivity. We perform simulations of nanoco mposites with Sr-ferrite as the hard\nphase and Fe or Ni as the soft phase, concentrating our eﬀorts on analyzing the eﬀects of ( i) the\nimperfect intergrain exchange and ( ii) the non-spherical shape of hard grains. We demonstrate tha t\n- in contrast to the common belief - the maximal energy produc t is achieved not for systems with\nthe perfect intergrain exchange, but for materials where th is exchange is substantially weakened.\nWe also show that the main parameters of the hysteresis loop - remanence, coercivity and the en-\nergy product - exhibit non-trivial dependencies on the shap e of hard grains, and provide detailed\nexplanations for our results. Simulation predictions obta ined in this work open new ways for the\noptimization of materials for permanent magnets.\nPACS numbers: 75.40.Mg, 75.50.Ww, 75.50.Tt, 75.60.-d\nI. INTRODUCTION\nMagnetic materials which can be used for the\nmanufacturing of permanent magnets belong to\nthe key materials in many high-technology applica-\ntions today1. Modern electromotors, actuators and\nmagnetic-ﬁeld-basedsensors(to name only a few ap-\nplications) require high-performance magnets. The\neﬀectivenessofthesemagnetsisusuallyestimatedby\nthe value of the maximal energy product ( BH)max\nachieved in the second quadrant of their B−Hhys-\nteresis loop. One of the most promising ways to\nobtain large values of this parameter is the usage\nof magnetic nanocomposites, i.e.materials which\ncombine a high coercivity of a magnetically hard\nphase (the phase made of a material possessing a\nlarge magnetocrystalline anisotropy) with the high\nsaturationmagnetizationoftheanother(soft)phase.\nAt present, best performance magnets are pro-\nduced basing on rare-earth metals (NdFeB, SmCo)\nand employing the precise fabrication control (see,\ne.g.2). Unfortunately, these magnets are relatively\nexpensive and subject to the availability and price\nﬂuctuations due to the high volatility of the rare\nearth elements market.\nAnother very important class of materials for per-\nmanent magnets is represented by nanocomposites\ncontaining ferrites as the hard phase. The energy\nproduct of Co-, Ba- and Sr-ferrite-based magnets is\nsuﬃcient for many applications of permanent mag-\nnets, e.g. in microwave devices, telecommunication,recording media, and electronic industry. Another\nimportant advantage of these materials have a much\nbetter temperature and corrosion resistance than\nNdFeB-based magnets (see, e.g. Ch. 12.2 in3). In\naddition, situation with the production of ferrite-\nbased magnets is more stable, and costs are much\nlowerdue to the wideravailabilityofthe correspond-\ning raw materials.\nThe speciﬁed performance of a nanocomposite\nmaterial can in principle be achieved by tailoring\nvarious parameters of a nanocomposite, such as rel-\native fractions of the soft and hard phases, size\nand shape of hard grains, mutual arrangement of\ngrains belonging to diﬀerent phases, and the qual-\nity of the intergrain boundaries. The development\nof new magnets of any type requires the thorough\nunderstanding of the relationship between their mi-\ncrostructure and magnetic properties4. Advanced\nstructural experimental techniques can provide a\nvery important information; well known examples\nare, e.g. the electron Bragg scattering diﬀraction\nstudies of the grain alignment in sintered NdFeB5or\nthe X-ray diﬀraction along with the high-resolution\ntransmission electron microscopy applied for the\nmeasurements of the grain size distribution in soft\nmagnetic alloys6.\nRecent publications of diﬀerent scientiﬁc collabo-\nrations have shown that in order to study the ques-\ntion ofthe microstructure-magnetismrelationshipin\ndetails, it is absolutely necessary to employ numeri-\ncal modeling - in particular, micromagnetic simula-tions, combining it with other experimental meth-\nods like 3D atome probe7, energy-dispersive X-ray\nspectroscopy8or magnetic neutron scattering9,10.\nThe usage of micromagnetic modeling in the de-\nvelopment process allows the a priori performance\noptimization of permanent magnets, by predicting\nmagnetic characteristics of a nanocomposite mate-\nrial before its actual manufacturing.\nIn general, micromagnetic simulations are per-\nfectly suited for modeling of magnetic compos-\nites, because the typical micromagnetic character-\nistic length - a few nanometers - allows to resolve\nvery well the magnetization distribution inside the\nnanocomposite grains with sizes of several tens of\nnanometers. However,correspondingsimulationsre-\nquire an enormous computational eﬀort due to two\nmain reasons. First, a very ﬁne mesh of ﬁnite el-\nements is required for the adequate approximation\nof each single grain as a geometrical object with a\ncomplicated shape. Second, a large number of soft\nand (especially) hard grains should be present in the\nsimulated volume, in order to study the magneti-\nzation reversal as a collective phenomenon and to\nobtain a suﬃciently accurate statistics for these dis-\nordered systems (see a detailed discussion of these\nissues in11–13).\nIn recent years, major simulation eﬀort was de-\nvoted to the understanding of rare-earth-based ma-\nterials, with the strategic goal ”to push the border”\nof the conventional (non-superconducting) mag-\nnets. In the ﬁrst line, a large amount of nu-\nmerical research was carried out for composites\nbased on NdFeB7,8,14–16and PrFeB17–19; some re-\nsults have been published also for SmCo and similar\ncompounds20.\nFor NdFeB-based materials, the question of the\ninternal magnetization structure (mainly vortex) in\na single grain have been studied14. The inﬂuence of\nthe soft phase concentration (Fe or FeB), the hard\ngrain size15and (very recently) magnetic behaviour\nof hard grains for several diﬀerent shapes16was in-\nvestigated. Special attention was paid to Nd-rich\ncompositions, where it has been shown that the en-\nrichment with Nd leads to the coercivity enhance-\nment due to the concentration of the additional Nd\non intergrain boundaries7,8.\nStudies of composites based on PrFeB and con-\ntaining Fe as the soft phase have been devoted to\nthe increasingroleofthe magnetodipolarinteraction\nwith the growingsoft phase fraction17, correlationof\nthe magnetization reversal of soft and hard grains18\nand the eﬀect of the hard grains alignment19. For\nSmCo-like materials diﬀerent mechanisms of the\nmagnetization reversal when changing the angle be-tween the anisotropy axis and the applied ﬁeld were\nidentiﬁed20.\nSomewhat apart from the main road lie the sim-\nulations of a highly interesting yet not really ap-\nplicable class of MnBi-based materials. Here the\ninﬂuence of the soft phase concentration (Co and\nFeCo) and the orientation degree of the hard grain\nanisotropy axes was studied numerically in21.\nIn contrast to the rare-earth-based composites,\nmaterials based on various ferrites as the hard phase\nhavenot been studied - up to ourknowledge - by mi-\ncromagnetic simulations, although this class of ma-\nterials becomes increasingly important for reasons\nlisted above. In the present paper we address this\nchallenge, starting with detailed studies of the eﬀect\noftheintergrainexchangecouplingandthe inﬂuence\nof the hard grain shape on the material properties in\nnanocomposites SrFe 12O19/Fe and SrFe 12O19/Ni.\nOur polyhedron-based micromagnetic algorithm\nprovides a high statistical accuracy of simulated re-\nsults, because we are able to handle systems con-\ntaining a few thousand grains, including the ability\nto resolve a possibly non-trivial magnetization dis-\ntribution in every grain. Due to the ﬂexibility of our\nmesh generation method, a nearly arbitrary grain\nshape can be adequately approximated, so that a\nsimulated sample may include the grains of diﬀerent\nshapes and sizes.\nThe paper is organized as follows: in Sec. II we\nexplain the mesh generation method which we use\nto create a polyhedron mesh for a system containing\nnon-spherical hard grains embedded into a magnet-\nically soft matrix. Evaluation of micromagnetic en-\nergy contributions in our methodology is also brieﬂy\npresented. Sec. III contains simulation results.\nHere, subsection IIIA is devoted to the analysis of\nthe inﬂuence of the intergrain exchange weakening\non the magnetization reversal, whereas subsection\nIIIB deals with the eﬀect of non-spherical shapes of\nthe hard grains. Both subsections contain a detailed\nphysical discussion of results obtained. We conclude\nwith the summary of our ﬁndings and possible per-\nspectives for the improvement of ferrite-based com-\nposites in Sec. IV.\nII. OUR MICROMAGNETIC\nMETHODOLOGY FOR SIMULATION OF\nNANOCOMPOSITES\nTo overcome the diﬃculties in modeling mag-\nnetic nanocomposites using standard micromagnetic\nmethods (ﬁnite diﬀerence and tetrahedral ﬁnite ele-\nments), we have developed13a novel micromagnetic\n2methodology based on a discretization of magnetic\nmaterials using polyhedrons of a special type. This\nmethodology combines the ﬂexibility of general ﬁ-\nnite element schemes for the geometrical description\nof a nanocomposite structure with the possibility to\nuse Fast Fourier Transformation for the calculation\nofthe most time consumingcontribution tothe total\nenergy - magnetodipolar interaction.\nA. Mesh generation for grains of a general\nshape\nOne of the main questions discussed in this pa-\nper is the inﬂuence of the non-spherical (spheroidal)\nshape of hard grains on the magnetic behavior\nof nanocomposites. For such a system, we had\nto introduce two additional steps into the mesh\ngeneration procedure described in our previous\npublications11–13. Namely, in the present work the\nmesh for hard grains is generated separately from\nthe soft phase mesh (Fig.1) (additional step 1), and\nthen both systems are merged (Fig. 2) (additional\nstep 2).\nFIG. 1. (color online) Examples of the spatial distribu-\ntion of hard crystallites (soft crystallites not shown) in\nsimulated samples for diﬀerent aspect ratio a/bof corre-\nsponding ellipsoids of revolution (see text for details).\nAsinourstandardmethodology,westartfromthe\ngeneration of mesh consisting of small nearly spheri-\ncal polyhedrons with the sizes less the characteristic\nmicromagnetic length. These polyhedrons will be\nused in micromagnetic simulations as corresponding\nﬁnite elements (see11–13for details).\nNext, we generate a system of non-overlapping el-\nlipsoidal particles (additional step 1) with sizes and\nshapes corresponding to the hard grains of our com-\nposite. We point out that the generation of an en-\nsemble of non-overlappingellipsoids is computation-\nally challenging: the evaluation of an overlapping\nFIG. 2. (color online) Example ofa microstructure (hard\nandsoftphases)usedinourmodelingofnanocomposites.\nof two ellipsoids and the introduction of the suitable\noverlappingcriterionrequire a development ofa spe-\ncial numerical scheme. We use a method described\nby Donev et al.22, which is based on the Perram-\nWertheim overlap potential and provides a suitable\nparameter describing the overlapping degree of two\nellipsoids. This parameter is then used in the model\nof interacting particles with a short-range repulsive\npotential, where initially ellipsoids are placed ran-\ndomly, but due to the nature of this potential the\nnumber of overlaps is continuously decreasing.\nAt the second additional step, these ellipsoids are\nmapped on the system of (much smaller) polyhe-\ndrons used as mesh elements in our micromagnetic\nsimulation. By this mapping all mesh elements with\ncenters inside ellipsoids, are assigned to the hard\nphase and the rest of elements - to the soft phase.\nAll mesh elements belonging to the same hard grain\nhave the same direction of the anisotropy axes; this\ndirection coincides with the rotational symmetry\naxis of the ellipsoid. Note, that the discretization\nof crystallites of the soft and hard phases is based\non mesh elements of the same size.\nB. Energy contributions and minimization\nprocedure\nIn our simulations we take into account all four\nstandard contributions to the total magnetic free\nenergy: energy in the external ﬁeld, magnetocrys-\ntalline anisotropy energy, exchange stiﬀness and\nmagnetodipolar interaction energies11–13.\nThe local energy parts - energy in the external\nﬁeld and the magnetocrystalline anisotropy energy -\narecomputedinastandardway,multiplyingthecor-\nresponding energy densities by the volume of ﬁnite\n3elements (polyhedrons in our case) and summing\nover all these elements. The magnetodipolar ﬁeld\nand energy are computed using the optimized ver-\nsion of the lattice Ewald method for disordered sys-\ntems. In this algorithm, the mapping of the initial\n(disordered) system of mesh elements on the trans-\nlationally invariant regular lattice allows to keep the\nhigh speed of the lattice method (fast Fourier trans-\nformation), at the same time making the mapping\nerrors negligibly small.\nIn this work, we concentrate ourselves in particu-\nlar on the eﬀect of the intergrain exchange. Hence\nwe remind that the exchange energy in our method-\nology is computed in the nearest neighbouring ap-\nproximation as\nEexch=−1\n2N/summationdisplay\ni=1/summationdisplay\nj⊂n.n.(i)2AijVij\n∆r2\nij(mimj),(1)\nwhereVij= (Vi+Vj)/2, ∆rijis the distance be-\ntween the centers of i-th and j-th ﬁnite elements\nwith volumes ViandVj. The exchange constant\nAijfor the homogeneous bulk material is equal\nto the corresponding exchange stiﬀness constant\nA, but is obviously site-dependent in composite\nmaterials12,13.\nFor the minimization of the total magnetic en-\nergy, obtained as the sum of all four contributions\ndescribed above, we use the simpliﬁed version of a\ngradient method employing the dissipation part of\nthe Landau-Lifshitz equationof motion for magnetic\nmoments23,24. The minimization is considered as\nconverged, when the condition for the local torque\nmax{i}|[mi×heﬀ\ni]|< εis fulﬁlled (here miis a nor-\nmalized magnetic moment of the i-th mesh element\nandheﬀ\niisthecorrespondingeﬀectiveﬁeld; thevalue\nε= 10−3was found to be small enough for our qua-\nsistatic minimization procedure).\nFurther details of our method can be found in13.\nIII. RESULTS AND DISCUSSION\nDue to the high performance of our methodology,\nwe are able to simulate bulk nanocomposites with\nhard grains of any prescribed shape, whereby the\nsimulated system may contain up to 600 hard grains\nwith an average discretization of 300 mesh elements\npro grain. Employing this algorithm, we could ob-\ntain systematic results with the high statistical ac-\ncuracy for two model systems: SrFe 12O19/Fe and\nSrFe12O19/Ni.For simulations of these nanocomposites we have\nused standard magnetic parameters of correspond-\ning materials (see, e.g.3, Chap. 5.3 and 11.6) which\nare summarized in the table below:\nSrFe12O19Fe Ni\nMs(G) 400 1700 490\nAnis. kind uniaxial cubic cubic\nK(erg/cm3)4.0·1065.0·105−4.5·104\nA(erg/cm) 0.6·10−62.0·10−60.8·10−6\nThe average grain volume for all phases was cho-\nsen to be equal to the volume of a spherical grain\nwith the diameter D= 25nm. The volume concen-\ntrationofthe hardphasein allpresentedsimulations\nwaschard= 40%.\nA. Eﬀect of the exchange weakening in\nSrFe12O19/Fe\nOne of the central questions for permanent mag-\nnets made of nanocomposite materials is the depen-\ndence of magnetic properties on the exchange weak-\nening between diﬀerent grains. This weakening is\nunavoidable in real systems, because it is nearly\nimpossible to obtain perfect intergrain boundaries.\nThe quality of these boundaries strongly depends\non the concrete method used for the manufactur-\ning of a nanocomposite and substantial eﬀorts has\nbeen devoted to obtaining materials with more per-\nfect intergrain boundaries (and especially bound-\naries between grains belonging to diﬀerent phases)\nin order to achieve better exchange coupling. How-\never, recently25is was demonstrated both experi-\nmentally and theoretically that the perfect inter-\ngrain exchange may strongly decrease the perfor-\nmance of a magnetic nanocomposite material, so\nthat this question requires a detailed theoretical\nstudy.\nThe exchange weakening in our methodology is\ndeﬁned bymultiplying the exchangeenergyofneigh-\nboring mesh elements belonging to diﬀerent crystal-\nlitesbyafactor0 ≤κ≤1. Thisexchangeweakening\ncoeﬃcient is introduced into the expression (1) for\nthe exchange energy as\nEexch=−1\n2N/summationdisplay\ni=1/summationdisplay\nj⊂n.n.(i)κ2AijVij\n∆r2\nij(mimj),(2)\nif neighboring magnetic moments iandjare located\nin diﬀerent grains (crystallites). From (2) it can be\n4seen that κ= 1 correspondsto the perfect intergrain\nexchange (equal to the exchange within a bulk ma-\nterial) and κ= 0 means no exchange interaction at\nall between diﬀerent grains.\nDependence of magnetic properties on this ex-\nchange weakening was studied for the composite\nSrFe12O19/Fe with approximately spherical hard\ngrains (obtained from the random placement of\nspheres with D= 25nm, see Sec. IIA).\nFIG. 3. (color online). Simulated hysteresis curves of the\nnanocomposite SrFe 12O19/Fe with spherical hard grains\nfor diﬀerent exchange weakening constant.\nThe overall trend is shown in Fig. 3, where the\nevolution of hysteresis curves by increasing the ex-\nchange coupling ( κ= 0.0→0.5) between grains is\ndemonstrated. Systems without ( κ= 0.0) or with a\nstrongly reduced ( κ= 0.05) exchange coupling be-\ntween grains exhibit the two-steps magnetization re-\nversal. The ﬁrst step - large jump on the hystere-\nsis loop in small negative ﬁelds (see the panel for\nκ= 0.05 in Fig. 3) - represents the magnetization\nreversal of the soft phase, which volume fraction is\nrelatively high. The second step - reversal of hard\ngrains in much higher ﬁelds - leads to the closure\nof the loop. Reversal of the hard phase occurs in\nﬁelds∼Hhard\nK, where the anisotropy ﬁeld is deﬁned\nasHK= 2K/Ms=βMs(β= 2K/M2\nsdenotes the\nreduced anisotropy constant). The large magnitude\nof the magnetization jump during the ﬁrst reversal\nstep is due to the dominating contribution of thesoft phase to the system magnetization: msoft=\ncsoft·Msoft/(chard·Mhard+csoft·Msoft)≈0.86.\nFor the detailed analysis of the magnetization re-\nversal in a magnetic composite it is very useful to\nplot hysteresis loops for the soft and hard phase sep-\narately. Such loops can be easily obtained from sim-\nulated magnetization conﬁgurations by summing up\ncontributions from ﬁnite elements belonging to ei-\nther soft or hard phase and calculating correspond-\ning total magnetizations of these phases.\nWe begin our consideration from systems with\nan absent or very low intergrain exchange coupling,\nwhere the dominant interaction is the magnetodipo-\nlar one. To clarify the eﬀect of this interaction, the\nabove mentioned magnetization reversal curves of\nhard and soft phases are plotted in Fig. 4 for the\ncomposite without any intergrain exchange coupling\n(κ= 0).\nIn order to understand the hysteretic behavior of\nboth phases, it is useful to calculate the reduced\nanisotropy constant β= 2K/M2\ns, which magnitude\ngives (roughly speaking) the relation of the magne-\ntocrystalline anisotropy ﬁeld to the magnetodipo-\nlar ﬁeld from the nearest neighbor in the system\nof spherical particles. Substitution of magnetic pa-\nrameters of our materials (see the table above) re-\nsults in the values βh= 50(≫1) for the hard phase\n(SrFe12O19) andβs= 0.34(∼1) for the soft phase\n(Fe). We note that the much higher value of βfor\nthe hard phase is due not only to its largeanisotropy\nconstant K(which is ’only’ 8 times larger than by\nFe), but mainly due to the much higher value of the\nsoftphasemagnetization MFe/MSrFeO= 4.25,which\ngives an additional factor of ≈18.\nConsidering the magnetization reversal of the soft\nphase ﬁrst, we note that this phase would exhibit\nin the absence of the magnetodipolar interaction\nthe ’ideal’ hysteresis loop for a system of non-\ninteracting particles with the cubic anisotropy con-\nstantKcub>0 (as it is the case for Fe). Such a loop\nhas the remanence j(0)\nR≈0.83 and the coercivity\nH(0)\nc≈0.33HK= 0.33βMs≈195Oe (see, e.g.26).\nThe relatively low value of the reduced anisotropy\nforoursoftphase βs(Fe) = 0.34meansthat themag-\nnetodipolar interaction can considerably modify the\ncorresponding’ideal’ hysteresis. This inﬂuence man-\nifests itself primarily in the smoothing of the ’ideal’\nloop26, as it can be seen in Fig. 4, where the loop\nfor the soft phase of our system is shown in red. The\nremanence jR≈0.836is nearly the same and the co-\nercivityHc≈260Oe increased by ≈30% compared\nto the non-interacting case.\nUnfortunately, we are not awareof any systematic\n5theoreticalstudiesofthe magnetodipolarinteraction\neﬀects in systems of ’cubic’ particles, except for the\npaper27, whereonlysimulationresultsfortheHenkel\nplots are shown; any quantitative comparison with a\ndetailed study of these eﬀects for ’uniaxial’ particles\npresented in28is meaningless due to very diﬀerent\nenergy landscapes for these two anisotropy types.\nFor this reason, we can only suggest that the nearly\nunchanged remanence (compared to the ’ideal’ sys-\ntem) is due to the interplay of the magnetodipolar\ninteractions within the soft phase and between the\nsoft and hard phases. The increase of Hcis most\nprobably due to the ’supporting’ action of the mag-\nnetodipolar ﬁeld from the hard phase onto the soft\ngrains. Magnetization of the hard phase in our sys-\ntem is rather low, so that the corresponding eﬀect is\nrelatively small.\nFIG. 4. (color online). Simulated hysteresis loops for\nSrFe12O19/Fe (with spherical hard grains) without the\nintergrain exchange ( κ= 0) presented for hard (solid\nblue line) and soft (solid red line) phases separately.\nDashed line represents the unsheared loop of the SW\nmodel with particle parameters as for SrFe 12O19, solid\ngreen line - the SW loop sheared according the aver-\naged internal ﬁeld (see text for details). External ﬁeld\nis normalized by the anisotropy ﬁeld of the hard phase\nHK=βhMh= 20kOe.\nThe non-interacting hardphase consisting of\ngrains with the uniaxial anisotropy (as for\nSrFe12O19) would reverse according to the ideal\nStoner-Wohlfarth (SW) loop29withjR= 0.5 and\nHc≈0.48HK≈10kOe shown in Fig. 4 with the\nthin dashed green line. The very large value of the\nreduced single-grain anisotropy βh(SrFeO) = 50 for\nthis phase means that intergraincorrelationsof hard\nphase magnetic moments are negligible. However,\nin our composite material hard grains are ’embed-ded’ into the soft phase. Hence, in order to prop-\nerly compare (at least in the mean-ﬁeld approxi-\nmation) the simulated hard phase loop - blue solid\nline in Fig. 4 - with the SW model, we have to\ntake into account the average magnetodipolar ﬁeld\n/angbracketleftHmd,z/angbracketright= (4π/3)/angbracketleftMsoft\nz/angbracketrightacting on a spherical par-\nticle inside a continuous medium with the average\nmagnetization of the soft phase /angbracketleftMsoft\nz/angbracketright.\nCorrection of the SW loop using this internal ﬁeld\n(which depends on the external ﬁeld via the corre-\nsponding dependence /angbracketleftMz(Hz)/angbracketright) leads to the loop\nshown with the thick solid green line in Fig. 4.\nIt can be seen that this corrected SW loop is in\na good agreement with the simulated hard phase\nloop. Remainingdiscrepanciesaredue to localinter-\nnal ﬁeld ﬂuctuations (always present in disordered\nmagnetic systems) which are especially pronounced\nin our composite due to the high diﬀerence between\nthe magnetizations of soft and hard phases.\nThis analysis reveals that the ﬁrst jump on the\nhard phase loop in small negative ﬁelds is due to\nthe abrupt change in the internal averaged dipolar\nﬁeld due to the magnetization reversal of the soft\nphase. The second jump - for Hz/Hk≈ −0.3 -\nis the manifestation of the singular behavior of the\nSW loop of the hard phase itself, which occurs for\nthe unsheared loop at Hcr=−Hk/2 (near this ﬁeld\nMz∼/radicalbig\n−(Hz−Hcr) forHz< Hcr30).\nIn summary, despite a relatively high saturation\nmagnetization Ms= 1180G, the corresponding\ncomposite without any intergrain exchange coupling\nwould have only a relatively small maximal energy\nproduct of ≈15kJ/m3(see Fig. 5b). The reason is\nits very small coercivity Hc≈250Oe, which is de-\ntermined entirely by the magnetization reversal of\nthe soft phase in small negative ﬁelds.\nBefore we proceed with the analysis of the eﬀect\nof the intergrainexchange coupling on the hysteretic\nproperties of a nanocomposite, an important me-\nthodical issue should be clariﬁed. Namely, we have\ntodetermine the maximalvalue ofthe exchangecou-\npling(maximalvalueof κ), forwhichoursimulations\ncan produce meaningful results.\nThe problem is that with increasing the cou-\npling strength, the interaction between the grains\nincreases, so that grains are starting to form clus-\nters, inside which magnetic moments of constituting\ngrains reverse nearly coherently. The average size of\nsuch a cluster /angbracketleftdcl/angbracketrightobviously growths with increas-\ningκ. In order to obtain statistically signiﬁcant re-\nsults, we have to assure that /angbracketleftdcl/angbracketrightis signiﬁcantly less\n(ideally much less) than the maximal system size ac-\ncessible for simulations. Otherwise we might end up\n6with the case where we are simulating the magne-\ntization reversal of a system consisting of a single\n(or very few) cluster(s), so that corresponding re-\nsults will be non-representative for the analysis of\nreal experiments.\nThe best quantitative method to determine /angbracketleftdcl/angbracketright\nis the calculation of the spatial correlation function\nof magnetization components perpendicular to the\napplied ﬁeld (in our case MxandMy): the average\nvalue of these components should be zero, and the\ndecay length of their correlation functions Cx(r) =\n/angbracketleftMx(0)Mx(r)/angbracketright(the same for My) would provide a\nmost reliable estimation of /angbracketleftdcl/angbracketright.\nHowever, taking into account a complex 3D char-\nacter of Cx,y(r), we have adopted another crite-\nrion to determine the approximate number of in-\ndependent clusters contained in our simulated sys-\ntem. Namely, as the ﬁgure of merit we have em-\nployed the maximal value of the perpendicular com-\nponent of the total system magnetization m⊥=/radicalBig\nM2x+M2y/Msduring the magnetization reversal.\nIf the system contains only one (or very few) clus-\nter(s), than for some ﬁeld during the reversal pro-\ncess this component should be large (close to 1), be-\ncause one cluster reverses nearly in the same fashion\nas a single particle, i.e., its magnetization rotates\nas a whole without signiﬁcantly changing its magni-\ntude. Hence at some reversal stage m⊥would un-\navoidably become relatively large. In the opposite\ncase, where a system contains many nearly indepen-\ndent clusters ( Ncl≫1), their components Mx,iand\nMy,i(i= 1,...,N), being independent variableswith\nzero mean, would averagethemselves out, leading to\nsmall values of m⊥.\nA simple statistical analysis based on the assump-\ntion of the independence of diﬀerent clusters shows\nthat the number of such clusters can be estimated\nasNcl≥1/m2\n⊥. This means that up to m⊥≈0.3\nwe produce statistically signiﬁcant results, because\nin this case Ncl≥10. Corresponding analysis shows\nthat for our systems (containing about ∼5·105ﬁ-\nnite elements) this is the case up to κ≈0.5, so\nbelow we show results only in this range of exchange\ncouplings.\nSimulation results showing basic characteristics\nof the hysteresis loop - remanence jR, coercivity\nHcand energy product Emax= (BH)max- for\nthe SrFe 12O19/Fe composite as functions of the ex-\nchange weakening κare presented in Fig. 5. We\nremind that for these simulations approximately\nspherical hard grains were used.\nFrom Fig. 5 it can be clearly seen that the rema-\nnencejRof this material depends on the intergrain(a)\n(b)\n(c)\nFIG. 5. (color online). (a)Remanence, (b)coerciv-\nity and(c)energy product of simulated nanocomposite\nSrFe12O19/Fe with spherical hard grains as a functions\nof exchange weakening on the grain boundaries. Inset\nin (a) represents the maximal value of perpendicular (to\nthe directions of applied ﬁeld) component of magnetiza-\ntion during the remagnetization process. Dashed lines\nare paths for the eye.\nexchange coupling relatively weak. The reason is\nthatjRis very high already for the fully exchange\ndecoupled composite ( jR(κ= 0)≈0.8). Such a high\nvalue, in turn, is due to the fact that the remanence\nis governed by the soft phase consisting of cubical\ngrains. The remanenceofthe non-interacting(ideal)\nensemble of such grains is j(0)\nR≈0.83. This high re-\nmanence can not be signiﬁcantly increased by the\nexchange interaction within the soft phase (as it is\nthecaseforthesystemof uniaxialparticleswithran-\ndomly distributed anisotropy axes, where j(0)\nR= 0.5;\nsee also31for the analysis of a corresponding 2D\n7system). Neither can this remanence be substan-\ntially decreased by the exchange coupling with hard\ngrains, because their magnetizationat Hz= 0is still\nnearly aligned along the initial ﬁeld direction due\nthe strong magnetizing ﬁeld from the Fe soft phase,\n(with its high magnetization MFe= 1700 G).\nIn contrastto jR, the coercivity Hcexhibits a pro-\nnounced maximum as the function of the exchange\ncoupling κ, resulting in the corresponding maximum\nof theκ-dependence of the maximal energy product\n(BH)max(κ). We will explain the reasons for the\nappearance of this maximum below, analyzing the\nhysteretic behavior of our nanocomposite for vari-\nousκ.\nFor the smallest non-zero κstudied here the mag-\nnetization reversal process is visualized in Fig. 6,\nwhere hysteresis loops for soft and hard phases are\nshown separately and the magnetization conﬁgura-\ntion is displayed for several characteristic external\nﬁelds. First, it can be clearly seen that the mag-\nnetizations of soft and hard phases reverse sepa-\nrately. The inspection of magnetization conﬁgura-\ntions shows that the reversal of magnetic moments\nstarts within the soft phase (see panel (a)) around\nthe hard grains which anisotropy axis are directed\n’favorably’(i.e. deviatestronglyfromtheinitialﬁeld\ndirection). Then the reversed area expands, occupy-\ning even larger regions of the soft phase (panel (b))\nuntil nearly the entire soft phase is reversed (panel\n(c)). Note that in the negative ﬁeld corresponding\nto this nearly complete reversal of the soft phase,\nthe majority of the hard phase is still magnetized\napproximately along the initial direction. Only in\nmuchlargernegativeﬁelds(rightdrawingofhystere-\nsis loops) the hard phase magnetization also starts\nto reverse (see panel (d)).\nWe emphasize here two important circumstances:\nalthoughtheexchangecouplingbetweenthesoftand\nhard phases is very weak ( κ= 0.05) and the concen-\ntration of the hard phase is moderate (40%), the\n’supporting’ action of the hard phase is enough to\nnearly double the coercivity of the soft phase and\nhence - of the whole system, when compared to the\ncase ofκ= 0 - see Fig. 5. At the same time, due\nto this low exchange coupling, hard grains reverse\nseparately from the soft phase and nearly separately\nfrom each other (see panel (c)), leading to a high\ncoercivity of the hard phase (right drawing of hys-\nteresis loops).\nFor the larger exchange coupling κ= 0.1 (see Fig.\n7) the ’supporting’ eﬀect of the hard phase increases\nthe coercivity of the soft phase even further (com-\npared to κ= 0.05). At the same time, this larger\ncoupling also leads to the much earlier reversal of\nFIG. 6. (color online). Magnetization reversal process\nfor the composite with exchange weakening κ= 0.05.\nFrom top to bottom: microstructure of the system\n(warm colors - soft, cold colors - hard grains); hystere-\nsis shown as separate curves for the soft (red) and hard\n(blue line) phases (note diﬀerent scales of the H-axis);\nmagnetization conﬁgurations shown as mz-maps for ﬁeld\nvalues indicated on the hysteresis plots shown above.\nthe hard phase, signiﬁcantly decreasing its coerciv-\nity - see hysteresis plots in Fig. 7. Magnetization\nreversal for this coupling starts in those system re-\n8FIG. 7. (color online). Magnetization reversal for the\ncomposite with the exchange weakening κ= 0.10 pre-\nsented in the same manner as in Fig. 6.\ngions where the hard phase is nearly absent (due to\nlocal structural ﬂuctuations) - see panel (b) in Fig.\n7 - and is much more cooperative compared to the\ncase ofκ= 0.05.\nThe resulting coercivity of the entire system is at\nits maximum, because the interphase coupling is,\non the one hand, large enough to prevent the soft\nphase from the reversal in small ﬁelds, but on an-other hand, small enough to enable to the reverse of\nthe hard phase in much higher negative ﬁelds than\nthe soft phase.\nFIG. 8. (color online). Magnetization reversal for the\ncomposite with the exchange weakening κ= 0.20 pre-\nsented in the same way as in Fig. 6. Simultaneous re-\nversal of the hard and the soft phases is clearly visible.\nWhen the intergrain exchange coupling is in-\ncreased further, magnetization reversal of the sys-\ntem becomes fully cooperative, so that the soft and\nhardphasesreversesimultaneously(inthesameneg-\native ﬁelds) - see hysteresis loops shown in Fig. 8\nforκ= 0.2. Spatial correlations between the mi-\ncrostructure and the nucleation regions for the mag-\nnetization reversal become weak, as it can be seen\nfrom microstructural and magnetic maps presented\nin this ﬁgure. It is also apparent that the correlation\ndistance of the magnetization conﬁguration strongly\nincreases, as it was noted in the discussion above.\nThe overallresult is the decreaseof the system co-\n9ercivity, because the soft phase causes the much ear-\nlierreversalofthehardphase,sothatthesupporting\neﬀect of the high anisotropy of the hard phase be-\ncomes smaller. However, for this relatively low value\nofκ= 0.2 this ’supporting’ eﬀect is still present:\nHc(κ= 0.2) is nearly twice as large as Hc(κ= 0).\nWhen the exchange coupling increases even fur-\nther, the magnetizationreversalbecomes completely\ndominated by the soft phase due to its larger mag-\nnetization and volume fraction. In particular, for\nκ= 0.5 both the coercivity and the energy product\nare nearly the same as for κ= 0. We note that hys-\nteresis loops for these two cases ( κ= 0 and κ= 0.5)\nlook qualitatively diﬀerent, but this physically im-\nportant diﬀerence (two-step vs one-step magnetiza-\ntion reversal) does not matter for the performance\nof the nanocomposite from the point of view of a\nmaterial for permanent magnets.\nThe non-monotonous dependence of the max-\nimal energy product on the exchange coupling\n(BH)max(κ) can be easily deduced from the depen-\ndenciesjR(κ) andHc(κ). When κincreases from 0\nto≈0.1, bothremanenceandcoercivityincrease,re-\nsultingintherapidgrowthof( BH)max. Forκ >0.1,\nthe small increase of the remanence (up to κ≈0.2)\ncan not compensate the large drop of coercivity, re-\nsulting in the overall decrease of the energy prod-\nuct. We point out here that such a behavior occurs\nonly when the dependence of the coercivity on the\ncorresponding parameter (in our case the exchange\nweakening κ) is really strong. The case when the co-\nercivity depends relatively weak on the parameter of\ninterest, is analyzed in detail in the next subsection.\nSummarizing this part, we have shown that, in\ncontrast to the common belief, there exist an op-\ntimalvalue of the interphase exchange coupling in\na soft-hard nanocomposite which provides the max-\nimal energy product. This optimal value obviously\ndepends on the fractionsof the soft and hard phases,\nbut it is very likely that the optimal coupling should\nbesigniﬁcantlylessthantheperfectcoupling( κ= 1)\nfor all reasonable compositions in this class of mate-\nrials.\nThis important insight opens a new route for the\noptimization of the permanent magnet materials.\nB. Eﬀect of the grain shape of the hard phase\ninSrFe12O19/FeandSrFe12O19/Nicomposites\nOne of the intensively discussed questions when\noptimizing the nanocomposite materials for perma-\nnent magnets is whether the materials containing\nthe hard grains with the non-spherical shape couldprovide an improvement of the energy product for\ncorresponding composites (see corresponding refer-\nences in the Introduction).\nThe standard argument in favor of the possi-\nble improvement of Emaxis the additional shape\nanisotropy of non-spherical particles. For an elon-\ngated(prolate)ellipsoidofrevolutionthisanisotropy\ncould increase the already present magnetocrys-\ntalline anisotropy (mc-anisotropy), thus enhancing\nthe coercivity of the hard phase and hence - the en-\nergy product. Below we will demonstrate that this\nline of arguments is not really conclusive and that\nthe grainshape eﬀect may be even the opposite - the\nenergy product can be larger for a material contain-\ningoblatehard grains.\nBefore proceeding with the analysis of our results,\nwe emphasize, that the relativecontribution of the\nshape anisotropy can be approximately the same for\nrare-earth and ferrite-based materials. The former\nmaterials have a much larger mc-anisotropy Kcr, so\nthat on the ﬁrst glance shape eﬀects for rare-earth\n’hard’ grains should be much smaller. But the the\nrelation between the shape anisotropy and the mc-\nanisotropy contributions is determined not only by\nthe value of Kcr, but by the reduced anisotropy con-\nstantβ= 2Kcr/M2\ns, which gives, roughly speaking,\nthe relation between the mc-anisotropy energy and\nthe self-demagnetizing energy of a particle.\nThe presence of the material magnetization in the\ndenominator of the expression for βmakes this con-\nstants for both material classes very similar. For ex-\nample, the mc-anisotropy Kcr≈4.6×107erg/cm3\nfor Nd 2Fe14B is more than one order of magnitude\nlarger than its counterpart Kcr≈4×106erg/cm3\nfor SrFe 12O19. However, the much lower magne-\ntization Ms≈400G of SrFe 12O19compared to\nMs≈1300G of Nd 2Fe14B makes the diﬀerence be-\ntween reduced anisotropies of these materials quite\nsmall:βNdFeB≈60, whereas βSrFeO≈50.\nIn the language of the anisotropy ﬁeld we have to\ncompare the values of the mc-anisotropy ﬁeld HK=\nβMs= 2Kcr/Mswith the values of the magnetiz-\ningmagnetodipolar ﬁeld, which attains its maximal\nvalueHmax\ndip= 2πMsfor a needle-like particle. Cor-\nresponding relation Hmax\ndip/HK=πM2\ns/Kcr= 2π/β\nis≈10.5 for Nd 2Fe14B and≈12.5 for SrFe 12O19.\nThis means that in the best case the eﬀect of\nthe shape anisotropy for both material classes can\nachieve≈20%, what would be a non-negligible im-\nprovement on a highly competing market of modern\npermanent magnet materials.\nUnfortunately, severalcircumstancesare expected\nto strongly diminish the shape anisotropy contribu-\n10tion. First, the estimate above holds for a strongly\nelongated particle; for ellipsoidal particles with a re-\nalistic aspect ratio a/b∼2−3 (ais the length of\nthe axis of revolution) the shape anisotropy ﬁeld is\nonly about half its maximal value. Second, this es-\ntimation holds for a single-domain particle, whereas\nstrongly elongated or nearly ﬂat particles acquire a\nmulti-domain state much easier than the spherical\nones, because the domain wall energy for strongly\nnon-spherical particles is much smaller, than for a\nsphere. Finally, the relation derived above is true\nonly for an isolated particle, and hard grains in\nnanocomposites are always embedded into a soft\nphase or are in a close contact with another hard\ngrains.\nFor these reasons we have performed a detailed\nnumerical study of the dependence of hysteresis\nproperties on the hard grain shape for nanocom-\nposite SrFe 12O19/Fe and - for comparison - for\nSrFe12O19/Ni . For this purpose we have simulated\nmagnetization reversal in these composites with the\nhard grains having the shape of ellipsoids of rev-\nolution (spheroids) with the aspect ratio a/b=\n0.33,0.5,1.0,2.0,3.0; aspect ratios a/b >1 corre-\nspond, as usual, to prolate spheroids. For all aspects\nratios the volume of a single hard grain was kept\nthe same (and equal to the volume of the approxi-\nmately spherical grains with D= 25 nm). Volume\nconcentration of the hard phase chard= 40% was\nthe same, as for simulations reported in the previ-\nous Sec. IIIA. The exchange weakening parameter\nκ= 0.1 was chosen close to the optimal value for\nspherical hard grains obtained above.\n1. Grain shape eﬀect for SrFe12O19/Fe\nFirst we discuss simulation results obtained for\nthe composite SrFe 12O19/Fe - see Figs. 9, 10 and\n11. In Fig.9, magnetization reversal curves for dif-\nferentaspectratios a/bareshown; both theloopsfor\nthe entire system and for the soft and hard phases\nseparately are presented. The most interesting ob-\nservation here is the pronounced diﬀerence between\nthe reversal curves of ’soft’ and ’hard’ phases for\na/b= 1 and nearly synchronous magnetization re-\nversalofboth phasesforotheraspectratiosshownin\nthe ﬁgure. This is a key feature for the understand-\ning of the system behavior and will be discussed in\ndetail below.\nOverall dependencies of basic hysteresis parame-\ntersjR,Hcand (BH)maxon the aspect ratio a/bis\npresented in Fig. 5. Both main parameters of the\nhysteresis - remanence jRand coercivity Hcexhibita/b = 0.33 a/b = 1.0\na/b = 2.0 a/b = 3.0\nH (kOe) H (kOe)Mz/Ms Mz/Ms\nFIG. 9. (color online). Simulatedhysteresis curvesof the\nnanocomposite SrFe 12O19/Fe for the exchange weaken-\ningκ= 0.1 and diﬀerentaspect ratios of hardcrystallites\nas indicated on the panels. Black loops -hysteresis of the\ntotal system, blue curves - upper part of the hysteresis\nloop for the hard phase, red curve - the same for the soft\nphase.\na highly non-trivial dependence on this aspect ratio,\nwhich should be carefully analyzed.\nThe dependence jR(a/b) shown in Fig. 10\nis clearly counter-intuitive, because normally one\nwould expect a higherremanence for a system con-\ntaining elongated particles - in our case for a/b >1\n- due to the positive shape anisotropy constant for\nsuch particles. The simulated dependence shows\nthe opposite trend - the remanence increases with\ndecreasing the aspect ration a/b, i.e.,jRbecomes\nlarger for a composite with oblatehard grains.\nThis behavior can be explained taking into ac-\ncount that hard ellipsoidal grains are mostly embed-\ndedintothesoftmagneticmatrix(softphase), which\nmagnetization is larger than that of the hard phase:\nMFe> MSrFe12O19. This means that hard grains\nrepresent magnetic ’holes’ inside a soft matrix, what\nmeans, in turn, that the total magnetodipolar ﬁeld\nacting on the magnetization of the hard grain, is\ndirected (on average) towards the initially applied\nﬁeld. With another words, this ﬁeld acts as a mag-\nnetizing ﬁeld, i.e. it increases the remanence of the\nhard phase.\nThe magnitude ofthis magnetizingﬁeld is propor-\ntional to the diﬀerence between magnetizations of\n11(a)\n(b)\n(c)\noblateprolate\nFIG. 10. (color online). (a)Simulated reduced re-\nmanence, (b)coercivity and (c)energy product of\nnanocomposites SrFe 12O19/Fe with diﬀerent aspect ra-\ntios (a/b) of hard grains. Inset in (a) shows the demag-\nnetizing factor in dependence on a/b. Dashed lines are\nguides for an eye.\nthe soft and hard phases and is of the order Hmag\ndip∼\nNdem·(MFe−MSrFe12O19) =Ndem·∆M. Foroursys-\ntem parameters ∆ M= 1300G, so that, taking into\naccount that Ndem∼π, we obtain Hmag\ndip∼4kOe.\nThis value is comparable to the mc-anisotropy ﬁeld\nof the hard grain itself ( HK(SrFe12O19) = 20kOe),\nso the eﬀect of this magnetodipolar ﬁeld can be sig-\nniﬁcant.\nTo explain the trend jR(a/b) seen in Fig. 11, it\nremains only to note that this magnetizing ﬁeld is\nlarger for oblatespheroids, for which it can achieve\nthe magnitude of 4 π∆Ms- the limiting case for a\nthin disk with the revolution axes along the mag-\nnetizing direction of the system. In contrast, for\nthe prolate spheroid Hmag\ndipbecomes weaker when\na/bincreases (spheroid becomes more prolate), be-cause the main contribution to this ﬁeld comes from\nthe soft phase regions near the ends of this prolate\nspheroid.\nThe result of this complicated interplay is the\nbetter alignment of magnetic moments of the hard\nphase consisting of oblate particles. This leads to\nthe higher remanence of the whole system for two\nreasons: ( i) the remanence of the hard phase itself\nis larger and ( ii) the ’supporting’ action of the hard\nphase on the soft phase - due to the interphase ex-\nchange coupling - is more signiﬁcant.\nThe explanation of the non-trivial dependence of\nthe coercivity on the aspect ration Hc(a/b) - with\nthe maximum between a/b= 0.5 anda/b= 1.0 -\nrequires a detailed understanding of the magnetiza-\ntion reversal mechanism in composites with partial\ninterphase exchange coupling.\nNamely, magnetizationreversalofthese nanocom-\nposites always occurs according to the following sce-\nnario: the soft phase switches ﬁrst, and then exhibit\na torque on the hard grains due to the interphase\nexchange interaction. For non-negligible interphase\nexchange this torque is the main interaction mech-\nanism between the phases and leads (together with\nthe applied ﬁeld) to the magnetization reversal of\nthe hard phase in larger negative external ﬁelds.\nIn order to understand, why the coercivity has\nits maximum for particles with a weak shape\nanisotropy, we have to recall that the interphase ex-\nchange interaction is a surface eﬀect and as such is\nproportional to the interphase surface area. In our\ncase this is the surface area of hard grains, which are\nmostly surrounded by the soft phase. This means,\nthat exchange torque which the soft phase exhibits\non the hard grains, is proportional to the surface\narea of these grains. Hence, this torque should be\nminimalforthehardgrainswiththesphericalshape,\nbecause the surface area of an ellipsoid of revolu-\ntion with the given volume is minimal for a/b= 1\n(sphere).\nFor this reason hard phase with grains having the\nshape close to spherical will have the maximal coer-\ncivity, i.e. reverse in the largest negative ﬁeld. Such\ngrains will be also able to ’support’ soft phase up to\nnegative ﬁelds larger than non-spherical hard grains\nwould do, leading to the largest coercivity of the\nwhole sample.\nTo provide further proof of this hypothesis, we\nhave plotted in Fig. 11 the coercivities of the hard\nand soft phases separately (see curves for Hhard\ncand\nHsoft\ncon the panel (a)) and the diﬀerence between\nthem∆Hconthepanel(b)asfunctionsoftheaspect\nratioa/b. The excellent qualitative agreement be-\ntween ∆Hc(a/b) and the inverse of the surface area\n12hard phasehard phase\nsoft phase(a)\n(b)\nFIG. 11. (color online) (a)Coercivities of the hard (blue\ncircles)Hhard\ncand soft (red circles) Hsoft\ncphases and (b)\ndiﬀerence ∆ Hcbetween these coercivities as functions of\nthe aspect ratio a/b; inset in (b) - inverse of the surface\narea of an ellipsoid of revolution in dependence on a/b\n. Dashed lines are guides for an eye. See text for the\ndetailed explanation.\nof an ellipsoid of revolution 1 /Sell(a/b) (see inset to\nthis panel) as the functions of a/bclearly shows that\nthe observed eﬀect is due to the surface-mediated\ninteraction, what in our case clearly means the in-\nterphase exchange interaction.\nWe ﬁnish this subsection with the explanation\nwhy the dependence of the maximal energy prod-\nuct on the aspect ratio Emax(a/b) (panel (c) in\nFig.10)foroursystemcloselyfollowsthecorrespond-\ning trend of the remanence jR(a/b) (see panel (a)),\nbut is not inﬂuenced by the dependence Hc(a/b)\n(panel (b)).\nTo understand this phenomenon, we recall that\ntheenergyproductisdeﬁnedasthemaximalvalueof\nthe product ( BH) within the second quadrant ofthe\nhysteresis loop, i.e. for external ﬁelds −Hc< H <0\n(here and below we omit for simplicity the index z\nbyH,BandM):\nEmax= max\n−Hc<H<0[B(H)·H] = max[( H+4πM(H))·H],\n(3)\nwhere it is important, that the energy product de-\npends on Hboth explicitly and implicitly - via the\ndependence M(H).Let us now assume that for some reference pa-\nrameter value (e.g. in our case for a/b= 1) with\nthe magnetization vs. ﬁeld dependence given by the\nfunction Mref(H) the product (3) reaches its maxi-\nmumE(0)\nmaxfor the ﬁeld value H0. This means that\nthe corresponding derivative of the energy product\ndE/dHvanishes at this point, leading to the condi-\ntion\nd\ndH[(H+4πMref(H))·H]|H=H0=\n=H0+2π/bracketleftBigg\nMref(H0)+H0dMref\ndH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=H0/bracketrightBigg\n= 0(4)\nIf the parameter in question changes (i.e. we take\nanother value of a/b), then the hysteresis loop also\nchanges, becoming Mnew(H) =Mref(H)+∆Mand\nthe maximum of the energy product is achieved at\nanother ﬁeld Hnew=H0+∆H. The new maximal\nenergy product then is\nEnew\nmax= (Hnew+4πMnew(Hnew))·Hnew].(5)\nAssuming that ∆ Mand ∆Hare small, we can\nexpand the functions Mref(H) and ∆M(H) in the\nvicinity of the point H0. Starting from Eq.5 and re-\ntaining only terms linear in ∆ Mand ∆H, we obtain\nforEnew\nmaxthe expression\nEnew\nmax=E(0)\nmax+4πH0∆M(H0)+\n+2∆H/braceleftBigg\nH0+2π/bracketleftBigg\nMref(H0)+H0dMref\ndH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=H0/bracketrightBigg/bracerightBigg\n(6)\nThe coeﬃcient in braces after ∆ His exactly the\nexpression (4) at the point where the reference en-\nergy product reaches its maximum and as such is\nequal to zero. Hence we are left with the expression\nof the new maximal energy product as\nEnew\nmax=E(0)\nmax+4πH0∆M(H0) (7)\nThis expression shows that the change of the en-\nergy product due to the variation of some exter-\nnal system parameter is controlled mainly by the\nchange of the magnetization curve (upper part of\ntheM−Hhysteresis loop) at the external ﬁeld\nH0where the reference energy product reached its\nmaximum. Obviously, this magnetization change\nis roughly proportional to the remanence change,\n13which explains the semiquantitative correspondence\nbetween the dependencies jR(a/b) andEmax(a/b).\nWe point out once more that this statement is true\nonlyifthe coercivityshift due tothe variationofthis\nparameter is relatively small. If this is not the case,\nthen the position of the vertical side of the rectangle\n(in the ( B−H) plane) used for the determination\nof (BH)maxcan strongly depend on the coercivity\nvalue, making the derivation of Eq. (6) invalid.\n2. Grain shape eﬀect for SrFe12O19/Ni\nThe second composite which we have used to\nstudy the grain shape eﬀect - SrFe 12O19/Ni - is\nqualitatively diﬀerent from the previous material\n(SrFe12O19/Fe) due to the much lower magnetiza-\ntion of the soft phase: Ms(Ni)≈490G. The idea\nbehind the usage of a soft phase with such a low\nmagnetization is that the coercivity of the resulting\nmaterial should be much higher due to the weaker\nresponse of the soft phase with a smaller magneti-\nzation to the external ﬁeld. This higher coercivity\nmight compensate the decrease of the net magneti-\nzation, resulting in a competitive energy product.\na/b = 0.33 a/b = 1.0\na/b = 3.0 a/b = 2.0\nFIG. 12. (color online).Simulated hysteresis curves of\nthe composite SrFe 12O19/Ni forκ= 0.1 and diﬀerent as-\npect ratios of hardcrystallites as indicated on thepanels.\nBlack loops - total hysteresis, blue curves - upper part\nof the hysteresis loop for the hard phase, red curve - for\nthe soft phase.(a)\n(b)\n(c)\nFIG. 13. (color online). (a)Remanence, (b)coer-\ncivity and (c)energy product of the nanocomposite\nSrFe12O19/Ni in dependence on the aspect ratio a/bof\nhard grains. Dashed lines are guides for an eye.\nSimulation results for SrFe 12O19/Ni with various\ngrain shapes are presented in Fig. 12 (hysteresis\nloops) and Fig. 13 (basic characteristics of the hys-\nteresis). As it can be clearly seen, the Ni-containing\ncomposite behaves itself qualitatively diﬀerent com-\npared to Fe-containing material.\nThe main new feature is - as expected - the higher\ncoercivity of both soft and hard phases. The consid-\nerably larger coercivity of the Ni phase ( Hc(Ni)≈\n1000Oe - see Fig. 12, red loops) compared to the\ncoercivity of the Fe phase in the previously stud-\nied composite ( Hc(Fe)≈600Oe - see Fig. 11a) is\nmainlyduetothelowermagnetizationofNi, asmen-\ntioned above.\nThe much larger coercivity of the hard phase (we\nremind that this phase is the same for both stud-\nied materials) for SrFe 12O19/Ni can be most prob-\nably explained by three reasons. First, due to the\n14lowerMs(Ni) the reversal of the soft phase starts\nfor the Ni-containing composite in higher negative\nﬁelds, what should by itself lead to larger Hhard\nc\nfor the hard phase also. However, this reason alone\ncould not be responsible for the more than fourfold\nincrease of Hhard\nc(Hhard\nc(SrFe12O19/Ni)≈4000Oe\nvsHhard\nc(SrFe12O19/Fe)<1000Oe).\nThesecondreasonisthatthemagnetodipolarﬁeld\nof the soft phase acting on hard grains is much\nsmaller for Ni- than for Fe-containing composites\n(due to the same much lower magnetization of Ni).\nAfter the reversal of the soft phase this magne-\ntodipolar ﬁeld is directed opposite to the initial ma-\nterial saturationand thus assiststhe reversalof hard\ngrains. Much smaller magnitude of this ﬁeld thus\nleads to much higher external ﬁeld required to re-\nverse the hard phase.\nFinally, the considerably smaller exchange con-\nstant of Ni compared to Fe (see Table 1) results in\nthe lower exchange torque acting on hard grains af-\nter the reversal of the soft phase, also decreasing the\ntotal torque acting on the hard phase and increasing\nits coercivity.\nThis qualitatively new situation - the non-\ncorrelated magnetization reversals of the soft and\nhard phases - leads to the another type of the co-\nercivity dependence on the aspect ratio a/bof hard\ngrains: coercivity Hc(a/b) decreases when this ratio\nincreases (see Fig. 13b), behaving itself in a very\nsimilar way as the remanence jR(a/b) (Fig. 13a).\nThe most likely explanation of this behavior is\nthe following: the degree of the magnetization align-\nment ofhardgrainsfortheir diﬀerentaspectratiosis\nnearly the same for various values of a/b. Hence the\nmain eﬀect on the soft phase reversal is due to the\ndiﬀerence in the magnetodipolar ﬁeld distributions\ncaused by hard grains having various shapes. Pro-\nlate ellipsoids of revolution produce a non-uniform\nmagnetodipolar ﬁeld which maximal value is higher\nthan for oblate ellipsoids. However, the ﬁeld of\nprolate particles is strongly concentrated near their\n’sharp’ ends, whereas the dipolar ﬁeld of oblate el-\nlipsoids (magnetized on average along their axes of\nrevolution) occupies a much larger region near their\n’ﬂat’ surfaces. For this reason the dipolar ﬁeld of\noblate hardgrainssupportsthe magnetizationofthe\nsoft phasein largerspaceregions,thus leadingto the\nincrease of the soft phase (and total) coercivity with\ndecreasing a/b, i.e. when hard grains become more\noblate.\nAs the result of the decrease of both jRandHc,\nthe energy product also decreases with increasing\na/b, as shown in Fig. 13c. We point out that al-\nthough the resulting behavior is qualitatively some-what similar to the case of the Fe-containing ma-\nterial (compare to Fig. 10) - the energy product\ndecreases with increasing aspect ratio - the physical\nreasonsfor this behavior are fundamentally diﬀerent\nfor these two composites.\nIt also interesting to note that despite the larger\ncoercivity of Ni-containing composite, its energy\nproduct remains smaller than for the Fe-containing\nmaterial due to its much lower net magnetization.\nThis relation can change for materials with diﬀerent\nfractions of soft and hard phases.\nIV. CONCLUSION\nIn this paper we have presented a detailed numer-\nical study for the dependence of magnetic proper-\nties of Sr-ferrite-based nanocomposites on two very\nimportant material parameters: ( i) exchange cou-\npling between various crystallites (what includes the\ncoupling between soft and hard grains) and ( ii) the\nshape of the hard grains.\nFirst, we have demonstrated - in contrast to the\ncommon paradigm - that the maximal energy prod-\nuctEmax= (BH)maxis a non-monotonous func-\ntion of the intergrain exchange coupling κand that\nthe optimal κ-value (for which the energy product\nreachesitshighestvalue)isfarbelowtheperfectcou-\npling. This non-monotonous character of the func-\ntionEmax(κ)isduetothecorrespondingdependence\nof the coercivity on the exchange coupling Hc(κ).\nSecond, we have studied the dependence of the\nhysteresis properties and the maximal energy prod-\nuct on the shape ofhard grainsfor twovery diﬀerent\nnanocomposite materials - SrFe 12O19/Fe (Ms(Fe)≈\n1700G) and SrFe 12O19/Ni (Ms(Ni)≈490G). Hard\ngrains have been assumed to have (approximately)\na shape of ellipsoids of revolution, which aspect ra-\ntio was varied in the range 1 /3≤a/b≤3 (a/b >1\ncorresponds to a prolate ellipsoid). We have shown,\nthat for both materials the aspect ratio dependence\nofthemaximalenergyproduct Emax(a/b)essentially\nfollows the corresponding dependence of the hys-\nteresis loop remanence jR(a/b) and have supported\nthis observation by analytical considerations. For\nboth materials, the maximal value of jR(a/b) - and\nhenceofEmax(a/b)-wasobtainedforthe oblatehard\ngrains with the smallest aspect ration a/b= 1/3\n(also in contrast with common expectations). Phys-\nical reasons for this behavior are revealed.\nFinally, we have also analyzed the dependence\nof the coercivity on the shape of hard grains\nHc(a/b) and have shown that this dependence for\n15the two composites under study is qualitatively dif-\nferent. For SrFe 12O19/Fe the function Hc(a/b) has\na pronounced maximum for approximately spher-\nical grains, whereas for SrFe 12O19/Ni coercivity\nmonotonously decreases with increasing a/b. This\ndiﬀerence is explained by analyzing the dominating\ninteraction mechanisms between the hard and soft\nphases in these materials.V. ACKNOWLEDGMENT\nFinancialsupport ofthe DFG projectBE2464/10-\n3 and the EU-FP7project ”NANOPYME”(310516)\n[www.nanopyme-project.eu] is greatly acknowl-\nedged. 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Parkin, Handbook of Mag-\nnetism and Advanced Magnetic Materials , Vol. 2, Mi-\ncromagnetism (Wiley, Chichester, 2007) edited by H.\nKronm¨ uller and S. Parkin.\n25A. Quesada, C. Granados-Miralles, A. L´ opez-Ortega,\nS. Erokhin, E. Lottini, J. Pedrosa, A. Bollero,\nA. M. Arag´ on, F. Rubio-Marcos, M. Stingaciu,\nG. Bertoni, C. de Juli´ an Fern´ andez, C. Sangregorio,\nJ. F. Fern´ andez, D. Berkov, and M. Christensen,\nAdvanced Electronic Materials 2, 1500365 (2016).\n26N. Usov and S. Peschany,\nJournal of Magnetism and Magnetic Materials 174, 247 (1997).\n27J. Garca-Otero, M. Porto, and J. Rivas,\nJournal of Applied Physics 87, 7376 (2000).\n28D.Berkov,Journal of Magnetism and Magnetic Materials 161, 337 (1996).\n29E. C. Stoner and E. P. Wohlfarth,\nPhilosophical Transactions of the Royal Society A: Mathema tical, Physical and Engineering Sciences 240, 599 (1948).\n30D. V. Berkov and S. V. Meshkov, Sov. Phys. JETP\n67, 2255 (1988).\n31D. V. Berkov and N. L. Gorn,\nPhysical Review B 57, 14332 (1998).\n16"    },    {        "title": "1510.05901v1.Reflecting__on__the_Nucleation_Curve__Negative_Surface_Energy_Stabilizes_Nickel_Ferrite_Nano_Particles_in_Nuclear_Reactor_Coolant.pdf",        "content": " \nReflecting (on) the  Nucleation Curve : Negative Surface Energy Stabilizes \nNickel Ferrite Nano -Particles  in Nuclear Reactor Coolant . \n \nC.J. O’Brien, Z s. Rák and D.W. Brenner  \nDepartment of Materials Science and Engineering  \nNorth Carolina State University,  Raleigh, NC 27695  \n \nClassical nucleation th eory was first developed over eight decades ago  (1-3),  and with \nrefinements and modifications since then  (4) it remains the primary tool for understanding the \nkinetics of phase transformations. In it s traditional  form, classical homogene ous nucleati on \ntheory gives the change in free energy ∆𝐺𝑇 as a phase transformation proceeds  as a sum of two \nterms. The first term is an interfacial energy  per unit area  𝛾 times the area between the two \nphase s. The second term is the difference in free  energy  of the two phases  Δ𝐺 times the volume \nof transformed material. Assuming that the transformation occurs through a spherical cluster  of \nradius r  and that  𝛾 is isotropic , the free energy change as the transformation progresses is given \nby    \nΔ𝐺𝑇=4𝜋𝑟2𝛾+ 4𝜋\n3𝑟3Δ𝐺.                                                    (1) \nA combination of positive surface energy and a negative Δ𝐺 yields a free energy barrier of   \n16𝜋𝛾3\n3(Δ𝐺)2 at a critical radius 𝑟∗=−2𝛾\nΔ𝐺 (Figure 1 (a)). In general, the magnitude of Δ𝐺 increases as \nthe temperature is lowered below that of the phase transition, which produces both a smaller \nnucleation barrier and critical cluster radius. For direction -dependent 𝛾 values , a Wulff \nconstruction  (5) can be used to determ ine appropriate volume and surface energies for a given \ncluster size .  \nBased on first principles calculations, Lodziana et al.  recently suggested that the \nexothermic  dissociative chemisorption  of water with particular surfaces of θ-alumin a can lead to \nnegative surface energies  (6, 7). Based on this result  and supporting experimental evidence, they  \nproposed that this negative surface energy  may be responsible for a thermodynamic stabilization \nof porous alumina  and may contribute  to alumina’s sintering resistance .  \nWe recently used a thermodynamics -informed first principles (TIFP) scheme  (8, 9)  to \ncalculate the temperature -dependent surface energies of nickel oxide NiO and  nickel ferrite \nNiFe 2O4, two compounds that are known to deposit  on the fuel rods in nuclear pressurized w ater \nreactors  (PWR s) (10, 11). As described in det ail elsewhere  (9), these calculations predict a \nnega tive surface free energy  for nickel ferrite  when formed from ions in solution under PWR \nconditions  of temperature  (~600K) , pressure  (155 bar) , and species concentration . Under these conditions the  thermodynamics of bulk nickel ferrite yields a positive change in free energy \nΔ𝐺 for formation of the solid from dissolved ions.  \n𝑁𝑖2++2𝐹𝑒2++4𝐻2𝑂𝑦𝑖𝑒𝑙𝑑𝑠→     𝑁𝑖𝐹𝑒2𝑂4+(6𝐻+)𝑎𝑞+(𝐻2)𝑎𝑞                       (2)  \nCombi ning this with a negative surface energy changes the sign of the nucleation relation Eq.  (1) \nso that the barrier in traditional nucleation theory becomes a well that  thermodynamic ally \nstabilizes dissolved clusters  (Figure 1 (b)). We call this a reflected nucleation curve. While a size \ndependence of phase stability, including the influence of aqueous  and humid conditions o n \nsurface energie s (12-15) is well established, the influence of negative surface energies on cluster \nstability under conditi ons where bulk thermodynamics gives di ssolution has not been previously \nrecognized .  \nIn the TIFP scheme e ffective chemical potentials  (ECPs)  𝜇0(𝑇) for the metals and \noxygen are determined by solving a system of linear equations of the form  \n∆𝑓𝐺𝐴𝑥𝐵𝑦𝑂𝑧0(𝑇,𝑃)=𝐸𝐴𝑥𝐵𝑦𝑂𝑧(0𝐾)−𝑧\n2𝜇𝑂2(𝑇)−𝑥𝜇𝐴0(𝑇)−𝑦𝜇𝐵0(𝑇),                           (3)  \nwhere ∆𝑓𝐺𝐴𝑥𝐵𝑦𝑂𝑧0 are experimental values of the Gibb’s free energy of formation and 𝐸𝐴𝑥𝐵𝑦𝑂𝑧 are \nenergies from Density Functional Theory (DFT) calculati ons at 0K.(8) The results presented here \nused ECPs that were  determined from a least squares fit to data for  NiO, ZnO, Fe 2O3, Fe 3O4, \nFeO(OH), Cr 3O4, CoFe 2O4, ZnFe 2O4 and NiFe 2O4.(8) ECPs for  water and solvated metal cations \nare determined from the expressions   \n(∆𝑓𝐺𝐻2𝑂0(𝑇,𝑃))\n𝑙=(𝜇𝐻2𝑂(𝑇,𝑃))\n𝑙−1\n2𝜇𝑂20(𝑇)−𝜇𝐻20(𝑇)                               (4) \nand  \n(Δ𝑓𝐺𝑀𝑛+0(𝑇,𝑃))𝑎𝑞=(𝜇𝑀𝑛+(𝑇,𝑃))𝑎𝑞−𝜇𝑀0(𝑇)+𝑛\n2𝜇𝐻20(𝑇)−𝑛 (𝜇𝐻+0(𝑇,𝑃))𝑎𝑞           (5) \nrespectively , and t he conventional chemical po tential form  \n𝜇𝐻+(𝑇)=𝜇𝐻+0(𝑇𝑟)+𝑅𝑇𝑙𝑛(10−𝑝𝐻)                                            (6) \nis used for the solvated proton.  This scheme avoids having to perform DFT calculations on H 2 \nand O 2 molecules, and provides a straight forward method for incorporating solvated phases into \nDFT calculations.   \nSurface energies were determined from DFT slab calculations that were carried o ut using  \nthe Vienna Ab-initio  Simulation Package  (16-18) and the generaliz ed gradient approximation \nwith the exchange -correlation functional of Perdew, Burke, and Ernzerhof   (19, 20) plus on -site \nCoulomb interactions (GGA+ U). The on-site Coulomb interactions were  implemented using the \nformulation of Dudarev, et al. (21) in which the single parameter, Ueff = U-J, describes  the \nCoulomb repulsion. Values of 4.5 eV  and 6.0 eV were used for Ueff for all Fe and Ni atoms , \nrespectively, in the oxide s (12, 22). Further details are given in references (8) and (9). Plotted in Figure 2 (a) is the energy of the nickel ferrite (111) surface as a function of \ntemperature under PWR conditions of pressure =  155 bar, pH = 7.2, and concentrations [𝑁𝑖2+]= \n1.66×10-14 and [𝐹𝑒2+]= 4.17×10-13 mol/kg  (23).  This was the lowest energy surface of the 36 \nsurfaces studied, and through a Wulff construction it is predicted to be the only surface that \nwould appear at equi librium, consistent with experiment  (24). Also plo tted in Figure 2 (b) is the \nchange in free energy from Eq.  (2) for the same conditions. Plotted in Figure 3  is the change in \nfree energy of an octahedral cluster , calculated with the data in Figure 2 , as a function of the \ncharacteristic length  at different water temperatures. As discussed above, a negative surface \nenergy and a positive change in bulk free energy from solvated ions to forming a solid cluster  \nyields reflect ed nucleation curve s that, instead of having free energy barrier s, have  free energy \nwells that stabilize formation of nickel ferrite clusters. Furthermore, the depth  of the wells  and \nsize of the clusters associated with these wells vary significantly with temperature similar  to the \nnucleation barriers and critical radii that generally become smaller the further the temperature is \nbelow the liquid -solid transition temperature.    \nThis result has potentially important implications for  measuring, understanding  and \ncontrolling the contribution of nickel ferrite to the porous metal oxides that form on the fuel rod \ncladding in PWRs . The current understanding of the deposition process is that species are \ndeposited from the coolant to the fuel cladding during subcooled nucleate boiling  by mic ro-layer \nevaporation and dryout, a proc ess by which evaporation into the vapor concentrates dissolved \nspecies  (25). To re duce concentrations of various species from the coolant  and hence mitigate \nthis process, PWRs use a Chemical and Volume Control System  (CVCS) that filters suspended \nparticulates (which  are thought t o originate primarily from corrosion  of the steam generator \ntubing ) and removes dissolved ions with a mixed bed d emineralizer  (26). The coolant \ntemperature and pressure in the CVCS is typically reduced from those in the reactor to avoid \ndamage to the demineralizing resins and the water pumps. Our results suggest filtration strategies \nshould consider not onl y relatively large suspended corrosion products, but also nanometer -scale \nnickel ferrite clusters that are predicted to be inherently stable  within the coolant.  The filtration \nshould also take into account changes in stable cluster sizes that result from reductions in the \ncoolant tempera ture in the CVCS. A t still lower temperatures, the size and stability of the \nclusters both decrease significantly (c.f. Figure 3 ) such that they may not be readily observed in \ncoolant after reactor cool down.  Therefore c oolant sampled during reactor operation but \nanalyzed at lower  temperatures may not show the  same relatively large clusters present during \nreactor operation.  Inste ad in situ  measurements may be needed to observe these solvated \nclusters .  \nThese calculations and their analysis in terms of classical nucleation theory have  \nsugg ested  a new fundamental relation – the reflected nucleation cu rve – that reveals a previously  \nunrecognized aspect  of the theory of phase stability . These calculations also suggest the presence \nof stable octahedral nickel ferrite clusters in PWR coolant that may not be observed in the \ncoolant outside of service conditions, and that should be conside red in designing strategies for \npurifying  PWR coolant  during reactor operation .  This research was supported by the Consortium for Advanced Simulation of Light Water \nReactors ( http://www.casl.gov ), an Energy  Innovation Hub ( http://www.energy.gov/hubs ) for \nModeling and Simulation of Nuclear Reactors under U.S. Department of Energy Contract No. \nDE-AC05 -00OR22725.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1.  (a) Traditional homogenious nucleation curve with a positive surface energy and \nnegative energy associated with the bulk leading to a nucleation barrier.  (b) The reflected \nnucleation curve with a negative surface energy and positivebulk term.  \n \n \n \n \n \n  \nFigu re 2.  The energy of the (111) surface of N iFe2O4 as a function of temperature, under \nconditions of pressure, pH, and concentrat ions typical of PWR coolant (solid red line).  The free \nenergy of reaction for forming NiFe 2O4, as described by Eq. (2)  (dashed bl ue line) . \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 3 . Change in free energy of an octahedral NiFe 2O4 cluster as a function of its \ncharacteristic length, a, at different temperatures.  \n \nReferences  \n1. R. Becker, W. Doring, Ann Phys -Berlin  24, 719 (Dec, 1935).  \n2. M. Volmer, Kinetik der Phasenbildung (Theodore Steinkopff, Dresden, 1939).  \n3. J. B. Zeldovich, Acta Physicochim Urs  18, 1 (1943).  \n4. V. I. Kalikmanov, Nucleation Theory . Lecture Notes in Physics (Springer, Dordrecht Heidelberg \nNew York London, 20 13), vol. 860.  \n5. G. Wulff, Z Krystallogr Minera  34, 449 (Mar, 1901).  \n6. Z. Lodziana, N. Y. Topsoe, J. K. Norskov, Nat Mater  3, 289 (May, 2004).  \n7. A. Mathur, P. Sharma, R. C. Cammarata, Nat Mater  4, 186 (Mar, 2005).  \n8. C. J. O'Brien, Z. Rak, B. D. W., Journal of Physics: Condensed Matter  25, 445008 (2013).  \n9. C. J. O'Brien, D. W. Brenner, to be submitted ,  (2013).  \n10. J. W. Yeon, I. K. Choi, K. K. Park, H. M. Kwon, K. Song, J Nucl Mater  404, 160 (Sep 15, 2010).  \n11. J. A. Sawicki, J Nucl Mater  402, 124 (Jul  31, 2010).  \n12. H. B. Guo, A. S. Barnard, Phys Rev B  83,  (Mar 8, 2011).  \n13. H. B. Guo, A. S. Barnard, J Phys Chem C  115, 23023 (Nov 24, 2011).  \n14. H. B. Guo, A. S. Barnard, J Mater Chem A  1, 27 (2013).  \n15. I. V. Chernyshova, S. Ponnurangam, P. Somasundara n, Phys Chem Chem Phys  15, 6953 (2013).  \n16. G. Kresse, J. Furthmuller, Comp Mater Sci  6, 15 (Jul, 1996).  \n17. G. Kresse, J. Furthmuller, Phys Rev B  54, 11169 (Oct 15, 1996).  \n18. G. Kresse, J. Hafner, Phys Rev B  47, 558 (Jan 1, 1993).  \n19. J. P. Perdew, K. Burke, M. Ernzerhof, Phys Rev Lett  77, 3865 (Oct 28, 1996).  \n20. J. P. Perdew, K. Burke, M. Ernzerhof, Phys Rev Lett  78, 1396 (Feb 17, 1997).  \n21. S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, A. P. Sutton, Phys Rev B  57, 150 5 \n(Jan 15, 1998).  \n22. A. Jain  et al. , Phys Rev B  84,  (Jul 12, 2011).  \n23. J. Henshaw  et al. , J Nucl Mater  353, 1 (Jul 1, 2006).  \n24. Y. Cheng, Y. H. Zheng, Y. S. Wang, F. Bao, Y. Qin, J Solid State Chem  178, 2394 (Jul, 2005).  \n25. H. Bindra, B. G. Jones, Colloid Surface A  397, 85 (Mar 5, 2012).  \n26. R. Prince, Radiation Protection at Light Water Reactors .  (Springer -Verlag, Berlin -Heidelberg, \n2012).  \n \n "    },    {        "title": "0901.3748v2.Structural_Origin_of_the_Metal_Insulator_Transition_of_Multiferroic_BiFeO3.pdf",        "content": "STRUCTURAL ORIGIN OF THE METAL-INSULATOR TRANSITION  OF \nMULTIFERROIC BiFeO 3 \n \nS. A. T. Redfern 1, J. N. Walsh 1, S. M. Clark 2, G. Catalan 1, and J. F. Scott 1 \n \n1Department of Earth Sciences, University of Cambrid ge, Downing Street, Cambridge \nCB2 3EQ, UK \n \n2Advanced Light Source, Lawrence Berkeley National L aboratory, 1 Cyclotron Road, \nBerkeley, CA 94720-8226, USA \n \nAbstract \n \nWe report X-ray structural studies of the metal-ins ulator phase transition in bismuth \nferrite, BiFeO 3, both as a function of temperature and of pressure  (931 oC at \natmospheric pressure and ca. 45 GPa at ambient temp erature).  Based on the \nexperimental results, we argue that the metallic γ-phase is not rhombohedral but is \ninstead the same cubic Pm3m structure whether obtai ned via high temperature or high \npressure, that the MI transition is second order or  very nearly so, that this is a band-type \ntransition due to semi-metal band overlap in the cu bic phase and not a Mott transition, \nand that it is primarily structural and not an S=5/ 2 to S=1/2 high-spin/low-spin \nelectronic transition.  Our data are compatible wit h the orthorhombic Pbnm structure for \nthe β-phase determined definitively by the neutron scatt ering study of Arnold et \nal .[Phys. Rev. Lett. 2009]; the details of this β-phase  had also been controversial, with \na remarkable collection of five crystal classes  (c ubic, tetragonal, orthorhombic, \nmonoclinic, and rhombohedral!) all claimed in recen t publications.  \nBismuth ferrite BiFeO 3 (BFO) has become the cornerstone of research in \nmagnetoelectric multiferroics [1, 2] due to it bein g a rare (though perhaps not unique [4, \n3]) example of a simple perovskite oxide with stron g ferroelectric polarization [5, 6] and \nmagnetic ordering [7, 8, 9] at room temperature, wi th interesting coupling between the \ntwo ferroic orderings [10, 11]. Yet in spite of all  the attention that it has received, \nseveral basic aspects still remain unresolved. One of these is the phase diagram, which \nis turning out to be rather complex, with several n ew phase transitions being reported \njust in the last year [12-20]. Bismuth ferrite is s imultaneously ferroelectric, \nantiferromagnetic and ferroelastic, that is, there are at least three ferroic order \nparameters involved in its phase transitions, and o ther crystallographic distortions, such \nas rotations of the oxygen octahedra, also play an important role in the functional \nproperties. Given the number of order parameters in volved, and the subtle coupling \nbetween them, it is perhaps not unreasonable that t he phase diagram should be so rich.  \n \nHere we would like to focus our attention on the hi gh-pressure and high-temperature \nends of the phase diagram, where a metal-insulator (MI) transition is known to occur \n[12, 13, 21, 22]. Optical and transport studies hav e previously shown that BFO becomes \nmetallic at 930 oC and ambient pressure [12], or at ca. 45 GPa and r oom temperature [13, \n22, 23]; however, the nature of this MI transition and the associated structural changes \nis still not clear, and conflicting models have bee n proposed.  Specifically, it is not at \nthis point clear whether the MI transition is of ba nd-type [12] or Mott type [13]. A \nband-type insulator has an even number of electrons  in the unit cell, and these fill \ncompletely the valence band, so that there is a gap  between them and the excited states \nin the conduction band. For a valence band insulato r to become a metal, there has to be \na structural transition whereby the number of formu la units (and therefore the number of \nelectrons) per unit cell changes. In a Mott-type in sulator, by contrast, the gap is due to \nelectrostatic repulsion between the conduction elec trons. A transition from Mott-\ninsulator to metal is thus one in which the size of  the electrostatic repulsion (the Mott-\nHubbard parameter U) becomes smaller than the width  of the conduction band [24, 25]. \nMott transitions are purely electronic and do not i n theory require a change in either \ncrystal structure or magnetic symmetry, although in  practice the coupling between \ncharge, spin and lattice means that other transitio ns tend to happen simultaneously [26]. \nThe challenge, often, is to find which comes first:  does the structural transition drive the \nelectronic one (band MI transitions) or is it the o ther way round (Mott MI transitions). \n \nBased on diffraction experiments, it is argued here  that the nature of the MI transition in \nBFO is the same irrespective of whether it is achie ved with temperature or with pressure, \nand is primarily due to a structural change from or thorhombic to cubic symmetry. The \nkey structural parameter is identified as the rotat ion of the oxygen octahedra, which \ndisappears in the cubic phase thereby enhancing the  orbital overlap between oxygen and \niron ions. \n \nPowdered BiFeO 3 was studied between room temperature and 1000 oC using a Bruker \nD8 Advance X-ray diffractometer in θ−θ  geometry. Diffraction patterns were collected \nbetween 20 and 90 oθ CuK α radiation using a rapid Vantec position sensitive detector. \nThis allowed patterns to be obtained on a time scal e of less than 10 minutes, with rapid \nscanning rates. Such rapid data collection is absol utely essential at the highest \ntemperatures of the experiment since BiFeO 3 is not chemically stable in air or vacuum \nat such temperatures, and upon entering the cubic p hase breaks down as a function of time.  However, with this precaution of rapid data collection, the samples were cycled 5 \ntimes each to >931 oC without decomposition.  We note that phase transi tions to higher \nsymmetry structures often trigger dissociation due to the higher entropy (e.g., the \nbreakdown of calcite on entering its high-T phase– [27] or the melting of ferroelectric \nLiNbO 3 on entering the paraelectric phase [28]). \n \nFigure 1 illustrates the resolution of the three mo st intense diffraction lines in the 2 θ -\nplot of XRD results, including the strongest (110) c line (subscript “c” here refers to \nindexing based on the primitive cubic unit cell).  We display XRD data as a function of \nd, since different X-ray excitation wavelengths were  used in the high-T and high-P \nexperiments.  Above ca. 1200K, there is a single pe ak, compared with a large splitting \nin the rhombohedral phase at ambient temperatures ( a small asymmetry arises from \nα1/α 2 source wavelengths). This highest temperature phas e ( γ phase) therefore appears \nto be cubic. Below 1200K there is a transition to a  β phase. The lattice constant data \npublished earlier [12] indicated that the β-phase is orthorhombic and not tetragonal (nor \npseudo-tetragonal), and that the β-γ (orthorhombic-cubic) phase transition is continuou s \nor very nearly so. Very recent neutron powder diffr action studies at high temperature \nhave confirmed the orthorhombic structure of the β-phase below the MI transition [15].  \n \nThe high pressure data were collected on BiFeO 3 loaded into a diamond anvil cell and \npressurised to around 50 GPa, pressures measured by  the fluorescence of ruby chips in \nclose proximity to the sample, using the high-press ure beam line 12.2.2 of the \nAdvanced Light Source, Lawrence Berkeley Laboratory . Data were collected onto Marr \nimage plates and converted into one dimensional dif fraction patterns using standard \nmethods, by Fit2D. Fig. 2 illustrates the most inte nse [110] XRD line as a function of \npressure (to 47 GPa). The same cubic/non-cubic tran sition seems evident. We note \nparenthetically that we clearly see the low-pressur e phase transitions near 5-10 kbar \nrecently reported by Haumont et al . [17], both as a change in the diffraction pattern  and \nas a slight contraction of the unit-cell volume nea r 7.5 GPa (see Figure 3), which is \nsimilar to, though smaller in magnitude than, the c ompression reported for the α-β  \n(rhombohedral-orthorhombic) transition as a functio n of temperature [12]. The \nevolution of the unit cell volume as a function of pressure is compared in figure 3 with \npreviously reported results from Gavriliuk et al, [ 13, 23]. Both sets of data agree \nquantitatively rather well. Importantly also, when plotted together, the two appear \nconsistent with a continuous evolution and no sharp  changes in unit cell volume, \nsuggesting a second order phase transition.  \n \nIn order to consider whether the cubic phase we inf er at high pressure is the same as that \nwhich we observe at high temperature, we show in Fi g.4 a wider range of XRD data. \nThe high-pressure data suffer from broadening due t o lower hydrostaticity at extreme \npressure, plus obviously the lattice parameter is c onsiderably smaller, but the patterns at \nboth high-pressure and high-temperature are identic al in relative peak positions and \nheights, and are both compatible with a primitive c ubic structure in both cases. We \nemphasize also the disappearance of the superlattic e peaks characteristic of the \northorhombic β phase. The disappearance of all superlattice peaks  indicates that the \ndiffraction pattern corresponds to a simple perovsk ite unit cell (i.e., there is no unit cell \ndoubling), indicating clearly that the high pressur e phase cannot be orthorhombic Pbnm, \nnor rhombohedral R3c, as both of these have unit ce lls that are multiples of the \nprimitive perovskite. This is also important regard ing the MI transition: simple cubic \nBFO has only one formula unit per unit cell, with a n odd number of valence electrons (there are 5 electrons in the d-shell of Fe +3 ), whereas orthorhombic and rhombohedral \nBiFeO 3 possess 2 formula units per unit cell, with an eve n number of electrons; \naccordingly, on cooling from cubic to orthorhombic BFO can become a band insulator. \n \nThe results therefore indicate the same sequence of  phase transitions as a function of \nincreasing temperature or increasing pressure. The first structural phase transition, α-β \n(rhombohedral to orthorhombic) is first order, as i ndicated by sharp volume contraction. \nIn the vicinity of this first order phase transitio n there can be phase coexistence of the α \nand β phases [13], which may have contributed to the pas t discrepancies about the \nnature of this β phase [14, 16, 29, 30]. The variety of crystal cla sses wrongly attributed \nto this phase is itself also remarkable, as the lis t includes cubic [29], tetragonal [30], \nmonoclinic [16] and rhombohedral [14]. While this m ay seem surprising, it is less so \nwhen put in the context of i) the difficulty in the  interpretation of the patterns due to \nphase coexistence, ii) the very high temperatures a t which the measurements are done, \nwhich contribute to sample decomposition and peak b roadening and iii) the low \nsensitivity of x-ray diffraction to oxygen position s, which are key.  \n \nContrary to the α-β transition, the β-γ phase transition at high temperature or high \npressure appears to be essentially continuous (seco nd order), which is significant \nbecause second order MI transitions cannot be Mott- type [24]. As a function of \nincreasing temperature or pressure, a decrease in o ptical bandgap and resistivity has \nbeen reported [12, 12]. Upon entering the orthorhom bic β-phase, the resistivity \ndecreases further, but BFO remains still semiconduc ting [12, 14]. When the cubic phase \nis finally reached, BFO becomes metallic [12]. The evidence thus suggests that the \nbandgap is directly linked to the crystallographic distortion, and that the structural \nchange may be sufficient to drive the metal-insulat or transition, rather than the other \nway round.  \n \nIn earlier high pressure studies, however, a differ ent scenario was proposed. It was \nnoted that the MI transition coincides with a chang e in the magnetic configuration of the \nFe 3+  ions from high spin to low spin [13], a finding al so supported by first principles \ncalculations [18]. Gavriliuk et al.  hence proposed [13] that strong electron-electron \nrepulsion (the Mott-Hubbard parameter U) in the hig h-spin phase could be responsible \nfor the opening of the bandgap that causes BFO to b e an insulator. In the low-spin phase \nthis electrostatic repulsion would be smaller, enab ling the bandgap to decrease, leading \nto a metallic state. Gavriliuk et al.  also mention the presence of Mott’s variable range  \nhopping [32] in the semiconducting phase as consist ent with this, although variable \nrange hopping is not itself a proof of a Mott-type phase transition, nor is it likely to exist \nat room temperature [33]: it is  unphysical to cons ider tunnelling over a length scale that \nis larger than the inelastic mean free path [34], w hich is of the order of a unit cell at \nroom temperature [35].  \n \nSeveral key aspects of Gavriliuk’s model --the exis tence of a high-spin to low-spin \ntransition at high pressure, the weakness of the ma gnetic interactions in the low-spin \nphase leading to paramagnetism at room temperature,  and the existence of a sizeable \ndensity of states at the fermi level (i.e., metalli city) in a paramagnetic low spin phase-- \nare also supported by ab-initio calculations [18]. On the other hand, the first principle \ncalculations [12, 18] also show the valence bands t o be too broad and strongly \nhybridized to be compatible with a Mott-Hubbard ori gin of the gap. There are also other \npoints that weight in favour of a band-type and aga inst a Mott-type phase transition: for  example, band-structure calculations show that cubi c BFO is a semimetal [12], which \nimplies that a structural phase transition to a cub ic phase is by itself enough to cause a \nband-type insulator-metal transition. Also Mott’s r equirement that the MI transition be \nfirst order is at odds with the observed continuous  evolution of the lattice parameter/unit \ncell volume. The Mott-type MI transition is in fact  defined as an isostructural transition \nfrom paramagnetic metal to paramagnetic insulator t ransition [24-26]. This is not the \ncase in BiFeO 3, where both a structural transition (from orthorho mbic to cubic \nsymmetry), and a magnetic phase transition (from an tiferromagnet to paramagnet [21]) \ntake place simultaneously at high pressure. Finally , The MI transition at high \ntemperature is not due to a change in spin ordering  (BFO is paramagnetic in both \nphases), so the high-spin to low-spin model of the transition does not apply to it; \nconversely, since the structural changes are the sa me as those as a function of pressure, \nthe MI transition mechanism is likely to also be th e same.  \n \nWe therefore believe that both the temperature-driv en and the pressure-driven MI \ntransitions are controlled by the crystallographic change and not by the electron-electron \nrepulsion. The key structural parameter is likely t o be the straightening of the Fe-O-Fe \nbond angle, which leads to increased orbital overla p between the oxygen p orbital and \nthe iron d orbital thereby facilitating charge transfer. This  angle is known to play a key \nrole in the functional properties of transition met al perovskite oxides [37], and its effect \non BFO is backed by experimetal results: as shown i n Figure 5, there is a clear inverse \ncorrelation between the evolution of the Fe-O-Fe bo nd angle and the bandgap. Using a \ncrude linear extrapolation from the rhombohedral ph ase, we estimate that a critical \nangle of ca. ~159 o degrees may be enough to eliminate the bandgap alt ogether and \ntrigger a metallic state at high temperature, even in the absence of a structural phase \ntransition. This critical angle is in fact amply su rpassed in the cubic phase, where it is \n180 o. The correlation between octahedral rotation and b andgap in BiFeO 3 is also \nsupported by recent studies of the ferroelectric do main walls, whose local structure is \ncharacterized by decrease in octahedral rotation le ading to decreased bandgap and \nincreased conductivity [38]. Finally, it is also wo rth mentioning that cubic SrFeO 3 is \nalso a metal and, although the Fe +4  oxidation state in this compound precludes direct \ncomparison with BFO, we think that the structural c orrelation is not a coincidence. \n \nIn summary, the diffraction results show the same o rthorhombic-cubic phase transition \nat high temperatures (930 oC) or pressure (~45GPa), and support the view that the \nmetal-insulator transition in BFO is a conventional  band-type transition triggered by the \nsymmetry change, with the key structural parameter being the Fe-O-Fe bond angle.  \n \nAcknowledgements.  We thank John Robertson, Jorge Iñiguez and Peter L ittlewood for \nuseful discussions regarding this manuscript. The f inancial support of the EC-STREP \nprogram MULTICERAL is also gratefully acknowledged.  \n     \n  \nReferences \n \n1.  M. Fiebig, J. Phys. D 38 , R123-R152 (2005). \n2.  W. Eerenstein, N. D. Mathur, and J. F. Scott, Natur e (London) 442 , 759-765 (2006) .  \n3.  A. Kumar, I. Rivera, R. S. Katiyar and J. F. Scott,  Appl. Phys. Lett.  92, 132913 \n(2008) \n4.  A. Kumar, G. L. Sharma, R. S. Katiyar, J. F. Scott,  arXiv:0812.3875 (2008). \n5.  J. Wang. et al., Science 299 , 1719-1722 (2003).  \n6.  D. Lebeugle, D. Colson, A. Forget, M. Viret, P. Bon ville, J. F. Marucco, S. Fusil, \n Appl. Phys. Lett. 2007, 91, 022907.  \n7.  G. A. Smolenskii and V. M. Yudin  Sov. Phys.-Solid St. 6, 2936 (1965). \n8.  P. Fischer, M.Polomska, I. Sosnowska and M. Szymans ki, J. Phys. C: Solid St. \nPhys., 13 (1980) 1931-40. \n9.  I. Sosnowska, T. Peterlin-Neumaier and E Steichele,  J. 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A., \nKunstscher C. A., and Kreisel J., arXiv:0811.0047. (2008). \n18.  O.E. Gonzalez-Vazquez, J. Iñiguez, arXiv:0810.0856v 1 (2008).  \n19.  M. K. Singh, R. Katiyar, and J. F. Scott, J. Phys. Condens. Matter 20, 252203 \n(2008). \n20.  M. Cazayous, Y. Gallais, and A. Sacuto, R. de Sousa , D. Lebeugle and D. Colson, \nPhys. Rev Lett. 101 , 037601 (2008).  \n21.  Gavriliuk A. G., Struzhkin V. V., Lyubutin I. S., H u M. Y., and Mao H. K., JETP \nLett. 82 , 224-227 (2005). \n22.  Gavriliuk A. G., Lyubutin I. S., and Struzhkin V. V ., JETP Lett. 86 , 532-536 (2007).  \n23.  A. G. Gavriliuk, V.V. Struzhkin, I. S. Lyubutin, I.  A. Troyan, JETP Lett. 86, 197 \n(2007). \n24.  Mott N. F., Rev. Mod. Phys. 40 , 677-683 (1968). \n25.  N. F. Mott, Metal Insulator Transitions (Taylor & F rancis,London, 1974). \n26.  R. G. Moore, Jiandi Zhang, V. B. Nascimento, R. Jin , Jiandong Guo, G.T. Wang, Z. \nFang, D. Mandrus, E. W. Plummer, Science 318, 615 ( 2007).  \n27.  Redfern S. A. T., Salje E. K.-H., and Navrotsky A.,  Contrib. Mineral Petrol. 101 , \n479-484 (1989). \n28.  G. A. Smolenskii, N. N. Krainik, N. P. Khuchua, V. V. Zhdanova, I. E. Mylnikova, \nPhys. Stat. Sol. 13 , 309 (1966). \n29.  Haumont R., Kreisel J., Bouvier P., and Hippert F.,   Phys. Rev. B 73, 132101 (2006). 30.  Kornev I. A., Lisenkov S., Haumont R., Dkhil B.,  a nd Bellaiche L., Phys. Rev. Lett. \n99, 22706 (2007). \n31.  Mott N. F., Phil. Mag. 19 , 835-852 (1969).  \n32.  Mott N. F. and Davis E., Electronic Processes in No n-Crystalline Materials \n(Clarendon Press, Oxford, 1979).  \n33.  Shklovskii B. I. and Efros A. L., Electronic Proper ties of Doped Semiconductors \n(Springer, Berlin, 1984), p.212.  \n34.  Littlewood P. B. (private communication).  \n35.  H. Frohlich and N. F. Mott, Proc. Roy. Soc. (London ) A171 , 496-504 (1939). \n36.  J. B. MacChesney, R. C. Sherwood, and J. F. Potter,  J. Chem. Phys. 43 , 1907 (1965). \n37.  J. B. Goodenough, Rep. Prog. Phys. 1915, 67  (2004).  \n38.  J. Seidel, L.W. Martin, et al ,  Nature Materials, doi:10.1038/nmat2373 (2009). \n39.  A. Palewicz, R. Przeniosło, I.Sosnowska and A. W. H ewat, Acta Crys. B 63 , 537 \n(2008).  \nFigures: \n \n \n \nFigure 1: X-ray diffraction patterns of the (110) c peak of BiFeO 3 as a function of \ntemperature through the α−β−γ  phase transitions. Splitting in the rhombohedral ( α) \nand orthorhombic( β) phases disappears in the cubic γ-phase, where the slight peak \nasymmetry arises from the CuK α1/α2 splitting of the radiation used. \n  \n \n \nFigure 2: Resolved X-ray diffraction doublets of (110) c in BiFeO 3 as a function of \npressure; above 47 GPa these become singlets. Altho ugh broadened, the cubic (110) \nof the γ-phase shows no sign of asymmetry. \n  \n \n \n \n \n \n \n \n \nFigure 3 : Unit cell volume as a function of pressure, both from the present study and \nfrom an earlier report by Gavriliuk et al [23]. The  data at high pressures is consistent \nwith a continuous evolution of the unit cell volume , indicative of a second order phase \ntransition.  \n \n  (a) \n(b) \n  \n \nFigure 4: [a] Comparison of X-ray patterns of BiFeO 3 at high temperatures (below) \nand high pressures (above), showing the same cubic structure in the γ-phase with no \nindication of lower-symmetry super-lattice reflecti ons. [b] Pressure evolution of the \ndiffraction patterns of BiFeO 3 reveal the presence of the orthorhombic β-phase at \nintermediate pressures, indicated by the superlatti ce peak around 3.4 Å, which \nvanishes in the highest-pressure patterns of cubic γ-BiFeO 3.   \n \n300 400 500 600 700 800 900 1000 155.5 156.0 156.5 157.0 \nT (K) Fe-O-Fe angle (degrees) \n1.2 1.4 1.6 1.8 2.0 2.2 2.4 Optical bandgap (eV) \n \nFigure 5 : Fe-O-Fe bond angles (extracted from ref [39]) and  optical bandgap (extracted \nfrom ref.[12] ) of BFO as a function of temperature . As the Fe-O-Fe bond becomes \nstraighter, the bandgap decreases, consistent with increased orbital overlap between Fe +3  \nand O 2-.  \n "    },    {        "title": "1909.02844v1.Unveiling_multiferroic_proximity_effect_in_graphene.pdf",        "content": "Unveiling multiferroic proximity eﬀect in\ngraphene\nFatima Ibrahim,\u0003,yAli Hallal,yDaniel Solis Lerma,yXavier Waintal,zEvgeny Y.\nTsymbal,{and Mairbek Chshiev\u0003,x\nyUniv. Grenoble Alpes, CEA, CNRS, SPINTEC, Grenoble, France\nzUniv. Grenoble Alpes, CEA, IRIG-PHELIQS, Grenoble, France\n{Department of Physics and Astronomy and Nebraska Center for Materials and\nNanoscience, University of Nebraska, Lincoln, NE, USA\nxUniv. Grenoble Alpes, CEA, CNRS, SPINTEC, 38000 Grenoble, France\nE-mail: fatima.ibrahim@cea.fr; mair.chshiev@cea.fr\nAbstract\nWe demonstrate that electronic and magnetic properties of graphene can be tuned\nvia proximity of multiferroic substrate. Our ﬁrst-principles calculations performed both\nwith and without spin-orbit coupling clearly show that by contacting graphene with bis-\nmuth ferrite BiFeO 3(BFO) ﬁlm, the spin-dependent electronic structure of graphene\nis strongly impacted both by the magnetic order and by electric polarization in the un-\nderlying BFO. Based on extracted Hamiltonian parameters obtained from the graphene\nband structure, we propose a concept of six-resistance device based on exploring mul-\ntiferroic proximity eﬀect giving rise to signiﬁcant proximity electro- (PER), magneto-\n(PMR), and multiferroic (PMER) resistance eﬀects. This ﬁnding paves a way towards\nmultiferroic control of magnetic properties in two dimensional materials.\n1arXiv:1909.02844v1  [cond-mat.mtrl-sci]  6 Sep 2019Spintronic devices possessing high speed and low-power consumption have opened new\nprospects for information technologies. As the spin generation, manipulation, and detection\nis the operating keystone of a spintronic device, realizing those three components simulta-\nneously stands as a major challenge limiting applications.1–4In this context, developing a\nsuitable spin transport channel which retains both long spin lifetime and diﬀusion length\nis highly desirable. Graphene stands as a potential spin channel material owing to its ex-\nceptional physical properties. Beside its high electron mobility and tunable-charge carrier\nconcentration, graphene has demonstrated room temperature spin transport with long spin-\ndiﬀusion lengths .5–15Accordingly, graphene spintronics became a promising direction of\ninnovation that attracted a growing attention in the scientiﬁc community .16,17\nMuch eﬀorts have been devoted to induce magnetism in graphene via diﬀerent means\n,18–33one of which is the exchange-proximity interaction with magnetic insulators .34–36\nTheoretically, this eﬀect was demonstrated using diﬀerent materials such as ferromagnetic\n,37,38antiferromagnetic ,39topological ,40and multiferroic41insulators where exchange-\nsplitting band gaps reaching up to 300 meV were demonstrated. Recently, a detailed study\nhas shown the inﬂuence of diﬀerent magnetic insulators on the magnetic proximity eﬀect\ninducedingraphene.42Ontheotherhand, experimentsonYIG/Gr,34,35,43,44EuS/Gr,45and\nBFO/Gr46,47demonstrated proximity induced eﬀect in graphene with substantial exchange\nﬁeld reaching 14T. However, combining both conditions of a high Curie temperature ( Tc)\nmagnetic insulator and a weak graphene doping stands as a major challenge which limits\npractical spintronic applications.\nMultiferroics, co-exhibiting a magnetic and ferroelectric order, constitute an interesting\nclass of magnetic insulators that bring about an additional parameter in play which is the\nelectric polarization. On one hand, proximity induced magnetism was reported in graphene\nusingmultiferroicmagneticinsulator39,41,48ignoringtheinﬂuenceofelectricpolarization. On\nthe other hand, the ferroelectrically-driven manipulation of the carrier density in graphene\nwas demonstrated .49However, the ferroelectric control of magnetic proximity eﬀect has not\n2-0,4-0,20,00,20,40,60,8\n Fe\n Bi\nGrP.head  Fe(Bi) displacement Å\nAtomic layerGrP.tail\nGrP.head\nGrP.tailP BiFe\nO(a) (b)Figure 1: (Color online) (a) The Gr P:head/BFO/Gr P:tailsupercell is shown in the lower panel.\nMagenta (Gold) balls designate Bi (Fe) atoms respectively and small red balls represent\nO atoms. A top view of the Gr/BFO interface is shown in the upper panel where one Fe\natom occupies a hollow site and the other two occupy top sites. (b) The layer-by-layer\nFe(Bi) displacement from their centrosymmetric positions shown by square (circle) symbols.\nThe blue (red) dashed lines correspond to the bulk values of the displacements for Fe(Bi).\nThe direction of the electric polarization originating from these atomic displacements is\nperpendicular to the interface, along the c-axis, and shown by an arrow.\nbeen addressed so far. In this letter, we report the multiferroic-induced proximity eﬀect\n(MFPE) in graphene proposing the concept of controlling electronic and magnetic proper-\nties of graphene via multiﬀeroic substrate. For this purpose, we considered bismuth ferrite\nBiFeO 3(BFO) whose room-temperature multiferroicity promotes it as a good candidate for\napplications .50–54Our ﬁrst-principles calculations demonstrate that by contacting graphene\nwith BFO, the spin-dependent electronic structure of graphene is highly inﬂuenced not only\nby the magnetic order but also by the ferroelectric polarization in the underlying BFO. These\nﬁndings propose additional degrees of control for proximity induced phenomena in graphene\nand perhaps in other two-dimensional materials.\nOurﬁrst-principlescalculationsarebasedontheprojector-augmentedwave(PAW)method\n55as implemented in the VASP package56–58using the generalized gradient approximation\nas parametrized by Perdew,Burke, and Ernzerhof.59,60A kinetic energy cutoﬀ of 550 eV has\nbeen used for the plane-wave basis set and a 9\u00029\u00021k-point mesh to sample the ﬁrst Bril-\n3louin zone. The supercell comprises of nine (Bi-O 3-Fe) trilayers of BFO ( 111) surface with\nFe termination sandwiched between two 4\u00024graphene layers as shown in Figure 1 (a). We\nﬁxed the in plane lattice parameter to that of BFO where the lattice mismatch in this super-\ncell conﬁguration is about 1:5%. This heterostructure provides the opportunity to compare\nsimultaneously the properties of two diﬀerent graphene layers relatively sensing opposite di-\nrections of the BFO polarization P. Since maintaining the polarization is a critical issue in\nferroelectric slabs, a thick BFO slab is used both to restore the electric polarization within\nthe bulk layers and to assure that the two graphene layers do not interact. At both Gr/BFO\ninterfaces, one Fe atom is placed at a hollow site whereas the other two atoms occupy top\nsites as shown in the top view of Figure 1 (a). Then, the atoms were allowed to relax in all\ndirections until the forces became lower than 1meV/Å. As the GGA fails to describe the\nelectronic structure of strongly correlated oxides, we have employed the GGA+U method to\nthe Fe- 3d orbitals .61We have optimized the value of Uusing the bulk unit cell of BFO and\nfound that Ueff= 4eV yields 2:44eV band gap and \u00064:15\u0016B/Fe magnetic moments which\nare in good agreement with experimental values .62–64\nBiFeO 3has a perovskite type cryctal structure and belongs to the polar space group R3c.\nThespontaneouspolarization PalongBFO(111)directionoriginatesfromthedisplacements\nof the Bi and Fe atoms from their centrosymmetric positions along the (111) direction .50,52,63\nTo examine Pof BFO after the formation of the Gr/BFO/Gr interfaces which accounts for\nboth the ionic and charge relaxation, we show in Figure 1(b) the Fe and Bi z-displacements\nfrom their centrosymmetric positions per atomic layer. It can be clearly seen that the two\nBFO/Gr interfaces have diﬀerent values of atomic displacements whereas in the bulk layers\nthe values are almost constant in good relevance to the bulk values (shown by dashed lines).\nThis infers that P, which arises from such non-centrosymmetric structure, is sustained in\nBFO and it is perpendicular to the interface and pointing from lower graphene layer, lying\nat the tail of Pand denoted hereafter by Gr P:tail, towards the upper one lying at the head of\nPdenoted by Gr P:head. A rough estimate of the z-averaged polarization can be deduced from\n40 4 8 12 16 20 24 28 32 36 40-20-15-10-505 Planar average\n Macroscopic average\n  Planar averaged electrostatic potential (eV)\nz (Å)Figure 2: (Color online) The calculated planar averaged electrostatic potential (dashed black\nline) and its macroscopic average (solid red line) across the Gr/BFO/Gr supercell. The\ninset shows the induced spatial charge distribution upon the formation of the two Gr/BFO\ninterfaces. The red (green) regions indicate charge accumulation (depletion), respectively.\nthe plot is obtained using an isovalue = 0:002e/Å.\nthe values of the local polarization based on Born eﬀective charges: P(z) =e\n\nPN\nm=1Z\u0003\nm\u000ezm\n; where N is the number of atoms, \u000ezmis the displacement of the mth atom from the\ncentrosymmetric position, \nis the volume of the unit cell, and Z\u0003\nmis the Born eﬀective\ncharge of the mth ion. In our supercell a value of P= 63\u0016C/cm2is estimated which\nreasonably compares to the calculated value in a bulk BFO unit cell P= 100\u0016C/cm2.50\nWe discuss now the formation of Gr/BFO/Gr interface. The BFO(111) slab is Fe3+\nterminated on both sides which makes the two surfaces polar with a nonzero net charge.\nFrom a macroscopic electrostatic point of view, this is equivalent to a slab having a polar\nsurface charge \u001bs= +1:5e=A= 88\u0016C=cm2on both surfaces and no charges inside the slab,\nwhereAis the surface area per Fe atom. On the other hand, assuming a uniform polarization\nPin the BFO slab whose direction is shown in Figure 1(a) yields surface polarization charges\n\u001bP= +Pand\u001bP=\u0000Pon theheadandtailsurfaces, respectively. Therefore, the whole\nBFO slab is equivalent to a slab with total bound charge \u001bhead=\u001bs+P= 151\u0016C=cm2on\n5 \n \nK\n-0,90-0,85-0,80-0,75\n  \nK\n-0,60-0,55-0,50-0,45-0,40\n  \nGrP .tailWithout SOC With SOC\n(a) (b) (d)\n(e)\nGrP .head(c)Figure 3: (Color online) (a) Calculated band structure for Gr/BFO/Gr heterostructure\nwithout including spin-orbit interactions. Spin up (spin down) bands are shown in blue\ndiamond (red cross) lines, respectively. (b), (c) are zoom around K point shown by the\nshaded areas in (a) corresponding to the Dirac cones for Gr P:tailand GrP:head, respectively.\n(d), (e) are the band structure calculated by including spin-orbit coupling shown for the same\nshaded region as in (b,c) for comparison. The dotted symbols and solid lines in (b, c, d, and\ne) correspond to the DFT calculated and tight-binding ﬁtted band structures, respectively.\ntheheadsurface and \u001btail=\u001bs\u0000P= 25\u0016C=cm2on thetailsurface. This dissimilarity in\nthe BFO surface charges leads to the formation of two signiﬁcantly diﬀerent interfaces with\ngraphene giving rise to two adsorption distances \u0001z(GrP:head\u0000BFO ) = 2:35Å compared\nto\u0001z(GrP:tail\u0000BFO ) = 2:7Å. In fact, graphene sheets adsorbed on both sides of the slab\naccumulate negative charges trying, ideally, to screen the positive bound charges on the\nBFO surfaces. This produces a strong electrostatic interaction between graphene and the\nBFO surfaces in particular at the headinterface where the bound charges are quantitatively\nlarger as shown in Figure 2. Consequently, (i) the GP:headrelaxes closer to the BFO surface\ncompared to GP:tailand (ii) strong relaxations are induced at the headBFO surface revelaed\nby the smaller polar displacements at the outermost layers, as shown in Figure 1(b), thus,\nreducing the eﬀective polarization at this surface.\nTo get more insights on the interaction at the Gr/BFO/Gr interfaces, the inset of Figure\n2 shows the induced charge distribution upon the formation of the interfaces. Negative\ncharges, represented by red regions, are accumulated at both Gr/BFO interfaces in accord\n6with the description we provided in the previous paragraph. However, the charges at the\nGrP:headare obviously larger than at the Gr P:tail. This is a direct implication of the stronger\nelectrostatic interaction at the headinterface which is responsible for the shorter interfacial\ndistance.\nTable 1: Extracted energy gaps and exchange splitting parameters of Gr P:headand GrP:tailat\nDirac point for Gr/BFO/Gr heterostructure. EGis the band-gap of the Dirac cone given in\nunits of meV. \u0001\"and\u0001#are the spin-up and spin-down gaps in meV, respectively. The spin-\nsplitting in meV of the electron and hole bands at the Dirac cone are \u000eeand\u000eh, respectively.\nEDin eV is the Dirac cone position with respect to Fermi level. \rsocdenotes the spin-orbit\nband opening at the avoided crossing of the spin-up and spin-down bands given in meV.\nThe hopping parameters used to ﬁt the tight-binding Hamiltonian to the DFT calculated\nband structure are denoted by t\"andt#for spin up and spin down given in eV. Those are\ndirectional dependent for Gr P:headand their three values are listed. tRis the strength of the\nRashba spin orbit coupling given in meV.\nEG \u0001\"\u0001#\u000ee\u000ehED\rsoct\"t#tR\nGrP:head-48.6 55 26 104 75 -0.85 4 2.66 2.3 8.7\n2.66 2.28\n2.61 2.32\nGrP:tail-34.04 6 1.5 -35 -40 -0.47 5 2.42 2.5 7.5\nWe discuss now the induced multiferroic-proximity eﬀect in graphene by BFO. As we\nhave demonstrated that the two graphene sheets exhibit diﬀerent interaction strengths with\nthe underlying BFO surface, the corresponding proximity eﬀect is expected to diﬀer. The\ncalculated band structure for Gr/BFO/Gr supercell, displayed in Figure 3 (a), reveals two\ndistinct graphene band dispersions highlighted by blue and red corresponding to spin up\nand spin down, respectively. However, both graphene sheets are negatively doped which is\nexpected due to accumulated negative charges on graphene side in response to the positive\nbound charges at both BFO surfaces. Following its weaker interaction with BFO, the Dirac\ncone corresponding to Gr P:tail, shown in Figure 3(b) lies in the bulk gap of BFO closer\nto the Fermi level. On the other hand, the stronger interaction at Gr P:head/BFO interface\nresults in a larger doping of the Dirac point, as seen in Figure 3 (c). The proximity of the\ninsulating BFO induces modiﬁcations in the linear dispersion of the graphene band structure\nopening a band gap at the Dirac point. This degeneracy lifting at the Dirac point is spin\n7dependent owing to the interaction with the magnetic BFO substrate. Interestingly, the\nspin-dependent band gaps and exchange splittings are inﬂuenced by the interaction strength\nat the BFO interface. Spin dependent band-gaps are found to be 55(26) meV for spin\nup (spin down) in Gr P:head, whereas smaller values of 6(1:5) meV are reported for Gr P:tail.\nMoreover, the spin splittings for Gr P:headare found to be 104(75) meV for electrons (holes),\nrespectively, compared to 35(40) meV for Gr P:tail. Figure 3 (d,e) show the evolution of\nthe graphene band structure upon adding spin-orbit coupling to the calculations. The main\nimpact of the spin-orbit interaction is inducing an additional band opening denoted by \rsoc\nat the spin up/spin down band crossings. We ﬁnd corresponding values of 4and5meV for\nGrP:headand GrP:tail, respectively.\nThe parameters obtained from the band structure are summarized in Table ??for both\nGrP:headand GrP:tail.EGand\u0001\"(#)represent the energy band gap and the spin dependent\nband gaps, respectively. The spin splitting of the electron and hole bands are denoted as \u000ee\nand\u000eh.EDindicateshowlargetheDiracpointdopingiswithrespecttoFermienergyand \rsoc\nis the spin-orbit coupling induced band opening. The negative value of EGindicates a spin\nresolved band overlap while spin splittings are deﬁned by spin-dependent energy diﬀerences\nat Dirac point with negative value indicating that spin-up bands are lower in energy than\nthat of spin-down bands. Due to the stronger interaction at the headinterface compared to\nthetail, the proximity-induced gaps and splittings are larger in Gr head. However, the spin\norbit coupling induced gap \rsocis rather smaller. We should note here that our calculated\nvalues are diﬀerent from those obtained in Ref39due basically to the diﬀerence in the k-mesh\nsize. As the band structure of graphene is highly sensitive to the k-mesh, we have used a\ndense 9\u00029\u00021k-mesh in our calculations.\nThe following tight-binding Hamiltonian describes the graphene’s linear dispersion rela-\ntion in proximity of a magnetic insulator:\n8H=X\ni\u001bX\nltl\u001bcy\n(i+l)1\u001bci0\u001b+h:c:+X\ni\u001b\u001b01X\n\u0016=0[\u000e+ (\u00001)\u0016\u0001\u000e]cy\ni\u0016\u001b[~ m:~ \u001b]\u001b\u001b0ci\u0016\u001b0\n+X\ni\u001b1X\n\u0016=0[ED+ (\u00001)\u0016\u0001s]cy\ni\u0016\u001bci\u0016\u001b;(1)\nwheretl\u001bis the anisotropic hopping connecting unit cells ito their nearest neighbors cells\ni+l.cy\ni\u0016\u001bcreates an electron of type ( \u0016= 0;1) corresponding to A and B sites, respectively,\non the unit cell iwith spin ( \u001b= 0;1) for spin up and spin down electrons, respectively.\n\u0001\u000e=\u000ee\u0000\u000eh\n2where\u000eeand\u000ehis the strength of the exchange spin-splitting of the electron and\nhole bands at the Dirac cone, respectively. ~ mis a unit vector that points in the direction of\nthe magnetization and ~ \u001bis the vector of Pauli matrices, so that ~ m:~ \u001b=mx\u001bx+my\u001by+mz\u001bz.\nEDis the Dirac position with respect to the Fermi level and \u0001s=\u0001\"+\u0001#\n2is the averaged\nstaggered sublattice potential. The Rashba spin orbit coupling term is written as ,65,66\nHSO=itRX\ni\u001b\u001b0X\nlcy\n(i+l)1\u001b[\u001bx\n\u001b\u001b0dx\nl\u0000\u001by\n\u001b\u001b0dy\nl]ci0\u001b0+h:c:; (2)\nwheretRis the Rashba spin orbit coupling strength and the vector ~dl= (dx\nl;dY\nl)connects\nthe two nearest neighbors.\nTo obtain the hopping values, the tight binding bands where ﬁtted in good accordance\nto the DFT bands as shown by solid lines in Figures 3(b-e). In the case of Gr P:head, it was\nnecessary to include direction dependent hopping parameters into the model. The values of\nthe hopping parameters used for both Gr P:headand GrP:tailare listed in Table ??.\nBased on the Hamiltonian parameters extracted from the graphene band structure, we\nemployed the tight-binding approach with scattering matrix formalism conveniently imple-\nmented within the KWANT package in order to calculate conductances and proximity re-\nsistances .67The system modeled is shown in 4(a) and comprises two identical proximity\ninduced multiferroic regions of width W= 39:6nm and length L= 49:2nm, separated by\n9-0,6 -0,5 -0,4 -0,3-10010\n  \n (1,6)\n (3,8)\n (1,8)\n (3,6)PMER %\nE - EF (eV)\n-40-2002040PMR %\n  \n (1,2)\n (3,4)\n (5,6) \n (7,8)\n-40-2002040\n  \n (1,5)       \n (3,7)  \n (2,6)     \n (4,8)     PER %\nV\nJ, Js\nL1 L2\nL L d\nP\n↑(↓)\nM\n→\nP\n↑(↓)\nM\n→(\n←)\n-0,6 -0,5 -0,4 -0,3246\n \n \n \n \n \n \n \n \n  Conductance (e2/h) \nE - EF (eV)(f)  Without SOC \n(g)   With SOC(a)\n(b)(c) \n(d) \n(e) \n-60-3003060Resistance %\n  \n PER PMR PMER\n-60-3003060\n  \n Resistance %\nPER PMR PMER1. P↑↑M\n2. P↑↑M\n3. P↓↓M\n4. P↓↓M\n5. P↑↓M\n6. P↑↓M\n7. P↓↑M\n8. P↓↑M\nFigure 4: (color online) Model spintronic device used to calculate the multiferroic proximity\nmagnetoresistance consisting of two multiferroic regions on top of a graphene sheet (a). The\nmultiferroic regions have a length L, widthWand are separated by a distance d. (b) The\nconductances calculated without including spin-orbit coupling for the diﬀerent conﬁgura-\ntions of electric polarization Pand magnetization Mof the two multiferroic regions. The\ncorresponding eight conductance states are explicitly given and indexed by numbers. (c, d,\ne) The calculated proximity electro (PER), magneto (PMR), and multiferroic (PMER) mag-\nnetoresistances, respectively, calculated without (closed symbols) and with (open symbols)\ninlcuding spin orbit coupling. The indices of the two conductance states used to obtain each\nproximity resistance curve are designated. The maximum values of the PER, PMR, and\nPMER calculated without (with) including spin orbit coupling are shown in f (g), respec-\ntively.\na distanced= 1:5nm of nonmagnetic region of graphene sheet with armchair edges. Both\nmagnetic graphene regions are connected to the leads L1andL2and modeled using the\nHamiltonian parameters. All the relative magnetization and polarization conﬁgurations are\nconsidered in this model device. The conductance Gin the linear response regime can be\nobtained according to:\nG\u001b;\u001b0\n\u000b;\u000b0=e\nhX\n\u001bZ\nT\u001b;\u001b0\n\u000b;\u000b0\u0012\u0000@f\n@E\u0013\ndE; (3)\n10whereT\u001b;\u001b0\n\u000b;\u000b0indicates spin-dependent transmission probability with ( \u000b;\u000b0) and (\u001b;\u001b0)\nbeing, respectively, the relative polarization and magnetization conﬁgurations in the multi-\nferroic regions. f=1\ne(E\u0000\u0016)=kBT+1is the Fermi-Dirac distribution in which \u0016andTindicate\nelectrochemical potential and temperature, respectively. It is important to mention that the\ntemperature smearing has been taken into account using the room temperature since the\nCurie termperature of BFO is well above it. In order to show the impact of polarization on\ntransport calculations, we choose to adjust the doping energy for the Gr P:headto be the same\nas for Gr P:tailbands. The conductance curves shown in Figure 4(b), which are explicitly\ndescribed in the legend and indexed by numbers, reveal six diﬀerent resistance states two\nof which are degenerate; those are ( 5and7) and ( 6and8). The conductance for a given\nenergy should be seen as if one could gate the whole device to bring the region of interest, in\nthe vicinity of the Dire cone splittings, to the Fermi level. We observe that the conductance\ncurves are splitted the most in the energy range aﬀected by proximity eﬀect which is around\n\u00000:47eV. Since the gaps and exchange splittings are much larger for Gr P:headcompared\nto GrP:tail, a diﬀerence in the energies and conductance values between the corresponding\nconductance states is observed.\nThe diﬀerent combinations of these conductance states give rise to three types of proxim-\nity resistances: proximity electroresistance (PER), proximity magnetoresistance (PMR), and\nproximity multiferroic resistance (PMER). We introduce the generalized formulas of these\nthree types of proximity resistances as follows:\nPER\u001b;\u001b0\n\u000b;\u000b0=G\u001b;\u001b0\n\u000b;\u000b\u0000G\u001b;\u001b0\n\u000b;\u0000\u000b\nG\u001b;\u001b0\n\u000b;\u000b+G\u001b;\u001b0\n\u000b;\u0000\u000b(4)\nPMR\u001b;\u001b0\n\u000b;\u000b0=G\u001b;\u001b\n\u000b;\u000b0\u0000G\u001b;\u0000\u001b\n\u000b;\u000b0\nG\u001b;\u001b\n\u000b;\u000b0+G\u001b;\u0000\u001b\n\u000b;\u000b0(5)\nPMER\u001b;\u001b0\n\u000b;\u000b0=G\u001b;\u001b\n\u000b;\u000b\u0000G\u001b;\u0000\u001b\n\u000b;\u0000\u000b\nG\u001b;\u001b\n\u000b;\u000b+G\u001b;\u0000\u001b\n\u000b;\u0000\u000b: (6)\n11Based on this formalism, sixteen diﬀerent conductance states are expected. However, due\nto symmetry in our considered model device we obtain G\u001b;\u0000\u001b\n\u000b;\u000b0=G\u0000\u001b;\u001b\n\u000b;\u000b0andG\u001b;\u001b0\n\u000b;\u0000\u000b=G\u001b;\u001b0\n\u0000\u000b;\u000b\nand, consequently, we end up with six conductance states G\u001b;\u001b0\n\u000b;\u000b0.\nThe calculated PER, PMR, and PMER are plotted in Figure 4(c),(d), and (e), respec-\ntively, in which the indices of the two conductance states used to obtain each proximity\nresistance curve are designated. Closed (open) symbol lines correspond to the calculations\nwithout (with) including SOC. Owing to the two degenerate conductance states, we ob-\ntain one (two) degenerate PMR (PMER) curves, correspondingly. The PER values range\nbetween\u000044%and+33%, PMR has values from \u000022%to+48%, whereas PMER ranges\nbetween +7%and+13%. We should note that including SOC doesn’t change our results\nqualitatively but rather decreases the values of the conductances and consequently the values\nof the diﬀerent types of proximity resistances as shown in Figure 4(f,g). This is basically\ndue to the mixing of the spin channels imposed by the spin-orbit interaction. Our ﬁndings\nlead to a concept of multi-resistance device and pave a way towards multiferroic control\nof magnetic properties in two-dimensional materials. Interestingly, recent experiments have\ndemonstratedtheelectriccontrolofmagneticproximityeﬀectatthegraphene/BFOinterface\n68which further enhances the possibility of realizing our proposed concept device.\nIn conclusion, we have demonstrated that the magnetic proximity eﬀect in graphene can\nbe tuned by the electric polarization existing in the multiferroic substrate. The presence\nof electric polarization together with the polar surface charges lead to diﬀerent interaction\nstrength at the Gr/BFO interface depending on the relative direction of the electric polar-\nization. Consequently, the spin-dependent band gaps and exchange splittings are impacted.\nThose ﬁndings suggest tuning the magnetic proximity eﬀect in graphene through altering\nthe direction or even the magnitude of the electric polarization. Such approach is accessible\nin multiferroic oxides where the interplay between electric and magnetic order oﬀers the pos-\nsibility of tuning the magnetization and polarization by applying electric or magnetic ﬁelds,\nrespectively.\n12We thank J. Fabian, K. Zollner and S. Roche for fruitful discussions. This project has\nreceived funding from the European UnionâĂŹs Horizon 2020 research and innovation pro-\ngramme under grant agreements No. 696656 and 785219 (Graphene Flagship).\nReferences\n(1) Chappert, C.; Fert, A.; Dau, F. N. 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New Journal of Physics 2014,16, 063065.\n(68) Song, H.-D.; Zhu, P.-F.; Yang, X.; Qin, M.; Ren, Z.; Duan, C.-G.; Han, G.; Liao, Z.-\nM.; Yu, D. Electrical control of magnetic proximity eﬀect in a graphene/multiferroic\nheterostructure. Applied Physics Letters 2018,113, 183101.\n20"    },    {        "title": "2304.10048v1.Control_of_conductivity_in_Fe_rich_cobalt_ferrite_thin_films_with_perpendicular_magnetic_anisotropy.pdf",        "content": "1 \n Control of conductivity in Fe-rich cobalt -ferrite  thin films with perpendicular \nmagnetic anisotropy  \n \nMasaya Morishita,1) Tomoyuki Ichikawa,1) Masaaki A. Tanaka,1,*) Motoharu  Furuta,1) Daisuke \nMashimo,1) Syuta Honda,2,3) Jun Okabayashi,4) and Ko Mibu1) \n \n1 Graduate School of Engineering, Nagoya Institute of Technology, Nagoya, Aichi 466 -8555, Japan  \n2 Department of Pure and Applied Physics, Kansai University, Suita, Osaka 564 -8680, Japan  \n3 Center for Spintronics Research Network, Graduate School of Engineering Science, Osaka University, Toyonaka, \nOsaka  560-8531, Japan  \n4 Research Center for Spectrochemistry, The University of Tokyo, Bunkyo -ku, Tokyo 113 -0033, Japan  \n* Corresponding  author  \n \n \nAbstract  \n We fabricated two types of cobalt -ferrite  (001) thin films,  insulative Fe-rich cobalt -ferrite  CoxFe3-xO4+δ \n(I-CFO) and  conductive Fe-rich cobalt -ferrite  CoyFe3-yO4 (C-CFO) , with perpendicular magnetic anisotropy  (PMA)  on \nMgO (001) substrates . Although the stoichiometric cobalt -ferrite  is known as an insulat ing material, it is found that the \nconductivity of Fe-rich cobalt -ferrites can be controlled by changing the source  materials and deposition conditions  in the \npulsed laser deposition technique . The I-CFO and C-CFO films exhibit  PMA through  the in -plane lattice distortion. We \ninvestigated the Fe -ion-specific valenc e states  in both I-CFO and C-CFO films by Mössbauer spectroscopy and X-ray \nmagnetic circular dichroism , and found that the difference in conductivity corresponds  to the abundance ratio of Fe2+ \nstate at the octahedral B-site (Oh) in the inverse spinel structure . Furthermore, first -principles calculations  reproduce the \nchanges in the density of states at the Fermi level depending on the  cation vacancies at the B-site, which explains the \ndifference in the conductivity between  I-CFO and C -CFO.  \n \nI. Introduction  \nSpinel -type cobalt -ferrite s (CoFe 2O4: CFO) have been investigated for a long time for magnetic applications as a \nmagnetic insulator  with high-frequency performance, magneto -strictive properties , and so on , in the forms of bulk, \npowder, and thin films , because of their insulating advantages  [1-13]. They have  an inverse spinel structure in AB 2O4 \nchemical formulation , with Fe3+ at the tetragonal A-site ( Td), and Fe3+ and Co2+ at the octahedral B-site ( Oh). Recent \ndevelopments in the spintr onics boost the application of thin CFO films stacked with other nonmagnetic and magnetic \nlayers, which are, for example , utilized as spin -filtering tunnel barriers  [14-19]. There are some reports that CFO (001) \nfilms exhibit perpendicular magnetic anisotropy (PMA) under the tensile epitaxial strain [20-35]. The materials with \nPMA are strongly demanded for the high-density recording techniques. Therefore, CFO films can be one of the best \ncandidate s for the PMA system without rare -earth elements. The PMA in the CFO films is thought to originate from Co2+ \n(3d 7) at the octahedral B-site (Oh) in the tetragonally distorted inverse spinel structure  [30]. Furthermore, it is reported \nthat the squareness of hysteresis curves , i. e., the abrupt ness in  magnetization reversal , is improved in Fe -rich CFO  films 2 \n [28]. The valence states of the cations in these stoichiometric and Fe -rich CFO  films  can be examined by site -specific \nanalyses such as X-ray magnetic circular dichroism (XMCD)  [25,35] and Mössbauer  spectroscopy (MS) [7,33]. For the \ncharge neutrality, the formal valence states are recognized as Co2+ (Oh), Fe3+ (Oh), and Fe3+ (Td) in stoichiometric CFO \nfilms  [9]. However, the substitution of Co2+ by Fe2+ promote s the formation of Fe2+ (Oh) sites, which induces the \nconductive properties in CFO  films . The conductivity in CFO films would arise depending on the compositions due to \nthe formation of Fe2+ states through the Co2+ → Fe2+ substitution , which is regarded as an approach to magnetite Fe3O4. \nOn the other hand, insulative CFO can be grown by  the Co2+ → Fe3+2/3□1/3 (□ = cation vacancies) substitution , which is \nan approach to insulative maghemite -Fe2O3 [4,33,36]. Here after, we refer to  insulative Fe -rich cobalt -ferrite \nCoxFe3-xO4+δ as “I-CFO ” and conductive Fe-rich cobalt -ferrite Co yFe3-yO4 as “C-CFO ”. The cation vacancies in I -CFO \nare expressed as excess oxygen  with the symbol of δ. \nThe I -CFO films with PMA can be used as spin -filtering tunnel barriers to create perpendicularly spin -polarized \nelectronic currents [19]. For this purpose, the epitaxial I -CFO (001) films with in -plane tensile strain have to be grown on \nconductive underlayers. In order to develop the fully epitaxial spin-filtering magnetic tunnel junctions  (MTJs) , the \ninterfac ial matching between conducti ve underlayers  and I-CFO insulating barriers has to be examined caref ully. From \nthe viewpoints of the lattice match  and PMA  stabilization , the C -CFO layers with PMA are highly promising  as \nconductive  electrode layers  under the I -CFO  barriers . Although the use of  Fe3O4, which is conductive through the \nelectron  hopping between Fe2+ and Fe3+ ions, as electrode layers of conventional MTJs with nonmagnetic barriers are \nreported recently  [37,38], the PMA properties  can also be appended to the oxide spintronics researches  by using the \nFe-rich I -CFO and C-CFO  films . In fact, the MTJ combining both I-CFO tunnel barriers and C-CFO electrode  layers  \nachieved a large tunnel magnetoresistance ratio of −20% at 100 K  [19]. In order to develo p I-CFO and C-CFO films with \nPMA for the high -quality Co-ferrite MTJs, the exact selective growth conditions for both I-CFO and C-CFO films and \ntheir detailed electronic states have to be examined precisely .  \nIn spite of the realization of large PMA in CFO films on MgO substrate s, it is known that the magnetic properties of \nepitaxial films of spinel ferrites are strongly affected by the existence of anti -phase boundary (APB). Until now, the APB \nin Fe 3O4 films grown on the MgO substrate s fabricated by various methods was examined [ 39-43]; the APB broadens  the \nabruptness in magnetization curves ( MH curves ) with the suppression  of saturation magnetization  Ms values. The APB in \nCFO films on the MgO substrate s is also discussed [ 4,25]. Although PMA was indicated by the MH curves in the se \nexperiments, the magnitude of the Ms is much smaller than that of bulk CoFe 2O4. Therefore, the existence of APB in the \nspinel -type lattice structure is inevitable and it would  affec t the MH curves. These facts have to be considered  also for the \nI-CFO and C-CFO cases.  \nIn this paper, we aim to establish the selective film growth methods between insulative Fe -rich cobalt -ferrite and \nconductive Fe-rich cobalt -ferrite films with PMA, and discuss the origin of conductive properties from the results of \ntransport measurements and site-specific magnetic spectroscopies of XMCD and MS.  \n \nII. Experimental  \nSingle -crystalline I -CFO (Co xFe3-xO4+δ) and C -CFO (Co yFe3-yO4) films of 20 nm in thickness with various Co \ncompositions of x (x = 0.00, 0.11, 0.23, 0.36, 0.43, 0.56, 0.66, 0.76, 0.87, and 1.00 ) and y (y = 0.00, 0.23, 0.42, and 0.66) 3 \n were grown on the MgO (001) substrates by a pulsed laser deposition ( PLD ) method using the source materials with \ndifferent pow der compositions. The source materials for the I -CFO, except x = 0, were prepared by mixing CoFe 2O4 and \nFe3O4 powder, and those for the C -CFO, except y = 0, were by mixing CoFe 2O4 and α-Fe powder. Pure Fe 3O4 powder \nwas used for t he source materials of the I-CFO with x = 0 and C -CFO with y = 0. The compositions between the Co and \nFe atoms were controlled by the ratios of the source materials and confirmed by an electron probe microanalyzer. After \nthe pressing process under 20 MPa, the I -CFO and C -CFO sources were baked in an atmospheric environment for 12 \nhours at the maximum temperature of 1100oC and 350oC, respectively. Note that the source materials w ere not perfectly  \nconsummated as the aiming I-CFO and C -CFO films  because the amount s of oxygen and cation vacancies are expected \nto be controlled  by the film growth conditions . \nThe PLD was performed using frequency -doubled Nd:YAG laser with a pulse width of 6 ns and a repetition rate of 30 \nHz. The energy density of the laser beam was cont rolled to 1 J/cm2 by an optical lens. The I -CFO films were grown in O 2 \npressure of 6 Pa at the substrate temperature of 300oC. The C -CFO films were grown in Ar pressure of 4 Pa at the \nsubstrate temperature of 300oC. The gas pressure conditions and substrate temperatures during the deposition were  \noptimized to obtain CFO films hav ing large in-plane tensile strain  in the (001) crystal orientation , which induces the \nPMA . The eclipse PLD method, where a shadow mask was placed  between the source  material and the substrates, was \nused in order to reduce the formation of droplets and particulates [44].  \nThe crystal structures of the prepared films were investigated by X-ray diffraction (XRD) using a Cu -Kα source. In \norder to investigate the in -plane and out -of-plane strains, reciprocal space maps of X-ray diffraction intensity were \nmeasured  for both I-CFO and C-CFO films. The transport properties of p repared films were inves tigated using the \nfour-probe methods with cooling by a helium cryostat. The MH curves  were measured using a superconducting quantum \ninterference device  (SQUID)  magnetometer. 57Fe Mössbauer spectroscopy using a radioactive 57Co source was conducted \nat room temperature  (RT)  by the conversion electron detection method.  The -rays were i rradiated from the film normal \ndirection and the emit ted elect rons were detected by a proportional  gas counter . X-ray absorption spectroscopy ( XAS ) \nand XMCD were performed at BL -7A in the Photon Factory at the High Energy Accelerator Research Organization \n(KEK -PF). For the XAS and XMCD measurements, the photon helicity was fixed, and a magnetic field of ±1.2 T was \napplied parallel along the incident polarized soft X-ray beam, to obtain signals defined as μ+ and μ− spectra. The total \nelectron yield  (TEY)  mode was adopted, and all the measurements were performed at RT. Since the TEY measurements \nare surface sensitive which detects the signals beneath 3 nm from th e sample surface, we prepared the samples for XAS  \nand XMCD by capping the surfaces by 1 -nm-thick Pd layer s deposited at RT.  \n \nIII. Results  \nA. X-ray diffraction  \nFigure 1(a) shows the XRD patterns of the typical I-CFO and C-CFO  films with various Co composition  x and y (x = \n0.00, 0.23, 0.36, 0. 66 and y = 0.00, 0.23, 0.42 , 0.66) , with the scattering vector perpendicular to the film plane . They \nexhibit clear I-CFO (008) and C-CFO (008) diffraction peaks . This means that the I-CFO and C-CFO films are highly \noriented along the [001] directions on the MgO (001) substrates . The I -CFO (008) peaks are shifted to lower angles \nsystematically with increasing x, indicating the extension  of the lattice along the c-axis. On the other hand, the C -CFO \n(008) peaks are shifted to hi gher angles systematically with increasing y, indicating the suppression of lattice along the 4 \n c-axis. The additional satellite peaks , which  originate from the interference effect, are detected beside the (008) peaks \nshowing  high crystalline qualities  of the films prepared by the PLD with a shadow mask . Figure 1(b) shows the Co \ncomposition dependence of the vertical lattice constant 𝑎⏊ of the I-CFO  and C-CFO  films estimated from the XRD \npatterns. The red and blue broken line s represent the linear relations in Begard law between bulk Fe3O4 and CoFe 2O4, and \nthat between bulk -Fe2O3 and CoFe 2O4, respectively. The out-of-plane lattice constant s of the I-CFO films decrease with \nthe decrease of the Co composition  x, while  those  of the C-CFO films increase with the decrease of the Co composition y. \nFigure 1(c) shows the reciprocal space maps around the MgO (113) peak in the horizontal Q (110) and vertical Q (001) \ndirections for the I-CFO  (x = 0.66) and C-CFO  (y = 0.66) films . The I-CFO (226) and C-CFO (226) peaks were observed \nnear the MgO (113) peak. The in -plane Q values of the I-CFO (226) and C-CFO (226) peaks coincide with the value of \nthe MgO (113) peak, which means that the in -plane lattice constants of the I-CFO and C-CFO films are twice that of the \nMgO substrates (8.42  Å). Note that the  tendency of the in -plane lattice constant, which is less dependent on the Co \ncomposition, is  consistent with previously reported results [ 19, 35]. From the results of the in -plane  𝑎|| and out -of-plane \n𝑎⏊ lattice constants , it is suggested that the I-CFO and C-CFO films have in -plane tensile strain and that the structures \nare tetragonally distorted  to settle  the lattice mismatch  with the MgO substrate s. \n \nB. Resistivity measurements  \nTemperature dependence of resistivity for the C-CFO films  grown on MgO (001) substrates with different Co \ncomposition  y is shown in Fig . 2(a).  With  increas ing y, the resistivity  of the C-CFO films  systematically increases. For \nthe I -CFO films, the resistivity was over 105 Ωcm at 300 K for all the Co composition x and they can be regarded as \ninsulator s. The resistivity of the C-CFO ( y = 0.0 0), Fe 3O4, film at 300 K is 0.03 Ωcm, which is in the same order as in the \nprevious studies [36, 45]. In order to confirm the hopping conduction through the thermal activation, temperature T \ndependence of the resistivity ρ is plotted by  the Arrhenius equation 𝜌=𝜌0exp(∆𝐸𝑘B𝑇⁄) and fitted with the \nleast-squares method, as shown in  Figs. 2(b)-(e). Here, 𝜌0 is the pre -exponential factor of the resistivity, ΔE is the \nactivation energy, and 𝑘B is Boltzmann constant. Except the low -temperature part s for y = 0.00 and  0.23, the resistivity \nfollows the Arrhenius plots, which means that the conduction mechanism of C -CFO is due to the electron hopping \nbetween the B-site Fe ions with different valence s. The deviation  of the experimental data from  the fitting line for y = \n0.00 and 0.23 originates from the Verwey transition. The Verwey transition is a dramatic change in the electrical transport \nproperties accompanied with  a structural transition from the cubic phase to the monoclinic phase around 120 K, which \nsuggests that a high -quality Fe 3O4 film is prepared [ 46,47 ]. Figures 2(f) and 2(g) exhibit  the composition y dependence of \n𝜌0 and ∆𝐸, respectively.  The ∆𝐸 corresponds to the activation energy of electron hopping between Fe2+ and Fe3+ ions. \nThe increase in resistivity with increasing Co in Fig.  2(a) is found to be  due to the increase in activation energy  of the \nelectron hopping.  \n \nC. Magnetic properties  \nFigures 3(a)-(d) show the results on magnetization measurements by a SQUID magnetometer with the magnetic field \napplied along the in-plane and out -of-plane directions  at 300 K  for the I-CFO ( x = 0.00, 0.43, and 0.87) and C -CFO ( y = \n0.23, 0.42, and 0.66) films . The PMA characteristics are obtained for all cases of  the I-CFO  and C -CFO fi lms except x = 5 \n 0.00 (-Fe2O3) and y = 0.00 (Fe 3O4). Figure 3(e) shows the Co composition x, y dependence  of squareness ratio Mr/Ms of \nI-CFO [33] and C-CFO  in the out -of-plane MH curve s. The squareness ratio in I-CFO is larger than that in  C-CFO for \nentire Co composition range s. In I-CFO  films , the squareness ratio decreases with the Co composition x. For C-CFO  \nfilms , on the other hand, the squareness ratio increases with Co composition y. The difference in tendency on the \nsquareness ratio as a function of Co composition can be explained by the strain magnitude  as a function of Co \ncomposition. As shown in Fig. 1(b), the out -of-plane lattice constants of I -CFO films increase and th ose of I-CFO films \ndecrease with the increase of Co composition . The in -plane lattice constants of both I-CFO and C -CFO films almost \nremain unchanged for  all the Co composition s. Therefore, the distortion of I -CFO films decrease s, and that of C -CFO \nfilms increase s with the Co composition x. The PMA energies estimated from  the area surrounded by the loops of \nout-of-plane and in -plane loop s for I-CFO ( x = 0.66) and C -CFO ( y = 0.66) are 1.0 and 0.3 MJ/m3, respectively . The \ndeterioration of the squareness in the MH curves and the changes of Ms imply the existence of APB in the films.  \n \nD. Mössbauer spectroscopy  \nFigure 4 shows the 57Fe Mössbauer spectra for the I -CFO ( x = 0.23) and C -CFO ( y = 0.23) films  taken at RT using \nthe conversion electron Mössbauer spectrometry  (CEMS ) mode . Both cases clearly show magnetic spectra , with no \ncomponent from α-Fe or α-Fe2O3. In the case of I -CFO, a six -line spectrum pattern with symmetric line height s is \nobserved , which suggests that the film is composed only of the Fe3+ ions without Fe2+ ions. This is consistent with the \nresistivity measurement because the  hopping conduction  is suppressed  without Fe2+ [48-50]. On the other hand, the \nspectrum for C-CFO represents an asymmetric and rather broad line shape due to the overlapping of two magnetic \ncomponents . These spectra are fitted using the Mössbauer  parameters of isomer shift ( δ), quadrupole splitting ( 2ε) and \nhyperfine field ( Bhf). The fit ted parameters are listed in Table  1. For C -CFO, two kinds of magnetic components \ncorresponding to the  A and B-sites in the spinel structure  are necessary for the fitting . The B-site components are detected \nas an Fe2.5+ sextet because the observation  time in Mössbauer spectroscopy is longer than the time scale of electron \nhopping between Fe3+ and Fe2+ sites. The combination of two magnetic components with different isomer shift and \nhyperfine fields clearly reproduce the asymmetric spectra. The obtained Mössbauer parameters are similar to those in the \nprevious reports on Mössbauer spectra of C oFe 2O4 [2,7,10,12,51,52 ] and Fe 3O4 [53,54]. From these site -specific analyses, \nthe Fe2+ components are thought to be essential for the conducti vity in C-CFO. \nThe intensity ratios of the six lines in both spectra are between 3:0:1:1:0 :3 and 3:2:1:1:2:3, which means that the \nfilms have perpendicu lar magnetic anisotropy but that the direction of magnetic moment is not perfect ly perpendicular to \nthe film plane  at zero external field . This is consistent with the magnetization measurements, where the s quareness ratios \ndo not reach 1. The intensity ratio of the 2nd and 5th peak is smaller for the I -CFO film, suggesting that the perpe ndicular \nanisotropy is stronger in the  I-CFO system.  \n \nE. XAS and XMCD  \nFigure 5 shows the XAS and XMCD of C-CFO  for Fe and Co L-edges  with different Co compositions . Spectra are \nnormalized by a n incident white line beam intensity before the absorption at the samples. The Fe and Co intensities are \nplotted in the same vertical scales because XAS roughly provide s the Fe and Co compositions. Because of the composition \nratio of Co:Fe  = 1:12 in y = 0.23, the XAS intensities of Co in this film are suppressed. With increasing the Co \ncompositions, the Co L-edge intensities are enhanced systematically. XAS and XMCD line shapes for the Fe L-edges show 6 \n distinctive features with differential line shapes due to the three kinds of Fe states  (Fe3+ in Oh, Fe3+ in Td, and Fe2+ in Oh). \nFor the Fe L-edges, although the difference in XAS is small, clear differential XMCD line shapes are detected. The Fe3+ \nstate with Td symmetry exhibits the opposite XMCD sign, which is common for the spinel ferrite compounds. The Fe2+ \ncomponent  at 708 .0 eV decreases in high Co compositions. In the case of Fe 3O4, the Fe2+ component is more enhanced  [38]. \nTherefore, the conductive properties are related to the amounts of Fe2+ states, whose quantitative analysis is performed by \nusing the ligand -field-multiplet (LFM)  calculation in later. The cases of I -CFO are also shown in Fig. 6, which is similar to  \nthe previous reports [35]. Large XMCD signals in Co L-edge  in spite of smal l XAS intensity  correspond to the saturated \nmagnetized states. Within the orbital sum rule, the large  orbital magnetic moments  (morb) come from the asymmetric \nXMCD line shapes. Since the Co site is almost identi fied as Co2+ (Oh) symmetry, the sum rules can be applicable for the \nCo XMCD spectra. We note that the sum rule analysis for the entire Fe L-edge XMCD cannot be applicable because it \ninclude s three kinds of components, which has to be deconvoluted into each component because XAS/XMCD originate s \nfrom the atomic excitation process  [55]. The spin and orbital magnetic moments for Co2+ sites in y = 0.23 sample are \nestimated as 1.32 and 0.63  µB, respectively, using an electron number of 7.1  [33]. Large morb originates from the orbital \ndegeneracy in  the d7 electron system in strained Co site s and contributes to the PMA [ 56]. The strained Co sites are also \nconfirmed by the extended x -ray absorption  fine structures [ 35]. \nFor the analysis of XMCD spectra, we employed the LFM cluster -model calculations including the configuration \ninteraction for the Fe sites in Co yFe3-yO4 of y = 0.23 and 0.6 6 as tetrahedral ( Td) TMO 4 and octahedral ( Oh) TMO 6 clusters, \nmodeled as a fragment of the spinel -type structures. The line shapes of Fe2+ (Oh), Fe3+ (Oh) and Fe3+ (Td) are calculated in \nthe previous reports in the I-CFO using the Coulomb interaction U of 6.0 eV [35] by considering the chemical shift of \nenergy position through the electron number . Here, we adopt this analysis for the estimations of the ratio in Fe2+ and Fe3+ \nintensities.  \nAs shown in Fig. 7, the spectral line shapes of XMCD can be reproduced by the LF M calculations qualitatively, at least \nthe peak positions, with three kinds of Fe states.  The XMCD spectr a for different valence s and symmetr ies are calculated \nby using the same parameters as the cases of  I-CFO [35]. The peak position of Fe2+ (709.0 eV) is composed of not only \nFe2+ but also Fe3+ (Oh) contribution. We fitted the spectra using these LFM -calculated spectra  by changing the ratios of \nthree components . Then, the ratios of Fe2+ (Oh), Fe3+ (Oh) and Fe3+ (Td) can be  estimated to be 1:1:1 for y = 0.23, and \n0.2:1:1 for y = 0.66. The fitting result s indicate that the intensity ratio of Fe3+ (Oh) and Fe3+ (Td) is not affected by the Co \nion substitution. Therefore, the site-specific analysis in XMCD clearly deduces the contributions of Fe2+ states , although it \nmight be overestimated , which play an essential role for the conduction mechanism in C -CFO films.  \n \nF. DFT calculation  \n   In order to investigate  the conductive properties of Co yFe3-yO4 films  from the viewpoint of the electronic structures , \nwe performed the first -principles density -functional -theory (DFT) calculations  with periodic boundary conditions  for an \noptimized Co 1Fe11O16 super -cell structure shown in Fig . 8(a), which was considered  as the C-CFO of Co0.25Fe2.75O4 (y = \n0.25) . The lattice constant for the unit cell of 8.35 Å was employed which  include s the strain estimated from the XRD \nexperiment s. The unit cell include s the A- and B-sites of 4 and 8 cations, respectively , and  these correspond to  4 Fe3+ ions \nat the A-sites, and 4 Fe3+ ions, 3 Fe2+ ions, and  1 Co2+ ions at the B-sites. The DFT calculation s were performed using the \nVASP code with the projector augmented wave ( PAW ) potentials [57] in the  generalized gradient approximation  (GGA)  - 7 \n perdew burke ernzerhof (PBE) [58,59] including the Coulomb repulsion energy U of 6.0 eV . The cutoff energy is set to \n400 eV and the crystal momentum k sampling mesh is set to (21, 21, 17).  \nFigure 8(b) shows the calculated spin -dependent density of state s (DOS)  for Co1Fe11O16 as C-CFO. The spin-down \nstates of the B-site in Fe cross the Fermi level  (EF) and contribute to the conductivity. The partial DOS of Co site is \nlocate d at deeper level and d oes not contribute to the formation of the  intensity at the  EF. The band gap between  occupied \nand unoccupied spin -up states is estimated to be  2.58 eV . The DOS clearly indicate s the half -metallic property. The Fe \nmagnetic moment of −4.15 µB for A-sites and 4.02 µB for B-sites can be  estimated.  The local magnetic moment of 2.70 µB \nof Co site is also  estimated, which is smaller than those of Fe sites and qualitatively consistent with that of the XMCD \nanalysis . \nWe also calculated the spin-dependent DOS for an optimized Co 1Fe10O16 super -cell structure as shown in Fig. 8(c), \nwhich is considered as the I-CFO  of Co0.25Fe2.50O4 or Co0.273Fe2.727O4.364 (x = 0.273, δ = 0.364). The calculated DOS is \nshown in Fig. 8(d) and the insulating property can be seen.  The band gaps between occupied and unoccupied states in  \nspin-up and spin-down cases  are estimated to be  2.67 eV and 2.4 0 eV , respectively , whose entire DOS is similar to the \nprevious results of I -CFO  [33]. Note that this spin-dependent difference in the band gaps can be applied for the \ngeneration of the spin-polarized electrons . In fact, we reported that  electrons  tunneling  through the I -CFO thin films \nwere spin -polarized because of the difference in tunnel probability of spin -up and spin -down electrons caused by the \ndifference in the band gaps  [19]. The half -metallic property is  guaranteed in the C -CFO , as in  the case of Fe 3O4, where \nB-site DOS contribute the conduction at the EF [38]. \n \nIV. Discussion  \nThe conduction mechanism , structural deformation, and advantages  in I-CFO  and C-CFO  can be discussed using the \nabove results . First , MS and XMCD reveal the existence of Fe2+ states in C-CFO, which is responsible for the  conduction \nas analogous to the case of Fe3O4. However, for I -CFO, small amounts of Fe2+ states are still detected in XMCD, which is \nnot detected in MS as shown in Fig. 4. It may come from the difference in probing regions; while the MS detects the bulk \ninformation  because of the larger escape depth  of the conversion electrons (~ 100 nm) in comparison with the film \nthickness (20 nm) , XMCD in soft X -ray detects the surface regions  beneath 3 nm from the sample surface . Since the \nvalence states of surface regions are slightly modulated and the surface  oxygen  reduction might occur. However, it is not \nsignificant but an offset factor for the detailed analysis of Fe2+ states  to estimate the chemical compositions . Therefore, \nsite-specific magnetic spectroscopies by MS and XMCD provide the complementary  information for the analyses of I-CFO \nand C-CFO.\nSecond , the conduction mechanism can be discussed by combining the resistivity and DFT calculation. As an analogous \nsystem with Fe3O4, the conduction in C -CFO is caused by the electron hopping between Fe2+ and Fe3+ in the B-site through \nthe activation type excitation. Tempe rature dependence of the resistivity in Fig. 2 is different from the small polaron \nformation and variable range hopping schemes [48]. The double -exchange mechanism can be anticipated in the t 2g states. \nHalf-metallic nature in the DFT calculations can also b e explained by the electron hopping mechanism, which is different \nfrom the noble metals and distinctive for conductive oxide materials such as p erovskite Mn oxide compounds  [60]. Yasui et \nal. reported that a large tunnel magnetoresistance (TMR)  effect was obtained for Fe 3O4/MgO/Fe MTJs and  that the large 8 \n TMR effect derives from the  half-metallic nature of Fe 3O4 [38]. Thus the half-metallic C-CFO films may become a good \ncandidate for the spin injector in  spintronics devices.  \nFinally , we discuss the selective growth of two types of Fe-rich CFO , C-CFO and I -CFO . The element - and \nsite-selective XMCD and MS clearly suggest the valance states of Fe2+ is an essential factor for  controlling the \nconductivity of CFO.  For the growth  of Fe-rich CFO  films , the v alance states of Fe ions can be controlled by changing \nthe oxygen flow rate, the source materials , and the  substrate temperature  during the growth  with the  deposition \natmosphere.  The deposition s under the  O2 pressure  with high growth temperature promote the oxidation of Fe2+ states . \nOn the other hand, the deposition s under the  Ar pressure with low growth temperature suppress the oxidation of Fe2+ \nstates . In the case of C -CFO, the XMCD results (Fig. 7) show  that the abundance ratio of Fe2+ ions in the B-site decreases \nwith increasing the Co composition y, which suggests that the Co2+ ions in CoFe 2O4 are replaced with Fe2+ ions.  The \nXRD result s (Fig. 2(b)) also display  that the vertical lattice constant  of C-CFO films increases with decreasing y. From \nthese results, the C-CFO can be recognized as  a magnetite -type CFO because their bond distances  are tuned  between \nCoFe 2O4 and Fe 3O4 according to the y values,  following qualitatively to the Begard law.  On the other hand, in the case of \nI-CFO, the XRD result show s that the vertical lattice constant of I -CFO films decr eases with decreasing y [33]. Therefore,  \nI-CFO can be considered as a maghemite -type CFO because of the tuning of  CoFe 2O4 and -Fe2O3 compositions \naccording to x values . The junction composed of  I-CFO tunnel barriers and C-CFO electrode  layers  can be applicable for \nthe spintronics researches and demonstrated for the spin-filtering effect with PMA  [19]. \n \nV. Summary  \nWe fabricated two types of Fe-rich cobalt -ferrite  thin films, insulative Co xFe3-xO4+δ (I-CFO) and conductive \nCoyFe3-yO4 (C-CFO) , with PMA. Although the stoichiometric cobalt -ferrite s are known as an insulating material, it is \nfound that the conductivity in Fe -rich CFO can be controlled by changing the source  materials and deposition conditions \nin the PLD  technique. The I-CFO and C-CFO films also exhibit the PMA through the in -plane lattice distortion. We \ninvestigated the Fe -ions-specific valence states in I-CFO and C-CFO films by Mössbauer spectroscopy and XMCD , and \nfound that the difference in conductivity corresponds to the abundance ratio of Fe2+ state at the octahedral ( Oh) site. \nFurthermore, first -princip les band -structure c alculation suggested that the electronic structures of C -CFO are \nhalf-metallic characteristics  and reproduce d the difference in the DOS  depending on the cation vacancies at the B-site in \ninverse spinel structures, which explains the difference in the conductivity between I -CFO and C -CFO. 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Red and blue broken lines represent the linear relations in Begard law between \nbulk Fe3O4 and CoFe 2O4, and that between bulk -Fe2O3 and CoFe 2O4, respectively.  (c) Reciprocal space maps around \nthe MgO(113) peaks for the I-CFO  (x = 0.66) and C-CFO  (y = 0.66) films. \n14 \n  \nFig. 2. (a) Electrical resistivity as a function of temperature for conductive CoyFe3-yO4 films, (b-e) Plot of lnρ versus 1/ T \nwith a linear fitting, ( f, g) Pre-exponential factor of the resistivity ρ0 and activation energy  ΔE as a function of Co \ncomposition y. \n15 \n  \nFig. 3. Out-of-plane and in-plane MH curve s of the CoxFe3-xO4+δ ((a) x = 0.00 , (b) x = 0.43 , and  (c) x = 0.87 ) and \nCoyFe3-yO4 ((d) y = 0.23 , (e) y = 0.42 , and  (f) y = 0.66 ) films at 300 K . (g) Squareness ratio  of remanent and saturat ion \nmagnetizations  Mr/Ms as a function of Co composition for  I-CFO and C-CFO films.  \n \n \n16 \n  \nFig. 4. Mössbauer  spectra for  insulative Co 0.23Fe2.77O4+δ (I-CFO) and conductive Co 0.23Fe2.77O4 (C-CFO) films  taken at \nRT. Dots represent the experimental data. Lines are fitting results. For C -CFO, two components are used and sum of \nthese are also shown.  \n \n  \n17 \n  \nTable 1 Mössbauer fitting parameters for I-CFO and C-CFO films.  Isomer shifts ( δ) were defined with respect to  α-Fe at \nRT. \n \n18 \n  \nFig. 5 XAS and XMCD of conductive Co yFe3-yO4 films for y = 0.23, 0.42, and 0.6 6. \n \n19 \n  \nFig. 6 XAS and XMCD of insulative Co xFe3-xO4+δ films for x = 0.23 and 0.66. \n20 \n  \nFig. 7 Ligand -field multiplet cluster -model calculation of Fe2+ and Fe3+ states. (a) Fragments of Oh and Td symmetries \nused in the calculation  drawn  by VESTA ( K. Momma and F. Izumi, 2011) . (b) Calculated XMCD spectra  of Fe3+ (Oh, Td) \nand Fe2+ (Oh). (c,d) XMCD spectra of experimental (dots) and LFM calculation (lines) for  y = 0.23 and 0.6 6. \n21 \n  \n \nFig. 8 (a) Crystal structure of the Co1Fe11O16 (C-CFO) used in first principles calculation. The white, dark green , light \ngreen  and blue spheres are O, Fe  (A-site), Fe (B-site) and Co atoms, respectively. (b) The local density of states for \nCo1Fe11O16 of spin up and spin down states . The Fermi energy was  set to  0 eV . (c) Crystal structure of the Co1Fe10O16 \n(I-CFO) used in first principles calculation. The white, dark green , light green  and blue spheres are O, Fe ( A-site), Fe \n(B-site) and Co atoms , respectiv ely. The dotted circle represents a cation defect. (d) The local density of states for \nCo1Fe10O16 spin up and spin down states . \n"    },    {        "title": "1712.06114v1.Low_temperature_ferromagnetic_properties__magnetic_field_induced_spin_order_and_random_spin_freezing_effect_in_Ni1_5Fe1_5O4_ferrite__prepared_at_different_pH_values_and_annealing_temperatures.pdf",        "content": "1\n \n \nL\now temperature \nferro\nmagneti\nc properties\n,\n \nmagnetic field induced spin order \nand random \nspin freezing effect \nin Ni\n1.5\nFe\n1.5\nO\n4 \nferrite; prepared at different pH va\nlue\ns and annealing \ntemperatures\n \n \nR.N. Bhowmik\n*\n \nand \nK.S. Aneeshk\numar\n \nDepartment of Physics, Pondi\ncherry University, R. Venkataraman Nagar,\n \nKalapet, Pondicherry\n-\n605014, India\n \n*\nCorresponding author: Tel.: +91\n-\n9944064547; Fax: +91\n-\n413\n-\n2655734\n \nE\n-\nmail: \nrnbhowmik.phy@pondiuni.edu.in\n \nABSTRACT\n \n \nWe \npresent \nthe\n \nlow temperature magnetic \nproperties in \nNi\n1.5\nFe\n1.5\nO\n4 \nferrite\n \nas the function \nof pH at which the material was prepared by chemical route \nand \npost \nannealing temperature. The \nmaterial\n \nis a \nferri/ferromagnet, but \nshowed \nmagnet\ni\nc\n \nblocking and \nrandom \nspin freezi\nng\n \nprocess \non lowering the measurement temperature\n \ndown to 5 K\n. The sample prepared at \npH ~12\n \nand \nanneal\ned at \n800 \n0\nC\n \nshowed\n \na sharp magnetization peak\n \nat 10\n5\n \nK\n; \nthe\n \nsuperparamagnetic blocking\n \ntemperature of the particles\n.\n \nThe magnetization peak \nremained \nin\ncomplete within measurement \ntemperature up to 350 K for rest of the samples, although peak \ntemperature\n \nwas brought down \nby \nincreasing applied \ndc \nfield. \nThe fitting of \ntemperature\n \ndependen\nce of\n \ncoercivity data \naccording to \nKneller\n′s l\naw \nsuggested \nrandom\n \norient\na\nt\ni\no\nn\n \no\nf\n \nferromagnetic \nparticles\n. \nThe \nfitting of \ns\naturation \nmagnetization \na\nc\nc\no\nr\nd\ni\nn\ng\n \nto \nBloch\n′s law \nprovided the \nexponent \nthat \nlargely \ndeviate\nd\n \nfrom \n3/2, a\n \ntypical value \nfor long range\nd\n \nferromagnet\n. \nA\nn abrupt increase \nof \nsatura\ntion \nmagnetization below \n50\n \nK \nsuggested \nthe \nactive role \nof frozen surface spins\n \nin low temperature magnetic properties\n. \nAC susceptibility \ndata elucidated the \nlow temperature spin freezing dynamics \na\nn\nd\n \ne\nx\nh\ni\nb\ni\nt\ne\nd\n \nt\nh\ne\n \nc\nh\na\nr\na\nc\nt\ne\nr\ns\n \no\nf\n \nc\nl\nu\ns\nt\ne\nr\n \ns\np\ni\nn\n \ng\nl\na\ns\ns\n \nin the samples depend\ni\nn\ng\n \non pH value and annealing temperature\n.\n 2\n \n \nKey words: Ferrite \nnano\nparticles, \nrandom\n \nspin freezing, ac susceptibility, \ncluster spin glass\n.\n \n1. \nINTRODUCTION\n \n \n \nSpinel ferrites, having general formula \nof \nMFe\n2\nO\n4\n \n(M = divalent metal ion, e.g. Ni, Co, \nCu\n, etc.), remained \nas \none of the most attractive magnetic oxides due to \na rich class of \nmagnetic \nand electronic \nproperties. \nThe interest for \nNickel ferrite (NiFe\n2\nO\n4\n) \nstem\ns\n \nfrom the fact that it is \na \nsoft ferri\n/ferro\nmagnet \nwith\n \nreasonably high\n \nmagnetization.\n \nThe magnetic properties of NiFe\n2\nO\n4\n \nare determined primarily by \nthe \nsuperexchange interaction\ns\n \nbetween \nNi and Fe ions \nin \ntetrahedral \n(A) \nand octahedral \n(B) \ns\nites of cubic spinel lattice\n \ns\ntructure\n \n[1\n-\n2\n]\n. \nWithout elemental \nsubstitution effect, \nt\nh\ne\n \nlong\n-\nranged fe\nrrimagnetic spin order in \nNiFe\n2\nO\n4\n \ncan \na\nl\ns\no\n \nbe\n \nperturbed in \nn\nanocrystalline \nform by \nsurface \neffects (\nspin disorder, anisotropy, \nand \nexchange \ninteraction\ns) \nand exchange of Ni and Fe ions\n \namong \nthe \nA and B sites\n.\n \nThis \nincreases \nmulti\n-\nfunctionalit\nies\n \nin \nn\nanocrystal\nline \nform of soft ferromagnetic \nferrites \nand make the\nm\n \nsuitable \nfor applications in the \nfield of \npower electronics (\ntransformer core\n)\n, recording \nmedia,\n \nmicrowave devices, \nand medical \ntreatment (\nhyperthermia, \nferrofluid\n) \n[\n3\n-\n5\n]\n. The \ntuning of \nferr\no\nmagnetic parameters \n(\ncoercivity, \nsaturation \nmagnetization, \nmagneti\nc\n \nblocking, \nand ferromagnetic\n \nsquareness\n) in a wide\n \ntemperature scale is important \nfor \napplying \nthe \nmagnetic \nmaterial\n. \nHence, it is necessary\n \nto \nunderstand the ba\nsic mechanisms that control the tuning of \nferro\nmagnetic parameters\n \nin magnetic \nmaterials\n.\n \nFrom literature reports, it is understood that t\nhe nature of \nsurface spin \nstructure\n, magnetic \nfrustration and inter\n-\nparticle\n \ninteractions\n \ndetermine the\n \nlow temperatur\ne magnetic properties\n \nof \nnanoparticles \nin the form of \nsuperparamagneti\nc blocking\n, \nspin glass, \nand \nexchange bias \neffect \n[\n6\n-\n9\n]\n.\n \nThe \nreduc\ntion of \nmagnetization, higher coercivity, and \nsurface spin \ndisorder \nin \nnanoparticles \nof \nNiFe\n2\nO\n4\n \nare rema\nrkably different from that in micron\n-\nsized \nparticles\n \n[\n2\n, \n1\n0\n]\n.\n \nCore\n-\nshell spin 3\n \n \nstructure \nhas been modeled \nto explain the \nm\na\ng\nn\ne\nt\ni\nc\n \nr\ne\nd\nu\nc\nt\ni\no\nn\n \nin NiFe\n2\nO\n4\n \nnanoparticles \n[\n1\n, \n1\n1\n]\n, \nwhere\n \nspins in \nthe \ncore \nare \nferri\nmagnetically\n \na\nligned \na\ns\n \nin bulk\n \nand shell \nform\ns\n \na \ndisordered\n \nspin\n \nstructure that \ncontrol\ns the magnetic \nreduc\ntion and exhibition of spin glass and superparamagnetic \nfeature\n. \nT\nhe \nexchange of Ni and Fe io\nns among \nA\n \nand \nB\n \nsites\n \nand surface spin canting\n \nwere \nidentified\n \nas other factors\n \nthat determine\n \nmagnetic reduction and spin glass feature\n \nin \nNiFe\n2\nO\n4\n \nnanoparticles\n \n[1\n, \n9\n]\n.\n \nLee et al. [1\n2\n] \nc\no\nn\ns\ni\nd\ne\nr\ne\nd\n \nt\nh\ne\n \nintermediate spin layer \nbetween core and \nshell\n \nfor\n \nthe \nmagnetic field \ninduced \nshift of blocking temperature.\n \nIt\n \ni\ns a \ngreat \nchallenge \nto \ndistinguish \nthe \nblocking of core spins \nfrom the freezing of \nsurface spin\ns\n \n[\n6\n]\n. Kodama et al. \n[\n1\n1\n]\n \nattributed \nthe s\npin glass \nlike feature\n \nin NiFe\n2\nO\n4\n \nnanoparticles\n \ndue to freezing of frustrated surface \nspins\n. \nIn \nNiO nanoparticles, \nWinkler et al. \n[\n13\n]\n \nproposed \nthat antiferromagnetic \ncore \nof NiO was \nblocked \nat higher te\nmperature \nand \ndisordered surface spin\ns\n \nshowed\n \ns\npin glass \nlike \nfreezing\n \na\nt\n \nlow \ntemperature\n. \nRecent studies have shown that \nblocking/\nfreezing\n \nphenomena \nand ferromagnetic \nproperties\n \nin \nnano\nparticles \nremarkably depend on the \nvar\niation of \nsynthesis condition\ns\n \n(pH\n \nvalue\n, \nannealing temperature, \nannealing atmosphere\n, etc\n)\n \nof chemical routed \nferrite \nsamples \n[\n9\n-\n10, 14\n]\n. \n \nW\ne \nreported \nthe \nr\no\no\nm\n \nt\ne\nm\np\ne\nr\na\nt\nu\nr\ne\n \nmagnetic properties of \nNi\n1.5\nFe\n1.5\nO\n4\n \nf\ne\nr\nr\ni\nt\ne\n \n[1\n5]\n \nwith \nthe variation pH value and annealing temperature. \nThe as\n-\nprepared material \nwas annealed \nin the \ntemperature range 500\n-\n1000 \n0\nC [\n16\n\n. \nUsing \ndielectric \nspectroscopy study\n \n[\n17\n],\n \nwe observed \na \ntransformation \nin electrical \ncharge dynamics\n \nfrom low tempe\nrature semiconductor state to high \ntemperature semiconductor state with an intermediate metal like state \nin the samples\n. The metal \nlike state \nwa\ns understood as \na\nn\n \neffect of \nt\nh\ne\n \ncrossover of localized hopping \nof electronic \ncharge \n(electrons) \nat low measureme\nnt temperature\ns\n \nto thermal\nly \nactivated \nlong range \nhopping at higher \ntemperatures.\n \nIn th\nis \nwork, we \nr\ne\np\no\nr\nt\n \nthe \nelectronic spin dynamics in \nNi\n1.5\nFe\n1.5\nO\n4\n \nsamples \nu\nn\nd\ne\nr\n \nm\na\ng\nn\ne\nt\ni\nc\n \nf\ni\ne\nl\nd\n \nfor measurement temperature down to 5 K. Our \nobjective is to study the e\nffects of 4\n \n \nthe \nvar\niation of \npH \nvalue \nduring chemical reaction and \np\no\ns\nt\n \nannealing temperature on low \ntemperature magnetic \nphenomena\n \n(\nbl\nocking\n/\nfreezing\n \nof electronic spin moment\n)\n \nand tuning of \nferro\nmagnetic parameters\n \nin \nNi\n1.5\nFe\n1.5\nO\n4\n \nferrite\n.\n \n2. EXPERIMENTAL\n \n2.1 Sample preparation\n \n \nDetails of the material \npreparation\n \nand characterization \nwere\n \nreported \nearlier \n[\n15\n-\n17\n\n.\n \nThe\n \nsamples\n \nwere \nprepared by chemical reaction of \nthe \nstoichiometric amount\n \nof Ni(NO\n3\n)\n2\n.6H\n2\nO and \nFe(NO\n3\n)\n3\n.9H\n2\nO \nsalts \nin solution \nat\n \n80 °C \nby mai\nntaining \npH \nat\n \n6, 8, and\n \n12.\n \nFinally\n, t\nhe \nchemical \nrouted (\nas\n-\nprepared\n)\n \nmaterial\n \nwas made into pellet\ns\n \nand annealed at \nselected\n \ntemperatures. The \nX\n-\nray diffraction pattern (CuK\nα\n \nlines with λ =1.5406 A°)\n \nand Raman spectra were used to \nconfirm\n \nthe \nform\nation of \nsingle \nphased cubic spinel structure\n. T\nhe material chemically reacted at \npH value 8\n-\n12 formed single phase \nupon\n \nanneal\ning\n \nthe as\n-\nprepared material in air\n. T\nhe sample \nprepare\nd at pH 6 formed \na \nminor amount of \nα\n-\nFe\n2\nO\n3\n \nwhen annealed in air\n.\n \nHowever, the \nsample \nwhen annealed \nunder \nvacuum (~10\n-\n5\n \nmbar\n)\n \nformed single\n-\nphased \ncubic spinel structure\n \nwith \nspace group Fd3m\n. \nThe samples \nare label\ned as NFpHX_Y, where X is pH and Y is annea\nling \ntemperature in degree centigrade. \nThe \nstructural information \n(\nlattice parameter and grain size\n)\n \nof \nthe single\n-\nphased \nsample\ns\n \nused in this work is indicated \nin \nFig.\n \n1\n \nfor \ninformation to readers.\n \n \n2.2 Measurement\n \n \nMagnetic properties of the samples were\n \nstudied using \nphysical properties measurement \nsystem (\nPPMS\n-\nEC2, Quantum Design, USA\n). \nThe t\nemperature dependent \ndc \nmagnetization\n \n(M)\n \nwas measured using \nzero field cooling (\nZFC\n)\n \nand \nfield cooling (\nFC\n)\n \nmode\ns\n. In ZFC mo\nde, the \nsample was cooled from \nhigher t\nemperature (\n32\n0 K\n)\n \nwithout applying \nexternal \nmagnetic field \ndown \nto \nthe\n \nlow\nest\n \nmeasurement \ntemperature (5 K). Then, magnetic measurement started in the 5\n \n \npresence of set magnetic field\n \n(say, 100 Oe)\n \nand MZFC(T) data were recorded \nduring \nthe \nincreas\ne of \ntempe\nrature (T)\n \nfrom 5 K to 300 K/320 K\n. In FC mode, the sample was cooled \nunder \n \nset \nmagnetic field \nfrom 320 K to low temperature \n(5 K) \nand M\nFC(T)\n \ndata were \nrecorded \nwithout \nchanging the magnetic field \nduring \nthe \nincreas\ne of \ntemperature \nup to \n320 K\n. The magnet\nic field \n(H) dependence of \ndc \nmagnetization (M(H)) was measured by zero field cooling the sample from \n3\n2\n0 K to \nthe \nmeasurement \ntemperature\n, which was set\n \ni\nn the range \n5\n-\n300\n \nK.\n \nThe ac susceptibility \n(real: \nc\n/\n \nand imaginary: \nc\n//\n \ncomponents) data \nwere recorde\nd in the \ntemperature range \n10 K\n-\n3\n4\n0 K \nby applying \nac magnetic field \n(\namplitude 1 Oe\n \nwith \nfrequenc\nies\n \nin the range 3\n7\n \nHz \n-\n10 kHz\n)\n.\n \n \n3. Results and discussion\n \n3.1. \nTemperature dependent magnetization\n \n \nFig. \n1\n \nshows the \ntemperature dependence of \nMZFC\n \nand MFC c\nurves at \ndc \nfield of 100 \nOe for the samples, chemically synthesized at pH values \n6\n \n(Fig. 1(a\n-\nd))\n, 8 \n(Fig.1 (e\n-\ng))\n,\n \n12\n \n(Fig.1 \n(h\n-\ni))\n \nand \nannealed at \ndifferent \ntemperature\ns\n. \nT\nhe samples\n, prepared at pH 6 and 8,\n \nexhibited \nthe\n \ncharacter\ns\n \nof ferro\n/ferri\nmagnetic\n \nnanoparticles \nwith splitting \nbetween MFC(T) and MZFC(T) \ncurves below 320 K\n, \nwhere \nMZFC(T) curve decreased \nand \nMFC(T) curve increase\nd\n \non \ndecreasing the \nmeasurement \ntemperature \ndown \nto 5 K\n.\n \nT\nhe M\nZFC\n \n(T) curves \nof these samples \nindicated \na broad maximum or \ni\nn\ncomplete maxim\num\n \nwithin \nthe m\neasurement temperature \nlimit \n320 K. However, magnetic gap between MZFC and MFC curves at lower temperature decreases \nand the peak appears to be \nshifted \nto \nhigher temperature on increasing the annealing temperature\n \n(and increas\ne of grain size) \nof the samples\n. \nThe magnetization of the samples prepared at pH 6 is \nfound higher than the samples prepared at pH 8 and 12\n. \nAn additional \nmagnetic \ns\nhoulder\n \nis \nclearly\n \nappeared \nat low temperature (\nbelow 30 K\n)\n \nfor the samples \nprepared at pH \n6 and annealed \nat higher \ntemperature \n(\n800\n-\n1000 °C\n). \nThe origin of low temperature magnetic feature \nwill be \ndiscussed using \nthe \nac susceptibility data\n.\n \nA dif\nferent type of magnetic feature is\n \nobserved for the \nsamples prepared at pH 12 and annealed at 800 \n0\nC\n \n(grain size \n\n \n6 nm\n)\n. This sample exhibited a \nwell defined superparamagnetic blocking \ntemperature (T\nB\n) at about 105 K, and splitting between \nMZFC and MFC curves below the blocking temperature\n \n(Fig. 1(\nh)). On increasing the annealing \ntemperature to 1000 \n0\nC,\n \nthe sample prepared at pH 12 \nwa\ns not able to achieve the blocking 6\n \n \ntemperature within 300 K\n \n(Fig. 1(i))\n.\n \nThis is the effect of increasing grain size \n(6 nm to 29 nm) \nin the material. \nFrom physics point of view, t\nhe FC curve is \nin \nquasi\n-\nequilibrium \nstate \ndue\n \nto \nlocal order\ning of spins or cluster of spins \nduring\n \nfield cooling process \nwhile \nthe \nZFC curve is \nin \nnon\n-\nequilibrium \nblocking \nstate \nwhen relaxation time \n(\n\n) \nof \nthe \nspins or cluster of spins \nis \ngreater \nthan the magneti\nc\n \nmeasurement time (\n\nm \n~10\n2\n \ns). Since\n \nM\nZFC\n(T) curves exhibited a broad \nmaximum or \nin\ncomplete maxim\num\n, it is difficult to \nexact\nly\n \ndetermine \nthe \nblocking temperature \nfor most of the samples.\n \nA\n \nbroad peak indicates \na \ndistribution of blocking temperature (T\nB\n)\n, \nwhich can be \nrelated to \ndistribution\n \nof \nmagnetic anisotropy \nconstant (K)\n \nand grain volume (V) by \nthe relation k\nB\nT\nB\n \n\n \n25KV\n \n[\n18\n]\n. In such case, \nthe \ntemperature derivative of MZFC\n(T)\n \ncurve\n \n(\n\u0000\u0000\u0000\u0000\u0000\n\u0000\u0000\n) \nwas \nfitted \nwith Gaussian shape\n \nto get t\nhe \ninformation of the \ndistribution of relaxation \ntime \nor anisotropy barrier of \nthe \nmagnetic particles below \nT\nB\n \n[\n19\n]\n. The \naverage \nblocking \ntemperature \nwas \nestimated from the intercept \nof \n\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\u0000\n \nvs. T curve\ns\n \non temperature axis where \n\u0000\u0000\u0000\u0000\u0000\n\u0000\u0000\n= 0\n.\n \nThe \nfit parameters, like peak position (inflection point\n \nof M\nZFC\n \n(T) curves below T\nB\n)\n, \nfull width at half maximum (FWHM) of the peak, and peak height\n) were \nused \nas \nthe initial \ninput \nparameters \nto \ndefine a distribution curve with \nmedian blocking temperature (T\nBm\n) \nand T\nBm\n \nis \nfound \nnear to \ninfl\nection point \n(T\nP\n) \nof\n \nthe MZFC(T) curve \nbelow \n(T\nB\n)\n.\n \nThe distribution curve was \nfitted with \nlog\n-\nnormal distribution \n(\nf\n(\nt\nB\nm\n)\n)\n \nfunction\n.\n \n                  \nf\n(\n\u0000\n\u0000\u0000\n)\n=\n\u0000\n√\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\u0000\u0000\n\u0000\n−\n\u0000\u0000\n\u0000\n\u0000\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n \n \n \n \n \n(1\n)\n \nA\nccuracy of the obtained parameters were \nc\nhecked \nby a\n \nquantitative \nanalysis \nof the MZFC\n(T)\n \ncurves\n \nusing\n \nthe \nfollowing equation \n[\n18\n,\n \n20\n]\n, where \nf\n(\n\u0000\n\u0000\u0000\n)\n \nhas been replaced by \nf\n(\n\u0000\n\u0000\n)\n.\n \n         \n\u0000\n\u0000\u0000\u0000\n(\n\u0000\n)\n=\n\u0000\n+\n \n\u0000\n\u0000\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n∫\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n(\n\u0000\n\u0000\n)\n\u0000\n\u0000\n\u0000\n+\n∫\n\u0000\n(\n\u0000\n\u0000\n)\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n \n(\n2\n) \n \nC is a constant \nthat takes into account \nthe \nresidual \nvalue of low temperature magnetization\n. \nT\nhe \nfirst term in\nside the \nbracket c\norresponds to \nthe \ncontribution from mutually interacting super\n-\nparamagnetic nanoparticles\n \nwith \nthe factor \n\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n \nis 25. \nT\nhe second term corresponds to \nthe \ncontribution from \nsuperparamagnetic\n \nparticles in \nrelaxed state\n \n(\n\n>\n\nm\n)\n. \nM\nsat\n \nis the s\naturation \nmagnetization\n \n(the value \nis taken \nfrom M(H) curves at 5 K/10 K)\n; H is the applied magnetic field \nduring ZFC measurement; \nK\neff\n \nis the effective anisotropy constant, \nt\n \nis the\n \nreduced temperature\n \n(\nT\n/\nT\nBm\n)\n \nand \nt\nB\n \nis \nth\ne reduced blocking temperature (\nT\nB\n/T\nBm\n)\n. \nThe example of typical distribution 7\n \n \n(\nf\n(\nt\nB\nm\n)\n) \ncurve and final fit of the respective MZFC(T) curve are shown for samples NFpH8_500 \n(Fig. 1(e)) and NFpH12_800 (Fig. 1(h)).\n \nThe total \nanisotropy \nenergy \nper unit volume \n(E) \nof a magnetic particle in the p\nresence of \nexternal magnetic field (H) and a\nt\n \nspecific measurement temperature \n(T) \nbelow the blocking \ntemperature (T < T\nB\n) \nis \ngoverned by \nthe \ncompetition between \nuniaxial \nanisotropy energy (E\nA\n= \nK\neff\n \nsi\nn\n2\nθ) and Zeeman energy (E\nH\n \n= \n-\nM\nsat\nHcos(θ \n–\n \nφ)\n)\n \n[\n \n21\n]\n, as defined below.\n \n  \n   \n      \n \nE\n(\nT\n,\nH\n)\n=\nK\n\u0000\u0000\u0000\n(\nT\n)\nsin\n\u0000\nθ\n−\nM\n\u0000\u0000\u0000\n(\nT\n)\nHcos\n(\nθ\n−\nφ\n)\n \n \n \n   \n(\n3\n)\n \nθ is the angle between magnetization vector and \nlocal \nanisotropy \neasy \naxis\n \n(EA)\n, and φ is the \nangle between \nH\n \nand \nEA\n. At higher fields, \nthe \nincrease of \nZeeman energy \nwill \nreduce the \neffective \nanisotropy barrier \nfor blocking \no\nf the particle\n \nand subsequently, \nT\nB\n \nwill shift \nto low\ner \nvalues\n.\n \nFig. \n2\n(a\n-\nc) demonstrate\ns\n \nthe applied field\n \ninduced shift of \nT\nB\n \n(reduction) \nfor the samples \nNFpH6_600\n \n(Fig.2(a), NFpH8_500 (Fig.2(b)), and NFpH12_800 (Fig.2(c). \nF\nirst order derivative \nof the MZ\nFC(T,\n \nH) \ncurves \nand final distribution curves used for fitting \nof \nthe \nfield dependence \nof \nMZFC(T) data are shown for NFpH8_500 (\nFig. \n2\n(\nd\n-\ne\n)\n) and NFpH12_800 (\nFig. \n2\n(\nf\n-\ng\n)\n)\n \nsamples.\n \nFig. 3 \nshows the \nvariation of \npeak parameters\n \n(\nposition: \nT\np\n, peak height, and\n \nFWHM\n) \nfrom \nGaussian fit of \ndMZFC(T)/dT \ncurves \nwith the applied field increment\n. \nThe general features \nare as follows, (1) the peak position \nof \ndMZFC(T)/dT curves \nshifts to lower temperature with the \nincrease of applied magnetic field \nand it can be taken as\n \nequivalent information of the \nfield \ndependence of T\nB\n, (\n2\n) the peak height increases with the increase of applied field\n, and \n(\n3\n) the \npeak width decreases with the increment of applied field\n. \nThe increase of peak height with the \ndecreas\ning \nwidth \nconfirms\n \nth\ne \nmagnetic field induced \nclustering of \nsmall\n \nmagnetic domains/\n \nparticles \nto form a larger sized multi\n-\ndomain particle\n. This leads to narrowing of \nthe \ndistribution \ncurve with reduced height, indicating the magnetic field induced reduction of \nanisotropy barr\nier \nand magnetic exchange interactions in the system \n[\n19\n]\n.\n \nThe fact is supported by the reduction of \narea (h\neight and width) of distribution curves (Fig. 2(e, g)) with the increase of applied magnetic \nfield. \nIn fitting of MZFC(T) data using equation (2), we have used the low temperature M\nsat\n \nof \nthe samples (34.1 emu/g\n,\n \n30.1 emu/g, 16.5 emu/g for \nNFpH6_600, NFpH8\n_500, \nNFpH12_800\n, \nrespectively\n). \nI\nn\n \nTable 1\n,\n \nw\ne\n \nh\na\nv\ne\n \ns\nh\no\nw\nn\n \nthe \nc\no\nn\ns\nt\na\nn\nt\n \nC\n \na\nn\nd\n \nt\nh\ne\n \nanisotropy constant per unit \nfield (K\neff\n/H) \nv\na\nl\nu\ne\ns\n \nr\ne\nq\nu\ni\nr\ne\nd\n \nt\no\n \nf\ni\nt\n \nt\nh\ne\n \nM\nZ\nF\nC\n(\nT\n)\n \nc\nu\nr\nv\ne\ns\n \na\nt\n \nd\ni\nf\nf\ne\nr\ne\nn\nt\n \nm\na\ng\nn\ne\nt\ni\nc\n \nfield. \nW\ne\n \no\nb\ns\ne\nr\nv\ne\nd\n \nt\nh\na\nt\n \nf\ni\nt\n \no\nf\n \nt\nh\ne\n \nM\nZ\nF\nC\n(\nT\n)\n \nc\nu\nr\nv\ne\ns\n \na\nr\ne\n \nn\no\nt\n \nm\nu\nc\nh\n \na\nc\nc\nu\nr\na\nt\ne\n \nf\no\nr\n \nN\nF\np\nH\n6\n_\n6\n0\n0\n \ns\na\nm\np\nl\ne\n,\n \ne\nx\nh\ni\nb\ni\nt\ne\nd\n \nm\no\nr\ne\n \np\ne\na\nk\n \nb\nr\no\na\nd\nn\ne\ns\ns\n.\n \nH\no\nw\ne\nv\ne\nr\n,\n \nK\neff\n/H) \nv\na\nl\nu\ne\ns\n \nf\no\nr\n \nN\nF\np\nH\n8\n_\n8\n0\n0\n \na\nn\nd\n \nN\nF\np\nH\n1\n2\n_\n8\n0\n0\n \ns\na\nm\np\nl\ne\ns\n \nd\ne\nc\nr\ne\na\ns\ne\nd\n 8\n \n \nw\ni\nt\nh\ni\nn\n \ne\nr\nr\no\nr\n \nb\na\nr\n \n(\nn\no\nt\n \ns\nh\no\nw\nn\n)\n \nw\ni\nt\nh\n \nt\nh\ne\n \ni\nn\nc\nr\ne\na\ns\ne\n \no\nf\n \nf\ni\ne\nl\nd\n \nm\na\ng\nn\ni\nt\nu\nd\ne\n.\n \nThis take\ns\n \ninto account the \neffect of \ndecreasing distribution curve area on increas\ning \nmagnetizatio\nn at higher field. \nSecondly, \nwidth and \nT\np\n \nfor\n \nthe NFpH8_500 sample (grain size\n \n\n \n10 nm\n) are higher\n \nin comparison to the \nvalues \nin \nNF\npH12\n_800\n \nsample\n \n(grain size\n \n\n \n6 nm\n). \nThe\n \nresults show that \nmagnetic \ndistribution \nin s\nuperparamagnetic blocking\n \nregime \nis \naff\nected by\n \ngrain size\n \nvariation\n; exhibiting\n \na \nnarrow \ndistribution \nof anisotropy barriers \nfor the samples prepared at low annealing temperature\n \nand \na \nlarge distribution \nf\nor \nthe \nsamples anneal\ned at higher\n \ntemperatur\ne\n.\n \nFig. \n3\n(\nd\n) \nillustrate\ns\n \nthe \nmagnetic field \ni\nnduced \nreduc\ntion of \nT\nB\n, estimated from \n\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\u0000\n \n= 0 on temperature axis, \nfor \nthe \nsamples \nNFpH6_600, NFpH8_500, and NFpH12_800. In order to \nidentify\n \nthe nature of magneti\nc\n \norder\n \nbelow T\nB\n, whether spin glass or superparamagnetic blocking or random \nspin \nfreezing\n \nof \nmagnetic \nparticles, the \nvariation of \nT\nB\n \nwith applied \nfield has been fitted with power law\n \n[\n22\n]\n: \nT\nB\n(H) = \na\n–\n \nb\nH\nn\n \nwith \nfield exponent (\nn\n) \n\n \n0.18, 0.14, and 0.22 for NFpH6_600, NFpH8_500, and \nNFpH12_800 samples, respectively\n \n(Fig. 3(e))\n.\n \nThe fittin\ng of field dependence of mean blocking \ntemperature (\nT\nB\nm\n(H)\n) gives the \nexponent (\nn\n) \n\n \n0.12 and 0.257 for \nNFpH8_500\n \nand \nNFpH12_800\n, respectively\n. \nAs discussed in Ref. [\n22\n-\n24\n] and  references there\nin\n, t\nhe\n \nn\n \nis \nexpected \n~ 0.67 \naccording to \nAlmeida\n-\nThouless lin\ne \nfor classical \nspin glass or typical \nsuperparamagnetic nanoparticles with single relaxation time (Neel\n-\nBrown model). \nThe \ntypical \nn\n \nvalue \nfor \nsystem of \ncluster spin glass coexists with ferromagnetic order\n \n(\ne.g., La\n0.5\nSr\n0.5\nCoO\n3\n)\n \nis ~ \n0.58\n.  The \nn\n \nvalue \nis ~\n \n0.48 for 3D Ising spin glass with random anisotropy\n. \nT\nhe exponent \n(\nn\n) \nvalues in our samples (\nn \n= 0.14\n-\n0.22)\n \nare much \nsmaller than the above specific class of \ndisordered magnetic systems\n. \nThe field exponent value covered a wide range 0.023\n-\n0.46 for the \nCo\n0\n.2\nZn\n0.8\nFe\n1.95\nHo\n0.05\nO\n4\n \nferrite, where \nspin glass and\n \nsuperparamagnetism \nare coexisting \nwith \nferrimagnetic order\n \nbelow the magnetic blocking temperature. \nHence, \nthe \ndivergence between \nM\nZFC\n \nand M\nFC\n \ncurves below T\nB\n \nin Ni\n1.5\nFe\n1.5\nO\n4\n \nferrite samples can be attrib\nuted to a random \nfreezing \nof spins \nor relaxation of magnetic \nclusters with a strong intra\n-\ncluster interaction\n \nalong \nlocal anisotropy axes. \nThe field induced magnetic spin ordering below room temperature is now \ndemonstrated from the magnetic field dependenc\ne of magnetization (M(H)) measurements.\n \n3.2 Field dependent magnetization\n \n \nM(H) data were measured under ZFC mode at selected temperat\nures in the temperature \nrange 5\n \nK\n-\n300 K by sweeping the magnetic field \nwithin ± \n60 kOe. \nHowever, \nM(H) data are \nshown with\ni\nn\n \nmagnetic field range \n± \n10kOe for clarity of \nthe information\n. \nFig. \n4\n(a\n-\nd\n) shows\n \na 9\n \n \ncomparative plot of M(H) data measured at 10 K and 300 K for the samples prepared at pH 6 and \nthermally annealed at different temperatures. The immediate observation is that\n \nthe samples at 10 \nK showed a wide loop in comparison to a narrow loop at 300 K, indicating a transformation of \nfeatures from medium hard magnet at low temperature to soft ferromagnetic character at room \ntemperature\n. Additionally, \nthe \nsoft ferromagnetic ch\naracter \ni\ns rapidly enhanced \n(with increasing \nmagnetization) \non increasing annealing temperature of \nthe \nmaterial chemically prepared at pH 6.\n \nSuch samples are useful for applications in transformer core and hyperthermia\n \n[21\n]\n.\n \nWe have not \nseen any appreciabl\ne shift of the M(H) loop measured under field cooling @70 kOe with respect \nto the M(H) loop measured under ZFC mode. A typical example is shown for NFpH6_600\n \nat 10 \nK, which\n, of course,\n \nshowed a minor enhancement of positive magnetization after field coolin\ng.\n \nThe \nM\n(\nH\n)\n \ndata at \nselected temperatures\n \nare shown for NFpH6_600 (Fig. 4(e)), \nNFpH8_500 \n(Fig. \n4(f)), \nand NFpH12_800 \n(Fig. 4(g))\n \nsamples\n.\n \nAs shown in Table 1, the ferro\nmagnetic parameters \n(magnetization, \nsquareness, \ncoercivity\n, anisotropy constant\n)\n \nof the \nsamples\n \nreduced \nwith the \nincrease of measurement temperature. The samples NFpH6_600 and NFpH8_500 \nclearly \nshowed \na loop \nat \nall measurement temperatures, where as NFpH12_800 does not show magnetic loop \nand retaining \n(M\nR\n) \nof high field magnetization after re\nducing the field to zero \nfor temperatures at \n\n \n200 K. \nThe magnetic squareness\n \n(S)\n, defined as the ratio of M\nR\n \nand spontaneous magnetization \n(M\nS\n), is an important parameter to estimate the retaining of high field magnetization. \nMagnetic \nsquareness of the pr\nesent system \nis reasonably good and it \nincreased up \nto the value 0.28\n-\n0.46 \non \nlowering the temperature down to 5 K\n.\n \nIn case of uniaxial anisotropy in magnetic material, the \nsquareness is expected close to 0.5 [\n25\n]. The small smaller value of squareness in \nNFpH12_800 \ncan be attributed to surface spin disorder effects. \nIn the absence of magnetic saturation at higher \nfields\n \n(up to 60 kOe)\n, \nthe M\ns\n \n(spontaneous magnetization)\n \nwas \ncalculat\ned using \nArrot plot (\nM\n\u0000\n \nvs. H/M) \n[\n15\n]\n \nof \ninitial M(H) curve \nand it is represented in the insets of Fig. 4(f\n-\ng).\n \nThe lack of \nmagnetic \nsatura\ntion indicates the randomly distributed\n \nmagnetic \nspins \nstructure and exchange \ninteractions in nanoparticles\n.\n \nThe inset of\n \nFig. 4(h) shows that spontaneous \nmagnetization \nin the \nmaterial at 5 K \nhas \nincreased with annealing temperature. \nIt is worthy to mention that \nM\nS\n \nat low \ntemperatures in our \nNi rich ferrite \nsamples \nis \ncomparable to the reported value for nickel ferrite \nnanop\narticles \n[\n1,\n \n25\n]\n. For \nNF\npH12\n_800\n \nsample\n, \nM\ns\n \nat 5 K\n \n(\n1\n3.2\n \nemu/g)\n \nis \nrelatively \nlow \ndue \nto \ncoexistence of \na significant \nfraction of \nsuperparamagnetic \ncomponent \nalong with ferromagnetic \ncomponent in \nsmaller grain size\nd\n \nsample \n[\n26\n]\n. \nIt may be noted (\nTa\nble 1\n) that \nM\ns\n \nof the samples 10\n \n \nat different measurement temperatures\n \nare \ns\nmaller than \nthe \nhigh field magnetization.\n \nThe \nsaturated magnetization (M\nsat\n) ha\ns\n \nbeen determined from \nthe \nfitting of high\n-\nfield M(H) curves \nusing the law of approach to saturation of \nmagnetization\n \n[\n19\n]\n.\n \n                 \n \nM\n(\nH\n)\n=\nM\n\u0000\u0000\u0000\n\u0000\n1\n−\n\u0000\n\u0000\n−\n\u0000\n\u0000\n\u0000\n\u0000\n+\nc\n\u0000\nH\n \n \n \n \n \n(\n4\n)\n \nHere, χ\nd\n \nis the high field induced \npara\nmagnetic susceptibility, \na\n \nand \nb\n \nare \nthe \nconstants. The term\n \n\u0000\n\u0000\n, which \ntakes into account \nthe existence of micro\n-\nstructural de\nfects \nin the system\n, is generally \nnegligible\n \n[\n27\n]\n. The K\neff\n \nwas calculated \nusing \nthe relation \n\u0000\n=\n \n\u0000\n\u0000\u0000\u0000\n×\n\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\n\u0000\n[\n14\n]. \nM(H) curves \nof \nthe samples \nNFpH6_600\n,\n \nNFpH8_500 \nand NFpH12_800\n \nwere \nfitted \nwith equation (\n4\n) and \nshown \nin Fig. 4(d) \nfor N\nFpH6_600 \nsample\n. \nTable\n \n1 shows t\nhe \nferromagnetic \nparameters \nobtained from fitting of M(H) curves using \neq\nuation \n(\n4\n).\n \nAmong the three samples, \nK\neff\n \nvalues \nare relatively high in NFpH8_500 and low in \nNFpH12_800\n.\n \nAlthough \nequation (\n4\n) is applicable \nfor \nNFpH12\n_800\n \nsample\n, \nwhere superparamagnetic component plays a major role on \ndetermining \nthe \nmagnetic properties\n, \nits M(H) curves \nare\n \nbest fitted \nwith the replacement of \nc\n\u0000\nH\n \nin equation (4)  by Langevin function for superparamagnetic component of magnetization\n \n[\n28\n]\n.\n \n      \n    \nM\n(\nH\n)\n=\nM\n \n\u0000\u0000\u0000\n_\n\u0000\n\u0000\n1\n−\n\u0000\n\u0000\n\u0000\n\u0000\n+\nM\n\u0000\u0000\u0000\n_\n\u0000\u0000\n\u0000\ncoth\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n−\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n \n \n(\n5\n)\n \n \nThe \nM\nsat_f\n \nand M\nsat_\nsp\n \nare contributions \nfrom \nferromagnetic and superparamagnetic \ncomponents \nto \nthe saturated \nmagnetization\n. \nT\nhe contribution from ferromagnet\nic component (\nM\nsat_f\n \n\n16.46 at \n5 K and 6.15 emu/g at 300 K\n) to saturated magnetization \nof NFpH12_800 sample \nis nearly same \n(\nM\nsat\n) as \ngot \nfrom fit\n \nof M(H) curves using equation (4\n). \nThe \ncontribution of \nsuperparamagnetic \ncomponent (\nM\nsat_\nsp\n)\n, as plotted in th\ne inset of Fig. 5(a), is remarkably high for \nNFpH12_800 \nsample\n.\n \nInterestingly, saturat\ned\n \nmagnetization \nrapidly increased \nbelow 50 K\n. I\nt \ncan be\n \nattributed \nt\no \nan \nadditional \nhigh field magnetic contribution from randomly froze\nn\n \nsurface spins \nat low \ntemperatur\nes [\n9\n, \n27\n]. \nOtherwise, saturat\ned\n \nmagnetization \ndecreases above 50 K. \nThe main reason \nof \nthe \ndecreas\ning \nmagnetization \ncan be\n \nattributed \nto \nthe breaking of ground state \nferro/\n \nferrimagnetic \nspin order by \nlow energy excitations\n \nof \nthe \nspins in\n \ncore as well as\n \nin \ndisordered \nsurface \nof \nferromagnetic \nnano\nparticles\n.\n \nTh\nis \nis \nrealized by fitting the \ntemperature dependence of \nsaturat\ned\n \nmagnetization \ndata\n \nusing Bloch\n′s law  \n[\n27, 29\n-\n30\n]\n. \n \n                  \nM\n\u0000\u0000\u0000\n(\nT\n)\n=\n \nM\n\u0000\u0000\u0000\n(\n0\n)\n(\n1\n−\n\u0000\nT\nα\n)\n \n \n \n          \n   \n \n(\n6\n)\n 11\n \n \n \nB\n \nis the \nBloch\n \nconstant\n.\n \nM\nsat\n(0) is the saturation magnetization at 0\n \nK\n. \nα is the exponent that is \ndependent on \nmagnetic \nspin \norder \nand typically 3/2\n \nfor long range\nd\n \nferromagnet\n. The constant \nB\n \nwas reported \n~ \n10\n-\n4\n-\n10\n-\n5 \nfor n\nanoferrites \nand \n~ 10\n-\n6\n \nfor bulk ferromagnet\ns\n. \nIn\n \nbulk \nferromagnetic \nferrites\n, α \nis \nreported \n~\n \n2 or less\n \n(\ne.g., \n\n \n2 \nfor \nCoFe\n2\nO\n4\n, NiFe\n2\nO\n4\n, Fe\n3\nO\n4\n, \nand \n\n \n1.5\n \nfor MnFe\n2\nO\n4\n)\n \n[\n27, 31\n]\n. \nIn nanomaterials\n, the temperature dependen\nt \nmagnetization de\nviated\n \nfrom Bloch\n′s\n \nlaw \nwith exponent α = 1.5\n, which \nvaries in a wide range 0.58\n-\n2.44 \n \n[\n9\n,\n \n25\n, \n30, \n32\n]\n.\n \nThe\n \nBloch\n′s law is \nvalid \nin our samples \nfor \nmeasurement \ntemperature \nabove\n \n50 K.\n \nIn NFpH12_800 sample, the \nM\nsat\n-\nf\n(T)\n \ndata\n \nare fitted with \nα\n \n\n1.39\n \nand \nB\n~\n1.94x10\n-\n4\n. \nHowever, \nM\nsat_sp\n(T\n) \ndata \ndo not follow Bloch´s \nlaw (inset of Fig. \n5\n \n(\na\n)).\n \nOn the other hand, \nM\nsat\n(T)\n \ndata \nof \nNFpH6_600 and NFpH8_500 \nsamples\n \nfollowed Bloch’s law\n \nwith noticea\nbly large values\n \nof \nα ~ 2.73 and 2.89\n \nwith \nB\n \n~\n2.25x10\n-\n8 \nand\n \n8.65x10\n-\n9\n, respectively.\n \nOur results suggest that magnetic spin interactions are \nsufficiently strong for the \nNFpH6_600 and NFpH8_500 samples\n \nwith larger grain size and the \nspin interaction is \nhigh\nly perturbed \nfor the \nNFpH\n12\n_\n8\n00 sample\n \nwith smaller grain size. \nThe \nmagnetic upturn \nat low temperature \ndeviated from the \nBloch law \nand it\n \nis termed as \nthe \nquantization of spin\n-\nwave spectrum \nin nano\nparticles\n, where surface spin freezing \n[\n30\n] or site \nexchang\ne of cations [\n9\n] \nplays an important role\n.\n \nTable 1 \nshow\ned\n \na monotonic decrease of \ncoercivity \n(\nH\nc\n) \nwith the \nin\ncrease \nof\n \nmeasurement \ntemperature. \nFig. \n5\n(\nb\n)\n \nshows a \ngood \nfit \nfor \nH\n\u0000\n(\nT\n)\n \ndata with Kneller’s \nlaw\n:\n \nH\n\u0000\n(\nT\n)\n=\nH\n\u0000\n(\n0\n)\n[\n1\n−\n(\n\u0000\nβ\n\u0000\n\u0000\n)\n\u0000\n] \n[\n14, 28, 33\n]\n. The best linear fit of \nthe \nH\nc\n(T)\n \ndata \nshowed \nk ~ 0.5 for NFpH6_600, k ~ 0.41 for NFpH8_500 and k ~ 0.21 for \nNFpH12_800\n, respectively\n. \nT\nhe \nobtained \nk\n \nvalue\ns are lying\n \nwithin the\n \nth\neoretically predicted range \nof \n0\n-\n1.5 \nfor randomly oriented magnetic nanoparticles \n[\n21\n]\n. The shape of M(H) loop also suggest that \nmagnetic particles \nin our samples \nare randomly oriented, where \nφ\n \nin equation (\n3\n) is n\non\n-\nzero [40].\n \nW\ne \nobserved two important points regarding micro\n-\nstructural change, viz., a decrease of grain size of \nthe material with the increase of pH value during chemical reaction, and an increase of grain size \nduring increase of annealing temperature \nof the chemical routed material.\n \nSuch micro\n-\nstructural \nchanges are significantly affecting the magnetic spin dynamics of particles\n, especially at surface \nspins\n.\n \nThe\n \ncoercivity (H\nc\n) of magnetic particles \nat low temperature (5K/10K) \nhas a noticeable \ndependen\nce on grain size\n \n(\nD\n)\n. I\nt followed (Fig. 5\n(b)\n) the relation\n \nof\n \n\u0000\n\u0000\n(\n\u0000\n)\n=\n\u0000\n+\n\u0000\n\u0000\n\u0000\n\u0000\n, proposed \nfor \nmulti\n-\ndomain \nstate\n \nof magnetic particles\n \n[\n10\n, \n27\n]\n. The fitted values of constants (\na\n \nand \nb\n)\n \nare \ntabulated \nin\n \nFig. 5.\n \nThe fit of \n\u0000\n\u0000\n(\n\u0000\n)\n \ndata \nat low tem\nperature \nis characteristically different \nfrom 12\n \n \nthe annealing temperature (grain size) dependence of coercivity measured at room temperature \n(300 K)\n \n[1\n5]. One possibility is that small\n-\nsized grains (magnetic domains) of the samples are \ngetting clustered \nat \nl\now measurement\n \ntemperature and responds\n \nlike \nfrom \nmulti\n-\ndomain particles. \nThe \nK\neff\n \nvs. 1/\nD\n \nplot \n(Fig. 5(c)\n) \nusing low temperature data (5\n \nK/10\n \nK) \nfollowed an empirical \nrelation \n\u0000\n\u0000\u0000\u0000\n=\n \n\u0000\n\u0000\n+\n \n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n[\n27\n,\n \n34\n]\n, where \nK\nv,\n \nand \nK\ns\n \nrepresent anisotropy contribution\ns\n \nfrom \n(core) bulk\n \nand surfac\ne \nparts of the \nnanoparticle\ns\n. \nWe have found negative intercept (\nK\nv,\n) with \npositive slope (6\n \nK\ns\n) for relatively large grain size \nwith multi\n-\ndomain nature \n(\nregime \n1) \nand it \nbecomes \nre\nverse (\nK\nv\n \nis positive\n. slope is \nnegative\n) for smaller grain size \nwith singl\ne domain \nnature \n(\nregime \n2) \nof the samples. \nIn regime 1, the effective anisotropy constant (\n\u0000\n\u0000\u0000\u0000\n), i.e., \nanisotropy energy per unit \nvolume/gram \nof \nthe \nmaterial\n, increas\ned \ndue to increas\ning \nsurface \nanisotropy contribution with the decrease of grain size. \nI\nn regime 2, \n\u0000\n\u0000\u0000\u0000\n \nis \ndecreas\ned \nfrom \nK\nv\n \ndue to decreas\ne of \nsurface anisotropy \ncont\nribution (\nincrease of \nspin \ndis\norder) \non \ndecreas\ning the \ngrain size in single domain range of magnetic particles\n \n(Fig. 5(d))\n.\n \nT\nhe \nmagnitude of K\nv\n \nlies in \nthe range (3.0\n-\n4.8)x10\n4\n \nemu Oe/g \nin \nregime 1 and in the \nrange (2.3\n-\n12)\n \nx10\n5\n \nemu Oe/g for \nregime 2. Simil\narly, the magnitude of 6K\ns\n \nlies in the range (1.7\n-\n19.8)x10\n6\n \nnm\n-\nemu Oe/g for \nregime 1 and in the range (1.4\n-\n11.4)\n \nx10\n6\n \nnm\n-\nemu Oe/g for regime 2.\n \n3.3 Magnetic dynamics using AC susceptibility analysis\n \nThe \nspin freezing phenomena \nhas been stud\nied \nfrom the \nrea\nl χ\n´(T) and imaginary \nχ\n´´(T) \nparts\n \nof ac susceptibility\n \nin the measurement temperature \nrange\n \n10 K\n-\n340 K \nfor selected samples\n.\n \nThe \nχ\n´(T) and \nχ\n´´(T) \ndata \n(Fig. \n6\n(a\n-\nb)) \nfor \nNFpH6_500\n \nsample \nshowed dispersion on \nincreasing \nthe driving frequency \n(\nf\n) \nfrom \n1\n37 Hz\n \nto 9037 Hz. \nThe \nχ\n´(T)\n \ncurves did not show any peak up to \n340 K, although magnitude\n \nof\n \nχ\n´(T)\n \ndecreased with the increase of frequency. The \nχ\n´´(T)\n \ncurves \nshowed a peak \nat temperature (T\nf\n) \nabove 250 K at low\ner\n \nfrequency (\n1\n37 Hz)\n. The \npeak is \nslowly \ntransform\ned \nin\nto a shoulder \nwith increasing magnitude \nof \nχ\n´´(T)\n \ncurves \nat higher \nfrequencies.\n \nThe ac susceptibility\n \ndata (Fig. \n6\n(c\n-\nd)) for NFpH8_500 sample also showed dispersion on \nincreasing the driving frequency (\nf\n) from 37 Hz to 9037 Hz. The \nχ\n´(T)\n \ncurves did no\nt show any \npeak up to 340 K, although its magnitude decreased with the increase of frequency. The \nχ\n´´(T)\n \ncurves showed a \nwell defined \npeak at temperature (T\nf\n) above 2\n00 K with increasing \nmagnitude \non increasing the \nfrequenc\ny of ac field\n.\n \nThe position (T\nf\n) \nof \nχ\n´´(T)\n \npeak in both the samples \nshifted to higher temperature with the increase of driving frequency and \nthe\n \npeak shift can be \nfitted with \na \ngeneral form of \nVogel\n–\nFulcher law \n[\n27\n, \n34\n]\n.\n 13\n \n \n                        \n    \n\u0000\n(\n\u0000\n\u0000\n)\n=\n\u0000\n\u0000\n\u0000\u0000\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n(\n\u0000\n\u0000\n\u0000\n\u0000\n\u0000\n)\n\u0000\n                                    \n \n       \n(\n7\n)\n \nThe constant\n \nT\nₒ is an effective temperature \n(< T\nm\n) that takes into account \nthe \ninteraction in spin \nglass system\n \nand it is zero for non\n-\ninteracting\n \nparticles/clusters\n \n(superparamagnetic system)\n. \nT\nhe \nT\nf\n(\nf\n) data for \nnon\n-\ninteracting superparamagnetic particles\n \nfollow\n \nNéel\n–\nArrhenius \nlaw (\n\u0000\n=\n\u0000\n\u0000\nexp\n \n(\n−\n\u0000\n\u0000\n/\n\u0000\n\u0000\n\u0000\n\u0000\n)\n)\n \nwith \nf\nₒ\n \nin the range 10\n9\n–\n10\n1\n2\n \nHz and T\n0\n \n= 0 K\n.\n \nThe \nln\nf\n \nvs. 1/(T\nf\n-\nT\n0\n) plot in \nthe \ninset of Fig. \n6\n(b) suggests that \nT\nf\n(\nf\n) \ndata for NFpH6_500 sample are best fitted with \ncharacteristic frequency \nf\n0\n \n\n1.8x10\n9\n \nHz\n,T\n0\n \n\n \n188 K \nand \nactivation energy E\na \n\n \n116 meV. The \n\u0000\n(\n\u0000\n\u0000\n)\n \ndata for NFpH8_500 sample (inset of Fig. \n6\n(d)) \nare best fitted \nwith \nf\n0\n \n\n \n1.\n2\nx10\n15\n \nHz,\n \nT\n0\n \n= \n0 \nK and E\na \n\n \n5\n6\n4\n \nmeV.\n \nOn the other hand, increase of the annealing temperature to 1000 \n0\nC \n(NFpH6_1000) \nfor the material prepared at pH 6 \nshowed a significant change in ac susceptibility \ncharacter\n.\n \nThe \nχ\n´(T)\n \ncurves (Fig. \n7\n(a)) sh\nowed a prominent shoulder at about 45 K \n(T\nm\n). \nT\nhe \nshoulder \nshifts to higher \ntemperature \non increasing the frequency from 37 Hz to 9037 Hz with \na \nnoticeably divergence of \nχ\n´(T)\n \ncurves (decreasing magnitude) at lower temperatures. \nIn contrast, \nt\nh\ne \nχ\n´´(T)\n \ncurves (Fig. \n7\n(b)) showed a sharp peak at the inflection point (T\nf\n) of the \nχ\n´(T)\n \ncurves \nbelow the respective T\nm\n. The divergence of \nχ\n´´(T)\n \ncurves \ndecreased \nat higher temperature and \nshowed a minor increasing trend with temperature, unlike a steady \nincrease in \nχ\n´(T)\n \ncurves. The \nfrequency dependence of T\nm\n, as estimated from the shoulder of \nχ\n´(T)\n, followed equation (7\n) with \nsubstitution of T\nf\n \nby T\nm\n \n(Fig. \n7\n(c))\n,\n \nand \nprovided the fit parameters \nf\n0\n \n\n2.8x10\n6\n \nHz, T\n0\n \n\n \n0 K and \nE\na \n\n \n40 meV.\n \nThese fit paramete\nrs are close to the values (\nf\n0\n \n\n2.7x10\n7\n \nHz, T\n0\n \n= 0 K and E\na \n\n \n38 \nmeV) obtained \nby \nfit\nting the \nT\nf\n(\nf\n) \ndata \nfrom \nχ\n´´(T)\n \npeaks (Fig. \n7\n(c)).\n \nThe observed \nf\n0\n \nand E\na \nfrom \nthe\n \nlow temperature shoulder of \nχ\n´(T)\n \ncurves or peaks of \nχ\n´´(T)\n \ncurves in NFpH6_1000 sample \nis \nnoticeably high in comparison to the values (\nf\n0\n \n\n10\n5\n \nHz, E\na \n\n \n6.5 meV) reported for spin glass \nlike transition at about 35 K in \nFe\n3\nO\n4\n \nnanoparticles\n \n[35\n]. \nThe peak temperature shift per decade \nof frequency change (\n∆\n\u0000\n\u0000\n\u0000\n\u0000\n∆\n\u0000\u0000\u0000\n \n(\n\u0000\n)\n) is found \n\n \n0.058, 0.087, and 0.293 for the samples NFpH6_500, \nNFpH8_500, and NFpH6_1000, respectively.\n \nIt \nis \ndiscussed in literature [\n6, 23, 27\n, \n36\n-\n3\n7\n] that \nthe typical value of  \n∆\n\u0000\n\u0000\n\u0000\n\u0000\n∆\n\u0000\u0000\u0000\n \n(\n\u0000\n)\n \nis \nexpected to be \nvery small (\n\n \n0.001) for classical \nspin\n-\ngl\nass\n,\n \nextremely large (\n\n \n0.1)\n \nfor superparamagnet\n,\n \nand intermediate for \ninsulating spin\n-\nglass (EuSr)S\n \n(\n\n0.06\n) and cluster spin glass (\n\n0.0\n32\n-\n0.044)\n \nsystems\n. The \nf\n0\n \nvalues are expected in the range \n10\n12\n \nHz\n-\n10\n14\n \nHz \nwith non\n-\nzero T\n0\n \nfor a typical spin glass. \nO\nn the other hand,\n \nf\n0\n \ncan be \nexpected 14\n \n \nin the range 10\n6\n \nHz\n-\n10\n7\n \nHz for cluster spin glass system\ns\n, where spin dynamics is relatively slow \ndue to intra\n-\nspin interactions inside the clusters [\n23\n].\n \nThe observed \nFrom application point of \nview, the magnetic nanopa\nrticles with \nlow temperature \nsuperparamagnetic blocking \nphenomenon\n \nand spin \nrelaxation time \nat \nroom temperature \nof the order of 10\n-\n14\n \ns \nare\n \nessential for biomedical \napplications and fast computing process \n[\n3\n8\n]\n. On the other hand, data storage medium \nand \nhyperthermia \napplications \nneed magnetic \nnanoparticles with \nreasonably \nstable \nmagnetization\n,\n \nlarge \nsquareness\n, and controlled inter\n-\nparticle interaction with \nblocking\n/\n \nfreezing\n \ntemperatu\nre \nwell above \n300 K\n \n[\n25\n]\n.\n \nThe results of the present work can be interesting for \ndesigning the ferrite \nnanomaterials for specific \napplications. \n \n \n4. \nSummary and c\nonclusion\ns\n \nThe present work shows that ferromagnetic properties and random surface spin freezi\nng \nphenomena in Ni rich ferrite (\nNi\n1.5\nFe\n1.5\nO\n4\n) particles can effectively be engineered by adopting a \ncombined technique of varying pH value during chemical reaction and annealing temperature. \nWe have carried out a detailed study for \nthe \nselected samples pr\nepared at pH 6, 8 and 12. \nThe \nrandom surface spin freezing or quantization of \nspin\n-\nwave spectrum \nin nano\nparticles\n \ncan be \nattributed \nas the cause of \nan\n \nabrupt increase \nof \nsaturat\ned \nmagnetization \nbelow\n \n50\n \nK\n; resulted in \na\n \ndeviation from Bloch law \nof magnetic\n \nspin wave theory \nthat obeyed for \nsaturat\ned \nmagnetization \nof the samples at \ntemperature \n\n \n50 K. \nHowever, the temperature exponent (α) \nis \nlargely deviated \nfrom 3/2, a typical value assigned according to mean field theory\n \nfor long ranged ferromagnet. \nIt \nis u\nnderstood that \nmagnetic spin interactions are sufficiently strong (large \nα and small Bloch \nconstant \nB\n) for the \nsamples\n \nwith larger grain size (prepared at pH 6 and 8) and the spin exchange \ninteraction is highly perturbed (small \nα and large \nB\n) for the \nsampl\ne\n \nwith smaller grain size \n(prepared at pH 12).\n \nThis resulted in \na decrease of \nferromagnetic component and \nincrease \nof \nsuperparamagnetic \ncomponent \nin saturated magnetization \nalong with \nlow coercivity, low \nferromagnetic squareness and low magnetic blocking t\nemperature\n \nfor the NFpH12_800 sample\n. \nThe increase of surface spin disorder (superparamagnetic component) in ferromagnetic particles \nof \nthe samples prepared\n \nat higher pH value is \nalso confirmed from the observation of \ndecreas\ned \nk\n \nvalue\n, obtained from fitti\nng of H\nC\n(T) data,\n \nand \nferromagnetic squarenes \nin our samples with the \nincrease of pH during chemical reaction\n.\n \nIn the present system, we have observed an interesting \nchange in the pattern of random spin freezing behavior. Among the three samples used for a\nc \nsusceptibility study, the sample (NFpH6_500) prepared at low pH value showed relatively high 15\n \n \nvalues of magnetization\n,\n \nspin freezing temperature and non\n-\nzero value of interaction parameter \nT\n0\n \nin comparison to the sample (NFpH8_500) prepared at pH 8 with i\nnteraction parameter T\n0\n \n= \n0. The obtained values of \nf\nₒ\n \nsuggest a relatively slow spin dynamics (\ninteracting spin \nfreezing \nphenomena) in NFpH6_500 sample in comparison to fast \n(non\n-\ninteracting) \nspin freezing \ndynamics in NFpH8_500 sample. The increase of ann\nealing temperature of the sample prepared \nat pH 6 to 1000 \n0\nC (NFpH6_1000) further slowed down the spin dynamics and confined to lower \ntemperature \nwith reduced activation energy \nin comparison to the same material annealed at 500 \n0\nC (NFpH6_500). \nThe striking\n \ndifference is that T\nf\n(\nf\n) data in ac susceptibility spectrum of \nNFpH6_500 sample are best fitted by T\n0\n \n= 188 K in comparison with T\n0\n \n= 0 K for NFpH6_1000 \nsample.\n \nThis implies that large amount of inter\n-\nspin/particle interactions exist in the sample with \nlo\nw annealing temperature and thermal annealing of the sample brings a major change in the \nmechanism of spin freezing phenomena. \nIn case of NFpH6_1000 sample, it is understood \nthat \nsmall magnetic \ndomains/grains\n \nare clustered \ndue to increase of annealing temp\nerature \nof the \nmaterial. The large\n-\nsized clusters respond\ned\n \nlike \nnon\n-\ninteracting superparamagnetic particles; \neach of them consisting of \nmulti\n-\ndomain\ns/grains,\n \nin the temperature and field dependence of \nmagnetization and ac susceptibility features, and spin\n \ndynamics becomes relatively slow due to \nstrong intra\n-\nspin interactions inside the clusters in comparison to \ninter\n-\nclusters interactions\n.  \n \nAcknowledgment\n \n \nThe authors thank CIF, Pondicherry University, for dielectric properties measurements. \nRNB thanks to\n \nUGC for supporting research Grant (F.No. 42\n-\n804/2013 (SR)) for the present \nwork.\n \nReferences\n \n[1] V. Šepelák, I. Bergmann, A. Feldhoff, P. Heitjans, F. Krumeich, D. Menzel, F.J. Litterst, S.J. \nCampbell, K.D. Becker, The Journal of Physical Chemistry C 111 (\n2007) 5026.\n \n[2\n] \nR\n.\n \nMalik, S. Annapoorni, S\n. \nLamba, V.R\n.\n \nReddy, A\n.\n \nGupta, P\n.\n \nSharma, \nand \nA\n.\n \nInoue, \nJ. \nMagn. Magn. 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Sinnecker, \na\nn\nd\n \nM.A. Novak, J. Magn. Magn. \nMater. 320 (2008) e327.\n \n[\n38\n] A.\nK. Gupta, \na\nn\nd\n \nM. Gupta, Biomaterials 26 (2005) 3995.\n \n \n \n \nFigure descriptions:\n \nFig. 1\n \n(a\n-\ni) Temperature dependence of MZFC(black) and MFC(red) curves at 100 Oe for the \nsamples prepared at pH 6, 8 and 12 (\nl\na\nt\nt\ni\nc\ne\n \np\na\nr\na\nm\ne\nt\ne\nr\n \na\nn\nd\n \ngrain size inside the bracket). The \nfitting of MZFC curve\n \nfor\n \nsome of the samples are shown \n(e, h) where distribution function f(t\np\n) \nwas obtained by fitting the peak of the temperature derivative of MZFC curve (dMZFC/dT), \nwhich is around the inflection point of the MZFC(T) curve below blocking temperature, T\nB\n.\n 18\n \n \nF\nig. 2 MZFC(T) curves at different magnetic fields (a\n-\nc) are fitted (lines) with a guidance to \npeak shift. The first order derivatives of MZFC(T) curves and distribution curves used for fitting \nof MZFC(T) data are shown for two samples (f, h).\n \nFig. 3 (a\n-\nc) \nVariation of the \nfit\n \nparameters from first order derivative of MZFC curves for\n \ntwo \nsamples with magnetic field (a\n-\nc). The field dependence of blocking temperature (d). \n \nFitted data \nof T\nB\n \n(e) and T\nBm\n \n(f) with the expression T\nB\n \n= \na\n \n-\n \nb\nH\nn\n.\n \nFig. 4 M(H) loops \nat 10 K and 300 K for the samples prepared at pH 6 and annealed at different \ntemperatures (a\n-\nd). Inset of (b) shows the ZFC and FC loops at 10 K. The M(H) loops at \ndifferent measurement temperatures are shown for 3 samples (e\n-\ng). Inset of (f\n-\ng) shows the \nA\nrrot \nplot. The inset of (h) shows the increment of spontaneous magnetization with annealing \ntemperature. The fit of initial M(H) curves using law of approach to saturation of magnetization \nis shown for two samples (h\n-\ni).\n \nFig. 5 The fit of saturated magneti\nzation of the ferromagnetic component of the samples with \nBloch law (a). The inset of (a) plots the temperature depe\nn\ndence of the saturated magnetization \ndue to superparamagnetic component for NFpH12_800 sample. Fit of the temperature \ndependence of H\nC\n \nwith\n \npower law (b). Linear fit (y = a +bx) of the H\nC\n \nand K\neff\n \nat low \nmeasurement temperature (10 K for pH 6 and 5 K for pH 8) with inverse of grain size of the \nsamples with fit parameters in the box. The fitted curves are shown by dotted lines.\n \nFig. 6 Temperat\nure dependence of ac susceptibility data for samples NFpH6_500 (a\n-\nb) and \nNFpH8_500 (c\n-\nd), measured at different frequencies. The frequency shift of the peak position \n(T\nf\n) in \nc\n’’\n(T) data are fitted with Vogel\n-\nFulcher law and shown as inset figures.\n \nFig. 7 T\nemperature depen\nd\nence of ac su\ns\nceptibility (\nc\n’ \nand \nc\n’’\n) data for NFpH6_1000 sample, \nmeasured at different frequencies (a\n-\nb). The frequency shift of shoulder in \nc\n’\n(T) and peak in \nc\n’’\n(T) data are fitted with Arrhenius law (c\n-\nd).\n \n \n \n \n \n 19\n \n \nTable 1\n \nThe f\nerromagneti\nc parameters of \nthe \nsamples \nNFpH6_600, NFpH8_500 and NFpH12_800 \nwere\n \nobtained by analyzing the \nMZFC(T) curves using equation (2) at different applied magnetic field \n(first 3 columns)\n,\n \nand by analyzing \nthe \nM(H) curves at different measurement temperatures\n \n(\nlast \n6 columns)\n \nand using equation (4)\n.\n \n \nH(kOe)\n \nC \n(emu/g)\n \nK\neff\n/H\n \n(emu/g)\n \nSample\n \nT \n(K)\n \nM\nS\n \n(emu/g)\n \nM\nsat\n \n(emu/g)\n \nH\nc\n(Oe)\n \nM\n\u0000\nM\n\u0000\n \nK\neff         \n(emu\n-\nOe/g)\n \n0.10\n \n0.65\n \n527\n \n \nNFpH6_600\n \n10\n \n28.\n6\n \n34.1\n \n532.5\n \n0.\n46\n \n542019\n \n0.25\n \n1.02\n \n8\n12\n \n50\n \n27.1\n \n33.4\n \n505\n \n0.47\n \n4415\n87\n \n0.50\n \n1.53\n \n788\n \n1.00\n \n3.75\n \n9\n17\n \n100\n \n26.76\n \n32.8\n \n360\n \n0.33\n \n405095\n \n1.50\n \n7.90\n \n1195\n \n150\n \n26.4\n \n32.9\n \n283.5\n \n0.23\n \n482310\n \n2.50\n \n11.20\n \n3920\n \n200\n \n25.4\n \n31.8\n \n141.5\n \n0.15\n \n427374\n \n5.00\n \n12.90\n \n4968\n \n250\n \n24.6\n \n30.7\n \n65.6\n \n0.09\n \n435604\n \n10.00\n \n15.14\n \n26790\n \n300\n \n23.9\n \n28.9\n \n21.16\n \n0.02\n \n334114\n \n \n0.\n10\n \n-\n0.34\n \n1615\n \n \nNFpH8_500\n \n5\n \n22.6\n \n30.1\n \n884\n \n0.44\n \n582338\n \n0.\n20\n \n0.56\n \n1186\n \n50\n \n22.38\n \n28.6\n \n631\n \n0.38\n \n503527\n \n0.\n50\n \n1.23\n \n635\n \n100\n \n22.24\n \n28.6\n \n357\n \n0.21\n \n540937\n \n0.\n75\n \n2.37\n \n545\n \n150\n \n21.9\n \n27.9\n \n168\n \n0.11\n \n498838\n \n1\n.\n00\n \n2.54\n \n435\n \n200\n \n20.8\n \n27.8\n \n119\n \n0.15\n \n54\n9206\n \n1\n.\n50\n \n4.58\n \n419\n \n250\n \n20.7\n \n26.4\n \n74.9\n \n0.06\n \n454682\n \n2\n.\n00\n \n5.97\n \n399\n \n300\n \n19.7\n \n25.1\n \n7.08\n \n0.01\n \n23897\n \n3\n.\n00\n \n9.39\n \n57\n0\n \n0.\n1\n0\n \n0.19\n \n224\n6\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nNFpH12_800\n \n \n5\n \n13.2\n \n16.5\n \n1122\n \n0.28\n \n239704\n \n0.\n20\n \n0.45\n \n118\n4\n \n25\n \n9.73\n \n14.8\n \n514\n \n0.25\n \n160406\n \n0.\n50\n \n0.92\n \n53\n4\n \n50\n \n9.59\n \n12.8\n \n160\n \n0.11\n \n101490\n \n0.\n75\n \n1.33\n \n39\n6\n \n75\n \n9.37\n \n12.4\n \n27.5\n \n0.028\n \n101768\n \n1\n.\n00\n \n1.76\n \n33\n2\n \n100\n \n7.74\n \n11.9\n \n6.66\n \n0.008\n \n101142\n \n1\n.\n50\n \n3.66\n \n35\n5\n \n150\n \n6.4\n \n10.6\n \n19.67\n \n0.018\n \n95079\n \n3\n.\n00\n \n5.52\n \n286\n \n200\n \n4.29\n \n9.28\n \n-\n \n-\n \n89789\n \n \n \n \n250\n \n1.86\n \n7.86\n \n \n \n80804\n \n \n \n \n300\n \n \n6.15\n \n \n \n62702\n \n \u0000\u0001\u0000\u0002\u0000\u0000\u0002\u0001\u0000\u0003\u0000\u0000\u0003\u0001\u0000\u0004\u0000\u0000\u0005 \u0006 \u0007 \b\n\t \n \u000b \f \r\n\u000e\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0016\u0017 \u0018 \u0019 \u001a \u001b \u001c \u001d \u001e \u001f  !\n\"#\"$\"\"$#\"%\"\"%#\"&\"\"'()*+*,-./01\n232422432522532622789:;<=>?@ABCDE\nFGHIJKLMNOPPPQRSTUVVTWXYZ[\\]^_`\nabc\nd\ne\nfghijklmnoppqrstuvvwxyz{|}~\n\n\n\n¡\n¢\n£¤¥¦§¨©ª«¬®®¯°±²³´´²µ¶·¸¹º»¼½¾¿ÀÁÂÃÄÅ¿ÆÇÈÉÊÈËÄÌÍËÈÎÈÊÈÏÎÈÏÐÈÑÒÓÔ¾ÕÃÖ×ÄÐØÆÄÏÎÓ¾ÕÃËÈÎÆÐÍËÙÈÚÄÌÂÛÛÜÈÒÑËÌÝÈÚÄÉÊ×ÈÚÊËÈÊÄËÈÎÄÌÞßàáâãäåæçèéãêêëìíÞãîãïíêíîãäåðîãëäñëòíëäñëåíêóíôîãìõíêö÷øóíùëêúùûüýþìÿî\u0000íùúîñúïíúùêóíñãïÞéíñ\u0001\u0002\u0003\u0004\u0005\u0006\u0007\b\t\u0003\n\u0005\u000b\u0007\u0005\u0003\u0002\u0003\f\r\u0004\u000e\u0002\r\u000f\u0010\u000e\r\u0006\b\u0011\u0010\b\u0012\u000e\r\u0006\b\u0011\t\u000e\n\u0013\n\u0014\u0015\u0016\u0017\u0018\u0019\u001a\u0016\u001b\u001c\u001d\u001e\u0019\u001f \u001b\u001a\u001a\u001b\u001c!\u001a\"\u001d#\u001d\u0016$\u0018 \u001a\u001d%#\u001d&\u0016\u001a'&\u001d\u001e\u001d&\u001b(\u0016\u001a\u001b(\u001d\u0018 )*+,-./01234567839:;<=>-=>?@/A.B3C=1>BDE1-C>ABFA>BCADC=1456729:-./01G1EA<GEA-H>BIC1JF1/@C./1;9\nK\nL\nM\nNOPQRSTUVWXYYZ[\\]^_`a_bcdefghijklmnopqrsrttuvwxyz{|}~                   \n¡¢£¤¥¥¦¥§¨©ª«¬®¯°±²³±´µ¶·´±¸¹º\n»¼½¾¿ÀÁÂÃÄÅÅÅÆÇÈÉÊËËÌËÍÎÏÐÑÒÓÔÕ\nÖ×ØÙÚÛÜÝÞßàááâãäåæçåèéêëìíîïðñòóôõöô÷øùú÷ôûüý\u0000 \u0001 \u0002 \u0001 \u0003 \u0004 \u0001 \u0003 \u0005 \u0001\n\u0006\u0007\b\t\n\n \u000b \n \n \f \n \n \r \n \n \u000e \n \n\u000f\u0010\n\u0011 \u0012\u0013 \u0014\n\u0015 \u0016 \u0015 \u0017 \u0015 \u0018 \u0019 \u0015\n\u001a \u001b \u001a \u001c \u001a \u001a \u001c \u001b \u001a \u001d \u001a \u001a \u001d \u001b \u001a \u001e \u001a \u001a\n\u001f\n !\" #\n$%\n&'() * + ,- . / 01 2 3 4 5 67 8 9 :; 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C ; D C E F F ? B? G D F B ? H A ? G I E ? : J ; KL MN O P ? F B ? H A ? G I Q : P E F D R F : P R A > C ? B E G\nS\nTUV WX Y Z [ \\ X ] ^ Y _\n` `ab c d e f e e gh i j f f h d k j f l m g g l h n j o p q e k a r s d c t\nu v w x\nyz { | } ~  ~\n"    },    {        "title": "1404.1573v1.Preparation_and_Characterization_of_Nano_particle_Substituted_Barium_Hexaferrite.pdf",        "content": "Preparation and Characterization  \n of Nano-particle  Substituted  Barium Hexaferrite  \nYomen Atassi∗, Iyad Seyd Darwish  and Mohammad Tally  \nDepartment of applied physics, Higher Institute for Applied Sciences and Technology, \nHIAST, P.O. Box 31983, Damascus, Syria.  \n \nAbstract : High density magnetic recording requires high coercivity \nmagnetic media and small particle size. Barium hexaferrite12 19BaFe O\nhas been considered as a leading candidate material because of its \nchemical stability, fairly large crystal anisotropy and suitable \nmagnetic characteristics. In this work, we present the preparation of the hexagonal ferrite \n12 19BaFe O  and o ne of its derivative; the Zn -Sn \nsubstituted hexaferrite by the chemical co -precipitation method. \nThe main advantage of this method on the conventional glass -\nceramic one, resides in providing a small enough particle size for \nmagnetic recording. We demonstra te using the X -ray diffraction \npatterns that the particle size decreases when substituting the \nhexaferrite by the Zn -Sn combination. This may improve the \nmagnetic properties of the hexaferrite as a medium for HD \nmagnetic recording  \n \nKey-words: Nanoparticle,  Barium hexaferrite, Substituted barium \nferrite, Magnetic recording.  \n \n1. Introduction    \nNanosized ferrite materials have attracted great attention in recent \nyears. They exhibit unusual physical and chemical properties significantly \n∗ corresponding author: yomen.atassi@hiast.edu.sy                                                             different from those of the bulk materials due to their extremely small size \nand large specific surface area [1,2]. More precisely, some magnetic properties, such as saturation magnetization and coercitivity, depend strongly on the \nparticle size, morpholog y and microstructure of the materials.  \nOn the other hand, High- coercivity magnetic materials with \nnanoparticle size are required for high density magnetic recording such as high \ndensity magnetic tapes and floppy disks. The hexagonal ferrite \n12 19BaFe O  \nconstitutes a promising candidate material due to its chemical stability, fairly \nlarge crystal anisotropy and suitable magnetic properties. In fact, it has been \ndemonstrated that barium ferrite tapes, which utilize very small particles D=40 -50 nm and of high coercivity \n1750 2060OecH= −  offer superior high-\ndensity recording performance [3].  \nIn this work, we present the preparation of the hexagonal ferrite 12 19BaFe O  and \none of its derivative; the Zn -Sn substituted hexaferrite by the chemical co -\nprecipitation method. The main advantage of this method on the \nconventional glass -ceramic one, resides in providing a small enough particle \nsize for magnetic recording. Althou gh, there are different other techniques \nthat may lead to nanosized particles like ball milling, sol -gel and \nhydrothermal techniques, the coprecipitation technique seems to be the most convenient for the synthesis of nanoparticles because of its simplicity  and \nbetter control over crystallite size. In the coprecipitation technique, common \nhydroxides of barium and iron are precipitated from their salts by adding the \nmetal salts solution to an alkaline medium, and crystallized to barium hexaferrite upon suitable heat treatment. The most common drawback of the coprecipitation technique is formation of agglomerated precipitate. The crystalline powders formed from these precipitates are nanocrystalline but agglomerated, which are difficult to disperse.  \nWe report h ere that using isopropyl alcohol to wash the precipitate may \novercome this problem. On the other hand, we demonstrate, using the X -ray \ndiffraction patterns that the particle size decreases when substituting the hexaferrite by Zn -Sn combination. This may enhance the properties of the \nhexaferrite to meet the requirements of high density magnetic recording.  \n \n2. Experimental Section  \nAll the used reagents were of analytical grade.  \n2.1. Preparation of barium hexaferrite  \nAqueous solutions of barium chloride and iron chloride in appropriate \nvolumetric amounts were used as starting materials in the synthesis of \nphase pure12 19 BaFe O . Precipitation of the desired powder was then achieved \nby the technique of acid- base titration.  \nIt was noticed that although a Ba/Fe molar ratio of 1/12 in aqueous \nsolution should be sufficient for the preparation of pure barium ferrite \naccording to stoichiometry, an excess of barium chloride was necessary because barium hydroxide is slightly soluble  in water (solubility product at \n25 C, 4\n22 (Ba(OH) .8H O) 2.55 10−= ×sK [4]). The molar ratio of Ba/Fe in \naqueous solution was taken as  1/11. The determination of the appropriate molar ratio was illustrated elsewhere [2,3]. \nThe product of the co -precipitation was filtered, washed repeatedly with \ndeionized water to remove the unwanted chloride ions.  The complete \nremoval of chloride ions was confirmed from the pH and conductivity measurements of the deionized water and the wash effluent. After washing, the precipitate was dewatered with isopropyl alcohol to remove the surface adsorbed water molecules, which might lead to the agglomeration of the particles during the drying and further processing. The powder was then \ndried at a temperature of \n110 Cfor 12 hours, and then divided into two \npatches. The first patch was pressed into pellets of 10 mm diameter and a \nthickness of 4 mm for dilatometer studies and the second patch was pressed into pellets (of 25 mm at 200 MPa) and toroids (with internal and \nexternal diameters 10 and 25 mm, respectively, and a thickness of about 8 mm) and sintered at  \n800 Cfor two hours.  \n2.2. Preparation of Zn -Sn  substituted hexaferrite  \nAqueous solutions of barium chloride, iron chloride, zinc chloride and tin chloride (IV) in appropriate volumetric amounts were used as starting materials in the synthesis of phase pure\n11.4 0.3 0.3 19 BaFe Zn Sn O . One should notice \nthat the standard  redox   potentials   of iron  and tin ions are \n3+ 2(Fe /Fe ) 0.770eVE+=and 4+ 2(Sn /Sn ) 0.159 eV+=E . So we use directly tin IV: \n4Sn+instead of 2Sn+in order to avoid the reduction of ferric ions to ferrous \nones by 2Sn+and the formation of 4Sn+.  The procedure of the preparation \nof Zn -Sn substituted hexaferrite is identical to the one described above \nparagraph 2.2.  \n2.3. Densification  \nIn order to determine the suitable sintering temperature,  dilato meter \nstudies were performed using (Setaram, TMA 92) at a heating rate of \n010 C / min in nitrogen.  \n2.4. X -ray diffraction pattern  \nA computer interface X -ray powder diffractometer (Philips) with Cu K \nradiation ( 0.1542nm ) was used to identify the crystalline phase.  \nThe data collection was over the 2 -theta range of 010 to 0100 in steps of\n00.02 / sec . The average crystallite size was related to the pure X -ray line \nbroadening using Scherrer's formula [5]. The Fourier method was applied to the (110) and (107) lines.  \nThe mean hexagonal diameter \nD and thickness t of particles have been \ndetermined from the apparent size (110)Fε  and (107)Fε [6]. In fact, Fεvalue is the ratio of the sample volume to the projection area of this volume  on the \nreflecting plane (hkl). This definition allows to deduce the mean hexagonal \ndiameter D from (110)Fε  [6]: \n2(110)\n3F Dε=  \nThe thickness was determined by the relation [6]:  \n(107) cosF tεϕ=  \nWhere ϕ is the angle between the (001) direction and the perpendicular to \nthe (107) plane, in our case 33ϕ=.   \n20 40 60050100150200\n  \n  \nAngle (2θ )Counts\n \nFig.1: X -ray pattern of barium hexagonal ferrite before heat treatment.  \n3. Results and discussions \n3.1. Barium hexagonal ferrite  \nThe XRD pattern of the precipitate before heat treatment shows that it's \namorphous, Fig. 1.  \nThe formation of barium hexaferrite was confirmed by the XRD pattern \nafter heat treatment and no other phases were apparently detectable, Fig. 2. All peaks matched well with the characteristics reflections of barium \nhexagonal ferrite, as it's indicated clearly in table 1. The XRD pattern presents a noticeable line broadening, indicating the fine -particle nature of \nthe hexagonal ferrite so prepared. The mean hexagonal diameter and \nthickness of particles were \n45.2 3.8nm D= ± and 31.1 1.5nmt= ± . \n(A)d\n thI \n(%) expI (%) hkl th2θ exp2θ \n2.782 100 78.5 107 32.2 32.2 \n2.627 98 100 114 34.1 34.1 \n2.424 60 53.2 203 37.1 37.1 \n2.947 55 45.6 110 30.3 30.3 \n2.900 32 12.7 008 30.8 30.8 \nTable 1: The five strongest diffraction peaks of barium hexagonal ferrite.  \nOn the other hand, the elemental analysis of hexaferrite after heat treatment \nconfirms Ba/Fe molar ratio of 1/12.  \n20 40 6005010015020025020 40 60\n  Counts\nAngle (2θ )\n \nFig.2: X -ray pattern of barium hexagonal ferrite after heat treatment.  The suitable temperature of  sintering was determined using a dilatometer \ncurve, Fig.3. \n \nFig.3: A dilatometer curve of a pellet of the green body.  \n3.2. Zn -Sn substituted barium hexaferrite  \nThe XRD pattern of the precipitate before heat treatment shows that it's \nalso amorphous, Fig. 4.  \n20 40 60050100150200\n  Counts\nAngle (2θ )\n \nFig.4: X -ray pattern of Zn -Sn substituted barium hexaferrite before heat \ntreatment.  \n  500\n0\n-500\n-1000\n-1500\n-2000\n200 400 600 800 1000DISP /µm  \n Temperature /C  The formation of the Zn- Sn barium hexaferrite was confirmed by the XRD \npattern after heat treatment, Fig. 5 . \n All peaks matched well with the characteristics reflections of barium \nhexagonal ferrite, as it's indicated clearly in table 2. There is still a small \nresidual amount of ferr ic oxide at  33. \n2refθ exp2θ \nthI (%) expI (%) hkl \n34.2 34 100 100 114 \n32.1 32.1 79.3 84 107 \n56.1 56.2 69 48 2011 \n54.5 55 62.1 52 217 \n62.4 63 48.3 52 220 \nTable 2: The five strongest diffraction peaks  of substituted barium hexferrite.  \nSubstituting hexagonal ferrite by the combination Zn- Sn results in peaks \nbroadening and intensities decreasing. This indicates a decrease in particle \nsize. The mean hexagonal diameter and thickness of particles were \n31.0 1.7nmD= ± and 15 0.7 nmt= ± . We notice a decrease in crystallite size when \ndoping the hexaferrite with Zn- Sn, and this is suitable for high density \nmagnetic recording.   \n20 40 60050100150200\n050100150200\n  Counts\nAngle (2θ)\n \nFig.5: X-ray pattern of Zn -Sn substituted barium hexaferrite after heat \ntreatment.  On the other hand, as the radii of the 2+Zn(0.74 A) and 4+Sn(0.71A) ions are \nlarger than that of 3+Fe(0.64 A), then the lattice parameters a and c are \nexpected to increase when substituting the hexagonal ferrite by the Zn -Sn \ncombination.  \nThis result is co nfirmed experimentally, as it's clearly indicated in table 3. \n/ca (A)\nc (A)\na  \n3.93 23.194 5.8994 Hexaferrite \n \n3.90 23.240 5.9571 Zn-Sn \nhexaferrite \nTable 3: Lattice parameters of  barium hexaferrite and its Zn -Sn substitution.   \n4. Summary  \nBarium hexaferrite and its Zn -Sn derivative have been successfully \nsynthesized by chemical coprecipitation method. Crystallite sizes, measured \nfrom X -ray patterns, indicate that it's more suitable for high density \nmagnetic recording applications to use the substituted hexaferrite.  \nAcknowledgement:  \nThe authors would like to thank Dr. A hmad Alfarra for the fruitful  \ndiscussions.  \nReferences  \n [1]. A. Animesh, C. Upadhyay and H.C. Verma,  Physics Letters A, \n311(2003)410.  \n [2]. D -H. Chen and Y -Y. Chen, Journal of Colloid and Interface Science, \n235(2001)9.  \n [3]. H.C. Fang, Z. Yang, C.K. Ong, Y. Li and C.S. Wang,  Journal of \nMagnetism and Magnetic  Materials, 187(1998)129.   [4]. D. R. Lide, Handbook of Chemistry and Physics, 71ST Edition, CRC \npress, 1991.  \n [5]. Z. Yang, H.-  X. Zeng, D. - H. Han, J. - Z liu and S. -L. Geng, Journal of \nmagnetism and magnetic materials,  115(1992)77.   \n[6]. M. Pernet, X. Obradors, M. Vallet, T. Hernandez, and P. Germi, IEEE \ntransactions on magnetics, No. 2, 24(1988)1898.  "    },    {        "title": "1312.3407v1.Physical_and_Structural_Design_of_Fast_Extraction_Kickers_for_CSNS_RCS.pdf",        "content": "Submitted to “Chinese Physics   C ” \n \nPhysical and Structural De sign of Fast Extraction \nKickers for CSNS/RCS \n \n WANG Lei( 王磊 )1 KANG Wen( 康文 )1 HAO Yao-Dou( 郝耀斗 )1 CHEN Yuan( 陈沅 )1                           \nHUO Li-Hua( 霍丽华 )1 \n1. Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  \n \nAbstract：China Spallation Neutron Source (CSNS) is a high intensity beam  facility being built now in China. Three kicker assemblies, \neight pulsed magnets, will be used in the CSNS rapid circle sy nchrotron (RCS). The physical and structural designs of eight kic ker magnets \nthat are grouped in 5 different types are presented. The results of OPERA-3D simulation show that magnet center field integral meet the \nphysics requirements of design by choosing a suitable magnet co il structure. Field uniformity for 60% width is ±0.7%. The ferri te magnet \nstructure and composition is introduced, and the high voltage feedthrough design, the installation of six magnets in long vacuu m cavity design \nis discussed. \n \nKey words:  China Spallation Neutron Source,  rapid circle synchrotron, extraction and feedthrough. \n \nPACS: 29.20.dk \n \n1 Introduction \n \nThe CSNS/RCS is composed of Linac, RCS and target \nstation. The components layout about extraction of the RCS which distribute in one of accumulator ring’s straight sections \nis shown in the Fig.1. It is a total of eight kicker magnet \nmodules to kick the circulating beam vertically such an angle into the downstream extraction septum (Lambertson magnet). Then the septum will deflect the beam horizontally an angle of \n15 degrees and make the beam leave the RCS completely [1]. \n \nFig.1. The components layout of the extraction. \nThe extraction kickers will work on 25 Hz rate with a flat \ntop of 600 ns and a rise time of 265 ns. So the ferrite that has high frequency response and little loss is used as the core material. The eight kicker magnets are installed inside three \nvacuum tanks that connect with vacuum chamber, but not put \nthe vacuum chamber into kicker magnet. Because the extraction kickers are high frequency AC magnets, so if place vacuum chamber into the magnet, that must use ceramic \nvacuum chamber structure, which will bring a series of \nproblems such as the bigger magnet aperture, a much higher \nvoltage, more expensive cost and so on. The high voltage \nfeedthrough with three electrode structure is designed to \nconnect kicker magnet with high voltage power supply reliably. \n \n2 Kicker specification  \n \nConsidering beta-function and acceptance about the beam, \nthe 8 kicker magnets have various apertures, effective lengths and relative heights to the beam. To simplify the kicker design, \nthe 8 kicker magnets are grouped in 5 different types[2]. The \nparameters of the kicker assembly with a typical magnet K1 are listed in Table 1.  Each magnet needs a separate power \nsupply [3]. \nTable 1. The required specifications of the kickers \nMax. field 558 G \nEffective Length of Magnet 0.52 m \nPeak Current 6660 A \nPeak Voltage 45 kV \nMagnet width 143 mm \nMagnet gap 155 mm \nTurns per coil 1 \nRise Time of Field <265 ns \nTop Time of Field >600 ns \nField uniformity(60% width) ±1% \nRepetition rate 25 Hz \n \n___________________________  \n1) E-mail: wanglei@ihep.ac.cn                                                                                                                  \n \n The magnet inductance is suggested to be measured \n                    hl NLc/02                             (1) \nWhere 0 is the initial permeability,  is the magnet \nwidth, h is the magnet gap and cl is the effective length of \nmagnet. \n \n3  Design of the kicker magnet \n \nThe fast kicker magnet is designed as rectangle frame \nmagnet with a ferrite core [4]. The cross section and 3 dimensional structure model of kicker magnet are shown in \nFig.2. \n \nFig.2. The structure of kicker magnet \nThe Ni-Zn type ferrite is used for this magnet ，which was \nsuccessfully developed after a lot of tests. This type ferrite \nprovides little outgassing rate, high frequency response and \nhigh resistivity, which are important for fast pulsed magnet in \nhigh vacuum environment and actual application. The inner \nsurfaces will be coated with TiN films to reduce secondary \nelectron yield. Along the direction of beam, ferrites assembly \ncompose kicker magnets with different effective length and \naperture. But the process for sintering a C- shaped ferrite block is very difficult, because the block size is too large. \nMoreover, the block thickness has a limit value, because thick \nferrite block  easily generates cracks inside it when sintered. In one experiment after another, we found that the block thickness maximum size of C- shaped ferrite should be smaller \nthan 55 mm. Early drying treatment of the ferrite block is also \nvery important. \nIn order to install the coil, magnet core can be split into \nupper and lower halves. In the middle of the two C- shaped \nferrite block, two 0.5mm copper strips that connect with \nvacuum tanks at last are used to carry the beam-induced image \ncurrent and also to reduce the coupling between the ferrite and \nthe beam. Ceramic plates are added in the magnet above and \nbelow, which are used for fixing on coil and electric insulation. The coil is made of a single–turn copper conductor. Cooling \ncoil is not needed because its power is very small. \nThe simulation results with OPERA-3D [5] about K1 are \nshown in Fig.3. If the core length is set to 460 mm, magnet \nwidth is set to 143 mm, magnet gap is set to 155 mm and \npowered by 6600 A current, the result after simulation shows that effective magnetic length is 520 mm, and the center field of the magnet is 558 G. Maximum field intensity around the \nferrite is 2584 G, that is located in its 4 corners. But the \nsaturation magnetic flux ferrite is 3000 G which has been obtained by measurement and experiment. Field uniformity for 60% width after many times simulation is ±0.7%. So the \nresult of simulation shows that its scheme design is to meet the \nphysical requirements. \n \n \nFig.3. The results of OPERA-3D simulation about K1 \nThe eight kicker magnets are installed inside three vacuum \ntanks that connect with vacuum chamber. The first two magnets are placed each into a vacuum tank, and other magnets are placed in the third vacuum tank that has about 3 \nmeters long (Fig.4). Before insertion, all the magnets are \nplaced on one long plate. According to their final position, each magnet is aligned and fixed on the plate with adjusting device. Then the plate, with the magnet on it, is horizontally \npushed into the tank by the sliding track. The sliding track is composed of a sliding bolt assembly and an adjusting bolt \nassembly (Fig.5). The sliding bolt assembly is used for smooth pushing the plate with the magnet on it. The adjusting bolt assembly which is collimated at first is used to carry and \nadjust the plate. Then at last, the plate is fixed reliably. \n \nFig.4. The 3 meters long vacuum tank  \nFig.5. Construction about the sliding track \n \nFig.6. The plug-in structure \nLead wire connection for feedthrough and the magnet coil \nuses a coaxial plug-in structure without any bolts and nuts. The inner conductor is connected with elastic contact part that \nfixed on the magnet coil, the outer conductor is connected \nwith hollow metal braid embedded in another part that fixed \non the magnet coil. This structure is not only very compact but \nalso easy to install and disassemble (Fig.6).  \nThe kicker magnet is powered by a pulsed power supply. \nThe 45 kV pulsed voltage coming from the power supply will \nbe transmitted into the magnet that is located in the vacuum \ntank by the feedthrough. The feedthrough was designed with a tri-polar structure, which isolate the ground of the pulsed \npower supply from the vacuum tank. Instead of the traditional \nuse of precious metals such as molybdenum and titanium, the inner and outer conductors in the feedthrough are made of oxygen-free copper that cost very little and are convenient for \nmachining. Fixing and insulating material between them is \nceramics. However, the expansion coefficient of copper and \nceramic is not same, so it is difficult to solder them together. \nOxygen free copper thickness should be as thin as possible \nwithout losing its rigidity. The ceramics that be welded with \ncopper conductor adopt a symmetrical structure that is two \nceramic pieces with a thin copper in the middle. That greatly \nimprove the success rate of the welding. So after giving full consideration to this point in design, it has been successfully \nwelded one. The developed high-voltage feedthrough is shown \nin Fig.7. In addition, the inductance is very small, only 0.15 microhenry. Inside and outside conductor of feedthrough can withstand 25kV after many high-voltage experiments. \n \nFig.7. The developed high-voltage feedthrough \nSix magnets fixed on a long plate need to be lifted and \npushed into the third vacuum tank. Because the plate has three meters long, a suitable lifting point has a great relationship to the overall deformation of the plate. Through simulation \nanalysis, the maximum deflection is 0.2mm at the optimal two \npoints for lifting the long plate (Fig.8). The optimal two points is located at 750mm and 2250mm along the plate. However, if the plate is lifted from both ends, then the maximum \ndeflection will be increased to 2.6mm, that will be dangerous \nfor all the magnets.   \n \nFig.8. Stress analysis about the long plate \n \n4 Vacuum quality criteria \n \nThe required vacuum quality for extraction kicker is in the \n1×10-6Pa range. But large quantities of ferrite and also ceramic \nplates are used in the tank. A total of eight ion pumps are installed outside of three assemblies to overcome the outgassing from these materials. The vacuum tanks and \nflanges are made of 304L stainless steel. And flanges are wire \nseal type with copper gaskets. All blind holes in the tanks are \ndrilled with venting paths to preventing the generation of air \nentrainment. Weldment about the vacuum tank will follow a principle that is an internal continuous welding and an external \nspot welding. Ferrite will be cleaned and baked to 120°C \nslowly before installation. After sealing, all the assemblies can be baked to 180°C in the ring. \n \nVacuum tank \nLong plate \nAdjusting bolt assembly Sliding bolt assembly  5 Conclusion \n \nExtraction kicker magnets are very important in CSNS/RCS. \nIts performance is directly related to the smooth operation of \nthe accelerator. Eight kicker magnets with 5 different types are \nused to realize the fast extraction out of RCS. Physical and structural designs of eight kicker magnets that located in three \nvacuum tanks are presented and analyzed. The sliding track \ncomposed of a sliding bolt assembly and an adjusting bolt \nassembly is introduced that is used for smooth pushing in the \nplate with the magnet on it. Eight kicker magnets are produced now, that will be completed in the near future. There is always work to be done. \n \nReferences \n \n1 C. Pai, et al, Proceedings of the 2003 Particle Accelerator \nConference: 2147-2149. \n2 Y.L.Chi, Visit Report From ISIS Internal notes, CSNS, \nSept.15, 2005. \n3 W.zhang, J.Sandberg et al, Proceedings of EPAC \n2004:1810-1812.  \n4 W. Kang et al, Transactions on Applied Superconductivity, \nVol.20, No.3, June 2010.  \n5 Vector Fields Inc. \n \n \n "    },    {        "title": "1302.2838v1.Anomalous_Surface_Segregation_Profiles_in_Ferritic_FeCr_Stainless_Steel.pdf",        "content": "Anomalous Surface Segregation Pro\fles in Ferritic FeCr Stainless Steel\nMaximilien Levesque\u0003\n\u0013Ecole Normale Sup\u0013 erieure, D\u0013 epartement de Chimie,\nUMR 8640 CNRS-ENS-UPMC, 24 rue Lhomond, 75005 Paris, France and\nUniversit\u0013 e Pierre et Marie Curie, CNRS UMR 7195 PECSA, 75005, Paris, France\nThe iron-chromium alloy and its derivatives are widely used for their remarkable resistance to cor-\nrosion, which only occurs in a narrow concentration range around 9 to 13 atomic percent chromium.\nAlthough known to be due to chromium enrichment of a few atoms thick layer at the surfaces,\nthe understanding of its complex atomistic origin has been a remaining challenge. We report an\ninvestigation of the thermodynamics of such surfaces at the atomic scale by means of Monte Carlo\nsimulations. We use a Hamiltonian which provides a parameterization of previous ab initio results\nand successfully describes the alloy's unusual thermodynamics. We report a strong enrichment in\nCr of the surfaces for low bulk concentrations, with a narrow optimum around 12 atomic percent\nchromium, beyond which the surface composition decreases drastically. This behavior is explained\nby a synergy between (i) the complex phase separation in the bulk alloy, (ii) local phase transitions\nthat tune the layers closest to the surface to an iron-rich state and inhibit the bulk phase separation\nin this region, and (iii) its compensation by a strong and non-linear enrichment in Cr of the next few\nlayers. Implications with respect to the design of prospective nanomaterials are brie\ry discussed.\nPACS numbers: 64.75.Nx, 68.35.bd, 61.66.Dk, 64.70.kd, 81.30.Bx\nThe iron-chromium alloy and its derivatives are in-\nexpensive, have satisfactory mechanical properties and\nabove all exhibit a remarkable resistance to corrosion: it\nis the most widely used class of alloy in the world. Its out-\nstanding corrosion resistance is known for a century1to\nonly occur in a narrow range of concentrations, around 10\natomic percent of chromium (at. % Cr)2. Their excellent\nproperties make them candidate materials for future fu-\nsion nuclear reactors3,4, one of the reasons that induced a\nconsiderable amount of work on the various aspects of the\nFe{Cr alloy both experimentally5,6and theoretically7.\nCorrosion resistance of stainless steels is due to the pas-\nsivation of the material by an inert, chromium rich layer\nat the interface between the alloy and the environment,\ni.e.at the surfaces. Passivation is a phenomena inher-\nent to how much Cr is located at the surfaces, which\nis a non-linear function of the bulk concentration8. In\naustenitic Fe{Cr, which only exists at high temperatures\nabove\u0019800 C and for less than \u001910 at. % Cr, the more\nchromium in the bulk, the more chromium in the surface\nand thus the more stainless the alloy. In ferritic Fe{Cr\nalloys, the picture is more complex. Without additive\nelements, the Cr content at which the alloy is passivated\nis narrow, from 9 to 13 at. % Cr, beyond which occurs\nan increase in the corrosion rate and a strong decrease in\nmechanical properties.\nThis important property of stainless steels has been\nextensively studied, but its complex origin at the atomic\nscale has remained a missing understanding, subject to\ncontroversial \fndings: How chromium causes passiva-\ntion, i.e. how it interacts and reacts with chemical el-\nements coming from the environment like dioxygen or\nhydrogen9, is out of the scope of this study. The reader\nis refered to Greeley et al.10for a review of surface chem-\nistry of metal surfaces at the atomic and electronic scale.\nSurface reaction requires nevertheless that Cr is presentin large enough quantity on the surface to form a few\natoms thick protective layer, e.g.of chromium(III) oxide\nCr2O3. How chromium atoms enrich the surfaces remains\nunclear. Venus and Heinrich11shew by angle-resolved\nAuger electron spectroscopy that Cr atoms deposited on\na whisker of Fe (100) migrates from the surface to the\n\frst few layers, in contradiction with the expected ten-\ndency. This surface-alloying has been clearly identi\fed to\nbe linked to anomalies in the magnetic properties of the\nCr/Fe system, speci\fcally the change in surface magneti-\nzation at low Cr coverage and the strong interactions be-\ntween surface Cr atoms12,13. Ropo et al. showed by First\nPrinciples that a pure Fe{Cr surface behaves like stain-\nless steels with respect to Cr enrichment14. They also put\nin evidence a competition between the relative stabilities\nof the surfaces and the complex thermodynamics of the\nbulk alloy. Later, ab initio calculations revealed that un-\nexpected interactions between subsurface Cr atoms and\nsurface Fe atoms15are at the origin of an anomalous16,17\nsegregation behavior of Cr in Fe in the dilute regime.\nAt temperatures of industrial and technological inter-\nest,i.e.between 300 and 600 K, the body-centered cu-\nbic (bcc) solid-solution of Fe{Cr shows a miscibility gap\nfrom 9\u000013 to 94\u000099 at. % Cr6. Inside, a phase-\nseparation occurs into an iron-rich bcc solid solution, \u000b,\nand chromium-rich bcc precipitates, \u000b0, as one would ex-\npect for a binary alloy that seemed to have a segrega-\ntion tendency, i.e.that mix solely for entropic reasons.\nHowever, both theoretical and experimental studies sub-\nverted this simple picture, showing favorable dissolution\nenergy of chromium in iron up to an anomalously high\n\u00197 at. % Cr18{20due to a competition between repulsive\nCr-Cr interactions and attractive Fe-Cr interactions21{24.\nThe increased chromium content leads to more frustrated\nmagnetic interactions in between Cr atoms that make\nthe dissolution exothermic at low concentrations, thenarXiv:1302.2838v1  [cond-mat.mtrl-sci]  12 Feb 20132\nendothermic. Several theoretical models7,25{29have suc-\ncessfully reproduced the sign change of the mixing energy.\nHere, the iron-chromium ferritic stainless steel is mod-\neled by the Hamiltonian proposed recently in Ref.29. It is\nspeci\fcally designed to reproduce both (i) the whole ex-\nperimental \u000b{\u000b0phase diagram at all temperatures and\ncompositions, and (ii) the change of sign of the mixing en-\nergies. It is also compatible with large-scaled simulations,\nbecause of its conceptual simplicity and its rigid bcc lat-\ntice nature: The internal energy \u0001 Hmix=\u0000\ncb(1\u0000cb)\nis a function of the bulk concentration in chromium, cb,\nand of the order energy \n described in terms of local-\nconcentration and temperature dependent pair interac-\ntions:\n\n =X\niz(i)\n2\u0010\n\u000f(i)\nAA+\u000f(i)\nBB\u00002\u000f(i)\nAB\u0011\n; (1)\nwherez(i)is the coordination number of shell iand\u000f(i)\njj0is\nthe pair interaction between atoms of type jandj0onith\nneighbor sites. In\ruenced by the strategy of Caro et al.25,\nthe order energy is advantageously expressed as a sim-\nple concentration and temperature dependent Redlich-\nKister30expansion:\n\n (x;T) = (x\u0000\u000b)\u0000\n\fx2+\rx+\u000e\u0001\u0012\n1\u0000T\n\u0012\u0013\n; (2)\nwherexis the local concentration and Tthe temperature.\nDiscrete mixing energies have been calculated ab initio\nin the whole range of concentrations and interpolated by\nEq. 2, whose coe\u000ecients \u000b; \f; \r; \u000e are given in Table I.\nCoe\u000ecient \u0012, also given in Table I, is the critical temper-\nature of the miscibility gap.\nHomo-atomic pair interaction energies in Eq. 1 are\ngiven by the experimental cohesive energy of the pure\nelements, given in Tab. I, according to Ecoh(j) =\n\u0000P\niz(i)\u000f(i)\njj. The expressions of the hetero-atomic inter-\nactions\u000f(i)\nAB(x;T) are then easily deduced from Eq. 1 and\n2. They are consequently simple parametric functions of\nthe temperature Tand local concentration x. This last\nquantity,x, around a pair including an atom on site i\nand an atom on another site jis naturally de\fned as\nx=Pr\nn=0Pz(n)\nk=1p(n)\nik+Pr\nn=0Pz(n)\nk=1p(n)\njk\n2Pr\nn=0z(n); (3)\nwherep(n)\nik= 1 when the kth site of the nth coordina-\ntion shell of site iis a Cr atom, and 0 if it is a Fe atom\nor an empty site, i.e. a site outside the surface. The\ninteraction range is restricted to second nearest neigh-\nbors with\u000f(2)\nij=\u000f(1)\nij=2, which has been found optimal.\nIt is worth emphasizing that the resulting bulk phase di-\nagram is in very good agreement with the most recent\nexperimental reviews5,6: While this model does not cap-\nture the extraordinarily complex electronic structure of\nthe bcc Fe{Cr alloys, it captures both the local nature\u000b\f(eV)\r(eV)\u000e(eV)\u0012(K)Ecoh.(Fe)Ecoh.(Cr)\n0.070\u00002:288 4.439 \u00002:480 1400 4.28 4.10\nTABLE I. Parameters of the local-concentration and temper-\nature dependent pair potential from Ref.29. Experimental\ncohesive energies from Ref.31are given in eV per atom.\nof the interactions and the associated energetics, without\nempirical parameters.\nThis letter focuses on the most stable surface of bcc\niron, which has the orientation (100)32. It is modeled\nby a stack of 100 layers of 400 atoms each, in periodic\nboundary conditions. Interactions between periodic im-\nages in directionh100iare prevented by a slab of vacuum.\nSpecial attention has been given to the choice of the size\nof the system in order (i) not to arti\fcially hide the sur-\nface e\u000bects by a too large volume/surface ratio, and (ii)\nnot to restrain the formation of precipitates by too small\nsystems, ie to give the system the ability to precipitate\nand have the precipitates to interact with the surfaces.\nIt induces that the bulk solubility limit near the surfaces\ncan be slightly di\u000berent from that of a pure bulk system.\nA perspective view of the supercell is shown in Fig. 1.a.\nAs stated above, the model does not capture the e\u000bect\nof the surface on the electronic structure of atoms in its\nvicinity, such as expected stronger bonds. It captures the\ne\u000bect of the reduced coordination in terms of energetics:\nEq. 3 implies that the more reduced the coordination, the\nmore the local energetics are dependent on the remain-\ning bonds. The various surface orientations only di\u000ber in\nthe number of surface-induced dangling bonds, and thus\nin the strength of the surface e\u000bects described below.\nConclusions are thus transferable to other orientations.\nImportantly, the e\u000bectiveness of our Hamiltonian allows\nto deal with a number of atoms that make it possible to\n\fnely tune the bulk and layer concentrations. Here, one\natom accounts for the bulk concentration by less than\n10\u00004at. % and the layer concentration by 5 \u000110\u00003at. %.\nThis point is crucial as recent ab initio calculations have\nbeen limited to few layers and few atoms per layer, impos-\ning large bulk and even larger planar concentrations. Our\nHamiltonian is sampled by Monte Carlo simulations us-\ning the Metropolis algorithm in the canonical and pseudo\ngrand-canonical ensembles33. The equilibrium state is\nconsidered reached after 104accepted permutations per\nsite.\nWe de\fne the planar concentration cp=PNp\ni=1qi=Np\nas the chromium content of each layer pparallel to the\nsurface, with NPthe number of atoms per layer (400\nhere) andqi= 1 if sitei\u001apcontains a Cr atom, qi= 0\notherwise.pranges from 0 for the top surface layer to 99\nfor the bottom surface layer.\nIn \fgure 2, we plot the concentration pro\fle cp(p) of\nFeCr at 300 K (\u00191=3Tc) and various bulk concentra-\ntionscbranging from 0.02 to 0.98. Special attention is\ngiven to the temperature range that is of industrial and\ntechnological importance, i.e.forcbbelow 0.3. The con-3\nFIG. 1. Snapshots of the simulation cell containing the (100)\nFeCr surface at 300 K for bulk concentrations cb= 0:5 (left)\nandcb= 0:15 (right). The top surface layer is at the top of\nthe \fgure. Fe atoms are shown in red and Cr atoms in blue.\nForcb= 0:15, only Cr atoms are shown. In both cases, the\nbulk phase separation \u000b\u0000\u000b0occurs.\n0 10 20 30 40 50p00.20.40.60.81cp\n0 1 2 3 4 5p00.050.10.150.2\ncpb) a)\nFIG. 2. a) and b) Concentration pro\fles of a FeCr surface at\n300 K. Index p= 0 indicates the top surface layer. Various\nbulk concentrations cbare indicated: 0.02 (black); 0.05 (red);\n0.10 (green); 0.15 (blue); 0.30 (yellow); 0.50 (brown); 0.90\n(gray); 0.98 (violet); b) Only the \frst six layers are shown.\nNote the change in scale.\ncentration pro\fles are highly non-linear functions of the\nbulk concentration.\nIn order to get insight to these concentration pro\fles,\nsemi-grand canonical Monte Carlo simulations have been\nperformed, where the total number of sites and the dif-\nference in chemical potential \u0001 \u0016between pure bcc-iron\nand pure bcc-chromium are kept \fxed, while the bulk\nconcentration is free33. The evolution of the surface and\nbulk concentrations with respect to \u0001 \u0016at 300 K are plot-\nted in Fig. 3. Three hysteresis loops are found, which are\nindicated in the \fgure by asterisks. They bring out three\nphase transitions. The \frst and stronger one, indicated\nby the black asterisk in Fig. 3.a., is an evidence of the\nwell-known bulk phase-separation \u000b\u0000\u000b0happening in\n-2 0 2 4\n∆µ (eV)00.20.40.60.81cb\n-1 0 1 2 3 4\n∆µ (eV)00.511.52\nc0+c1b) a)\n**** *FIG. 3. a) Evolution of the bulk concentration with the chem-\nical potential \u0001 \u0016at 300 K. b) Evolution of the sum of the\nconcentrations of the top surface and subsurface layers, ver-\nsus the same chemical potential. In both \fgures, the black\nasterisk indicates the bulk phase separation, the red and blue\nasterisks show the transitions in the subsurface and surface\nlayers, respectively.\nbcc Fe-Cr alloys and discussed in the introduction. Note\nthat the bulk solubility limit at low chromium concentra-\ntion is slightly a\u000bected by the presence of the surface. It\ncauses the large variations in the density pro\fles in Fig. 2\nmore than 10 layers away from the surface for cb'0:12.\nSnapshots of systems that undergo phase-separation at\nthese concentrations are shown in Fig. 1. At higher dif-\nference in chemical potential, as indicated in Fig. 3.b.,\ntwo less visible phase transitions occur. Each transition\nis localized in a single layer. The \frst one corresponds to\nthe subsurface layer transiting from pure Fe to pure Cr\n(indicated by the red asterisk in Fig. 3.a., followed by an\naccompanying transition in the surface layer (blue aster-\nisk in Fig. 3a. and b.). The change in concentration of\nthe two \frst layers is abrupt and discontinuous. It also\ngives insight to the emptiness in Cr of these layers. First,\nthe di\u000berence in surface chemical potential of the two el-\nements, which is proportional to the di\u000berence in surface\nenergies of Fe and Cr, implies that Fe recover the layers\nwhere bonds are dangling. Indeed, surface energies of Fe\nare always lower than that of Cr for a given orientation.\nThey range from 2.2 to 3.4 J/m2and 3.2 to 4.2 J/m2for\niron and chromium, respectively34{36. Secondly, and in\nrelation, the chemical potential of surface atoms is much\nmodi\fed by the surface, which explains why the alloy do\nnot phase separate at the same concentration than in the\nbulk. One could see them as two new alloying elements\nonly present in the surfaces.\nAs shown in Fig. 2.b and 4.a., the concentration of the\nthird layer, c2, increases quickly with cb, contrary to that\nof the \frst two layers discussed above: c2gets from 0 to\n0.2 whencbgoes from 0 to 0.1, which represents a relative\nincrease inc2of about 100 % at cb\u00190:1. Importantly, c2\nis at this point greater than the bulk solubility limit, so\nthat phase separation would occur in the absence of the4\n0 0.1 0.2cb-0.500.51(c2−cb)/cb\n0 0.2 0.4cb00.020.040.060.080.1\ncsa) b)\nFIG. 4. a) Relative evolution of the concentration of the third\nlayer,c2, versus the bulk concentration, cb, at 300 K. b) Av-\nerage surface concentration, \u0016 cs= (c0+c1+c2+c3)=4, as a\nfunction of cbat 300 K (black squares), 450 K (red triangles)\nand 600 K (green stars). A black shaded line indicates the\noptimal concentration copt\nb\u00190:12 at which ferritic FeCr steel\nis the most corrosion resistant.\nsurface: its presence changes here the very nature of the\nFe and Cr atoms, particularly their complex magnetic in-\nteractions as shown experimentally12and theoretically15,\nand consequently the perturbated alloy's thermodynam-\nics. For higher bulk concentrations, c2decreases sharply\nand becomes even depleted in Cr with respect to the bulk\natcb'0:13.\nFinally, and importantly, we plot in Fig. 4.b the sur-\nface concentration, \u0016 cs, de\fned as the average concentra-\ntion of all layers in direct contact with the surface, i.e.\n\u0016cs=Pimax\ni=1ci=imaxwithimax= 4 for orientation (100), as\na function of cbat 300 K, 450 K and 600 K. Two regimes\nare clearly identi\fed: (i) For bulk concentrations under\n0.12, the surface concentration increases with cb, up to\na narrow maximum between 0.09 and 0.12. As soon as\ncb= 0:07, surface composition exceeds the bulk solubil-\nity limit, where phase-separation would occur in the bulk\nalloy. (ii) For larger bulk compositions, there is a discon-\ntinuity incs, which is reduced to a \rat regime cs\u00190:05,\nequivalent to that of a \u000b\u0000\u000b0phase-separated bulk. In-\ndeed,\u000b0precipitation occurs as expected and discussed\nabove to explain the large variations in the density pro-\fles of Fig. 2, and illustrated in Fig. 1.\nBetween 300 K and 600 K, the pro\fles shown in Fig. 4\nare almost temperature-independent. It re\rects a sub-\ntle compensation between the temperature dependence\nof the order energy, more speci\fcally of the energy asso-\nciated with magnetism, and the entropic e\u000bects. It was\nidenti\fed in the bulk as the cause of the anomalously\nsteep solubility limit of the Fe{Cr alloy at low temper-\natures, and identi\fed by Williams as an e\u000bect of mag-\nnetism29,37.\nAn atomistic explanation of the thermodynamic origin\nof the narrow optimum in corrosion resistance of stain-\nless steels emerges from the above results. The di\u000berence\nin surface chemical potential between Fe and Cr induces\na strong Cr depletion in the \frst layers, where atoms\nhave dangling bonds. This result can be understood as\na surface e\u000bect resulting from the surface energies of Fe\nbeing always lower than that of Cr. The sub-surface lay-\ners balance this local depletion by a strong enrichment\nin Cr, leading to three distinct regimes: (i) at low bulk\nconcentrations, the ordering energy drives chromium into\nsolution, far away from the surface: the resistance to cor-\nrosion is low and increases with Cr concentration. (ii) in\na narrow range of bulk concentrations between 0.07 and\n0.12, enough Cr is present to strongly enrich the surface\nin average but not to exceed the \u000b-\u000b0solubility limit38:\nthe Cr content in the surface is maximum. (iii) at higher\nbulk concentrations, the bulk solubility limit is exceeded\nand most of available chromium bulk-precipitates in the\n\u000b0phase, depleting the surfaces: stainlessness is lost.\nIn light of the above results, in order to improve the\nstainlessness of the Fe{Cr system and its derivative al-\nloys, one may consider adding alloying elements that in-\ncrease the solubility of Cr in Fe, without altering the\nanomalous thermodynamics of the Fe{Cr surfaces. The\nsharp decrease in the surface concentration would thus\nhappen, nevertheless, but at higher bulk concentrations.\nLarger surface concentrations would be reached, induc-\ning better protection against corrosion. One could also\nincrease the quantity of chromium atoms at the surfaces\nwithout increasing the content in the bulk. This could for\ninstance be done by the localized addition of chromium-\nrich nanoscale precipitates or dispersoids. 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Leseigneur, Acta Mater. 60, 7150 (2012)."    },    {        "title": "1910.06831v1.3D_Field_Simulation_Model_for_Bond_Wire_On_Chip_Inductors_Validated_by_Measurements.pdf",        "content": "© 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media,\nincluding reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to\nservers or lists, or reuse of any copyrighted component of this work in other works.3D Field Simulation Model for Bond Wire On-Chip\nInductors Validated by Measurements\nKatrin Hirmer and Klaus Hofmann\nIntegrated Electronic Systems Lab\nTechnische Universit ¨at Darmstadt, Germany\nMerckstr. 25, 64283 Darmstadt\nfKatrin.Hirmer,Klaus.Hofmann g@ies.tu-darmstadt.deThorben Casper and Sebastian Sch ¨ops\nComputational Electromagnetics\nTechnische Universit ¨at Darmstadt, Germany\nDolivostr. 15, 64293 Darmstadt\nfcasper,schoepsg@temf.tu-darmstadt.de\nAbstract —This paper proposes 3D ﬁeld simulation models\nfor different designs of integrated bond wire on-chip inductors.\nTo validate the simulation models, prototypes for three designs\nwith air and ferrite cores are manufactured and measured.\nFor air core inductors, high agreement between simulation and\nmeasurement is obtained. For ferrite core inductors, accurate\nmodels require an exact characterization of the ferrite material.\nThese models enable the prediction of magnetic ﬁeld inﬂuences\non underlying integrated circuits.\nIndex Terms —Bond wire inductor, electromagnetic simulation,\nmodeling.\nI. I NTRODUCTION\nInductors are critical components in most power electronic\nand radio-frequency (RF) circuits. However, their integration\nwith the goal of high inductances on small areas remains\nan ongoing research topic [1]. A common approach for RF\nintegrated circuits (ICs) is to realize inductances by 2D in-\ntegrated planar spiral inductors [2]. These inductors are easy\nto implement and compatible for every process. However, this\napproach requires a high area consumption [3]. Moreover, the\nmagnetic ﬂux is perpendicular to its substrate causing possible\ninterferences with the circuit within the IC.\nFor power applications, high inductances and high quality-\nfactors (Q-factors) are required. Thus, the trend to miniatur-\nization leads to an increasing research on power supply on\nchip (PwrSoC) with focus on inductors and transformers [1],\n[4], [5]. To obtain the required high inductances for power\napplications, discrete semiconductors are more promising than\n2D spiral integrated inductors [4]. In particular, the fabrication\nof on-chip inductors and transformers using bond wires around\na magnetic core is a promising approach [6]–[8]. Since the\nbonding process is a common subsequent process step after\nIC fabrication, this approach results in low additional costs\nand a low production time [9]. In addition, these components\nare less area consuming yielding a high power density and\nefﬁciency.\nTo reduce the number of prototypes, IC designers rely on\nsimulation tools that require reliable inductor and transformer\nmodels. This work proposes bond wire on-chip inductors to\nbe designed with the help of 3D ﬁeld simulation. To test\ndifferent geometric and material parameters without having\nto fabricate additional prototypes, simulation models were\n(a)\n (b)\nFig. 1: (a) Prototype and (b) simulation view of a closed core\nmagnetic inductor (model C).\nset up and validated using measurements of corresponding\nprototypes, such as the one in Fig. 1.\nThe paper is structured as follows. First, Section II in-\ntroduces the simulation model to yield the resistance and\ninductance of an inductor. Then, Section III presents the\nprototypes and the measurement setup. Section IV discusses\nthe results focusing on the comparison between measurement\nand simulation and Section IV concludes the paper.\nII. S IMULATION MODEL\nTo avoid the costly fabrication of prototypes, simulation\nmodels were set up to predict the devices’ behavior. In\nthis paper, the major quantities of interest are the resistance\nand the inductance of the on-chip inductor. This allows to\nconsider appropriate stationary simpliﬁcations of Maxwell’s\nequations [10]. First, to compute the DC resistance, one needs\nto solve the electrokinetic problem\nr\u0001(\u001b(x)r'(x)) = 0; (1)\nwith magnetic boundary conditions and, e.g., '= 0 and\n'=Vat the contacts. Here, 'is the electric scalar po-\ntential and \u001bis the electric conductivity. Then, the current\nI=\u0000R\nA\u001br'dxcan be computed, where Ais the cross-\nsectional area of a bond wire. Therefore, the DC resistance\nyieldsR=V=I. Secondly, the inductance can be calculated\nfrom the magnetostatic ﬁeld that is obtained from\nr\u0002\u0010\n\u0017(x)r\u0002~A(x)\u0011\n=\u001f(x)I; (2)arXiv:1910.06831v1  [physics.app-ph]  15 Oct 2019Table I: Fabrication parameters for the different solenoid\ninductor models. Here, Nis the number of windings and A\nis the size of the footprint.\nModelN A inmm2Core\nModel A 9 3.8 Air\nModel B 9 3.8 Bar ferrite\nModel C 18 16.8 Closed ferrite\nwhere\u0017is the reluctivity, ~Ais the magnetic vector potential,\nand\u001fis a winding function that distributes the current I\naccordingly. From ~A, the magnetic ﬁeld ~Hand the magnetic\nﬂux density ~Bare obtained by ~B=\u0017\u00001~H=r\u0002~A. Then,\nthe magnetic energy Wis calculated by\nW=1\n2Z\n\n~B(x)\u0001~H(x) dx;\nwhere \nis the volume in which the energy shall be evaluated,\nand the inductance reads L= 2W=I2. Therefore, computing\nthe resistance and the inductance of an on-chip inductor boils\ndown to solving (1) and (2). For this purpose, CST EM\nSTUDIO is used.\nIII. P ROTOTYPES AND MEASUREMENT SETUP\nThree different bond wire inductor models and one bond\nwire transformer model were fabricated on a printed cir-\ncuit board (PCB). Model A is an air core solenoid in-\nductor with N= 9 windings and a total footprint of\nA= 2:0 mm\u00021:9 mm , see Fig. 2a. Model B differs from\nmodel A only in an additional ferromagnetic core material as\nshown in Fig. 2b. Model C consists of a closed ferromagnetic\ncore with two bond wire inductors in series that are placed\non opposite legs of the ferromagnetic core. Each of the two\ninductors consists of N= 9windings and model C has a total\nfootprint ofA= 4:2 mm\u00023:0 mm as shown in Fig. 1. Table I\nsummarizes the different parameters for the three models.\nFinally, the transformer model was realized by omitting the\nwires that connects the two serial inductors in model C.\nThe solenoid inductors and the transformer were fabricated\nusing gold wires with a diameter of 25µmon a 1:55 mm\nPCB with 35µmthick and 100µmwide copper traces and\nan electroless nickel electroless palladium immersion gold\n(ENEPIG) surface plating. For models B and C, the magnetic\ncore consists of a ferrite foil which was trimmed using\na laser cutter. Its initial relative permeability is given by\n\u0016r;i= 150\u000620% at13:56 MHz [11].\nThe impedance of the inductors were measured by an Agi-\nlent E5062A network analyzer which is capable of measuring\nS-parameters up to 3 GHz . Input and output reﬂections are\nreduced by a matching network built around the inductor\n(device under test) ZX, see Fig. 3. To avoid reﬂections, resistor\nvalues ofR1= 62 \n; R2=R3= 470 \n , andR4= 58 \nwere chosen such that an input and output impedance of\n50 \n was obtained. To compensate parasitic inﬂuences of the\nPCB and the cables, a differential measurement method was\n(a)\n(b)\nFig. 2: Prototype and simulation model of (a) an air core\ninductor (model A) and (b) a ferrite core inductor (model B).\nused. Therefore, a matching circuit with a reference resistor\nRref= 5 \n is connected in parallel to the test channel.\nRelating the measured forward voltage gain S21at the test\nchannel to the one at the reference channel denoted by S31,\nthe impedance ZXis determined by\nZX=RrefS21\nS31: (3)\nFor frequencies far below the resonance frequency of the\ncoil, the inductance can be calculated by\nLX(f) =Im(ZX)\n2\u0019f: (4)\nThe Q-factor is given by [12]\nQ(f) =Im(ZX)\nRe(ZX): (5)\nThe DC resistances RDCof the prototypes were measured with\na 4-wire sensing Keysight 34465A multimeter.\nIV. R ESULTS AND DISCUSSION\nIn this section, the results of the measurements and simu-\nlations are presented. First, the complex impedance ZXhas\nbeen measured at an input power of 0 dBm . It is shown\nfor the three different models with respect to frequency in\nFig. 4. As expected, the frequency dependent losses of the\ncore material [11] result in an increase of Re( ZX(f)) at higher\nfrequencies for models B and C. On the other hand, only\na slight increase with frequency is observed for the coreless\nmodel A. The inductances and Q-factors were extracted from\nS-parameter measurements and are shown with respect to\nfrequency in Fig. 5 and Fig. 5b, respectively. The ferrite core\nof models B and C leads to a signiﬁcant increase in inductance,R1R2\nZXR3\nR4\nport 2port 1R2\nRrefR3\nR4\nport 3Fig. 3: Test setup for differential measuring of the solenoid\ninductors.\n0 50 1000204060\nFrequency fin MHzIm(ZX) in ΩA\nB\nC\n(a)\n0 50 100050100\nFrequency fin MHzRe(ZX) in ΩA\nB\nC (b)\nFig. 4: Measured (a) imaginary and (b) real impedances of\nmodels A, B, and C.\nsee Fig. 5. However, due to the losses in the core, only a small\nincrease of the Q-factor is observed, see Fig. 5b.\nFor the simulations, CST EM STUDIO was used to solve\n(1) and (2) and to compute the resistance and the inductance of\nthe three different models. In the simulation of all considered\nmodels, electric boundary conditions were applied at a distance\nof6 mm from the model itself and a current of 1 A was\napplied as the excitation. To take into account the uncertainty\nin the core’s material parameters [11], a parameter sweep for\nmodel B was carried out to compute the inductance for the\nnominal and for the corner values of the core’s thickness. In\nFig. 6, the results for model B are shown with respect to the\nrelative permeability. Fixing the core’s thickness to a value\nof120µm, a numerical optimization was applied to adjust the\nsimulation model to the measurement results for model B. The\noptimization yielded a permeability of \u0016r= 209:83resulting\nin a simulated inductance of 147:42 nH with a relative error\nof0:11 % with respect to the measured value. With these\nvalues for the thickness and the permeability, model C and\nthe transformer have been evaluated. To allow an assessment\nof the parasitic inﬂuences of the inductors, Fig. 7 shows the\nﬁeld plots for the magnetic ﬂux density ~Bfor models A and B.\nThe results from measurement and simulation are sum-\nmarized in Table II. The relative errors between measured\n1001011020100200300\nFrequency fin MHzInductance Lmin nHA\nB\nC(a)\n1001011020246\nFrequency fin MHzQ-factorA\nB\nC\n(b)\nFig. 5: (a) Inductances Lmand (b) Q-factor extracted from\nthe measurements of models A, B, and C with respect to\nfrequency.\n100 200 300 400100120140160\nRelative permeability µrInductance Lsin nH\ntFe=80µm\ntFe=100µm\ntFe=120µm\nFig. 6: Simulated inductance Lsof model B with respect to\nthe core’s thickness tFeand its relative permeability \u0016r.\n(a)\n (b)\nFig. 7: Magnetic ﬂux density of (a) model A and (b) model B.Table II: Measured (index m) and simulated (index s) results for the solenoid inductors. The measured inductance has been\nextracted at 10 MHz .\nModelLminnHLsinnH\u000f(L)in%Rmin\nRsin\n\u000f(R)in%QMaxfQMaxinMHz\nModel A 23.74 23.77 0.13 1.08 0.89 17.59 4.63 62.11\nModel B 147.26 147.42 0.11 1.12 0.89 20.54 6.23 11.27\nModel C 347.27 1155.72 232.80 2.30 1.83 20.44 5.18 10.27\n0 1 2 3 4 5−0.500.5\nTime tin µsVoltage Vin V\nIn\nOut\nFig. 8: Measured input and output voltage of a 1:1 transformer\nwith respect to time at an input power of \u000020 dBm .\nand simulated resistances are all below 25 % . The remaining\nerrors are assumed to stem mainly from the soldering joints\nthat are not modeled in the simulation. While the relative\nerrors of the inductance of models A and B are below 1 %,\nthe simulated value for model C is much higher than the\nmeasured value. One possible reason for this deviation stems\nfrom non-idealities such as edges resulting from the laser\ncutting process. However, the major modeling error is assumed\nto result from a weak coupling within the sintered magnetic\ncore due to a uniaxial anisotropy of the ferromagnetic foil, as\nalso observed for sputtered ferrite cores in [13].\nDespite the assumed weak coupling, a 1:1 transformer was\nsuccessfully built. The measured input and output voltages for\nthis transformer are shown in Fig. 8 giving a coupling factor\nof approximately 0:10at1 MHz , whereas the simulated value\nfor ideal isotropy is about 0:34.\nPrototypes for on-chip inductors and transformers have been\nfabricated on printed circuit boards for cost-effective and\nreliable simulation models which enable system simulation for\nIC designers. Measurements of the air core inductor matched\nwell with the simulation results. For inductances with closed\nmagnetic core, the anisotropy of the material is assumed to be\nof great signiﬁcance. This results in a very weak coupling\nof two inductors when placed on two opposite sides of a\nclosed core. Nevertheless, the coupling is sufﬁcient to allow\nthe implementation of a transformer. Future work includes\nto investigate the error in the inductance of model C and\nthe inﬂuence of the magnetic ﬁelds on any underlying ICs.\nUltimately, design rules for the usage of bond wire inductors\ncan be derived from the ﬁeld results.ACKNOWLEDGMENT\nThe authors thank Victoria Heinz, Isabel Kargar, David\nRiehl, and Timo Oster for their passionate work setting up\nthe prototypes and simulation models. This work is supported\nby the Excellence Initiative of the German Federal and State\nGovernments and the Graduate School CE at Technische\nUniversit ¨at Darmstadt.\nREFERENCES\n[1] D. Dinulovic et al . High inductance thin-ﬁlm transformer for\nhigh switching frequency. In 2018 IEEE Applied Power Elec-\ntronics Conference and Exposition (APEC) , pages 560–564, 2018.\ndoi:10.1109/APEC.2018.8341067.\n[2] S. S. Mohan et al . Simple accurate expressions for planar spiral\ninductances. IEEE Journal of Solid-State Circuits , 34(10):1419–1424,\n1999. doi:10.1109/4.792620.\n[3] J.-M. Yook et al . High-quality solenoid inductor using dielectric ﬁlm\nfor multichip modules. IEEE Transactions on Microwave Theory and\nTechniques , 53(6):2230–2234, 2005.\n[4] C. ´O. Math ´unaet al. Review of integrated magnetics for power supply on\nchip (PwrSoC). IEEE Transactions on Power Electronics , 27(11):4799–\n4816, 2012. doi:10.1109/TPEL.2012.2198891.\n[5] D. Dinulovic et al . On-chip high performance magnetics for point-\nof-load high-frequency dc-dc converters. In 2016 IEEE Applied Power\nElectronics Conference and Exposition (APEC) , pages 3097–3100, 2016.\ndoi:10.1109/APEC.2016.7468306.\n[6] C. R. Sullivan et al . Integrating magnetics for on-chip power: A\nperspective. IEEE Transactions on Power Electronics , 28(9):4342–4353,\n2013. doi:10.1109/TPEL.2013.2240465.\n[7] A. Camarda et al . Design and optimization techniques of over-chip\nbond-wire microtransformers with ltcc core. IEEE Journal of Emerging\nand Selected Topics in Power Electronics , 6(2):592–603, 2018.\n[8] E. Macrelli et al. Modeling, design, and fabrication of high-inductance\nbond wire microtransformers with toroidal ferrite core. IEEE Transac-\ntions on Power Electronics , 30(10):5724–5737, 2015.\n[9] Z. J. Shen et al. On-chip bondwire inductor with ferrite-epoxy coating:\nA cost-effective approach to realize power systems on chip. In 2007\nIEEE Power Electronics Specialists Conference , pages 1599–1604, 2007.\ndoi:10.1109/PESC.2007.4342235.\n[10] J. D. Jackson. Classical Electrodynamics Third Edition . Wiley, 1998.\n[11] Laird. MULL5040-200 Datasheet . (2019-09-05).\n[12] T. Brander. Trilogie der induktiven Bauelemente . W¨urth Elektronik, 4\nedition, 2008.\n[13] J. M. Wright et al. Analysis of integrated solenoid inductor with closed\nmagnetic core. IEEE Transactions on Magnetics , 46(6):2387–2390,\n2010. doi:10.1109/TMAG.2010.2044982."    },    {        "title": "2101.04883v2.An_empirical_approach_to_measuring_interface_energies_in_mixed_phase_bismuth_ferrite.pdf",        "content": " 1 An empirical approach to measuring interface energies in mixed -phase bismuth ferrite  \nStuart R. Burns*,†,#, Oliver Paull†, Ralph Bulanadi†,⊥, Christie Lau†, Daniel Sando*,†, J. Marty \nGregg‡, and Nagarajan Valanoor† \n†School of Materials Science and Engineering, UNSW Sydney, NSW 2052, Australia  \n‡Centre for Nanostructured Media, School of Mathematics and Physics, Queen’s University \nBelfast, University Road, Belfast BT7 1NN, UK  \n#Present address: Department of Chemistry, University of Calgary, 2500 University Drive NW, \nCalgary AB T2N 1N4, Canada  \n⊥Present address: Department of Quantum Matter Physics, University of Geneva, 24 Quai \nErnest -Ansermet, CH -1211, Geneva 4, Switzerland  \n*stuart.burns@ucalgary.ca; daniel.sando@unsw.edu.au  \n \nKEYW ORDS  \nbismuth ferrite, thin films, strain engineering, thickness effects, energetics  \n \nABSTRACT  \nIn complex oxide heteroepitaxy, strain engineering is a powerful tool to obtain phases in thin films \nthat may be otherwise unstable in bulk. A successful example of this approach is mixed phase \nbismuth ferrite (BiFeO 3) epitaxial thin films. The coexistence of a tetragonal -like (T -like) matrix \nand rhombohedral -like (R -like) striations provides an enhanced electromechanical response, along \nwith ot her attractive functional behaviors. In this paper, we compare the energetics associated with \ntwo thickness dependent strain relaxation mechanisms in this system: domain walls arising from \nmonoclinic distortion in the T -like phase, and the interphase bound ary between the host T -like \nmatrix and tilted R -like phases. Combining x -ray diffraction measurements with scanning probe \nmicroscopy, we extract quantitative values using an empirical energy balance approach. The \ndomain wall and phase boundary energies are  found to be  113 ± 21 and 426 ± 23 mJ.m-2, \nrespectively. These numerical estimates will help us realize designer phase boundaries in \nmultiferroics, which possess colossal responses to external stimuli, attractive for a diverse range \nof functional applicati ons.  2  \nINTRODUCTION  \nBismuth ferrite (BiFeO 3; BFO) is a multiferroic perovskite oxide which forms in the R3c space \ngroup, extensively studied for its room temperature ferroelectric, electromechanical , and \nantiferromagnetic properties  [1,2] . Although BFO was first synthesized  in epitaxial thin film form \nin 2003  (Ref.  [3]), a dramatic surge in interest – particularly reg arding the effects of epitaxial \nstrain – occurred in 2009 upon the discovery of a giant axial ratio phase of BFO (“T -like” or “T’ \nBFO”, with c/a = 1.23), induced by large (> 4%) compressive misfit strains  [4]. Under \nintermediary strain states, the material can also be grown with a mixed p hase microstructure, i.e. \npartially forming bulk-like tilted phases (R -like; R’) within a matrix comprising the T -like phase \nof BFO  (Ref.  [5]). This “mixed phase BFO” is commonly observed when the film is grown on \nlanthanum aluminate (LaAlO 3; LAO) substrates, to thick nesses above ~25 nm. In mixed phase \nBFO, electrical switching transitions between both structural (T’ and R’) and polar variants (+T’ \nand –T’) [6], doping  [7], enhanced response from electromechanical switching  [8–13], pinning  \nmechanisms  [14], optical effects  [15,16] , ferroelectric tu nnel junctions  [17], and deterministic \ncontrol of phases  [18,19]  have been studied extensively. These films routinely display an increas e \nin mixed phase population upon increasing film thickness, driven by strain relaxation. \nInterestingly, in some cases where defects are incorporated throughout the film volume, the \nformation of the mixed phase can be completely inhibited  [20–22]. This complex equilibrium \nprovides a wide scope of stimuli which can locally tailor the functionality of the system.  \nTo accurately describe these stimuli and the associated energy landscape, we require a numerical \nhandle of the physical parameters governing the strain relaxation process. Strain relief in mixed \nphase BFO is  manifested through the formation of new interfaces, either domain walls (DWs) or \ninterphase boundaries  (Fig. 1) [23]. However, an explicit value of the energy density of either of \nthese remains unknown. Having a knowledge of these energy values would help guide studies of \nnew phase space in BFO and similar epitaxial thin film multiferroics.  \n  3  \nFigure 1.  Schematic of various types of strain -relaxation mechanisms in strongly epitaxially -strained BFO \nfilms grown on L AO substrates. (a,b) only T’/T’ twin DWs, which  form  along {11̅𝑎̅} planes  (where a is \nrelated to the elastic compliances)  [24], are observed for films with thickness below about 25 nm . (c,d) \nT’/T’ twin DWs as well as  T’/R’ mixed phase boundaries are observed for films with thickness above about \n25 nm. The arrows denote the loc al direction of ferroelectric polarization.  \n \nHere, we analyze the energy landscape associated with mixed phase BFO as a function of film \nthickness within the framework of an energy balance approach and a modification of the general \nKittel’s law  [25]. Piezor esponse force microscopy (PFM) and atomic force microscopy (AFM) \nwere used to calculate the periodicities of monoclinic domain walls within the T -like matrix \n(referred to as T’/T’ twins) and the boundaries between coexistent structural phases (T’/R’ phase \nboundaries), respectively. To obtain these periodicities, a substantial  amount of data was collected \nand processed to ensure significant statistical weigh. X -ray diffraction (XRD) techniques were \nused to quantitatively demonstrate the thickness -dependent s train. Following thermodynamic \nanalyses based on elastic potential s, we deduce  values of the  energ y associated with each of these \nstrain relief mechanisms.  \na)\nb)c)\nd)d 25 nm d 25 nm 4 METHODS  \nThin film fabrication  \nBiFeO 3 thin films were deposited on LaAlO 3 (001) substrates by pulsed laser deposition (PLD) , \nusing conditions reported previously  [26]. A ceramic target of Bi 1.1FeO 3 was ablated at a repetition \nrate of 10 Hz using a KrF excimer laser (waveleng th of 248 nm) in an oxygen partial pressure of \n 0.1 Torr. The laser fluence and film growth rate were measured to be ~1.8 J cm-2 and 0.04 Å \npulse-1, respectively. The substrate was held at a temperature of 590 °C at a distance of  5 cm \nfrom the target. After a predetermined number of pulses dependent on film thickness, each sample \nwas cooled at a rate of 20 °C min-1 in an oxygen pressure of 5 Torr.  \n \nX-ray diffraction and structural characterization  \nStructural characterization by x -ray diffraction, including high -angle θ-2θ scans and low -angle \nreflectometry (not shown), was carried out using K α-1 radiation ( 𝜆=1.5406  Å) on a Philips \nMaterials Research Diffractometer (MRD), equipped with a two -bounce ( 220) Ge \nmonochromator. To determine the phase composition (R’ and T’ phases) and the in - and out -of-\nplane lattice parameters, reciprocal space maps (RSMs) were collected on the same  diffractometer  \nusing a 1D detector (PIXcel). The lattice parameters were c alculated by two dimensional Gaussian \npeak -fitting the film peaks in the RSMs at each thickness.  The uncertainty values for the lattice \nparameters were calculated from the uncertainty in the position of the peak centers of the 2D \nGaussian fits, combined wi th the statistical uncertainty when calculating the lattice parameter from \nthe various film spots from the two separate in -plane RSM s (namely 013 and 103).  \n \nAtomic force microscopy and piezoresponse force microscopy  \nTopography imaging and ferroic domain ma pping were  carried out on an Oxford Instruments \nAsylum Cypher in contact AFM and PFM modes. Due to the  fact that T’ BFO is monoclinic M C \n(Ref. [23]), with an in -plane component of polarization , the PFM data  were collected with the \nlateral displacement channel. Vertical displacement  (not shown) provided no contrast  in the \ndomain images in the regions of T’/T’ walls. The AFM and PFM images were analyzed with \nWSxM software  [27] through both line profiling and image flooding to calcula te the periodicities \nof T’/T’ and T’/R’ interfaces.  \n  5 RESULTS AND DISCUSSION  \nWe study here a series of BFO films grown on LAO (001).  This substrate  induces  a misfit strain \nof approximately -4.5%. As previously reported, the film grows in a pure T’ phase at the growth \ntemperature (590 °C), but during post -growth cooling to room temperature, the material relaxes to \na mixture of tilted tetragonal -like and rhombohedral -like phases , of which the volume phase \nfractions are thickness dependent  [28–30]. The films  had thicknesses of 15 -120 nm , as measured \nby x-ray reflectometry (XRR) for thinner films (thickness d < 50 nm) (not shown), and  the growth \nrate extrapolated to estimate the th icknesses for samples with d > 50 nm. This sample set \ndemonstrated the typically observed pure T -like phase in thinner films, while a mixed phase \nmicrostructure was present a t higher  thicknesses.  \nTo establish a framework which can account for the two afor ementioned strain relieving \nmechanisms, we must  first calculate the strain energies and then periodicities for the T’/T’ twins \nand the mixed phase interfaces. Thus, we begin with detailed XRD investigations to address the \nstrain energies.  \nFigure 2a, b)  present the results of  XRD reciprocal  space maps (RSMs) of the 120 nm -thick \nsample , around the 001 ( symmetric) and 103 (asymmetric) reflections  (the full maps for all film \nthicknesses are provided in Figs. S1 -S4) [31]. The phases are labelled according to the following \nconvention: T’ is the tetragonal -like phase that forms the matrix of the film (the smooth , flat \nregions in Fig. 1a), T’ tilt (R’ tilt) is the tilted T -like (secondary strained R -like) phase which is \nobserved within the mixed -phase regions as striations on the film surface, and R’ relaxed  is the almost \nfully relaxed R -like phase (not observe d in the AFM images). For the remainder of this paper, for \nsimplicity we use T’/T’ to denote the domain walls separating the T -like ferroelastic domains, and \nT’/R’ to denote the interphase boundary between the T’ tilt and R’ tilt phases.  \n  6  \nFigure 2.  a) Symm etric x -ray diffraction RSM for the 120 nm -thick film near the (001) reflection of LAO \nand BFO. b) Asymmetric RSM near the (103) reflection for the 120 nm -thick film, analyzed to calculate \nin-plane lattice parameters. The three -fold splitting of the T’ peak is a signature of  a monoclinic M C crystal \nstructure. The dashed lines in both RSMs indicate the theoretically calculated Q x value of the corresponding \nLAO reflection. c), d), e), f) , g), h), i)  a, average in -plane,  b, and c lattice parameters , tetrago nality, a/b, and \nmonoclinic angle β, of the T -like phase as a function of thickness, respectively. Note the position of the \nLAO pseudo -cubic lattice parameter (3.791 Å) with respect to the average in -plane parameters. Three \nthickness regimes are apparent, as discussed in the main text, and are shaded purple (~15 to ~40 nm), yellow \n(~40 to ~80 nm) and pink (~80 to ~120 nm).   \n \n In the XRD RSMs ( Fig. 2a, b) several  film diffraction peaks are observed, which correspond \nto the various phases related to the striations on  the sample surface. We first consider the T’ phase. \nAs mentioned above, it is well established that T -like BFO typically forms as a M C monoclinic \nstructure  [29,32] . A signature of such crystallograp hic symmetry is the three -fold splitting of the \nT-like peak for the asymmetric (103) reflection , as labelled in Fig. 2b). Combining symmetric and \n-0.01 0.00 0.010.160.170.180.190.200.21Qz (r.l.u.)\nQx (r.l.u.)\n-0.24 -0.20 -0.160.480.520.560.60Qz (r.l.u.)\nQx (r.l.u.)R’\nT’T’tiltR’relaxed\nLAOR’\nT’LAO001\n103\nT’tilt\n0 40 80 120\n3.793.813.833.85Thickness (nm)aTBFO (Å)\n3.733.753.773.79bTBFO (Å)\n0 400 800 12004.624.644.664.68cTBFO (Å)\nThickness (Å)3.763.783.803.82ave in plane (Å)\nLAO0 40 80 120\n1.221.231.24Thickness (nm)tetragonality\n1.001.011.021.03aTBFO/bTBFO\n0 400 800 1200888990b (deg)\nThickness (Å)a)\nb)c)\nd)\ne)\nf)g)\nh)\ni) 7 asymmetric RSMs measured on the full thickness series of films allows us to calculate we \ncalculated  the in-plane and out -of-plane lattice parameters of the T -like host matrix.  These  \nparameters are shown in Fig. 2c), e), f), as well as the average in -plane lattice parameter  𝑎+𝑏\n2 (Fig. \n2d), the tetragonality  2𝑐\n𝑎+𝑏 (Fig. 2g ), the distortion 𝑎/𝑏 (Fig. 2 h) and the monoclinic angle  β (Fig. \n2i).  \nAbove 10 -15 nm, t he thickne ss-dependent behavior of the unit cell in the T -like phase can be \nseparated into three regimes:  \n1) ~15 to ~40 nm  (shaded in purple ): The a and b lattice parameters for films in this \nthickness range , i.e., prior to nucleation of mixed phase, clearly  deviate  from the pseudo -\ncubic lattice parameter of the LAO  substrate  (3.791 Å); however, the average in -plane \nparameter ( Fig. 2d) is still consistent with that of the  substrate.  This phenomenon is \nconsidered to be a strain relaxation mechanism: instead of creating misfit dislocations  to \nrelieve strain , the film forms T’/T’ domain walls. In this thickness range, the distortion \nincre ases with thickness  (Fig. 2h ). The out -of-plane lattice parameter, c, remains \nvirtually unchanged  (Fig. 2f ). \n2) ~40 to ~80 nm (shaded in yellow ). For thickness values at which one would expect the \nmixed phase striations  to form, the c lattice parameter increa ses, and the tetragonality of \nthe unit cell  also increases  (Fig. 2g). The a and b parameter both decrease gradually . \n3) ~80 to ~120 nm (shaded in pink ). When a significant volume of mixed phase striations \nis formed in the thickest films, the a and c lattice parameters remain constant while the \nb parameter continues to decrease  slightly. This is seemingly caused by the striations \ncompressing the T -like un it cell. One could suggest that the freedom along the b axis in \nthe T -like phase is linked to the preferential nucleation direction of the secondary phase.\n  \nWe also used XRD RSMs to determine the lattice parameters of the relaxed R’ phases  which \nappear wit hin the mixed phase striations , by combining the symmetric and asymmetric scans. This \nwas only possible for films of thickness above 70 nm, since for thinner films, the XRD peaks for \nthe secondary R’ phase  were too weak to obtain reliable parameters.  Generally speaking, for the \nfilms 70 -120 nm -thick, t he lattice parameters of the R’ phase  tend only to  marginally change for \nthe thickest sample. The in -plane parameter slightly increases, while the out -of-plane parameter  8 falls, with both migra ting towards the BFO bulk pseudo -cubic lattice parameter  [33] of 3.965 Å \n(Fig. S5)  [31]. \nHaving characterized the lattice parameters of each film,  next we  estimate d the volume fractions \nof the R’ and T’ phases as a function of the thickness, particularly at the local scale. This thickness \ndependence  of the mixed phase microstructure is more distinct when imaged through AFM. A FM \nimaging was carried out across the sample series in order to establish the ratio of mixed phase for \na given film thickness  (representative data for films 25 -120 nm thick are given in Figs. S6 -\nS9)  [31]. Alongside this, an image processing technique was applied to quantify the density of the \nphase interfaces  – boundaries between the stripe -like structures and their host matrix. Figure S10 \npresents 5  x 5 μm2 topography scans of sample surfaces for three film thicknesses  [31], alongside \nimages highlighting the approximate phase interfaces created with the microscopy ana lysis \nsoftware described in Ref. [27].  \n \nFigure 3.  Average interface density (blue  upward triangles ) and R’ -T’ phase ratio (orange  downward \ntriangles ) as a function of film thickness. The shading is in reference to the thickness regimes discussed in \nthe main text and in Fig. 2. The solid lines are guides to the eye.  \n \nFigure 3 presents  the average R’ phase percentage and interface density calculated and \nnormalized  to the film volume. Note that this is remarkably consistent with the thickness -\n0.00 0.04 0.08 0.12051015\n Interface area/unit volumeInterface area/unit volume  (mm-1)\nThickness  (mm)No Data Selected\n0102030\n Mixed phase ratio\nMixed phase ratio (%)0 40 80 120Thickness (nm)Thickness (nm) 9 dependent thermodynamic treatment of Ouyang and Roytburd  [34,35] . The weighted average \nlattice parameter of the complete microstructure as a function of thickness is plotted in Fig. \nS11 [31]. This was calculated by combining the data collected from XRD and AFM and shows the \nexpected behavior  of strain relaxation in the system – a decrease of t he c parameter and increase \nin average in-plane lattice parameter above a critical thickness  of about 40 nm . \nNext, we describe how the periodicities for both the T’/T’ domain walls and the mixed phase \ninterfaces were calculated. In T -like BFO, ferroelastic  monoclinic domain structure has been \npreviously reported  [36–39]. These periodic walls , arising from the monoclinic distortion of the \nT-like unit cell , partially minimize elastic self -strain  energy  [40]. PFM  was used to image this \ndomain structure, with the amplitude shown in Fig. 4a) and Fig. 4b) for 40 and 80 nm thick films, \nrespectively. In all cases the out -of-plane PFM phase was homogeneous, consistent with the out -\nof-plane polarization direction pointing downwards (not shown). Here , one can see the presence \nof tilted phas e striations and ferroelastic DWs in the host T -like phase. All thicknesses studied here \nexhibited these ferroelastic DWs, implying that they are not as energetically costly as mixed phase \ninterfaces.  \nThese T’/T’ domain walls are aligned along the <110> pc crystallographic directions (as expected \nfor a M C symmetry) and terminate at the boundaries between the T -like regions and the mixed \nphase striations. The mixed phase striations themselves are tilted by up to 15 degrees from the high \nsymmetry [100] or [010 ] directions ( Fig. 4b) [41,42] . \n \nFigure 4. PFM amplitude images (5 x 5 μm2) of; a) 40 nm -thick  and b) 80 nm -thick films, where tilted  \nphase structure and ferroelastic DWs are present.  \n 10 Now that both the strain and periodicity for both types of interfaces have been obtained as a \nfunction of thickness, we proceed with the empirical analysis.  For this, we start with the key \nexpression employed in this work, Kittel’s scaling law for ferroic domain walls  [43], given in the \ngeneral form as  \n 𝑤2= 𝛾\n𝑈𝑑 (1) \nwhere 𝑤 is domain width, 𝛾 is the domain wall energy, 𝑈 is the domain energy per unit volume , \nand 𝑑 is the film thickness. The form of U is dependent on the ferroic order parameter associa ted \nwith the domain. When empirically adopting this expression, two quantitative results must arise \nfrom experiments: 1) an energy term related to the order parameter and, 2) a periodicity of the \ndiscontinuity manifesting in the ferroic to reduce free ener gy. In this case, we exclusively consider \nelastic energy terms.  \n By modifying the general Kittel’s law expression to add a second  strain relieving \nmechanism, we can incorporate the role of phase interfaces in the analysis alongside the \nferroelastic domain walls. As is typical for a Kittel expression, we start from free energy, 𝐹: \n 𝐹=𝐴𝑇(𝑈𝑇/𝑇𝑤𝑇/𝑇+𝛾𝑇/𝑇𝑑\n𝑤𝑇/𝑇)+(1−𝐴𝑇)(𝑈𝑇/𝑅𝑤𝑇/𝑅+𝛾𝑇/𝑅𝑑\n𝑤𝑇/𝑅) (2) \nwhere 𝐴𝑇 and (1−𝐴𝑇) are the percentage fractions of T -like and mixed phase regions, \nrespectively, 𝑈𝑇/𝑇 is the elastic potential energy arising from the T -like monoclinic domain walls \nand 𝑈𝑇/𝑅 is the elastic potential energy arising from the formation of mixed phase regi ons. The \nsymbols 𝑤, 𝛾 and 𝑑 have their usual meanings (given  above), and the subscripts 𝑇/𝑇 and 𝑇/𝑅 \ndenote which strain relieving interface the variables refer to T -like domain walls  or mixed phase \nboundaries, respectively. Taking partial derivative s to describe energetic equilibrium, we arrive at \nthe following relations:  \n 𝜕𝐹\n𝜕𝑤𝑇/𝑇=𝐴𝑇(𝑈𝑇/𝑇−𝛾𝑇/𝑇𝑑\n𝑤𝑇/𝑇2)=0; (3) \n 𝜕𝐹\n𝜕𝑤𝑇/𝑅=(1−𝐴𝑇)(𝑈𝑇/𝑅−𝛾𝑇/𝑅𝑑\n𝑤𝑇/𝑅2)=0. (4) \n  All of our samples exhibit T’/T’  twins, therefore 𝐴𝑇 ≠0. When there is little to no mixed \nphase in the sample, (1−𝐴𝑇)=0, and the equation reduces to the ‘classic’ Kittel’s law, Equation  11 (1). From this free energy expression, one can thus calculate the energy of formation for both the \nT’/T’ twins ( 𝛾𝑇/𝑇) and mixed phase microstructure ( 𝛾𝑇/𝑅). \nReturning to the experiments, we consider  next the PFM and AFM derived periodicities for both \nT’/T’ twins and mixed phase boundaries. Figure 5  displays the periodicities of both strain relieving \nentities as a function of the square root of thickness (with intercept = 0), as a linear fitting of this \nplot would indicate that the scaling mechanism  is following Kittel’s law. As there are distinct \nthickness regions with different behaviors , we isolated the four thinnest film s to calculate the \nperiodicity of T’/T’ twin walls. Using a similar line of reasoning, the four thickest films were \nisolated to calculate the periodicity of the phase boundaries. In this way, we identify two separate \nregimes to which we applied our energy analysis, each following a dense domain wall model. \nRationale for using only these four end points in each case is provided later.  \n \nFigure 5.  Periodicity against square root of film thickness of both T -like twin domain walls (top) and mixed \nphase boundaries (bottom) in the sample series of BFO//LAO (001). The linear fit for each data set indicates \nthese strain relieving entities follow Kittel’s  scaling law (w ∝d1/2).   \nWe denote the linear fit gradients as M𝑇/𝑇 (blue fit, ≈ 4.03 x 10-4) and M𝑇/𝑅 (red fit, ≈ 2.93 x 10-\n4) for the T -like domain walls and mixed phase boundaries, respectively. The gradients are related \nto the energy terms through the following equations, where 2π is a shape factor  [44] typically used \nin Kittel expressions:  \nT’/T’ twins\nT’/R’ phase boundaries 12  M𝑇/𝑇=2𝜋(𝛾𝑇/𝑇\n𝑈𝑇/𝑇)12⁄\n  (5) \n M𝑇/𝑅=2𝜋(𝛾𝑇/𝑅\n𝑈𝑇/𝑅)12⁄\n (6) \nThe elastic energy term 𝑈𝑇/𝑇 was derived from the elastic misfit between the substrate and both \nthe a and b lattice parameters:  \n 𝜀𝑎T=𝑎T−𝑎sub\n𝑎T (7) \n 𝜀𝑏T=𝑏T−𝑎sub\n𝑏T (8) \n 𝑈𝑇/𝑇=𝑌\n1−𝜈(𝜑𝜀𝑎T2+[1−𝜑]𝜀𝑏T2) (9) \nwhere 𝑎T and 𝑏T are the in -plane lattice parameters, 𝑎sub is the substrate lattice parameter of 3.791 \nÅ, 𝑌 is the Young’s modulus (186 GPa, calculated from Ref. [34]), 𝜈 is the Poisson’s ratio (0.35 ; \nRef. [45]), and 𝜑 is the phase fraction of a oriented domains in the region of T -like phase, which, \nif assuming equal T’/T’ domain populations, we take 𝜑=12⁄.  \nSimilarly, the energy 𝑈𝑇/𝑅 was calculated by consideration of the average in -plane (IP) elastic \nmisfit strains in the region of mixed phase striations:  \n 𝜀IPT=IPT−𝑎sub\nIPT  (10) \n 𝜀IPR=IPR−𝑎sub\nIPR (11) \n 𝑈𝑇/𝑅=𝑌\n1−𝜈(𝜑𝜀IPT2+[1−𝜑]𝜀IPR2) (12) \nwhere IPT and IPR are the average in -plane lattice parameters for the T -like and R -like striations, \nand in this case 𝜑 is the ratio of T -like to R -like phases in the areas containing striations, which \nwe assume will reduce to 𝜑=12⁄, given previous literature  [28] identifying the phases of each \nportion of these needle -like structures.  \nThe value of T’/T’ domain wall (or phase boundary) energy , 𝛾𝑇/𝑇 (or 𝛾𝑇/𝑅) is extracted from the \nlinear fits in Fig. 5:  13  𝛾𝑇/𝑇(𝑇/𝑅)=𝑈𝑇/𝑇(𝑇/𝑅)𝑀𝑇/𝑇(𝑇/𝑅)2\n4𝜋2 . (13) \nFrom this analysis, the average energy for forming T’/T’ twin domain walls is  found to be  113 \n± 21 mJ.m-2, while for forming mixed phase T’/R’ boundaries it is  426 ± 23 mJ.m-2. These values, \nas expected, are consistent with the experimental observation of a critical thickness at which mixed \nphases form – only at this thickness does the film overcome the cost of forming phase boundaries \nto relieve strain, whereas the more energe tically favorable method of partial strain relief (forming \nT’/T’ twins) occurs at all the thicknesses (above ~15 nm) across our sample range.  These values \nof domain wall and interphase boundary energy are also of the same order of various types of \ndomain walls in PbTiO 3 and BiFeO 3, as reported in Table 1.  \n \nMaterial and domain wall \nvariant  Domain wall energy \n(mJ/m2) Reference  \nBaTiO 3 - 180° wall  7.2 - 16.8 \n [46] PbTiO 3 - 90° wall  29.4 - 35.2 \nPbTiO 3 - 180° wall  132 - 169 \nBiFeO 3 - 71° wall  363 - 436 \n [47] BiFeO 3 - 109° wall  205 - 1811  \nBiFeO 3 - 180° wall  829 \nBiFeO 3 - T’/T’ wall  113 ± 21  \nThis work  \nBiFeO 3 - T’/R’ phase boundary  426 ± 23  \n \nTable 1 . Reported values of domain wall energy fo r BaTiO 3, PbTiO 3 (first principles calculations) , and \nBiFeO 3 (first principles calculations and this work).  \n \n \nDISCUSSION AND CONCLUSIONS  \nOur empirical approach to determine domain wall and interphase boundaries in the mixed -phase \nBFO system comes with some caveats, which we briefly discuss here. Recall that our treatment  \nuses two key quantitative results from experiments:  i) a strain energy term related to the order \nparameter, and ii) a periodicity discontinuity, which we ascribe to the system changing the type of \ninterface nucleated in  order to reduce the elastic energy. As there  is no change of phase in the out- 14 of-plane PFM signal  with increasing thickness , we surmise that these  domains do not form to  \nminimize depolarization  field. Therefore, we have disregard ed the electrostatic energ ies and \nassume  that the formation of  T’/T’ occurs  to minimize  the internal self -strain energy arising from \nthe monoclinic distortion of the T -like unit cell. This explains why we exclusively treat the free \nenergy in -elastic energy terms.  \nNext, we point out  that we have not accounted for the curvature of the domain walls. This would \nrequire incorporating a gradient term (in the vicinity of the wall) in the strain energy expression, \nonly possible experimentally by precise mapping of the atomic level displacem ents as a function \nof the distance from the wall. To the best of our knowledge, such information would only be  \naccessible  using aberration  corrected transmission electron microscopy  which is beyond  the scope \nof this paper. Therefore, we only fitted the first four data points to the T’/T’ twin walls as we are \ncertain with only these four samples the strain state is  governed only by the T’/T’ twins and not \nthe mixed phase  formation. In the same vein, only the  last four  data points are fitted to extract the \nenergy for the T’/R’ interphase boundaries. It is only in these last four samples that we truly find \nthe dense domain state, which allows us to reasonably disregard the gradient terms  [34,48] . \nFinally, for simplicity, our approach does not take into account  the presence of other possible \ndomain arrangem ents, such as nanotwins which can form parallel to the substrate interface, \nreported by Pailloux et al.  [49]. The laser used for PLD grown in that work was Nd:YAG, while \nfor our samples we used an excimer laser. It has been shown that the strongly different growth \nconditions for the two ablation laser types result s in BFO films with different crystallinity, \nmosaicity, and domain structures  [50,51] . Similarly, it is possible that the Nd:YAG laser promote s \nthe formation of such exotic nanotwin defects  in BFO//LAO. T o the best of our knowledge, \nexcim er grown BFO//LAO samples have not been reported to show such nanotwin structures.  \nIn summary, we have explored the energetics of formation of T’/T’ and interphase boundaries \nin the mixed -phase BiFeO 3 thin film system. Combining XRD, AFM, and PFM technique s, we \ncharacterized a series of BFO//LAO (001) films with thicknesses of 15-120 nm. From the XRD \ndata we extracted the lattice parameters for both the T -like and mixed phase regions, allowing the \nestimation of elastic strain energies. The periodicities of the T’/T’ twin ferroelastic  domain walls \nand mixed phase T’/R’ striations were determined through PFM and AFM analyses, respectively. \nThe energy of formation for domain walls and interphase boundaries were calculated to be 𝛾𝑇/𝑇 ≈ \n113 ± 21 mJ.m-2 and 𝛾𝑇/𝑅 ≈ 426 ± 23 mJ.m-2, respectively. These  results strengthen our   15 understanding of these strain relieving microstructures and provide numerical guidelines for the \nengineering of new exotic phases in tailor -made and dimensionally -confined  [52] multiferroic  \nsystems . \n \nAUTHOR INFORMATION  \nCorresponding Author s \n*stuart.burns@ucalgary.ca; daniel.sando@unsw.edu.au   \nPresent Addresses  \n# Department of Chemistry, University of Calgary, 2500 University Drive NW, Calgary AB T2N \n1N4, Canada  \n⊥ Department of Quantum Matter Physics, University of Geneva, 24 Quai Ernest -Ansermet, CH -\n1211, Geneva 4, Switzerland  \nAuthor Contributions  \nThe manuscript was written throug h contributions of all authors. All authors have given approval \nto the final version of the manuscript.  \nNotes  \nThe authors declare no competing financial interest.  \nACKNOWLEDGMENT S \nThis research was partially supported by the Australian Research Council Centre of Excellence \nin Future Low -Energy Electronics Technologies (Pro ject No. CE170100039) and funded by the \nAustralian Government. D.S. and V.N. acknowledge the support of the Australian Research \nCouncil through Discovery Grants. S.R.B. acknowledges current funding from the Canada First \nResearch Excellence Fund, and partia l funding from the UNSW Science PhD Writing Scholarship. \nS.R.B. and O.P. thank AINSE Limited for providing financial assistance (Award - PGRA). The \nauthors thank Ekhard Salje, Alina Schilling and Marios Hadjimichael for helpful discussions.  \n  16 REFERENCES  \n[1] G. Catalan and J. F. Scott, Physics and Applications of Bismuth Ferrite , Adv. Mater. 21, \n2463 (2009).  \n[2] S. R. Burns, O. Paull, J. Juraszek, V. 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The device uses a magnetic tunnel junction (MTJ) , which is adjacent to a non -magnetic \nmetallic and a ferrite film.  This film stack is heated or cooled by a Peltier element which creates a bi -\ndirectional magnonic pulse in the ferrite film.  Conversion of magnon s to spin current occurs at the \nferrite -metal interface, and the resulting spin -transfer torque is used to achieve  sub-nanosecond \nprecessional switching of the ferromagnetic free layer  in the MTJ. Compared to  electric current driven \nSTT-MRAM with perpendicular magnetic anisotropy  (PMA) , thermoelectric STT -MRAM reduces the \noverall magnetization switching energy by more than 40% for nano -second switching, combined with a \nwrite error rate (WER) of less than 10-9 and a lifetime of 10 years or higher . The combination of higher \nthermal activation energy, sub -nanosecond read/write speed, improved tunneling magneto -resistance  \n(TMR)  and tunnel barrier reliability make thermo electric STT -MRAM a promising choice for futur e non -\nvolatile memory application s.  2 \n Over the last few decades, the trend towards increased adoption of embedded memory  to increase \nthe bandwidth of high performance processors  and mobile system -on-chips (SoCs) , has prompted \nresearch in novel memory techn ologies  [1-4].  Among them , spin -transfer torque magnetic random access \nmemory (STT -MRAM) has stimulated significant interest due to its unique combination of properties, \nincluding data non -volatility, unlimited endurance, negligible static leakage power, high performance and \nhigh integration densities  [5-7].  Unlike magnetic field -driven MRAM [4], STT -MRAM offers lower \nswitching current, simpler cell architecture, reduced manufacturing cost and excellent scalability (with a \nbit-cell size as small as 6F2), making it a competitive choice for future technology nodes [6-8].   \nAn electric current passing through a pair of ferromagnetic electrodes separated by a metallic \nspacer or tunnel barrier exerts pseudo -torque on the magnetic moment of the individual electr odes [9-10].  \nThe magnitude of this spin -transfer torque is proportional to the electric current [9-12].  Switching using \nthis torque requires a pulse length of at least 5 -10 ns [13-14], unless the device is overdriven with currents \nsignificantly higher th an the switching threshold.  For applications to MRAM, such high currents are \nincompatible with the requirement to avoid breakdown of the tunnel barrier during its lifetime (say,  10 \nyears).  For slower switching speeds, breakdown of the barriers is  typica lly avoided  by limiting the write \nvoltage across the barrier to around 400 mV and using a tunnel barrier with resistance -area (RA) product \nin the range of 5 – 10 \n m2 [15].  These l ower RA barriers are more susceptible to breakdown and have \nlower tunnelin g magneto -resistance (TMR), which reduces the read margin.  Lowering the switching \nthreshold (for example by making the free layer thinner) would reduce the activation energy, causing poor \nretention due to thermally activated switching.  Hence, it is diffi cult to achieve fast sub -nanosecond \nswitching of the free layer using spin-transfer torque,  without increasing errors due to tunnel barrier \nbreakdown, read, and thermal instabilities.  \nRecently, Slonczewski proposed initiating spin -transfer torque by heat f lux flowing through the \nfree layer and an insulating reference ferrite [16]. This torque is generated by the creation  or annihilation  \nof magnons in the ferrite and subsequent conversion to electron spin current at the interface between the 3 \n ferrite and a no n-magnetic metal film.   Depending on the direction of heat flow, this spin -transfer torque \ntends to align or anti -align the free layer magnetic moment with the ferrite magnetic moment. With proper \nsuppression of heat energy carried by phonons inside the f errite [17-18], the proposed magnonic spin -\ntransfer torque has a quantum yield of almost two orders of magnitude higher than achievable using \nconventional electric current through the magnetic tunnel junction (MTJ) [16]. This enhanced quantum \nyield opens u p the possibility of designing n ew spintronic devices with low power and high speed of \noperation. However, s ince the sign of the thermagnonic spin -transfer torque depends on the direction of \nheat current, magnonic spin -transfer initiated by Joule heating o f a resistor adjacent to the MTJ is not \ndirectly applicable to MRAM, where bi -directional switching of the free layer is required.  \nIn this article, we propose an alternative MRAM design  in which the storage layer is efficiently \nswitched by  magnonic ally gen erated  spin-transfer torque .  An insulating ferrite film with uniaxial \nmagnetic anisotropy collinear to that of the nearby  free layer is used to flip the free layer moment bi -\ndirectionally during write. The free layer can be switched to be either parallel or anti -parallel to the ferrite \norientation, depend ing principally on the direction of heat current flowing through the insulating ferrite \nand the free layer separated by a thin metal spacer [16 ].  By using a Peltier element in the device stack, it \nis pos sible to simply reverse the direction of heat flow , and hence , obtain bidirectional writing.  \nConversion from magnons to electrical spins occurs at the ferrite -metal spacer boundary.  \nFigure 1 shows the schematic of the proposed  device configuration to uti lize thermoelectrically \ncontrolled magnonic spin -momentum transfer for bi -directional magnetic switching.  The uniaxial \nmagnetic anisotropy of the ferrite film  is perpendicular to the x -y plane and collinear to those of reference \nand free magnetic layers [ all along z -direction].  For this structure, we estimate the thermo electrically \ninitiated magnonic spin -transfer torque by considering the exchange coupling between the ferrite moment \n(MFerrite) and conductive s -electrons in the adjacent metal spacer throu gh interfacial atomic monolayer \nwith paramagnetic 3d -electron -spin moments ( ∑) [16, 19 ].  In Slonczewski’s model, σ and F are \nrespectively the average thermal moment of the s -electron -spins per unit area of the metal spacer and an 4 \n effective molecular field  at the ferrite -metal interface. MRef and mFree are respectively the magnetic \nmoments of the ferromagnetic pinned and free layers, separated by a thin tunnel barrier ( typically MgO).  \nThe transient response of σ can be understood  by considering its coupling  to the monolayer moment ∑ \nand solving the Bloch equation appropriately [16, 20 ].  \nA temperature differential δT between the ferrite spins including the magnetic monolayer and the \nadjacent non-magnetic metal develops an interfacial heat current.  A steady  flow of heat current from the \nheat source to the thermal sink yields a non -vanishing time rate of σ, defined as the thermagnonic spin -\ntransfer torque per unit free layer area.  In practice, the thermal conductance due to interfacial phonon \nscattering, and  ferrite -to-monolayer and monolayer -to-metal heat transfer efficiencies jointly determine \nthe effective spin -momentum transfer due to ferrite magnons [21-22].  The magnitude of the \n \nFig. 1  Schematic of the proposed thermo electric STT-MRAM device using bipolar ther magnonic current for bi -\ndirectional free layer switching   5 \n thermagnonic spin -transfer torque along z -direction (perpendicular to the f ilm plane, x -y) can be \nestimated as [16]:            \n2\nd sd\nz.S.(S+1).N .(J ρ) .F(T)τ δT3. .T\n                 \n(1) \nwhere S is the  spin quantum number of the paramagnetic metallic atoms in the magnetic \nmonolayer , Nd is the  number of magnetic ions or atoms per unit area o f the magnetic monolayer , F(T) is \nthe m olecular -field exchange splitting of the magnetic ions sitting at the ferrite -metal interface at \ntemperature T , δT is the  effective temperature differential across the ferrite -metal interface with proper \nsign dependen t on the direction of heat flow , Jsd is the on -site s -to-d-electron  exchange coupling , ρ is the  \ns-electron density per atom , respectively , and \n is the  Dirac constant.  \nIt is worth noting that the sign of the perpendicular magnonic torque τz depends on the direction \nof heat current flow across the ferrite -metal interface.  In the model proposed in Fig. 1, a positive  δT \n(equal to  TFerrite - TMS) tends to align free layer moment with MFerrite. A negative δT anti-aligns mFree with \nMFerri te. In our proposed device, the sign and magnitude of δT is controlled using thermo -electric Peltier \neffect, thus giving the ability to achieve fast nanosecond bi-directional switching.  6 \n  \n \nFig. 2(a)  Steady state temperature di stribution in the thermoelectric STT -MRAM device under Peltier heating, \nenhanced by joule heating. The metal strip (MS) L 3 is at a higher temperature than the Barium Ferrite. δT > 0K  \n \n \n \nFig. 2(b) Steady state temperature distribution in the thermoelectric STT -MRAM device under Peltier cooling, \nhindered by joule heating. The metal strip (MS) L 3 is at a lower temperature than the  Barium Ferrite. δT < 0K  7 \n The detailed device s tructure of the proposed thermo electric STT -MRA M using magnonic \ncurrent for bi -directional free layer switching is shown in Fig. 2. For the magnetic tunnel junction, we use \nthe vertical stack  structure as described and fabricated in [23]. The MTJ with PMA is placed vertically \nabove a  non-magnetic metal lic spacer, adjacent to a ferrite  film grown on amorphous carbon substrate. A \nthin layer of peltier material , such as Nb -doped Strontium Titanate, sandwiched between two metal straps \nL2 and L 3 is placed right above the magnetic tunnel junction. .   Th e com posite stack forms a  \nthermoelectric circuit, generating bi directional  magnonic current through the free magnetic layer  [24]. A \nbipolar electric potential between L 1 and L 2 sends a current I P through the Peltier element . On assertion of \na positive potential  at L2 relative to L1,  the metal strap L 2 heats , converting it to a heat source as \ndesignated in Fig. 1. Conversely , a negative potential at L 2 relative to L 1 cools down L 2, transforming it \ninto a heat sink.  Changing the polarity of the applied voltage (VP) between L1 and L2 thus develops a bi -\ndirectional heat flux flowing normally across the metal (L 3)-Barium F errite interface as shown in Fig. 1  \nand 2(a, b), and in turn  creates a bipolar magnonic current essential for bi -directional magnetic switching  \nof the free layer .  As we shall see, a  magnonic current with higher spin -torque efficiency in combination \nwith a n efficiently designed Peltier circuit helps in scaling down the current I P in thermo electric STT -\nMRAM. The design with thermo electric circuit on top provides a broader choice of magnetic and \nelectrode materials, since the ferrite (which is typically grown at high temperatures) is  grown first , below \nthe magnetic and Peltier stack  [25].  \nIn thermoelectric STT -MRAM, the Peltier current I P required for  heating/cooling of the metal \nstrap L 2 during write essentially determines the overall switching energy dissipation.   To estimate the \nelectro -thermal efficiency of the proposed device structure (Fig. 1), we solve for transient ther moelectric \nconduction cou pled with Landau -Lifshitz -Gilbert -Slonczewski (LLGS) equation in a single domain \nmagnetic landscape [16, 26 -27].  First, we solve the Fourier heat transfer equation [28] coupled with \nPoisson’s equation self -consistently in field variables temperature ‘T’ and electric potential ‘V’, using a \ncommercial finite element analysis  package [29-30]. For all subsequent micro -thermal analysis of the 8 \n proposed device structure shown in Fig. 2, the geometry parameters are chosen as follows: Individual \nlayer thicknesses are amorphous carbon(100nm) /BaFe  (20nm)/ Cu  (10nm)/ free layer (3nm)/ MgO  \n(1.2nm)/ reference layer (6nm)/ PtMn (25nm )/ Ta  (60nm )/ Cu(10nm)/ SrTiO 3(40nm )/Cu( 50nm ),  and the \ndiameter of  the circular MTJ pillar is 50nm, and the   Peltier current  IP is describ ed by a rectangular pulse \nof amplitude V P and pulse width PW  of 0.15ns.   Fig. 3 shows the transient response of the temperature \ndifference δT , across the ferrite -metal interface . The thermal time constant of the proposed nanostructure \ncauses a gradual decay  in δT  as predicted in Fig. 3.  \n \nFig. 3 Transient respon se of heating/cooling in thermo electric STT -MRAM under short duration Peltier current \n(IP) pulse; The thermal time constant of the system ensures gradual decay of the temperature profile from  \n|δTMAX|, achieved using a short duration voltage pulse of magnitude V P and pulse width PW.  9 \n In a recent communication [31], the time -resolved heat -flow dynamics in ferromagnetic thin films \nhave been studied experimentally.   A large magnonic contribution to the total heat current flowing normal \nto the film surface has been reported over a broad temperature range.  Subject to a pulsed heating /cooling , \nthe local magnon temperature rises /diminishes  instantaneously due to fast (~ps) therma l energy transfer \nwith phonons. Once the required temperature differential is develop ed across the ferrite -metal interface, \nmagnonic spin -transfer torque acts i nstantaneously on the storage layer to switch its magnetization.      \nAssuming a transient δT  profile as predicted in Fig. 3 , we solve LLG S equation in the presence of \nthermally induced stochastic magnetic noise, with thermagnonic spin -transfer torque term s included , as \nmodeled in Eq. 1 . Subject to a maximum temperature differential δTMAX of ±8K  across the metal strip L 3 \nand ferrite interface, the P -AP and AP -P switching characteristics are shown in Fig. 4. Sub -nanosecond \n \nFig. 4 Bi-directional magnetic switching  using bipolar thermagnonic current for δT= ± 8K with free layer \nanisotropy perpendicular to the film plane (along z -axis). The simula tion parameters are chosen as: E a=80kT, \nMsat=850emu/cm3, H ku2=880emu/cm3, vol= 50x5 0x3nm3, S= 2.5; N d=4x1018/m2; interfaci al spin moment in ferrite \n(FT)=3 0meV; T=300K; On -site sd exchange coupling (J sd)=-0.5eV; Density of S -electrons per atom (ρ)=0.15 eV-\n1.spin-1.atom-1; \n 10 \n switching delay is achieved, with n o electric current flowing through the thin tunnel barrier.  In an electric \ncurrent driven STT-MRAM with PMA, the critical switching current density J E for a delay of 1 ns and \nwith a switching failure probability of 10-9 has been estimated to be close to 14 MA/cm2 at 300K  [32].  In \norder to avoid voltage -induced breakdown, the RA product of the tunnel barrier would have  to be reduced \nto well below 5 Ω-μm2.  In practice it has been difficult to achieve such low RA barriers which are \npinhole -free [33].  Thus, in conventional STT-MRAM, sub -nanosecond precessional switching is difficult  \nto achieve in combination with a lifetime o f ~10 years, as the tunnel barrier breaks down due to the \nexcessive current requirement for switching  [15].  To eliminate the stochastic thermal incubation delay \nduring free layer switching, we assume a n angular tilt (θ Tilt=250) in ferrite magnetic anisotr opy relative to \nthat of the free layer [ 34-36].  The non -collinear alignment of the ferrite and free layer anisotropies ensure \nthe reproducible (error -free) switching of the free layer magnetization under thermagnonic spin -transfer \ntorque.  The proposed thermoelectric STT -MRAM offers sub -nanosecond, low power and reproducible \nmagnetic switching, with lifetime of 10 years or higher. Compared to electric current driven STT -MRAM \nwith perpendicular magnetic anisotropy (PMA), thermoelectric S TT-MRAM reduces the overall \nmagnetization switching energy by more than 40% for nano -second switching, combined with a n \nestimated  Write Error Rate (WER) of less than 10-9. \nIn an STT -MRAM bit cell, t he presence of an access device  in series with the MTJ dur ing read \nreduces the effective TMR of the bit  cell, which can be expressed as:  \nMTJ\nCELL\nACCESS\nPTMRTMR =R(1+ )R\n (\n(2.a) \nAP P\nMTJ\nPR -RTMR =R\n (\n(2.b) 11 \n RACCESS , R AP and R P are the resistance of the access device during read, and anti -parallel and \nparallel re sistances of the MTJ , respectively. To ensure a high cell TMR during read, one needs to \nmaximize both R P and TMR MTJ for a fixed width of the access transistor.  This can be done by increasing \nthe thickness of the tunnel barrier in the MTJ pillar.  In a stat e-of-the-art STT -MRAM, higher switching \ncurrent density electrically limits the use of a thicker tunnel barrier and T OX is typically kept below 1.2nm \nfor a WER less than 10-9. In the proposed thermoelectric STT -MRAM, as no electric current is required to \nflow through the tunnel barrier, a thicker tunnel barrier (T OX ~ 1.5nm) can be incorporated. Higher RA \nproduct enhances the bit cell TMR and significantly reduces the disturb failure probabilities during read \n[12, 37]. Fig. 5(a-c) shows the effect of tunnel barrier thickness (T OX) and voltage on R AP, R P, TMR and \ntunneling current density (J E) of a n MTJ with MgO as tunnel barrier  [37-38]. As shown in Figs. 5(a) and \n(b), the differential resistances in parallel and anti -parallel states of an  MTJ increases almost by an order, \nwhen T OX is changed from 1.2nm to 1.5nm.  The thicker tunnel barrier provides an enhanced guard band \nagainst ‘soft’ oxide breakdown [15] and memory disturb failures, in addition to achieving almost 40% \nhigher MTJ TMR  than the conventional STT -MRAM.   \nIn this article, we propose  a new genre of STT-MRAM , using thermoelectrically controlled \nmagnonic spin -transfer torque for low power, fast , reliable and error -free magnetic switching. A bipolar \nelectric current through a thin f ilm Peltier element generates a bidirectional magnonic current at a ferrite -\nmetal interface placed in contact w ith a ferromagnetic free layer. Conversion of magnonic current into \nspin-transfer torque is used to achieve sub -nanosecond bidirectional switchin g of the free layer. \n \nFig. 5  Bias-voltage dependence of parallel and anti -parallel differential resistances (dV/dI) of an MTJ with (a) \nTOX=1.2nm and (b) TOX=1.5nm. (c) A thicker tunnel barrier helps in boosting up the MTJ TMR.   \n \n \n 12 \n Compared to the conventional STT -MRAM with PMA, the proposed therm oelectric  STT-MRAM \nreduces the magnetizat ion switching energy by almost 40% for a switching delay of 1 ns and WER  less \nthan 10-9. The combination of higher thermal activa tion energy, sub -nanosecond read/write speed, \nimproved TMR , and tunnel barrier reliability make thermo electric STT -MRAM a promising choice for \nfuture non -volatile memory application s.  \n \nReferences:  \n1. Changhyuk, L. et al.  A 32 -Gb MLC NAND Flash Memory With Vt h Endurance Enhancing Schemes in 32 nm \nCMOS. Solid -State Circuits, IEEE Journal of  , vol.46, no.1, pp.97 -106, Jan. 2011  \n2. Cheng,  H. Raoux, S. Jordan -Sweet , J. L. The crystallization behavior of stoichiome tric and off -stoichiometric \nGa-Sb-Te materials for p hase-change memory . Appl. Phys. Lett . 98, 121911 (2011)  \n3. Barth, J.  et al. 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Cambridge University Press, 1995  \n \n \n \n "    },    {        "title": "2001.02829v2.High_temperature_thermal_cycling_effect_on_the_irreversible_responses_of_lattice_structure__magnetic_properties_and_electrical_conductivity_in_Co___2_75__Fe___0_25__O___4_δ___spinel_oxide.pdf",        "content": "High temperature thermal cycling effect on the irreversible respon ses of lattice structure,\nmagnetic properties and electrical conductivity in Co 2.75Fe0.25O4+δ spinel oxide \n            Rabindra Nath Bhowmik*a, Peram D Babub, Anil Kumar Sinhac,d, and Abhay Bhisikarc\naDepartment of Physics, Pondicherry University, R.V. Nagar, Kalapet, Pondicherry 605014,\nIndia\nbUGC-DAE Consortium for Scientific Research, Mumbai Centre, Bhabha Atomic Research \nCentre, Trombay, Mumbai-400085, India \ncHXAL, SUS, Raja Ramanna Centre for Advanced Technology, Indore- 452013, India\ndHomi Bhabha National Institute, Anushakti Nagar, Mumbai -400 094 India\n*Corresponding author: Tel.: +91-9944064547; E-mail: rnbhowmik.phy@pondiuni.edu.in\nABSTRACT : \nWe report high temperature synchrotron X-ray diffraction (SXRD), magnetizat ion and current-\nvoltage (I-V) characteristics for the samples of Co 2.75Fe0.25O4 ferrite. The material was prepared\nby chemical reaction of the Fe and Co nitrate solutions at pH  ∼ 11 and subsequent thermal\nannealing. Physical properties of the samples were measured by cycling the temperature from\n300 K to high temperature (warming mode) and return back to 300 K (cooling mode). The lat tice\nstructure showed sensitivity to high measurement temperatures. Magne tization curves showed\ndefect induced ferromagnetic phase at higher temperatures and superparamagnetic  blocking of\nthe ferrimagnetic particles near to 300 K or below. Electrical conduct ivity exhibited thermal\nhysteresis  loop at higher measurement  temperatures.  The  samples  exhibi ted  new  form (not\nstudied so far) of surface magnetism in Co rich spinel oxides and irre versibility phenomena in\nlattice structure, magnetization and conductivity on cycling the measurement tempera tures.\n1Key words: Co rich spinel oxide; Synchrotron X-ray diffraction; Bi-phased magnetic mat erial;\nthermal dependent irreversible properties.\n■ INTRODUCTION\nThe spinel oxides are strongly spin correlated electronic system. They are defined by formula\nunit AB 2O4, where metal ions occupy two inequivalent lattice sites A and B with tetrahedrally\nand octahedrally coordinated oxygen ions, respectively in cubic crystal wi th space group Fd\n m\n[1-2]. The distribution of Co and Fe ions at A and B sites determine magnetic and electrical\nproperties in spinel structure. Co rich spinel oxides (Fe 3-xCo xO4; 1<x<3) are interesting for their\nability to tune properties by altering the synthesis route, heat treatm ent, and Co content in the\nequilibrium and non-equilibrium cubic spinel structure [3-11]. Structural phase of Co rich spinel s\nthat lie in the miscibility gap of the Fe 3O4-Co 3O4 phase diagram is kinetically sensitive to Co\ncontent [1-3, 9, 11]. The annealing in the temperature range 900–950  0C form stable phase.\nOtherwise, cubic spinel phase of Co rich spinel ferrites shows spinodal decomposition (Co-rich\nand Fe-rich phases). The spinodal decomposition is also formed in thin films of Co 1.8Fe1.2O4 (Co\nrich and Fe rich phases) [12] and Mn ferrite (Mn rich and Fe rich pha ses) [13]. The self-\ncomposite material is new for studying heterogeneoud chemical structure in  oxides, which is\ncharacteristically different from hybrid composite of two different compounds [14]. \nWe cite some of the works that reported high temperature structure and prope rties of\nspinel pxides. For example, CuFe 2O4 spinel oxide showed a fully inverted tetragonal phase in the\ntemperature range 2 K to 600 K and a cubic spinel phase at 660 K [15]. Bot h the phases\ncoexisted up to 700 K and cubic phase alone is stabilized at higher  temperatures. Magnetically,\nferrimagnetic structure of CuFe 2O4 disappeared at T N  ∼750 K. The Mn xFe3−xO4  oxide at room\n2temperature formed cubic phase for x ≤ 1 and tetragonal distortion for 1 < x ≤ 3 [16]. FeMn 2O4\nshowed a structural phase transition at ∼ 595 K from (low temperature) tetragonal phase to (high\ntemperature) cubic phase with ferrimagnetic transitions at ∼ 373 K and ∼ 50 K [17]. O’Neill et\nal. [18] performed neutron diffraction to study the thermal cycling effect on cati on order-disorder\nin Mg 2TiO 4. The distribution of Mg2+ and Ti4+ ions was found in ordered state (inverse structure\nwith all Ti4+  ions occupy half of the B sites) when temperature warmed from 300 K t o 1173 K\nand a disordered state (migration of a fraction of Ti4+ to A sites) at higher temperatures. The rate\nof reordering of the cations between A and B sites was faster during cooling process and high-\ntemperature disordered state is not preserved after cooling back to 300 K . The high temperature\nXRD patterns suggested a correlation between cell parameter and di stribution of cations in\nMgFe 2O4 spinel oxide during thermal cycling (warming ↔ cooling) of measurements [19]. \nThe intrinsic point defects, introduced during thermal cycling, generally perturb  the structure\nof cations from their equilibrium state and modify the properties [20-22]. Despi te of showing\nmany unusual low temperature magnetic properties in Co rich spinel oxides  [2, 3, 6, 9, 11, 23],\nthe high temperature study is limited [10]. In this work, we studied hi gh temperature properties\n(structure, magnetism and electrical conductivity) for a highly Co rich  spinel oxide Co 2.75Fe0.25O4\nand discussed the role of thermal cycle induced irreversible effects on properties. We performed\nthe measurements in air to realize the naturally expected defec ts in spinel oxides. The materials\nwith inhomogeneous lattice structure, magnetic spin order and electronic structure are expected\nto be interesting for studying first order magnetic phase transition, and a chieving high magneto-\ncaloric effect, high catalytic and surface chemical kinetics [11, 24-26].\n■ EXPERIMENTAL SECTION\n3Sample preparation. The spinel oxide Co 2.75Fe0.25O4 was prepared through chemical reaction of \nthe required amounts of Co(NO 3)2.6H 2O and Fe(No 3)3.9H 2O salts. The salts were dissolved in \ndistilled water to yield a transparent solution at pH ∼1.4. NaOH solution with initial pH ∼13 \nwas used as the precipitating agent and added into the nitrate solution until pH reached to  ∼ 11. \nThe reaction temperature was maintained at 80 °C for 4h with continuous stirring. The pH was \nmaintained at 11 by adding required amount of NaOH solution during chemical reaction. The \nproducts were allowed to cool down to room temperature and allowed to precipitate at the \nbottom of a Borosil beaker. The transparent solution from the top portion was removed carefully \nand remaining product was washed several times with distilled water and each time  it was dried \nat 100°C. Finally, the resultant powder was heated at 200-250 °C in a beaker to confirm the \ncomplete removal of the bi-product of NaNO 3, which formed a white coating on the wall of \nbeaker and black coloured (magnetic) powder was collected at the centre of the beaker whe n \nplaced on a Rotamantle. The collected black powder was made into several pellet s and annealed \nat selected temperatures in the range 200-900o C for 6h in air. The prepared samples were \ndenoted as CF_20, CF_50and CF_90 for annealing temperature at 200 0C, 5000C and 900 0C, \nrespectively. The heating and cooling rate was maintained @ 5 °C/min. \nSample characterization. The synchrotron X-ray diffraction (SXRD) patterns were recorded in \nthe 2θ range 5-40 ° for samples CF_20 and CF_90 using synchrotron radiation facilities at angle \ndispersive X-ray diffraction beam line 12 of Indus-2, Indore, India. The wave length was fixed at\n0.7820 Å and 0.7600 Å for the samples CF_20 and CF_90, respectively. Photon energy and the \nsample to detector distance for SXRD were calibrated by using SXRD pattern of LaB 6 NIST \nstandard. The high temperature SXRD measurements were performed in the temperature range \n(300 K to 873 K) in air with temperature variation @ 20 °C/min and temperature stabilization for\n4nearly 2 minutes before measurement at each temperature. The dc magnetization [M(T)]  in t he \nwarming mode (300 K-950 K) and cooling mode (back to 300 K) were measured using physical \nproperties measurement system (PPMS-EC2, Quantum Design, USA). The field dependent \nmagnetization [M(H)] data were recorded within magnetic field range ± 70 kOe at selec ted \ntemperatures in the range 300 K-900 K during warming mode after completing the M(T)  \nmeasurement cycle. The current-voltage (I-V) curves for CF_50 and CF_90 samples were \nmeasured in the temperature range 300-623 K using Keithley 6517B high resistance meter. The \ndisc shaped samples ( ∅= 10 mm, t ∼0.5 mm) were placed between Pt electrodes of a home-made \nsample holder to make a Pt/sample/Pt device structure. The current passing through the sampl e \nwas measured by sweeping the dc voltage within ± 50 V at each measurement temperature.  \n■ RESULTS\nStructural Properties.  Figure 1 shows SXRD pattern of the CF_20 and CF_90 samples at\nselected temperatures. The Y-axis has been re-scaled for some pa tterns to compare between\nwarming and cooling modes. SXRD pattern at all measurement temperatures (300-873 K) show\n5cubic spinel structure (space group Fd\n m) for both the samples [5, 7]. The crystalline planes\ncorresponding to cubic spinel structure are indexed at the top of Figure 1(a-b). In  CF_20 sample,\nthe usual peak intensity ratio is maintained for temperatures up to  773 K. At higher temperatures\n(825K and 873K), a sharp peak same as (531) plane sits on the amorphous background a t 2θ\nrange 32.5-34.5o  (inset of Figure 1(a)). The background introduces a butterfly type wing; an\nindication of structural distortion. Subsequently, a preferential orientation of  the (511) and (440)\nplanes showed unusual increase of intensity in contrast to usually obse rved highest intensity for\n(311) plane. The SXRD patterns possibility of any impurity phases, such as α-Fe 2O3 and CoO [2,\n4, 8]. But, a non-equilibrium lattice structure is expected due to low annealing tem perature (200 0\n6C) [5, 27]. On the other hand, CFO_90 sample (Figure 1(b)) showed sharp crystalli ne peaks. The\ncubic spinel structure is maintained at 300 K and 373 K during warming m ode. In addition to\ncubic spinel phase, an exceptionally high intensity (200) peak of CoO phase (cubic structure with\nspace group Pm3m) [4-5] is seen at 2 θ ∼ 20 0 for temperature at 473 K (inset of Figure 1(b)). The\nCoO phase is grain oriented along (200) direction. Intensity of the CoO phase decreased at 573 K\nand finally disappeared at  ≥ 673 K during warming mode. The CoO phase reappeared during\ncooling mode with less intensity in the range 773-573 K and is absent i n the temperature range\n473-300 K. A close  scrutiny of the  shape and asymmetric  nature  of the  peaks  indicates  a\n7possibility of two-phase cubic spinel structure. Figure 2 shows Rietveld  refinement of the SXRD\n8patterns at 300 K and 873 K of the CF_20 and CF_90 samples in warming  mode. Details of the\nrefinement process are included in supplementary information. The SXRD pat terns have been\nfitted using single phase and two phase models of spinel structure. T he single phase is modeled\nby full occupancy of 8a sites for Co2+ ions and 16 sites are co-occupied by Co3+ and Fe3+ ions.\nThe difference in the fit with single phase and two phase components is shown in the insets of\nFigure 2 for (333) peak. The two-phased model is best suited for CF_20 sampl e. The difference\nbetween two phase components of spinel structure is negligible for CF_90 sample. The peak of\nCoO phase in CF_90 sample was excluded during fit of cubic spinel phase. The two phase model\nfor CF_20 sample is fitted by assuming Co rich phase (phase 1), where 8a sites are fully\n9occupied by Co ions and Fe content at 16d sites is assigned less amount than the assigned value\nfor single phase, and Fe rich phase (phase 2), where Co and Fe are all owed to co-occupy both at\n8a and 16d sites and the total Fe content is more than the assigned value for single phase model. \nTable 1 provides parameters obtained from fitting of SXRD data at 300 K using two-phase cubic\nspinel structure for CF_20 and single phase model for CF_90 samples at 300 K. Rest of the fit\nparameters at 300 K and 873 K are shown in Table S1 (supplementary). We  found that oxygen\ncontent and composition of metal ions nearly matched to chemical c omposition of the samples at\n300 K, where as composition at 873 K shows excess O atoms per formula unit of the spinel\nstructure. The fraction of spinel phase components for CF_20 sample has been  calculated from\nstructural refinement (consistent to average value of the calculation from intensity ratio using\n(311), (400), (333) peaks). The fraction of spinel phase and CoO phase in CF_90 sample has\nbeen calculated from intensity ratio of the (311) peak for spinel pha se and (200) peak for CoO.\nFigure 3 schematically shows temperature variation of the fraction of  phases for CF_20 and\nCF_90 samples during warming and cooling modes. The Co-rich phase dominates than Fe-rich \nTable 1. Structural parameters from Rietveld refinement parameters of SXRD data at 300 K \n(warming mode) \n(a) single phase model of cubic spinel structure for  CF_20 sample and CF_90 sample\nAtoms\n(sites)CF_20 sample CF_90 sample\n   Wyckoff positions     B Occupancy    Wyckoff positions     B Occupancy\n       X                   Y                   Z          X                 Y                Z\nCo(8a)   0.12500        0.12500         0.12500 0.97 1.000    0.12500       0.12500        0.12500 0.97 1.000\nFe(8a)   0.12500        0.12500         0.12500 0.97 0.000    0.12500       0.12500        0.12500 0.97 0.000\nFe(16d)   0.50000        0.50000         0.50000 0.97 0.250    0.50000       0.50000        0.50000 0.97 0.250\nCo(16d)   0.50000        0.50000         0.50000 0.97 1.750    0.50000       0.50000        0.50000 0.97 1.750\nO (32e)  0.26020(33) 0.26020(33) 0.26020(33) 0.78 3.962(29)  0.26020(33) 0.26020(33) 0.26020(33)  0.78 4.049(29)\nCell parameter s: a = 8.10075 (33) Å, V  = 531.588 (37) Å3\n Rp: 3.52, R wp: 4.54,  R exp: 2.28, χ2:  3.96Cell parameters: a = 8.16673(12) Å, V = 544.683(14) Å3\n Rp: 9.44, R wp: 12.4,  R exp: 10.94, χ2:  1.28\nb) Two-phase model of cubic spinel structure for CF_20 sample\nAtoms\n(sites)Co- rich phase Fe- rich phase\n   Wyckoff positions     B Occupancy    Wyckoff positions     B Occupancy\n10       X                  Y                 Z        X                   Y                Z\nCo(8a)  0.12500          0.12500       0.12500 0.97 1.000    0.12500       0.12500      0.12500 0.97 0.850\nFe(8a) 0.12500          0.12500        0.12500 0.97 0.000   0.12500        0.12500      0.12500 0.97 0.150\nFe(16d) 0.50000          0.50000        0.50000 0.79 0.050   0.50000        0.50000      0.50000 0.79 0.190\nCo(16d) 0.50000          0.50000        0.50000 0.79 1.950(0)   0.50000        0.50000      0.50000 0.79 1.750\nO (32e) 0.26119(28)  0.26119(28) 0.26119(28) 0.72 3.947(23)  0.26250(48) 0.26250(48) 0.26250(48)  0.72 4.640(54)\nCell parameters: a = 8.10992 (30) Å, V = 533.396(34) Å3\nCo rich phase (Bragg R-factor: 4.57, phase fraction: 65.69%)\nFe rich phase (Bragg R-factor: 4.95, phase fraction: 34.31%)Cell parameters: a = 8.17176 (71) Å, V =545.692 (82)Å3\n Rp: 4.04, R wp: 5.08,  R exp: 2.39, χ2:  4.51\nspinel phase in CF_20 sample. The difference between Co-rich and Fe -rich phases decreased\nduring cooling mode. This gives rise to a loop, indicating the gain of Fe rich phase by cost ing the\nCo-rich phase. A thermal  hysteresis in the formation of CoO phase or int ermediate  kinetic\ninstability in spinel structure is observed for CF_90 sample, where CoO  phase (maximum ∼ 80%\nat 473 K in warming mode) has reduced during cooling mode (maximum  ∼ 17 % at 573 K).\nFigure  4(a-f) shows  the  temperature  variation  of structural  parameters  (latt ice  constant  ( a),\noxygen parameter ( u) and  δO (excess oxygen)) obtained using single phase model. All these\nparameters showed irreversible paths during thermal cycling of the measurem ent, irrespective of\nCF_20 (two spinel phase structure) or CF_90 (single spinel phase + interme diate CoO phase)\n11samples. The lattice parameter shows usual thermal expansion, but hi gher values of the lattice\n12 \nparameter during cooling mode suggest an effect of the exchange of Co and  Fe ions between A\nand B sites. Generally, an increasing population of Co ion at B s ites reduces lattice parameter in\nspinel oxides [2-3, 11]. The application of two-phase model shows a large difference bet ween the\n13lattice parameters of Co rich phase (phase 1) and Fe rich phase (phas e 2) in CF_20 sample (inset\nof Figure 4(a)). It may be noted that lattice parameter of the single phase model is close to the\n        \nvalues for Co-rich phase using two-phase model. The parameter, u, defines a displacement of O\natoms with reference to regular tetrahedron at 8a sites and octahedron at 16d sites. The ideal\nvalue of u is 0.25, but usually it lies between 0.24 and 0.275 for spinel oxides [28-29]. F igure 4\n14(b, e) show a bit scattered values of u (0.259-0.264) for CF_20 sample, whereas a systematic\ndecreasing trend (range: 0.261-0.264) at higher temperatures is found for CF_90 sample.  The\ndecreasing trend of u indicates the expansion of B sites at the expense of A sites. T his is realized\nby calculating the M-O (metal-oxygen) bond lengths at A sites (R tet) and B sites (R oct) using the\nformula R tet = a\n (u-\n) and R oct = a\n  [28] for CF_90 sample. The M-O bond\nlength at A sites decreases by ∼ 1.53% (1.963 to 1.933 Å) on increasing the temperature from\n300 K to 873 K by increasing the M-O bond length ∼ 1.19% (1.936 to 1.959 Å) at B sites. The\nchange of bond length shows chemical disorder (site exchange of Co and Fe ions or the variation\nof charge state of metal ions) at higher measurement temperature s. In order to understand the\ntemperature variation of ion defects, we have defined the difference of O content ( δO) = O cal-Otheo\nper formula unit of the spinel structure, where O cal is the calculated value from refinement using\nsingle phase model and O theo is the theoretically assigned value 4. The  δO curves are always\npositive  and a  rapid increase  at  higher temperatures  shows  absorption  of oxygen in spinel\nstructure. The absorption is remarkably high for CF_20 sample. Interestingly , oxygen content re-\ngained the normal value of the samples at room temperature after thermal cycling.             \nThe temperature induced structural irreversibility is also realized by  analyzing the SXRD peak\nparameters (intensity, position (2 θ) and full width at half maximum ( β)), which were determined\nby Lorentzian shape fit. Figure S1(supplementary) shows intensity of selec ted peaks at different\ntemperatures, normalized by the peak intensity at 300 K in warming mode. Intensity of the\ncrystalline planes corresponding to spinel structure of CF_20 sample and C F_90 sample reduced\n15during cooling mode in comparison to that during warming mode. Intensity of t he planes [(111),\n(311) and (400)] at lower scattering angles decreased at higher tempera tures. The intensity of\n(111) plane approached to nearly zero at 873 K for CF_20 sample. This peak is highly sensitive\n16to surface defects in spinel oxide [21, 26]. The Intensity of (111) peak i s high for CF_90 sample.\n17Intensity  of  the  crystalline  planes  [(422),  (333),  (440)]  at  higher  scattering  angle s  initially\ndecreases in the temperature range 300 K-473 K, followed by an unusual incre ase with a broad\nmaximum around 673 K and decreases above 800 K for both the samples during wa rming mode.\nInterestingly, intensity of these peaks showed an increasing trend on  decreasing the temperature\nduring cooling mode for CF_20 sample, unlike a decreasing trend for CF_90 sample.  This shows\n18characteristic differences in irreversibility effect as the funct ion of annealing temperature of the\nas-prepared  material.  The  crystallite  size  (<D>) and  micro-strai n  (ε) were  calculated  using\nWilliamson-Hall equation:\n , where \n  is wavelength of X-ray radiation.\nFigure 5(a-b) shows the fit of \n  vs. \n  data at 300 K (warming mode). The change of\ncrystallite  size  (Figure  5(c-d))  and  micro-strain  (Figure  5(e-f))  is  considerabl y  small  in\ntemperature range 300-873 K, but there is a thermal cycling effect. Crysta llite size and micro-\nstrain in CF_20 sample increased with temperature. As an effe ct of high temperature annealing,\nCF_90 sample showed larger crystallite size (23.0-23.4 nm) and small er micro-strain (0.073-\n0.077) in comparison to smaller crystallite size (1.95-2.25 nm) and large r micro-strain (0.16-\n0.21) of CF_20 sample. \nMagnetic properties. Figure 6(a-c) shows the temperature dependence of magnetization curves\nduring field warming (MFW(T)) and field cooling (MFC(T)) modes under magneti c field at 100\nOe or 500 Oe. The MFW(T) curve of CF_50 sample showed superparamagnetic blocking of\nferrimagnetic particles/clusters below temperature (T m) at 375 K for applied field 100 Oe [5, 10].\nThe combined plot of low temperature magnetization and high temperature MFW(T) curves\n(insets of Figure 6 (a, c)) showed blocking temperature below room temperature  for CF_20\nsample ( ∼ 270 K for field 100 Oe) and CF_90 sample ( ∼ 230 K for field 500 Oe and at 300 K for\nfield 100 Oe). The MFW(T) curves gradually decreased above the superparamagne tic blocking\ntemperature. Surprisingly, an increment in MFW(T) curves appeared above a typical temperature\n19            \nT0MFW that depends on annealing temperature of the samples (e.g., 684 K, 690 K a nd 700 K for\nthe samples CF_20, CF_50 and CF_90, respectively). A local broad maximum is observed at\nabout 880 K for all samples. Height of the magnetization maximum at 880 K dominates over the\nlow  temperature  maximum  at  T m (  ≤ 300  K)  for  CF_90  sample,  whereas  magnetization\nmaximum at T m dominates over the high temperature maximum in CF_20 and CF_50 sample s.\n20The  unusual  properties  in  MFW(T) curves  at  higher temperatures  is  an  exam ple  of defect\ninduced ferromagnetism, as reported in Fe deficient Fe 3-xTixO4 due to γ-Fe 2O3 surface layer [21].\nAn extrapolation of the MWF(T) curve for CF_90 sample indicates a T C at about 980 K, which is\nin the range of T C (∼ 920 K) for γ-Fe 2O3 (maghemite) [30]. According to Readman and O'reilly\n[21], the adsorption and ionization of O atom result in the creation of ne w sites at A and B\nsublattices. The created A sites will be filled with Co2+ ions, and each new B sites will be filled\nby Fe3+/Co3+ ions. Under oxygen rich environment, a fraction (z) of the Co2+ ions at A sites can\nbe converted into Co3+ ions through the mechanism Co2+ + \nO → zCo3+ + (1-z) Co2+ +\nO2-. A\nfraction of the Fe3+/Co3+ ions at B sites, highly mobile at high temperatures, diffuses int o the\nsurface of grains by creating corresponding vacancies at B sites and forms a skin of  γ-Fe 2O3\n(maghemite) (or Co doped  γ-Fe 2O3) phase. The interior and skin of the grains rapidly become\nhomogenous within B sites due to migration of vacancies and Fe3+/Co3+ ions. At the same time,\npopulation of cations at A sites becomes homogeneous by the electrons transfe r between interior\nCo2+ ions and surface Fe3+ ions. This contributes different character in MFC(T) curves, which\nfollowed different paths without any local magnetization maximum during fi eld cooling mode of\nmeasurement down to 300 K. The MFC(T) curves showed a sharp increase  below a typical\ntemperature T CMFC (∼705 K for CF_20 and CF_50 samples, and 820 K for CF_90 sample). A\nwide gap between MFC(T) and MFW(T) curves shows substantial change i n the magnetic spin\norder during cooling mode. The M(H) curves (Figure 6 (d-f)) at selected tempera tures in the\nrange 300- 900 K and measured in the field range +70/50 kOe to -5 kOe showe d ferrimagnetic\nfeatures with hysteresis loop and lack of magnetic saturation at higher fields. The inset s of Figure\n216 (e-f) showed enhancement of M(H) curves at 300 K after in-field MF W(T) and MFC(T)\nmeasurements (post thermal cycling) than the M(H) curves recorded before high  temperature\nmeasurement (pre-thermal cycling). The magnetic parameters (M S, M R, H C) of the samples,\ncalculated using M(H) curves, at different temperatures are shown in Figure 7(a-c). A  linear fit of\nthe high field M(H) curve on M axis at H = 0 Oe gives the M S (spontaneous magnetization)\nvalue,  whereas  the  intercept  of  M(H)  curve  on  M  axis  at  H  =  0  Oe  gives  the  remanent\nmagnetization (M R) and the intercept of M(H) curve on negative H axis for M = 0 gives the\ncoercivity (H C). The M S(T) values (with units emu/g and  µB/f.u in Figure 7(a)) decreased at\nhigher temperatures. The rate of decrement became slow above 700 K and t here was no unusual\npeak above 700 K. The M S  values below 600 K are found higher for the samples with low\ntemperature annealing (CF_20 and CF_50) than the CF_90 sample. However, MR of the CF_20\nsample indicted lower values than the values for CF_50 and CF_90 sam ples (Figure 7(b)). The\nHC value increased with annealing temperature of the samples. An inc rement of the H C (Figure\n7(c)) at higher measurement temperatures may be associated with ons et of an additional strain\ninduced magnetic phase above 600 K. In order to understand the nature of magnet ic order, we\nused Arrot plot [25] (see M2  vs. H/M curves in Fig. 7(d-f))). An extrapolation of the linear or\npolynomial fit of high field M2  vs. H/M curves on positive M2 axis at H/M = 0 corresponds to\nsquare of spontaneous magnetization (M S2(T)). The M S values from Arrot plot analysis (not\nshown) are close to the values obtained from direct linear fit of the hi gh field M(H) curves. The\nM2 vs. H/M plot showed a linear curve at higher fields and a sharp decrea se at lower fields with\npositive slope for all the samples at 300 K. An upward curvature at highe r fields and a negative\nslope at lower fields are observed at temperatures ≥ 350 K/375 K for CF_20 and CF_50 samples\n22               \nwith low annealing temperature, and ≥ 650 K in case of CF_90 sample. According to Banerjee\ncriterion [31], the slope of H/M vs. M2 plot is positive for second-order transition and negative\nfor first-order phase transition at the boundary between paramagnetic and ferrom agnetic phases.\nThe negative slope continuously increased with measurement temperature above 300 K.\n23Current-voltage characteristics and Electrical conductivity: The thermal cycling effect on I-\nV curve measurement was studied for CF_50 and CF_90 samples during warming (W) and \n                 \ncooling  (C)  modes.  After  completing  the  measurement  during  first  cycle  (W1C1),  the\nmeasurement was repeated during second cycle (W2C2). Figure 8 (a-b) shows I-V characteristi cs\nof the CF_50 sample at selected temperatures during W1C1 and W2C2 c ycles. The CF_90\nsample showed similar I-V characteristics and the data during W2C2 cycle are shown in Figure 8\n24(c). The I-V characteristics at positive bias voltage are identical t o the characteristics at negative\nbias voltage. We analyzed I-V characteristics at different temperatures  using power law (I = I 0Vn)\nto understand space charge effect. The exponent ( n) was obtained from the slope of log-log plot\nof I-V curves. The I-V curves at all temperatures are best fitted with  two slopes. The fit and\nexponent ( n) values are given in supplementary information (Figure S2). The slope (n1) at  low\nvoltage regime is smaller (1-1.2 for CF_50 sample and 1-1.1 for CF_90 sample ) in comparison\nto slope (n2) at high voltage regime (> 10 V). The  n1(T) curves showed small irreversibility\neffect in comparison to  n2(T) curves. The higher values (1-1.8)  of n2(T) curves show some\ncontribution from interfacial space charges at higher voltage regime [32]. How ever, n1 values\nclose to 1 suggest Ohmic behaviour at low voltage regime. It confirms proper  electrical contact\nbetween  electrode  and  disc-shaped  sample.  The  resistance  (R=  V/I)  of  the  samples  was\ncalculated  from the I-V curves at bias voltages  +5 V, +20 V and +40 V. The temperature\ndependence of resistance (R(T)) curves of the samples showed high resistance (low conducti vity)\nduring warming mode and low resistance (high conductivity) during cooling mode. A s shown in\nFigure 8(d-e), the irreversible feature and magnitude of the resistance during W1C 1 cycle of the\nCF_50 sample appeared larger than the W2C2 cycle. By comparing the  results in W2C2 cycle,\nresistance  of the  CF_50 sample is  found smaller  (higher conductivity)  tha n  that  in CF_90\nsample.  Interestingly,  R(T)  curves  transformed  from  an  unusual  metal  like  state  to  usual\nsemiconductor state above the temperature T MSW during warming mode and semiconductor to\nmetal like state below the temperature T SMC during cooling mode. The temperature T MSW (∼ 443\nK) in warming mode of the W1C1 cycle for CF_50 sample is higher than  the value ( ∼ 373 K) in\nW2C2 cycle and the same value (373 K) is found for CF_90 sample. The t emperature T SMC is\nfound at about 330 K for both the samples and irrespective of the repetition  of cycles. The\n25positive temperature coefficient of resistance may not represent a t rue metallic state as free\nelectrons are not generally expected in spinel oxides [33-35]. However, trapping  and de-trapping\nof the electronic charges (electrons and holes) in the defect induced localized states at the edges\nof conduction (CB) and valence (VB) bands can contribute to metal like state in spinel oxide\n[34]. A detailed discussion has been made later by considering the defe cts. The R(T) curves in\nthe semiconductor state were fitted with Arrhenius law: R(T) = R 0exp (E a/kBT). The fit of the lnR\nvs 1000/T data at 5 V are shown in the insets of Figure 8(a-c). The act ivation energy (E a) of the\nCF_50 sample is found to be notably high 3.413 eV during warming mode in compa rison to\n0.635 eV in the cooling mode of W1C1 cycle. In the W2C2 cycle, activation energy decreased to\n0.679 eV and 0.617 eV during warming and cooling modes, respectively. The ac tivation energy\nof the (high temperature annealed) CF_90 sample in W2C2 cycle was found 0.99 eV and 0.67 eV\nduring  warming  and  cooling  modes,  respectively. The  activation  energy  in  our  samples  is\ncomparable to the values in spinel oxides [33, 36]. \n■ DISCUSSION\nNow, we understand a correlation between lattice structure, magnetism  and conductivity\nin the present spinel oxide by referring some of the literature reports. A defe ct free III−II spinel\noxides  is  supposed  to be  magnetically  ferrimagnet  (FIM)  and antiferromagnet  (AFM),  and\nelectrically insulator if all divalent cations (Fe2+, Co2+) occupy at A sites and all trivalent cations\n(e.g., Fe3+, Co3+) occupy B sites. It becomes magnetically disorder and electrica lly conductive\nunder site exchange of the cations [37]. The semiconductor properties in Co xFe3-xO4 ferrite can be\neither  p type (hole hopping through Co2+–O2-Co3+ superexchange paths for x > 1) or  n type\n(electron hopping through Fe3+–O2-Fe2+ superexchange paths for x < 1) [11, 36]. The ideal\nmagnetic spin order and charge conduction are substantially modified in  the presence of defects\n26[20].  Huang  et  al.  [22]  proposed  different  intrinsic  point  defects  (V Co(B),  V Fe(A),  V Fe(B),  V O,\nCo(B) Fe(A), Co(B) Fe(B), Fe(B) Co(B), [Co(B) Fe(A), Fe(A) Co(B)]) for CoFe 2O4. The V Co(B) is the vacancy\nof Co ions at B sites; V Fe(A) is the vacancy of Fe ions at A sites; Co(B) Fe(A) represents the vacancy\nof Fe ions at A sites substituted by Co ions at B sites, and [Co(B) Fe(A), Fe(A) Co(B)] represents\nexchange between Co ion at B site and Fe ion at A site. The creation of vacancies in spinel\nstructure is site specific. The energy required for creating cation defects at B sites are high in\nmetal (Co/Fe) rich condition and energy values can increase up to  4.72 eV and 3.69 eV for V Fe(B)\nand V Co(B), respectively. The energy for forming V Fe(A) is high under all conditions and it can go\nup to 7.20 eV, 5.68 eV and 3.69 eV under metal rich, Co rich and oxygen ri ch conditions,\nrespectively. The V Co(A)  is not energetically favoured, but energy required to form [Co(B) Fe(A),\nFe(A) Co(B)] through exchange of Fe(A) and Co(B) ions is small ( ∼0.65 eV) and favoured than\nother defects. \nRietveld refinement confirmed non-stoichiometry in metal/oxygen ratio. The composi tion\nof our system is close to Co 3O4 [(Co2+)A[Co3+Co3+]BO4], where divalent Co ions occupy 1/8 of the\n(A sites) tetrahedral holes and trivalent ions occupy 1/2 of the (B sit es) octahedral holes. Rest of\nthe lattice spaces is vacant and expansion of the lattice space  at higher temperature allows\nadsorption of oxygen atoms. The chemical formula unit can be modified as A 1-x1B2-x2[]yO4 to\npreserve the oxygen packing in the presence of cation vacancies or AB 2O4+δ to incorporate excess\noxygen  [21,  38].  The  excess  oxygen  content  ( δO)  per  formula  unit  increases  at  higher\nmeasurement temperatures in the present oxide, irrespective of bi-phased or single-phased spinel\nstructure. Of course, non-equilibrium structure of CF_20 sample adsorbed more oxygen atoms\nand equilibrium spinel structure of CF_90 sample contained small amount of  excess oxygen.\nConsidering complex lattice structure of the spinel oxide, certain rest rictions in the distribution\n27of Co/Fe ions at A and B sites were followed for simplicity. The structural information may not\nbe affected much by a simple distribution of Co and Fe ions, beca use both Co and Fe ions have\nnearly equal number of electrons (atomic scattering factor). It may lim it an exact determination\nof the Co/Fe ratio between A and B sites. However, changes in the Co/Fe ratio between A and B\nsites  can  be  realized  qualitatively  from  the  irreversibility  effe ct  in  phase  fraction,  lattice\nparameters, magnetization, and conductivity difference during thermal cycling [18-19]. \nNow, we discuss the role of defects on modified magnetism and elect rical conductivity.\nAccording to Bean-Rodbell theory [39], distorted spin-lattice structure introduce s strain induced\nenergy. Net free energy in the system is lowered by distorting t he lattice in a direction that\nincreases magnetic exchange energy. A linear relation between the e xchange energy constant (λ)\nand Curie temperature (T C) with the change of cell volume (V) and strength of magnetoelasti c\ncoupling (β) can be expressed as  λ = λ 0(1+\n ) and T C = T 0(1+\n ) [25]. The coupling\nbetween structural disorder and spin order introduces first order magnetic phas e transition [24-\n25], where free energy corresponding to meta-stable and stable states provides two minima in\nM(T, H) curves. The MFW(T) curves correspond to meta-stable state (cons ists of clamped/rigid\nstructure with small lattice distortion at lower temperatures and  free structure with strong lattice\ndistortion at higher temperatures) and MFC(T) curves represent a stable state. The FM-PM\ntransition temperature (T CMFW) in MFW(T) curves is expected higher than the PM-FM transition\n(TCMFC) in MFC(T) curves. We define T 0MFW (∼ 680 K-700 K) as the Curie temperature of\nclamped structure and T CMFW (∼ 980 K) as the Curie temperature of free structure or defect\ninduced ferromagnetic  state. In the inter-mediate state between T CMFW and T CMFC, a thermal\n28hysteresis (MFC(T) <MZFC(T)) is observed during cooling mode because the s ystem remains in\nPM state until temperature reached close to \"true\" Curie point (T CMFC) to re-establish its ferro/\nferrimagnetic state. The T CMFC values are found close to T 0MFW for the samples (CF_20 and\nCF_50) with low annealing temperature and a substantial difference (T CMFC> T 0MFW) is noted for\nCF_90 sample. This is due to minor changes in meta-stable state  for those samples, where\nmeasurement temperature (up to 950 K) is higher than the annealing tem perature (473 K for\nCF_20 sample and 773K for CF_50 sample). In CF_90 sample, meta-stabl e state is not much\naffected because measurement temperature (950 K) is less than it s annealing temperature 1173\nK. The magnetization below 600 K is smaller for CF_90 sample, whos e lattice structure is close\nto Co 3O4 with low magnetic (AFM) state and T N ∼ 30-40 K [40-41], except small replacement of\nCo by Fe atom. The higher moment in CF_20 and CF_50 samples is cont ributed by more lattice\ndefects [20, 27]. The local ordering of 3d electronic spins of Co and Fe ions in doubly and triply\ndegenerated e g and t 2g levels of octahedral sites control the magnetic spin order and charge\nhopping process at B sites of the spinel structure [1]. The magnetic moment  in defect free spinel\nstructure is determined by the difference of ionic moments in A and B s ites (M = M B- M A). The\nCo2+ ion (moment ∼ 2.52 µB) at A sites contains three unpaired electrons at t 2g level and at B sites\ncontains one unpaired electron at t 2g level and two unpaired electrons at e g level, respectively.\nEach Fe3+ ion (moment ∼ 3.91 µB) at A site or B site contains five unpaired electrons at e g (two)\nand t 2g (three) levels. On the other hand, each Co3+ ion at B sites results nearly zero moment due\npaired six electrons at t 2g level. The calculated magnetic moment for Co2+  and Fe3+ ions in Ref.\n[1] are relatively smaller than the usual values from spin contribution alone (3 µB for Co2+ and 5\nµB for Fe3+). The smaller value of magnetic moments for Co2+ and Fe3+ ions in the highly Co rich\n29spinel oxide samples is expected due to the fact of strong hybridization of unoccupied 3d orbital\nof Fe/Co with 2p orbital of O. The DFT calculations show simil ar DOS contours for defective\nand defect-free spinel structure of CoFe 2O4 [22], except extra levels are introduced within band\ngap in the defective structure. It may be mentioned that an O ion vacancy (V O) produces two\nelectrons ( n type conductivity) per defect site (O O  → [V O0  +2e′] +\n O2(g)), where as a Co2+ ion\nvacancy (V Co) produces two holes ( p type conductivity) per defect site (Co Co→ [V Co0+2h•] +\nCo(s)) . In the presence of V Co(B)0 (also V Fe0 both at A and B sites), some occupied levels with spin\nup (↑) states at the Fermi energy and unoccupied levels with spin up  states are created in the\nband gap. In contrast, V O0 induces extra donor like level occupied by two electrons in the upper\npart of band gap in spinel oxide. The net moment in spinel structure decre ases in the presence of\nVFe(B) and Co(B) Fe(B) defects and increases in the presence of V Fe(A) defects. High concentration of\ndefects in Co rich spinel oxide also increases band gap. The exception ally high activation energy\n(3.4 eV) during warming mode of the W1C1 cycle measurements for CF_50 sa mple could be\nattributed to formation of V Fe(A)  defects under high oxygen rich condition. The reduction of its\nactivation  energy  during  W2C2  cycle  and  also  in  cooling  modes  suggest s  a  possibility  of\ndifferent Co/Fe ratio between A and B sites or changes in the fract ion of Co-rich/ Fe-rich spinel\nphases. It may be possible the system slowly approaches to equilibriu m structure under repeated\nthermal cycling, where small activation energy ( ∼0.62-0.99 eV) is needed for the formation of\n[Co(B) Fe(A), Fe(A) Co(B)] defects under site exchange of Co and Fe ions [22]. \nFinally, we suggest that the capacity of adsorbing oxygen atoms during warmi ng mode of\nhigh temperature measurement and subsequent release during cooling mode could be promising\nfor studying catalytic properties of surface atoms or surface engineering in Co rich spinel oxide.\n30This property is highly sensitive to surface response along (111) directi on in spinel structure [26]\nand structural changes in defective spinels during heating cycles [38]. The  structure of Co 3O4\nspinel oxide along (111) direction has been modeled as a sequence of Co3+/Co2+ layers separated\nby layers of O ions. The surface can by either be A-terminated (by Co2+ and Co3+ ions) or B-\nterminated (by Co3+ ions) or oxidized on the A (AO) or B (BO) surfaces. In case of the AO and\nBO terminated surfaces the band gap contains extra states located near to Fermi level. The extra\nstates are partially filled by d-orbitals of Co ions and p-orbitals of O ions. The number of extra\nstates increases for systems with A- or B-terminated surface. T he strongly delocalization of\nelectron states, which almost fill the forbidden gap, gives rise to metal like character and such\nbehavior most probably affected the temperature dependence of resistance curves in our s amples.\nThe Co2+ ions at A and AO-terminated surfaces have magnetic moment close to  that in bulk. The\nCo3+ ions are nonmagnetic in bulk, but they have a magnetic moment close to t hat of Co2+ ions at\nthe B-terminated surface and substantially smaller (non-zero) at the B O surface. The A and AO\nsurfaces  contribute  AFM  properties  owing  to  A-A  superexchange  interactions,  unli ke  FM\nproperties in B and BO surfaces due to B-B superexchange interactions. Alt hough beyond the\nscope of the present work, but we believe that a detailed study of high temperature prope rties can\nbring out a nice correlation between surface chemistry of metal ions and ferromagnetism [37-40],\nwhich is not expected in defect free (conventional) AFM or weak ferrimagnetic spinel oxides.\n■ CONCLUSION\nThe structure of Co 2.75Fe0.25O4+δ (δ: 0-0.68)  spinel oxide becomes non-stoichiometric during\nhigh temperature measurement and re-gains stoichiometric value of oxygen on reversing back to\nroom temperature. The high temperature measurement indicated non-equilibrium lattice structure\nby showing a preferential orientations of (511) and (440) planes of cubic spinel structure for the\n31sample at low annealing temperature (200 0C) and an additional phase of CoO with preferential\n(200) plane orientation with thermal hysteresis for the sample annealed  at 900  0C.  The lattice\ndistortion is more prominent for the samples at low temperature annealing in comparison to the\nsample annealed at 900 0C. The samples exhibited irreversible effect in the structure, magne tic\nand electrical properties between warming and cooling modes of the me asurements. The meta-\nstable magnetic state exhibited two magnetic transitions during fiel d warming mode. The defect\ninduced ferromagnetic transition at higher temperature is suppressed during fie ld cooling mode\nand a ferrimagnetic state is stabilized below a low Curie tempera ture. The coupling between\nstructural disorder (defects) and magnetic spin order introduces a first order phase transition. We\nhave proposed in graphic abstract a possible correlation between lattice  structure, magnetism and\nelectrical conductivity, and promising catalytic application of the studied Co rich s pinel oxide. \n■ ASSOCIATED CONTENT\nAdditional figures are shown in supplementary information. Table S1 shows the  information of\nRietveld refinement using SXRD pattern at 873 K and adopting the approa ch of either variation\nof Oxygen content at 32e sites or variation of Co content at 16d sites . Figure S1 shows the\ntemperature variation of the SXRD intensity for selected peaks (normal ized) of the CF_20 and\nCF_90 samples during thermal cycling process of SXRD measurements. F igure S2 shows the fit\nof I-V curves using power law and temperature variation of the power law exponent during\nthermal cycling process of measurements.\n■ ACKNOWLEDGMENTS\n We acknowledge the Research grant from UGC-DAE- CSR (No M-252/2017/1022), Gov.  of\nIndia. We thank RRCAT-Indus 2, Indore and UGC-DAE- CSR Mumbai Centre for provi ding\n32high temperature synchrotron x-ray diffraction and magnetization measurement  facilities. RNB\nthanks Dr. Archana Sagdeo and Mr. M.N. Singh for assisting in SXRD measurements.\n■ REFERENCES\n[1] Walsh, A.; Wei, Su-H.; Yan, Y.; Al-Jassim, M.M.; and Turner, J.A. St ructural, magnetic, and \nelectronic properties of the Co-Fe-Al oxide spinel system: Density-functional theory \ncalculations.  Phys. Rev. B 2007 , 76, 165119. \n[2] Hayashi, K.; Yamada, K.; Shima, M. Compositional dependence of magnetic anisotropy in \nchemically synthesized Co 3-xFexO4 (0 ≤  x  ≤ 2). Jpn. J. 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H.; Ansaldo, E. J.; Morris, G. \nD.; Andreica, D.; Amato, A. Spatial inhomogeneity of magnetic moments in the cobalt oxi de \nspinel Co 3O4. Phys. Rev. B 2007 , 75, 054424.\n[41] Roth, W.L. The magnetic structure of Co 3O4. J. Phys. Chem. Solids 1964 , 25, 1-10.\n37Table of contents\n1. Graphic abstract\n2.  Synopsis\n3. Supplementary information\n4. Table S1\n5. Figure S1\n6. Figure S2\nSynopsis\nSynchrotron XRD, magnetization and current-voltage curves were measured from 300  K to\nhigh temperature and back to 300 K for investigation of the irreversibili ty effect on structure and\nphysical properties. The material was prepared by chemical route and sui tably heat treated to\nform single phase and bi-phased materials. A correlation between latt ice structure, magnetism\nand electrical conductivity, and also a promising catalytic applic ation of the Co rich spinel oxide\nhave been schematically proposed. \n381 \n SUPPLEMENTARY INFORMATION  \nHigh temperature thermal cycling effect on the irreversible response s of lattice structure, \nmagnetic properties and electrical conductivity in Co2.75Fe0.25O4+ spinel oxide  \n            R.N. Bhowmik*a, P.D. Babub, A.K. Sinhac,d, and Abhay Bhisikarc \n \naDepartment of Physics, Pondicherry University, R.V. Nagar, Kalapet, Pondicherry 605014, \nIndia  \nbUGC -DAE Consortium for Scientific Research, Mumbai Centre, Bhabha Atomic Research \nCentre, Trombay, Mumbai -400085, India  \ncHXAL, SUS, Raja Ramanna Centre for Advanced Technology, Indore - 452013, India  \ndHomi Bhabha National Institute, Anushakti Nagar, Mumbai -400 094 India  \n*Corresponding author: Tel.: +91 -9944064547; E -mail: rnbhowmik.phy@pondiuni.edu.in  \n \n The cubic spinel structure of Co 2.75Fe0.25O4 spinel oxide is modeled close to normal spinel \nstructure [(Co) tet[Co3+Co3+]octO4] of Co 3O4 with Wyckoff positions at tetrahedral (8a) sites ( 1/8, \n1/8, 1/8) fully occupied by Co2+ ions (or minor occupancy of Fe3+ ions), at octahedral (16d) sites \n(1/2, 1/2, 1/2) co -occupied by Co3+ and Fe3+ ions, and at 32e sites  occupied by oxygen (O2-) ions. \nThe lattice structure was refined using single phase and two -phase models. Some of the fit \nparameters (e.g., isotropic thermal displacements ( B), oxygen parameter ( u), occupancy of the \nCo atoms at tetrahedral sites and Fe at octahedral sites) were suitably refined. The final \nrefinement was carried out by adopting two approaches. In the first approach, occupancy of O \natoms at 32e sites was allowed to vary from the value 4 and fixed the 16d sites occupancy of Co \natoms and Fe atoms to 1.75 and 0.25, respectively . In the second approach, 16d sites occupancy \nof Co atoms allowed to vary and fixed occupancy of Fe atoms to 0.25 at 16 sites and occupancy \nof O atoms to 4 at 32e sites. In case of two phase model for CF_20 sample, we defined Co rich 2 \n phase as phase 1 (where Co content at 16d sites is assigned more amount than the assigned value \nfor single phase) and the Fe rich phase as phase 2 (where Co content at 16d sites is less and total \nFe content at 8a and 16d sites is more than the assigned value for single phase model). The \noccupancy of Co and Fe atoms at 8a sites and Fe atoms at 16d sites are suitably fixed and final \nrefinement of two phase model was per formed either by making occupancy of Co atoms at 16d \nsites fixed (1.95 for Co rich phase and 1.75 for Fe rich phase) and O atoms variable or occupancy \nof Co atoms at 16d sites variable and O atoms fixed.  We noted that the structural parameters \n(lattice par ameter ( a), oxygen parameter ( u)) and the refinement parameters (R p, Rwp, Rexp, 2) do  \nnot differ much whether occupancy of O atoms at 32e site is fixed to 4 and occupancy of Co \natoms at 16 sites allowed to vary or occupancy of Co atoms at 16d sites is fi xed to 1.75 and \noccupancy of O at oms at 32e site allowed to vary . Table S1 summarizes structural parameters \nfrom refinement of SXRD patterns at 300 K using single phase for CF_90 sample and  two-phase \nmodel at 873 K for CF_20 sample.  Figure S1 shows the temperature variation of the SXRD \nintensity  for selected peaks (normalized) of the CF_20 and CF_90 samples during thermal \ncycling process of SXRD measurements. The exponent parameters from fit of I -V curves using \npower law (I = I 0Vn) are given in Figure S 2. 3 \n \n0.00.51.0\n0.00.51.0\n0.51.0\n0.51.00.00.51.0\n300 400 500 600 700 800 9000.51.00.51.0\n0.51.0\n0.51.0\n0.51.00.51.0\n3004005006007008009000.51.0cooling \n  \n(111)warming\n  \n(400) \n (422) \n (333/511)\n  \n(311)\n \nTemperature (K)(440)coolingwarming\n  \n  \n(111)\n  \n (311)\n  \n (400) \n (333/511) \n (422)Normalized peak intensityCF_90 sample\n \nTemperature (K)(440)CF_20 sampleNormalized peak intensity\nFigure S1 Temperature dependence of SXRD peak intensity in warming and cooling \nmodes for the samples of CF_20 (left side) and CF_90 (right side). 4 \n \n1.01.21.4\n300 400 500 6001.01.21.41.61.81 1010-710-510-3\n1 1010-710-610-510-4\n300 400 500 6001.01.21.41.61.81.01.11.2(c)\nexponent (n1)  \n Warming\n Coolingexponent (n1)\nFigure S2. Current vs. voltage curves at 300 K and 573 K in log-log scale to show power \nlaw fit (a-b). Temperature variation of the exponent values at low and high voltage \nregimes for the samples CF_50 (c-d) and CF_90 (e-f), respectively.(e)\nexponent (n2) exponent (n2)\nCurrent (A)  \nn2= 1.222(10)\nn1 = 1.030(6)\nn2 = 1.167(15) 300W\n 573WCurrent (A) \nn1= 1.136(6)CF_50 sample\n(a) (b)\n  \nn2 = 1.412(4)n1= 1.000(5)\nn2 = 1.656(9) 300W\n 573W\nBias voltage (V)n1= 1.015(4)CF_90 sample\n(f)\n  Warming\n Cooling(d)\n   Table S1 (supplementary) . Rietveld refinement parameters  \n(a) Refinement of SXRD data at 873 K fitted with single cubic spinel structure  for CF_90 sample  \nAtoms  \n(sites)  occupancy of O fixed to 4 and Co(16d) variable  occupancy of Co(16d) fixed to 1.75 and O  variable  \n   Wyckoff positions      B Occupancy     Wyckoff positions      B Occupancy  \n       X                 Y                   Z          X                     Y                   Z    \nCo(8a)   0.12500          0.12500       0.12500  0.97 1.000     0.12500           0.12500        0.12500  0.97 1.000  \nFe(8a)  0.12500          0.12500        0.12500  0.97 0.000    0.12500            0.12500        0.12500  0.97 0.000  \nFe(16d)  0.50000          0.50000        0.50000  0.97 0.250    0.50000            0.50000         0.50000  0.97 0.250  \nCo(16d)  0.50000          0.50000        0.50000  0.97 1.738( 11)   0.50000            0.50000        0.50000  0.97 1.750  \nO (32e)  0.26127 (50)  0.26127( 50)  0.26127(50)  0.78 4.000(0)  0.25910(73)  0.25910(73)  0.25910(73)     0.78 4.069 (64) \nCell parameter ( a) = 8.18980 (19) Å, volume ( V) = 549.313 (23)Å3 \n Rp: 10.5, Rwp: 13.8,  R exp: 12.57 , 2:  1.20 Cell parameter ( a) = 8.18966 (19) Å, volume ( V) = 549.285 (22)Å3 \n Rp: 10.3, Rwp: 13.7,  R exp: 12.56 , 2:  1.19 \n \n (b) Refinement of SXRD  data at 300  K (warming mode) using two-phase model of cubic spinel structure  for CF_20 sample  (O fixed)  \nAtoms  \n(sites)  Co rich phase ( Co at 16d sites variable)  Fe rich phase ( Co at 16d  sites variable ) \n   Wyckoff positions      B Occupancy     Wyckoff positions      B Occupancy  \n       X                 Y                   Z          X                      Y                   Z    \nCo(8a)   0.12500          0.12500       0.12500  0.97 1.000     0.12500           0.12500        0.12500  0.97 0.850  \nFe(8a)  0.12500          0.12500        0.12500  0.97 0.000    0.12500            0.12500        0.12500  0.97 0.150  \nFe(16d)  0.50000          0.50000        0.50000  0.79 0.050   0.50000            0.50000        0.50000  0.79 0.190  \nCo(16d)  0.50000          0.50000        0.50000  0.79 2.026(7)   0.50000            0.50000        0.50000  0.79 1.602(17)  \nO (32e)  0.26047(27)  0.26047( 27)  0.26047(27)  0.72 4.000   0.26212(59)  0.26212(59)  0.26212( 59)    0.72 4.000  \nCell parameter (a) = 8.11062 (30) Å, volume (V) = 533.535 (34)Å3 \nCo rich phase (Bragg R -factor: 4.7 7, phase fraction: 6 4.81%) \nFe rich phase (Bragg R -factor: 6.21, phase fraction: 35.19%) Cell parameter (a) = 8.17313 (73) Å, volume (V) = 545.965 (85)Å3 \n Rp: 4.12, Rwp: 5.20,  R exp: 2.39, 2:  4.72 \n "    },    {        "title": "1911.07496v1.Chemical_synthesis_and_magnetic_properties_of_monodisperse_cobalt_ferrite_nanoparticles.pdf",        "content": "Chemical Synthesis and Magn etic Properties of Monodisperse \nCobalt Ferrite Nanoparticles  \n \nZ. Mahhouti1,2,3 *  , H. El Moussaoui1 , T. Mahfoud1, M. Hamedoun1 , M. El Marssi3 , A. \nLahmar3 , A. El Kenz2,  and A. Benyoussef1,2,4 \n1 MAScIR Foundation, Institute of Nanomaterials and Nanotechnologies, Materials and \nNanomaterials Center, B.P. 10100, Rabat, Morocco  \n2 LaMCScI , URAC 12, Département  de Physique, Faculté des Sciences, Université \nMohammed V, B.P. 1014, Rabat, Morocco  \n3 LPMC EA2081, Université de Picardie Jules Verne 33 Rue Saint Leu, 80000 Amiens, \nFrance  \n4 Hassan II Academy of Science and Technology, Rabat, Morocco  \n \n \nAbstract  \nIn this work, a successful synthesis of magnetic cobalt ferrite (CoFe 2O4) nanoparticles \nis presented. The synthesized CoFe 2O4 nanoparticles have a spherical shape and highly \nmonodisperse in the selected solvent.  The effect of different reaction conditions  such as \ntemperature, reaction time and varying capping agents on the phase and morphology is studied. \nScanning transmission electron m icroscopy  showed that the size of these  nanoparticles can be \ncontrolled  by varying reaction conditions.  Both X-ray diffraction and energy dispersive X -ray \nspectroscopy corroborate the formation of CoFe 2O4 spinel structure with cubic  symmetry . Due \nto optimized reaction parameters, each nanoparticle was shown to be  a single magnetic domain \nwith diameter ranges from 6 nm to 16 nm. Finally, the magnetic investigations showed that the \nobtained nanoparticles are superparamagnetic with a small coercivity  value of about 315 Oe \nand a saturation magnetization of 58 emu/g at room temperature . These  results make  the cobalt \nferrite nanoparticles promising for advanced magnetic  nanodevice s and biomagnetic \napplications . \nKeywords:  \nFerrite; Spinel; Ferrite; Nanoparticles; Magnetic properties;   \n \n* Corresponding author: zakaria.mahhouti@etud.u -picardie.fr   \n1. Introduction  \n The magnetic nanoparticles with spinel structure MFe 2O4 (M = Fe, Co, Mn, Zn, Ni ...) \nhave been widely studied for their  properties compatible with  various  applications ranging from \ndata st orage to biomedical applications [1, 2, 3, 4, 5]. Recently, a  special interest is devoted to \nmagnetic  nano -object materials  [6, 7], because they endorsed interesting magnetic properties, \nwith the possibility of tailoring their functionalities, by controlling the shape and morphology. \nParticularly, magnetic nanoparticles ( MNPs) c an be tuned in a straight  forward manner by  the \ncontrol of the  size, monodispersity, chemical composition, as well as the adequate synthesis \nroute , which is desirable for advanced magnetic  nanodevice s or magnetic hyperthermia.  \nIt is worthy to mention that a monodomain nan oparticle has a permanent magnetic \nmoment, which is the sum of all magnetic moments of the atoms constituting  it. However, \nduring the structuring of the magnetic monodomain, the reduction of the total number of atoms \n(on the nanometric scal e) leads to an increase in the contribution of surface atoms that do not \nhave the same environment as in the core of the nanoparticle.  \nThe critical diameter d C from which the particle can be considered as a magnetic \nmonodomain is defined by  Frey et al.  [1]: \n𝑑𝐶= 36√𝐴𝐾𝑒𝑓𝑓\n𝑀𝑠2𝜇0  (𝐸𝑞.1) \nWhere Keff is the effective anisotropy and A is the exchange constant. 𝜇0 is the vacuum \npermeability and 𝑀𝑆 is the saturation magnetization. 𝑑𝐶 is in the range  of 10 -100 nm.  \nThe contribution of s urface e ffects affect the magnetic prop erties of the material  [8]. \nIndeed, in addition to the core spins as in the bulk mat erial, the nanoparticles have  surface spins, \nthat creating supplementary interactions.  Therefore,  for controlling  the physical and chemical \nproperties of nanoparticles, it is necessary to control the size, morphology, monodispersity , and \nchemical composition of the nanoparticles.  \nFor instance , cobalt ferrite  has a ferromagnetic behavior at ambient conditi ons, with \nhigh magnetic  coercivity  [8], a high Curie temperature at the vicinity of 793 K, strong \nmagnetocrystalline anisotropy  [8, 9], as well as  a large magnetostriction coefficie nt [10]. These \nproperties are very attracting for advanced technological devices , namely  in data storage and  in \nthe biomedical applications. Magnetic order in cobalt ferrite arises from the superexchange interaction between the cations  located in  tetrahedral and octahedral sites through the oxygen \nanion. The induced antiferromagnetic coupling between the Fe3+ cations  in the tetrahedral sites \nand the Co2+ and Fe3+ cations  in the octahedral sites is strong; although  another weak  \nantiferromagnetic coupling is present between tetrahedral  Fe3+ cations  [11, 12]. In addition, a  \nweak ferromagnetic coupling also exists between the cations of the o ctahedral sites . The two \nlast couplings are masked by the interactions between tetrahedral and octahedral sites.  \nTo date , several synthesis methods of MNPs  have been developed  [13, 14, 15, 16, 17,  \n18] in an effort to improve the magnetic properties by  control ling the size, the mo rphology , and \nthe composition of the  obtained  nanoparticles. Among these different routes of synthesis, we \nhave found the co-precipitation s, solvothermal , hydrothermal and thermal decomposition which \nare the most effective ones. The co -precipitation method has  been used to synthesize  crystals \nwith different morphologies includin g spherical, cubic and nanorods [19]. Using solvothermal \nand h ydrothermal methods , nanocrystals of iron oxide have been grown as  spheres and \nhexagons  [16, 17]. Thermal decomposition method has produced monodispersed nanoparticles \nof spinel ferrite with a narrow size distribution and good c rystallinity  [22].  In this respect,  the \npresent work  report s on the synthesi s of CoFe 2O4 NPs by decomposition of acetylacetonate \nprecursors at high temperature . Among many  advant ages of this synthesis route, the ability to \ncontrol the particle size, size distribution, shape, and phase purity. The thermal decomposition \napproach has been chosen because  the synthesis system is simple with one type of complex es, \none type of  ligand s and a high boiling point organic solvent.  The obtained nanoparticles  are \nmonodisperse with  varied  morphologies and sizes . \nThe CoFe 2O4 nanoparticles synthesized with  the standard  protocol were characterized \nusing many experimental techniques such as ThermoG ravimetric A nalysis ( TGA ), X-Ray \nDiffraction ( XRD ), Fourier Transform InfraRed spectroscopy ( FT-IR), Scanning Transmission \nElectron Microscopy (STEM), Energy Dispersive X -ray Spectroscopy (EDS) and Magnetic  \nProperty Measurement System ( MPMS ) SQUID magnetometer .  \nOur main research topic in this work is especially the development of low cost, \nflexibility, and ease of chemical synthesis of CoFe 2O4 nanoparticles (NPs).  To deep \nunderstanding  how some synthesis parameters affect the nucleation and growth steps;  the \ndecomposition temperature, reflux time, nature of solvents, the quantity of surfactants were \ninvestigated.  Therefore , the CoFe 2O4 nanoparticles obtained by varying the experimental conditions were characterized by STEM in order to describe the influence of synthesis \nparameters on the size and shape of NPs.  \n2. Experimental  section  \n2.1. Chemicals  \nThe synthesis was carried out using commercially available reagents. The starting \nPrecursors were  iron(III) acetylacetonate (Fe(acac) 3, 99.99%), and cobalt(II) acetylacetonate \n(Co(acac) 2, 99%). The used solvents were Benzyl ether (98%, boiling point: 298 °C), absolute \nethanol (100%), and hexane (98.5%). For the surfactants and reductant  we used oleic acid (90%, \nboiling point: 360 °C), oleylamine (70%, boiling point: 350 °C), and 1,2 -hexadecanediol (90%).  \nAll the chemicals were purchased from Sigma -Aldrich Ltd. and were used as received without \nfurther purification.  \n2.2. Synthesis of coba lt ferrite nanoparticles  \nInto a 100 mL three -necked flask under nitrogen flow , we placed 4  mmol of Fe(III) \nacetylacetonate, 2 mmol of Co(II) acetylacetonate, 20 mmol of 1,2 -hexadecanediol, 12 mmol \nof oleic acid, 12 mmol of oleylamine , and 40 ml of benzyl ether. That is to say in proportions \nfive times higher for the hexadecanediol compared to the Fe(III) acetylacetonate  and six times \nhigher for the surfactants (oleic acid and oleylamine) compared to the Fe(III) acetylacetonate . \nThermal controlling is  carried out using a thermocouple probe to control the temperature and \nthe duration of the high temperature treatment.  The reaction mixture was magnetically stirred \nand de gassed at room temperature for 60 min , then was heated and kept  at 100 °C  for 30 min \nto remove water. Subsequently, t he temperature was increased  and kept  at 200 °C, for 30 min , \nthen, heated (to reflux) and kept at 300 °C for 60 min. The final mixtur e is cooled to room \ntemperature and purified  three times  with ethanol and hexane . A black magnetic precipitate is \nobtained after magnetic settling. The precipitate is redispersed in 20 ml of hexane and a \nferrofluid composed of surfaced  CoFe 2O4 nanoparticles is obtained.  Figure 1 illustrates the \nthermal decomposition process. It is int eresting to note herein that the presence of the used \nsurfactants help s the good dispersion of the obtained NPs in hexane. However,  the presence of \nhexadecanediol helps to initiate the reaction by promoting the decomposition of the metal \nprecursor’s acetyl acetonat es. The choice of benzyl ether as an appropriate solvent for this \nprocess because its boiling temperature (298 °C) is higher than the decomposition temperature \nof precursors . The equation of the reaction is as follows  [23]: Co(C5H7O2)2 + Fe(C5H7O2)3→CoFe 2O4 + CH3COCH 3 + CO2 (𝐸𝑞.2) \nAccording to the equation ( Eq. 2), in the presence of oleylamine, oleic acid and 1,2 -\nhexadecanediol, thermal decomposition of acetylacetonates of cobalt and iron produced cobalt \nferrite nanoparticles, releasing acetone and carbon dioxide as by-products.  \n \nFigure  1: Illustration of thermal decomposition method . \nVarious synthesis parameters described above in the initial protocol have been modified \nin order to know the influence of synthesis parameters on the shape and size of nanoparticles: \nthe decomposition temperature, the duration of the heat treatment or the quantity of the reagents. \nThis also allowed to better understand the role of reagents such as hexadecanediol, oleic acid \nor oleylamine in the synthesis.   \n2.3. Characterization techniques  \nIn order to  get information about the mas s loss of CoFe 2O4 NPs, thermogravimetric \nanalysis (TGA, TA Instrument Q500) was used to know the percentage and the degradation \ntemperature of the organic molecules  on the surface of nanoparticles.  The sample was analyzed \nunder an inert atmosphere, the heating rate is 10°C/min, the temperature range is between 25 -\n600°C and the mass used is between 10 and 30 mg.  \nFourier Transform - Infrared Spectroscopy (FT -IR) spectra were recorded in the region from \n250 to 4000 cm-1 by using ABB Bomem FTLA2000 on KBr -dispersed sample pellets.  In order \nto avoid the signal saturation effects , the studied powders are diluted with KBr (transparent to \ninfra-red radiation ), and compressed into a disk with a diameter of  1 cm , in the form of pellets \nconsisting of 30 mg of KB r and 1 mg of the sample. The spectra was  recorded between 400 and \n4000 cm-1 and processed using the Win -IR software.  \nX-ray powder diffraction  (XRD)  patterns of  the nan oparticle assemblies were collected on a \nBruker D8 Discover diffractometer under C uK1 radiation ( =1.5406 Å) at 25°C . Scanning \nangle 2θ ranging from 10° to 100° with a step of 0.1°. The objective of this  analysis is the \ndetermination of the phases present in the samples, verification of the absence of secondary \nphases, the calculation of the unit cell parameter as well as the determination of the particle \nsize.  \nScanning transmission electro n microscopy ( STEM) studies and associated energy dispersive \nX-ray spectroscopy (EDS ) microanalysis were performed using a FEI electronic microscopy \noperating at 30 KV.  The nanoparticles were dispersed on holey carbon grids for STEM \nobservation.  EDS chemical analysis we re also carried out on several zones to determine locally \nthe quantity of the elements.  \nThe magnetic properties of the CoFe 2O4 nanoparticles were studie d at various temperatures \nusing a Quantum Design MPMS -XL-7CA SQUID magnetometer with a magnetic field strength \nup to 6 T.  The principle of this measurement is based on the displacement of the sample within \na set of superconducting detection coil.  During the  movement of the sample t hrough the coils \nat a given temperature and magnetic field, the magnetic moment of the sample induces an \nelectric current in the sensing coils. Any change of this current in the detection circuit induces  \na change of magnetic flux; therefore, by moving the sample on either side of the detection coils, \nthe magnetic flux is integrated. A flux tran sformer is used to transmit the signal to the SQUID.  \n3. Results and discussion  \n Figure 2 represents the mass loss of the synthesi zed CoFe 2O4 NPs as a function of \ntemperature under an inert atmosphere. As can be seen in Thermogravimetric  analysis (black \ncurve) and differential thermal analysis (blue curve), a mass loss of about 5 % is detected below \n300 °C  (573 K) , which  can be attributed to solvent remainders and adsorbed humidity. \nHowever,  near to 350  °C (623 K)  a mass loss of about 12.5%  is clearly identified as the thermal \ndegradation of the surfactants (oleic acid and oleylamine) on the surface of the nanoparticles  \n(the boiling point between 250°C (523 K)  and 360 °C  (633 K) ). Moreover,  no other peaks are \nobserved in the range of  test temperature, which means that there is no phase change of the material after  heating at high temperature (T ≤ 600 °C (873 K) ). This diagram confirms thermal \nstability and negligible structure leaching . \n \nFigure 2: Thermogravimetric analysis (black curve) and differential thermal analysis (blue \ncurve) of the synthesized CoFe 2O4 nanoparticles at atmospheric pressure.  \nFigure 3 represents the FT -IR analysis to identify the presence of functional groups of \norganic molecules surrounding the NPs as well as the vibrational modes of metal -oxygen bonds \nfor spinel structure.  As shown in the figure, t he principal vibrational modes of metal -oxygen \n(M-O) bonds are present between 300 and 670 cm-1 that correspond to metals in a tetrahedral \nor octahedral configuration for spinel structures. In general, bands of M -O bonds in the \noctahedral sites appear at 380 -450 cm-1, whereas they are around 540 -600 cm-1 for tetrahedral \nsites. In our case, a band of M -O bonds in the octahedral sites appear at 400 cm-1, whereas the \nband of M -O bonds in the tetrahedral sites appear s at 591 cm-1. \n \nFigure 3: The i nfrared spectrum of the synthesized  CoFe 2O4 nanoparticles.  \nFurther, the absorption bands observed in the range 670 -3700 cm-1 (see Figure 3) \ncorrespond to the vibration bands of the surfactant groups. The bands at 2924 and 2851 cm-1 \ncan be assigned to the asymmetrical and symmetrical stretching of CH 2 groups, chara cteristic \nof the hydrocarbon chain of the used surfactants. Two bands at 1453 and 1408 cm-1 are observed \nand correspond to the asymmetric and symmetric elongations of the carboxylate groups (COO- \nstretching). In addition, vibrational modes observed at 3440  and 1611 cm-1 correspond to the \nangular deformations of the amine groups (NH stretching and NH 2 bending, respectively), \nanother band appearing at 702 cm-1 correspond to CH 2 wagging.  \nIn order to verify the spinel structure and to estimate the particle size, X -ray diffracti on \nmeasurement  was carried out. The X -ray diffraction pattern of the synthesized CoFe 2O4 (Figure \n4) shows that the obtained diffraction peaks correspond well to the spinel structure (J CPDS No. \n04-016-3954) with face -centered cubic phas e. Traces of any other phases, kind of detectable \nimpurities  or intermediate phase  were not observed.  It is worthy to note that all other \nnanoparticles synthesized with a modification of the reaction conditions (see below) led to the \nsimilar phase purity and no clear difference could be spotted from the diffractograms  \nFigure 4: X-ray diffraction pattern of the synthesized CoFe 2O4 nanoparticles.  \nThe broad diffraction peaks obtained are expected for such small crystalline domains. \nThe Scher rer’s formula all ows estimating  the crystalli te size by taking the full width at half \nmaximum  (FWHM) of the main diffraction peak (311).  \n𝐷=0.9𝜆\n𝛽𝑐𝑜𝑠 (𝜃) (𝐸𝑞.3) \nWhere  is the X -ray wavelength,  is the broadening at half the maximum intensity (FWHM) \nof the hkl peak (in our case the 311 peak) and  is the Bragg an gle of this peak . Generally,  β is \ncorrected according to the formula √βx2−βsi2 , where 𝛽𝑥 is the experimental FWHM and 𝛽𝑆𝑖 is \nthe FWHM of a standard silicon sample . The average  crystalli te size estimated is  11.2 nm. This \nvalue can be used to calculate the specific surface area  using the  formula : 𝑆𝑆𝐴 =𝐴\n𝑉.𝜌 , where \nA is the surface area, V is the volume of nanoparticle (sphere in our case) and ρ is the theoretical \ndensity of CoFe 2O4 which is 5259*103 g.m-3 [24]. The specific surface area of the synthesized \nCoFe 2O4 nanoparticles is 102.04 m2.g-1. \nAccording to the images obtained by STEM (Figure 5), the synthesized CoFe 2O4 \nnanoparticles are spherical and highly monodispersed. The statistical analysis by ImageJ \n(macro particle size analyzer), allowed to obtain the histogram giving rise the information about \nthe size distribution of the nanoparticles. We can deduct from the histogram that the average \ndiameter of NPs is around 10.7  nm. \nIt is worth noting that the mean size calculated from STEM images is in good agreement \nwith that estimated from XRD. This agreement supports that these nanoparticles are single \ncrystals. The slight difference observed between the two techniques could be explained by the \nfact that for XRD only the largest particles are counted, whereas in STEM, the size distribution \nwith an average diameter is obtained on a limited number of particles (around 5 50 NPs).  \nFigure 5: STEM image  (a) and size distribution histogram (b) of the synthesized CoFe 2O4 \nnanoparticles . \nTo verify the formation of CoFe 2O4 phase, energy dispersive X -ray spectroscopy (EDS) \nwas used to analyze the chemical composition  as shown  in table 1 . In order to carry out the \nanalysis under good condition, we removed the organic molecules surrounded the surface of \nthe nanoparticles by a heat treatment at 400 °C for 2 hours. EDS results indicated that the ratio \nof Co/Fe is 1/2, which agree well  with the ratio of initial metal precursors. Thus, the final Co/Fe \ncomposition could be readily controlled.  This conclusion is in good agreement with the result \nobtained by Lu et al . [23]. \n \n \nElement  Line s.  Mass [%]  Mass Norm[%]  Atom [%]  \nOxygen  K-Serie  27.52  27.83  57.84  \nIron K-Serie  45.97  46.50  27.68  \nCobalt  K-Serie  25.37  25.67  14.48  \n  98.86  100.00  100.00  \n \nTable 1: EDS spectra for CoFe 2O4 nanoparticles.  \nOne of the main characteristics of CoFe 2O4 nanoparticles is that they are magnetic. In \norder to study their magnetic properties, measurements were carried out on CoFe 2O4 NPs with \nand without o rganic molecules. Notice that in order to  study CoFe 2O4 NPs without organic \nmolecules, we removed the orga nic molecules surrounded the surface of the nanoparticles by a \nheat treatment at 400 °C for 2 hours.  \nFigure 6 shows the magnetization curves as a function of the magnetic field at 300K and \n10K of CoFe 2O4 NPs. As it is clearly seen from the plots, a difference in saturation \nmagnetizations of about 12% is spotted between CoFe 2O4 NPs with and without organic \nmolecules. Therefore, the contribution of the organic molecules is estimated at 12%.  This value \nagree s well with that one found by TGA measurement (12.5%).  Figure 6: Magnetization as a function of magnetic field of CoFe 2O4 nanoparticles with and \nwithout magnetic field at 10 K (bottom) and 300 K (top).  \n The transition from the superparamagnetic state to the blocked state takes place at a \ntemperature  called blocking temperature ( TB). This depends on the material, the size of the \nparticles  and also  on the presence o f interparticle magnetic interactions . To determine ( TB), the \nevolution of the magnetization as a function of the temperature is performed unde r a constant \nmagnetic field of 1 00 Oe. Figure 7 shows the zero -field-cooled (ZFC) and field -cooled (FC) \ncurves of cobalt ferrite nanoparticles. The CoFe 2O4 NPs have a blocking temperature at the \nvicinity of 300K, which is close to the room temperature.  The difference between ZFC and FC \nmagnetizations below  TB is caused by the energy barriers of the magnetic anisotropy  [25]. The \nmagnetic anisotropy constant  K of the CoFe 2O4 nanoparticles can be estimated using the \nformula  K=25𝐾𝐵𝑇𝐵𝑉−1, where K B is the Boltzman n constant, T B is the blocking temperature \nof the samples, and V is the volume of a single particle. The calculated magnetic anisotropy \nconstant K of our sample is found equal to1.6*105 J.m-3. This estimated value is comparable \nwith the value reported in the literature for the bulk  [8]. \n \nFigure 7: ZFC/FC curves of cobalt ferrite nanoparticles measured at temperatures ranging \nfrom 10 K to 300 K and with an applied magnetic field of 1 00 Oe.  \nAt room temperature (300 K), the behavior of dispersed nanoparticles is \nsuperparamagnetic (see Figure 8), their magnetization curve is reversible. At low temperature, \naround  10 K, the ferrofluid contain s nanoparticles is frozen and the intrinsic characteristics of \nthe nanoparticles are found. The magnetization curve exhibits a large hysteresis loop, similar  \nto a hard magnetic material, and with a coercivity of about 18.6 kOe, suggesting the presence \nof pa rticles in a blocked and non -equilibrium state. In contrast, the coercivity value of CoFe 2O4 \nNPs is only 315 Oe at 300 K  due to the additional thermal activation energy which  decreasing \nthe exchange interaction between spin moment . At 10 K, t he coercivity values are in the same \norder of magnitude as those of CoFe 2O4 nanotubes  [14] and nanowires  [26] fabricated by \nelectrospinning, nanorods synthesized by microemulsion [27] and nanoparticles synthesized by \nco-precipitation method  [28]. \n \nFigure 8: Hysteresis loops of the CoFe 2O4 nanoparticles measured at 10 K and 300 K.  \nFinally, the values of saturation magnetization, Ms, obtained for CoFe 2O4 NPs in this \nwork, are in the range of 58 to 65 emu/g. These values are slightly smaller  than those of the \nbulk CoFe 2O4 ranging between 80 and 85 emu/g  [8, 13]. This magnetization reduction may be \nexplained by the magnetic moment disorder at the particle surface.  \nIt is interesting to not e herein that t he size and the shape of the obtained particles depend \non various parameters such as the quantity of precursors, the temperature or the duration of the \nheat treatment  [13, 18]. More than that, t hey depend on the oleic acid/ oleylamine ratio and the \npresence or absence of hexadecanediol.  \nThe fact of the matter,  when the amount of the surfactants increases four times compared \nto the initial protocol, the obtained nanoparticles are smaller  (d = 7.3  2.1 nm) , but they are \npolydisperse  as seen in figure 9.(a). Another parameter  likely  to reduce the particle size is to \nwork under more dilute conditions  [29]. For example, a  twofold dilution co mpared to the \noriginal protocol, leading to  a diameter comparable to the first one ( d = 7.8  1.6 nm ), but with  \na lower polydispersity  of NPs than those obtained by increasing the amount of surfactants  (see \nFigure 9.b). On the contrary, for increasing the particle size,  we have increased the duration of \nthe heat treatment , Figure 9.c shows that  the size of CoFe 2O4 NPs increased from 10.7  nm to \n13.2 nm when the duration of heat treatment increased from 60 min to 1 20 min . This result is \nin good agreement with the work of Perez -Mirabet et al.  [30]. In the meanwhile, another  \nparameter  could  involve  in the control of NPs size,  is the nature of the solvent used during the \nreaction. Baaziz et al. [31] carried out the synthesis with both polar and non-polar solvents \nhaving different boiling temperatures . With the non -polar solvents, the  authors found that the  \nsize of the nanoparticles increase d almos t linearly when the boiling temperature  of the solvent \nincrease d, they suggested that  the growth step of the particles depends on the temperature of \nthe reaction. However, using the  polar solvents, the size of the nanoparticles deviate d from this \nlinear growth . The authors  concluded  that the nature of the solvent has an influence on the \nnucleation and growth steps of the nanoparticles . That was  related to the stability of the formed \nmetal complexes which depends on the interactions with the solvent and its functional group. \nBasing in these conclusions, we replaced the benzyl  ether ( Bp= 298 °C) by octadecene ( Bp= \n318 ° C) to see the influence of the solvent on size and morphology of NPs . We found that t he \nnanoparticles obtained in octadecene have a higher size than those obtained  before  in benzyl  \nether. This result confirms the conclusion of Baaziz et al. [31] which says that  the nature of the \nsolvent has an influence on the nucleation and growt h steps of  nanoparticles .  \nFigure 9: STEM image of the synthesized CoFe 2O4 nanoparticles.  \nConcerning the indispensability of the hexadecanediol, Crouse et al. [32] reported  that \nthe absence of hexadecanediol  does not influence th e particle size, but  affect s NPs \npolydispersity . The higher  is the concentration of hexadecanediol, the greater the particle size \ndistribution  is, and with a linear  dependence. In the other hand, Moya et al. [33] reported that \nhexadecanediol favors the decomposition of acetylacetonate precursors, and therefore \nnucleation of the particles at lower temperatures . So, the CoFe 2O4 nanoparticles obtained \nwithout hexadecanediol have a defective crystallographic structure. In this context,  we have \nsynthesized CoFe 2O4 NPs without hexadecanediol, t he STEM image (Figure 9.d) show s that \nthe synthesized particles without hexadec anediol are  slightly larger, more polydisperse than \nthose synthesized with hexadecanediol , and have a poorly defined morphology; which is \ndifferent from the results described above. The role of hexadecanediol in the synthesis is not \nclearly defined; complementary  supporting evidence are  needed  to shed more light  on its \ninfluence . Continuously, in  order to study the role of each surfactant in the synthesis, CoFe 2O4 \nNPs were synthesized in the absence of each one . Without oleic acid , the particles are very \nsmall and more aggregated  (Figure 9.e), whereas  without oleylamine the particles have a poorly \ndefined morphology and are highly polydisperse  (Figure 9.f). We surmise then that  oleic acid \nis a surfactant that stabilizes nanoparticle s, and oleylamine provides the basic medium \nnecessary to form oxides of the spinel structure.  \n4. Conclusion s \nIn summary, the structural and magnetic properties of CoFe 2O4 nanoparticles  are \npresented . The thermal decomposition process allow ed us to obtain  spherical and monodisperse \ncobalt ferrite NPs surfaced by organic molecules and stabilized in an organic solvent. Using \nSTEM analysis, w e found that t heir size and shape could be controlled by varying certain \nparameters such as the synthesis temperature, the quantity , and nature of reagents .  EDS and \nXRD measurements confirm ed the formation of CoFe 2O4 nanoparticles with spinel structure.  \nThe magnetic investigations revealed a blocking temperature very close to the room \ntemperature , attesting then the room temperature  superparamagnetic  behavior of the CoFe 2O4 \nNPs with  a small coercivity value of about 315 Oe. Otherwise, At 10 K, CoFe 2O4 nanoparticles \nshow the intrinsic characteristic behavior similarly to CoFe 2O4 nanotubes, nanowires or \nnanorods.   \nThe results obtained in this work are likely to offer useful information about the \npreparation and the role of different parameters in this synthesis route of CoFe 2O4 NPs, \npromising for application  in magnetic nanodevice s and biomagnetic applications . \nAcknowledgments  \nThis work has been carried out with the support of the Ministry of Higher Education, Scientific \nResearch,and Professional Training (Enssup) (Morocco) and the National Center for Scientific \nand Technological Research(CNRST) through the grant Number: PPR15,  and by the European \nH2020 -MC-RISE -ENIGMA action (N°778072) and FEDER.   References  \n1.  Frey NA, Peng S, Cheng K, Sun S (2009) Magnetic nanoparticles: synthesis, functionalization, \nand applications in bioimaging and magnetic energy storage. 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Wakabayashi1, Yoshiharu Krockenberger1, Naoto Tsujimoto2, Tommy Boykin1,†, Shinji Tsuneyuki2, Yoshitaka Taniyasu1 & Hideki Yamamoto1 1NTT Basic Research Laboratories, NTT Corporation, Atsugi, Kanagawa 243-0198, Japan 2Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan †on leave from University of Central Florida  Magnetic insulators have been intensively studied for over 100 years, and they, in particular ferrites, are considered to be the cradle of magnetic exchange interactions in solids. Their wide range of applications include microwave devices1 and permanent magnets2. They are also suitable for spintronic devices owing to their high resistivity3, low magnetic damping4,5 and spin-dependent tunneling probabilities6. The Curie temperature (TC) is the crucial factor determining the temperature range in which any ferri/ferromagnetic system remains stable. However, the record TC has stood for over eight decades in insulators and oxides (943 K for spinel ferrite LiFe5O87). Here we show that a highly B-site ordered double-perovskite, Sr2(SrOs)O6 (Sr3OsO6), surpasses this long standing TC record by more than 100 K. We revealed this B-site ordering by atomic-resolution scanning transmission electron microscopy. The density functional theory (DFT) calculations suggest that the large spin-orbit coupling (SOC) of Os6+ 5d2 orbitals drives the system toward a Jeff = 3/2 ferromagnetic (FM) insulating state8-10. Moreover, the Sr3OsO6 is the first epitaxially grown osmate, which means it is highly compatible with device fabrication processes and thus promising for spintronic applications.     The B-site ordered double-perovskite A2BB’O6 family includes lots of fascinating magnetic materials such as half-metals11-13, multiferroic materials14, antiferromagnetic (AFM) materials15 and magnetic insulators8,16-18. The A site is usually occupied by an alkaline-earth or rare-earth element, and B and B’ are transition metal elements. Explorations of magnetism have mainly focused on varying the combination of transition metal elements at B and B’ sites, and it has been believed that having them occupied by two different transition metal elements is a prerequisite for a magnetic order at high temperatures12. Some 4d or 5d element-containing double-perovskites, e.g., Sr2FeMoO611 (TC = 415 K), Sr2CrReO613 (TC = 634 K) and Sr2CrOsO617 (TC = 725 K), reach a point of FM instability at high temperatures, although the majority of double-perovskites show an AFM order or weak spin-glass behavior19. In contrast to those 4d or 5d double-perovskites that follow the above-mentioned criteria, we discovered FM ordering above 1000 K in a novel insulating double-perovskite Sr3OsO6, in which only one 5d transition metal element occupies the B sites. Remarkably, the TC of Sr3OsO6 (~1060 K) is about ten times higher than the previous highest magnetic transition temperature in double perovskites including only one transition element (Sr2ScOsO6, AFM, TN = 110 K20). The density functional theory (DFT) calculations suggest that the large SOC of Os6+ 5d2 orbitals drives Sr3OsO6 toward the Jeff = 3/2 FM insulating state8-10. They also elucidate that the canted FM order has the lowest total energy.     High-quality B-site ordered double-perovskite Sr3OsO6 films (300-nm thick) were epitaxially grown on (001) SrTiO3 substrates in a custom-designed molecular beam epitaxy (MBE) setup that provides a precise flux even for the high-melting-point element Os (3033°C) (METHODS). Maintaining a precise flux rate for each constituent cation (Os and Sr) with a simultaneous supply of O3 is essential for avoiding deterioration of the magnetic properties; a deviation of only 2% from the optimal Os/Sr ratio is fatal (METHODS).     High-resolution scanning transmission electron microscopy (STEM) and transmission electron diffraction (TED), combined with high-resolution reciprocal space mapping (HRRSM) and reflection high-energy electron diffraction (RHEED), ascertained a cubic double-perovskite structure12,16. As schematically shown in Figs. 1a and 1d (viewed along [100] and [110] directions), Sr- or Os-occupied, fully Sr-occupied, fully Os-occupied and fully oxygen-occupied columns exist. The  2 STEM images overtly demonstrate that these columns are arranged in a spatially ordered sequence. Since the intensity in the high-angle annular dark-field (HAADF)-STEM image is proportional to ~\"# (n ~ 1.7-2.0, and Z is the atomic number), in Fig. 1b, the brightest and gray spheres are assigned to Sr- (Z = 38) or Os- (Z = 76) occupied and fully Sr-occupied columns, respectively. In Fig. 1e, the brightest and gray spheres are assigned to fully Os-occupied and fully Sr-occupied columns, respectively. In contrast to HAADF-STEM, annular bright-field (ABF)-STEM images allow for oxygen discrimination in addition to Sr and Os. Accordingly, fully oxygen-occupied columns were also detected, which are labeled O (insets of Figs. 1c and 1f). The energy dispersive X-ray spectroscopy (EDS)-STEM intensity profiles along the [100] direction shown in Figs. 1e and 1f complementarily confirm the above elemental assignments. The peak positions in the EDS profile of the Os L shell (oxygen K shell) agree well with the Os (O) positions determined by STEM. The STEM observation revealed the rock-salt type order of Os6+ — the hexavalent state of Os is confirmed by X-ray photoemission spectroscopy (XPS) measurements (METHODS) — to an excellent extent, and this ordering is driven by the large difference in the electronic charges and ionic radii between Sr2+ and Os6+.21 Accordingly, there are no Os-O-Os paths. Therefore, advanced mechanisms need to be considered as the Goodenough-Kanamori rules22, which well predict magnetic interactions between two next-nearest-neighbor magnetic cations through a nonmagnetic anion, do not cover the theoretical framework exigent for the FM order in Sr3OsO6.     Figures 2a and 2b show the temperature dependence of magnetization versus the magnetic field (M-H) of a Sr3OsO6 film. The hysteretic response of the Sr3OsO6 film shows a soft magnetic behavior with the small coercive field of ∼100 Oe at 1.9 K (Fig. 2b), and the saturation magnetization at 70000 Oe (Fig. 2a) decreases with increasing temperature. The saturation magnetization persists up to 1000 K [limit of measurement range (METHODS)], indicating TC > 1000 K. Figure 2c shows the magnetization versus temperature (M-T) curve with H = 2000 Oe. In Fig. 2b, we also plot the spontaneous magnetization as a function of temperature. The TC value, estimated from the extrapolation of the M-T curve to the zero magnetization axis, is about 1060 K (Fig. 2c). This is the highest TC among all insulators and oxides.     While such high TC is common for systems with free charge carriers, e.g., Fe3O4 and Co, their absence in Sr3OsO6 requires other exchange paths. The temperature dependence of resistivity (%) for a Sr3OsO6 film is shown in Fig. 3a. The electronic charge carriers [5d electrons in the Os6+ state] move by hopping between localized electronic states, and this is supported by ln(%) ∝ T-1/4 [variable range hopping (VRH) model] (Fig. 3b) along with the high resistivity value [%(300 K) = 75 Ωcm]: other mechanisms, e.g., ln(%) ∝ T-1/2 [Efros-Shklovskii Hopping (ESH) model] and ln(%) ∝ T-1 [thermal activation (TA) model], are not supported by the electronic transport response. Figure 3c shows electron energy loss spectroscopy (EELS) spectra of a Sr3OsO6 film; we measured three different spots as indicated in the cross-sectional STEM image (inset of Fig. 3c) with a spot size of ~4 nm. An EELS spectrum corresponds to the loss function ()−1/-, where - is a complex dielectric function. The three EELS spectra are almost identical, indicating that electronic states are uniform in the entire Sr3OsO6 film. The band gap (indicated by the black arrow), at which EELS intensities start to increase23, is ~2.65 eV. These results indicate that Sr3OsO6 is an insulator with a band gap of ~2.65 eV. Accordingly, models based on the double exchange interaction and direct exchange interaction, where itinerant electrons are driving the magnetic order, can be ruled out as the origin of the emergent ferromagnetism.     The saturation magnetization of Sr3OsO6 (~49 emu/cc at 1.9 K) is significantly smaller than that for typical magnetic metals; e.g., Nd2Fe14B (~1280 emu/cc), SmCo5 (~860 emu/cc) and AlNiCo 5 (~1120 emu/cc)24, and typical ferrites; e.g., CoFe2O4 (~430 emu/cc), Y3Fe5O12 (~170 emu/cc) and LiFe5O8 (~390 emu/cc)7. The small saturation magnetization unique to Sr3OsO6 allows for small stray fields and low-energy spin-transfer-torque switching25, which are advantageous for high-density-integration and low-power consumption spintronic devices. The small saturation magnetization is most likely associated with the low composition ratio of Os in Sr3OsO6. The  3 saturation magnetic moment of Os at 1.9 K was estimated to be 0.77 .//Os, which is smaller than the expected value of the spin-only magnetic moment (g0(0+1)=2.83\t.//Os) for the Os6+ (5d2 6787) state with S = 1. This indicates that we have to take the SOC into account, which is often the case with 5d systems9,26, to understand the electronic states and physical properties of Sr3OsO6.     We analyzed the electronic and magnetic states of Sr3OsO6 by DFT with SOC (METHODS). It was revealed that the canted FM order (Fig. 4a) has the lowest total energy among all possible magnetic arrangements. However, the energy differences between the canted FM order and the collinear FM order (Extended Data Fig. 9a) or the AFM order (Extended Data Fig. 9b) are very small (~3.6 meV per atom or ~1.4 meV per atom, respectively), implying a competition among these orders. Note that such canted magnetic order is recently reported in other Os containing double perovskites27,28. The extended superexchange paths (Os-O-O-Os and Os-O-Sr-O-Os), which are well recognized to drive the magnetic order in Os containing double-perovskites29,30, are one possible origin of the observed ferromagnetism in Sr3OsO6. Our GGA + U + SOC calculation indicates that the electronic structure of Sr3OsO6 with the canted FM order has a gap (~0.37 eV) at the Fermi energy (EF), equivalently an insulating state (Figs. 4b and 4c). With U + SOC, the t2g↑ states are split into effective total angular momenta Jeff = 3/2 (doublet) and Jeff = 1/2 (singlet) states with opening a gap. The Jeff = 3/2 states are fully occupied with two 5d electrons per Os6+, resulting in the insulating state. We note that a metallic grand state is obtained for the canted FM order when only SOC (without U) is taken into account. This is because the band dispersions of Jeff = 3/2 and Jeff = 1/2 states are greater than the spin-orbit splitting. The DFT calculation on the element-specific partial density-of-state (PDOS) (Fig. 4c) also revealed that the Jeff = 3/2 and Jeff = 1/2 bands around EF are mostly composed of the Os 5d orbitals and the O 2p component is very small, implying that d-p hybridization is negligibly small. The calculated magnetic moment of osmium is 1.56 .//Os, which coincides better with the experimentally obtained value (0.77 .//Os) as compared with the spin-only magnetic moment (2.83\t.//Os). Although further work is required to reveal the underlying mechanisms driving Sr3OsO6 into the robust FM order, and also to achieve a quantitative agreement of the band gap between experiment (~2.65 eV) and calculation (~0.37 eV), our calculation provides information on the magnetic arrangement at the grand state and how the energy gap is open by the interplay between electron correlation (U) and SOC.     Our current findings in epitaxial Sr3OsO6 films — an extraordinarily high TC of 1060 K, Jeff = 3/2 insulating state, rock-salt type Os6+ order and small magnetic moment — enrich the family of ferri/ferromagnetic insulators. Although the underlying electronic exchange mechanisms driving the robust FM order in Sr3OsO6 remain murky, applications of Sr3OsO6 to oxide-electronics beyond the current ferrite technology are feasible.  Author Contributions Y.K. and Y.K.W. prepared the samples. Y.K.W. performed experimental measurements and data analysis. N.T. and S.T. carried out the electronic-structure calculations. Y.K.W. wrote the paper. All authors contributed to the manuscript and the interpretation of the data.  Acknowledgements We thank Ken-ichi Sasaki for a valuable discussion. We thank Ai Ikeda for her help with the X-ray diffraction and resistivity measurements. We thank Hiroshi Irie for his help with the resistivity measurements. We thank Kazuhide Kumakura for his installation of MPMS3 SQUID-VSM oven option.  Author information Correspondence and requests for materials should be addressed to Y.K.W. (wakabayashi.yuki@lab.ntt.co.jp).      4  References 1 Naito, Y . & Suetake K. 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B 93, 161116 (2016).  6 Figures and figure legends \n Figure 1 Atomic-resolution STEM images of a Sr3OsO6 film. a, Schematic diagram of the Sr3OsO6 viewed along the [100] direction. b, c, HAADF-STEM (b) and ABF-STEM (c) images near the center of the Sr3OsO6 layer along the [100] direction. d, Schematic diagram of the Sr3OsO6 viewed along the [110] direction. e, f, HAADF-STEM (e) and ABF-STEM (f) images near the center of the Sr3OsO6 layer along the [110] direction. The insets in e and f show enlarged views, and the corresponding graphs show EDS-STEM intensity profiles along the [001] direction. In all figures, purple, yellow, red and blue dotted circles indicate Sr- or Os- occupied, fully Sr-occupied, fully Os-occupied and fully oxygen-occupied columns, respectively.  \n1 nmabcAlong [100]defAlong [110]\nO\nO1 nm1 nm1 nm\n[001]⨀[110][001]⨀[100][001]⨀[110][001][100]⨀\n432EDS Intensity (a.u.)Sr K shell Os L shell \n43.53EDS Intensity (a.u.)O K shellSr+ OsSrO\nSrOOs 7  Figure 2 Magnetic properties of a Sr3OsO6 film. a, In-plane M-H curves at 1.9 to 1000 K for a Sr3OsO6 film. Here, H was applied to the [100] direction. b, Close-up near the zero magnetic field in a. c, M-T curve with H = 2000 Oe applied to the [100] direction for a Sr3OsO6 film. Spontaneous magnetization deduced from Fig. 3(b) as a function of temperature is also shown. \n 8  Figure 3. Resistivity and dielectric properties of a Sr3OsO6 film. a, %-T curve for a Sr3OsO6 film. b, Logarithm of % versus T-1/4 plot, corresponding to the VRH model. The insets of b show the logarithm of % versus T-1/2 and T-1 plots, corresponding to the ESH and TA models, respectively. The black dashed lines in b are guides for the eye. c, The EELS spectra of a Sr3OsO6 film measured at the spots indicated in the cross-sectional STEM image (inset). The background was corrected with a power-law fit from 2 to 2.3 eV .  ab\n123456\n101102103104105106ρ (Ωcm)400300200100T (K)\nIn(ρ)0.300.280.260.24T-1/4 (K-1/4)In(ρ)0.0080.0060.004T-1 (K-1)In(ρ)0.080.06T-1/2 (K-1/2)VRHTAESH123456\n123456c\n[001]Sr3OsO6SrTiO3⨀[110]50 nm1231000050000Intensity (arb. u.)5.04.54.03.53.02.5Energy Loss (eV) Position 1 Position 2 Position 3 9  Figure 4. Electronic-structure calculations. a, Schematic diagram of the magnetic ground state (canted FM order) of Sr3OsO6 obtained from the DFT calculation. In a, red spheres and blue arrows indicate Os atoms and magnetic moments of Os atoms, respectively, and the Sr and O atoms are omitted for the simplicity. b, The band structures for Sr3OsO6 with the canted FM order calculated by GGA + U + SOC. c, The element-specific partial density-of-state (PDOS) for the canted FM order calculated by GGA + U + SOC. d, Schematic energy diagrams for the Os 5d2 configurations. In d, only PDOS for Os is taken into account and the contributions by Sr and O are omitted for the simplicity.  \n 10 METHODS Growth of Sr3OsO6 on SrTiO3. We grew the high-quality epitaxial B-site ordered double-perovskite (001) Sr3OsO6 films (300- or 250-nm thick) on (001) SrTiO3 substrates (CrysTec GmbH) in a custom designed molecular beam epitaxy (MBE) system31,32 (Extended Data Fig. 1a). After cleaning with CHCl3 (10 min, 2 times) and acetone (5 min) in ultrasonic cleaner, the SrTiO3 substrate was introduced in the MBE growth chamber. After degassing the substrate at 400°C for 30 minutes and successive thermal cleaning at 650°C for 30 min, we grew a Sr3OsO6 film. The growth temperature was 650°C. The oxidation during the growth was carried out with O3 gas (non-distilled, ~10% concentration) from a commercial ozone generator, and the resultant chamber pressure during the growth was ~4.5 × 10-6 Torr. After the growth, films were cooled to room temperature under ultra-high vacuum (UHV). The MBE system is equipped with multiple e-beam evaporators (Hydra, Thermionics) for Sr and Os. The electron impact emission spectroscopy (EIES) sensor (Guardian, Inficon) is located next to the sample holder in the same horizontal plane. The sensor head is equipped with a filament, which generates thermal electrons for the excitation of Sr and Os atoms. Optical band-pass filters are used for element-specific detection of the excited optical signals, since the emitted light spectra are characteristic for Sr and Os. The EIES sensor is equipped with photomultipliers (PMTs) located outside of the vacuum chamber that amplify optical signals. The Sr and Os fluxes measured by EIES were kept constant (Extended Data Fig. 1b) by the proportional-integral-derivative (PID) control of the evaporation source power supply. We optimized the flux ratio of Sr and Os to obtain a Sr3OsO6 film with a high saturation magnetic moment. Extended Data Fig. 2 shows the in-plane M-H curves at 300 K for Sr3OsO6 films grown with a different flux ratio of Sr and Os. The saturation magnetic moment of the film grown with the flux ratio of Sr:Os = 2.05:1 is ten or more times larger than those for the films grown with the flux ratio of Sr:Os = 2.05:1.02 and 2.05:0.98. This means that the magnetic properties of Sr3OsO6 films are very sensitive to the Sr/Os ratio and that well-controlled Sr and Os fluxes during the growth are important for the high saturation magnetic moment. Therefore, in this study, we fixed the flux ratio of Sr:Os = 2.05:1.     The cubic crystal structure of Sr3OsO6 is illustrated in Extended Data Fig. 3a. Extended Data Figs. 3b and 3c show reflection high-energy electron diffraction (RHEED) patterns of a Sr3OsO6 thin film surface, where the sharp streaky patterns with clear surface reconstruction indicate the growth of the Sr3OsO6 film proceeded in a layer-by-layer manner, leading to the high crystalline quality of the film. Notably, [01l] diffractions are not seen (Extended Data Fig. 3b) due to extinction rules, indicating the formation of a cubic B-site ordered double-perovskite12,16. The cubic structure model is further evidenced by high-resolution X-ray reciprocal space mapping (HRRSM) (Extended Data Fig. 3d): the in-plane and out-of-plane lattice constants of Sr3OsO6 are identical within the resolution limits (8.24±0.03 Å and 8.22±0.03 Å, respectively). It is therefore reasonable that the Sr3OsO6 films are epitaxially but not coherently grown on the SrTiO3 (3.905 Å) substrate. Transmission electron microscopy. High-angle annular dark-field (HAADF) and annular bright-field (ABF) scanning transmission electron microscopy (STEM) images were taken with a JEOL JEM-ARM 200F microscope. Electron energy loss spectroscopy (EELS) spectra of a Sr3OsO6 film were recorded from a spot with ~4 nm diameter also with a JEOL JEM-ARM 200F microscope.     Extended Data Figs. 3e-3g show cross-sectional high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images of a Sr3OsO6 film taken along the [100] direction. Since the intensity in the HAADF image is proportional to ~\"# (n ~ 1.7-2.0, and Z is the atomic number)33, the HAADF intensity is dominated by Sr (Z = 38) and Os (Z = 76) ions, whereas oxygen (Z = 8) is scarcely discernible. At a glance one can recognize that a single-crystalline Sr3OsO6 film with an abrupt substrate/film interface has been grown epitaxially on a (001) SrTiO3 substrate, as expected from the RHEED. Misfit dislocations at the Sr3OsO6/SrTiO3 interface (Extended Data Fig. 3g) are due to the ∼5% larger lattice constant of the perovskite  11 Sr3OsO6 lattice (8.23 Å/2 = 4.115 Å) than that of SrTiO3 (3.905 Å). The cubic crystal structure of Sr3OsO6 was also confirmed by the STEM analysis. In addition to the [100] direction (Extended Data Figs. 3e-3g), the epitaxial growth of the Sr3OsO6 layer on the SrTiO3 substrate was also confirmed in STEM images taken along the [110] direction (Extended Data Figs. 4a-4c). The rock-salt type order of Os6+ confirmed in the main text (Fig. 1) is observed to an excellent extent (Extended Data Figs. 4d and 4e). Chemical composition of a Sr3OsO6 film. Extended Data Fig. 5a shows the depth profile of the chemical composition of a Sr3OsO6 film (250-nm thick) estimated from Rutherford backscattering spectroscopy (RBS). The chemical composition of the Sr3OsO6 layer is uniform (Sr:Os:O = 2.7±0.1:1.15±0.05:6.15±0.4). The concentrations of Os and oxygen are slightly larger than those for an ideal composition (Sr:Os:O = 3:1:6). This slight difference may originate from the non-stoichiometry and existence of the very small amount of paramagnetic OsO234, which was observed in the X-ray diffraction (XRD) measurements, as described later. To exclude the possibility of the contamination by magnetic impurities, we performed EDS measurement for a Sr3OsO6 film (Extended Data Fig. 5b). There are no peaks except for Sr, Os, Ti, oxygen and C, which confirms the absence of magnetic impurities. X-ray diffraction. 2<-< and reciprocal space map XRD measurements of the Sr3OsO6 films were performed with a Bruker D8 diffractometer using monochromatic Cu K=1 radiation at room temperature. In Extended Data Fig. 5c, we show the <-2< XRD pattern for a Sr3OsO6 film. In addition to the diffraction peaks of the SrTiO3 substrates, (002) and (004) diffractions from Sr3OsO6 are clearly observed. (001) and (003) diffractions from Sr3OsO6 are not seen due to the extinction rules. Note that traces of OsO2, which is known as a paramagnetic metal35, are detected as indicated by *. The XRD intensities of OsO2 is about 700 times smaller than those of Sr3OsO6 and segregation of OsO2 is not discernible in the STEM images, indicating that volume fraction of OsO2 (paramagnetic metal) is negligibly small. Therefore, Sr3OsO6 predominates the magnetic response of the film. X-ray photoemission measurements. XPS is one of the most powerful methods to determine the valence of Os in compounds36,37, since the 4f7/2 core level binding energies in Os compounds with well-defined oxidation states are known. ULV AC-PHI Model XPS5700 with a monochromatized Al K= (1486.6 eV) source operated at 200 W was used for the experiment. The scale of binding energy was calibrated against the C 1s line (284.6 eV). Extended Data Fig. 6 shows the Os 4f spectrum of a Sr3OsO6 film at 300 K. The observed 4f7/2 binding energy (54.1 eV) is close to the reported values for those of Os6+ states (53.2-53.8 eV) and far from those of Os2+ states (49.7 eV), Os3+ states (50.4-51.0 eV), Os4+ states (51.7-52.3 eV) and Os8+ states (55.9-56.3 eV)36,37. Accordingly, the hexavalent state of Os (Os6+) is supported. Note that a shoulder structure at ~ 53 eV may originate from a surface layer formed due to the slightly hygroscopic nature of Sr3OsO6 as the sample was transferred to the XPS apparatus not in vacuo but in atmosphere. Resistivity measurements. Resistivity was measured using the four-probe method in a Physical Property Measurement System (PPMS) Dynacool sample chamber. The Ag electrodes deposited on a Sr3OsO6 surface were connected to an Agilent 3458A Multimeter. Magnetic measurements. The magnetization measurements for Sr3OsO6 films were performed with a Quantum Design MPMS3 SQUID-VSM magnetometer. Using a quartz sample holder (oven sample holder), we measured the M-T curves with increasing temperature from 1.9 (300) to 300 (1000) K with H = 2000 Oe applied along the [100] direction. In the M-T measurements, M was measured with increasing temperature after the sample was cooled to 1.9 (300) K from 300 (1000) K without a magnetic field. We also measured M-H curves at 1.9-300 K (400-1000 K) using the quartz sample holder (oven sample holder).     To check the accuracy of the measurement temperature in the MPMS SQUID-VSM magnetometer, we measured the magnetic properties of a pure Ni reference plate (Quantum Design Part Number: 4505-155). The M-H curves at 300 and 1000 K show FM and paramagnetic response,  12 respectively (Extended Data Fig. 7a), and the magnetization of Ni rapidly increases between 623 and 629 K (Extended Data Fig. 7b). These results indicate that TC of Ni is between 623 and 629 K. This is consistent with the TC value in the literature (627 K)38. Thus, the error of the measurement temperature in the MPMS SQUID-VSM magnetometer is less than ± 4 K.      Extended Data Fig. 7c shows the in-plane M-H curves at 1.9, 300 and 1000 K for a SrTiO3 substrate. They show only a linear diamagnetic response at 300 and 1000 K. The nonlinear magnetic response near the zero magnetic field at 1.9 K indicates the existence of a tiny amount of a paramagnetic impurity in the SrTiO3 substrate. In Figs. 2a and 2b, the linear diamagnetic response of the magnetic moment for the SrTiO3 substrate was subtracted from the raw M data.     Extended Data Fig. 7d shows the M-T curve with H = 2000 Oe for the oven sample holder without a sample. The curve shows a dip structure at around 800 K. This means that the dip structure in the M-T curve at around 800 K for the Sr3OsO6 film (Fig. 2c) is an unavoidable experimental artifact.     The magnetic properties of Sr3OsO6 at 300 K did not change much after it was heated to 1000 K as shown in Fig. 8a. This means that heating to 1000 K does not affect much its magnetic properties.     Although the Sr3OsO6 films were epitaxially grown on the SrTiO3 substrate, the shapes of the in-plane M-H curves measured with H applied to the [100] and [110] directions are identical (Extended Data Fig. 8b). This indicates that the in-plane magnetic anisotropy of the Sr3OsO6 film is negligibly small. This small magnetic anisotropy might be related to the misfit dislocations (Extended Data Figs. 3e-3g), which often decrease the magnetic anisotropy of magnetic insulators39,40. The electronic-structure calculations. The electronic-structure calculations were based on density functional theory (DFT). The calculations were performed by using Vienna Ab initio Simulation Package (V ASP)41,42 with the projector augmented-wave (PAW)43,44 method and the Parder-Burke-Ernzerhof (PBE)45 functional of the generalized gradient approximation (GGA)46. To describe the localization of Os 5d electrons accurately, we used the DFT + U calculations47. The value of the screened Coulomb interaction U = 3 eV was used for the Os atoms. This value is comparable to reported values for Os containing double perovskites (2-4 eV)9,10,18,48. The contribution of the spin orbit coupling was included in our calculations. The crystal structure was optimized for the conventional unit cell (40 atoms) of Sr3OsO6 whose lattice constant was fixed to the experimental value 8.23 Å. We performed the optimization until all forces on the atoms become smaller than 10-5 eV/Å with a \u0002-centered 2\u00012\u00012 k-point grid and cut-off energy of 800 eV . The total energies and electronic structures are calculated with the optimized crystal structures resulting in the canted FM ground state (Fig. 4a).     By comparing with the total energies of the magnetic ground state (canted FM order) (Fig. 4a), collinear FM order [Extended Data Fig. 9(a)] and the AFM order [Extended Data Fig. 9(b)], we found that the energy differences between the canted FM order and the collinear FM order, and between the canted FM order and the AFM order are very small (~3.6 meV per atom and ~1.4 meV per atom, respectively), implying a competition among these orders.  References for METHODS 31 Naito, M. & Sato, H. Stoichiometry control of atomic beam fluxes by precipitated impurity phase detection in growth of (Pr,Ce)2CuO4 and (La,Sr)2CuO4 films. Appl. Phys. Lett. 67, 2557 (1995). 32 Yamamoto, H., Krockenberger, Y . & Naito, M. Multi-source MBE with high-precision rate control system as a synthesis method sui generis for multi-cation metal oxides. J. Cryst. Growth 378, 184 (2013). 33 Nellist, P. D. & Pennycook, S. J. Advances in Imaging and Electron Physics p. 147 (Elsevier, 2000). 34 Greedan, J. E., Willson, D. B. & Haas, T. E. The metallic nature of osmium dioxide. Inorg. Chem.   13  7, 2461 (1968). 35 Yen, P. C., Chen, R. S., Chen, C. C., Huang, Y . S. & Tiong, K. K. Growth and characterization of OsO2 single crystals. J. Cryst. Growth 262, 271 (2004). 36 White, D. L., Andrews, S. B., Faller, J. W. & Barrnett, R. J. The chemical nature of osmium tetroxide fixation and staining of membranes by X-ray photoelectron spectroscopy. Biochim. Biophys. Acta 436, 577–592 (1976). 37 Hayakawa, Y . et al. X-ray photoelectron spectroscopy of highly conducting and amorphous osmium dioxide thin films. Thin Solid Films 347, 56–59 (1999). 38 Kouvel, J. S. & Fisher, M. E. Detailed magnetic behavior of nickel near its Curie temperature. Phys. Rev. 136, 1626 (1964). 39 Margulies, D. T. et al. Origin of the anomalous magnetic behavior in single crystal Fe3O4 films. Phys. Rev. Lett. 79, 5162 (1997). 40 Wakabayashi, Y . K. et al. Electronic structure and magnetic properties of magnetically dead layers in epitaxial CoFe2O4/Al2O3/Si(111) films studied by X-ray magnetic circular dichroism. Phys. Rev. B 96, 104410 (2017). 41 Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Physical Review B 47, 558 (1993). 42 Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical Review B 54, 11169 (1996). 43 Blochl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994). 44 Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758 (1999). 45 Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865 (1996). 46 Perdew, J. P. Accurate Density Functional for the Energy: Real-Space Cutoff of the Gradient Expansion for the Exchange Hole. Phys. Rev. Lett. 55, 1665 (1985). 47 Dudarev, S. L., Savrasov, S. Y ., Humphreys, C. J. & Sutton, a. P. Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys. Rev. B 57, 1505 (1998). 48 Kanungo, S., Yan, B., Felser, C. & Jansen, M. Active role of nonmagnetic cations in magnetic interactions for double-perovskite Sr2BOsO6 (B = Y ,In,Sc). Phys. Rev. B 93, 161116 (2016).  14  Extended Data Figure 1 Multi-source oxide MBE setup and fluxes. a, Schematic illustration of our multi-source oxide MBE setup. EIES: Electron Impact Emission Spectroscopy. QCM: Quartz Crystal Microbalance. RHEED: Reflection High-Energy Electron Diffraction. PMT: Photomultiplier Tube. b, Sr and Os fluxes measured by EIES during the growth.  \nFilamentVaporLight emissionElectrons\nSignal processor and Microcontroller(For real-time feedback control) Evaporation source power supplyOptical filterPMTdetector\nHeaterSubstrate\nO3Shutter\ne-beamevaporatorsO*\nEIES\nSrOs\nVacuumchamberEIES sensor headQCM\t1\t\n86420Flux (µmol·m-2·s-1)\n20151050 Time (min.) Sr Osa\nb 15  Extended Data Figure 2 Magnetic properties of Sr3OsO6 films grown with a different flux ratio of Sr and Os. a, In-plane M-H curves at 300 K for Sr3OsO6 films grown with a different flux ratio of Sr and Os. Here, H was applied to the [100] direction. b, Close-up near the zero magnetic field in a.    ab\n-40-2002040Magnetization (emu/cc)1000050000-5000-10000Magnetic Field (Oe)300 K Sr:Os = 2.05:1 Sr:Os = 2.05:1.02 Sr:Os = 2.05:0.98 (×5)-20-1001020Magnetization (emu/cc)-400-2000200400Magnetic Field (Oe)300 K Sr:Os = 2.05:1 Sr:Os = 2.05:1.02 Sr:Os = 2.05:0.98 (×5) 16  Extended Data Figure 3 Crystal structure, RHEED, X-ray HRRSM and HAADF-STEM for a Sr3OsO6 film. a, Schematic diagram of the B-site ordered double-perovskite Sr3OsO6. The yellow, red and blue spheres indicate Sr, Os and O ions, respectively. b, c, RHEED patterns of an epitaxial Sr3OsO6 film surface, where the incident electron beams are parallel to [100] (b) and [110] (c). d, X-ray HRRSM of a Sr3OsO6 film around the SrTiO3 (103) reflection. e, Cross-sectional HAADF-STEM images of a Sr3OsO6 film taken along the [100] direction. f, Magnified image near the interface in e. g, Magnified image near the interface in f. In g, a misfit dislocation at the Sr3OsO6/SrTiO3 interface is indicated by the yellow arrow.  \n 17  Extended Data Figure 4 STEM images for a Sr3OsO6 film. a, Cross-sectional HAADF-STEM image of a Sr3OsO6 film taken along the [110] direction. b, Magnified image near the interface in a. c, Magnified image near the interface in b. d, e, HAADF-STEM images of the Sr3OsO6 layer taken along the [110] (d) and [100] (e) directions.  \n 18  Extended Data Figure 5 Chemical composition, EDS spectrum and !-2! XRD pattern of Sr3OsO6 films. a, Depth profile of the chemical composition of a Sr3OsO6 film (250-nm thick) estimated from RBS. b, EDS spectrum of a Sr3OsO6 film, which was taken from a wide area (1 × 1 mm2). c, #-2# XRD pattern for a Sr3OsO6 film. Traces of OsO2 are detected as indicated by \u0002\u0001   6040200Concentration (%)2001000Depth (nm) O Sr Os Ti3210Intensity (arb. u.)10987654321Energy (kev)OsTiTiOsSrOsSrOCSr,Os\n103104105106 Intensity (arb. u.)6050403020102θ (degree)Sr3OsO6(002)Sr3OsO6(004)SrTiO3(002)SrTiO3(001)**abc 19  Extended Data Figure 6 XPS of a Sr3OsO6 film. The Os 4f XPS spectrum of a Sr3OsO6 film at 300 K. 1.00.80.60.4Intensity (arb. u.)605550Binding Energy (ev)4f7/24f5/2 20  Extended Data Figure 7 Magnetic properties of a Ni reference plate and a SrTiO3 substrate, and experimental artifact from oven sample holder. a, M-H curve at 300 and 1000 K for a Ni plate. Here, H was applied to the in-plane direction. b, M-T curve with H = 200 Oe applied to the in-plane direction for a Ni plate. The inset of b shows a close-up near the Curie temperature. c, In-plane M-H curve at 1.9, 300 and 1000 K for a SrTiO3 substrate. Here, H was applied to the [100] direction. d, In-plane M-H curve at M-T curve with H = 2000 Oe for the oven sample holder without a sample.   ba\n-400-2000200400Magnetization (emu/cc)-20000-1000001000020000Magnetic Field (Oe)Ni Reference 300 K 1000 K20010006406306203002001000Magnetization (emu/cc)1000900800700600500400300Temperature (K)Ni Reference 200 Oe\n-0.0002-0.000100.00010.0002Magnetic Moment (emu)400000-40000Magnetic Field (Oe)SrTiO3 substrate 1.9 K 300 K 1000 K-0.00002-0.000010Magnetic Moment (emu)1000800600400Temperature (K)Oven sample holder FC 2000 Oecd 21  Extended Data Figure 8 Magnetic properties of a Sr3OsO6 film after heating and Magnetic properties of a Sr3OsO6 film with H applied to the [100] and [110] directions. a, In-plane M-H curves of a Sr3OsO6 film at 300 K before and after the sample was heated to 1000 K. Here, H was applied to the [100] direction. b, In-plane M-H curve at 1.9 and 300 K for a Sr3OsO6 film. Here, H was applied to the [100] and [110] directions.  -1.0-0.500.51.0Magnetic Moment (a.u.)-40000040000Magnteic Field (Oe)Sr3OsO6in plane [100] 1.9 K [110] 1.9 K [100] 300 K [110] 300 K-1.0-0.500.51.0Magnetization (arb. u.)-4000-2000020004000Magnetic Field (Oe) Before 1000 K After 1000 Kba 22  Extended Data Figure 9 Schematic diagram of the magnetic orders. a, b, Schematic diagram of the collinear FM order (a) and the AFM order (b). In a and b, red spheres and blue arrows indicate Os atoms and Os magnetic moments, respectively, and the Sr and O atoms are omitted for the simplicity. \n"    },    {        "title": "1311.1177v1.Diffusion_behavior_in_diluted___Fe_Cr___alloys__An_environment_for_H_diffusion_in_ferritic_steels.pdf",        "content": "arXiv:1311.1177v1  [cond-mat.mtrl-sci]  5 Nov 2013Diﬀusion behavior in Nickel-Aluminium and Aluminium-Uran ium\ndiluted alloys\nViviana P. Ramunni\nCONICET - Avda. Rivadavia 1917,\nCdad. de Buenos Aires, C.P. 1033, Argentina. and\nDepartamento de Materiales, CAC-CNEA,\nAvda. General Paz 1499, 1650 San Martín, Argentina.∗\n(Dated: June 23, 2021)\nAbstract\nImpurity diﬀusion coeﬃcients are entirely obtained from a l ow cost classical molecular statics\ntechnique (CMST). In particular, we show how the CMST is appr opriate in order to describe the\nimpurity diﬀusion behavior mediated by a vacancy mechanism . In the context of the ﬁve-frequency\nmodel, CMST allows to calculate all the microscopic paramet ers, namely: the free energy of vacancy\nformation, the vacancy-solute binding energy and the invol ved jump frequencies, from them, we\nobtain the macroscopic transport magnitudes such as: corre lation factor, solvent-enhancement\nfactor, Onsager and diﬀusion coeﬃcients. Also, we report fo r the ﬁrst time the behavior of diﬀusion\ncoeﬃcients for the solute-vacancy paired specie. We perfor m our calculations in diluted NiAl and\nAlUf.c.c. alloys. Our results are in perfect agreement with ava ilable experimental data for both\nsystems and predict that for NiAl the solute diﬀuses through a vacancy interchange mechanism , while\nfor theAlUsystem, a vacancy drag mechanism occurs\nPACS number(s): Diﬀusion, Numerical Calculations, Vacanc y mechanism, diluted Alloys, NiAl and\nAlU systems.\nPACS numbers:\n∗Electronic address: vpram@cnea.gov.ar; This work was part ially ﬁnanced by CONICET - PIP 00965/2010.\n1I. INTRODUCTION\nThe low enrichment of U−Mo alloy dispersed in an Almatrix is a prototype for new ex-\nperimental nuclear fuels [1]. When these metals are brought into contact, diﬀusion in the\nAl/U−Mointerface gives rise to interaction phases. Also, when subj ected to temperature\nand neutron radiation, phase transformation from γUtoαUoccurs and intermetallic phases\ndevelop in the U −Mo/Al interaction zone. Fission gas pores nucleate in these new phases\nduring service producing swelling and deteriorating the al loy properties [1, 2]. An important\ntechnological goal is to delay or directly avoid undesirabl e phase formation by inhibiting inter-\ndiﬀusion of AlandUcomponents. Some of these compounds are believed to be respo nsible\nfor degradation of properties [3]. On the other hand, there i s an experimental work [4], that\nargues that these undesirable phases have not inﬂuence on th e mobility of UinAl, based on\nthe results of the eﬀective diﬀusion coeﬃcients calculated from the best ﬁt of their permeation\nexperimental curves.\nAnother technique to study the diﬀusion of Uranium into Alum inum was based on the max-\nimum rate of penetration of uranium into aluminum as functio n of the temperature [5]. From\nthis perspective, the authors also report the activation en ergy values of Uranium mobility. In\navoiding interdiﬀusion, Brossa et. al. [6] studied the eﬃci ent diﬀusion barriers that should have\na good bonding eﬀect and exhibit a good thermal conductivity at the same time. In this work,\ndeposition methods have been developed and the diﬀusion beh avior of the respective couples\nand triplets has been evaluated by metallographic, micro-h ardness and electron microprobe\nanalyses. The practical interest of a nickel barrier is show n by several publications concerning\nto the diﬀusion in the systems AlNi,NiUandAlNiU . The knowledge of the binary system is\nthe only satisfactory basis for the study of the ternary syst em, these binary systems are treated\nbrieﬂy before proceeding to the ternary. The study of the NiAl binary system was, limited\nto solid samples of the sandwich-type, clamped together by a titanium screw and diﬀusion\ntreatments have been carried out. Results from this work, ha ve inspired as to also study the\nNiAl together with the AlUsystem.\nTherefore it is important to watch carefully and with specia l attention the initial microscopic\nprocesses that originate these intermetallic phases. In or der to deal with this problem we started\nstudying numerically the static and dynamic properties of v acancies and interstitials defects\nin theAl(U) bulk and in the neighborhood of a (111)Al/(001)αUinterface using molecular\ndynamics calculations [7, 8]. Here, we review our previous w orks [7, 8], performing calculation\nof three diﬀusion coeﬃcients, namely: the solvent self diﬀu sion coeﬃcient, the solute tracer\ndiﬀusion coeﬃcient and one more, never before calculated in the literature for this alloy, of the\n2Uranium-vacancy paired specie. With this purpose we use ana lytical expressions of the diﬀusion\nparameters in terms of microscopical magnitudes. We have su mmarized the theoretical tools\nneeded to express the diﬀusion coeﬃcients in terms of micros copic magnitudes as, the jump\nfrequencies, the free vacancy formation energy and the vaca ncy-solute binding energy. Then we\nstarts with non-equilibrium thermodynamics in order to rel ate the diﬀusion coeﬃcients with\nthe phenomenological L-coeﬃcients. The microscopic kinetic theory, allows us to w rite the\nOnsager coeﬃcients in term of the jump frequency rates. At th is point we follow the procedure\nof Okamura and Allnat [9], and Allnatt and Lidiard [10].\nThe jump frequencies are identiﬁed by the model developed fu rther by Le Claire in Ref. [11],\nknown as the ﬁve-frequency model for f.c.c lattices. The met hod includes the jump frequency\nassociated with the migration of the host atom in the presenc e of an impurity at a ﬁrst nearest\nneighbor position. All this concepts need to be put together in order to correctly describe the\ndiﬀusion mechanism. Hence, in the context of the shell appro ximation, we follow the technique\nin Ref. [10] to obtain the corresponding transport coeﬃcien ts which are related to the diﬀusion\ncoeﬃcients through the ﬂux equations. A similar procedure f or f.c.c. structures was performed\nby Mantina et al. [12] for Mg,SiandCudiluted in Albut using density functional theory\n(DFT). Also, for b.c.c. structures, Choudhury et al. [13] have calculated the self-diﬀusion and\nsolute diﬀusion coeﬃcients in diluted αFeNi andαFeCr alloys including an extensive analysis\nof the phenomenological L-coeﬃcients using DFT calculations. Also the authors discu ss about\nthe risk induced by radiation on based FeNi andFeCr alloys.\nIn the present work, we do not employ DFT, instead we use a clas sical molecular statics\ntechnique, the Monomer method [14]. This much less computat ionally expensive method allows\nus to compute at low cost a bunch of jump frequencies from whic h we can perform averages\nin order to obtain more accurate eﬀective frequencies. Also , for the ﬁrst time in the literature,\nwe have calculated the diﬀusion coeﬃcient of the paired solu te-vacancy specie by exploring all\nthe possibilities of the solute mobility, either via direct exchange solute-vacancy mechanism or\nby a vacancy drag mechanism in which the solute-vacancy pair migrates as a complex defect.\nAlthough we use classical methods, we reproduce the migrati on barriers for NiAl using the\nSIESTA code coupled to the Monomer method [15] using pipes of UNIX for the communications.\nWe proceed as follows, ﬁrst of all we validate the ﬁve-freque ncy model using the NiAl system\nas a reference case for which there are a large amount of exper imental data and numerical\ncalculations [16, 17]. Since, the AlUandNiAl systems share the same crystallographic f.c.c.\nstructure, the presented description is analogous for both alloys. The full set of frequencies\nare evaluated employing the echonomic Monomer method [14]. The Monomer [14] is used to\ncompute the saddle points conﬁgurations from which we obtai n the jumps frequencies deﬁned\n3in the 5-frequency model. Here, the inter-atomic interacti ons are represented by suitable EAM\npotentials [7] for the AlUbinary system. For the case of the NiAl system, our results are in\nexcellent agreement with the experimental data for both, th e solvent self-diﬀusion coeﬃcient\nand the solute tracer diﬀusion coeﬃcient [16, 17]. In this ca se we found that AlinNiat diluted\nconcentrations migrates as free species, conﬁrmed by a weak binding between Alwith vacancies\n(V). Comparison of the available experimental data of the diﬀu sion coeﬃcient of Udiluted in\nAl, with the diﬀusion coeﬃcient of the paired U+Vcomplex, show an excellent agreement.\nFrom theoretical evidence here presented, and from experim ental data in [4], we can infer that\nin this alloy a vacancy drag mechanism is likely to occur. Mag nitudes as, the strong uranium-\nvacancy binding, the values of the vacancy wind at high tempe ratures and negative values of\nthe cross L-coeﬃcient, (give us magnitudes that)lead us to this conclu sion.\nThe paper is organized as follows. In Section II we brieﬂy int roduce a summary of the\nmacroscopic equations of atomic transport that are provide d by non-equilibrium thermody-\nnamics [10, 18]. In this way an analytical expression of the d iﬀusion coeﬃcients in binary alloys\nin terms of Onsager coeﬃcients is presented. In section III, we describe the kinetic theory of\nisothermal diﬀusion process with an emphasis on the magnitu des used later. This allows to\nexpress the Onsager coeﬃcients in terms of the frequency jum ps following the procedure of\nAllnatt as in Ref. [10]. Section IV, is devoted to give the way to evaluate the Onsager phe-\nnomenological coeﬃcients following the procedure of Okamu ra and Allnat [9] in terms of the\njumps frequencies in the context of a multi-frequency model . In Section V, we present expres-\nsions to evaluate the self diﬀusion coeﬃcient in terms of so c alled solvent enhancement factor\nat ﬁrst order in the solute concentration ( cS), and the solute diﬀusion coeﬃcient is calculated\nat zero order in cS. Finally, in Section VI we present our numerical results usi ng the theoretical\nprocedure here summarized and showing a perfect accuracy wi th available experimental data,\nthat is, for the NiAl system. The last section brieﬂy presents some conclusions.\nReaders trained in this theory, can directly jump to section V.\nII. THEORY SUMMARY: THE FLUX EQUATIONS\nIsothermal atomic diﬀusion in multicomponent systems can b e described by the theory of\nirreversible processes, in which the main characteristic i s the rate of entropy production per\nunit volume S[10],\nTS=N/summationdisplay\nk/vectorJk./vectorXk, (1)\n4whereTis the absolute temperature, Nthe number of components in the system, /vectorJkdescribes\nthe ﬂux vector density, while /vectorXkis the driving force acting on component k. A linear expression\nfor the ﬂux vector /vectorJkin terms of the driving forces, involves the Onsager coeﬃcie ntsLij,\n/vectorJk=N/summationdisplay\niLki/vectorXi. (2)\nThe second range tensor Lijis symmetric ( Lij=Lji) and depends on pressure and temperature,\nbut is independent of the driving forces /vectorXk. From (2) the 1stFick’s law, which describe the\natomic jump process on a macroscopic scale, can be recovered . On the other hand, on each k\ncomponent, the driving forces may be expressed, in abscense of external force, in terms of the\nchemical potential µk, so that [10],\n/vectorXk=−T∇/parenleftigµk\nT/parenrightig\n. (3)\nWhere the chemical potential µkis the partial derivative of the Gibbs free energy with respe ct\nto the number of atoms of specie k,\nµk=/parenleftbigg∂G\n∂Nk/parenrightbigg\nT,P,N j/negationslash=k=µ◦\nk(T,P)+kBTln(ckγk). (4)\nwithγk, the activity coeﬃcients, deﬁned in terms of the activity ak=γkckandckthe concen-\ntration of specie k. For an isothermal diﬀusion process mediated by a vacancy me chanism, and\nby making use of the elimination of the dependent ﬂuxes,\nN/summationdisplay\niJi= 0⇒N/summationdisplay\nk=1Lki= 0. (5)\nFor the particular case of a binary diluted alloy (A,S)containing NAhost atoms, NS, solute\natoms (impurities), NVvacancies after some algebra we arrive at the ﬂux expression s,\nJA=LAA(XA−XV)+LAS(XS−XV); (6)\nJS=LSA(XA−XV)+LSS(XS−XV); (7)\njV=−(JA+JS). (8)\nNow we come back to the ﬂux equations (6-8) where we will intro duce the chemical potential\nequations (4) in the driving forces (3). In this way we obtain the generalized 1stFick’s law,\nwhich includes cross eﬀects:\nJA=−/parenleftbiggLAA\ncA−LAS\ncS/parenrightbigg\nkBT/parenleftbigg\n1+∂lnγA\n∂lncA/parenrightbigg\n∇cA, (9)\nJS=−/parenleftbiggLSS\ncS−LAS\ncA/parenrightbigg\nkBT/parenleftbigg\n1+∂lnγS\n∂lncS/parenrightbigg\n∇cS, (10)\n5Hence, for a binary system the diﬀusion coeﬃcient of the solv ent and the solute Sare:\nDA=kBT\nN/parenleftbiggLAA\ncA−LAS\ncS/parenrightbigg\nφA=D⋆\nAφA, (11)\nDS=kBT\nN/parenleftbiggLSS\ncS−LSA\ncA/parenrightbigg\nφS=D⋆\nSφS. (12)\nand\nDV=kBT\ncV(LAA+LSS+2LAS). (13)\n.DA,DSare commonly known as the intrinsic diﬀusion coeﬃcients, wh ileD⋆\nAandD⋆\nSare the\nisotopic tracers diﬀusion coeﬃcients that are the magnitud es experimentally measured. DSis\nthe vacancy diﬀusion coeﬃcient. In the spite of Gibbs-Duhem relation,\n/summationdisplay\nkNkXk= 0, (14)\nthe thermodynamic factors φA, φSare equal:\nφA=/parenleftbigg\n1+∂lnγA\n∂lncA/parenrightbigg\n=φS=/parenleftbigg\n1+∂lnγS\n∂lncS/parenrightbigg\n=φ0. (15)\nWe are interested in diluted alloys, that is, in the limit cS→0whereφ0= 1. The solute\ndiﬀusion coeﬃcient is calculated directly from the intrins ic one through the expression,\nD⋆\nS=DS=1\ncS/parenleftbiggkBT\nNLSS/parenrightbigg\n;cS→0, (16)\nwhile for the solvent D⋆\nA, is calculated from (11).\nIn the next sections, we express these last Onsager coeﬃcien ts in terms of microscopical\natomic jump frequencies.\nIII. THE KINETIC EQUATIONS\nIn this section we present a brief description of the applica bility of the master equation to\natomic transport in metals in terms of the spatial distribut ion of atoms and defects [10]. The\ntheory provides speciﬁc results to evaluate the atomic Onsa ger transport coeﬃcients for systems\nin which there is an attractive interaction between solute a nd vacancies. The solute-vacancy\npair is identiﬁed by the subscripts p,q. Where p,qdenotes the sites in the lattice where the\nsolute and vacancy are locate respectivelly. By conﬁgurati on we mean any distinct orientation\nof the pair .\nWe suppose that the solute-vacancy defect changes from ptoqby thermal activation at a\nrateωqp. These transition are taken to be Markovian, i.e, the ωqpdepend on the initial and ﬁnal\nconﬁgurations but are independent of all previous transiti ons. We denote the number density\n6of defects which are in conﬁguration pat timetbynp(t). For a closed set of conﬁguration p\nthe rate equations for the densities np(t)are,\n∂np(x,t)\n∂t=−/summationdisplay\nq/negationslash=pωqpnp+/summationdisplay\nq/negationslash=pωpqnq. (17)\nThe ﬁrst term represents the rate at which the vacancy in pleave the site to all the other\nconﬁgurations q(q/ne}ationslash=p) ﬁrst neighbors of p. The second is the rate at which the vacancy\nreachespfromq. Herenp(t) =ndpp(t)wherendis the defect density independently of its\nconﬁguration and pp(t)is deﬁned as the fraction of all defects that are in conﬁgurat ionpat\ntimet. In matrix notation equation (17) is,\ndpp(x,t)\ndt=−Ppp(t), (18)\nwhereppis a column matrix whose elements are the probabilities of oc cupation, pp≡ {p1,p2,...},\nwhilePis deﬁned as follows:\nPqp=−ωqp; (q/ne}ationslash=p) (19)\nPpp=/summationdisplay\nq/negationslash=pωqp. (20)\nOne feature of this equation, that we will be used later, is th at we can solve equation (17) in\nterms of a reduced matrix Q, which can be obtained from Psuch that its matrix elements Qpq\nare\nQpq=Ppq−Ppq, (21)\nnow the indexes pandqonly take positive values, that is, jumps that involve a drif t in the\npositively deﬁned sense of the principal crystal axis, whil e jumps in the opposite direction are\ndenoted by overlineq . In this way, Qis an×ndimension square matrix, where nis the number\nof diﬀerent conﬁgurations in the positive principal crysta l axis minus 1, as we will see in next\nsection. Under thermal equilibrium P≡P(0), is given by statistical thermodynamics as,\np(0)\np=exp(−E(0)\np/kBT)\n/summationtext\nγexp(−E(0)\nγ)/kBT; (∀p) (22)\nin which E(0)\nγis the Gibbs energy of the system in state γ. Under the same conditions we write\nin the steady state (dpp(x,t)\ndt= 0), the principle of detailed balance which gives us the usefu l\nrelation,\nω(0)\nqpp(0)\np=ω(0)\npqp(0)\nq; (∀p,q) (23)\nthat is,\nω(0)\nqp\nω(0)\npq= exp(E(0)\np−E(0)\nq). (24)\n7Where supraindex (0)denote magnitudes in the thermodynamical equilibrium.\nFrom the master equations, it is possible to calculate mean v alues and second momenta of the\nbasic kinetic quantities, (ex. xcoordinate of a tracer atom) in the thermodynamic equilibri um.\nAlso it is very useful to perform averages in regions that are small in a macroscopic sense, but\nlarge enough to contain many lattice points. Then, solving t he master equation in the linear\nresponse approximation it is possible to obtain formal expr ession for the transport coeﬃcients\nand to verify the Onsager relations. In this way, the macrosc opic ﬂux equations and the\ntransport coeﬃcients may be expressed in terms of averaged m icroscopic variables. This is\nindeed a generalization of the Einstein relations for the Br ownian motion. Also, it permits to\nexpress the Onsager coeﬃcients in terms of the jump frequenc ies. This procedure is described\nin details in [10].\nIt is now very useful to introduce the expressions derived by Franklin and and Lidiard\n[22] for the Onsager coeﬃcients and kinetic theory, in terms of the reduced Qmatrix. The\nauthors wrote equations for the ﬂuxes JSandJD(Dcan be vacancies, V, or interstitials, I) in\nterms of thermodynamical forces, which are precisely of the form required by non-equilibrium\nthermodynamics, then up to second-virial coeﬃcients. The, let the Onsager coeﬃcient obtained\ndeﬁned as,\nkBT\nNLKM=1\n2/summationtext\np,qaK\nqpaM\nqpωqpp(0)\np+/summationtext\nd,paK\ndpaM\ndpωdpp(0)\np\n+1\n2δKMp(0)\nK(1−zfp(0)\nK)/summationtext\nr(aK\nr0))2ωK(0)\nr0\n−2/summationtext(+)\np,qvK\nq(Q−1)qpvM\npp(0)\np.(25)\nThe velocity function is deﬁned as,\nvM\np=/summationdisplay\nuaM\nupωup+/summationdisplay\npaM\ndpωdp. (26)\nThe subscripts K,M each of which may be either A,SorD, whereA,Srepresent the solvent\nor solute atoms, while Dstand for the defects that may be either vacancies or interst itials. We\nuse the same labeling p,qandufor the paired species as before, and rfor unpaired species that\ncan be of type S(free solute) or D(free defects). While the label d(second term) takes into\naccount dissociative jumps, that is, it runs on sites that af ter the jump the defect is unpaired\nwith the solute. The assumption at which the species are rega rded to be paired or free may\nbe ﬁxed arbitrarily. The jump distance in a p→qtransition are represented by aK,M\nqp, they\ntake account the movement of the both, the KandMspecies. Similarly, aK(0)\nr0are the jump\ndistance of the free species, and aK,M\ndp, stand for the distance of dissociative jumps.\nThe ﬁrst term on the right side in (25), is the uncorrelated co ntribution of transitions from\none paired conﬁguration to another. The second term gives th e sum of the two corresponding\n8contributions from dissociation and association transiti on (equal by detailed balance, hence\nno factor 1/2as in the ﬁrst term), while the third term is the uncorrelated contribution from\nthe free-free transition (corrected by the term in zffor the fact that some movements of the\nunassociated pair may result in the formation of an associat ed pair, the contribution of which\nhave already been accounted for in the second term).\nCorrelated movements are represented in the fourth term, wh ich contain the Qmatrix.K\nmeans not K(i.e., ifK=V, thenK=Sand vice versa). Note that the summation only\nruns over those pair conﬁgurations in the (+)set, that is, jumps that involve a drift in the\npositively deﬁned sense of the principal crystal axis. Alth ough the summations contained in\nthe velocity analogue vM\npare over all conﬁgurations which can be reached in one transi tion\nfrom a conﬁguration plying in the (+)set. We note that the relevant point is to obtain the\nreducedQmatrix from Kinetic theory.\nBelow we apply the formalism following the procedure descri bed by Allnat and Lidiard [10]\nto calculate the Onsager coeﬃcients LAA,LSSandLAS=LSAand therefore the tracer diﬀusion\ncoeﬃcients D⋆\nAandD⋆\nS.\nIV. THE L-COEFFICIENTS IN THE SHELL APPROXIMATION\nHere we assume that the perturbation of the solute movement b y a vacancy V, is limited\nto its immediate vicinity, hence we adopt an eﬀective ﬁve fre quency model à la Le Claire [11]\nfor f.c.c. lattices, to understand the eﬀect of diﬀerent vac ancy exchange mechanisms on solute\ndiﬀusion. In such a model, the frequencies jumps ωqpare now denoted only with one index ωi\n(i= 0,1,2,3,4). In Figure 1 the jump rates are indicated as ωi(i= 1,2,3,4). We suppose\nthat all gradient potential and concentration are along a pa rticular crystal principal axis ˆx.\nConsidering only jumps between ﬁrst neighbors, for them, w2implies in the exchange between\nthe vacancy and the solute, w1when the exchange between the vacancy and the solvent atom\nlets the vacancy as a ﬁrst neighbor to the solute (positions d enoted with circled 1 in ﬁgure 1).\nThe frequency of jumps such that the vacancy goes to sites tha t are second neighbor of the\nsolute is denoted by ω3(sites with circled 2). The model includes the jump rate ω4for the\ninverse of ω3. Jumps toward sites that are third and forth neighbor of the s olute are denoted by\nω′\n3andω′′\n3respectively while ω′\n4andω′′\n4are used for their respective inverse frequency jumps.\nThe jump rate ω0is used for vacancy jumps among sites more distant than forth neighbors\nof the solute atom. In this context, that enables associatio n (ω4) and dissociation reactions\n(ω3), i.e the formation and break-up of pairs, the model include free solute and vacancies to\nthe population of bounded pairs. It is assumed that a vacancy which jumps from the second\n9FIG. 1: The ﬁve-frequency model of a solute-vacancy pair in a f.c.c. lattice.\nto the third shell, with ω0, never returns (or does so from a random direction). As in Ref . [13]\nwe express\nω⋆\n3= 2ω3+4ω′\n3+ω′′\n3, (27)\nand\nω⋆\n4= 2ω4+4ω′\n4+ω′′\n4. (28)\nThis procedure allows us to usufruct Boquet equations [19], and the technique develloped by\nAllnatt and Lidiard to evaluate the transport coeﬃcients of dilute solid solutions [10]. The\nsix symmetry types of vacancy sites that are in the ﬁrst coord ination shell (ﬁrst neighbor of\nthe solute) or the second coordination shell (sites accessi ble from the ﬁrst shell by one single\nvacancy jump) are listed in Table I (using the same notation a s in Ref. [9]) and plotted in\nFigure 2. As usual [19], sites that are equally distant from t he solute atom Sat the origin,\nand that have the same abscissa (x-coordinate in Fig.2) shar e the same vacancy occupation\nprobability ni,ni. Table II resumes the here employed notation. Here, we denot e the sites\nprobability with nijwhere for i/ne}ationslash= 0there is only one index ithat is given in crescent order\nin the distance to the solute atom S. Also, non overlined indexes imply in a positive abscissa,\nwhile overlined ones idenote sites with negative xcoordinate. For the sites in the x= 0plane\n(i= 0), the sites are denoted with two subindexes n0j, where the second index jis given in\ncrescent order of the distance to the solute atom S. Table II denotes the number of diﬀerent\ntypes of sites and the distance of them to the xaxis. With this classiﬁcation, the basic kinetic\nequations (17) for the ﬁrst coordinated shell approximatio n [9] in the steady state are written\n10TABLE I: Symmetry types for f.c.c. lattice for vacancy-sites at the ﬁrst 4 nearest −neighbor separation\nfrom the impurity Sat the origin (Ref. [9]). The forth and ﬁfth columns denote th e velocity functions\nof the solvent v(A)\npand solute atoms v(B)\nprespectively devided by the spacing parameter a(see text\nbelow).\nSymmetry type ivacancy-sites (Ref. [19]) n.n.s. v(A)\np/a v(S)\np/a\n1 (1,1,0),(1,1,0),(1,0,1),(1,0,1) 1(2ω1−3ω⋆\n3)ω2\n2 (2,0,0) 24(ω⋆\n4−ω0) 0\n3 (2,1,1),(2,1,1),(2,1,1),(2,1,1) 32(ω⋆\n4−ω0) 0\n4 (1,2,1),(1,2,1),(1,2,1),(1,2,1) 3(ω⋆\n4−ω0) 0\n(1,1,2),(1,1,2),(1,1,2),(1,1,2)\n5 (2,2,0),(2,2,0),(2,0,0),(2,0,2) 4(ω⋆\n4−ω0) 0\nTABLE II: Probability of occurrence of the vacancy at a site of the subset nj.\nnij(Ref. [19]) n5n4n3n2n1n01n02n0n1n2n3n4n5\n#of sites 4 8 4 1 4 4 4 4 4 1 4 8 4\nseparation 2a a√\n20a√\n5a a√\n2 2a a√\n2a a√\n50a√\n2 2a\nas,\n∂n1\n∂t=−(2ω1+ω2+7ω⋆\n3)n1+ω⋆\n4n2+2ω⋆\n4n3+2ω⋆\n4n4+ω⋆\n4n5+2ω1n0+ω4n01+ω2n1= 0,\n∂n2\n∂t= 4ω⋆\n3n1−(8ω0+4ω⋆\n4)n2+4ω0n3= 0,\n∂n3\n∂t= 2ω⋆\n3n1+ω0n2−(10ω0+2ω⋆\n4)n3+2ω0n4+ω0n5= 0,\n∂n4\n∂t=ω⋆\n3n1+ω0n3−(9ω0+2ω⋆\n4)n4+ω0n5+ω⋆\n3n0+ω0n01+ω0n02= 0,\n∂n5\n∂t=ω⋆\n3n1+4ω0n3+2ω0n4−(8ω0+ω⋆\n4)n5= 0, (29)\n∂n0\n∂t= 2ω1n1+2ω⋆\n4n4−(4ω1+7ω⋆\n3)n0+2ω⋆\n4n01+ω⋆\n4n02+2ω1n1+2ω⋆\n4n4= 0,\n∂n01\n∂t=ω1n1+2ω0n4+2ω0n0−(8ω0+4ω⋆\n4)n01+ω3n1+2ω0n4= 0,\n∂n02\n∂t= 2ω0n4+ω3n0+ω0n01−(11ω0+ω⋆\n4)n02+2ω0n4= 0,\n...\nwhere the vertical dots denotes the analogous sets of equati ons for the overlined indexes\n∂ni/∂t= 0. Hence, the matrix Pdeﬁned in (20) that stands from equation (29) is such\nthatP∈R13×13. Then the reduced matrix Q, whose elements are Qpq, can be obtained from\n11\u0001\n\u0002\u0002\u0002\u0002\u0001\u0003\u0001\u0004\u0001\u0005\n\u0001\u0006\u0001\u0007\u0003\n\u0001\u0007\u0006\n\u0001\u0007\u0001\u0005\n\u0001\u0004\n\u0001\u0003\u0001\b\u0001\b\n\u0001\u0006\n\u0002\nFIG. 2: The coordinated shell model in f.c.c. lattice (see Re f. [19]). The diﬀerent types of symmetries\nshown are detailed in Table II. In the ﬁgure, blue bullets are the ﬁrst twelve neighbors sites to the\nsoluteSat the origin. In green the 42 subsequent sites. In red, the th ird coordinated shell from which\nthe vacancy never returns to the second shell.\nPas,\nQpq=Ppq−Ppq, (30)\nnow the indexes pandqtake only positive values, such that Qis a ﬁve dimension square matrix,\nso that\nQ=\n(2ω1+2ω2+7ω⋆\n3)−4ω⋆\n42ω⋆\n4 −ω⋆\n4−ω⋆\n4\n−ω⋆\n3 (4ω⋆\n4+ω0)−ω0 0 0\n−2ω⋆\n3 −4ω0(2ω⋆\n4+ω0)−ω0−ω0\n−2ω⋆\n3 0 −2ω0(2ω⋆\n4+ω0)−2ω0\n−ω⋆\n3 0 −2ω0−ω0(ω⋆\n4+ω0)\n(31)\n12The matrix element (Q−1)11of the inverse matrix of Qdeﬁnes a factor named F, introduced\nby Manning [23],\n(Q−1)11= (2ω1+2ω2+7ω⋆\n3F)−1. (32)\nThe quantity 1−Fis the fractional reduction in the overall frequency of jump s from a ﬁrst-shell\nsite to a second-shell site caused by returns of vacancy to ﬁr st-shell sites,\n7(1−F) =10ǫ4+B1ǫ3+B2ǫ2+B3ǫ\n2ǫ4+B4ǫ3+B5ǫ2+B6ǫ+B7(33)\nwhereǫ=ω⋆\n4/ω0and Table III shows the Bicoeﬃcients calculated by Koiwa [10, 24] and that\nwill be employed in the present calculations.\nTABLE III: Coeﬃcients in the expression for Ffor the ﬁve\nfrequency model calculated by Koiwa [24].\nB1B2B3B4B5B6B7\nRef. [24] 180.3 924.3 1338.1 40.1 253.3 596.0 435.3\nFor evaluating the L-coeﬃcients, we shall need both the site fraction cpof solute atoms which\na vacancy among their znearest-neighbor sites, also the fraction of unbounded vac anciesc′\nV=\ncV−cpand of unbound solute atoms c′\nS=cS−cp. These are related through the mass action\nequation, namely\ncp\nc′\nVc′\nS=zexp(−Eb/kBT) =ω⋆\n4\nω⋆\n3. (34)\nWithEbthe binding energy of the solute vacancy pair related to ω⋆\n4/ω⋆\n3by the use of detailed\nbalance. Then, if the pairs and free vacancies are in local eq uilibrium and, since the fraction\nof solute cSwill be much greater than cVand thus also cp, we can express the equilibrium\nconstant Kas,\ncp\ncV−cp=zcSexp(−Eb/kBT)≡KcS, (35)\nand equivalently\ncp=cV/parenleftbiggKcS\n1+KcS/parenrightbigg\n. (36)\nThe Onsager coeﬃcients can be entirely obtained from equati on (25) in terms of the concen-\ntration of free and paired species, and in terms of the jump fr equency rates ωi. For the case of\nbinary alloys in f.c.c. lattices, symmetry arguments and sp acial isotropy implies that the only\nneeded coeﬃcients are LAA,LSSandLAS. In this respect, the velocity terms are depicted in\nthe forth and ﬁfth column of Table I. For the case where the Ons ager coeﬃcients are expressed\nin terms of the ﬁve-frequency model, the only required eleme nts ofQ−1is(Q−1)11all the other\n13elements appearing in LASandLAAcan be eliminated [10]. Hence, the Onsager coeﬃcients\n(25) are [9]\nLAA=Ns2\n6kBT/braceleftig\n12c′\nV(1−7c′\nS)ω0+cp(A(0)\nAA+A(1)\nAA)/bracerightig\n(37)\nLAS=LSA=cpA(1)\nAS (38)\nLSS=Ns2cpω2\n6kBT/braceleftbigg\n1−2ω2\nΩ/bracerightbigg\n(39)\nwheres=aA/√\n2is the jump distance, aAthe lattice parameter of solvent A. The concentration\nof free solute and vacancy defects are denoted with c′\nSandc′\nVrespectively. While,\nΩ = 2(ω1+ω2)+7ω⋆\n3F. (40)\nWe deﬁne the solute correlation factor fSfor the bracket in (39) as,\nfS= 1−2ω2\n2(ω1+ω2)+7ω⋆\n3F. (41)\nCompleting the deﬁnitions in (37) and (38) with,\nA(0)\nAA= 4ω1+14ω⋆\n3 (42)\nA(1)\nAA=1\nΩ/bracketleftbig\n−2(3ω⋆\n3−2ω1)2+14ω⋆\n3(1−F)/parenleftbiggω0−ω⋆\n4\nω4/parenrightbigg\n×/braceleftbigg\n(3ω⋆\n3−2ω1)−2(ω1+ω2+7ω⋆\n3/2)/parenleftbiggω0−ω⋆\n4\nω⋆\n4/parenrightbigg/bracerightbigg/bracketrightbigg\n(43)\nA(1)\nAS=ω2\nΩ/bracketleftbigg\n2(3ω⋆\n3−2ω1)+14ω⋆\n3(1−F)/parenleftbiggω0−ω⋆\n4\nω⋆\n4/parenrightbigg/bracketrightbigg\n. (44)\nIn order to calculate the self diﬀusion coeﬃcients D⋆\nAandD⋆\nSwe must replace the L-\ncoeﬃcients expressions (37,38,39) in (11,12), that for the diﬀusion coeﬃcients.\nV. EXPRESSIONS FOR D⋆\nA,D⋆\nSANDD⋆\npCOEFFICIENTS\nA comparison between experimental data and the present simu lations are possible with the\nknowledge of the two tracer diﬀusion coeﬃcients D⋆\nAandD⋆\nS. ForD⋆\nAor equivalently LAAit\nis necessary to consider the motion of the tracer atom A⋆via a vacancy mechanism caused\nby both, vacancies at ﬁrst neighbors of Sor at the unperturbed lattice sites. The tracer self-\ndiﬀusion coeﬃcient D⋆\nA(cS)of the specie Ain a diluted alloy with a concentration cSof solute\natomsS, can be written in terms of the self diﬀusion coeﬃcient D⋆\nA(0), of the specie Ain pure\nf.c.c. lattice as,\nD⋆\nA(cS) =D⋆\nA(0)(1+bA⋆cS), (45)\n14at ﬁrst order in cS. The solvent enhancement factor, bA, is obtained in terms of the properties of\nthe solute-vacancy model. On the other hand, for the pure sol ution, the self diﬀusion coeﬃcient\nD⋆\nA(0)is given by [11],\nD⋆\nA(0) =a2\nAc0\nVf0ω0. (46)\nwhereaAis the solvent lattice parameter, f0= 0.7815is the correlation factor for the self-\ndiﬀusion in f.c.c. lattices, and c0\nVis the vacancy concentration at the thermodynamical equi-\nlibrium. This former is such that,\nc0\nV= exp/parenleftigg\n−EV\nf\nkBT/parenrightigg\n, (47)\nwhereTis the absolute temperature, EV\nfis the formation energy of the vacancy in pure A. The\nentropy terms are set to zero, which is a simplifying approxi mation. So that, inserting (47) we\nget\nD⋆\nA(cS) =a2\nAf0ω0exp/parenleftigg\n−EV\nf\nkBT/parenrightigg\n. (48)\nWe assume cS→0then, we use pure lattice parameters for all our calculation s. The solute-\nenhancement factor bA⋆, is obtained by replacing (37) in (11) up to ﬁrst order in the s olute\nconcentration. In the particular case of the ﬁve-frequency model, the expressions for the On-\nsager coeﬃcients are (37,38,39). Hence, as in Ref. [10, 11], we get,\nbA⋆=−19+4ω1+14ω⋆\n3\nω0/parenleftbiggω⋆\n4\nω⋆\n3/parenrightbigg\n+1\n(ω1+ω2+7ω⋆\n3F/2){\n−14(1−F)(ω0−ω⋆\n4)2×(ω1+ω2+7ω⋆\n3/2)\nω0ω⋆\n4(49)\n+ω⋆\n4(3ω⋆\n3−2ω1)2+14ω⋆\n3(1−F)×(3ω⋆\n3−2ω1)(ω0−ω⋆\n4)\nω0ω⋆\n3/bracerightbigg\n.\nIn the diluted limit ( cS→0),D⋆\nSis identical to the intrinsic diﬀusion coeﬃcient DSgiven by\n(16)\nDS=D⋆\nS=kBT\nnSLSS. (50)\nIntroducing LSSin (39) and the detailed balance equation (34) in (50), we obt ain an expression\nfor the tracer solute diﬀusion coeﬃcient,\nD⋆\nS=a2ω2/parenleftbiggcp\n3cS/parenrightbigg\n×/braceleftbiggω1+7ω3F/2\nω1+ω2+7ω3F/2/bracerightbigg\n=a2ω2/parenleftbiggcp\n3cS/parenrightbigg\n×fS. (51)\nIn (51) we introduce the solute correlation factor fSas,\nfS=/braceleftbiggω1+7ω3F/2\nω1+ω2+7ω3F/2/bracerightbigg\n. (52)\n15whereFwas previously deﬁned in (32). In the Le Claire description, D⋆\nScan also be expressed\nas,\nD⋆\nS=a2\nAfSω2exp/parenleftigg\n−EV\nf+Eb\nkBT/parenrightigg\n. (53)\nFor the drift of solutes in a vacancy ﬂux we shall make contact with the alternative phenomenol-\nogy oﬀered by Johnson and Lam [29]. In terms of thermodynamic forces, which are precisely\nof the form required by non-equilibrium thermodynamics, up to second-virial coeﬃcients, the\nﬂux of solute atoms JBis expressed as\nJS=−Dp∇Cp+σVc′\nSDV∇c′\nV, (54)\nThe coeﬃcients DpandDVare interpreted as diﬀusion coeﬃcients of pairs and free vac ancies,\nrespectively, while σVis a sort of cross section for vacancies to induce solute moti on. When\nwe insert the appropriate chemical potential gradients (se e Franklin and Lidiard [30]) into the\nthermodynamic ﬂux equation (7), we ﬁnd that (7) is equivalen t to (54) if\nDp=kBT\nNcpLSS (55)\nand\nσVDV=kBT\nNLAS+2LSS\ncp/parenleftbigg12ω⋆\n4\nω⋆\n3/parenrightbigg\n. (56)\nWe see that for a vacancy mechanism, solute atoms may only mov e when they are paired with\na vacancy and it is reasonable therefore that DSshould be equal to ( cp/cS) as (12) and (55)\nrequire. To obtain σVfrom (56), we need the full expressions for LAAandLASin (37) and (38).\nIf we take this to be the vacancy diﬀusion coeﬃcient in the per fect solvent lattice, i.e. 4a2\nAlω0,\nwe then obtain\nσV=2ω2\nω0×[(3+7F)ω⋆\n4+7(1−F)(ω0−ω⋆\n4)]\nω1+ω2+7ω⋆\n3F. (57)\nWe proceed to show the results obtained by direct applicatio n of the previous theory, to the\nstudy of the diﬀusion of impurities in dilute alloys mediate d by a vacancy mechanism.\nVI. RESULTS\nWe present our numerical results for NiAl andAlUsystems. The interatomic interactions\nare represented by suitable EAM potentials [7, 25, 26] for bi nary systems. For AlU, the cross\npotential has been ﬁtted taking into account the available ﬁ rst principles data [25, 26]. Lattice\nparameters, formation energies and bulk modulus for each in termetallic compound are well\nreproduced. We obtain the equilibrium positions of the atom s by relaxing the structure via\nthe conjugate gradients technique. The lattice parameters that minimize the crystal structure\n16energy are respectively aNi= 3.52Å andaAl= 4.05Å forNiandAlsolvents. For all the cal-\nculations we used a christallyte of 2048 atoms, eventually i ncluding one substitutional Alatom\ninNiand one substitutional Uatom inAlbulk and a single vacancy in both defective systems.\nThe current calculations have been performed at T= 0K. In this case, the entropic barrier is\nignored. Our calculations are carried out at constant volum e, and therefor the enthalpic barrier\n∆H= ∆U+p∆Vis equal to the internal energy barrier ∆U.\nIn Table IV, we present our results for the vacancy formation energy (EV\nf) in pure hosts of\nNiandAlcalculated as,\nEV\nf=E(N−1)+Ec−E(N), (58)\nwhere,E(N)for the perfect lattice of Natoms,E(N−1)is the energy of the defective system,\nandEcthe cohesion energy. The migration barrier of the vacancy in perfect lattice ( EV\nm), is\ncalculated with the Monomer method [14], and the activation energyEQas,\nEQ=EV\nf+EV\nm. (59)\nTABLE IV: Energies and lattice parameters for the pure AlandNif.c.c. lattices. The ﬁrst column\nspeciﬁes the metal, vacancy formation energy EV\nf(eV)are shown in the second column. The third\ncolumn displays the migration energies EV\nm, calculated from the Monomer method [14]. In the forth\ncolumn we show the lattice parameter aA(Å). The last column displays the activation energy EQ(eV).\nReference Latt.EV\nf(eV)EV\nm(eV)aA(Å)EQ(eV)\nPresent work Al 0.649 0.65 4.05 1.299\n[31] Al 0.675 0.63 4.05 1.305\npresent work Ni 1.56 0.85 3.52 2.41\n[17] Ni 1.40 1.28 3.52 2.65\nFor the case of a diluted alloy, we may consider the presence o f solute vacancy complexes,\nCn=S+Vnin which n= 1st,2nd,3rd,...(see the insets in Fig. V) indicates that the vacancy\nis an−nearest neighbors of the solute atom S. The binding energy between the solute and the\nvacancy for the complex Cn=S+Vnin a f.c.c. matrix of Natomic sites is obtained as,\nEb={E(N−2,Cn)+E(N)}−{E(N−1,V)+E(N−1,S)}, (60)\nwhereE(N−1,V)andE(N−1,S)are the energies of a crystallite containing ( N−1) atoms\nof solvent Aplus one vacancy V, and one solute atom Srespectively, while E(N−2,Cn)is\nthe energy of the crystallite containing ( N−2) atoms of Aplus one solute vacancy complex\n17Cn=S+Vn. With the sign convention used here Eb<0means attractive solute-vacancy\ninteraction, and Eb>0indicates repulsion.\nWe calculate the migration energies Emusing the Monomer Method [14], a static technique to\nsearch the potential energy surface for saddle conﬁguratio ns, thus providing detailed information\non transition events. The Monomer computes the least local c urvature of the potential energy\nsurface using only forces. The force component along the cor responding eigenvector is then\nreversed (pointing “up hill\"), thus deﬁning a pseudo force t hat drives the system towards saddles.\nBoth, local curvature and conﬁguration displacement stage s are performed within independent\nconjugate gradients loops. The method is akin to the Dimer on e from the literature [28], but\nroughly employs half the number of force evaluations which i s a great advantage in ab-initio\ncalculations.\nBinding energies in Tables V and VI are displayed, respectiv ely forNiAl andAlU. Relative\nto theNiAl system, a weak energy interaction, Eb, between that vacancy and solute can be\nobserved in almost all the nearest neighbor conﬁgurations. Also, a weak attractive interaction\nexists between the vacancy an the Alsolute atom only at 1stand4thnearest neighbor conﬁgu-\nrations, while is repulsive for the rest of the pairs. The sam e behavior is observed for the AlU\nsystem but in this case, the binding energy of the pair ( U+V) at ﬁrst neighbor position, is\nstrongly attractive. Tables V and VI also display the diﬀere nt type of solute vacancy complex\nCn=S+Vnwith its binding energies Eb. Also, the same tables, depict the possibles conﬁgu-\nrations and jumps that involve the corresponding Cn=S+Vncomplex with the corresponding\njump frequencies. This jumps imply in a migration of the vaca ncy whose energies are shown\nfor the direct as well as for the reverse jumps. Relative to th e migration barriers, we see that,\nforNiAl and from Tables V, the vacancy migration barriers E←\nmare close to that in the perfect\nlatticeEV\nm= 0.85eV.\n18TABLE V: Jumps and frequencies in NiAl. The ﬁrst column denotes Cn=S+VnwhereVnmeans\nthat the vacancy is nnearest neighbor of the solute. Binding energy Ebis shown in the second column.\nThe jumps are depicted in the third column, while the forth co lumn describes the jump frequency ωi\nand the conﬁgurations involved in each jump. Migration ener giesEmfor direct and reversed jumps\nare written in the ﬁfth and sixth column respectively.\nCn=S+VnEb(eV)Conﬁg.(Fn) ωi E→\nm(eV)E←\nm(eV)\nC1 -0.06 C1ω1/d47/d47C1ω1/d111/d111 0.94 0.94\nC1S -0.06 C1Sω2/d47/d47C1Sω2/d111/d111 0.85 0.85\nC2 0.03 C1ω3/d47/d47C2ω4/d111/d111 0.85 0.77\nC3 0.03 C1ω′\n3/d47/d47C3\nω′\n4/d111/d111 0.86 0.77\nC4 -0.001 C1ω′′\n3/d47/d47C4\nω′′\n4/d111/d111 0.82 0.76\nC5 −0.001 C2ω⋆\n0/d47/d47C5\nω⋆\n0/d111/d111 0.84 0.87\nC6 −0.001 C4ω⋆\n0/d47/d47C6\nω⋆\n0/d111/d111 0.84 0.84\nC7 −0.001 C2ω43/d47/d47C5ω34/d111/d111 0.84 0.87\n19TABLE VI: Jumps and frequencies in AlU. The columns description is the same as in Table V.\nCn=S+VnEb(eV)Conﬁg.(Fn) ωi E→\nm(eV)E←\nm(eV)\nC1 -0.139 C1ω1/d47/d47C1ω1/d111/d111 0.81 0.81\nC1S -0.139 C1Sω2/d47/d47C1Sω2/d111/d111 0.48 0.48\nC2 0.004 C1ω3/d47/d47C2ω4/d111/d111 0.61 0.47\nC3 0.037 C1ω′\n3/d47/d47C3\nω′\n4/d111/d111 0.65 0.48\nC4 0.019 C1ω′′\n3/d47/d47C4\nω′′\n4/d111/d111 0.73 0.58\nC5 0.015 C2ω⋆\n0/d47/d47C5\nω⋆\n0/d111/d111 0.59 0.58\nC6 -0.003 C4ω⋆\n0/d47/d47C6\nω⋆\n0/d111/d111 0.63 0.65\n20For theAlUthe migration barriers are quite diﬀerent from that in perfe ct lattice for associ-\nation jumps, except for c6. In comparison with the NiAl case, the jump C1ω′′\n3/d47/d47C4\nω′′\n4/d111/d111 , involves\nmore than one atom, i.e. is a multiple jump as indicated in the ﬁgure in Table VI. In table\nVII, we show the migration barriers for more distant neighbo rs pairs than the forth, with the\npurpose to ﬁnd out from where the jump frequency are similar t o that of the perfect crystal ω0.\nIn order to obtain the jump frequencies, we assume that the ju mps are thermally activated\nTABLE VII: Jumps beyond the second coordinated shell. The col umns denotes the same notation as\nin Table V, binding energies are shown in the second column. T he third column denoted the frequency\nrate, where the supra indexes(⊥,∓)onω0implies vacancy jumps perpendicular to, backward or forwar d\n∓ˆxrespectively. Migration energies for the direct and backwa rd jumps ares shown in column four and\nﬁve respectively\nCn=S+VnEb(eV)ωiE→\nm(eV)E←\nm(eV)\nC7 0.002ω⊥\n0C7→C10 0.61 0.64\nC8 0.015ω−\n0\nC8→C11 0.64 0.61\nC9 0.002ω+\n0C12→C12 0.61 0.64\nFIG. 3: Vacancy jumps beyond the second coordinated shell. T he supra indexes(⊥,∓)onω0implies\nvacancy jumps perpendicular to, backward or forward ∓ˆxrespectively.\nand then the frequencies ωican be expressed as,\nωi=ν0exp(−E→\nm/kBT). (61)\nwhereE→\nmare reported in Tables V and VI for both systems. For the prefa ctor in (61), we use\na constant attempt frequency ν0= 6×1012Hz, taken from Ref. [31] for pure Al. We also use,\nin terms of the Wert model [32], a temperature dependent atte mpt frequency [27] as,\nν0(T) =kBT\nh, (62)\n21wherehis the Planck constant. Also in Tables V and VI, the migration barriers and the\ncorresponding rate frequency for each jump are shown. For bo thNiAl andAlU, Table VIII\npresents the calculated frequencies at two diﬀerent temper atures. We adopt the Wert model as\nin Ref. [17, 27], i.e, a temperature dependent pre-exponent ial factor from(61).\nTABLE VIII: Vacancy jump frequencies rate ωicalculated with a temperature dependent attempt\nfrequency ν0(T), at two diﬀerent temperatures in NiAl andAlUalloys. The symbol (⋆) indicates that\nwe are calculating the eﬀective frequencies ω⋆\n3andω⋆\n4.\nNiAl AlU\nT1= 800K T2= 1700KT1= 300K T 2= 700K\nωiωi(Hz)ωi(Hz) ωi(Hz)ωi(Hz)\nω07.36×1071.07×10111.63×1024.25×108\nω11.99×1075.79×10101.87×10−22.33×107\nω27.36×1071.07×10115.17×1045.01×109\nω⋆\n37.36×1071.06×10111.50×1023.59×108\nω⋆\n42.40×1081.87×10116.24×1045.03×108\nIt is clear that the inclusion of UinAlhas signiﬁcant inﬂuence on the solvent frequency\njumps that the inclusion of AlinNi. This fact may be a consequence of the marked diﬀerence\nbetween the solute and solvent atomic numbers, ZU−ZAl= 92−13 = 79 forUdiluted in Al,\nwhile it is ZAl−ZNi= 13−28 =−15forAlinNi.\nOnce we have calculated the jump frequencies, then the solut e correlation factors fSand\nthe solvent enhancement factors bAcan be obtained. We present our results in Table IX, where\nwe show for diﬀerent temperatures, the solvent-enhancemen t factorb⋆\nAcalculated from (49),\nwithf0= 0.7815, and the solute-correlation factor fSfrom (41), for both F= 1andF/ne}ationslash= 1\napproximations. Also, table IX, resumes the jump frequenci es ratios calculated according to\nthe ﬁve-frequency model of solute-vacancy interaction for pre-exponential factor depending on\nthe temperature.\nThe solute-correlation factor ( fS) withTand calculated from (52) and (41) in the F= 1\nandF/ne}ationslash= 1approximations. They are shown in Table IX and Figures 4 and 5 , forNiAl and\nAlUrespectively. The factor Fobtained from equation (33) is also shown.\nConcerning to the solvent-enhancement factors, ( bA), calculated from (49), the results are\nshown together with fSin Table IX, also in Figures 6, 7, respectively for NiAl andAlU, as a\nfunction of the temperature. In the Le Claire approximation s and for F/ne}ationslash= 1,bNiandbAlare\npositive with T. ForF/ne}ationslash= 1,b⋆\nAdecrease for both NiandAlsolvents with respect to the Le\n22TABLE IX: Solvent enhancement and solute correlated factors forNi,Al andAl,U at diﬀerent tem-\nperatures, for both F= 1andF/ne}ationslash= 1approximations. For the solvent enhancement factor bA(columns\ntwo and three), and for the solute correlated factor fS(columns four and ﬁve). The last tree columns\ndescribe the jump frequency ratios of the solute −vacancy interaction.\nAlloyT/K bF=1\nNi∗ bF/negationslash=1\nNi∗ fF=1\nAlfF/negationslash=1\nAlω2\nω1ω⋆\n3\nω1ω⋆\n4\nω0\nNiAl 700 30.22 24.96 0.78 0.68 3.78 3.79 3.28\n800 31.24 25.11 0.79 0.69 3.69 3.69 3.26\n900 25.47 21.05 0.79 0.70 3.19 3.19 3.86\n1000 21.38 18.05 0.79 0.71 2.84 2.83 2.57\n1100 18.34 15.75 0.79 0.72 2.58 2.57 2.36\n1200 16.00 13.93 0.80 0.72 2.38 2.37 2.20\n1300 14.16 12.46 0.80 0.73 2.23 2.22 2.07\n1400 12.66 11.25 0.80 0.73 2.11 2.09 1.96\n1500 11.43 10.24 0.80 0.73 2.01 1.99 1.88\n1600 10.40 9.37 0.80 0.73 1.92 1.91 1.81\n1700 9.52 8.63 0.80 0.74 1.85 1.84 1.74\nAlloyT/K bF=1\nAl∗ bF/negationslash=1\nAl∗ fF=1\nU fF/negationslash=1\nUω2\nω1ω⋆\n3\nω1ω⋆\n4\nω0\nAlU 3009.5×1034.4×1031.0×10−22.9×10−32.8×1058.0×1023.8×102\n3503.6×1031.8×1032.2×10−26.5×10−34.6×1042.9×1021.6×102\n4001.8×1039.4×1023.9×10−21.2×10−21.2×1041.4×1028.3×101\n4501.0×1035.6×1026.0×10−21.9×10−24.2×1037.7×1014.9×101\n5006.4×1023.6×1028.0×10−22.9×10−21.8×1034.9×1013.3×101\n5504.4×1022.6×1020.11 4.2×10−29.3×1023.4×1012.4×101\n6003.3×1021.9×1020.14 6.7×10−25.3×1022.5×1011.8×101\n6502.5×1021.5×1020.17 7.8×10−23.3×1021.9×1011.4×101\n7001.9×1021.2×1020.20 9.2×10−22.1×1021.5×1011.2×101\n7501.6×1021.0×1020.23 0.11 1.50×1021.3×1011.0×101\nClaire approximation (i.e., F= 1). It must be taking into account that this diﬀerence will be\nhighly diminished in the diﬀusion coeﬃcient because the enh ancement factor is multiplied by\nthe solute concentration cS, which is low for diluted alloys.\nThe Onsager and Diﬀusion coeﬃcients were calculated assumi ng a solute mole fraction of\n23FIG. 4: Solvent correlation factor fAl⋆in theNiAl system as a function of the temperature for both,\nF= 1(circles) and F/ne}ationslash= 1(squares) approximations. The Ffactor is denoted with up triangles.\nFIG. 5: Same as ﬁgure 4 for the AlUsystem.\ncS= 4.9×10−4, for both alloys, which corresponds to nAl= 4.53×1019cm−3atoms/cm3\nforNiAl andnU= 3.01×1019cm−3atoms/cm3forAlUsystem. Once calculated LASand\nLSS, and following the reasoning in Ref. [13], we also calculate the vacancy wind coeﬃcient\nG=LAS/LSS=−(1+LVS/LSS). The results are presented in Figures 8 and 9, for NiAl and\nAlUsystems respectively. We see that if G <−1,LVBis positive, then the vacancy and the\nsolute diﬀuse in the same direction as a complex specie [13]. This transport phenomena could\noccur in the AlUcase, due to the strong binding of the U+V1pair, while is unlikely to occur\nforAlinNiby the opposite argument. The vacancy wind parameter veriﬁe sG >−1forNiAl\nin both, F= 1andF/ne}ationslash= 1approximations, while for Al,U the behavior changes drastically\ndepending on the case. If F= 1,Gremains positive, but for F/ne}ationslash= 1,G >−1as shown in\nFig. 9 in the temperature range [300−650]◦C, this being an indication that a vacancy drag\n24FIG. 6: Solvent-enhancement factor bNiin theNiAl system as a function of the temperature for both,\nF= 1(circles) and F/ne}ationslash= 1(squares) approximations.\nFIG. 7: Solvent-enhancement factor bAlin theAlUsystem as a function of the temperature for both,\nF= 1(circles) and F/ne}ationslash= 1(squares) approximations.\nmechanism can occurs for AlU.\nThe full set of L-coeﬃcients for F/ne}ationslash= 1, are displayed in Figs. 10 and 11, against the\ninverse of the temperature for the NiAl andAlUrespectively. We see that for the NiAl case\ntheL-coeﬃcients follow an Arrhenius behavior, which implies a l inear relation between the\nlogarithm of L-coeﬃcients against the inverse of the temperature (see Fig . 10). While for the\nAlUcase, at high temperatures, we can appreciate a slight devia tion from the Arrhenius law\n(see Fig. 11). In Figure 11, the cross LAlU=LUAlcoeﬃcient is negative for all the range of\ntemperature. Now, we are in position to obtain the diﬀusion c oeﬃcients D⋆A(0),D⋆B(0)and\nDp, for the paired specie. First, we present the ratio of calcul ated tracer diﬀusion coeﬃcients\nD⋆\nS/D⋆\nAas a function of the inverse of the temperature for the NiAl andAlUin Figures 12\n25FIG. 8: Ratio of the vacancy-Onsager coeﬃcients of AlinNicalculated from eqs.(38,39) vs1/Tfor\nboth,F= 1(circles) and F/ne}ationslash= 1(squares).\nFIG. 9: Ratio of the vacancy-Onsager coeﬃcients of UinAlcalculated from eqs.(38,39) vs1/Tfor\nboth,F= 1(circles) and F/ne}ationslash= 1(squares).\nand 13, respectively.\nThe calculated D⋆\nAandD⋆\nSforF/ne}ationslash= 1, using the equations (48) and (51), are shown in Figures\n14 and 15 respectively for NiAl andAlU. The diﬀusion coeﬃcient of the paired specie, Dp,\ncalculated from (55) is also shown. It is import to perform a c omparison between theoretical\nresults obtained in present work with reliable experimenta l data. We have veriﬁed that the\ntracer self diﬀusion coeﬃcient D⋆\nA(cS)for a diluted alloy is practically equal to that for the pure\nsolventD⋆\nA(0)(i.e.,D⋆\nA(cS)≃D⋆\nA(0)). Hence, we can test our results for D⋆\nA(cS)with the best\nestimative of the diﬀusion parameter for pure solvent, D⋆\nL(A), taken from Campbell et al. [16].\nThe authors, have been used weighted means statistics to det ermine consensus estimators which\nrepresents best the available experimental data. They use a Gaussian distribution to represent\n26FIG. 10: Vacancy-Onsager coeﬃcients vs1/Tfor theNiAl system for F/ne}ationslash= 1. Squares denote LAlAl,\nempty circles denote LNiNiwhileLNiAlis described with ﬁlled circles.\nFIG. 11: Vacancy-Onsager coeﬃcients vs1/Tfor theAlUsystem for F/ne}ationslash= 1. Squares denote LUU,\nempty circles denote LAlAlwhileLUAlis described with ﬁlled circles.\nthe experimental error used to determine the best estimates of the parameters common to all\nof the included studies in the parameter D⋆\nL(A), the self-diﬀusivity of species Ain pureAgiven\nincm2s−1. The best estimate is given through an expression of the form ,\nD⋆\nL(A) =D0\nAexp(−QA/RT), (63)\nwhereD0\nAandQAfrom Ref. [16] are dysplayed in Table X for pure NiandAl,R= 8.314472\nJ/mol K is the ideal gas constant and Tis the absolute temperature and represented by solid\nlines in Figures 14 and 15.\nAs can be observed, D⋆\nL(A)ﬁts perfectly with the values of D⋆\nAcalculated in the present work.\nFor the case of NiAl alloys, in Fig. 14 experimental data of the solute diﬀusion c oeﬃcient are\nplotted with stars and cruxes respectively for T= [914−1212]◦C[36] andT= [1372−1553]◦C\n27FIG. 12: Ratio of the tracer diﬀusion coeﬃcient D⋆\nS/D⋆\nAinNiAlvs1/Tfor both, F= 1(circles) and\nF/ne}ationslash= 1(squares) approximations.\nFIG. 13: Ratio of the tracer diﬀusion coeﬃcient D⋆\nS/D⋆\nAinAlU)vs1/Tfor both, F= 1(circles) and\nF/ne}ationslash= 1(squares) approximations.\n[37]. As we can observe the accuracy with the calculated solu te diﬀusion coeﬃcient D⋆\nAlis\nastonishing, showing that the here employed procedure give s excellent results for calculating\nthe diﬀusion coeﬃcients in diluted f.c.c. alloys. The diﬀus ion coeﬃcient for the paired specie\nAl+VinNi,Al is also shown.\nA little more attention we devote to AlUsystem. In the literature, we have found experi-\nmental values for the Udiﬀusion coeﬃcient at inﬁnite dilution in Al[4]. The authors ﬁt their\nown experimental results solving numerically the diﬀusion equation\n∂C(x,t)\n∂t=D∂2C(x,t)\n∂x2, (64)\nwith boundary condition x= 0;C(0,t) =S0, whereS0is the maximum solubility of the\n28TABLE X: Parameters involved in the expression for the self-d iﬀusion consensus ﬁt D⋆\nL(A), where the\nparameter Aindicates NiorAlhosts. The ﬁrst column denotes the reference where the value s were\ntaken from. The solvent lattice is indicated in the second co lumn. The third and fourth columns\ndenote the preexponential factor D0\nAand the activation energy QAfor equation (63) respectively. The\nrange of temperatures of the description is referred in colu mn ﬁve, while the relative error of the self\ndiﬀusion coeﬃcient is shown in column six. The last column st ands for experimental or theoretical\nresults. The values were taken from Campbell work [16] and re ferences therein.\nRef. Lattice D0\nA(cm2s−1)QA(KJ/mol)T(◦C) error type\n[33, 34] Ni 1.9 279 .5 [773 −1023] 16% exp.\n[35] Al 0.137 123 .5 [90 −930] 15% exp.\n[16] Ni 1.1 279 .35 [769 −1667] −D⋆\nL(Ni)\n[16] Al 0.292 129 .7 [357 −833] −D⋆\nL(Al)\ndiﬀusing specie in the alloy ( U→Al). They propose a solution for equation (64) as,\nC(x,t) =S0[1−erf(x/2√\nDt]. (65)\nIn Table XI we show the results of the ﬁt of experimental data t aken from [4], against we will\ncompare our theoretical results. Also, the authors argue th at at inﬁnite dilution the dissolution\nof precipitates do not disturb the Uprocess diﬀusion in Al. In Figure 15, we establish a\nTABLE XI: Solubility and diﬀusion of UinAl.D×108cm2s−1(S0×1010at).\nT(◦C) 1.5%Wt 0.75%Wt0.15%Wt500×10−6%Wt\n620 1 .60±0.20 1.5±0.15 1.56±0.15 1 .62±0.16\n(20±10) (25±15) (30±15) ( 20±10)\n600 0 .78±0.08 0.68±0.07 0.70±0.15 0 .65±0.07\n(10±5) (10±5) (12±7) ( 15±5)\n580 0 .55±0.12 0.70±0.12 0.44±0.15 0 .67±0.10\n(15±7) (20±10) (6±4) ( 10±5)\n560 0 .40±0.10 0.35±0.10 0.31±0.10 0 .33±0.10\n(8±4) (10±5) (5±2) ( 6±3)\ncomparison with experimental data in Table XI for an Uranium dilution of 500×10−6%Wt,\nwhich corresponds to CU= 6.57×10−5. We see that, experimental values are in perfect\n29agreement with Dp, contrarily to the NiAl for which our calculations and calculations in Ref.\n[17] reveal a weak Aluminium-vacany binding, then experime ntal values of solute diﬀusion goes\nwithD⋆\nAl.\nThe diﬀusion of Uranium into Aluminum was also calculated in a study of the maximum rate\nof penetration of uranium into aluminum in the temperature r ange200−390◦Cas described\nin a report from the literature [5]. The maximum values calcu lated in [5] for the penetration\ncoeﬃcient was, KT=x2/t= 0.075,0.50,6.1×10−6inch2/hrat temperatures of 200,250and\n390◦C, respectively. The activation energy Qfrom the expression K=K0exp−Q/RTin the\ntemperature range T= [200−390]isQ= 14.300in calories per mole, Rthe gas constant\nin calories per 1/◦Cper mole, and Tthe absolute temperature. KOis the proportionality\nconstant. The plot lnKvs1/Tprovides a convenient basis for expressing and comparing\npenetration coeﬃcients.\nNot shown here, but also performed, we recalculate all the mi croscopical parameters for a\ncrystallite containing 100atoms using classical molecular static technique, includi ng one solute\natom and a vacancy at ﬁrst neighbor sites of the solute. We rep roduce all the migration barriers\nand therefore the jump frequency rates.\nIn summary, for pure NiandAlmaterials, a large amount of experimental data are availabl e\nin the literature, which have been summarized by Campbell [1 6] in a best conﬁdence estimation\nof the self diﬀusion coeﬃcient. Our calculations for pure ho sts match perfectly well with this\nbest estimation when a temperature dependent ν0is assumed, although results for a constant\nvalue ofν0, also gives accurate results.\nConcerning with diluted alloys, our results are in excellen t agreement with experiments\n[17, 33–35] for the tracer diﬀusion coeﬃcient in NiAl. For the diﬀusion behavior in AlU, we\nonly found in the literature the work by Housseau et al. [4]. O ur results when compared with\nthe experimental data [4], suggest that the diﬀusion behavi or is mainly due to a vacancy drag\nmechanism.\nVII. CONCLUDING REMARKS\nIn summary, We propose a general mechanism based on ﬁrst prin ciples for obtaining diﬀusion\ncoeﬃcients.\nThe ﬂux equations permits to relates the diﬀusion coeﬃcient s with the Onsager tensor.\nNon equilibrium thermodynamics allows to write this Onsage r coeﬃcients in terms of jump\nfrequencies. In this way we could write expressions for the d iﬀusion coeﬃcients only in terms of\nmicroscopic magnitudes, i.e. the jump frequencies. This la st ones have been calculated thanks\n30FIG. 14: Tracer diﬀusion coeﬃcients of Al(D⋆\nAlin open squares) and Ni(D⋆\nNiin open circles) in the\nalloy. Solid line represents the best estimative of the pure Niself-diﬀusion coeﬃcient D⋆\nL(Ni)taken\nfrom Campbell work [16]. Available experimental data, for t heAldiﬀusion coeﬃcient in the alloy, are\ndisplayed with stars [36] and cruxes [37].\nFIG. 15: Tracer diﬀusion coeﬃcients of U(D⋆\nUin open squares) and Al(D⋆\nAlin open circles) in the\nalloy. Solid line represents the best estimative of the pure Alself-diﬀusion coeﬃcient D⋆\nL(Al), taken\nfrom Campbell work [16]. Available experimental data, for t heUdiﬀusion coeﬃcient in the alloy, are\ndisplayed with triangles [4].\nto to the economic static molecular techniques namely the mo nomer method.\nThe ﬁve frequency model has also been of great utility in orde r to discriminate the relevant\njump frequencies for both the the Le Claire approximation ( F= 1) and one more accurate when\nF neq1is considered. Hence, we have calculated the full set of phen omenological coeﬃcients\nfrom which the full set of diﬀusion coeﬃcients are obtained t hrough the ﬂux equation.\nAlthough in this work we have performed the treatment for the case of f.c.c. latices where\n31the diﬀusion is mediated by vacancy mechanism, a similar pro cedure can be adopted for other\ncrystalline structures or diﬀerent diﬀusion mechanism (fo r example, interstitials).\nWe have exempliﬁed our calculations for the particular case s of binary NiAl andAlUf.c.c.\ndiluted alloys.\nWhen a temperature dependent attempt frequency is consider ed the agreement between\nexperimental data and numerical calculations is excellent while, when we assume a constant\nattempt frequency is also in very good agreement, but under e stimate the experimental value.\nNegative enhancement factor as observed Alsolvent, this could promote an enhancement of\nthe solvent diﬀusion coeﬃcient for less diluted alloys.\nFinally, the F= 1andF/ne}ationslash= 1approximations yield practically to the same results for\ntheLijandD⋆in both systems here studied. Diﬀerences were observed for t he ratios D⋆\nB/D⋆\nA,\nLAB/LBB,fBandbA⋆, evidently not reﬂected in the self and solute diﬀusion coeﬃ cients, despite\nthe notorious diﬀerences observed for the strong/weak attr active interaction between the solute\nU/Al-vacancy diluted in Al,Ni hosts respectively.\nThe vacancy tracer diﬀusion coeﬃcient for the NiAl andAlUsystem were compared with\navailable experimental data obtaining an excellent agreem ent with the here described theory.\nCalculations for the diﬀusion coeﬃcient of the paired speci e, shows that a vacancy drag mech-\nanism could occur for AlUwhenF/ne}ationslash= 1, but is unlikely to occur for NiAl in both, F= 1and\nF/ne}ationslash= 1.\nThis opens the door for future works in the same direction whe re similar procedure will be\nused that includes interstitial defects.\nAcknowledgements\nI am grateful to A.M.F. Rivas and Joaquín Guillén, for commen ts on the manuscript. Also, I\nam grateful to Martín Urtubey for the Figure 2. This work was p artially ﬁnanced by CONICET\nPIP-00965/2010 and the CNEA/CAC - Gerencia Materiales.\n[1] http://www.rertr.anl.gov/\n[2] A.M. Savchenko, A.V. Vatulin, I.V. Dobrikova, G.V. Kula kov, S.A. Ershov, Y.V. Konovalov, Proc.\nof the International Meeting of the RERTR, S12-5 (2005).\n[3] M.I. Mirandu, S.F.Aricó, S.N.Balart, L.M. Gribaudo, Mat . Charact., 60, 888 (2009).\n[4] N. Housseau, A. Van Craeynest, D. Calais; Diﬀusion et sol ubilite du plutonium et de l’uranium\ndans l’aluminium; Journal of Nuclear Materials 39-2; 189-193(1971).\n32[5] T.K. Bierlein and D.R. Green; The diﬀusion of Uranium into aluminium ;Phys-\nical Metallurgy Unit Metallurgy Research Sub-Section ; October 6, 1955. URL:\nwww.osti.gov/scitech/servlets/purl/4368963\n[6] F. Brossa, H.W. Sheleicher and R. Theisen; Proceeding of the symposium: New Nuclear Materials\nincluding non-metalic Fuells ; Prague - Juli 1-5, 1963 Vol II. Edited by I.A.E.A. 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Martin, Journal of Metals 8, 567, (1956).\n34"    },    {        "title": "2008.13634v3.A_collection_of_definitions_and_fundamentals_for_a_design_oriented_inductor_model.pdf",        "content": "arXiv:2008.13634v3  [physics.app-ph]  20 Oct 2020A collection of deﬁnitions and fundamentals for a\ndesign-oriented inductor model\n1stAndr´ es Vazquez Sieber\n* Departamento de Electr ´onica\nFacultad de Ciencias Exactas, Ingenier ´ıa y Agrimensura\nUniversidad Nacional de Rosario (UNR)\n** Grupo Simulaci ´on y Control de Sistemas F ´ısicos\nCIFASIS-CONICET-UNR\nRosario, Argentina\navazquez@fceia.unr.edu.ar2ndM´ onica Romero\n* Departamento de Electr ´onica\nFacultad de Ciencias Exactas, Ingenier ´ıa y Agrimensura\nUniversidad Nacional de Rosario (UNR)\n** Grupo Simulaci ´on y Control de Sistemas F ´ısicos\nCIFASIS-CONICET-UNR\nRosario, Argentina\nmromero@fceia.unr.edu.ar\nAbstract —This paper deﬁnes and develops useful concepts\nrelated to the several kinds of inductances employed in any c om-\nprehensive design-oriented ferrite-based inductor model , which is\nrequired to properly design and control high-frequency ope rated\nelectronic power converters. It is also shown how to extract\nthe necessary parameters from a ferrite material datasheet in\norder to get inductor models useful for a wide range of core\ntemperatures and magnetic induction levels.\nIndex Terms —magnetic circuit, ferrite core, major magnetic\nloop, minor magnetic loop, reversible inductance, amplitu de\ninductance\nI. I NTRODUCTION\nFErrite-core based low-frequency-current biased inductor s\nare commonly found, for example, in the LC output ﬁlter\nof voltage source inverters (VSI) or step-down DC/DC con-\nverters. Those inductors have to effectively ﬁlter a relati vely\nlow-amplitude high-frequency current being superimposed on\na relatively large-amplitude low-frequency current. It is of\nparamount importance to design these inductors in a way that\na minimum inductance value is always ensured which allows\nthe accurate control and the safe operation of the electroni c\npower converter. In order to efﬁciently design that speciﬁc type\nof inductor, a method to ﬁnd the required minimum number\nof turnsNminand the optimum air gap length goptto obtain a\nspeciﬁed inductance at a certain current level is needed. Th is\nmethod has to be based upon an accurate inductor model, for\nwhich certain inductances and properties need to be deﬁned\nand explained. Also, these inductance deﬁnitions needs to b e\nparametrized, among other things, according to the speciﬁc\nferrite material employed in the core.\nThe problem of designing such kind of inductors has been\nwidely treated in literature [1], [2], [3], [4]. At the same\ntime, there are many well established deﬁnitions of core\npermeability and inductance [4], [5], [11] according to the\nactual inductor operating condition. However, it seems tha t\nthis variety of inductance deﬁnitions can be better exploit ed\nin order to enhance the inductor design process. In this\npaper, some speciﬁc inductance deﬁnitions are revisited an d\npresented under a suitable context for the power electronic\nFig. 1. General magnetic circuit\npractitioner. A design-oriented inductor model can be base d\non the core magnetic model described in this paper which\nallows to employ the concepts of reversible inductance Lˆrev,\namplitude inductance Laand initial inductance Li, to further\ndevelop an optimized inductor design method. Those induc-\ntance deﬁnitions rely on their respective core permeabilit ies,\nwhich in this paper are also revisited, contextualized and\nobtained for two speciﬁc ferrite materials: TDK-EPCOS N27\nand TDK-EPCOS N87.\nThis paper is organized as follows. In section II, a general\nmagnetic core model is described and its associated perme-\nabilities are introduced. In section III, deﬁnitions of sev eral\ntypes of inductances are presented. Section IV shows how to\nobtain the previously deﬁned permeabilities from the ferri te\nmaterial datasheet. Section V presents some useful propert ies\nof the reversible inductance that could be needed to justify the\nselection criterion of the inductance value as well as Nminand\ngopt. Finally, conclusions are presented in section VI.\nII. M AGNETIC CIRCUIT MODEL\nIn this section, we obtain a model for the general magnetic\ncircuit considered in Figure 1 using an approach depending o nintegration along the mean magnetic path into the ferrite co re,\nLcand into the air gap, Lg[5]. Suppose that current i(t)can be\neasily decomposed into a) a component denoted as iLF(t)at a\nrelatively low frequency fLFand b) a component denoted as\niHF(t)at a relatively high frequency fHF, withfLF≪fHF.\nAt a certain time ˆtLF,iLFreaches its peak value ˆiLFand\nthen we have\ni(t)≈ˆiLF+iHF(t)t∈/bracketleftbigg\nˆtLF−1\nfHF,ˆtLF+1\nfHF/bracketrightbigg\n(1)\nIn such a situation, Ampere’s law relates the frequency com-\nponents of current i(t)with their corresponding magnetic ﬁeld\nstrengthHcomponents as follows\n/contintegraldisplay\n/vectorH(t,l)/vectordl=/integraldisplay\nLc∪Lg/bracketleftBig\nˆHLF(l)+HHF(t,l)/bracketrightBig\ndl\n=N/bracketleftBig\nˆiLF+iHF(t)/bracketrightBig\nwhere/vectordlis the path vector, parallel to /vectorH. Separating the\nfrequency components yields\n/integraldisplay\nLcˆHLF(l)dl+/integraldisplay\nLgˆHLF(l)dl=NˆiLF (2)\n/integraldisplay\nLcHHF(t,l)dl+/integraldisplay\nLgHHF(t,l)dl=NiHF(t)\n/integraldisplay\nLc∆HHF(l)dl+/integraldisplay\nLg∆HHF(l)dl=N∆iHF (3)\nwhere∆HHFis the amplitude of the ﬁeld strength excursion\ndue to∆iHF, the amplitude of the high-frequency current\nexcursion during1\nfHF.\nThe magnetic induction B(t,l) =ˆBLF(l) +BHF(t,l)\nand its peak-to-peak variation ∆BHF determine the peak\ninduction ˆB(l) =ˆBLF(l)+∆BHF(l)\n2. These are related to their\ncorresponding ﬁeld strength ˆHLF(l),HHF(t,l),∆HHF(l)\nandˆH(l)according to the medium permeability. Having the\nair gap paramagnetic properties, along Lgsimply hold\nˆBLF(l)\nˆHLF(l)=BHF(t,l)\nHHF(t,l)=∆BHF(l)\n∆HHF(l)=ˆB(l)\nˆH(l)=µ0 (4)\nwhereµ0is the vacuum permeability. In the magnetic core\npathLc, those relationships depend on the shape of the ferrite\nmagnetization curve which is shown in Figure 2. It is also\nthe speciﬁc major loop that characterizes the behaviour of t he\nferrite when its magnetic induction evolution spans the two\nextreme points ±Bs, beingBsthe saturation induction. In any\ninductor, although this situation can be reached with a curr ent\niLFhaving a sufﬁciently high ˆiLF, the actual ˆiLFhas to be set\nwell below that value since beyond that induction level the c ore\nferrimagnetic properties become severely affected. Start ing\nfrom a demagnetized core, as ˆiLFis gradually increased\nfrom zero towards the maximum value causing saturation,\nthe tipping points ( ˆBLF,ˆHLFand−ˆBLF,−ˆHLF) of the ever\nincreasing LF-major loops describe a LF-commutation curve\nwhich is also partly shaped by the current magnitude of\n∆BHF, due to the memory properties of the ferrite material.\nFig. 2. Ferrite magnetization curve\nFor any of the points pertaining to that LF-commutation\ncurve, the LF-amplitude permeability µLF\nais deﬁned as\nµLF\na/parenleftBig\nˆBLF,∆BHF/parenrightBig\n=1\nµ0ˆBLF\nˆHLF/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆BHF\nbecause it relates only the low frequency amplitude of the\nmagnetic induction and ﬁeld strength in the ferrite materia l\nwhen the low frequency iLFtakes also its amplitude value\nˆiLF. The magnetic induction generated by iLFwill not vary\nmuch from ˆBLFwhiletis into the time span deﬁned in (1).\nHence during1\nfHF,iHF(t)will produce an approximately\nclosed minor magnetic loop of amplitude ( ∆BHF,∆HHF)\nstarting and ending in the neighbourhood of ˆBLF, as it is\nshown in Figure 2. The incremental permeability µˆ∆at that\nquasi-static induction level ˆBLFis then deﬁned as\nµˆ∆/parenleftBig\nˆBLF,∆BHF/parenrightBig\n=1\nµ0∆BHF\n∆HHF/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆBLF\nConsequently, on Lcholds\nˆHLF(l) =ˆBLF(l)\nµ0µLFa/parenleftBig\nˆBLF,∆BHF/parenrightBig (5)\n∆HHF(l) =∆BHF(l)\nµ0µˆ∆/parenleftBig\nˆBLF,∆BHF/parenrightBig (6)\nIf∆BHF is made sufﬁciently small, then µˆ∆and the\nLF-commutation curve start to be practically independent o f\n∆BHF. At this point, on the one hand the existing linearrelationship between BHFandHHFis captured by the so-\ncalled reversible permeability at ˆBLF,µˆrev\nµˆrev/parenleftBig\nˆBLF/parenrightBig\n= lim\n∆BHF→0µˆ∆/parenleftBig\nˆBLF,∆BHF/parenrightBig\nOn the other hand, the LF-commutation curve tends to the\nregular commutation curve and their respective amplitude\npermeabilities are related as\nµa/parenleftBig\nˆB/parenrightBig\n=1\nµ0ˆB\nˆH= lim\nˆBLF→ˆB\n∆BHF→0µLF\na/parenleftBig\nˆBLF,∆BHF/parenrightBig\n=µa/parenleftBig\nˆBLF/parenrightBig\nNow making ˆBLF→0due toˆiLF→0, the initial\npermeability µiis deﬁned as\nµi= lim\nˆBLF→0µa/parenleftBig\nˆBLF/parenrightBig\n=µˆrev/parenleftBig\nˆBLF= 0/parenrightBig\nNote that in the core, the relationship between BandH\ndepends not only on the ferrite magnetic characteristics bu t\nalso on the way in which Bevolves with time.\nIn the magnetic circuit of Figure 1, a closed surface\nS=Sc∪Sgthat intersects both LcandLgpaths will satisfy\naccording to Gauss’ law that/integraldisplay/integraldisplay\nSc/vectorB(t,l)/vectordS=/integraldisplay/integraldisplay\nSg/vectorB(t,l)/vectordS (7)\nwhereScis a core cross-section perpendicular to Lc,Sgis\nthe remaining surface of Scrossing the air gap and /vectordSis the\narea vector of S. The left side of (7) is the core magnetic ﬂux\nΦcsince the magnetic induction there is mainly concentrated\nintoScbecauseµa,µˆ∆≫1. All the magnetic induction in\nthe air gap will then pass through Sg, thus the right side of\n(7) is the air gap magnetic ﬂux Φg. Consequently, Φc= Φg=\nˆΦLF+ΦHF(t).Lcpasses perpendicular through the center of\nSc, so the magnetic induction along Lcwill be approximately\nan average of that existing inside Scand equal to\nˆBLF(l)+BHF(t,l) =ˆΦLF+ΦHF(t)\nAc(l)(8)\nwhereAc(l)is the area of Scat a certain point l∈Lc. Around\nthe air gap, the magnetic induction is far more nonuniform\ninSgthan inScdue to the fringing ﬂux. Thus, the mean\ninduction on Lgcan be quite different from the actual values\nat the edges of the gap, but being a paramagnetic region, it\nsufﬁces to propose an effective gap area Age(l)withl∈Lg,\nas if all the induction were there concentrated.\nˆBLF(l)+BHF(t,l) =ˆΦLF+ΦHF(t)\nAge(l)(9)\nNote that Age(l)is approximately equal to Acg, the core cross-\nsection in contact with the air gap, if its length is much smal ler\nthan the linear dimensions characterizing Acg. Given that the\nwinding turns Nembrace practically all Φc, it follows that the\nlinkage ﬂux Ψis\nˆΦLF+ΦHF(t) =ˆΨLF+ΨHF(t)\nN(10)The peak linkage ﬂux ˆΨand magnetic induction ˆB(l)in the\nferrite core are\nˆΨ =ˆΨLF+∆ΨHF\n2(11)\nˆB(l) =ˆΨ\nNAc(l)\nIII. I NDUCTANCE DEFINITIONS\nCombining (2), (3), (4), (5), (8), (9) and (10) yield the LF-\namplitude inductance, LLF\naand the incremental inductance,\nLˆ∆\nLLF\na/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=ˆΨLF\nˆiLF/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n∆ΨHF\n=N2\nRLFca/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n+Rg(12)\nLˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=∆ΨHF\n∆iHF/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆΨLF\n=N2\nRcˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n+Rg(13)\nwith\nRLF\nca/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=/integraldisplay\nLcdl\nµ0µLFa/parenleftBigˆΨLF\nNAc,∆ΨHF\nNAc/parenrightBig\nAc(l)\n(14)\nRcˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=/integraldisplay\nLcdl\nµ0µˆ∆/parenleftBigˆΨLF\nNAc,∆ΨHF\nNAc/parenrightBig\nAc(l)\n(15)\nRg=/integraldisplay\nLgdl\nµ0Age(l)(16)\nRLF\nca,Rcˆ∆andRgare the core LF-amplitude reluctance, core\nincremental reluctance and the air gap reluctance respecti vely.\nConsidering the situation where ∆ΨHF→0,µaandµˆrev\ndeﬁne the amplitude and reversible inductances at ˆΨLF,La\nandLˆrev, as\nLa/parenleftBig\nˆΨLF/parenrightBig\n= lim\nˆΨLF→ˆΨ\n∆ΨHF→0LLF\na/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=N2\nRca/parenleftBig\nˆΨLF/parenrightBig\n+Rg(17)\nLˆrev/parenleftBig\nˆΨLF/parenrightBig\n= lim\n∆ΨHF→0Lˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=N2\nRcˆrev/parenleftBig\nˆΨLF/parenrightBig\n+Rg\nRca/parenleftBig\nˆΨLF/parenrightBig\n=/integraldisplay\nLcdl\nµ0µa/parenleftBigˆΨLF\nNAc/parenrightBig\nAc(l)(18)\nRcˆrev/parenleftBig\nˆΨLF/parenrightBig\n=/integraldisplay\nLcdl\nµ0µˆrev/parenleftBigˆΨLF\nNAc/parenrightBig\nAc(l)(19)RcaandRcˆrevare the core amplitude reluctance and the core\nreversible reluctance respectively.\nLaandLˆrevusually have dissimilar values at a same\nˆΨLFand vary differently as ˆΨLFincreases from zero to\nrelatively high values. It is then important to ﬁnd a common\nsituation to relate and relativize their current values wit h.\nIn a demagnetized material, µaandµˆrevcoincide at the\norigin which means that LaandLˆrevconverge to the initial\ninductance Li\nLi= lim\nˆΨLF→0La/parenleftBig\nˆΨLF/parenrightBig\n=Lˆrev/parenleftBig\nˆΨLF= 0/parenrightBig\n=N2\nRci+Rg\n(20)\nRci=/integraldisplay\nLcdl\nµ0µiAc(l)(21)\nbeingRcithe core initial reluctance. Note that only µidoes\nnot vary with the core cross-sectional area Ac(l)along the\nmagnetic path. However, µias well as µaandµˆrevdo depend\nheavily on the core temperature, as is modeled in the next\nsection.\nIV. P ERMEABILITY MODELS\nThe dependence of Rcain (18) and Rciin (21) from core\ntemperature Tcand magnetic induction ˆBLF=ˆB, in each\nspeciﬁc part of the core, is addressed when the correspondin g\nfunctions µa/parenleftBig\nˆBLF,Tc/parenrightBig\nandµi(Tc)are extracted from the\nferrite material datasheet [12] [13]. Permeability µais a func-\ntion of magnetic induction amplitude ˆBand core temperature\n(3-D lookup table) and permeability µiis a function of core\ntemperature (2-D lookup table). To get the best accuracy in t he\ninductor model, the µacurve given by the ferrite manufacturer\nshould have been obtained at a frequency close to fLF.\nThe temperature and induction dependence of Rcˆrevin\n(19) is subjected to ﬁnd µˆrev/parenleftBig\nˆBLF,Tc/parenrightBig\n. In [6] it is con-\ncluded that the commutation curve coincides with the so-cal led\ninitial magnetization curve for soft ferrite materials, th at is\n(BDC,HDC) = (ˆB,ˆH). This means that µˆrevis equal to\nDC-biased µrevwhich can be extracted from a graph or as a\nfunction of DC-bias ﬁeld strength HDC. That curve may not\nbe given in datasheets for a particular ferrite material or f or\nthe core temperatures at which µˆrevhas to be obtained, but\neven if it were available it should be put in terms of ˆBLFto\nbe employed in (19). To overcome these limitations, we use a\npermeability model directly relating DC-biased µrevwith DC-\nbias magnetic induction BDC[7], where all its parameters at\nthe desired core temperature can be entirely obtained from\nany ferrite datasheet, in the way it is next explained. This\napproach has been experimentally validated for many ferrit e\nmaterials operating at different temperatures [8] and it is\ncurrently employed by major ferrite manufacturers [9].\nLet us ﬁrst consider the empirical models that curveﬁt the\nupper (u) and lower (l) branches of the dynamic magnetizatio n(B-H) curve of Figure 2 [7],\nHu(B) =B\nµ0µc1\n1−/parenleftBig\nB\nBs/parenrightBigau−Hc (22)\nHl(B) =B\nµ0µc1\n1−/parenleftBig\nB\nBs/parenrightBigal+Hc (23)\nwith the positive parameters: coercive ﬁeld strength Hc, coer-\ncive permeability µcand squareness coefﬁcients auandalfor\neach branch. Supposing that al≈auand being ˆB=ˆBLF,\nµˆrev/parenleftBig\nˆBLF,Tc/parenrightBig\ncan be expressed as [7]\nµˆrev/parenleftBig\nˆBLF,Tc/parenrightBig\n=\n\n1+(al−1)/parenleftBigˆBLF\nBs/parenrightBigal\n/bracketleftBig\n1−/parenleftBigˆBLF\nBs/parenrightBigal/bracketrightBig21\nµc\n+bo/parenleftBig\n1−ˆBLF\nBs/parenrightBig/bracketleftBig\n2−/parenleftBig\n1−ˆBLF\nBs/parenrightBigao/bracketrightBig\n\n−1\n(24)\nao=boBs\nµ0Hcbo=1\nµi−1\nµc\nApart from µi,µˆrevdepends on TcthroughBs,Hc,alandµc\nand thus (23) has to be numerically ﬁtted for each particular\nTc. The ﬁtting data is obtained from the 3-D lookup table\nH(B,T=Tc)based on the corresponding curves from the\nferrite material datasheet [12] [13].\nThe starting guess points for the ﬁtting process are ex-\ntracted from 2-D lookup tables B∗\ns(T),H∗\nc(T),a∗\nl(T)and\nµ∗\nc(T).B∗\ns(T)andH∗\nc(T)are built to linearly interpolate the\ntwo saturation induction and coercive ﬁeld strength values\n(Bs1,Bs2,Hc1andHc2respectively), that are stated at\nthe two corresponding temperatures T1,T2, in the datasheet\nof the magnetic material. Lookup tables a∗\nl(T)andµ∗\nc(T)\nare conformed in the following way. Let H11(B11,T1)and\nH12(B12,T1)be the ﬁeld strength at two different induction\nlevels from the lower branch of the B-H curve at temperature\nT1given by the datasheet. The value B11could be from the\n”linear” region of the curve, while B12could be taken from\nthe ”knee” between ”linear” and ”saturation” regions of the\ncurve at temperature T1. The estimations of coefﬁcients al\nandµcfrom (23) at temperature T1,a∗\nl1andµ∗\nc1respectively,\nare found numerically solving\n1−/parenleftBig\nB11\nBs1/parenrightBiga∗\nl1\n1−/parenleftBig\nB12\nBs1/parenrightBiga∗\nl1=H12−Hc1\nH11−Hc1B11\nB12\nµ∗\nc1=1\nµ0B11\nH11−Hc11\n1−/parenleftBig\nB11\nBs1/parenrightBiga∗\nl1\nUsing the B-H curve at temperature T2,a∗\nl2andµ∗\nc2can\nbe also obtained following a similar reasoning. Finally, a∗\nl(T)\nandµ∗\nc(T)are built to linearly interpolate a∗\nl1,a∗\nl2andµ∗\nc1,µ∗\nc20 50 100 150 20000.050.10.150.20.250.30.350.40.450.5\nH [A/m]B [T]\n  \nN27 100oC\nN27 25oC\nN87 100oC\nN87 25oC\nFig. 3. Curve-ﬁtting of the lower branch of the magnetizatio n loops (ﬁrst\nquadrant) for materials N27 and N87. Dashed lines are interp olated data from\ndatasheet\nrespectively. Figure 3 shows the ﬁtting goodness of (23)\nfor TDK-EPCOS ferrite materials N27 and N87 while their\ncorresponding parameters for using (24) are summarized in\nTable I. To get the best accuracy in the inductor model, the\nmagnetization curve given by the ferrite manufacturer shou ld\nhave been obtained at a frequency close to fHF.\nTABLE I\nREVERSIBLE PERMEABILITY µˆrev MODEL PARAMETERS (EQUATION (24))\nMaterial Tc[oC]alHc[A/m ]µcµiBs[T]\nN27 100 1.25 18.12 14079 3231 0.4165\nN27 25 2.00 24.35 11154 1700 0.4895\nN87 100 8.00 10.94 4330 3976 0.3925\nN87 25 3.78 21.17 6014 2210 0.4803\nV. C ONSIDERATIONS ON THE REVERSIBLE INDUCTANCE\nRecall that if ∆ΨHF→0we can consider\nLˆrev/parenleftBig\nˆΨ/parenrightBig\n=Lˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\n=Lˆrev/parenleftBig\nˆΨLF/parenrightBig\nIt is important to remark that in this situation Lˆrevis the\nminimum value of reversible inductance arising along the\nwhole symmetric major loop having ±/parenleftBig\nˆHLF,ˆBLF/parenrightBig\nas tip-\nping points. This is in fact proved by taking into considerat ion\nthe upper branch of the particular ±/parenleftBig\nˆHLF,ˆBLF/parenrightBig\nmajor loop\nwhich can be described in terms of (22)-(23) as [6]\nHLF\nu(B) =Hu(B)+/bracketleftBig\nˆHLF−Hu(ˆBLF)/bracketrightBigβ\n/bracketleftBig\nHl(ˆBLF)−ˆHLF/bracketrightBigα (25)\nα=B−ˆBLF\n2ˆBLFβ=B+ˆBLF\n2ˆBLFand its ﬁrst derivative, the inverse of the so-called differ ential\npermeability µd[4]\ndHLF\nu\ndB=1\nµd\nSinceµrev< µd[4] it is sufﬁcient to show thatdµd\ndB<0\nto conclude that Lrevis decreasing when Btakes values\nfrom zero to ˆBLFand hence Lˆrevis the absolute minimum.\nAccordingly,\ndµd\ndB=−µd2d2HLF\nu\ndB2\nd2HLF\nu\ndB2=au/parenleftBig\nB\nBs/parenrightBigau/bracketleftBig\n1−/parenleftBig\nB\nBs/parenrightBigau\n+au+au/parenleftBig\nB\nBs/parenrightBigau/bracketrightBig\nµ0µc/bracketleftBig\n1−/parenleftBig\nB\nBs/parenrightBigau/bracketrightBig3\nB\n+\nlnHl(ˆBLF)−ˆHLF\nˆHLF−Hu(ˆBLF)\n2ˆBLF\n2/bracketleftBig\nˆHLF−Hu(ˆBLF)/bracketrightBigβ\n/bracketleftBig\nHl(ˆBLF)−ˆHLF/bracketrightBigα\n(26)\nˆHLFis inside the area delimited by the largest major loop,\nthe magnetization curve, that is described by (22)-(23). Co n-\nsequently, (26) is positive for B∈[0,ˆBLF]and thus µdis\never decreasing for increasing values of B >0.\nNow suppose that gradually ∆ΨHFis increased and ˆΨLFis\ndecreased in such a way that ˆΨremains unchanged, implying\nthatˆBandˆHare unmodiﬁed in all parts of the core. This\nscenario brings into existence increasingly asymmetric mi nor\nloops in the B−Hplane with tipping points\nˆB=ˆBLF+∆BHF\n2\nˆH=ˆHLF+kH∆HHF\nˆB−∆BHF=ˆBLF−∆BHF\n2\nˆH−∆HHF=ˆHLF−(1−kH)∆HHF\nbeingkH∈[0.5,1)the magnetic ﬁeld symmetry factor.\nConsidering that along a general magnetic loop µdincreases\nasBdecreases, it can be stated that\nˆB−dB\ndH/vextendsingle/vextendsingle/vextendsingle/vextendsingleˆB∆HHF≥ˆB−∆BHF\nand hence\nµˆrev/parenleftBig\nˆB/parenrightBig\n≤µd/parenleftBig\nˆB/parenrightBig\n≤µˆ∆/parenleftBig\nˆBLF,∆BHF/parenrightBig\nInside the minor loop, we can deﬁne the minor-loop amplitude\npermeability µMN\na as\nµMN\na=1\nµ0ˆB−ˆBLF\nˆH−ˆHLF=1\nµ0∆BHF\n2kH∆HHF=1\n2kHµˆ∆\nand to note that it holds µˆrev/parenleftBig\nˆB/parenrightBig\n< µˆrev/parenleftBig\nˆBLF/parenrightBig\n, since\nwhenBis far from the origin, µˆrevdecreases as Bincreases.\nConsequently,\nˆBLF+µ0µˆrev/parenleftBig\nˆBLF/parenrightBig\nkH∆HHF≥ˆBLF+∆BHF\n2and hence\nµˆrev/parenleftBig\nˆBLF/parenrightBig\n≥µMN\na≤µˆ∆\nIf the minor loop keeps some degree of symmetry around/parenleftBig\nˆBLF,ˆHLF/parenrightBig\n, i.e.kH≈0.5, thenµMN\na≈µˆ∆and hence\nµˆrev/parenleftBig\nˆB/parenrightBig\n≤µˆ∆/parenleftBig\nˆBLF,∆BHF/parenrightBig\n≤µˆrev/parenleftBig\nˆBLF/parenrightBig\nHowever, that minor loop with tipping points\n/parenleftBig\nˆB,ˆH/parenrightBig\n;/parenleftBig\nˆB−∆BHF,ˆH−∆HHF/parenrightBig\nand an enclosing symmetric major loop with tipping points\n/parenleftBig\nˆB,ˆH/parenrightBig\n;/parenleftBig\n−ˆB,−ˆH/parenrightBig\ncoincide in the vicinity of their uppermost tipping point/parenleftBig\nˆB,ˆH/parenrightBig\n[10] [7]. Hence, µˆrev(ˆB)at the existing minor loop\nis equal to µˆrev(ˆB)at that hypothetical major loop. In fact,\nµrev(BDC)[7], from which (24) is particularly derived,\ndepends on the current magnetic induction value regardless\nits previous evolution [11]. Note that from the minor loop\nstandpoint, ˆΨis given by (11), but it can be also put in terms\nof the amplitude permeability as\nˆΨ =La/parenleftBig\nˆΨ/parenrightBig\nˆi\nˆi= max\nt/parenleftBig\nˆiLF+iHF(t)/parenrightBig\nand thus it is valid\nLa/parenleftbigg\nˆΨLF+∆ΨHF\n2/parenrightbigg\nˆi=LLF\naˆiLF+∆ΨHF\n2\nLLF\na=LLF\na/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\nOn that enclosing symmetric major loop, characterized by\nparameters ˆΨMJ,ˆiMJ\nLFandˆiMJ, we have that\nˆΨMJ=La/parenleftBig\nˆΨMJ/parenrightBig\nˆiMJ=ˆΨ\nˆiMJ=ˆiMJ\nLF=ˆi\nConsequently, for ∆ΨHF>0we get\nLˆrev/parenleftBig\nˆΨMJ/parenrightBig\n=Lˆrev/parenleftBig\nˆΨ/parenrightBig\n< Lˆ∆< Lˆrev/parenleftBig\nˆΨLF/parenrightBig\n(27)\nLˆ∆=Lˆ∆/parenleftBig\nˆΨLF,∆ΨHF/parenrightBig\nVI. C ONCLUSION\nThe objective of this paper is to provide a collection of basi c\ndeﬁnitions and properties to ground a comprehensive ferrit e-\ncore based low-frequency-current biased inductor model fo r\nan optimized design method, which is a fundamental tool to\nproperly design and control any type of electronic power con -\nverter. The same procedures followed to extract the require d\nparameters of N27 and N87 materials can be easily adapted\nto obtain that data for other ferrite materials.ACKNOWLEDGMENT\nThe ﬁrst author wants to thank Dr. Hernan Haimovich for\nhis guidance and constructive suggestions.\nREFERENCES\n[1] C. Wm. T. McLyman, Transformer and Inductor Design Handbook ,\n3rd ed. New York, USA: Marcel-Dekker, 2004.\n[2] N. Mohan, T. M. Undeland and W. P. Robbins, Power Electronics.\nConverters, Applications, and Design , 3rd ed. New York, USA: John\nWiley & Sons, 2003.\n[3] M. K. Kazimierczuk, High-Frequency Magnetic Components , 2nd ed.\nWest Sussex, England: John Wiley & Sons, 2014.\n[4] A. Van den Bossche and V . C. Valchev, Inductors and Transformers for\nPower Electronics , 1st ed. Boca Raton, USA: Taylor & Francis, 2005.\n[5] E. C. Snelling, Soft Ferrites. Properties and Applications , 1st ed.\nLondon, England: Iliffe Books Ltd, 1969.\n[6] M. Esguerra, Computation of minor hysteresis loops from measured\nmajor loops , Journal of Magnetism and Magnetic Materials, no. 157/158,\npp. 366-368, Elsevier Science B.V ., 1996.\n[7] M. Esguerra, Modelling Hysteresis loops of Soft Ferrite Materials , in\nProc. of the International Conference on Ferrites ICF 8, Kyo to, Japan,\npp. 220-220, Sep. 2000.\n[8] M. Esguerra, M. Rottner and S. Goswami, Calculating Major Hysteresis\nloops from DC-biased Permeability , in Proc. of the International Con-\nference on Ferrites ICF 9, San Francisco, USA, Aug. 2004.\n[9] M. Esguerra, Magnetics Design Tool for Power Applications ,\nBodo’s Power Systems, pp. 42-46, Apr. 2015. [Online]. Avail able:\nwww.bodospower.com/restricted/downloads/bp 2015 04.pdf.\n[10] R. G. Harrison, Modeling High-Order Ferromagnetic Hysteretic Minor\nLoops and Spirals Using a Generalized Positive-Feedback Th eory, in\nIEEE Transactions on Magnetics, vol. 48, no. 3, pp. 1115-112 9, March\n2012.\n[11] C. Heck, Magnetic Materials and their Applications , 1st ed. London,\nEngland: Butterworth, 1974.\n[12] EPCOS AG, SIFERRIT material N27 , Ferrite and\nAccessories, May 2017. [Online]. Available: www.tdk-\nelectronics.tdk.com/download/528850/d7dcd087c9a2dbd 3a81365841d4aa9a5/pdf-\nn27.pdf\n[13] EPCOS AG, SIFERRIT material N87 , Ferrite and\nAccessories, Sep 2017. [Online]. Available: www.tdk-\nelectronics.tdk.com/download/528882/71e02c7b9384de1 331b3f625ce4b2123/pdf-\nn87.pdf"    },    {        "title": "2301.13788v1.Synthesis_and_characterization_of_PEG_coated_Zn___0_3__Mn__x_Fe___2_7_x__O__4__nanoparticles_as_the_dual_T1_T2_weighted_MRI_contrast_agent.pdf",        "content": "1 \n Synthesis and characterization of PEG -coated Zn 0.3Mn xFe2.7-xO4 nanoparticles as \nthe dual T 1/T2-weighted  MRI contrast agent   \nBahareh Rezaei, Ahmad Kermanpur*, Sheyda Labbaf \nDepartment  of Materials Engineering, Isfahan University of Technology , Isfahan 84156 -83111, Iran  \n     \nAbstract  \nSuper -paramagnetic nanoparticles (NPs) have been widely explored as magnetic resonance imaging  \n(MRI ) contrast agents because of a combination of favorable magnetic properties, biocompability and \nease of fabrication.  MRI using traditional T 1- or T 2-weighted single mode contrast -enhanced \ntechniques may yield inaccurate imaging results.  In the present work,  a T1/T2 dual mode contrast agent \nbased on the super -paramagnetic zinc -manganese ferrite (Zn 0.3Mn xFe2.7-xO4, x= 0, 0.25, 0.75 and 1 ) \nNPs with small core size and a hydrophilic  PEG  surface coating  is reported . The TEM, TGA and FTIR \nresults confirmed  the formation of a uniform coating on the NPs surface. The MRI analysis revealed \nthat the Zn 0.3Mn 0.5Fe2.2O4 NPs ha d the maximum image contrast compared to other zinc-manganese \nferrite samples. Cell viability evaluations revealed that the coated and uncoated  particles did not \ninhibit  cell growth pattern . The present  PEG -coated Zn 0.3Mn 0.5Fe2.2O4 NPs can be utilized as  a suitable \nT1/T2-weighted MRI contrast agent  for better diagnostic of abnormalities in the organs or tissues . \n \nKeywords  \nMagnetic Resonance Imaging (MRI); Super -paramagnetic nanoparticles ; Zn0.3Mn xFe2.7-xO4 \nnanopartic les; Polyethylene Glycol (PEG) coating  \n \n1. Introduction  \n The most potent  and painless test that gives extremely clear images of the internal organs in \nthe body is the magnetic resonance imaging (MRI) scan  [1, 2]. Based on the magnetic relaxation \nprocesses of water protons on soft tissue of nearly every internal structure in the human body  [1, 3-5], \nthis method is a sort of diagnostic test that generates detailed images  and functional in formation in a \nnon-invasive and real -time monitoring manner [6, 7] . It is a distinguished device since there is no \nionizing radiation during the imaging process and obviously reduces harmful side effects [2, 4, 8, 9] . \nHowever , this test  typically  provides poor anatomical details , and cl inicians have some difficulties to \ndistinguish between normal and abnormal tissues due to its low sensitivity  [9, 10] . Hence, the clinical \n                                                           \n* Corresponding author; Tel. (+98)3133915738; Fax (+98)3133912752; Email: ahmad_k@iut.ac.ir  2 \n domains urgently require more reliable MR images. There is a potential to create more accurate and \ncrisper images by adding contrast agents, which enables physicians to detect organs or in-vivo systems \nmore clear . This opens up a wide range of MRI applications for therapeutic medicine in addition to \ndiagnostic radiology.  Despite the fact of shorter circulation time of Gd3+ ions as a T 1-weighted MRI \ncontrast agent, which renders them useless for high -resolution and/or targeted MRI [9, 11]  and many \nconcerns about potential trace deposition  of Gd ions in the body, known as Nephrogenic Systemic \nFibrosis (NSF)  [12-14], which is a rare disease that frequently develops in patients with severe renal \nfailure or aft er liver transplantation  [15], Gd-based contrast agents can shorten the T 1 relaxation time \neffectively and prov ide brighter images in the regions of interest [16]. Following the increased \nawareness of this side effect , researchers have much more emphasis on alternative methods based  on \nMn-based complexes [15]. Although no scientific relationship has been proved between the NSF side \neffect and Mn so far, the metal is still known to pose some toxicity when inhaled. However, small \namounts are essential to human health, but overexposure to free Mn  ions may result in the \nneurodegenerative disorder known as ‘ Manganism’ with symptoms similar Parkinson’s disease [11]. \nUnlike Gd3+ and Mn2+ chelates , iron oxide nanoparticles ( NPs) have achieved great  attention \ndue to the outstanding properties th ey exhibit at the n ano-metric scale.  A large number of benefits \nincluding  biocompatibility, superparamagnetic behavior at room temperature, high saturation \nmagnetization that can be tailored by size, shape, composition and assembly, tunable cellular uptake, \nbiodispersibility, and large surface area s that make them a goo d candidate for polymer coating, \nconjugation with targeting molecules and other probes for achieving targ eting and multimodal agents  \n[17, 18]  is reported for the iron oxide NPs . Super -paramagnetic NPs can be employed as T 2-weighted \nMRI contrast agents since they are more sensitive in the micro - or nano-molar range than Gd \ncomplexes  [17]. Clinical MR imaging  applications often use iron oxide -based NPs with stro ng \nmagnetic moments as T 2-weighted MRI contrast agents. The limited usage of i ron oxide NPs as T 1 \ncontrast agents is due to their high transverse to longitudinal relaxivity ratio  [19]. However, the use of \nsuperparamagnetic NPs in MRI is constrained  by a negative contrast effect and magnetic susceptibility \nartifacts. Because the signal is frequently confused with signals from bleeding, calcification, or metal \ndeposits and the susceptibility artifacts alter the background image, the resulting dark sig nal in T 2-\nweighted MRI may be exploited to mislead clinical diagnosis  [18]. The T1-weighted MRI contrast \nagents, however, have advantages over T 2-weighted MRI contrast agents. These advantages include \nbetter imaging quality, brighter images that can more effectively distinguish between normal and \nlesion tissues, and also the ability to provide better resolution for b lood imaging. Nonetheless , in T 1-3 \n weighted MR imaging, some normal tissues (such as fatty tissue) may be mistaken for bright lesions \nthat have been increased by T 1 contrast agents  [20]. Ther efore, efforts to integrate T 1 and T 2 imaging \nto prevent probable MRI art ifacts and produce superior clinical images have been made as a result of \nthe rising demand  in the clinical diagnosis for both T 1- and T 2-weighted MR images. [18, 21] . \nAdditionally, when several organ scans are required, injecting one dosage offers unmatched benefits \nto patients and doctors [16]. Super -paramagnetic NPs have the potential to exhibit significant dual \nT1/T2 relaxation performances when their sizes are decreased to less than 10 nm, according to some \ntheoretical investigations [21-24]. Recently, super -paramagnetic iron oxide -gold composite NPs is \nsynthesized by a green method [ 25]. It is shown that  the NPs exhibit ed a high relaxivities ratio (r 2/r1) \nof 13.20, indicating the potential as a T 2 contrast agent.   \nSurface modification is often practical to provide better stability under physiological \nconditions and prolong blood stream  circulation time, thereby increasing MR  imaging  quality  [26]. \nThis surface modification  is known to restrict the uptake of plasma proteins ( i.e., corona proteins), \nwhich lowers the likelihood that macrophages will recognize and remove them  [27]. In order to \novercome the aforementioned difficulties, polymeric coatings on the surf ace of magnetic NPs are \nrecommended [28]. In a recent work [ 29], iron oxide ferrofluid is synthesized by thermal \ndecomposition using poly (maleic anhydride -alt-1-octadecene, noted as PMAO) as a phase \ntransferring ligand. The results have demonstrated that the magnetic particles were fully covered at \nhigh coverage by long non -magnetic polymeric chains. It is shown that t his ligand could  improve the \nferrofluid stability up to as long as 6 months. The MR imag es in solution and in rabbit using the \nprepared PMAO -coated  magnetic NPs had the best contrast effect on T 2 weighted maps.    \nPolyethylene glycol (PEG) is a highly water soluble, hydrophilic, biocompatible, non -\nantigenic, and protein -resistant polymer that  is easily eliminated through the kidney s and is not \nabsorbed by humans' immune systems among all forms of polymeric coatings . PEG has also been \nfrequently employed for linking anticancer medications to proteins to prolong their half -life, as well \nas for o rgan preservation  [30. It also functions as an antibacterial, non -toxic lubricant and binder that \nis frequently used in a variety of medicinal applications  [31, 32]. Additionally, PEG -capped magnetic \nNPs have demonstrated promise as effective and efficient magnetic hyperthermia candidates as well \nas multifunctional nano -carriers for the encapsulation of hydrophobic medicines [28]. In our previous \nwork, we successfully synthesized Zn0.3Mn xFe2.7-xO4 (x=0, 0.25, 0.5, 0.75 and 1) NPs by a one -step \ncitric acid -assistant hydrothermal method  and reported the effect of citric acid concentration , pH of \nthe medium and the amount of Mn addition on the structure, purity , and magnetic properties of the 4 \n synthesized  NPs [33]. According  to the author’s knowledge , citric acid -assistant hydrothermal \nsynthesis of PEG -6000 coat ed Zn0.3Mn 0.5Fe2.2O4 NPs as a dual mode T1/T2 imaging contrast  agent  \nhave not been previously reported . In the present study, PEG surface coating is applied  on the surface \nof the zinc-manganese ferrite NPs and the n physiochemical properties of the optimized sample  is \nthoroughly investigated. Th e mono -dispersed magnetic PEG -coated and uncoated  Zn-Mn ferrite NPs  \ncontaining different level s of Mn content is synthesized and the  MR imaging of the NPs in the presence \nof external magnetic field is investigat ed.  \n2. Materials and Experimental Techniques  \n2.1. Materials  \nAll raw materials, including Fe (NO 3)3.9 H 2O, NH 4OH 25%, Zn (NO 3)2.4H 2O, Mn (NO 3)2.4H 2O and \nC6H8O7.H2O (citric acid), CH 3OH, and PEG (MW=6000 g/mol) were purchased from Merck Co. with \nminimum purity of 99%.  \n2.2. Synthesis of Mn -Zn NPs  \nIn order to s ynthesi ze Zn0.3Mn xF2.7-xO4 NPs, where x is the molar fraction  of manganese ions (Mn2+) \nfrom 0 to 1 , various amounts of manganese iron nitrate, zinc nitrate and manganese nitrate were \ndissolved in 25 ml of distilled water. A reddish brown slurry was formed after adding a solution of \n25% NH 4OH which was added for the purpose of adjusting the pH of the media to 10.  The resulting \nslurry  was then washed with the d eionized distilled water three times . Following the addition of the \ncitric acid  (CA) , the mixture was rapidly stirred for 30 minutes before being placed to a 350 ml Teflon -\nlined autoclave with a 65% fill level.  The autoclave was kept at 185  °C for 15 h and then cooled to \nroom temperature [33]. Table  1 shows the experimental conditions of the synthesized samples. The \nuncoated  samples were coded as NCZMX in which X is the molar fraction of Mn2+ ions.   \nTable 1: The hydrothermal process parameters and  the co rresponding sample codes in the present work  \nSample code  Temperature ( ℃) Time (h)  Citric acid (mmol)  pH Molar fraction of Mn2+(x) \nNCZM  185 15 3.5 10.5 0 \nNCZM25  185 15 3.5 10 0.25 \nNCZM50  185 15 3.5 10 0.5 \nNCZM75  185 15 3.5 10 0.75 \nNCZM100  185 15 3.5 10 1 5 \n 2.3. Coating of Mn-Zn NPs  \n15 mg of NCZM50 and NCZM25 NPs were added to 1 ml deionized distilled water and then placed \nin an ultrasonic bath for 30 min. A polymeric solution containing 3 wt% PEG was dissolved in 1.5 ml \nof deionized distilled water  and stirred for 30 min. The prepared magnetic ferro-fluid  placed on a \nmagnetic stirrer and then, the PEG solution were slowly added. This mixture was stirred at room \ntemperature for another 1 h at ambient temperature (25 °C). Finally, the coated NPs were m agnetically \ncollected, washed with distilled water and dried in a vacuum oven at 40  °C for 24 h. The synthesized \ncoated NPs are named as CZM25 and CZM50.  \n2.4. Cell viability  \nThe MCF -7 cells were cultured in Dulbecco’s modified Eagle’s medium DMEM (Gibco 12800, UK) \nsupplemented with 10% fetal bovine serum, 100 U/ml penicillin, 100 μg/ml streptomycin and 2 mM \nL-glutamine at 37 °C in a humidified atmosphere of 5% CO 2. The MG -63 osteoblast -like-cells were \nseeded at a density of 10,000 cells/well in a 96 well plate and cultured with complete medium \ncontaining NPs at concentrations of 50, 100 and 250 g/ml. MCF -7 cells were exposed to particles \nfor 24 h, after which Alamar Blue cytotoxicity assay was conducted and absorbance was measured at \n450 nm using a micro -plate reader. The results represent the mean values ± SD of two individual \nexperiments each performed in quadruplicate. Differences between groups were determined by \nstudent’s t test with values of p<0.05 considered significant [34, 35]. \n2.5. Characterizations  \nPhilips diffractometer, MPD -XPERT model, using CuKα radiation (λ = 1.5406 Å), was used for phase \nidentification. Estimation of the average crystallite size (L) of the samples, using the full width at half \nmaximum value (β) obtained from the spinel peaks lo cated at every 2θ in the pattern, was  carried out \nby the modified Scherer’s formula. According to Scherer's modified formula, Lnβ (β in radians) is \nplotted against Ln(1/cosθ). A linear plot is obtained using the  linear regression  which is defined as Eq. \n(1). The intercept of the line would be Ln(kλ/L) (k=0.9); the value of L (mean crystallite size) can be \nobtained using all the peaks : [33, 36]. \n𝐋𝐧𝛃 =𝐋𝐧((𝟎.𝟗𝟒𝛌\n𝐋)+𝐋𝐧(𝟏\n𝐜𝐨𝐬𝛉)) (1) \nThe miller indices of the planes w ere extracted from the cards in the X ’Pert software. Then, the mean \nlattice parameter w as calculated based on  Eq. (2) [37]: 6 \n                                                                                                         (2) \nThe shape, size, an d size distribution of NPs were investigated  using  transmission electron microscopy  \n(TEM)  with energy of 200 kV  at Arya Rastak company in Tehran . A droplet of diluted magnetic flux \nwas placed on a carbon coated copper mesh and placed at room temperature to allow water to \nevaporate. The average particle size of the produced zinc -manganese ferrite NPs from the TEM and \nSEM data  was calculated by measur ing the diameter of at least 100 NPs with ImageJ software . The \ndata were fitted by a log -normal distribution curve and then the mean size was obtained.  \nFourier transform infrared spectra (FTIR) were recorded in the range of 4000 -400 cm-1 to detect \nfunctional groups.   \nSaturation magnetization (M s) values were conducted  from the high field part of the measured \nmagnetization curves, where the magnetization curve becomes linear and line’s slope reaches  to zero .  \nColloidal properties of the aqueous magnetic ferro -fluid s were investigated using a Zeta Potential \nEstimator  to measure the surface charge of NPs , hydrodynamic size, zeta potential and poly-dispersity \nindex of NPs (in pH=7 ) under different conditions.  \nThermo -gravimetric analysis (TGA) was u sed to investigate the presence of polymer coating on the \nsurface of NPs.  \nMRI tests were performed with a 1.5 T clinical MRI instrument with a head coil working at 37  ℃. For \nT1 and T 2-weighted MRI of in-vitro  cells at 1.5 T, the following parameters were adopted: [Mat \n(320*192), FoV (184*230), and TR (407)], [Mat (256*192), FoV (260*260), and TR (7)], [Mat \n(320*192), FoV (184*230), TR (2570)].  In order to simulate the physiological state of the body, PBS \nsolut ion and water was used to create a positive and negative contrast in the images.  \n7 \n  \nFig. 1. I mage of the prepared instrument for MRI imaging.  \n3. Results and Discussion  \n3.1. Structural properties  \nFig. 2. shows XRD pattern of the NCZM50 NPs in which the diffraction peaks are in good agreement \nwith planes (220), (311), (222), (400), (422), (511), (440), (620), (533) and (444) representing \nsynthesis of pure spinel phase without the need for any calcination step. The crystallite size of the \nsample w as estima ted as 22 nm . \n \nFig. 2. The XRD pattern of the NCZM50  sample.        \nSurface coating is important in preventing NPs from agglomeration in physiological environment \nwhich also act as a barrier, effectively shielding the magnetic core against the attack of chemical \n8 \n species in the aqueous solution . Here, PEG was utilized to coat the optimized NPs. The FT -IR spectra \nof the  pure NCZM50, the PEG -coated CZM50 NPs and the PEG are shown in Fig.  3. For the pure \nNPs, at around 3300 cm-1, a strong wide band exists which is attributed to the O -H stretching vibrations \nof water molecules which are assigned to –OH group of CA  absorbed by NCZM50 NPs (a structural \nbond). The stretching vibration of C -H corresponds to t he peak at ~2925 cm- 1 [38, 39]. The absorption \nband at 1690 -1760 cm-1 is due to the vibration of asymmetric carboxyl group ( -COOH) [28, 40]. \nHence, it is suggested that  CA binds to the NPs surface through carboxylate groups of citrate ions  \n[28]. Furthermore, Fe-O stretching band as the characteristic peak of magnetite NPs  was located at \naround 520 cm−1 which is attributed to the Fe -O stretching vibration bond in tetr ahedral sites and the \nabsorption band in the 437 cm-1 corresponds to a Fe -O vibrating bond in octahedral sites of ferrite \nphase  [41]. Hydroxyl groups ( -OH) of PEG are linked to the carboxyl group ( -COOH) of citric acid  \n(CA)  for coating of Zn 0.3Mn 0.5Fe2.2O4 NPs. As it can be seen in  Fig. 3, the highest peak for PEG  curve  \nshowed a  very small shift in PEG -coated sample. The peak at 1105 cm-1 for pure PEG were shifted to \nlower frequencies which is a proof of C-O-C and C -O-H groups bonding with Zn0.3Mn 0.5Fe2.2O4 NPs. \nThe absorption band at 288 4 cm-1 can also be due to the H -C bonds stretching vi brations of the \npolymeric chain. The peaks corresponding to the bonds, C-H and C -O-C are the strong evidence to \nshow that the synthesized magnetite NPs surface has been coated with PEG  [38, 40].  \n \nFig. 3. The FT-IR spectra of the pure NCZM50  and PEG -coated CZM50  NPs along with the  PEG coating  and \ncitric acid . \n9 \n  \nThe presence of PEG layer on the NPs surface was also characterized by TGA which is presented in \nFig. 4. The first stage of weight loss at a temperature about 32 -35 °C can be related to the removal of \nwater molecules (hydroxy l ions) that are physically absorbed to the surface of the NPs. This weight \nloss in the uncoated  sample is 2.45% and in the coated sample is equal to 2.15%. The comparison of \nthe first weight loss in the two samples shows that the total water loss of the NPs is more than coated \nNPs which is due to the total absence of water from the magnetic material structure  [42]. The second \nstep, starting at about 50 -300 °C, results from the loss of organic groups that were conjugated to the \nsurface of the particles . PEG desorption and subsequent evaporation were  the causes of this weight \nloss. When 7.5 mg of PEG 6000 were used, the weight loss for particles was almost 24%, indicating \n76% iron oxide in the polymer -coated NPs.  Weight losses less  than 15 –20% can imply that the \ncoverage of particle surface by the polymer is not complete  [40]. \n \nFig. 4. The TGA result  of the NCZM50 and CZM50 samples.  \n3.2. Microstructural analysis  \nFig. 5 shows TEM micrograph and particle size distribution curve  of the coated and uncoated  samples . \nBy using ImageJ software to measure the diameter of at least 100 NPs, the average particle size and \nthe standard deviation was determined . As it can be seen, t he synthesized NPs exhibit a rather uniform \nsize dist ribution, shape, and morphology . The mean particle siz e of the coated NPs is a bit greater than \nthat of the uncoated  ones. It can be seen that NPs have become more dispersed after applying the \n10 \n coating in an aqueous medium. The average size of NPs obtained from the results of the TEM images \nbefore and after co ating was 6.9±1.54 nm and 9.25±1.6 nm, respectively, which indicates that the \npolymer coating is applied on the surface of NPs at a low thickness  [39]. \n  \nFig. 5. (a, c) TEM images and (b, d) particle size distribution histogram of the (a, b) uncoated  NCZM50 and \n(c, d) coated CZM50 NPs.  \n3.3. Stability and colloidal properties  \nThe colloidal stability of magnetic fluids of NCZM50 sample was investigated using a zeta potential \nmeasurem ent at pH=7 and various time points . The result  indicated that the NCZM50 NPs  sample had \na mean zeta potential of -48.86 ± 0.70 mV and a mean hydrodynamic size of 104 nm.  According to \nthe results, the strong negative charge of the NPs (caused by the presence of citrate ions on their \nsurface) and the steric and electrostatic forces ensure their long -term stability in aqueous media  [43]. \nc d 11 \n Poly-dispersity index (PDI) of NCZM50 sample was found to be  0.306. PDI is a parameter for \ndetermining the particle size distribution of different NPs, which is obtained from  photon correlation \nspectroscopic analysis. It is a dimensionless number calculated  from the autocorrelation function and \nranges from a value of 0.01 up to 0.7 for mono -dispersed and greater than 0.7 for poly -dispersed \nparticles [44]. In general, the particle  size between 10 and 100 nm have the longest circulation time; \nby contrast, it has been reported that particles of more than 200  nm tend to be immediately destroyed \nby one of the MPS organs [43] and tend to be eliminated by the RES [9, 45], those with diameters <10 \nnm are removed mainly by renal filtration , and particles larger than 400 nm (minimum diameter of \ncapillaries) will be  filtered by the lung [46].  \n  \n3.4. Magnetic properties  \nThe particle size and magnetization saturation values of different NPs are presented in Table 2. The \nroom temperature M –H curve for NCZM50 and CZM50 samples  is shown in Fig. 6. No hysteresis \nloop can be seen and the value of magnetization sharply increases  with the external magnetic field \nstrength.  The M –H curve has an S -shape at the low field region, and the high field side of the curve is \nalmost linear with the external field  [47]. Saturation magnetization for the NCZM50 and CZM50 NPs \nis 55 emu/g and 38 emu/g respectively. The difference in particle size and the softening of the \nmagnetization caused by the presence of PEG can both be used to explain this mismatch [38]. The \nmagnetization curve of the CZM50 sample  also revealed a negligible remnant magnetization at zero \nfield, reflecting the super -paramagnetic behavior  of the f erro-fluid . Since magnetic powder has a \ndiameter much below the 20 nm cut -off expected for magnetite to show super -paramagnetic behavior, \nthe lack of hysteresis at ambient temperature is consistent with this theory  [48].  \n \nTable 2.  The size and Ms values of different NPs  \nCode Chemistry  Size (nm)  Ms (emu/gr)  \nNCZM0  Zn0.3Fe2.7O4 14.5±2.7  47 \nNCZM25  Zn0.3Mn0.25Fe2.45O4 23.6 ±2.3 47 \nNCZM50  Zn0.3Mn0. 5Fe2.2O4 6.9±1.5 55 \nNCZM75  Zn0.3Mn0. 75Fe1.95O4 11.3 ±2.3  41 \nNCZM100  Zn0.3Mn1Fe1.7O4 6.7±2.4 37 \nCZM50  PEG coated - Zn0.3Mn0.5Fe2.2O4 9.3±1.6 38  \n 12 \n  \nFig. 6. The M vs H curves of the synthesized NCZM50 and CZM50 NPs.  \n \n3.5. MRI analysis  \nMRI examination of the body can be performed with several coil types, depending on the design of \nthe MRI unit and the information required. Fig s. 7(a-b) show T 1– and T 2-weighted MR images of \nFe3O4 and Zn -Mn ferrite solutions recorded on a 1.5 -T MRI scanner at room temperature at different \nconcentrations (0.1, 0.15 and 0.2 mg/ml) . As it can be seen, both T 1 and T 2-weighted MR images show \na strong dependence of signal intensity on manganese concentrations and among the Fe 3O4 control \nsample , Zn-based and Mn -Zn-based super -paramagnetic NPs, Mn -Zn ferrites represent a better MRI \ncontrast [49]. This is due to the fact that Mn2+ with five unpaired electrons, after Gd3+, is the most \npowerful cation used as a MRI contrast agent  [50]. Due to their greater paramagnetism and five \nunpaired electrons, divalent manganese ions (Mn2+) have been shown to be a successful method of \nincreasing the r1 of ultra -small iron oxide NPs . A peculiar mixed spinel structure, a greater saturation \nmagnetization (Ms), and a high r2 of manganese doped iron oxide NPs result from the doped Mn2+ \nwith a higher magnetic moment (B=5.92) being able to fill both the tetrahedral (Td) and octahedral \n(Oh)  sites in the crystal lattice. The doped Mn2+ and ultra -small iron oxide NPs also exhibit synergetic \n13 \n enhancement, which will further enhance both r1 and r2 of Mn -iron oxide NPs, according to the \nembedding logic. The Mn-iron oxide NPs may therefore make sup erior candidates for dual -contrast \nCA [20]. Indeed, it has recently been discovered that decreasing iron oxide NPs below 10 nm improves \ntheir effectiveness as T 1 contrast agents, suggesting that this approach could be employed to create \ndual contrast agents. The utility of these NPs as T 1 contrast agents is unfortunately limited by the low \nr2/r1 values caused by the large decrease in r2 that occurred along with th e increase in r1. To get over \nthis restriction, alloy -based NPs which has a high Ms  are a suitable candidate  to achieve NPs with \nhigh MRI sensitivity  [51]. The addition of Mn2+ and Zn2+ divalent cation ions to the spinel ferrite \nstructure causes the mass magnetization of the material to rise, which enhances the magnetic \ncharacteristics.  Therefore, t he higher contrast in Zn 0.3Mn 0.5Fe2.2O4 NPs with higher saturation \nmagnetization  can be justified  [52]. As it is presented in Fig.  7, NCZM50 sample  with core  diameters \nabout 6.7±1.54 nm and saturation  magnetization  about 55 emu/g is capable of producing  dual positive \nand negative contrast in images [26, 53]. However, the length of the polymer ch ain, which relates to \ncoating thickness, has a substantial impact on relaxivity as well. According to computer simulations, \nthe physical exclusion of protons from the super -paramagnetic iron oxide  magnetic field and the \nprotons' residence period within the  coating zone compete to decide the influence of coating thickness \non relaxivity .  \nAs it can be seen in the Fig. 8, the surface coatings also affect the relaxivity of NPs. Laconte et al. \nreported that the increased coating thickness would dramatically decrease the r2 and r 1 relaxivity of \nmono -crystalline magnetic  NPs. Therefore  it is important to note that both the chemistry  of coating \nand its thickness affect the value of  r2 and r 1 in which as the coating thickness increases, the ratio r 2/r1 \ndecreases. This is due to the inner hydrophobic layer excluding water, while the outer hydrophilic \nPEG layer allows water to d iffuse within the coating zone [53] . \n   14 \n  \nFig. 7. (a) T1-weighted and (b) T2-weighted MR images of the uncoated  Fe3O4 and Mn -Zn ferrite NPs at \ndifferent concentration s indicated by different numbers: (1) Fe 3O4 control sample, (2) NCZM0, (3) NCZM25, \n(4) NCZM50, (5) NCZM75, and (6) NCZM100 . \n    \n \nFig. 8. (a) T 1-weighted and (b) T2-weighted MR images of un-coated and uncoated  sample s indicated by \ndifferent numbers: (1) Fe 3O4 control sample, (2) NCZM50, and (3) CZM50 . \n \n3.6. Cell viability  \nCytotoxicity evaluations of the uncoated  and coated NPs were investigated by evaluating their \ncytotoxicity using MCF -7 cell line. The results of Alamar blue cytotoxicity assay are presented in Fig.  \n9. According to the results, a similar trend is observed in the activity of cells affected by diff erent \nconcentrations of NPs after 24 h compared with the control group. In general, coated and uncoated  \nparticles did not negatively change the cell growth process, and did not result significant reduction in \ncell viability. In fact, a better growth was ob served in the presence of coated NPs.  \n \n15 \n  \nFig. 9. The cytotoxicity assays performed on MCF -7 cells in the presence of coated and uncoated  NPs after \n24 h.  \n4. Conclusions  \nMono -dispersed Zn 0.3Mn 0.5Fe2.2O4 NPs with an average size of about 6.9±1.5 nm were successfully \nsynthesized by a  facile, one step citric acid -assisted hydrothermal method. The NPs were stabilized \nwith a layer of hydrophilic PEG and exhibited long -term colloidal stability in aqueous media at pH=7. \nThe magnetic properties of the uncoated  and coated Zn -Mn ferrite  NPs were measured as 55 and 38 \nemu/g, respectively, showing super -paramagnetic behavior at room temperature.  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Kuzmakb\n                                 \n                                      I.Javakhishvili Tbilisi State University,\naE.Andronikashvili Institute of  Physics.\nbInstitute for Magnetism, National Academy of Science of Ukraine\n                                         \n*Corresponding author.\nE-mail address: mamni @iphac.ge\n(G.Mamniashvili)\nABSTRACT\nKeywords : NMR spectra; magnets; single-pulse echo; two-pulse echo; magnetic pulse \n     By the method of additional pulsed magnetic field influence in different magnetic \nmaterials (half metals, manganites, lithium ferrite, cobalt) it is established the analogy of time \ndiagrams of magnetic pulse influence on single- and two pulse echoes in magnets when the \ndistortion mechanism of single-pulse echo formation is effective, and the absence of such \nanalogy in the case of lithium ferrite where the multipulse mechanism of single-pulse echo \nformation is effective.\n        It is shown also that the timing and frequency diagrams of magnetic pulse influence on the \ntwo-pulse echo signals , corresponding to the symmetric and asymmetric magnetic pulse \napplications in the studied magnets, are defined by their domain walls parameters and could \nserve for their qualitative and quantitative characterization.\nInvestigation of nuclear spin echo in magnetically ordered crystals at excitation by \nadditional pulsed magnetic field of comparatively low (as compared with anisotropy fields) \nintensity turned out to be an effective tool to observe effects connected with a hyperfine field (HF) \nanisotropy, carry out quantitative estimations of local inhomogeneities and domain walls (DW) \nmobility, what is interesting for optimization of operation of different type of magnetoelectronic \ndevices [1].\nSo, in work [2] the influence of pulsed magnetic field on Eu151nuclear spin echo signals of \nnuclei, arranged in DW of europium garnet Eu 3Fe5O12, was explained for the first time by HF field \nanisotropy. In work [3] it was considered the influence of pulsed magnetic field on nuclear spin \necho in DW of spinel ferrites and spin echo signals in thin magnetic films.\nIt was studied two cases of magnetic pulse arrangement – the symmetric and asymmetric \nones relative to the second radio-frequency (RF) pulse in the two-pulse method. At symmetric \narrangement the decrease of echo intensity in samples with a DW originated NMR is explained by \nconsecutive excitation of nuclei stepwisely changing their location in DW, but at asymmetric one –\nby the inhomogeneous shifts of NMR local frequencies due to the HF field anisotropy. It was \npresented experimental dependences of echo intensities on amplitude and, correspondingly, on \nduration of magnetic pulse.\nIn addition, it was shown that, while recording spin-echo signal intensity dependence on a \nmagnetic pulse amplitude H d, one could define the value of magnetic field amplitude shifting DW \non its width.2\nBesides it, using the echo-signal intensity dependence at asymmetric excitation by \nmagnetic pulse on a magnetic pulse duration one could calculate the density of inhomogeneous \nshifts distribution of nuclear frequencies caused by inhomogeneous component of HF field.\nFrequency measurements demonstrated a non-uniform degree of influence of an\nasymmetric magnetic pulse on the echo in different parts of the resonance line of 59Co in cobalt\nthin magnetic films [3]. The influence was weakest in the fcc phase range (212 MHz) and strongest \nin the hcc phase range (218 MHz). This could serve as additional characteristics of investigated \nmagnets, as example, in respect of stacking faults or impurities influence. The frequency spectra of \nthe symmetric pulse influence could also turn out to be useful for the investigation of NMR line \nnature in magnets.\nThese results were found out their development in work [4] where it was investigated the \ninfluence of asymmetric magnetic field pulses (dephasing) on spin echoes of 59Co nuclei in Y2Co17\nand (Y 0.9Gd0.1)2Co17. It was revealed the influence of Gd ions on the anisotropic component of the \nHF field in a narrow frequency band of a wide NMR spectrum. The difference between dephasing \nspectra in Y2Co17 and (Y 0.9Gd0.1)2Co17shows that we have the useful method to study the changes \nof magnetic properties of Y2Co17 during the substitution by Gd. Most interesting peculiarity of \ndephasing method is a possibility to resolve of specific behavior of Gd ions in a very narrow \nfrequency range what could be related with one Co-position. Really, the fact that Gd ions stipulate \nsuch strong change of anisotropic component of HF filed on 59Co nuclei in a very narrow \nfrequency band makes it possible to conclude that they prefer some definite crystallographic \npositions. This way one could achieve larger resolution of NMR spectra.\nFor practical applications it is also perspective the effect of formation of additional echo \nsignals at influence of magnetic pulse applied in combination with a RF pulse [5,6].\nAs a matter of fact the single-pulse echo (SPE) – a resonance response of nuclear spin-\nsystem on a single radio-frequency (RF) pulse, in a number of magnets behaves mainly similarly \nto the usual two-pulse echo (TPE).\nFor investigations of the nature of this analogy in the work [7] it was studied the role of RF \npulse front distortions in the formation of SPE signal, when distorted edges of RF pulse play role \nof two RF pulses in the Hahn mechanism of two-pulse echo formation. \nTo clarify the role of pulse distortions in the formation of SPE it was used a technique of \nsuppressing of the influence of the RF pulse edges on the nuclear spin system. This was achieved \nby applying a dc magnetic field pulse of width dτ and amplitud e Н dwhose timing could be varied \nin respect to RF pulse. Earlier, this type of excitation has been used in a two-pulse echo experiment \nto estimate the static field required to shift a domain wall through a distance equal to its thickness \nd [8].   \nIn Fig.1 it is presented Fig.4 from [7] to illustrate the experimental layout and results \nobtained in work [7].\nAs it is seen from Fig.1, the influence of magnetic pulse for the both excitation cases of \necho signals is similar. On this basis in the cited work it was made conclusion that the SPE signal \nin the investigated sample (lithium ferrite) is formed by the distortion mechanism when the \ndistorted RF pulse edges play a role of RF pulses in the two-pulse method.\nFurther on, in the work [7] it was carried out more detailed analysis of magnetic pulse \ninfluence of the TPE intensity. As it has already been noted, the main effect of applied magnetic \npulse to multidomain magnets is the displacement of DW which is reversible at small pulse \namplitudes. Therefore, if the magnetic pulse is superposed on one of the RF pulses, it changes the \nlocation of the resonating nuclei in the DW (y-direction) with respect to its center thereby reducing 3\nthe nuclear enhancement factors for 180oC Bloch walls accordingly the known dependence \nsech(y/d)ηηo , where oη is the maximum enhancement at the center of the wall. Therefore, if in \nthe absence of the magnetic pulse the turning angles i of two RF pulses 1,2 1,2 γhητ α  were equal \nto each other in order to maximize the intensity of TPE [7,9], after the application of magnetic \npulse in coincidence with one of RF pulses it follows that turning angles would differ significantly \nwhat should result in a fast reduction of TPE amplitude.\nBesides it, due to the dependence of nuclear resonance frequency on location in the DW in \nsystems with anisotropic HF field [10], the application of magnetic pulse in the interval between \nRF pulses, or after the second pulse would also result in the decrease of TPE signal. This effect \n(the dephasing effect) arises because the frequency shift partially destroys the phase coherence \nreducing the effectiveness of the rephasing process. \nOn the basis of aforementioned considerations farther on in work [7] it has been made the \nconclusion that regardless of the temporal location of the magnetic pulse the TPE amplitude \ndecreases and the TPE intensity decrease is much larger in the case when H dis applied coincident \nwith one of the RF pulses.\nThe interest of present authors to the considered problem is connected with our own \nprevious investigations of the SPE formation mechanism in lithium ferrite [11,12], what didn’t \nreveal any contribution of the distortion mechanism to the SPE in this material which was formed \nby the multipulse mechanism. It has been shown that the SPE signal disappears in the limit of \nsingle-pulse excitation. Besides it, the controllable additional external frequency distortion of   \nexciting RF pulse edge similarly to work [13] resulted only in the decrease of SPE signal. Along \nwith it a similar influence considerably (several times) enhances the SPE signal in a number of \nother materials, e.g., Co and half-metallic Со2MnSi and NiMnSb, where, besides it, the SPE is \nobserved also in the single-pulse excitation mode.  \nFurther on, in work [7] at investigation of influence of the magnetic pulse timing in respect \nto the SPE signal, as it has already been noted, it was made conclusion that the decrease of TPE \nintensity would be the most strong when H dis applied coincident with RF pulses. But allowing for \nresults of work [3] one could suppose that the influence of magnetic pulse on the TPE signal in \nmagnetically soft lithium ferrite with a small HF field anisotropy and high mobility of DW could \ndiffer from the one for magnetically hard materials with a relatively small DW mobility and large \nHF field anisotropy, such, for example, as cobalt.\nTo illustrate these considerations it was carried out experiments to study the influence of \npulsed magnetic field with amplitude up to 500 Oe and duration of several μsusing NMR \nspectrometer, a pulsed magnetic field source, and samples described in details in work [7,5], \ncorrespondingly. Measurements were made on cobalt, lithium ferrite, half metals, such as NiMnSb, \nСо2MnSi and manganites, attracting currently great interest from the point of view of their \napplications in spintronics [14], and on cobalt thin magnetic films.\nThe obtained results confirm the main conclusion of work [7] that there is an analogy in the \ninfluence of magnetic pulse on SPE and TPE in systems where the echo signal is formed by the \ndistortion mechanism. Let us present for illustration the corresponding dependences for cobalt and \nNiMnSb in Fig. 2-3. But in the case of lithium ferrite, when the echo signal is formed by the \nmultipulse mechanism, the magnetic pulse influence picture on the SPE signal is different, Fig.4.\nIn this case the MP influence is similar to the MP influence on three-pulse stimulated echo \nwhen the MP is applied between the first and the second RF pulses, Fig.5. The observed influence \npicture corresponds to the stimulated character of the SPE formation in lithium ferrite [12,15]. 4\nAs it has already been pointed out, frequency diagrams of the symmetrical MP influence \ncould be useful, along with ones of asymmetric MP application [3,4] for the characterization of the \ninvestigated magnets. Let us show this an example of polycrystal cobalt and lithium ferrite, Fig, 6 \nand 7, correspondingly, and for half metal NiMnSb, Fig.8.\nIn the case of half metallic Co 2MnSi it was received the frequency diagrams of MP \ninfluence for 55Mn and 59Co nuclei, showing the significant difference in the degree of local HF \nfields anisotropy of corresponding nuclei, Fig.9. For visualization, let us present here as well  time \ndiagrams of MP influence for both cases, Fig. 10.\nAs per TPE excitation, the picture of pulse influence in polycrystal Co and thin Co films \nqualitatively differs from that one observed in materials with a relatively large DW mobility and a\nsmall HF field anisotropy and it is also different from the dependence assumed in [7].    Therefore, \ntime diagrams of magnetic pulsed field influence on TPE could be used for the characterization of \nmagnetic materials. For illustration, let us compare the time diagrams of MP influence on TPE \nsignals in cobalt, Fig.2a, with that of NiMnSb, Fig. 3a, for which there is an analogy with lithium \nferrite, Fig.4a.\nIn addition, the frequency diagrams of symmetric pulse influence reflect the peculiar \nphysical properties of materials and could be used for their characterization as ones for asymmetric \npulse [3,4]. \nIn conclusion, it is established by the method of magnetic pulse influence the analogy \nbetween the timing diagrams of magnetic pulse influence of TPE and SPE of magnets where SPE \nis formed by the distortion mechanism, in difference with the case of lithium ferrite where SPE is \nformed by multipulse mechanism.\nIt was shown also that the magnetic pulse influence timing and frequency diagrams could \nbe used for additional sensitive characterization of magnetic materials in both cases of symmetric \nand asymmetric excitations along with usual NMR spectra. \n     This work was supported by the STCU Grant Ge-051 (J).5\nREFERENCES  \n1. Kurkin M.I., Turov B.I. NMR in magnetically ordering materials and its application. M.: Nauka. \n1990. 244 p. \n2. Belotitski V.I., Chekmarev V.P. Abstracts of Papers presented at All Union Conf. on Physics of \nMagnetic Phenomena.  Physicotechnical Institute of Low Temperature, Academy of Sciences of \nthe Ukrainian SSR, Kharkov. 1979. 123 p.\n3. Rassvetalov L.A., Levitski A.B. Influence of a pulsed magnetic field on the nuclear spin echo in \nsome ferromagnets and ferrimagnets. Sov. Phys. Solid State Phys. 1981. v.23, N 11. pp. 3354-\n3359. \n4. Маch owska E., Nadolski S. Echo defocusing in Y 2Со17.  Solid State Commun. 1988. v. 68. N 2, \npp. 215-217.\n5. Аkhalkatsi A.M., Mamniashvili G.I., Sanadze Т I. The nuclear spin-echo signals under combined \naction of magnetic field and RF pulses. Appl. Magn. Reson. 1998. v. 15/3-4, pp. 393-399.\n6. Akhalkatsi A.M., Zviadadze M.D., Mamniashvili G.I., Sozashvili N.M., Pogorely A.N., Kuzmak \nO.M. Multipulse analogs of single-pulse echo signals in multidomain magnetics. \nPhys.Met.Metallogr. 2004. v.98, N.3, pp.1-9. \n7. Kiliptari I.G., Tsifrinovich V.I. Single-pulse nuclear spin echo in magnets. Phys. Rev. B. 1998. \nv. 57. pp. 11554-1154.\n8. Dean R.H., Urwin R.J. The use of nuclear spin-echoes to measure hyperfine field distributions \nin ferromagnets. J. Phys. C. 1970. v. 3. pp. 1747-1752.\n9. Kiliptari I.G., Kurkin M.I. A possibility of restoring the gain distribution function of nuclei \nspins.  Phys.Met.Metallogr.1992. v.74, N.2. pp. 136-139.\n10. Searle C.W., Kunkel H.P., Kupca S. and Maartense I. NMR enhancement of a modulating field \ndue to the anisotropic component of the hyperfine field in hcp Co and YCo 5 . Phys. Rev. B. 1977. v. \n15. N 7, pp. 3305–3308.\n11. Akhalkatsi A.M., Mamniashvili G.I., Ben-Ezra S. On mechanisms of single-pulse echo\nformation  in multidomain magnetic materials. Phys. Lett. A 291. 2001. pp. 34-38.\n12. Akhalkatsi A.M., Mamniashvili G.I., Gegechkori T.O., Ben-Ezra S. On formation mechanism \nof 57Fe single-pulse echo in lithium ferrite. Phys. Met. Metallogr. 2002. v. 94. N 1. pp. 1-7.\n13. Tsifrinovich V.I., Mushailov E.S., Baksheev N.V., Bessmertnyi E.A., Glozman E.A., Malt’sev\nV.K., Novoselov O.V., Reingardt A.E. Nuclear single-pulse in ferromagnets. Sov. Phys. JETP v. \n61, pp. 886 \n14. Moodera J.S. Half metallic materials. Phys. Today. 2001. v. 54. N 5. pp.39-44. \n15. Shakhmuratova L.N., Fowler D.K., Chaplin D.H. Fundamental mechanisms of single-pulse \nNMR echo formation.  Phys. Rev. A. 1997. v. 55. N 4. pp. 2955-29676\nFig. 1.\nFig. 2.\nFig. 3. 7\nFig. 4.\n                        Fig. 5.\n8\nFig. 6.\nFig. 7. \nFig. 8.\nFig. 9.9\nFig. 10. 10\nFIGURE CAPTIONS\nFig. 1. Timing diagrams of the intensity dependence of the two-pulse echo\n , single-pulse echo \n and FID’s original maximum ( ) on the temporal location of a dc magnetic pulse of H d=5 Oe \nlasting for a time interval d along the corresponding time scales for the 57Fe NMR in lithium \nferrite. The data for two-pulse echo were taken with 1= 2= 0.8 s, = 21 s,d = 3 s, while \nthe single-pulse echo data correspond to 1 = 25 s,d = 5 s. Resonance frequency 74.0 MHz, \nT=77 K. \nFig. 2. Timing diagrams of the intensity dependence of the two-pulse echo (a) and single-pulse \necho (b) on the temporal location of the H dmagnetic pulse with duration d in polycrystal cobalt \nat:\na)1= 2= 1.6 s,   = 10 s,d = 2.4 s, H d= 100 Oe, f NMR= 216.5 MHz;\nb)  = 17 s,d = 1 s, H d=30 Oe, f NMR=216.5 MHz.\nFig. 3. Timing diagrams of the intensity dependence of the two-pulse echo (a) and single-pulse \necho (b) on the temporal location of the H d magnetic pulse with duration d in half metal NiMnSb \nat:\na) 1= 2= 2 s, = 11 s,d = 3 s, H d=150 Oe, f NMR=300 MHz;\nb)  = 10 s,d = 1 s, H d=30 Oe, f NMR=300 MHz.\nFig. 4. Timing diagrams of the intensity dependence of the two-pulse echo (a) and single-pulse \necho (b) on the temporal location of the H d magnetic pulse with duration din lithium ferrite \n(Li0.5Fe2.35Zn0.15O4) at:\na) 1 = 1 s,2 = 1.4 s, = 7 s,d = 1.7 s, H d=28 Oe, f NMR=74 MHz;\nb)  = 20 s,d = 2 s, H d=10 Oe, f NMR=71.5 MHz.\nFig. 5 Timing diagrams of the intensity dependence of the three-pulse stimulated echo on the \ntemporal location of the magnetic pulse in lithium ferrite (Li0.5Fe2.35Zn0.15O4) at:\n1 = 0.8 s,2 = 1 s,3 = 1 s,d = 3 s, H d=40 Oe, 12 = 20 s,23 = 40 s,\n  \n   fNMR=71 MHz,  \n   fNMR=72 MHz, \n  fNMR=73 MHz.\nFig.6. Frequency dependence of the influence of the symmetric (\n ) and asymmetric (\n ) magnetic \npulses on two-pulse echo in cobalt at: \n1 = 1.1 s,2 = 1.2 s, = 10s,d = 2s, H d=100 Oe. I o – echo amplitude at H d=0. \nFig.7. Frequency dependence of the effect of the symmetric (\n ) and asymmetric (\n ) magnetic \npulses on two-pulse echo in lithium ferrite (Li0.5Fe2.35Zn0.15O4) at:\n1 = 0.8 s,2 = 0.9 s, = 14 s,d = 1 s, H d=15 Oe; I o – echo amplitude at H d=0. \nFig.8. Frequency dependence of the effect of the symmetric (\n ) and asymmetric (\n ) magnetic \npulses on two-pulse echo in NiMnSb at:\n1 = 2 = 2 s, = 10 s,d = 3 s, H d=150 Oe; I o – echo amplitude at H d=0. \nFig. 9. Frequency dependence of the effect of the symmetric (\n ) and asymmetric (\n ) magnetic 11\npulses on two-pulse echoes in  Co 2MnSi  for 59Со NMR ( а) and 55Mn NMR(b) at: \na) 1 = 1.1 s,2 = 1.4 s, = 10 s,d = 2 s, H d=550 Oe (59Со ЯМР );\nb) 1 = 0.8 s,2 = 1 s, = 13 s,d = 4 s, H d=190 Oe (55MnЯМР );\nIo – echo amplitude at H d=0. \nРис.10. Timing diagrams and intensity dependence of the two-pulse echo on the temporal \nlocation of the magnetic pulse with duration din Co2MnSi for 59Со NMR ( а) and 55Mn NMR (b) \nat:\na) 1 = 1.1 s,2 = 1.4 s, = 10 s,d = 2 s, f=145,5 MHz, H d=550 Oe;\nb) 1 = 2 = 3 s, = 7 s,d = 2 s, H d=300 Oe, f=354 MHz."    },    {        "title": "1701.08993v1.Heterodiffusion_coefficients_in_α_iron.pdf",        "content": " 1 \nHeterodiffusion coefficients  in \n -iron \n \nVassiliki Katsika -Tsigourakou* and Efth imios  S. Skordas  \nDepartment of Solid State Physics, Faculty of Physics, University of Athens,  \n Panepistimiopolis, 157 84 Zog rafos, Greece  \n \n \nAbstract   \nThe diffusion of tungsten  in \n -iron is important for the application of ferritic -\niron alloys to thermal power plants . These data,  over a wide temperature range across \nthe Curie temperature , have been re cently reported . We show that the se diffusion \ncoefficients can be satisfactory reproduced in terms of the bul k elastic and expansivity \ndata by means of  a thermodynamical model that interconnects point defects \nparameters with bulk qualities.  \n \n \n  \n \nPACS: 61.72.Jd, 62.20. D e, 66.30.Fq, 66.30. -h, 61.72.Bb  \n \n \nKeywords: Diffusion;  Bulk modulus; Gibbs energy ; Activation energy; Point defects  \n                                                 \n* Email: vkatsik@phys.uoa.gr   2 \n1. Intr oduction   \n \nIn a recent review  [1], the models that interconnect point defect parameters with \nbulk properties have be en presented . Chief among these, the so called \ncB  model [2 -\n7], which suggests  that the defect Gibbs energy gi (where i denotes the corresponding \nprocess, i.e., defect formation, f, migration, m, or self -diffusion activation, act)  is \nproportional to the isothermal bulk modulus \nB  and the mean volume \n  per atom. \nAfter investigating a large variety of solids, it was finally concluded  [1] that the  \ncB  \nmodel leads  to results that are in agreement with the experimental data.  \nThe above review  [1] was crossed with the publication by Takemoto et al.  [8] of \ntracer diffusion coefficients of 181W in \n -iron in the temperature range between 833 \nand 1173 K using serial  sputter -microsectioning method. These data in high purity \n -\niron, over a wide temperature range across the Curie temperature (T C=1043 K), were \nconsidered to be important primarily  in two respects: First, it is  well known that a \nsmall addition of large -size elements, such as W, Nb and Mo, into iron increase s \nconsiderably creep strength of the steels. Second, earlier measurements were limited \nto temperatures above 973 K, while data at lower temperatures, in parti cular around \n900 K are necessary for the application of ferritic -iron allo ys to thermal power plants.  \nIt is the object of this paper to investigate whether the \ncB  model can reproduce \nthese important diffusion data.  \nWe clarify that , as already mentioned in Ref. [1], the aforementioned “elastic” \nmodels (in the sense that gi is interrelated with bulk elastic data) have recently \nattracted the interest  in view of the following facts:  A challenging suggestion has been \nforwarded  [9, 10 ] that these “elastic” models may provide a basis  for the \nunderstanding of the non -Arrhenius temperature dependence of the viscosity of the \nglass forming liquids when the glass transition  [11] is approached.  Furthermore , it  3 \nwas found  [12] that, in a certain c lass of high T C-superconductors, the formation \nenergy for Schottky defects follows the expectations of the \ncB  model. Finally we \nnote that, when applying uniaxial stress in ionic crystals electric signals are produced \nwhich have par ameters that are consistent with the \ncB  model  [13]. This is important \nfor understanding the generation of precursory  electric signals that are measured \nbefore seismic events [14-16].  \n \n \n2. The d iffusion coefficients  \n \nThe diffusion coefficient D, if a single diffusion  mechanism is operating  in \nmono -atomic crystals , is described in terms of the activation Gibbs energy gact, as [7]: \n2exp( )act\nBgDfkT\n  (1) \nwhere f is a numerical constant depending on the diffusion mechanism an d the \nstructure , \n stands for the lattice constant, \n  the attempt frequency which for the \nself-diffusion activation process is of the order of the Debye frequency \nD  and \nBk  the \nusual Boltzmann constant .  \nThe activation entropy sact and the activation enthalpy hact are defined  [7] in \nterms of gact as follows:  \nact\nact\nPdgsdT\n     (2) \n,act\nact act\nPdgh g TdT\n, and hence  \nact act acth g Ts   (3) \nIf the plot  \nnD\n  versus 1/T is linear, both hact and sact are temperature \nindependent  and then Eq.(1) can be written as:   4 \n0exp( )act\nBhDDkT\n  (4) \nwhere D0 is given by  \n2\n0 exp( )act\nBsDfk\n  (5) \nLet us now write  D in terms of the \ncB  model . Since the defect Gibbs energy gi \nis interconnected with the bulk properties of the solid through the relation:  \nact actg c B\n  (6) \nwhere cact is a dimensionless constan t, by substituting  Eq. (6) into equation ( 1) we get  \n2exp( )act\nBcBDfkT\n  (7) \nThis relation , enables the calculation of D at any temperature provided that elastic and \nexpansivity data are available and that cact has been determined from a single \nmeasurement (i.e., once  the value D1 has been found experiment ally at a temperature \nT1, the value of cact can be determined since the pre -exponential factor \n2f  is \napproximately known  [17] because \n  can be roughly estimated as it will be explained \nbelow). T he values of sact and hact can then be directly calculated at any temperature \nby means of the following equations that result upon inserting Eq.( 6) into Eq.(2) and \n(3): \n()act i\nPdBs c BdT  \n  (8) \n()act i\nPdBh c B T B TdT    \n  (9) \nwhere  is the thermal (volume) expansion co efficient.  \n \n \n3. Application to the case of W diffusing in \n -Fe    5 \n \n We now apply Eq.(7) to the case of tungsten diffusing in \n -Fe. Co ncerning \nthe attempt frequency, \n , we consider  that for a given matrix and mechanism, it \ndepends roughly on the mass of the diffusant according to the approximation:  \n21\n\n\n\njm\nDj\nmm\n\n  (10) \nwhere  \nmm, \njm  denote the mass of the matrix ( m) and the d iffusant ( j), respectively \n(i.e., Fe  and W in the present case) and \nD  91012s-1. Concerning the elastic data, the \nadiabatic bulk modulus has been measured in the region 298 to 1173 K by Dever  [18] \nand is converted to the isotherm al one, B, by means of the expansivity and specific \nheat data given in the literature (see Ref. [17] and references therein). The \ndetermination of cact is now made at the temperature T1=973 K for which Takemoto et \nal. [8] reported two measurements for D, i.e., 1.9110-18 and 1.8110-18 m2/s. Hence, \nwe use here  their average value, i.e., D1=1.8610-18 m2/s and also consider that –\naccording to the elastic data mentioned above - B=133.3 GPa  at this temperature ; \nfurtherm ore, we take into account  that \n2.89Å (cf. recall that \n =\n3/2) and \nassume that the diffusion proceeds via monovacancies, thus f=0.727. By inserting \nthese values into Eq.(7), we find that cact has a value between 0.21 and 0.22 after \nconsidering plausible experimental error s in the quantities used in the calculation.  \nOnce cact is known, we can now compute D for every temperature by \nincorporating the appropriate data of B and \n  into Eq. (7). The calculation was made \nat all t emperatures (between 833 K and TC) at which experimental D values have been \nreported  by Takemoto et al.  [8] by using th e B and \n  values resulting from a linear \ninterpolation of the corresponding experimental values given in Table 1 of Ref. [17]. \nThes e calculated D values are insert ed with stars in Fig. 1, while the experimental  6 \nvalues are shown with open circles. Note that, s ince cact was taken as 0.21 or 0.22, as \nmentioned above, two calculated D values are depicted for each tempe rature. An \ninspect ion of this figure reveals  \n that the experimental D values lie more or less \nbetween the calculated ones.  \nWe now calculate  hact, for example at the temperature T=993 K, in which \nB=132. 2 GPa, 510-5K-1 and \n12.1310-30m3. Furthermore, we consider that \n(dB/dT)= -0.0624 GPa/K as it results from a least squares fitting to a straight line of \nthe B-values given in Table 1 of Ref. [17] between 973 and 1043 K . Inserting these \nvalues into Eq.(9) , we find hact=3.0 and 3.13 eV for cact=0.21 and 0.22 respectively, \nwhich are in excellent agreement with the experimental value  [8] hact=(3.00.2) eV.  \n \n \n4. Discussion   \n \nWe now discuss the following  empirical fact mentioned in Ref. [8]. Studying the \ndiffu sion of transition elements, such as Ti, V, Cr, Co, Ni, Nb, Mo and W in \nparamagnetic \n -iron, the activation enthalpy hact, j was found to increase linearly with \n(rsolute-rFe)/rFe, where  rsolute and  r Fe are the radi i of the solut e atom for the coordination \nnumber eigh t and of iron matrix lattice. In other words, atomic size affects the \nactivation enthalpies for  diffusion of transition elements in paramagnetic \n -iron. This \nis strikingly reminiscent of an e arly finding in alkali halides doped with divalent \ncations, in which electric dipoles of the form “divalent cation plus one cation \nvacancy” are produced  [19]. These dipoles, upon applying an external electric field, \nchange their orientation in space mainly  through jumps of the cation vacancy between \nneighboring sites to the divalent impurity (cf. these dipoles contribute of course to the  7 \nstatic dielectric constants  \nS, but even in their absence (i.e., in the case of “pure” \nalkali ha lides) \nS  varies upon changing the temperature or pressure, mainly due to the \nvolume dependence of the ionic polarizability  which is interrelated with B [20]). The \nactivation  enthalpy for this (re)orientation process, which is of c ourse governed by the \nvacancy migration, was found  [21] to increase linearly with the ionic radi us of the \ndivalent cations, when the latter have rare gas electron configuration.  \nThe \ncB  model cannot give any direct explanation for the aforementioned \neffect , either in paramagnetic \n -iron or in alkali halides. Only an indirect guess for \nthe effect in  \n-iron could be made along the following line s: First,  let us make the \nreasonable assumption that  a diffusant having  larger atomic size corresponds to a \nlarger activation volume act, j. Second, b y inserting Eq.(6) into the relation  [1] \nact\nact\nTdg\ndP\n, we find : \n,,1act j act j\nTdBcdP  \n\n  (11) \nwhich, when combined with Eq.(9), leads to the conclusion that th e ratio act, j / hact, j \nis a bulk quantity, i.e.,  \n1act\nT\nact\nPdB\ndP\ndB hBTBTdT\n\n\n\n  (12) \nand h ence should be the same for various diffusants j in the same matrix. Thus, on the \nbasis of \ncB  model, we can guess that a diffusant with larger atomic size should also \nhave a large r activation enthalpy . A quantitative assessment in terms of the atomic \nradius cannot be made.  \n \n  8 \n5. 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Dyre,  AIP Conference Proceedings 832 (2006)  113. \n[10] J.C. Dyre, T. Christensen, and N.B. Olsen, J. No n-Cryst. Solids 532 (2006)  4635 . \n[11] J.S. Langer, P hysics  Today , February 2007, p. 8;  \nJ.S. Langer, Phys. Rev. E 73 (2006)  041504.  \n[12] H. Su, D.O. Welch, and Winnie Wong -Ng, Phys. Rev. B 70 (2004)  054517.  \n[13] P. Varotsos, N. Sarlis, and M. Lazaridou, Ph ys. Rev. B 59 (1999)  24. \n[14] P. Varotsos  and K. Alexopoulos, Tectonophysics 110 (1984 ) 73; \nP. Varotsos  and K. Alexopoulos, Tectonophysics 110 (1984) 99;  \nP. Varotsos, K. Alexopoulos, K. Nomicos and M. Lazaridou, Tectonophysics  152 \n(1988)  193; \nN. Sarlis, M.  Lazaridou, P. Kapiris and P. Varotsos, Geophys. Res. Lett. 26 \n(1999)  3245.  \n[15] P. A. Varotsos, N.V. Sarlis, H.K. Tanaka and E .S. Skordas, Phys. Rev. E 71 \n(2005)  032102 ;  \nP. Varotsos, K. Eftaxias, M. Lazaridou, G. Antonopoulos, J. Makris and J. \nPoliyianna kis, Geophys. Res. Lett. 23 (1996) 1449.  \n[16] P.A. Varotsos, N.V. Sarlis, E.S. Skordas,  S. Uyeda , and M . Kamogawa , Proc \nNatl Acad Sci USA 108  (2011) 11361 ;  \nP.A. Varotsos, N.V. Sarlis, and E.S. Skordas, Phys. Rev. Lett. 91 (2003)  148501 ; \nP.A. Varotsos,N.V.Sarlis,E.S.Skordas,H. K.Tanaka,M.S.Lazaridou,Phys.Rev . \nE 73(2006)031114;  \nP.A. Varotsos,N.V.Sarlis,E.S.Skordas,H.K.Tanaka,M.S.Lazaridou,Phys.Rev.  \nE 74(2006)021123.  \n[17] P. Varotsos, and K. Alexopoulo s, Phys. Rev. B 22 (1980)  3130 .  11 \n[18] D.J. Dever, J. Appl. Phys. 43 (1972)  3293.  \n[19] D. Kostopoulos, P. Varotsos, and S. Mourikis, Can. J. Phys. 53 (1975)  1318 ;  \nP.A. Varotsos, N.V. Sarlis, and E.S. Skordas, Phys . Rev. E 66 (2002) 011902  \n[20] P.A. Varotsos , Phys. Status Solidi ( B) 90 (1978)  339; \nP. Varotsos, Phys. Status Solidi (B) 100 (1980)  K133. \n[21] P. Varotsos, and D. Miliotis, J. Phys. Chem. Solids 35 (1974)  927.  12 \nFIGURE  and FIGURE  CAPTION  \n \n \n \nFig. 1 . Tungsten diffusing in \n -iron. Diffusio n coefficient s, D, as measured in Ref. [8] \n(circles)  at various  temperatures, T, vs 1000/T. The upper and the lower value \ncalculated for each temperature by means of the \ncB  model are shown by  stars. "    },    {        "title": "1309.7483v1.High_efficiency_GHz_frequency_doubling_without_power_threshold_in_thin_film_Ni81Fe19.pdf",        "content": "arXiv:1309.7483v1  [cond-mat.mtrl-sci]  28 Sep 2013High-eﬃciency GHz frequency doubling without power thresh old in thin-ﬁlm Ni 81Fe19\nCheng Cheng1and William E. Bailey1\nMaterials Science and Engineering Program, Department of A pplied\nPhysics and Applied Mathematics, Columbia University, New York,\nNY 10027\nWe demonstrate eﬃcient second-harmonic generation at moderat e input power for\nthin ﬁlm Ni 81Fe19undergoing ferromagnetic resonance (FMR). Powers of the gene r-\natedsecond-harmonicareshowntobequadraticininputpower, wit hanupconversion\nratio three orders of magnitude higher than that demonstrated in ferrites1, deﬁned\nas ∆P2ω/∆Pω∼4×10−5/W·Pω, where ∆ Pis the change in the transmitted rf\npower and Pis the input rf power. The second harmonic signal generated exhibit s\na signiﬁcantly lower linewidth than that predicted by low-power Gilbert damping,\nand is excited without threshold. Results are in good agreement with an analytic,\napproximate expansion of the Landau-Lifshitz-Gilbert (LLG) equa tion.\n1Nonlinear eﬀects in magnetizationdynamics, apart frombeing offun damental interest1–4,\nhave provided important tools for microwave signal processing, es pecially in terms of fre-\nquency doubling and mixing5,6. Extensive experimental work exists on ferrites1,4,6, tradi-\ntionally used in low-loss devices due to their insulating nature and narr ow ferromagnetic\nresonance (FMR) linewidth. Metallic thin-ﬁlm ferromagnets are of int erest for use in these\nand related devices due to their high moments, integrability with CMOS processes, and\npotential for enhanced functionality from spin transport; low FMR linewidth has been\ndemonstrated recently in metals through compensation by the spin Hall eﬀect7. While some\nrecent work has addressed nonlinear eﬀects8–10and harmonic generation11–13in metallic\nferromagnets and related devices14–16, these studies have generally used very high power\nor rf ﬁelds, and have not distinguished between eﬀects above and b elow the Suhl instabil-\nity threshold. In this manuscript, we demonstrate frequency dou bling below threshold in\na metallic system (Ni 81Fe19) which is three orders of magnitude more eﬃcient than that\ndemonstrated previously in ferrite materials1. The results are in good quantitative agree-\nment with an analytical expansion of the Landau-Lifshitz-Gilbert (L LG) equation.\nFor all measurements shown, we used a metallic ferromagnetic thin ﬁ lm structure, Ta(5\nnm)/Cu(5 nm)/Ni 81Fe19(30 nm)/Cu(3 nm)/Al(3 nm). The ﬁlm was deposited on an oxi-\ndized silicon substrate using magnetron sputtering at a base press ure of 2.0 ×10−7Torr. The\nbottom Ta(5 nm)/Cu(5 nm) layer is a seed layer to improve adhesion a nd homogeneity of\nthe ﬁlm and the top Cu(3 nm)/Al(3 nm) layer protects the Ni 81Fe19layer from oxidation.\nA diagram of the measurement conﬁguration, adapted from a basic broadband FMR setup,\nis shown in Fig.1. The microwave signal is conveyed to and from the sam ple through a\ncoplanar waveguide (CPW) with a 400 µm wide center conductor and 50 Ω characteristic\nimpedance, which gives an estimated rf ﬁeld of 2.25 Oe rms with the inpu t power of +30\ndBm. We examined the second harmonic generation with fundamenta l frequencies at 6.1\nGHz and 2.0 GHz. The cw signal from the rf source is ﬁrst ampliﬁed by a solid state am-\npliﬁer, then the signal power is tuned to the desirable level by an adj ustable attenuator.\nHarmonics of the designated input frequency are attenuated by t he bandpass ﬁlter to less\nthan the noise ﬂoor of the spectrum analyzer (SA). The isolator limit s back-reﬂection of\nthe ﬁltered signal from the sample into the rf source. From our ana lysis detailed in a later\nsection of this manuscript, we found the second harmonic magnitud e to be proportional to\n2FIG. 1. Experimental setup and the coordinate system, θ= 45◦; see text for details. EM:\nelectromagnet; SA: spectrum analyzer. Arrows indicate the transmission of rf signal.\nthe product of the longitudinal and transverse rf ﬁeld strengths , and thus place the center\nconductor of CPW at 45◦from H Bto maximize the Hrf\nyHrf\nzproduct. The rf signal ﬁnally\nreaches the SA for measurements of the power of both the funda mental frequency and its\nsecond harmonic.\nFig.2(a) demonstrates representative ﬁeld-swept FMR absorptio n and the second har-\nmonic emission spectra measured by the SA as 6.1 GHz and 12.2 GHz pea k intensities as\na function of the bias ﬁeld H B. We vary the input rf power over a moderate range of +4\n- +18 dBm, and ﬁt the peaks with a Lorentzian function to extract t he amplitude and\nthe linewidth of the absorbed (∆ Pω) and generated (∆ P2ω) power. Noticeably, the second\nharmonic emission peaks have a much smaller linewidth, ∆ H1/2∼10 Oe over the whole\npower range, than those of the FMR peaks, with ∆ H1/2∼21 Oe. Plots of the absorption\nand emission peak amplitudes as a function of the input 6.1 GHz power, shown in Fig.2(b),\nclearly indicate a linear dependence of the FMR absorption and a quad ratic dependence\nof the second harmonic generation on the input rf power. Taking th e ratio of the radiated\nsecond harmonic power to the absorbed power, we have a convers ion rate of 3.7 ×10−5/W,\nas shown in Fig.2(c).\nSince the phenomenon summarized in Fig.2 is clearly not a threshold eﬀe ct, we look into\nthe second-harmonic analysis of the LLG equation with small rf ﬁelds , which is readily de-\nscribed in Gurevich and Melkov’s text for circular precession relevan t in the past for low-M s\n3FIG. 2. Second harmonic generation with ω/2π= 6.1 GHz. a) left panel : 6.1 GHz input power\n+17.3 dBm; right panel : 6.1 GHz input power +8.35 dBm. b) amplitudes of the ω(FMR) and\ngenerated 2 ωpeaks as a function of input power Pω; right and top axes represent the data set\nin log-log plot (green), extracting the power index; c) rati o of the peak amplitudes of FMR and\nsecond harmonic generation as a function of the input 6.1 GHz power; green: log scale.\n4ferrites18. For metallic thin ﬁlms, we treat the elliptical case as follows. As illustra ted in\nFig.1, the thin ﬁlm is magnetized in the yzplane along /hatwidezby the bias ﬁeld H B, with ﬁlm-\nnormal direction along /hatwidex. The CPW exerts both a longitudinal rf ﬁeld hrf\nzand a transverse\nrf ﬁeld hrf\nyof equal strength. First consider only the transverse ﬁeld hrf\ny. In this well es-\ntablished case, the LLG equation ˙m=−γm×Heff+αm×˙mis linearized and takes the\nform \n˙/tildewidermx\n˙/tildewidermy\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx\n/tildewidermy\n+\nγ/tildewiderhrf\ny\n0\n (1)\n, whereγis the gyromagnetic ratio, αis the Gilbert damping parameter, ωM≡γ4πMs, and\nωH≡γHz. Introducing ﬁrst order perturbation to mx,yunder additional longitudinal hrf\nz\nand neglecting the second order terms, we have\n\n˙/tildewidermx+˙/tildewidest∆mx\n˙/tildewidermy+˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidermx+/tildewidest∆mx\n/tildewidermy+/tildewidest∆my\n+\nγ/tildewiderhrf\nz/tildewidermy\n−γ/tildewiderhrf\nz/tildewidermx\n+\nγ/tildewiderhrf\ny\n0\n(2)\nSubtracting (1) from (2) and taking/tildewiderhrfy,z=Hrf\ny,ze−iωt,/tildewidemx,y= (Hrf\ny/Ms)e−iωt/tildewideχ⊥,/bardbl(ω), the\nequation for the perturbation terms is\n\n˙/tildewidest∆mx\n˙/tildewidest∆my\n=\n−α(ωH+ωM)−ωH\nωH+ωM−αωH\n\n/tildewidest∆mx\n/tildewidest∆my\n+Hrf\nzHrf\ny\nMse−i2ωt\nγ/tildewiderχ/bardbl(ω)\n−γ/tildewiderχ⊥(ω)\n(3)\nSinceχ⊥is one order of magnitude smaller than χ/bardbl, we neglect the term −γ/tildewiderχ⊥(ω).\nIn complete analogy to equation (1), the driving term could be viewed as an eﬀective\ntransverse ﬁeld of Hrf\nz(Hrf\ny/Ms)/tildewiderχ/bardbl(ω)e−i2ωt, and the solutions to equation (3) would be\n/tildewidest∆mx= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ⊥(2ω)e−i2ωt,/tildewidest∆my= (Hrf\nzHrf\ny/M2\ns)/tildewiderχ/bardbl(ω)/tildewiderχ/bardbl(2ω)e−i2ωt. We\ncan compare the power at frequency fand 2fnow that we have the expressions for\nboth the fundamental and second harmonic components of the pr ecessing M. The time-\naveraged power per unit volume could be calculated as /angbracketleftP/angbracketright= [/integraltext2π\nω\n0P(t)dt]/(2π/ω), P(t) =\n−∂U/∂t= 2M∂H/∂twhere only the transverse components of MandHcontribute\nto P(t). Using the expression for /angbracketleftP/angbracketright,MandH, we have Pω=ωH2\ny,rfχ(ω)′′\n/bardbland\nP2ω= 2ωH2\nz,rf(Hrf\ny/Ms)2|˜χ(ω)/bardbl|2χ(2ω)′′\n/bardbl, from which we conclude that under H Bfor FMR\nat frequency f=ω/(2π), we should see a power ratio\nP2ω/Pω= 2(Hrf\nz/Ms)2χ(ω)′′\n/bardblχ(2ω)′′\n/bardbl (4)\nWithMs= 844 Oe, α= 0.007 as measured by FMR for our Ni 81Fe1930 nm sample and\n2.25 Oe rf ﬁeld amplitude at input power of 1 W for the CPW, we have a ca lculated 2 f/f\n5power ratio of 1.72 ×10−5/W, which is in reasonable agreement with the experimental data\n3.70×10−5/W as shown in Fig.2(c). To compare this result with the ferrite exper iment in\nref.[1], we further add the factor representing the ratio of FMR ab sorption to the input rf\npower, which is 3 .9×10−2in our setup. This leads to an experimental upconversion ratio of\n1.44×10−6/W in ref.[1]’s deﬁnition (∆ P2ω/Pω\nin2), compared with 7 .1×10−10/W observed\nin Mg 70Mn8Fe22O (Ferramic R-1 ferrite).\nExamining Eq.(4), we noticethatthereshouldbetwo peaksintheﬁeld -swept 2femission\nspectrum: the ﬁrst coincides with the FMR but with a narrower linewid th due to the term\n|˜χ(ω)/bardbl|2, and the second positioned at the H Bfor the FMR with a 2 finput signal due to\nthe term χ(2ω)′′\n/bardbl. The second peak should have a much smaller amplitude. Due to the ﬁe ld\nlimit of our electromagnet, we could not reach the bias ﬁeld required f or FMR at 12.2 GHz\nunder this particular conﬁguration and continued to verify Eq.(4) a t a lower frequency of 2.0\nGHz. We carried out an identical experiment and analysis and observ ed an upconversion\neﬃciency of 0.39 ×10−3/W for the 4.0 GHz signal generation at 2.0 GHz input, again in\nreasonable agreement with the theoretical prediction 1.17 ×10−3/W. Fig.3 demonstrates the\ntypical line shape of the4 GHzspectrum, in which the input 2 GHzpowe r being +18.9 dBm.\nA second peak at the H Bfor 4 GHz FMR is clearly visible with a much smaller amplitude\nand larger linewidth than the ﬁrst peak, qualitatively consistent with Eq.(4). A theoretical\nline (dashed green) from equation (4) with ﬁxed damping parameter α= 0.007 is drawn to\ncompare with the experimental data. The observed second peak a t the 2fresonance H B\nshows a much lower amplitude than expected. We contribute this diﬀe rence to the possible\n2fcomponent in the rf source which causes the 2 fFMR absorption. The blue line shows\nthe adjusted theoretical line with consideration of this input signal impurity.\nSummary : We have demonstrated a highly eﬃcient frequency doubling eﬀect in thin-\nﬁlm Ni 81Fe19for input powers well below the Suhl instability threshold. An analysis of\nthe intrinsically nonlinear LLG equation interprets the observed phe nomena quantitatively.\nThe results explore new opportunities in the ﬁeld of rf signal manipula tion with CMOS\ncompatible thin ﬁlm structures.\nWe acknowledge Stephane Auﬀret for the Ni 81Fe19sample. We acknowledge support\nfrom the US Department of Energy grant DE-EE0002892 and Natio nal Science Foundation\n6FIG. 3. 4 GHz generation with input signal at 2 GHz, +18.9 dBm. A second peak at the bias ﬁeld\nfor 4 GHz FMR is clearly present; red dots: experimental data ; dashed green: theoretical; blue:\nadjusted theoretical with input rf impurity. See text for de tails.\nECCS-0925829.\nREFERENCES\n1W. P. Ayres, P. H. Vartanian, and J. L. Melchor, J. Appl. Phys. 27, 188 (1956)\n2N. Bloembergen and S. Wang, Phys. Rev. 93, 72 (1954)\n3H. Suhl, J. Phys. Chem. Solids. 1, 209 (1957)\n4J. D. Bierlein and P. M. Richards, Phys. Rev. B 1, 4342 (1970)\n5G. P. Ridrigue, J. Appl. Phys. 40, 929 (1969)\n6V. G. Harris, IEEE Trans. Magn. 48, 1075 (2012)\n7V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. B aither, G. Schmitz\nand S. O. Demokritov, Nature Mat. 11, 1028 (2012)\n8A. Berteaud and H. Pascard, J. Appl. Phys. 37, 2035 (1966)\n9T. Gerrits, P. Krivosik, M. L. Schneider, C. E. Patton, and T. J. Silv a, Phys. Rev. Lett.\n98, 207602 (2007)\n10H. M. Olson, P. Krivosik, K. Srinivasan, and C. E. Patton, J. Appl. Ph ys.102, 023904\n(2007)\n711M. Bao, A. Khitun, Y. Wu, J. Lee, K. L. Wang, and A. P. Jacob, Appl. Phys. Lett. 93,\n072509 (2008)\n12Y. Khivintsev, J. Marsh, V. Zagorodnii, I. Harward, J. Lovejoy, P . Krivosik, R. E. Camley,\nand Z. Celinski, Appl. Phys. Lett. 98, 042505 (2011)\n13J. Marsh, V. Zagorodnii, Z. Celinski, and R. E. Camley, Appl. Phys. Le tt.100, 102404\n(2012)\n14M. Yana, P. Vavassori, G. Leaf, F.Y. Fradin, and M. Grimsditch, J. M agn. Magn. Mater\n320, 1909 (2008)\n15V. E. Demidov, H. Ulrichs, S. Urazhdin, S. O. Demokritov, V. Besson ov, R. Gieniusz, and\nA. Maziewski, Appl. Phys. Lett. 99, 012505 (2011)\n16C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, and J. Q. Xia o, Appl. Phys.\nLett.99, 232506 (2011)\n17S. E. Bushnell, W. B. Nowak, S. A. Oliver, and C. Vittoria, Rev. Sci. In strum.63, 2021\n(1992)\n18A. G. Gurevich and G. A. Melkov, Magnetization Oscillation and Waves ( CRC, Boca\nRaton, 1996)\n8"    },    {        "title": "0711.4720v1.Realization_of_XNOR_and_NAND_spin_wave_logic_gates.pdf",        "content": "arXiv:0711.4720v1  [cond-mat.other]  29 Nov 2007Realization of XNOR and NAND spin-wave logic gates\nT. Schneider,∗A.A. Serga, B. Leven, and B. Hillebrands\nFachbereich Physik und Forschungsschwerpunkt MINAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\nR.L. Stamps and M.P. Kostylev\nSchool of Physics, University of Western Australia, Crawle y, WA 6009, Australia\nWe demonstrate the functionality of spin-wave logic XNOR an d NAND gates based on a Mach-\nZehnder typeinterferometer which has arms implemented as s ections of ferrite ﬁlm spin-wave waveg-\nuides. Logical input signals are applied to the gates by vary ing either the phase or the amplitude\nof the spin waves in the interferometer arms. This phase or am plitude variation is produced by\nOersted ﬁelds of dc current pulses through conductors place d on the surface of the magnetic ﬁlms.\nAlthough commonly used for data storage applications, there have been relatively few attempts to employ magnetic\nphenomenaforperforminglogicaloperations. Thesuggestedcon ceptsincludethe controlofdomainwallmovement[1],\nofmagnetoresistanceofindividualmagneticelements[2], andofam agnetostaticﬁeldofasetofmagneticnanoelements\n[3]. Yet another concept is using spin-wave interferometers. It wa s discussed theoretically in Refs. [4, 5, 6], but there\nwas only one experimental demonstration of spin wave logic gate fun ctionality [7], where an one-input NOT gate was\nimplemented in a interferometer-like geometry. In the present wor k we experimentally demonstrate the functionality\nof more complicated logic gates based on spin waves.\nThe fabricated prototype of a XNOR logic gate is a direct extension o f the NOT gate from Ref. [7] which was\nbased on a Mach-Zehnder interferometer. For its implementation t he reference interferometer arm of the NOT gate\nis replaced by an arm identical to the signal arm. Controlling phases a ccumulated by the spin waves in both arms\nallows one to perform the XNOR operation.\nDemonstrating the functionality of a NAND logic gate is a considerable step forward in the development of spin\nwave logic compared to the NOT and XNOR gates. Firstly because the NAND function belongs to a class of universal\nfunctions which means that combining NAND gates allows one to const ruct gates of other types. Secondly because for\nits implementation, we use here a new physical principle: direct contr ol of spin wave amplitudes in the interferometer\narms.\nFigure 1(b) showsthe principle setup of an exclusive not OR (XNOR, a lsocalled logicalequality) gate. It consistsof\ntwo arms of a spin-wave Mach-Zehnder interferometer implemente d as ferrite ﬁlm structures. Spin waves are inserted\nin both arms using microstrip antennas connected to a common micro wave pulse source, thus guaranteeing the same\nphase in both arms. The spin waves are phase-coherently detecte d using microstrip antenna detectors. The signals of\nboth arms are brought to interference electronically. The phase a ccumulated by the spin waves on their paths through\nthe two arms is controlled by applying dc currents I1andI2to the arms. Figure 1(a) shows phase inserted due to a\ncurrent in an interferometer arm. One sees a linear dependence of the accumulated phase on the current. One also\nsees that the phase characteristics in both arms are identical.\nThe currents I1andI2serve aslogicalinputs, where a logical zero is represented by I= 0 A and a logical one by the\ncurrentIπ, necessary to create a phase shift of π. The microwave pulses at the physical input of the Mach-Zehnder\ninterferometer represent rate pulses. The amplitude of the micro wave interference pattern at the interferometer\nphysical output serves as the logic output. Destructive interfer ence (i.e., zero intensity) represents a logical zero and\nconstructive interference (i.e., high intensity) represents a logica l one. It is assumed that both arms are identical and\nthus both spin waves reach the output with identical amplitude and p hase if no current is applied. If we now apply a\ncurrent of strength Iπto one of the conductors (i.e., logical one to one input, zero to the o ther) one of the spin waves\nis shifted in phase by πwhich will lead to destructive interference (logical zero). If the cu rrents in both conductors\nare identical (i.e., logical zero or one to both inputs) the interferen ce will remain constructive (logical one). This\nbehavior is summarized in the inset to Fig. 1(b). It resembles a XNOR g ate.\nThe physical mechanism underlying the control of the spin-wave ph ase by a dc current through a conductor can\nbe explained as follows. Due to the Oersted ﬁeld of the conductor th e dispersion curve is shifted along the frequency\naxis and thus the carrier wavenumber of the spin-wave pulse is chan ged. This results in a change of spin-wave phase.\nThe accumulated phase is linearly proportional to the current stre ngth. As shown in Ref. [7], in the case of wide\nconductors it is possible to shift the spin wave phase by πwithout introducing a noticeable decrease in the output\n∗Electronic address: tschneider@physik.uni-kl.de2\nspin wave amplitude due to reﬂection back from the region of induced ﬁeld inhomogeneity.\nTo simplify demonstration of functionality of the XNOR gate we used a n up-scaled prototype, compared to the\ncommon sizes of magnetic elements used in other logic schemes [2, 3]. T he physical principles underlying the device\nperformance would remain practically the same if the prototype is imp lemented with micrometer sizes. As a medium\nfor spin wave propagation we use two 6 /F1m thick and 1.5 mm wide yttrium iron garnet (YIG) spin-wave waveguide s\n(see Fig. 1). In YIG ﬁlms spin waves can travel over several tens o f millimeters due to its extremely low magnetic\ndamping. Conventional Permalloy ﬁlms would allow for spin wave propag ation over distances up to several tens of\nmicrometers and are well suited for miniature devices.\nAlso for the sake of simplicity the physical input and output of the Ma ch-Zehnder interferometer are implemented\nas microwave microstrip antennas. To transform microwave signals into spin-wave pulses in the YIG stripes and back\ninto microwave signals microwave transducers are utilized in the form of 50 /F1m wide microstrip antennas placed 8\nmm apart. (Note that the interferometer physical input and outp ut can be implemented as all-spin wave waveguides,\nas shown in Ref. [4]. This will make them compatible with a spin wave data b us, as recently suggested in Ref. [8].\nIn such a structure the microstrip transducers are not needed.) Wire loops (wire diameter is 0.5 mm) for spin wave\nphase control are placed between the input and the output microw ave transducers. The spin-wave pulses in both\ninterferometer arms are 20 ns in length. The microwave carrier fre quency of the pulses is 7.132 GHz and the applied\nbias magnetic ﬁeld is 1850 Oe. Control current pulses of 990 ns dura tion are applied to the conductors. Observe\nthat the length of the current pulses can be reduced to the length of the spin-wave pulses at the cost of an increased\ncurrent necessary to create the πshift due to the then shorter interaction time.\nFigure 1(c) demonstratesthe microwavesignal at the output of t he prototype gate for diﬀerent input conﬁgurations.\nThe expected behavior is clearly visible.\nIn our previous works [9, 10] we show that one can control the sp in wave amplitude by inserting a highly localized\nmagnetic ﬁeld inhomogeneity. The inhomogeneity can be created by a dc current through a narrow conductor placed\non the surface of a ferrite ﬁlm. A conductor with a width of 100 /F1m produces a highly localized Oersted ﬁeld.\nIndependently of the sign of this additional Oersted ﬁeld with respe ct to the applied static ﬁeld, a spin-wave pulse can\nbe reﬂected eﬃciently from such a strong inhomogeneity. In this wo rk we choose the current direction which produces\nan Oersted ﬁeld in the direction opposite to the applied ﬁeld. By applyin g a relatively small current it is possible to\nshift the dispersion curve so that the carrier frequency of the sp in wave pulse incident onto the inhomogeneity is no\nlonger inside of the frequency band of spin waves. Spin waves canno t propagate through this prohibited zone, they\ncan only tunnel. This introduces a strong back reﬂection of the incid ent pulse and one observes a strong change in\nthe intensity of the spin wave [9]. It is important to notice that one c an reduce the spin wave amplitude to nearly\nzero and thus use this setup as a spin-wave switch. The functionalit y of such a switch is experimentally demonstrated\nin Fig. 2(a).\nThe realization of a logic NOT AND (NAND) gate is demonstrated in Fig. 2 (b). The setup mainly consists of a\nMach-Zehnder interferometer, but this time the phase shifters in the arms are replaced by switches. Similar to the\nXNOR gate the logical output is implemented by the interference sign al, while the inputs are implemented by the\ncurrents. Logical zero is represented by I0= 0A, and logical one by the current IS, necessary to suppress the spin\nwave pulse transmission.\nExperimentally measured output interferometer pulses are shown in Figure 2(c). If a current is applied to only\none of the switches (logical one to one input, zero to the other) a m icrowave pulse of a large intensity is transmitted\n(logical one at the output). The same takes place if no current is ap plied to both arms (logical zero to both inputs).\nTo ensure that the intensity of the output microwavepulse is the sa me in this case as in the two other cases of a logical\none at the output an additional permanent phase shift of 2 π/3 is introduced in one of the interferometer arms. In this\nprototype we use an external coaxial microwave phase shifter. H owever a current controlled spin wave phase shifter\nsimilar to the one implemented in the XNOR gate prototype (Fig. 1(b)) may be easily integrated into the ferrite ﬁlm\nstructure instead. Any other possibility to create an additional ph ase shift (e.g., make the spin-wave propagation\npaths in the arms diﬀer by 2 π/3 by slightly increasing the length of one of the arms or using a thin per manent magnet\nto apply a small additional bias magnetic ﬁeld to one of the arms) is also possible. A current applied to both switches\n(logical one to both inputs) leads to a nearly complete suppression o f the output signal (logical zero). The described\nbehavior (summarized in the inset of Fig. 2(b)) is the one of a NAND ga te.\nIn conclusion, we experimentally demonstrated the functionality of universal NAND and a XNOR logical gate using\nspin-waves propagating in a magnetic ﬁlm interferometer based on Y IG. By changing the used spin-wave waveguides\n(e.g., to Permalloy) a downscaling of the presented devices down to m icrometer-scale dimensions should be possible\nand would open up a new approach to logic structures on the microme ter scale.3\nAcknowledgments\nSupport by the Deutsche Forschungsgemeinschaft (Graduierte nkolleg 792), the Australian Research Council, and\nthe European Community within the EU-project MAGLOG (FP6-5109 93) is gratefully acknowledged.\nThe work and results reported in this publication were obtained with r esearch funding from the European Commu-\nnity under the Sixth Framework Programme Contract Number 5109 93: MAGLOG. The views expressed are solely\nthose of the authors, and the other Contractors and/or the Eu ropean Community cannot be held liable for any use\nthat may be made of the information contained herein.\n[1] R.P. Cowburn and M.E. Welland, Science 287, 1466 (2000).\n[2] G. Reiss and D. Meyners, Appl. Phys. Lett. 88, 043505 (2006).\n[3] A. Imre, G. Csaba, L. Ji, A. Orlov, G.H. Bernstein, and W. P orod, Science 311, 205 (2006).\n[4] R. Hertel, Phys. Rev. Lett. 93, 257202 (2004).\n[5] S.V. Vasiliev, V.V. Kruglyak, M.L. Sokolovskii, and A.N . Kuchko, J. Appl. Phys. 101, 113919 (2007).\n[6] S. Kim, K. Lee, and S. Choi, 10th Joint Intermag-MMM Confe rence, Poster CU-10, Baltimore (2007).\n[7] M.P. Kostylev, A.A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Appl. Phys. Lett. 87, 153501 (2005).\n[8] A. Khitun and K.L. Wang, Superlattices and Microstructu res38, 184-200 (2005).\n[9] S.O. Demokritov, A.A. Serga, A. Andre, V.E. Demidov, M.P . Kostylev, B. Hillebrands, and A.N. Slavin, Phys. Rev. Lett .\n93, 047201 (2004).\n[10] M.P. Kostylev, A.A. Serga, T. Schneider, B. Leven, B. Hi llebrands, and R.L. Stamps, Phys. Rev. B 76, 184419 (2007).4\n(a)\n0 100 200 300 400 500 600 700 800 900 1000 11000.00.51.01.52.02.5Channel□1\nChannel□2\nI□[mA]HPhase□[Units□of ]/c112\n0 0□(0) 1□(I/c112)0 1□(I/c112) 0□(0)\n1 1□(I/c112) 1□(I/c112)1 0□(0) 0□(0)OutputInputs\nA (I1)□□□□□□B□(I2)\n0 0(0) 1(I/c112)0 1(I/c112) 0(0)\n1 1(I/c112) 1(I/c112)1 0(0) 0(0)OutputInputs\nA (I1)□□□□□□B□(I2)(b)\n350 400 450 500 350 400 450 500 350 400 450 500\nTime□[ns]350 400 450 500=0\n01 =1\n00 =0\n10 =1\n11\nIntensity□[a.u.](c)I1I2I\nYIG\nInput□microstripeOutput□microstripe\nSubstrate\nFIG. 1: XNOR gate. (a) Inserted phase versus current for the c urrent controlled spin wave phase shifters (CPS) used to\nconstruct the XNOR gate prototype. It is clearly visible tha t the phase shifts in both arms (channels) are identical. The inset\nshows the phase shifter geometry. (b) Spin-wave XNOR gate ge ometry. The currents I1andI2represent the logical inputs\n(0A corresponds to 0,Iπcorresponds to 1), the spin-wave interference signal represents the logica l output. Inset: Truth table\nfor an XNOR gate. (c) Gate output signals for input signals sh own in the diagrams.5\n350 400 450 500\nTime□[ns]350 400 450 500\nTime□[ns]I0=□0□mA IS=□1200□mA\nIntensity□[a.u.]\n350 400 450 500 350 400 450 500 350 400 450 500\nTime□[ns]350 400 450 500&1\n10 &0\n11 &1\n01 &0\n01\nIntensity□[a.u.]1 0□(0) 1□(IS)1 1□(IS) 0□(0)\n0 1□(IS) 1□(IS)1 0□(0) 0□(0)OutputInputs\nA (I1)□□□□B□(I2)\n1 0(0) 1(I )1 1(I ) 0(0)\n0 1(I ) 1(I )1 0(0) 0(0)OutputInputs\nA (I1)□□□□B□(I2)\n/c68/c102/c32/c61/c32/c50/c112/c47/c51I1I2I2(a)\n(b)\n(c)\nFIG. 2: NAND gate. (a) Demonstration of a spin-wave switch. L eft part: Output signal without applied current. Right part :\nOutput signal with applied current. Suppression of the outp ut pulse is clearly visible. (b) Geometry of a spin-wave NAND\ngate. The currents I1andI2represent the logical inputs (0A corresponds to 0,IScorresponds to 1); the spin-wave interference\nsignal represents the logical output. Inset: Truth table fo r a NAND gate. (c) Gate output signals for input signals as sho wn\nin the diagrams."    },    {        "title": "1903.09590v2.Spin_wave_coupling_to_electromagnetic_cavity_fields_in_dysposium_ferrite.pdf",        "content": "Spin-wave coupling to electromagnetic cavity \felds in dysposium ferrite\nM. Bia lek,1,\u0003A. Magrez,1and J.-Ph. Ansermet1\n1Institute of Physics, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), 1015 Lausanne, Switzerland\n(Dated: September 15, 2021)\nCoupling of spin-waves with electromagnetic cavity \feld is demonstrated in an antiferromagnet,\ndysprosium ferrite (DyFeO 3). By measuring transmission at 0.2{0.35 THz and sweeping sample\ntemperature, magnon-photon coupling signatures were found at crossings of spin-wave resonances\nwith Fabry-P\u0013 erot cavity modes formed in samples. The obtained spectra are explained in terms of\nclassical electrodynamics and a microscopic model.\nPACS numbers: 71.36.+c, 76.50.+g, 78.30.-j, 75.50.Ee\nCoupling of matter and electromagnetic radiation1is\na topic of great interest in solid state physics research\nbecause of their hybrid quantum nature.2In the THz\nrange, phonon-polaritons are a well-know example of a\nlight-matter coupling. Recently, polaritons in the THz\nregion were shown with intersubband transitions,3,4cy-\nclotron resonance5and plasmons6in two-dimensional\nelectron gases, as well as with intermolecular transitions\nin organic materials.7In these systems, a strong-coupling\nregime can be achieved when losses are smaller than the\nexchange rate between light and matter,8giving rise to\nthe vacuum Rabi splitting. Polaritons are composite\nparticles, which are studied in basic research on quan-\ntum optics,2and can be considered for use in quantum\ncomputing and quantum memories.9{13The coupling of\nelectromagnetic cavity-modes to magnons was researched\nintensively in ferromagnets at GHz frequencies,12,14,15\nmeeting with expectations of energy-e\u000ecient spintronic\ndevices.16Coupling of magnons with superconducting\nqubits was also investigated.17,18The Purcell enhance-\nment and the vacuum Rabi splitting were demonstrated\nin ferromagnetic materials.19{22\nIt is interesting to investigate magnon-photon coupling\nin antiferromagnetic materials1in view of their high-\nfrequency spin dynamics,23{27comparing to that of fer-\nromagnets. As this phenomenon is readily taken into ac-\ncount by classical electrodynamics, it was accounted for\nin the analysis of optical investigations of antiferromag-\nnetic materials, for instance in Ref.28{32. Some experi-\nmental reports focused on characterization of interaction\nof antiferromagnetic magnons with photons in FeF 233,34,\nNiO35, TmFeO 336and in ErFeO 3.37,38Most of these re-\nports use classical electrodynamics. A microscopic pic-\nture was developed in Ref. 37. Here, we confront the\nclassical electrodynamic model with predictions of a mi-\ncroscopic model by estimating, quantitatively from our\ndata, the strength of the interaction of antiferromagnetic\nspin-waves with electromagnetic cavity modes in dyspro-\nsium ferrite DyFeO 3(DFO).\nDysprosium ferrite is an orthogonally-distorted per-\novskite. It shows antiferromagnetic ordering below a N\u0013 eel\ntemperature of about 640 K39. Dzyaloshinskii-Moriya\nleads to a weak ferromagnetismIn DFO, the spin cant-\ning allows two antiferromagnetic resonance modes to beexcited: the quasi-ferromagnetic (qFMR) and the quasi-\nantiferromagnetic (qAFMR),28,29,39{41which are excited\nby the magnetic component of radiation.42\nOur measurements were performed above 400 K, where\nboth resonances, monotonously soften with rising tem-\nperature. We used polycrystaline samples for their\nisotropic properties, as large anisotropic properties in\ncrystals might hinder the coupling.43We have created\ndisk-shaped samples of 1.0 and 0.6 mm thicknesses, thus\nhaving di\u000berent cavity-mode spectra. A sample was\nplaced in a furnace, which allowed to control temperature\nup to 700 K. By using quasioptical methods, we measured\ntransmission with our continuous-wave THz spectrome-\nter based on frequency extenders to a vector network\nanalyzer (VNA).44This complex signal S21(f;T;H ) is a\nfunction of frequency f, temperature Tand magnetic\n\feldH. We report its power amplitude in dB units\nand its phase in degrees. In order to extract a signal\nrelated to magnetic resonances, we measured transmis-\nsion through a sample at di\u000berent temperatures T. We\nobtained temperature-di\u000berential spectra by subtracting\naveraged spectra measured at subsequent temperatures:\n@S21\n@T= \u0001T\u00001(S21(f;T+ \u0001T;0)\u0000S21(f;T; 0));(1)\nwith \u0001T= 1 K. We used this technique to measure\nspin-wave resonances in bismuth ferrite at high45and\nlow temperatures.43\nIn classical electrodynamics, the dispersion of electro-\nmagnetic radiation in a material with a resonance is mod-\ni\fed. This leads to creation of two polariton states split\nby a frequency representing coupling of the resonance\nwith the electromagnetic radiation.1In DFO, we model\nmagnetic susceptibility \u0016c(f;T;H ) using lorentzian dis-\ntributions\n\u0016c(f;T;H ) = 1 +MX\nm=1\u0001\u0016m(T;H)f2\nm(T;H)\nf2m(T;H)\u0000f2\u0000if\rm(T;H);\n(2)\nwhere, for the m-th magnetic resonance, fmis its fre-\nquency,\rmits width and \u0001 \u0016mis its input to the zero-\nfrequency magnetic susceptibility. In the case of DFO,\nM= 2, with m= 1 corresponding to the qFMR and\nm= 2 to the qAFMR.arXiv:1903.09590v2  [cond-mat.mes-hall]  12 Dec 20192\nFerrites have dielectric functions \u000fc(f;T), which, in our\nexperimental frequency range, i.e. far from phonons res-\nonant frequencies, can be approximated as linear around\na convenient point ( T0= 400 K,f0= 0:3 THz)\n\u000fc(f;T) =\u000f00+a(f\u0000f0) +b(T\u0000T0): (3)\nFor our polycrystalline samples, we assume an\ne\u000bective46,47magnetic susceptibility asp\u0016=pp\u0016c+(1\u0000\np) and e\u000bective dielectric function asp\u000f=pp\u000fc+(1\u0000p),\nwhere the factor p= 0:64 is a volume fraction of mate-\nrial to air in our pelletized samples.43This value is the\nmaximum density of random-packed hard spheres.48This\nassumption allows us to obtain values of \u000fcand\u0016cthat\nare comparable with literature values for single crystal\nsamples.29,49The complex wave vector is k= 2\u0019fp\u000f\u0016=c,\nwherecis the speed of light in vacuum. Transmission of\nelectric \feld t(f;T;H ) though a slab of a thickness dand\nin\fnite lateral dimensions is:50\nt(f;T;H ) =(1\u0000r2)eikd\n1\u0000r2ei2kd; (4)\nwherer2= (p\u000f\u0000p\u0016)2=(p\u000f+p\u0016)2is the square of the re-\n\rection coe\u000ecient at the vacuum-material interface. For\nfrequencies far away from resonances, Eq. 4 implies an\ninterference pattern, related to subsequent cavity modes\nof a slab, called Fabry-P\u0013 erot modes. At 300 GHz, in\nour polycrystalline DFO samples with the refractive in-\ndexp\u000f\u0016\u00193:6, the wavelength is about 300 \u0016m. With\nrising temperature,p\u000f\u0016increases, which shortens the\nperiod of the interference pattern. The magnetic sus-\nceptibility takes into account the e\u000bect of the magnetic\nresonance. When its frequency is close to a mode of a\ncavity, both the magnon and the cavity lines are altered\nby the matter-photon interaction. Since, the magnetic\nresonance frequencies have a much stronger temperature\ndependence than that of the interference pattern, they\ncross several interference pattern minima as temperature\nrises. We calculated\n\r\r\r\r@S21\n@T\r\r\r\r= \u0001T\u0000120log10\r\r\r\rt(f;T+ \u0001T;0)\nt(f;T; 0)\r\r\r\r(5)\nto \ft amplitude of temperature-di\u000berential spectra,\nwhere \u0001T= 1 K is the temperature step. We calculated\nphase of temperature-di\u000berential spectra using\narg(@S21=@T) = arg(t(f;T+ \u0001T;0)\u0000t(f;T; 0)):(6)\nThe experimental temperature-di\u000berential signal am-\nplitude for the 1.0-mm-thick DFO sample is shown in Fig.\n1a. These data show clearly that the resonant lines are\ndistorted when they cross the sequence of sample-cavity\nmodes. The amplitude and the widths of the resonances\nare altered because of their interaction with the electro-\nmagnetic standing waves. This is accounted for by Eq.\n5 as shown in Fig. 1c. Thus, we \fnd that this model\nreproduces most of the important features of Fig. 1a.\nThe \ftting parameters are parameters of the simpli\feddielectric function (Eq. 3) and of the magnetic suscep-\ntibility (Eq. 2). We assumed that resonance frequencies\nhave a temperature dependence described by a power law\nf\u0003\nm(1\u0000T=TN)\fm, applicable when approaching the N\u0013 eel\ntemperature TN,51wheref\u0003\nmhas a unit of frequency and\n\fm\u00191\n3in the case of DFO. To improve the quality of\nour \fts, we assumed that magnetic resonances have linear\ndependences of their widths and amplitudes on temper-\nature. These formulas and values of \ft parameters are\ngiven in the supplementary materials.\nWe measured the temperature-di\u000berential phase of\ntransmitted electric \feld, as presented in Fig. 1b. Despite\nhigh noise, features predicted in Fig. 1d are observed in\nthe experiment. The phase reveals clearly interactions\nbetween electromagnetic waves and magnetization dy-\nnamics. In Fig. 2a, a spectrum obtained at a temper-\nature between crossings points shows that the qAFMR\ncan be accounted for with a harmonic model. In Fig. 2b,\nthe spectrum obtained at the temperature of the crossing\nwith thel= 6 cavity mode shows a structure that can-\nnot be explained using a single oscillator. It is due to two\npolariton states which are not well-separated, i.e. they\nare in a weak coupling regime.2We used this phase pre-\ndiction to estimate the cavity mode-magnetic resonance\ncoupling strength. Thus, we superimposed on the Fig.\n1d predictions of the harmonic coupling model1,12,52\nf\u0006=1\n2\u0010\nf(l)+fm\u0006q\n(f(l)\u0000fm)2+ 4\u00142f(l)\u0011\n;(7)\nwheref\u0006indicates upper and lower polariton frequencies,\nf(l)is thel-th cavity mode frequency, fmwithm= 1;2\nare the resonances frequencies. The coupling strength is\ngiven by12\n\u0014=gs\u0016B\n2hr\n\u00160h\n2p\u001a; (8)\nwheregs= 2,\u001ais density of resonators and p= 0:64 is\nthe mass \flling factor of our polycrystalline sample.43,48\nEquation 8 assumes that a magnon is coupled to a single\nelectromagnetic cavity mode with a coupling strength53\ngs\u0016BB0=2h, whereB0=p\n\u00160hf(l)=2V(l)is the magnetic\ncomponent of vacuum \ructuations.54Amplitude of these\nis only\u00195\u000110\u000011T for an electromagnetic mode at\n0.3 THz taking the volume V(l)\u001916\u001910\u00009m3of the\n1-mm-thick sample. This is about 3 orders of magni-\ntude smaller than the amplitude of the THz \feld in our\nexperiment. However, in an ensemble of Nresonators\nthat collectively interact with a cavity mode, coupling\nis increased by a factorp\nN.12,37,52,53,55Thus, the col-\nlective coupling strength \u0014depends only on oscillators\ndensity\u001a=N=Vcand the ratio pof a crystalline vol-\numeVcto a cavity volume V(l)12,37. We take\u001a=1\n3\u001aFe,\nwhere\u001aFe= 1:76\u00021028m\u00003is density of iron atoms in\nDFO.56This factor re\rects the fact that in a polycrys-\ntalline material, on average, only 1 =3 of magnetic dipoles\nare excited by linear polarized electromagnetic wave. Un-\nder this assumption \u0014= 1:78\u0002104Hz1=2, which results3\n(c)\n 400  450  500  550  600\ntemperature T [K] 200 220 240 260 280 300 320 340 frequency f [GHz]\n-4-3-2-1 0 1 2 3 4\nabs(∂S 21/∂T) [10-2dB/K](a)\n 200 220 240 260 280 300 320 340 frequency f [GHz]\n-4-3-2-1 0 1 2 3 4\nabs(∂S 21/∂T) [10-2dB/K]\n(d)\n 400  450  500  550  600\ntemperature T [K] 200 220 240 260 280 300 320 340 frequency f [GHz]\n-3-2-1 0 1\narg(∂S 21/∂T) [10-1deg/K](b)\n 200 220 240 260 280 300 320 340 frequency f [GHz]\n-3-2-1 0 1\narg(∂S 21/∂T) [10-1deg/K]\nFIG. 1. Experimental temperature-di\u000berential spectra for the 1.0 mm-thick DFO sample: (a) amplitude, (b) median-shifted\nphase with dashed black lines showing coupled modes. Fit of the model to the amplitude data: (c) magnitude, (d) phase. In\nthe segment (d), green dashed lines show uncoupled spin-wave and cavity modes and purple lines show coupled modes.\n-2-1 0 1 2\n 200  240  280  320(a)\nl=7 l=8qAFMRarg(∂S 21/∂T) [10-1deg/K]\nfrequency f [GHz]T = 610.0 K\nl=6\n 200  240  280  320(b)\nl=7 l=8qAFMR\nf+f-\nfrequency f [GHz]T = 618.0 K\nl=6\nFIG. 2. Examples of phase spectra extracted from Fig. 1b.\n(a) spectrum at a temperature not showing light-matter in-\nteraction, (b) temperature close to a crossing point. Arrows\nmark positions of cavity modes, qAFMR and upper and lower\npolaritons. The violet lines show \fts obtained using data in\nthe entire temperature range (Fig. 1d), thus better re\recting\naverage properties of the sample.\nin a splitting 2 \u0014pf(l)\u001919:5 GHz with a cavity mode of\nf(l)= 0:3 THz.\nUsing Eq. 7, we calculated modes undergoing subse-quent interactions with the same coupling strength \u0014.\nWe determined f(l)from the condition c(l\u00001=2) =\n2<(p\u000f\u0016)f(l)dfor minimum of arg( @S21=@T). In the\ncase of 1-mm-thick sample, the lowest visible mode at\n\u0019230 GHz has l= 6. Resonant modes frequencies were\nobtained from the same \ft to the experimental @S21=@T\nmagnitude. For the 1.0-mm-thick sample, the tempera-\nture dependence of both coupled and uncoupled modes\nis presented in Fig. 1d.\nIn summary, we have observed coupling of spin-waves\nwith electromagnetic \felds in high-temperature anti-\nferromagnet DyFeO 3. 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Eibsch utz, Acta Crystallographica 19, 337 (1965)."    },    {        "title": "1102.4234v2.Contributions_of_Al_and_Ni_segregation_to_the_interfacial_cohesion_of_Cu_rich_precipitates_in_ferritic_steels.pdf",        "content": "arXiv:1102.4234v2  [cond-mat.mtrl-sci]  1 Apr 2011Manuscript\nContributions of Al and Ni segregation to the interfacial co hesion\nof Cu-rich precipitates in ferritic steels\nYao-Ping Xie and Shi-Jin Zhao∗\nInstitute of Materials Science, School of Materials Scienc e and Engineering,\nShanghai University, Shanghai, 200444, China\n(Dated: March 31, 2021)\nAbstract\nWe characterise the inﬂuence of the segregation behaviours of two typical alloying elements,\naluminium and nickel, on the interfacial cohesive properti es of copper-rich precipitates in ferritic\nsteels, with a view towards understanding steel embrittlem ent. The ﬁrst-principles method is used\nto compute the energetic and bonding properties of aluminiu m and nickel at the interfaces of the\nprecipitates and corresponding fracture surfaces. Our res ults show the segregation of aluminium\nand nickel at interfaces of precipitates are both energetic ally favourable. We ﬁnd that the inter-\nfacial cohesion of copper precipitates is enhanced by alumi nium segregation but reduced by nickel\nsegregation. Opposite roles can be attributed to the diﬀeren t symmetrical features of the valence\nstates for aluminium and nickel. The nickel-induced interf acial embrittlement of copper-rich pre-\ncipitates increase the ductile-brittle transition temper ature (DBTT) of ferritic steels and provides\nan explanation of many experimental phenomena, such as the f act that the shifts of DBTT of\nreactor pressure vessel steels depend the copper and nickel content.\nPACS numbers: 81.40.Np, 68.35.Dv, 64.75.Jk, 81.40.Cd\n∗Electronic address: shijin.zhao@shu.edu.cn\n1I. INTRODUCTION\nIt has long been known that the copper content in steels leads to pr ecipitation hardening.\nCopper is an element commonly occurring in steels either as an intentio nally added alloying\nspecies or as an impurity. Nanoscale copper-rich precipitates are u tilised to provide sub-\nstantial precipitation hardening for high-strength low-alloy steels , which possess excellent\nimpact toughness, corrosion resistance, and welding properties [1 –3]. In contrast, copper-\nrich precipitates induce hardening and embrittlement eﬀects in reac tor pressure vessel steels\n(RPV) after neutron irradiation [4–6], thereby limiting the operation al life of nuclear power\nplants. Therefore, understanding the properties of copper-ric h precipitates is desirable.\nManyinvestigationshaveprovidedinsightintothehardeningmechan ismthatresultsfrom\ncopper-richprecipitationinferriticsteels. Molecular dynamics simula tionhassuggested that\nthe major source of precipitation hardening is the dislocation core- precipitate interaction\n[7–9]. Dislocation core-precipitate interactions tend to induce the lo ss of screw dislocation\nslip systems and the transformation of the copper phase for large r body center cubic (BCC)\ncopper-rich precipitates (d >3.3 nm) while inducing polarised-to-nonpolarised transitions of\nscrew dislocation core structures in precipitates for very small BC C copper-rich precipitates\n(1.5 nm <d<3.3 nm) [10, 11]. In addition to the diameter of precipitates, the temp er-\nature and dislocation line characteristics are important to the disloc ation core-precipitate\ninteraction [12]. Experimental evidence relating to the dislocation co re-precipitate inter-\naction has been observed using transmission electron microscope ( TEM) experiments. In\nsitu TEM demonstrates dislocations pinned and curved with obtuse b ow-out angles by BCC\ncopper-rich precipitates with d ∼2 nm [13, 14]. TEM observations also demonstrate that\nthe transformation of copper phase and the appearance of disloc ation loops are induced by\nprecipitates with d ∼4 nm [15]. The bow-out angle of the dislocation can further be used\nto estimate macroscopic hardening [15].\nMany experiments have speciﬁcally studied the copper-rich precipit ation embrittlement\neﬀect on RPV steels and model alloys. Positron annihilation experimen ts suggest strongly\nthat copper-rich precipitates are responsible for irradiation-indu ced embrittlement [16]. The\ninﬂuence of nickel content on the embrittlement of RPV steels has a lso been found to be\nimportant[17–22], and the nickel and copper content in steels have a synergistic eﬀect on\nthe embrittlement tendency [23–26]. It has been reported that RP V steels with high nickel\n2content have a higher ductile-brittle transition temperature (DBT T) shift than low-nickel\nsteels with same copper content irradiated using same neutron ﬂue nce [23]. However, the\nshifts in the DBTT are small for irradiated low-copper, high-nickel R PV steels that do\nnot contain copper-rich precipitates [24]. Furthermore, a param etric study of model alloys\nafter neutron irradiation (at a neutron ﬂuence of 72 ×1018m−2) showed that DBTT shifts\nincrease with nickel content and that the shift is visible from a thres hold copper content of\napproximately 0.08 at.% [25, 26]. These results demonstrate clear ev idence for a relation\nbetween embrittlement and copper and nickel contents. The inﬂue nces of nickel on copper-\nrich precipitates have therefore attracted signiﬁcant attention .\nMicrostructure experiments show that nickel can occur in the cop per precipitates. Atom\nprobe experiments on thermally aged model alloys demonstrate tha t the nickel is located in\nthe core region of the BCC copper-rich precipitates during the initia l growth stage and is\nrejected from the core to the interfacial region during growth an d coarsening [27]. Obser-\nvations on neutron-irradiated RPV steels demonstrate a higher nic kel concentration in the\ncopper precipitates than in thermally aged model alloys [27, 28]. Rec ently, a series of atom\nprobe experiments have revealed the growing and coarsening beha viour of copper-rich pre-\ncipitates in concentrated multicomponent alloys with high strength [2 9–35]. The nickel and\nother alloying elements have been observed to segregate at the int erface of the precipitates.\nMoreover, phase-ﬁeld [36] andLanger-Schwartz [37] simulations of copper precipitates nucle-\nation also indicated nickel segregation at the interface of copper p recipitates during growth.\nFirst-principles calculation is very eﬃcient in predicting the embrittlem ent potential and\naids in understanding the mechanism of the impurity eﬀect based on e lectronic structure\n[38–43]. These ﬁndings motivated us to perform a careful, ﬁrst-pr inciples investigation of\nthe contribution of segregation, especially of nickel, on the interfa cial cohesive property of\nprecipitates to better understand the eﬀects of these precipita tes on embrittlement.\nIn this paper, we study aluminium and nickel as typical alloying element s to examine the\neﬀect of segregation at the interface of copper-rich precipitate s towards steel embrittlement.\nAfter calculating the segregation energies, we characterise the e ﬀect of aluminium and nickel\non the interfacial cohesion of copper-rich precipitates and attem pt to explain their bonding\nproperties in terms of their electronic structures. Finally, we discu ss the roles of aluminium\nand nickel in the embrittlement of ferritic steels. Our results sugge st that the nickel-induced\ninterfacial embrittlement process increases the DBBT of ferritic s teels.\n3II. METHODS AND MODELS\nOurﬁrst-principles calculations arebased ondensity functional th eory(DFT)[44,45]and\nperformed using the Vienna Ab-initio Simulation Package (VASP) [46] w ith a plane wave\nbasis set [47, 48]. The electron exchange and correlation is describe d within the generalised\ngradient approximation (GGA) [49, 50], and the interaction between ions and electrons is\ndescribed using the projector augmented wave method (PAW) [51 ]. The PAW potentials we\nchose are treated by considering Fe3d4s, Cu3d4s, Al3s3p and Ni3d 4s as valence states. All\ncalculations include spin polarisation. The structural relaxations of the ions are calculated\nby conjugate-gradient (CG) algorithm.\nWe model the coherent (001) interfaces between copper-rich pr ecipitates and the ferritic\nmatrix by designing (2 ×2×10) multi-layered supercells composed of BCC copper and\nBCC iron (Fig. 1). The lattice constants of the supercells are set us ing the theoretical value\nfor BCC iron. Our BCC iron value, 2.83 ˚A, is reasonably consistent with the experimental\nvalue, 2.86 ˚A [52]. These multilayer structures have a distance of 10 atomic layer s between\neach interface, and this is determined by considering the balance of avoidance of the inter-\nface interaction and computational expense. The three structu res illustrated in Figs. 1a-c\ncorrespond to the interfaces of precipitates with copper concen trations of 100, 75, and 50\nat.%, respectively. The k-points sample for these supercells are Mo nkhorst-Pack grids (6 ×\n6×1).\nWe create initial conﬁgurations of supercells modelling the alloying elem ent (M = Al,\nNi) in diﬀerent sites by substituting M for the iron or copper atoms fr om (2×2×10)\nsupercells. We substitute M for the iron atom at site 1, to simulate th e system of M in the\nbulk of the ferritic matrix. The distance from site 1 to the interface is ﬁve atomic layers,\nsuﬃcient to avoid the interaction between M and the interface. Site s 2 and 3 are used\nto simulate M segregation at the interface toward the matrix and to ward the precipitated\nphase, respectively. Site 4 is used to simulate M in the precipitated ph ase.\nThe formation energy is one of quantities typically used to describe t he segregation be-\nhaviour. The formation energy (EM) of M in the crystal is deﬁned as:\nEM=EM+CR−ECR−(EM−EA) (1)\nwhere E M+CRandECRare the total energies of the supercell with a substitution defect\n4M and defect-free supercell, while E Mand E Aare the energies per atom of equilibrium\npure-element reference states. The formation energy is depend ent on the reference states,\nwhich induce arbitrariness in the result [53]. It is usually diﬃcult to choo se the correct\nform of reference states when one calculates the formation ener gy to predict the segregation\nbehaviour.\nHere, a more eﬃcient quantity, the segregation energy, is used to predict the segregation\nbehaviour. The segregation energy (EM\nX) of M at site X can be written as:\n△EM\nX=Etot\nM−X−Etot\nM−matrix (2)\nwhere Etot\nM−Xand Etot\nM−matrixare the total energies of the supercells for M at site X in\nprecipitated phase and that for M in the matrix, respectively. A neg ative segregation energy\nindicatesthatMcantransferfromthematrixtositeX,whereasap ositivesegregationenergy\nindicates that M prefers to dissolve within the matrix. The segregat ion energy reﬂects the\ncompetitive capacity of trapping M between the matrix and the prec ipitated phase. This\nstrategy has previously been used to predict partition behaviours between cementite and\nferrite [53], and the method has been veriﬁed to be appropriate.\nThe embrittlement property of the interface can be obtained from the Griﬃth work sepa-\nratingtheinterface. BasedontheRice-Wangmode[54], theGriﬃthw orkisalinear function\nof the diﬀerence in segregation energy for the alloying element M at t he interface ( △EM\nI)\nand that at the fracture free surface ( △EM\nF),△EM\nI-△EM\nF. An alloying element M with\npositive△EM\nI-△EM\nFwill reduce the cohesion of the interface and induce an embrittlemen t\npotency, or vice versa. For the embrittlement property, the seg regation energies ( △EM\nF) at\nthe fracture free surface are needed. We construct an isolated fracture free (001) ferritic\nmatrix by subtracting the copper-rich phase from (2 ×2×10) supercells representing the\ninterface to calculate △EM\nF(Fig. 2).\nIn addition to energetic properties, we also analyse electronic stru ctures to provide in-\nsights into the bonding properties. We focus on the charge density diﬀerences within the\nregion of the interface and the fracture free surface as shown in Fig. 2. The charge density\ndiﬀerences(Fig. 3) are obtained by subtracting the superimposed charge density from the\nself-consistent charge density of the relaxed structure. Fig. 3 s hows the charge accumulation\nand depletion, which indicate the interatomic interactions.\n5III. RESULTS AND DISCUSSIONS\nA. Segregation energy\nBecause the composition of copper-rich precipitates in ferritic ste els has a wide range,\nwe present the segregation behaviour properties at the interfac es of copper-rich precipitates\nwith diﬀerent compositions. We have computed the interfaces of BC C precipitates with 100,\n75, and 50 at.% copper. The segregation energies of aluminium and nic kel at the interfaces\nof precipitated phases with diﬀerent compositions are listed in Tables 1 and 2. Apart from\nthe total segregation energy ( △EM\nX), we also present the chemical and mechanical contri-\nbutions to the segregation energy. The chemical contribution to s egregation is represented\nas△Echem,M\nX, which is calculated based on the unrelaxed structure. The mechan ical contri-\nbution to segregation is represented as △Emech,M\nX, i.e., the diﬀerence in segregation energy\nbetween the unrelaxed and relaxed structures. The relaxation en ergies of the supercell,\n△Esc, are also used when estimating the lattice distortion.\n1. Aluminium segregation\nThe segregation energies (see △EAl\nXin Table 1) of aluminium at the interfaces of the\nprecipitated phases are predicted to be negative, indicating that t he presence of aluminium\natoms at the interfaces is energetically favourable compared to th at in the ferritic matrix.\nMoreover, the segregation energies of aluminium at the interfaces are lower than those in\nthe core regions of the precipitates, indicating that the presence of aluminium atoms at\nthe interface is more favourable than that in the core region. Thes e results indicate that\nthe interface between the matrix and the precipitated phase can t rap aluminium atoms,\nconsistent with three-dimensional atom probe (3DAP) experiment s [29–31, 33].\nThe interfacial segregation energy at the interface for precipita tes with 100 at.% cop-\nper is larger than that for the precipitated phase with 50 at.% coppe r by 1.56 eV. This\nresult indicates that interfacial segregation is strongly dependen t on the composition of the\nprecipitated phase and increases with its copper concentration. T he strong dependence of\ninterfacial segregation on the composition of precipitates plays an important role in experi-\nmental phenomena in which the concentration of aluminium at the inte rface of precipitates\nincreases in ferritic steels during thermal treatment processes [2 9, 30, 33].\n62. Nickel segregation\nWe ﬁnd that the nickel segregation behaviour is similar to that of alum inium. The\nsegregation energies (see △ENi\nXin Table 2) of nickel in the interfaces are negative, indicating\nthat the presence of nickel atoms in the interfaces is energetically favourable compared to\nthat in the ferritic matrix. However, the most favourable sites dep end on the composition of\nthe precipitated phase. The most favourable sites are in the core r egion for the precipitated\nphase with 50 at.% copper, but at the interface for the precipitate d phase with 100 and 75\nat.% copper. These results indicate that nickel atoms can partition at the precipitates with\n50 at.% copper and segregate into the interfaces of the precipitat ed phase with 100 and 75\nat.% copper in a ferritic matrix. Therefore, nickel will segregate int o the core region of the\nprecipitates at the initial formation stage and be pushed away from the core towards the\ninterface of the precipitates for the following growth stage. This is consistent with phase\nﬁeld [36] and Langer-Schwartz [37] simulations. These phenomena w ere also observed for\nthe copper precipitates in RPV steels using 3DAP [27, 28].\nThe segregation energy of nickel at the interface also depends on the composition of the\nprecipitated phase. The segregation energy at the interface of t he precipitated phase with\n100 at.% copper is larger than that of 50 at.% copper by 0.22 eV, a diﬀe rence much lower\nthan that of aluminium. This result indicates that the nickel concent ration at the interface\nof copper-rich precipitates increases more slowly than that of alum inium with increasing\ncopper concentration during the growth of precipitates. This tre nd has previously been\nobserved in 3DAP experiments [55].\n3. Chemical and mechanic contributions\nAs shown in Tables 1 and 2, the values of △Echem\nXapproach those of △EX, so long as the\n△Emech\nXis very small. These values indicate that chemical energy is the main co ntributor\nto the segregation behaviour of aluminium and nickel. △Escand△EXare of comparable\nmagnitude, implying thattherelaxationenergyperatomismuchsmalle r thantheinterfacial\nsegregation energy. The average relaxation energies per copper atom for precipitated phases\nwith 100, 75 and 50 at.% copper are 0.012 eV, 0.027 eV, and 0.013 eV, w hereas the smallest\nsegregation energies of aluminium and nickel are 0.15 eV and 0.20 eV, r espectively. These\n7values reﬂect the fact that the energy gains of the interfacial se gregation of aluminium and\nnickel are much larger than that of lattice distortion.\nIt is necessary to discuss the factors aﬀecting the relaxation ene rgy of lattice distortion.\nThe relaxation energy is mainly determined by mechanical stability and the mixing eﬀects of\nthe BCC FeCu metastable alloy. The mechanical stability and mixing eﬀe ct both decrease\nwith copper concentration (at copper concentrations >50 at.%) [56, 57]. Higher mechanical\nstability reduces the relaxation energy, whereas larger mixing eﬀec ts enhance the relaxation\nenergy. Therefore, the relaxation energy of the precipitated ph ase with 75 at.% copper be-\ncome the largest among the three structures studied due to the c ompromise formed between\nthe eﬀects of mechanical stability and mixing.\nB. The Griﬃth work is inﬂuenced by segregation\nWe take the interface between the pure copper and ferritic matrix as a typical mode to\nanalyse the Griﬃth work. The Griﬃth work is the energy separating a n interface against\nthe atomic cohesion. The inﬂuence of alloying element segregation on the Griﬃth work can\nbe estimated by the value of △EM\nI-△EM\nF. We have calculated the segregation energies\n(△EM\nI) at the interface of precipitates in ferritic steels previously. Now, we calculate the\nsegregation energies ( △EM\nF) of aluminium and nickel at the fracture free surfaces, and the\nresults are -0.78 and -0.51 eV, respectively. The value of △EM\nI-△EM\nFfor aluminium and\nnickel at the interface between pure copper and the ferritic matr ix are -0.93 eV and 0.07 eV,\nrespectively. As a result, the segregation of aluminium will enhance t he interfacial cohesion,\nwhereas the segregation of nickel will reduce the interfacial cohe sion.\nFirst, we discuss the electronic structures of aluminium and nickel s egregation at the\ninterface. The charge diﬀerences of aluminium and nickel at the inte rface are presented in\nthe left-hand column of Fig. 3, showing that the charge accumulate s in the interval region\nbetween atoms and is depleted in the inner atomic shells. The charge d epletion region for\naluminium atoms is larger than that of iron and copper atoms, becaus e the 3s3p electrons\nof aluminium are more delocalised than the 3s3d electrons of iron and c opper atoms. The\n3p electrons of aluminium ﬁll into the degeneration state of p x, py, pz, therefore the charge\ndepletion region for the aluminium atom has a higher symmetry patter n. In contrast, the\ncharge depletion region for the nickel atom is similar to that of iron an d copper atoms,\n8because the valence states of nickel also include 3s3d. Because th e delocalised electrons can\nbe aﬀected more by surrounding atoms, the segregation energy o f aluminium is larger than\nthat of nickel and more sensitive to the composition of the precipita ted phase.\nWenowattempttounderstandtheeﬀect ofsegregationonGriﬃth workbycomparingthe\nchemical bonding in the fracture free surface to that in the interf ace. The charge diﬀerences\nof aluminium and nickel at the fracture free surface are presente d in the right-hand column\nof Fig. 3. Because the geometrical symmetry is broken for the fre e surface, the orbital\npzof aluminium state hybridises with the d z2of iron and becomes lower in energy. The\nonly one p-electron of aluminium ﬁlls into p z, and the p xyis left unoccupied. This results\nstronger vertical bonding (Fe2-Al) and weaker lateral chemical b onding (Fe1-Al and Fe3-Al)\nof the free surface compared to that of the interface. The weak ening eﬀect of aluminium\non lateral bonding contributes to a lower segregation energy at th e free surface than that\nat the interface (by 0.93 eV). Therefore, △EAl\nI-△EAl\nFfor the segregation of aluminium is\nnegative.\nThe alteration of the chemical bonding of nickel is totally diﬀerent to that of aluminium.\nThe spatial distribution of nickel at the fracture free surface is s imilar to that at the interface\ndue to the d electron. The charge accumulations in the interval reg ionof Fe1-Ni, Fe2-Ni, and\nFe3-Ni for the free surface are all greater than that for the int erface, indicating a stronger\nchemicalbondingbetweentheseatomsatfreesurface. Theenha ncedchemical bondingarises\nfrom the contraction of bond lengths at the free surface. Becau se the chemical bonding at\nthe free surface is stronger than at the interface, the segrega tion energy at the free surface\nis larger than at the interface (by 0.07 eV). △ENi\nI-△ENi\nFfor the segregation of nickel is\nconsequently positive.\nC. The embrittlement trend\nThe inﬂuence of alloying element M on an interfacial cohesive propert y is determined\nby both△EM\nIand△EM\nI-△EM\nF. We plot △EM\nIand△EM\nI-△EM\nFfor precipitated phases\nwith diﬀerent compositions in Fig. 4, showing that the values of △EAl\nIand△EAl\nI-△EAl\nF\nstrongly depend on the composition of the precipitated phase. The values of △EAl\nI-△EAl\nF\nare positive for precipitated phases of 50 ∼65 at.% copper and negative for precipitated\nphases of 65 ∼100 at.% copper. Obviously, there is a wide range of compositions for the\n9precipitates whose interfacial cohesion are enhanced by aluminium s egregation. The values\nof△EAl\nIfor these precipitates are all larger than those of the precipitate s whose interfacial\ncohesion is reduced by aluminium segregation. Therefore, aluminium s egregation plays a\nprominent role in enhancing interfacial cohesion for copper-rich pr ecipitates in ferritic steels.\nIn contrast, the values of △ENi\nIand△ENi\nI-△ENi\nFweakly depend on the composition of\nthe precipitated phase. The value of △ENi\nIvaries weakly with the composition of the precip-\nitated phase. The values of △ENi\nI-△ENi\nFare all negative. These results indicate that nickel\nsegregation can reduce the interfacial cohesion of copper-rich p recipitated phases of any\ncomposition. Undoubtedly, nickel segregation plays a constant ro le in reducing interfacial\ncohesion for copper-rich precipitates in ferritic steels.\nBCC copper-rich precipitates play important roles in dislocation pining and misﬁt growth\nin ferritic steels. Dislocation pinning will strengthen the ferritic matr ix, whereas the misﬁt\ngrowth will improve the ductility. The inﬂuence of the misﬁts produce d by copper precip-\nitates on ductile-brittle transformation has been proven to be con siderable [58, 59]. The\ninterfacial cohesive property and the amounts of precipitates pr esent contribute importantly\nto ductile-brittle transformation. The ductile-brittle transition de pend the competition be-\ntween fracture stress and ﬂow stress. Flow stress increases wit h decreasing temperature.\nWhen the ﬂow stress is larger than the fracture stress at lower te mperatures, the ferritic\nmatrix is brittle; and when the ﬂow stress is smaller than the fractur e stress at higher tem-\nperatures, the ferritic matrix is ductile. Therefore, the DBTT can be altered by fracture\nstress, which is determined by the interfacial cohesion of precipita tes. We can now predict\nthat the segregation of aluminium can lower the DBTT due to the enha ncing fracture stress\nof copper precipitates, whereas the segregation of nickel can inc rease the DBTT due to the\nreducing fracture stress of copper precipitates.\nThe nickel-induced interfacial embrittlement of copper-rich precip itates explains the ob-\nservation that the DBTT of low-carbon, copper-precipitation-st rengthened steels increase\nwith nickel and copper content [58]. This eﬀect also accounts for t he observation that the\nshifts of DBTT in RPV steels after neutron irradiation are enhanced by the copper and\nnickel content [23, 24]. Furthermore, it can account for the obse rvations that the inﬂuence\nof copper content on DBTT decrease is progressive when the nicke l content decreases and\nthe inﬂuence of nickel content on DBTT disappears for model alloys after neutron irradia-\ntion (at a neutron ﬂuence of 72 ×1018m−2) with copper contents below 0.08 at.% (0.1 wt.%)\n10[25, 26].\nIV. CONCLUSION\nWe have presented DFT-GGA calculations investigating the segrega tion behaviours of\naluminium and nickel at the interface of precipitates in ferritic steels . We have examined\nthe segregation energies of aluminium and nickel at the interface an d within the core regions\nin precipitated phases with diﬀerent compositions. Our results show that aluminium and\nnickel can segregate at the interface of precipitates in ferritic st eels, in agreement with 3DAP\nexperiments. Moreover, we also ﬁnd that the interfacial segrega tion of aluminium is more\nsensitive to the composition of copper-rich precipitated phases th an those of nickel. The\nmost energetically favourable site of nickel segregation depends t o the composition of the\ncopper-rich precipitated phase. These detailed segregation beha viours are also consistent\nwith 3DAP experiments.\nWe also calculated the contributions of aluminium and nickel segregat ion behaviours to\nGriﬃth work to predict the interfacial cohesive properties of prec ipitates and found that\nthere are aluminium-induced ductility and nickel-induced embrittlemen t eﬀects at the in-\nterface of precipitates. Aluminium-induced ductility arises mainly fro m the 3p electron of\naluminium that causes weaker Fe-Al bonding at the fracture surfa ce than at the interface.\nNickel-induced embrittlement, however, is due mainly to the 3d electr ons of nickel resulting\nin enhanced Fe-Ni bonding at the fracture surface compared to t hat at the interface. Finally,\nwe have used the nickel-induced embrittlement at the interface of t he copper precipitates\nin ferritic matrices to explain the experimental observation that th e DBTT of low-carbon,\ncopper-precipitation-strengthened steels and RPV steels increa se with their nickel and cop-\nper content. These studies suggest a possibility of improving the du ctility of ferritic steels\nfrom modifying interfacial cohesive properties of copper-rich pre cipitates by segregation of\nalloying elements.\nThe authors thank Professor B. X. Zhou, Mr G. Xu and Q. D. Liu for critical discus-\nsions. This work is ﬁnancially supported by National Science Foundat ion of China (Grant\nNo. 50931003, 51001067), Shanghai Committee of Science and Te chnology (Grant No.\n09520500100), Shanghai Municipal Education Commission (Shu-Gu ang project, Grant No.\n09SG36)andShanghaiEducationDevelopment FoundationandSha nghaiLeadingAcademic\n11Discipline Project (S30107). The computations were performed at Ziqiang Supercomputer\nCenter of Shanghai University and Shanghai Supercomputer Cen ter.\n[1] S. K. Dhua, D. Mukerjee, and D. S. Sarma, Metall. Mater. Tr ans. A, 32, 2259 (2001).\n[2] S. Vaynman, D. Isheim, R. P. Kolli, S. P. Bhat, D. N. Seidma n, and M. E. Fine, Metall.\nMater. Trans. A, 39, 363 (2008).\n[3] X. Yu, J. L. Caron, S. S. Babu, J. C. Lippold, D. Isheim, and D. N. Seidman, Acta. Mater.,\n58, 5596 (2010).\n[4] W. J. Phythian, and C. A. English, J. Nucl. Mater., 205, 16 2 (1993).\n[5] E. A. Kuleshova, B. A. Gurovich, Y. I. Shtrombakh, Y. A. Ni kolaev, and V. A. Pechenkin, J.\nNucl. 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Chung,\nLangmuir 26, 16254 (2010).\n14TABLE I: Segregation energies (in eV) of Al atoms at the inter faces and core regions of Cu-rich\nprecipitated phasesinaferriticmatrix, decompositions i ntochemical andmechanical contributions,\nand the relaxation energy of the supercell.\nSite(X) 1 2 3 4\nα-Fe/prue BCC-Cu\n△Echem,Al\nX0 -1.78 -1.52 -1.18\n△Emech,Al\nX0 +0.07 +0.07 +0.06\n△EAl\nX 0 -1.71 -1.45 -1.12\n△Esc -0.60 -0.53 -0.53 -0.54\nα-Fe/BCC-CuFe( 75 at.% Cu)\n△Echem,Al\nX0 -1.07 -0.93 -0.60\n△Emech,Al\nX0 -0.01 +0.02 -0.07\n△EAl\nX 0 -1.08 -0.91 -0.67\n△Esc -0.84 -0.85 -0.82 -0.91\nα-Fe/BCC-CuFe( 50 at.% Cu)\n△Echem,Al\nX0 -0.12 -0.13 0.14\n△Emech,Al\nX0 -0.03 -0.03 -0.01\n△EAl\nX 0 -0.15 -0.16 0.13\n△Esc -0.28 -0.31 -0.31 -0.29\n15TABLE II: Segregation energies (in eV) of Ni atoms at the inte rfaces and core regions of Cu-rich\nprecipitated phasesinaferriticmatrix, decompositions i ntochemical andmechanical contributions,\nand the relaxation energy of the supercell.\nSite(X) 1 2 3 4\nα-Fe/pure BCC-Cu\n△Echem,Ni\nX0 -0.39 -0.42 -0.05\n△Emech,Ni\nX0 -0.05 -0.01 -0.02\n△ENi\nX 0 -0.44 -0.43 -0.03\n△Esc -0.43 -0.48 -0.44 -0.41\nα-Fe/BCC-CuFe( 75 at.% Cu)\n△Echem,Ni\nX0 -0.37 -0.33 -0.15\n△Emech,Ni\nX0 -0.03 -0.04 -0.03\n△ENi\nX 0 -0.40 -0.38 -0.18\n△Esc -0.73 -0.76 -0.78 -0.76\nα-Fe/BCC-CuFe( 50 at.% Cu)\n△Echem,Ni\nX0 -0.22 -0.24 -0.29\n△Emech,Ni\nX0 -0.02 -0.03 +0.01\n△ENi\nX 0 -0.20 -0.27 -0.28\n△Esc -0.24 -0.22 -0.27 -0.23\n16FIG. 1: The atomic structures of the (001) interfaces betwee n BCC Cu-rich precipitated phases\nand ferritic matrix. The precipitated phases are Cu-Fe allo ys with (a) 100 at.% (i.e., pure Cu),\n(b) 75 at.%, and (c) 50 at.% Cu concentration, respectively. Red and blue balls denote Cu and\nFe atoms, respectively. The atoms marked by Arabic numerals denote the segregation sites in\n1, the matrix; 2, the region of interface toward the matrix; 3 , the region of interface toward the\nprecipitated phase; and 4, the core region of the precipitat ed phase.\nFIG. 2: Side view of the atomic structures of M segregation at (a) the interface between the BCC\nCu-rich precipitated phase and the ferritic matrix and (b) i ts corresponding fracture free surface.\n17FIG. 3: The valence charge density diﬀerences of Ni at the inte rface (top left), Ni at the fracture\nfreesurface(top right), Al at theinterface (bottom left), andAl at thefracturefreesurface(bottom\nright). Contours increase successively by a factor of 10−2/au3. Blue, light blue, and purple lines\ndenote charge depletion and pink lines denote charge accumu lation. Regions displayed correspond\nto the square cell depicted in Fig. 2.\n18/s48/s46/s53/s48 /s48/s46/s55/s53 /s49/s46/s48/s48/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s69/s110/s101/s114/s103/s121/s40/s101/s86/s41\n/s67/s117/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s40/s97/s116/s46/s41\nFIG. 4: △EM\nI(solid symbols) and △EM\nI−△EM\nF(blank symbols) for Ni (squares) and Al (dots)\nsegregating at the interfaces of the copper-rich precipita ted phases with diﬀerent copper concen-\ntrations in the ferritic matrix.\n19"    },    {        "title": "1706.05000v1.Preliminary_corrosion_studies_of_IN_RAFM_steel_with_stagnant_Lead_Lithium_at_550_C.pdf",        "content": "1 \n  \nPreliminary corrosion studies of IN-RAFM steel with stagnant Lead Lithium  at 550 C  \n \nA.Sarada Sree, Hemang S.Agravat , Jignesh Chauhan  and E.Rajendrakumar  \n \nInstitute for Plasma Research, Bhat, Gandhinagar -382428.  \n \n \nE-mail : sarada.sree@gmail.com  \nPhone :  +917923964032 . \nFax     :  +917923962277  \n \n \nAbstract  \nCorrosion of Indian RAFMS (reduced activation ferritic martensitic steel) material with  liquid \nmetal, Lead Lithium (  Pb-Li)  has been studied  under static condition , maintaining   Pb-Li at 550 \nC for different time durations, 2500 , 5000 and 9000 h ours. Corrosion rate was calculated from \nweight loss measurements. Microstructure a nalysis was  carried out  using  SEM and  chemical \ncomposition by  SEM -EDX measurements .  Micro  Vickers  hardness  and tensile testing were \nalso carried out. Chro mium was  found leaching from the near surface regions and surface \nhardness was found to decrease  in all the three cases . Grain boundaries were a ffected . Some \ngrains got detached fro m the surface   giving rise to pebble like structures  in   the surface \nmicrographs. There was  no significant re duction in the tensile strength,  after exposure to liquid \nmetal.  This paper discusses  the experimental details and the results obtained.  \n  2 \n 1. Introduction  \nIn Indian LLCB ( Lead Lithium cera mic breeder) test blanket module program [1] concept , Pb-\nLi is used as coolant, neutron multiplier and  tritium breeder.  Indian RAFMS  (reduced activation \nferritic m artensitic steel)  is the candidate structural  material  for this  test blanket module. \nCompatibility of the structural material with the liquid metal is one of  the prime concern s for the \nsuccessful operation of TBM in ITER  [2-4]. Corrosion in the form of dissolution, inter  granular \npenetration and imp urity transfer to the liquid metal   or from the liquid metal to the structural \nmaterial, can cause wall thinning , leading to loss of mechanical integrity. Corrosion b y the liquid \nmetal could  be the  limiting   factor for the life of the structural material in ITER  (international \nthermo nuclear experimental reactor) . Therefore compatibility of the structural material  with \nliquid metal  need to be studied in stagnant as well as in dy namic conditions  to understand the \ndifferent factors contributing to corrosion .  Corrosion  and compatibility  of  ferritic m artensitic \nsteels  [5-31] and  austenitic steels [ 32-35] with Pb -Li    have been carried out  in static and \nflowing conditions .  \nTherefore, to study the effect of corrosion of IN -RAFMS  in Lead Lithium ( Pb-Li) envi ronment , \ninitially a static experiment was planned, in which  IN -RAFMS sample coupons we re exposed to  \nstatic  liquid metal   maintained at 823 K (550 C) . Samples were  taken out after 2500,  5000 and \n9000 hours and analyzed using SEM (Scanning electron microsc ope). Chemical composition was \nmeasured on the cross section of the samples by SEM/EDX  (energy dispersive X -ray analysis ) \nmeasurements . Hardness and tensile testing were also carried out on exposed samples.  This \npaper discusses the experiment al details  and the results obtained.  \n2. Experimental set up   \nExperimenta l set up is shown in figure 1. Lead Lithium c hunks are loaded in  a  cylindrical  \nchamber   m ade of SS316 L. A thermo well was  fixed on the side of the cha mber. A ‘K’ type \nthermocouple was  introduce d into the thermo well, which extends up to the center of the \nchamber. A  3 KW  heater  coil  was wound over the circumference of the  chamber to melt the \nLead -Lithium chunks.  \n 3 \n  \nFig.1.  Experimental setup  for studying corrosion of IN -RAFMS with static Pb -Li at 550 C . \nTop flange of the chamber has the provision for the following.  \na. to evacuate the chamber  \nb. to introduce Argon gas into  the chamber to maintain positive pressure over liquid metal \nmelt.  \nc. a movable feed through arrangement to hold the samples . \nIN-RAFMS  flat and tensile samples were   fixed to a sample holder , attached at the end of the \nfeed through . Chemical composition of IN -RAFMS material  is given in table 1. Initially samples  \nwere  held above the Pb -Li chunks.  The chamber is evacuated upto 10-3 m.bar, raising the \ntemperature of the chunks upto 200 C. Then the vacuum system was isolated and  Argon gas was \nintroduced into the chamber.  Then  the temperature was further raised up to  550 C.  Once all the \nchunks were melted and main tained at a temperature of 550 C , the sample holder was lowered  \nwith the feed  through arrangement  to dip the samples in the liquid metal melt . A positive \npressure of 1.5 bar is maintained over the liquid melt throughout the experiment.  \nSamples  were  remov ed from the chamber after  exposing them to the liquid metal for  2500, \n5000 and 9000 hours. To remove the adherent   Pb-Li, the samples were cleaned , using cl eaning \nsolution consisting of  Acetic acid  (CH 3COOH), Hydrogen peroxide (H 2O2) and Ethyl Alcohol  \n4 \n (C2H5OH) in 1:1:1: ratio. The samples were cleaned and weighed   until similar consecutive \nreadings were obtained. Weight measurements were taken using Sartorius precision weighing \nbalance with a precision of ± 0.01 mg. After cleaning , the samples were cut and molds were \nprepared for me tallographic examination.  The molds  were ground  using different grade SiC  \npapers and finally polished with Alumina powder to achieve mirror  finishing.  \nThe samples were analyzed using SEM  (scanning electron micro scope ). Change in chemical \ncomposition was determined  with   SEM -EDX   (energy dispersive X -ray analysis) \nmeasurement.  Line scan analysis  was carried out on the cross section of the samples .  Surface \nmicrographs were obtained using scann ing electron microscope, model S440i , from LEO \ncorporation, UK.  Micro vickers hardness  and tensile strength measurements were also carried \nout on  the samples . Hardness measurements were carried out using a Mitutoyo HM 211, Micro \nVickers hardness tes ting machine. Tensile testing was carried out using Instron make, 5982 \nUniversal testing machine.  \nTable 1:  Chemical composition of IN - RAFMS (wt%)  \n \nCr C Mn V W Si P S Ta Nb Mo Ni Fe \n9.15 0.08 0.53 0.24 1.37 0.026  <0.002  0.002  0.08 <0.001  <0.002  0.004  Bal. \n \n3. Results  \n \n3.1. Weight loss measurements  \nWeight of the samples was measured before and  after exposure to liquid metal  at different time \nintervals.  Cor rosion rate is calculated from the weight loss measurements and  given  in table 2. \nTable 2 :  Corrosion rate after exposure to Pb -Li \n \nSr. \nNo. Exposure  time \n(hours)  Dissolution rate  \ng/(m2 x year)  Corrosion rate \n(m/year)  \n1 2500  292.42  37.68  \n2 5000  276.85  35.67  \n3 9000  184.72  23.80  \n 5 \n From weight loss measurements, corrosion rate  of IN -RAFMS was  found to be   ~ 40 m/year .  \nCorrosion rate estimated from weight loss measurements  was almost  found  to be   same  for  \n2500 and 5000 hours exposure s. Decrease in the corrosion rate  for  9000 hours  exposure  could \nbe due to saturation of liquid metal . This could  be possible,  due to the formation of saturated \nlayer of dissolved elements near the sample surface.  \n \n3.2. SEM surface micrographs  \nAfter exposure to liquid metal, the samp les lost their metallic luster and the surface became dull. \nSurface micrographs of exposed  samples for 2500, 5000 and 9000 hours were shown in figure  \n2(a), (b) and (c) respectively. Surface got deteriorated with increasing exposure time.  \n \nFig.2. SEM sur face micrographs of samples exposed to  stati c Pb-Li at 550 C. \nExposure time : (a) 2500 hours ,  (b) 5000 hours, (c ) 9000 hours.  \n \nIn all the exposed samples, pebble like structure can be seen. Similar  type of surface \nmorphology was o bserved by V.Tsisar et a l., [30] when  EP -823 steels were  exposed to \nNitrogen a dded Lithium maintained at 600 C. This type of structure is due to dislodging of \ngrains/ sub  grains from the surface. Initially, w hen the surface of the structural material is \nexposed to  Lead -Lithium eutectic  at 550 C , corrosion occurs on the surface and the bonding \nbetween grains / sub  grains become wea k. Slowly the grains/ sub  grains get detached from the \nsurface and go into the eutectic , giving rise to pebble like structure.  \n3.3. SEM cross sect ion micrographs   \nSEM cross section micrographs after exposure of the samples to static  liquid metal  at 550 C  are \nshow n  in figure 3(a),  (b) and (c) for the case of 2500, 5000 and 9000 hours  respectively .  \n6 \n Groove like structure  was observed in all the thr ee cases , showing  internal corrosion of the \nmaterial. After 5000 hours, a detached layer was observed. Chemical composition ,  measured in  \nthe  detached layer revealed that it is part of the matrix and contained mainly iron and chromium.  \n \nFig.3.  SEM cross  section micrographs  after exposure to  static Pb -Li at 550 C . \n   Exposure time:  (a) 2500 hours, (b) 5000 hours and (c) 9000 hours.  \n3.4. SEM EDX measurements on cross section  \nSEM EDX was carried out on the cross section of the samples exposed for 2500, 500 0 and 9000 \nhours. Chemical composition  was measured depth  wise and shown in figure 4 (a), (b) and (c) for \nthe above mentioned  three exposure times . In all the three cases, chromium is  found  leaching  \nfrom the near surface r egions and tungsten is found slightly high near the surface. In case of \n5000 and 9000 hours exposure, chromium is found l eaching from a greater depth ~20  m as \ncompared to ~5 m in case of 2500 hours.  \n \nFig.4.  EDX line scan on cross section of  IN-RAFMS  after exposure  to static Pb -Li at 550 C . \nExposure time:  (a) 2500 hours, (b) 5000 hours and (c) 9000 hours.  \n7 \n 3.5. Micro Vickers Hardness measurement   \nMicro Vickers hardness was measured on the cross section of the samples up  to a depth of 100 \nm and shown in figure 5  for all the three exposure times . \n \nFig.5.  Depth profiles of Micro Vickers hardness of exposed IN -RAFMS samples to static Pb -Li \nat   550 C . Exposure time:  (a) 2500 hours, (b) 5000 hours and (c) 9000 hours.  \nIt is observed that hardness decreased in the near su rface regions up  to a depth of 15  m in all \nthe three cases. After 2500 and 5000 hours exposure to liquid met al, hardness decreased up  to \n~170  HV. After 9000 hours exposure, surface hardness decreased drastically up  to 80 HV. \nDecrease in surf ace hardness was also observed for 9Cr -ODS steels exposed to static Li and Pb -\nLi at  600o C  for 250 hours by Y.Li et al., [22] and  for JLF-1 steel exposed to Li by Qi Xu et \nal.,[3 1]. \n3.6.Tensile testing   \nTensile testing was carried out at room temperature   on fresh and exposed samples  using Instron  \nmake  5982 universal testing machine  at a strain rate of  3x10-3/sec. The results are tabulated in \ntable 3.  After exposure to Pb -Li, no significant change in the tensile strength was observed as \ncompared to  unexposed  sample.  \n8 \n  \n \nTable 3 : Tensile strength after exposure to Pb -Li \nSl. No  Exposure time  \n(hours)  Tensile strength  \n(MPa)  \n1.  Zero  (unexposed) ~651 \n2.  2500  652.52  \n3.  2500  661.57  \n4.  5000  635.67  \n5.  5000  640.04  \n6.  9000  632.26  \n7.  9000  691.25  \n \n4. Result s and discussion  \n \nDissol ution of the alloying elements seems to be the main corrosion mechanism . From \nSEM/EDX measurements on the cross section of the samples, chromium was found selectively \ngetting leached out   from the  near surface regions , ~15 microns from th e surface. Chromium \ncontent reduced from 9% to (5 -6)% after exposur e to liquid metal.   Tungsten was found high \nnear the surface. This coul d be due to less solubility of t ungsten in the liquid metal.   \nFrom the surface micrographs, pebble like s tructures were  seen on the surface.  When the surface \nof the structural material is in contact with Pb -Li eutectic, corrosion occurs on the surface and \nthe bonding betw een the grains/sub grains bec omes weak, leading to detachment of some \ngrains/sub grains fro m the su rface which  pass into the eutectic.  9 \n From the Micro Vic kers hardness measurements it was observed that the surface hardness \ndecreased up  to a depth of 15 m for all the exposure times. Surface hardness decreased up  to \n170 HV after  2500 and 5000 h ours exposure and up  to 80 HV  after 9000 hours exposure to \nliquid metal. From weight loss measurements, estimated corrosion rate  was ~40 m/year.  Thus \nweight loss and decrease in surfac e hardness could  be  due  to the dissolution of chromium in  \nthe liqui d metal.  Tensile strength   of exposed samples  to liquid metal remained almost same as \nthat of fresh IN -RAFMS sample s. It c an be said that  r eduction of hardness in the near surface \nlayers did not affect the tensile strength of the material.  \nAcknowledgement s \nAuthors woul d like to acknowledge Mr. L. Narendrasinh L. Chauhan  for supporting SEM \nmeasurements.  \nReferences  \n[1] Paritosh  Chaudhuri et. al.,  Fusion Eng. Des.  89 ( 2014 ) 1362 -1369  \n[2] B.A. Pint, J.L. Moser, P.F. Tortorelli, Fusion Eng. Des. 81 (2006) 901 –908. \n[3] C.P.C.  Wong, J.F. Salavy, Y. Kim, I. Kirillov, E.R. Kumar, et al., Fusion Eng. Des. 83 (2008)    \n850–857. \n[4] T. Muroga, T. Tanaka, M. Kondo, T. Nagasaka, Q. Xu, Fusion Sci. Technol. 56 (2009) 897 –   \n901. \n[5] Omesh Chopra  and Dale Smith,  J. Nucl. Mater. 122 –123 (1984) 1219 -1224.  \n [6] P.F.Tortorelli and J.H.Devan, J. Nucl. Mater. 141 –143 (1986) 592 –598.  \n[7] O.K. Chopra, D.L. Smith, J. Nucl. Mater. 155 –157 (1988) 715 –721. \n[8] H.U. Borgstedt, G. Frees, Z. Peric, Fusion Eng. 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Mater. 155 –157 (1988) 710 –714 \n [33] V.Coen, P.Fenici, H.Kolbe, L.Orecchia, T.Sasaki, J .Nucl.Mater. 110(1982) 108 -114. \n [34] H.U.BORGSTEDT, Gunter Frees and Gunter Drechsler J. Nucl. Mater. 141 –143 (1986) \n561-565. \n [35] O.K. Chopra. and D.L.Smith,  J.Nucl.Mater.141 -143 (1986) 566 -570. "    },    {        "title": "1006.5080v1.Epitaxial_strain_effects_in_the_spinel_ferrites_CoFe2O4_and_NiFe2O4_from_first_principles.pdf",        "content": "arXiv:1006.5080v1  [cond-mat.mtrl-sci]  25 Jun 2010Epitaxial strain eﬀects in the spinel ferrites CoFe 2O4and NiFe 2O4from ﬁrst principles\nDaniel Fritsch∗and Claude Ederer\nSchool of Physics, Trinity College Dublin, Dublin 2, Irelan d\n(Dated: October 23, 2018)\nThe inverse spinels CoFe 2O4and NiFe 2O4, which have been of particular interest over the past\nfew years as building blocks of artiﬁcial multiferroic hete rostructures and as possible spin-ﬁlter\nmaterials, are investigated by means of density functional theory calculations. We address the\neﬀect of epitaxial strain on the magneto-crystalline aniso tropy and show that, in agreement with\nexperimentalobservations, tensile strain favors perpend icular anisotropy, whereas compressive strain\nfavors in-plane orientation of the magnetization. Our calc ulated magnetostriction constants λ100of\nabout−220 ppm for CoFe 2O4and−45 ppm for NiFe 2O4agree well with available experimental\ndata. We analyze the eﬀect of diﬀerent cation arrangements u sed to represent the inverse spinel\nstructure and show that both LSDA+ Uand GGA+ Uallow for a good quantitative description of\nthese materials. Our results open the way for further comput ational investigations of spinel ferrites.\nPACS numbers:\nI. INTRODUCTION\nSpinel ferrites CoFe 2O4and NiFe 2O4are insulating\nmagnetic oxides with high magnetic ordering temper-\natures and large saturation magnetizations.1This rare\ncombination of properties makes them very attractive\nfor a wide range of applications. Recently, particular\nattention has been focused on the possible use of spinel\nferrites as magnetic components in artiﬁcial multiferroic\nheterostructures2–5or as spin-ﬁltering tunnel barriers for\nspintronics devices.6–8\nFor these applications, the corresponding materials\nhave to be prepared either in the form of thin ﬁlms,\ngrown on diﬀerent substrates, or as components of more\ncomplex epitaxial heterostructures.2,9–11Due to the mis-\nmatch in lattice constants and thermal expansion coeﬃ-\ncients between the thin ﬁlm material and the substrate,\nsigniﬁcant amounts of strain can be incorporated in such\nepitaxial thin ﬁlm structures, depending on the speciﬁc\ngrowth conditions and substrate materials. This epitax-\nial strain can then lead to drastic changes in the proper-\nties of the thin ﬁlm material. Indeed, a reoriention of the\nmagnetic easy axis under diﬀerent conditions has been\nreported for CoFe 2O4,12–14and a strong enhancement\nof magnetization and conductivity has been observed in\nNiFe2O4thin ﬁlms.10,15,16\nIn order to eﬃciently optimize the properties of thin\nﬁlm materials, it is important to clarify whether the ob-\nserved deviations from bulk behavior are indeed due to\nthe epitaxial strain or whether they are induced by other\nfactors, such as for example defects, oﬀ-stoichiometry,\nor genuine interface eﬀects. First principles calculations\nbased on density functional theory (DFT),17–19can pro-\nvide valuable insights in this respect by allowing to ad-\ndress each of these eﬀects separately.\nHere we present results of DFT calculations for the\nstructuraland magneticpropertiesofepitaxiallystrained\nCoFe2O4and NiFe 2O4, with special emphasis on strain-\ninduced changes in the magneto-crystalline anisotropy\nenergy (MAE). Our results are representative for (001)-oriented thin ﬁlms of CoFe 2O4and NiFe 2O4, grown on\ndiﬀerent lattice-mismatched substrates. Our results pro-\nvide important reference data for the interpretation of\nexperimental observations in spinel ferrite thin ﬁlms and\nin heterostructures consisting of combinations of spinel\nferrites with other materials, such as perovskite struc-\nture oxides.\nWe ﬁnd a large and strongly strain-dependent MAE\nfor CoFe 2O4, and a smaller but also strongly strain-\ndependent MAE for NiFe 2O4. We discuss the inﬂuence\nof diﬀerent cation arrangements within the inverse spinel\nstructure and analyze the diﬀerence in the structural and\nmagneticproperties due to diﬀerent exchange-correlation\nfunctionals used in the calculations. From our calcula-\ntions we obtain the magnetostriction constants λ100for\nboth CoFe 2O4and NiFe 2O4, which agree well with avail-\nable experimental data.\nThis paper is organized as follows. Sec. IIA gives\na brief overview over the properties of CoFe 2O4and\nNiFe2O4thatareimportantforthepresentworkandalso\nsummarizes results of previous DFT calculations. The\nbasic equations governing the magnetoelastic properties\nof cubic crystals are presented in Sec. IIB, followed by\na detailed description of the structural relaxations per-\nformed in this work in Sec. IIC, and a summary of fur-\nthercomputationaldetails in Sec. IID. The resultsofthe\nbulk structural properties will be presented in Sec. IIIA,\nwhereas the eﬀect of strain on the structural properties\nis analyzed in Sec. IIIB. The eﬀect of strain on the MAE\nis discussed in Sec. IIIC. Finally, in Sec. IV a summary\nof our main conclusions is given.\nII. BACKGROUND AND COMPUTATIONAL\nDETAILS\nA. Spinel structure and previous work on ferrites\nThe spinel structure (space group Fd¯3m, general for-\nmulaAB2X4) contains two inequivalent cation sites, the2\ntetrahedrally-coordinated Asite (Tdsymmetry, Wyck-\noﬀ position 8a), and the octahedrally-coordinated Bsite\n(Ohsymmetry, Wyckoﬀ position 16d). In the normal\nspinelstructure, all Asites are occupied by one cation\nspecies (divalent cation), whereas all Bsites are occu-\npied by the other cation species (trivalent cation). On\nthe other hand, in the inverse spinel structure the triva-\nlent cations occupy all Asites as well as 50 % of the\nBsites, whereas the remaining 50 % of the Bsites are\noccupied bythe divalent cations. Ifthe distribution ofdi-\nvalent and trivalent cations on the Bsites is completely\nrandom, all Bsites remain crystallographically equiva-\nlent and the overall cubic Fd¯3msymmetry is preserved\n(see Fig. 1(a)).\nBoth CoFe 2O4and NiFe 2O4crystallize in the inverse\nspinelstructure,eventhoughforCoFe 2O4theinversionis\ntypically not fully complete, i.e. there is a nonzero Co2+\noccupationonthe Asite. Thereby,the exactdegreeofin-\n(a)\n(b)\nFIG. 1: (Color online) Inverse spinel structure of spinel fe r-\nrites. Oxygen cations are depicted as small red spheres,\nand the coordination polyhedra surrounding cation sites ar e\nshaded. (a) Random cation distribution on the octahedrally\ncoordinated Bsites, i.e. all Bsites remain equivalent. (b)\nImmacation arrangement used throughout this work. In-\nequivalent Bsites are represented by diﬀerent shadings of the\ncorresponding octahedra.version depends strongly on the preparation conditions.1\nTo the best of our knowledge, no deviations from cu-\nbic symmetry have been reported for either system, i.e.\nthe Co2+/Ni2+cations are believed to be randomly dis-\ntributed on the Bsites.20\nAccording to the formal d5andd7electron conﬁgura-\ntionscorrespondingtoFe3+andCo2+, respectively,these\nions can in principle exhibit both high-spin and low-spin\nstates, but only the high-spin states are experimentally\nobserved in both spinel ferrites. For the d8electron con-\nﬁguration of Ni2+no such distinction exists. The mag-\nnetic moments of the Asite cations are oriented antipar-\nallel to the magnetic moments of the Bsite cations: the\nso-called N´ eel-type ferrimagnetic arrangement.21Thus,\nthe magnetic moments of the Fe3+cations on the Aand\nBsites cancel each other exactly, and the net magneti-\nzation is mainly due to the divalent Bsite cations, i.e.\neither Co2+or Ni2+. This results in a magnetic moment\nper formula unit close to the formal values of 3 µBand\n2µBin CoFe 2O4and NiFe 2O4, respectively. Deviations\nfrom these values can be either due to orbital contribu-\ntions to the magnetic moments, or due to incomplete\ninversion and oﬀ-stoichiometric cation distribution.\nBoth CoFe 2O4and NiFe 2O4are small gap insulators,\nbut information on the experimental gap size is very lim-\nited. Waldron used infrared spectra to obtain threshold\nvalues of 0.11 eV and 0.33 eV for the electronic transi-\ntions in CoFe 2O4and NiFe 2O4, respectively,22whereas\nJonker estimated the energy gap in CoFe 2O4to be\n0.55 eV, based on resistivity measurements along with\nother methods.23\nTheoretical calculations of the electronic structure of\nspinel ferrites so far have been focused mostly on mag-\nnetite (Fe 3O4). This material can be viewed as parent\ncompound for the spinel ferrites, including CoFe 2O4and\nNiFe2O4, which are obtained by substituting the Fe2+\ncation in magnetite by a diﬀerent divalent 3 dtransition\nmetal cation. In an early work, P´ enicaud et al.per-\nformed DFT calculations within the local spin-density\napproximation (LSDA) for magnetite and the respective\nCo-, Ni-, Mn-, and Zn-substituted ferrites.24The use of\nLSDA leads to half-metallic band-structures for all sys-\ntems except NiFe 2O4, in contrast to the insulating char-\nacter observed experimentally. (We note that the case\nof magnetite is somewhat more involved than that of the\nother spinel ferrites, since magnetite exhibits a metal-\ninsulator transition at ∼120 K.) It was later shown by\nAntonov et al.that insulating solutions for Co-, Ni-,\nand Mn- substituted Fe 3O4can be obtained within the\nLSDA+Uapproach.25The same was found by Szotek et\nal.using a self-interaction-corrected LSDA approach.26\nThe latter study also addressed the energetic diﬀerence\nbetween normal and inverse spinel structures with dif-\nferent valence conﬁgurations. The electronic structure\nof NiFe 2O4was also calculated within a hybrid func-\ntional approach, where a large band-gap of 4 eV was\nobtained by using 40 % of Hartree-Fock exchange in the\nexchange-correlation energy functional.27Recently, Per-3\nronet al.investigated diﬀerent magnetic arrangements\nfor NiFe 2O4in both normal and inverse spinel structures\nusing both LSDA and the generalized gradient approx-\nimation (GGA), and found the inverse spinel structure\nwith N´ eel-type ferrimagnetic order to be energetically\nmostfavorable,28in agreementwith the experimentalob-\nservations.\nCalculations of the MAE in strained CoFe 2O4and\nNiFe2O4have been reported by Jeng and Guo.29,30How-\never, due to the use of the LSDA, these calculations were\nbased on half-metallic band-structures for both materi-\nals. Furthermore,noinformationonstructuralproperties\nor the inﬂuence of the speciﬁc cation arrangement used\nin the calculation were given.\nB. Magnetoelastic energy of a cubic crystal\nIn this work we are concerned with the eﬀect of epi-\ntaxial strain on the structural and magnetic properties\nof CoFe 2O4and NiFe 2O4, i.e. with the elastic and mag-\nnetoelastic response of these systems. Here, we therefore\ngive a brief overviewoverthe general magnetoelasticthe-\nory for a cubic crystal, and present the most important\nequations that are used in Sec. III to analyze the results\nof our ﬁrst principles calculations.\nThe magnetoelastic energydensity f=E/Vofa cubic\ncrystal can be written as:31\nf=fK+fel+fmel, (1)\nwhere the three individual terms describe the cubic (un-\nstrained) magnetic anisotropy energy density fK, the\npurely elastic energy density feland the coupled magne-\ntoelastic contribution fmel, respectively. To lowest order\nin the strain tensor εijand in the direction cosines αiof\nthe magnetization vector, these terms have the following\nforms:\nfK=K(α2\n1α2\n2+α2\n2α2\n3+α2\n3α2\n1), (2)\nfel=1\n2C11(ε2\nxx+ε2\nyy+ε2\nzz)\n+1\n2C44(ε2\nxy+ε2\nyz+ε2\nzx)\n+C12(εyyεzz+εxxεzz+εxxεyy),(3)\nfmel=B1(α2\n1εxx+α2\n2εyy+α2\n3εzz)\n+B2(α1α2εxy+α2α3εyz+α3α1εzx).(4)\nHere,Kdenotes the lowest order cubic anisotropy con-\nstant,C11,C12, andC44are the elastic moduli, and B1\nandB2are magnetoelastic coupling constants.\nThe bulk modulus Bis deﬁned as:\nB=V0/parenleftbigg∂2Etot\n∂V2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n(V=V0), (5)whereEtotis the total energy and V0is the equilibrium\nbulk volume. Using Eqs. (1)-(4) the bulk modulus of\na cubic crystal can be expressed in terms of the elastic\nmoduliC11andC12:\nB=1\n3(C11+2C12). (6)\nIn this workwe investigate the eﬀect of epitaxial strain\nthat is induced in thin ﬁlm samples by the lattice mis-\nmatch to the substrate. This situation can be described\nby a ﬁxed in-plane strain εxx=εyy= (a−a0)/a0, where\nais the in-plane lattice constant of the thin ﬁlm material\nanda0is the corresponding lattice constant in the bulk.\nThe resulting out-of-planestrain εzz= (c−a0)/a0can be\nobtained from Eqs. (1)-(4) together with the condition of\nvanishingstressfortheout-of-planelatticeconstant c, i.e.\n∂f\n∂εzz= 0. In the demagnetized state εzzis related to the\napplied in-plane strain via the so-called two-dimensional\nPoisson ratio ν2D:32\nν2D=−εzz\nεxx= 2C12\nC11. (7)\nAfter calculating both ν2Dand the bulk modulus using\nDFT, the elastic moduli C11andC12can thus be ob-\ntained from Eqs. (6) and (7).\nIn Sec. IIIC we monitor the diﬀerences in total en-\nergy for diﬀerent orientations of the magnetization as a\nfunction of the in-plane constraint εxx. Using expres-\nsion (7) for ν2Dand taking the energy for orientation of\nthe magnetization along the [001] direction as reference,\ni.e. ∆fhkl=f001−fhkl, one obtains:\n∆f/angbracketleft100/angbracketright=−B1(ν2D+1)ǫxx,\n∆f/angbracketleft110/angbracketright=−B1(ν2D+1)ǫxx−1\n4K,\n∆f/angbracketleft111/angbracketright=−2\n3B1(ν2D+1)ǫxx−3\n4K,\n∆f/angbracketleft101/angbracketright=−1\n2B1(ν2D+1)ǫxx−1\n4K.(8)\nThus, the epitaxialstrain-dependenceofthese energydif-\nferences is governed by the magnetoelastic coupling con-\nstantB1and the two-dimensional Poisson ratio ν2D.\nSince the constant B1is not directly accessible by\nexperiment, the linear magnetoelastic response is typ-\nically characterized by the magnetostriction constant\nλ100, which is related to B1and the elastic moduli C11\nandC12:\nλ100=−2\n3B1\nC11−C12. (9)\nλ100characterizes the relative change in length (lattice\nconstant) along [100] when the material is magnetized\nalongthisdirection,comparedtotheunmagnetizedstate.4\nC. Structural relaxations for the inverse spinel\nstructure\nAs described in Sec. IIA the distribution of divalent\nand trivalent cations on the octahedrally coordinated B\nsites in the inverse spinel structure is assumed to be\nrandom for both CoFe 2O4and NiFe 2O4. On the other\nhand, the periodic boundary conditions employed in our\ncalculations always correspond to a speciﬁc cation ar-\nrangement with perfect long-range order. Even though\na “quasi-random” distribution of divalent and trivalent\ncations could in principle be achieved by using a very\nlarge unit cell, the required computational eﬀort would\nbe prohibitively large. For simplicity, we therefore re-\nstrict ourselves to the smallest possible unit cell which\ncontains two spinel formula units, i.e. four Bsites. Dis-\ntributing two Fe atoms on two of these sites, and ﬁlling\nthe other two sites with either Co or Ni, lowers the sym-\nmetry from spacegroup Fd¯3m(#227) to Imma(#74)\nindependent of which of the four B sites are occupied by\nFe (see Fig. 1(b)). Diﬀerent choices (“settings”) simply\nlead to diﬀerent orientations of the orthorhombic axes\nrelative to the Cartesian directions. It will become clear\nfrom the results presented in Sec. III that the speciﬁc\ncationarrangementusedinourcalculationsdoesnotcrit-\nically aﬀect our conclusions.\nWithin the lower Immasymmetry (and setting 1, see\nbelow), the tetrahedrally coordinated Asites are located\non Wyckoﬀ position 4e (0 ,1\n4,z), whereas the octahe-\ndrally coordinated Bsites split into Wyckoﬀ positions\n4b (0,0,1\n2) and 4d (1\n4,1\n4,3\n4). In addition, the oxygen po-\nsitions split into Wyckoﬀ positions 8i ( x,1\n4,z) and 8h\n(0,y,z).\nIn order to minimize the eﬀect of this artiﬁcial sym-\nmetry lowering, and to obtain results that are as close\nas possible to the average cubic bulk symmetry seen in\nexperiments, we apply the following constraints during\nour structural relaxations. For the calculations corre-\nsponding to the unstrained bulk case, we constrain the\nlattice parameters along the three cartesian direction to\nbe equal, a=b=c, and we ﬁx the Asite cations to their\nideal cubic positions, corresponding to z(4e) =1\n8. On\nthe other hand, since the oxygen positions are charac-\nterized by one free structural parameter already within\ncubicFd¯3msymmetry (Wyckoﬀ position 32e ( u,u,u)),\nwe do not apply any constraints to the 8i and 8h posi-\ntions within Immasymmetry. The relaxed bulk struc-\nture is then found by relaxing all internal positions for\ndiﬀerent volumes and ﬁnding the volume that minimizes\nthe total energy. For the relaxations corresponding to\na certain value of epitaxial strain, we constrain the two\nin-plane lattice constants to be equal, and then vary the\nout-of-plane lattice constant cwhile relaxing all internal\ncoordinates (except for z(4e) =1\n8).\nWhile for the unstrained bulk structure all possible\nBsite cation distributions that can be accommodated\nwithin the primitive fcc unit cell of the spinel struc-\nture lead to the same Immasymmetry, diﬀerent cases(a) (b)\nFIG. 2: Cation distribution within diﬀerent x-yplanes for dif-\nferent values of z, corresponding to “setting” 1 (a) and “set-\nting” 3 (b) (see main text for details). Only layers contain-\ningBsite cations are shown. Divalent cations (Co2+, Ni2+)\nare depicted as ﬁlled black circles, trivalent cations (Fe3+) as\nopen black circles, and oxygen anions as black crosses. Note\nthat we use the convention where the origin is located at the\nmidpoint between two Asites.\ncan be distinguished once an epitaxial constraint is ap-\nplied to the resulting structure. To illustrate this, the\ncationdistributionsfortwodiﬀerent settingsaredepicted\nin Figs. 2(a) and 2(b). The Bsite cation distribution\n(and the oxygen positions) within diﬀerent x-yplanes\nare shown, corresponding to diﬀerent “heights” z. The\ndiﬀerent settings simply correspond to diﬀerent orienta-5\ntions of the orthorhombic Immacrystal axes relative to\nthe original cubic axes.\nIt can be seen from Fig. 2 that the Bsites are arranged\nin an interconnected network of chains along /angbracketleft110/angbracketright-type\ndirections. In setting 1 (Fig. 2(a)), the corresponding\nchains within the same x-yplane contain alternating di-\nvalent and trivalent cations. In contrast, for setting 3\n(Fig.2(b))each x-yplanecontainsonlyoneuniquecation\nspecies, which then alternates between adjacent planes\nalong the zdirection (see also Fig. 1(b)). In setting 1\nthe same alternatingplanesareorientedperpendicular to\nthexdirection, whereas for setting 2 (not shown) these\nplanesareorientedperpendiculartothe ydirection. Ifan\nepitaxial constraint is applied within the x-yplane, set-\nting 3 becomes diﬀerent from settings 1 and 2, with the\ndiﬀerence being the orientation of the “substrate plane”\nrelative to the planes deﬁned by the cation order.\nIn view of this, we have performed all calculations cor-\nresponding to epitaxially strained systems for both set-\nting 1 and setting 3. The diﬀerences between the results\nobtained for the two diﬀerent settings then represent a\nmeasure for the sensitivity of these results from the spe-\nciﬁc cation arrangement used in our calculations.\nD. Other computational details\nAll calculations presented in this work were performed\nusing the projector-augmented wave (PAW) method,33\nimplemented in the Vienna ab initio simulation pack-\nage (VASP 4.6).34–37Standard PAW potentials supplied\nwith VASP were used in the calculations, contributing\nnine valence electrons per Co (4s23d7), 16 valence elec-\ntrons per Ni (3p64s23d8), 14 valence electrons per Fe\n(3p64s23d6), and 6 valence electrons per O (2s22p4). A\nplane wave energy cutoﬀ of 500 eV was used, and the\nBrillouin zone was sampled using diﬀerent k-point grids\ncentered at the Γ point. A 5 ×5×5k-point grid was\nused for the structural optimization and all total energy\ncalculations, whereas a ﬁner 7 ×7×7 grid was used\nto calculate densities of states (DOS). The tetrahedron\nmethod with Bl¨ ochl corrections was used for Brillouin\nzone integration.33We have veriﬁed that all quantities of\ninterest, in particular the MAEs, are well converged for\nthe used k-pointgridandenergycutoﬀ. All structuralre-\nlaxations were performed within a scalar-relativistic ap-\nproximation, whereasspin-orbit coupling was included in\nthe calculation of the MAEs.\nAs already noted in Sec. IIA, CoFe 2O4and NiFe 2O4\nare small gap insulators, whereas half-metallic or,\nin the case of NiFe 2O4, only marginally insulating\nband-structures have been obtained in previous LSDA\ncalculations.24In the present work we therefore use the\nLSDA+Uand GGA+ Uapproach,38which is known to\ngive a good description of the electronic structure for\nmany transition metal oxides.39We employ the Hubbard\n“+U” correction in the simpliﬁed, rotationally invari-\nant version of Dudarev et al.,40where the same valueTABLE I: Optimized bulk lattice constants a0and bulk\nmoduliBfor CoFe 2O4and NiFe 2O4, calculated using the\nLSDA+U, GGA, and GGA+ Uexchange-correlation func-\ntionals in comparison with experimental data.\nCoFe2O4 NiFe2O4\na0[˚A]B[GPa] a0[˚A]B[GPa]\nLSDA+U 8.231 206.0 8.196 213.1\nGGA 8.366 211.0 8.346 166.2\nGGA+U 8.463 172.3 8.426 177.1\nExp. (Ref. 42) 8.392 185.7 8.339 198.2\nExp. (Ref. 43) 8.35 — 8.325 —\nUeﬀ=U−J= 3 eV is used for all transition metal\ncations. The corresponding results are compared to pure\nGGA calculations, using the GGA approach of Perdew,\nBurke,andErnzerhof.41(Werestrictedthecomparisonto\npure GGA since Perron et al.28presented some evidence\n(for NiFe 2O4) that LSDA might not be appropriate to\nproperly describe these materials.)\nValues for the local magnetic moments and atom-\nprojected DOS are obtained by integration of the ap-\npropriate quantities overatom-centred spheres with radii\ntaken from the applied PAW potentials (1.164 ˚A (Fe),\n1.302˚A (Co), and 1.058 ˚A (Ni)), respectively.\nIII. RESULTS AND DISCUSSION\nA. Unstrained bulk structures\nWe ﬁrst present our results for the unstrained bulk\nstructures. The calculated lattice constants and bulk\nmoduli for both CoFe 2O4and NiFe 2O4using diﬀerent\nexchange-correlation functionals are summarized in Ta-\nble I. It can be seen that for both CoFe 2O4and NiFe 2O4\nthe use of LSDA+ Uleads to an underestimation of the\nlattice constant and an overestimation of the bulk mod-\nulus compared to the experimental values, whereas the\nopposite is the case for GGA+ U. The corresponding de-\nviations ( ∼1-2 % for the lattice constants) are typical for\ncomplex transition metal oxides (see e.g. Refs. 44–46).\nInterestingly, the lattice constants calculated within\npure GGA match the experimental values almost per-\nfectly for both CoFe 2O4and NiFe 2O4. However, this is\nsomewhat fortuitous and probably due to a cancellation\nof errors, as can be seen by the large discrepancies in\nthe bulk moduli. It will become clear in the following,\nthat the “+ U” correction is necessary in order to obtain\na good description of the electronic structure for both\nCoFe2O4and NiFe 2O4.\nAs discussed in Sec. IIC the cation arrangementin our\nunit cell lowers the symmetry to orthorhombic Imma,\nwith 4 independent parameters describing the positions\noftheoxygenanionsatWyckoﬀpositions8hand8i, com-\npared to one parameter for Wyckoﬀ position 32e in the\ncubic space group Fd¯3m. Table II lists the correspond-6\nTABLE II: Calculated Wyckoﬀ parameters (setting 1) for\nthe oxygen anions 8h ( x,1\n4,z) and 8i (0 ,y,z) for CoFe 2O4\nand NiFe 2O4using diﬀerent exchange-correlation function-\nals. The last line lists the corresponding parameters resul ting\nfrom Wyckoﬀ position 32e ( u,u,u) within Fd¯3msymmetry, ¯ u\nis obtained from these relations byaveraging overrecalcul ated\nuvalues for each dataset.\nCoFe2O4 8i 8h\nx z y z ¯u\nLDA+U0.235 −0.498 0.009 −0.257 0.255\nGGA 0.240 −0.496 0.008 −0.255 0.255\nGGA+U0.234 −0.499 0.007 −0.259 0.255\nNiFe2O4 8i 8h\nx z y z ¯u\nLDA+U0.237 −0.495 0.010 −0.258 0.256\nGGA 0.239 −0.496 0.009 −0.257 0.255\nGGA+U0.235 −0.496 0.008 −0.258 0.256\nFd¯3m3\n4−2u u−3\n42u−1\n2−u\ning Wyckoﬀ parameters obtained from our structural op-\ntimizations. It can be seen that diﬀerences between dif-\nferent exchange-correlation functionals are rather small.\nThe last line in Table II indicates the relation be-\ntweenthe Wyckoﬀpositions8iand8hin the Immaspace\ngroup(setting 1)and Wyckoﬀposition32e( u,u,u)in cu-\nbicFd¯3msymmetry (assuming that no actual symmetry\nbreaking occurs). These relations allow us to obtain an\naverageWyckoﬀ parameter ¯ u, by calculating the value of\nucorresponding to each of the four calculated Wyckoﬀ\nparameters x(8i),z(8i),y(8h), and z(8h) for each data\nset and subsequent averaging. The resulting values for ¯ u\nagree very well with available experimental data, which\nare 0.256 for CoFe 2O4and 0.257 for NiFe 2O4.43Further-\nmore, the values calculated from the individual 8h and\n8i Wyckoﬀ parameters deviate only very little from the\naverage values, which indicates that the lower symmetry\nused in our calculation has only a negligible eﬀect on the\ninternal structural parameters.\nFigs. 3 and 4 show the calculated spin-decomposed\ndensities of states (DOS) for CoFe 2O4and NiFe 2O4, re-\nspectively, within the optimized bulk structure and using\nboth pure GGA and GGA+ U. Both the total DOS per\nformula unit and the projected DOS per ion for the d\nstates of the various transition metal cations are shown,\nthe latter separated into t2g/egande/t2contributions,\nrespectively. The DOS calculated within LSDA+ U(not\nshown) do not show any signiﬁcant diﬀerences compared\nto the ones calculated using GGA+ U.\nFrom the projected DOS it can be seen that all tran-\nsition metal cations are in high spin states, with one\nspin-projection completely occupied, and that the cu-\nbic component of the crystal ﬁeld on the octahedrally-\ncoordinated ( Oh) sites lowers the t2gstates relative to\ntheegstates, whereas on the tetrahedrally-coordinated\n(Td)sitesthe estatesareslightlylowerin energythanthe\nt2states. Due to the N´ eel-type ferrimagnetic order, the\nlocal majority spin direction on the Tdsites is reversedFIG. 3: (Color online) Total and projected DOS per formula\nunit for CoFe 2O4. Left (right) panels correspond to GGA\n(GGA+U) calculations. The dstates of Co( Oh) (upper pan-\nels), Fe(Oh) (middle panels), and Fe( Td) (lower panels) are\nseparated into t2g(green/dark grey) and eg(red/black) con-\ntributions for the Ohsites and into e(green/dark grey) and t2\n(red/black) contributions for the Tdsites. The total DOS is\nshown as shaded grey area in all panels. Majority (minority)\nspin projections correspond to positive (negative) values .\n-505DOS [eV-1]\n-505\nDOS [eV-1]\n-505DOS [eV-1]\n-505\nDOS [eV-1]\n-8 -6 -4 -2 0 2\nE-EFermi [eV]-505DOS [eV-1]\n-8 -6 -4 -2 0 2\nE-EFermi [eV]-505\nDOS [eV-1]\nrelative to the Ohsites.\nIt is apparent that within GGA CoFe 2O4turns out\nto be a half-metal, in contrast to the insulating behav-\nior found in experiment.23This is similar to what has\nbeen found in previous LSDA calculations.24The half-\nmetallicity is due to the partial ﬁlling of the minority\nt2gstates of Co(O h), which in turn results from the for-\nmald7conﬁguration of the Co2+cation (see upper left\npanel of Fig. 3). This apparent deﬁciency of the GGA\napproach is corrected within the GGA+ Ucalculation as\ncan be seen in the right part of Fig. 3. We note that\nthis is very similar to the case of rocksalt CoO, which\nalso contains Co2+with ad7electron conﬁguration that\nleads to a metallic solution in pure LSDA,47whereas ap-\nplication of the DFT+ Uapproach leads to an insulating\nstate in agreement with the experimental observations.387\nFIG. 4: (Color online) Total and projected DOS per formula\nunit for NiFe 2O4. Left (right) panels correspond to GGA\n(GGA+U) calculations. The dstates of Co( Oh) (upper pan-\nels), Fe(Oh) (middle panels), and Fe( Td) (lower panels) are\nseparated into t2g(green/dark grey) and eg(red/black) con-\ntributions for the Ohsites and into e(green/dark grey) and t2\n(red/black) contributions for the Tdsites. The total DOS is\nshown as shaded grey area in all panels. Majority (minority)\nspin projections correspond to positive (negative) values .\n-505DOS [eV-1]\n-505\nDOS [eV-1]\n-505DOS [eV-1]\n-505\nDOS [eV-1]\n-8 -6 -4 -2 0 2\nE-EFermi [eV]-505DOS [eV-1]\n-8 -6 -4 -2 0 2\nE-EFermi [eV]-505\nDOS [eV-1]\nThe Hubbard correction splits the occupied and unoc-\ncupied parts of the minority spin t2gstates of the Co2+\ncations (see upper right panel of Fig. 3), thereby opening\nan energy gap. In addition, the local spin splitting on\nthe Fe cation is drastically enhanced, shifting the local\nmajority spin dstates towards the bottom of the valence\nband. We note that once the value of Ueﬀon the Co sites\nis large enough to push the corresponding unoccupied\nminority spin t2gstates above the lowest minority-spin\nFe(Oh) states, the width of the band gap is determined\nby the diﬀerence in energy between these lowest unoc-\ncupied minority-spin Fe states and the highest occupied\nminority spin t2gstates of Co( Oh). Therefore, a fur-\nther increase of Ueﬀon the Co sites does not signiﬁcantly\nchange the size of the band gap. Similarly, the band gap\ndepends only weakly on the speciﬁc value of Ueﬀon theTABLE III: Calculated magnetic moments (in µB) for bulk\nCoFe2O4and NiFe 2O4using diﬀerent exchange-correlation\nfunctionals. The total magnetic moment per formula unit\namounts to 3 µB(2µB) for CoFe 2O4(NiFe 2O4), respectively.\nCoFe2O4 Co(Oh) Fe( Oh) Fe( Td)\nLSDA+U +2.52 +3.99 −3.82\nGGA +2.43 +3.66 −3.45\nGGA+U +2.62 +4.10 −3.98\nNiFe2O4 Ni(Oh) Fe( Oh) Fe( Td)\nLSDA+U +1.49 +4.00 −3.82\nGGA +1.36 +3.71 −3.46\nGGA+U +1.58 +4.11 −3.97\nFe sites.\nThe gap size of 0.9 eV obtained within GGA+ Ufor\nthe chosen values of Ueﬀis comparable to the 0.63 eV ob-\ntained by Antonov et al.using LSDA+ UwithUeﬀ=4.0\neV for the Co(O h) andUeﬀ=4.5 eV for the Fe(O h) and\nFe(Td) cations,25and also agrees well with the value\nof 0.8 eV reported by Szotek et al.utilizing a self-\ninteraction corrected LSDA approach.26\nNi2+formally has one additional electron compared to\nCo2+, leading to a fully occupied minority spin t2gmani-\nfold for the Ni2+cation within a cubic ( Oh) crystal ﬁeld.\nAccordingly, NiFe 2O4exhibits a tiny gap of ∼0.1 eV be-\ntween the occupied minority spin t2gstates of Ni( Oh)\nand the unoccupied minority t2gstates of Fe(O h) even\nin pure GGA (see left panels of Fig. 4). However, the\nuse of GGA+ Uleads to a signiﬁcant enlargement of this\nenergy gap to a more realistic value of 0.97 eV for the\nchosen values of Ueﬀ. This is in good agreement with\nband gaps of 0.99 eV and 0.98 eV reported by Antonov\net al.25and Szotek et al.,26respectively. Similar to the\ncase of CoFe 2O4the Hubbard correction also leads to a\nstrong enhancement of the local spin splitting on the Fe\nsites in NiFe 2O4.\nTable III shows the local magnetic moments of the\ntransition metal cations per formula unit calculated\nwithin the three diﬀerent approaches. The total mag-\nnetic moment is independent of the applied exchange-\ncorrelationfunctional, and equalto the integervalue that\nfollowsfromtheformalelectronconﬁgurationofthetran-\nsition metal cations and the N´ eel-type ferrimagnetic ar-\nrangement (3 µBfor CoFe 2O4and 2µBfor NiFe 2O4).\nThis is a result of the either half-metallic or insulating\ncharacter of the underlying electronic structures. Never-\ntheless, as can be seen in Table III, the Hubbard correc-\ntion has a signiﬁcant inﬂuence on the spatial distribution\nofthemagnetizationdensityandtheuseofGGA+ U(and\nLSDA+U) leads to more localized magnetic moments\ncompared to GGA, indicated by the increased magnetic\nmoments corresponding to the diﬀerent cation sites.\nThe results presented in this section indicate that for\na realistic and consistent description of the structural,\nelectronic, and magnetic properties of both CoFe 2O4and\nNiFe2O4, a Hubbard correction to either LSDA or GGA8\n-4 -2 0 2 4\nεxx [%]0.900.951.001.051.10 c/asetting 1\nsetting 2\nsetting 3\nFIG. 5: (Color online) Calculated c/aratio of NiFe 2O4as a\nfunction of epitaxial strain εxxobtained from GGA+ Ucalcu-\nlations and diﬀerent cation arrangements (“settings”) on t he\nOhsites.\nis required. In the following we will therefore present\nonly results obtained within the LSDA+ Uand GGA+ U\napproaches.\nB. Epitaxial strain and elastic properties\nFig. 5 shows the relaxed c/aratio of NiFe 2O4as\nfunction of the epitaxial constraint εxx, obtained from\nGGA+Ucalculations as described in Sec. IIC. The case\nof CoFe 2O4is very similar. Two important things can\nbe seen from this. First, due to the orthorhombic Imma\nsymmetry of the chosen cation arrangement on the Oh\nsites, the c/aratio is not exactly equal to 1 at zerostrain.\nHowever, this eﬀect is clearly negligible compared to the\nchangesin c/ainduced by epitaxial strains of order ∼1 %\nand therefore does not aﬀect our further analysis. Sec-\nond, the slope of the c/aratio, which characterizes the\nelastic response of the material, is nearly completely un-\naﬀected by the diﬀerent cation arrangements.\nFrom the data shown in Fig. 5 we can therefore ob-\ntain the 2-dimensional Poisson ratio ν2D(Eq. (7)), which\nrelates in-plane and out-of-plane strains. Together with\nthe bulk moduli listed in Table I, we can then determine\nthe two elastic constants C11andC12from Eqs. (6) and\n(7).\nThe calculated 2-dimensional Poisson ratios ν2Dand\nelastic constants C11andC12, together with the bulk\nmoduli already presented in Table I, are listed in Ta-\nble IV, and are compared with experimental results from\nRef. 42. It can be seen that, as already pointed out, the\nspeciﬁccationarrangementhasnearlynoinﬂuenceonthe\nvalue of ν2Dand thus C11andC12. On the other hand,\nthe speciﬁc choice of either LSDA+ Uor GGA+ Uhas a\nnoticeableeﬀect. Similartothe caseofthe bulkmodulus,\nwe observe an overestimation (underestimation) of the\nelasticconstants C11andC12intheLSDA+ U(GGA+U)TABLE IV: Bulk modulus B, 2-dimensional Poisson ratio\nν2D, and elastic coeﬃcients C11andC12for CoFe 2O4and\nNiFe2O4, obtained for diﬀerent exchange-correlation poten-\ntials and diﬀerent cation arrangements (“setting” s), in com-\nparison to experimental data. The experimental ν2Dhas been\nevaluated from Eq. (7) using the experimental elastic con-\nstants.\nCoFe2O4s B[GPa] ν2DC11[GPa] C12[GPa]\nLSDA+U1 206.0 1.191 282.0 167.9\n3 1.185 282.7 167.6\nGGA+U1 172.3 1.147 240.8 138.1\n3 1.132 242.5 137.3\nExp.42185.7 1.167 257.1 150.0\nNiFe2O4S B[GPa] ν2DC11[GPa] C12[GPa]\nLSDA+U1 213.1 1.172 294.4 172.5\n3 1.167 295.1 172.2\nGGA+U1 177.1 1.115 251.2 140.0\n3 1.106 252.2 139.5\nExp.42198.2 1.177 273.1 160.7\nFIG. 6: (Color online) CoFe 2O4GGA+Utotal energy dif-\nferences for orientation of the magnetization along variou s\ncrystallographic in-plane directions with respect to the [ 001]\ndirection (black solid lines): /trianglesolid[100],/triangledownsld[010],◭[110], and\n◮[1¯10]. Red broken lines denote crystallographic directions\nwhich include also out-of-plane components, namely [101]\n(dashed-dotted line), [011] (dashed-double dotted line), and\n[111] (dashed line). Left (right) panels contain the result s\ncorresponding to setting 1 (setting 3).\n-4 -2 0 2 4\nεxx [%]-6.0-4.0-2.00.02.0∆E [meV]\n-4 -2 0 2 4\nεxx [%]0.02.04.06.08.0\ncalculations. The same general trend holds for the 2-\ndimensional Poisson-ratioofCoFe 2O4, while for NiFe 2O4\nthe use of LSDA+ Ualso slightly underestimates ν2D.\nOverall the deviations are only within a few percent of\nthe experimental data (1-2 % for CoFe 2O4and up to\n6 % for NiFe 2O4), and we therefore conclude that both\nLSDA+Uand GGA+ Uallow for a good description of\nthe strain response of CoFe 2O4and NiFe 2O4.\nC. Magnetoelastic coupling\nThe calculated MAEs, deﬁned as the energy diﬀer-\nences for various orientations of the magnetization rel-\native to the energy for orientation of the magnetization\nparallel to the [001] direction, are depicted in Fig. 6 for9\nFIG. 7: (Color online) NiFe 2O4LSDA+U(GGA+U) total\nenergy diﬀerences for orientation of the magnetization alo ng\nvarious crystallographic in-plane directions with respec t to\nthe [001] direction are shown in the upper (lower) panels\n(black solid lines): /trianglesolid[100],/triangledownsld[010],◭[110], and ◮[1¯10]. Red\nbroken lines denote crystallographic directions which inc lude\nalso out-of-plane components, namely [101] (dashed-dotte d\nline), [011] (dashed-double dotted line), and [111] (dashe d\nline). Left (right) panels contain the results correspondi ng to\nsetting 1 (setting 3).\n-0.50.00.51.01.5∆E [meV]\n-1.0-0.50.00.5\n-4 -2 0 2 4\nεxx [%]-0.50.00.51.0∆E [meV]\n-4 -2 0 2 4\nεxx [%]-1.0-0.50.00.5\nCoFe2O4(GGA+Uonly)andinFig.7forNiFe 2O4(both\nLSDA+Uand GGA+ U). It can be seen that the calcu-\nlated MAEs for CoFe 2O4are roughly ﬁve to six times\nlarger than in NiFe 2O4. Furthermore, to a good approxi-\nmation, the calculated energy diﬀerences exhibit a linear\ndependence on strain. Deviations from this linear behav-\nior are most pronounced for the case of NiFe 2O4within\nsetting 3. Since the pure elastic response shown in Fig. 5\ndoes not exhibit any signiﬁcant non-linearities, we con-\nclude that higher order magnetoelastic terms are respon-\nsible for the slightly non-linear behavior of the MAE in\nthis case.\nIt can also be seen, that in all cases the strain depen-\ndence, i.e. the slopeofthe variouscurvesshowninFigs.6\nand 7, is largest for the in-plane versus out-of-plane en-\nergy diﬀerences, i.e. for orientation of the magnetization\nalong the [100], [010], [110], and [1 ¯10] directions (com-\npared to the [001] direction), consistent with Eqs. (8).\nThus, tensilestrainfavorsperpendicularanisotropy(easy\naxis perpendicular to the “substrate”) and compressive\nstrain favors in-plane orientation of the magnetization,\ni.e.B1>0. For suﬃcient amount of strain the easy axis\nof magnetization will therefore always be oriented either\nin-plane or out-of-plane, consistent with various experi-\nmental observations in thin CoFe 2O4ﬁlms under tensile\nstrain.13,14,48\nAccording to the phenomenological magnetoelastictheory for a cubic crystal discussed in Sec. IIB, in partic-\nularEqs.(8), thestraindependenceofthein-planeversus\nout-of-plane anisotropy should be stronger by a factor of\n2 compared to the anisotropy corresponding to [101] or\n[011] orientation of the magnetization, and by a factor of\n3/2 compared to [111] orientation. We note that these\nratiosarevery well observedby the calculated anisotropy\nenergies shown in Figs. 6 and 7. This indicates that the\nstrain-dependence of the calculated anisotropy energies\nis rather independent of the speciﬁc cation arrangement,\nwhich allows us to obtain the magnetoelastic constants\nof CoFe 2O4and NiFe 2O4from our calculations.\nOn the other hand, we recall that due to the spe-\nciﬁc cation arrangementused in the calculations, and the\nresulting symmetry lowering from cubic to orthorhom-\nbic, the three cubic axes are not equivalent even for\nzero strain. For setting 1 the yandzdirections are\nequivalent, but diﬀerent from x, whereas for setting 3\nxandyare equivalent but diﬀerent from z. This is\nreﬂected in the calculated anisotropy energies for zero\nstrain (εxx= 0), which in setting 1 are largest between\nz/yand thexdirection. (Note that for zero strain both\nsettings are completely equivalent apart from a rotation\nof the coordinate axes by 120◦around the [111] direc-\ntion.) The anisotropy induced by the symmetry-lowering\nto orthorhombic is therefore at least of the same magni-\ntude as the cubic anisotropy in the disordered inverse\nspinel structure, and the calculated anisotropy energies\nfor zero strain are therefore not representative for the in-\nversespinel structurewith randomdistributionofcations\non theOhsites, i.e. the cubic anisotropyconstant Kcan-\nnot be determined from our calculations.\nFrom the data shown on Figs. 6 and 7, we thus obtain\nthe magnetoelastic coeﬃcients B1by using the appropri-\nate equation out of Eqs. (8) (together with the calculated\nvalues for ν2D) for each calculated energy diﬀerence in-\ndividually, and then average over the resulting values for\nB1within the same setting and for the same exchange-\ncorrelation functional. From the so-obtained magnetoe-\nlastic coeﬃcients B1we then calculate the linear magne-\ntostriction coeﬃcient λ100from Eq. (9) using the elastic\nmoduli determined in Sec. IIIB. The resulting values for\nbothB1andλ100are listed in Table V.\nIt can be seen that overallthe calculated magnetostric-\ntion constants are in very good agreement with avail-\nable experimental data, despite the diﬃculties related to\nthe speciﬁc choice of cation arrangement and exchange-\ncorrelation functional. In particular the large diﬀerence\nin magnetostriction between CoFe 2O4and NiFe 2O4is\nwell reproduced by the calculations. For NiFe 2O4the\ndiﬀerence in the calculated values for λ100due to the\nuse of either LSDA+ Uand GGA+ Uis larger than the\neﬀect of the diﬀerent cation settings. For CoFe 2O4the\ncation arrangement seems to have a larger inﬂuence on\nλ100than for NiFe 2O4. This is consistent with the large\nspread in the experimentally obtained magnetostriction\nfor CoFe 2O4, where diﬀerent preparation conditions can\nlead to diﬀerences in cation distribution/inversion or10\nTABLE V: Magnetoelastic coupling constant B1and mag-\nnetostriction constant λ100for CoFe 2O4and NiFe 2O4using\ndiﬀerent exchange-correlation functionals and cation arr ange-\nments(“setting” s)in comparison with available experimental\ndata.\nCoFe2O4 NiFe2O4\nsB1 λ100 B1 λ100\n[MPa] ( ×10−6) [MPa] ( ×10−6)\nLSDA+U1 — — 10.03 −54.9\n3 — — 9.65 −52.3\nGGA+U1 29.23 −189.7 6.72 −40.3\n3 39.74 −251.7 6.08 −35.9\nExp. −225.0a−50.9d\n−250.0b−36.0e\n−590.0c\naPolycrystalline CoFe 2O4(Ref. 49).\nbSingle crystals with Co 1.1Fe1.9O4composition (Ref. 50).\ncSingle crystals with Co 0.8Fe2.2O4composition (Ref. 50).\ndSingle crystals of NiFe 2O4(Ref. 51).\neSingle crystals with Ni 0.8Fe2.2O4composition (Ref. 50).\nslightlyoﬀ-stoichiometriccompositions. Inaddition,only\nvery few measurements have been performed on single\ncrystals, whereas for polycrystalline samples only a su-\nperposition of λ100andλ111is measured.\nTherathergoodagreementbetweenourcalculatedval-\nues forλ100and the available experimental data demon-\nstrates that in principle a quantitative calculation of\nmagnetostriction in spinel ferrites is feasible, in spite of\nthe diﬃculties related to the inversespinel structure with\nrandom cation distribution on the Bsite, and the usual\ndiﬃculties regarding an accurate description of exchange\nand correlation eﬀects in transition metal oxides.\nIV. SUMMARY AND OUTLOOK\nIn summary, we have presented a systematic ﬁrst prin-\nciples study of the eﬀect of epitaxial strain on the struc-\ntural and magneto-structural properties of CoFe 2O4and\nNiFe2O4spinel ferrites. Special care was taken to as-\nsess the quantitative uncertainties resulting from diﬀer-\nent treatments of exchange-correlation eﬀects and dif-\nferent cation arrangements used to represent the inverse\nspinel structure.\nIt has been shown, in agreement with earlier works,\nthat “beyond LSDA/GGA” methods are required for a\nproper description ofthe electronic and magnetic proper-\nties of spinel ferrites CoFe 2O4and NiFe 2O4. The “+ U”approachusedinthepresentworkleadstoarealisticelec-\ntronic structure and good quantitative agreement with\navailable experimental data for lattice, elastic, and mag-\nnetostrictive constants.\nWe ﬁnd that the speciﬁc cation arrangement used to\nrepresentthe inversespinel structure has only little eﬀect\non the structural properties. The corresponding eﬀect on\nthe magnetoelastic constants is also weak, with a some-\nwhat stronger inﬂuence in the case of CoFe 2O4. The\nlatter fact is consistent with the considerable spread in\nthe reported values for the magnetostriction constant for\ndiﬀerent samplesofthis material. Ingeneral, the starting\npoint for the “+ U” correction, i.e. either LSDA or GGA,\nhas a somewhat stronger inﬂuence on the calculated ma-\nterials constants than the diﬀerent cation settings.\nHowever, in spite of these uncertainties, the overall\nagreement between our results and experimental data is\nvery good. In particular, the calculated magnetostric-\ntion constants λ100of about −45 ppm for NiFe 2O4and\nabout−220 ppm for CoFe 2O4fall well into the spectra\nof available experimental values obtained from diﬀerent\nsamples (see Table V). Consistent with the negative sign\nofλ100, the easy magnetization direction changes from\nin-plane for compressive epitaxial strain to out-of-plane\nfor tensile strain. This gives further conﬁrmation that\nthe reorientation of the easy axis observed experimen-\ntally in thin ﬁlms of CoFe 2O4under diﬀerent conditions\nis indeed predominantly strain-driven.13,14,48\nIn summary, our results indicate that a quantitative\ndescription of both structural and magnetoelastic prop-\nerties in spinel ferrites is possible within the DFT+ U\napproach, which opens the way for future computational\nstudies of these materials. Such calculations can then\nprovide important information regarding the eﬀect of\ncation inversion and oﬀ-stoichiometry, which can be used\nto optimize magnetostrictionconstants and anisotropyin\nspinel ferrites.\nAcknowledgments\nThis work was supported by Science Foundation Ire-\nland under Ref. SFI-07/YI2/I1051and made use of com-\nputational facilities provided by the Trinity Centre for\nHigh Performance Computing (TCHPC) and the Irish\nCentre for High-End Computing (ICHEC). D.F. ac-\nknowledges fruitful discussions with M. 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LGEP – SUPELEC, Plateau de Moulon, 11 rue Joliot -Curie, 91192 Gif Sur Yvette  \n \n \nAbstract  : The temperature dependence of core loss in cobal t substituted Ni-Zn-Cu \nferrites was investigated. Co 2+  ions are known to lead to a compensation of the ma gneto-\ncrystalline anisotropy in Ni-Zn ferrites, at a temp erature depending on the cobalt content \nand the Ni / Zn ratio. We observed similar behaviou r in Ni-Zn-Cu and it was found that \nthe core loss goes through a minimum around this ma gneto-crystalline anisotropy \ncompensation. Moreover, the anisotropy induced by t he cobalt allowed a strong decrease \nof core loss, a ferrite having a core loss of 350 m W/cm 3 at 80°C was then developed \n(measured at 1.5 MHz and 25 mT). This result repres ents an improvement of a factor 4 \ncompared to the state of art Ni-Zn ferrites. \n \n \nKeywords  : NiZnCu ferrites, cobalt substitution, core loss versus temperature, permeability  \n \n \n \n \n \n \n \n  21 Introduction \n \nNickel-zinc-copper ferrites are essential materials  because of their high permeability in MHz \nrange. Moreover, their low sintering temperature ma kes them suitable for the realization of \nintegrated components in power electronics. As for nickel-zinc ferrites, cobalt substitution is an \nefficient technique to decrease the permeability [1 ] and the magnetic losses of nickel-zinc-copper \nferrites [2]. The effect of cobalt is to allow the pinning of the domain wall by inducing a magnetic \nanisotropy [3]. The consequence is a decrease of ta n δµ at high frequency [4] and also an \nimprovement of core loss [2]. The aim of this paper  is to study the effect of cobalt substitution on \ncore losses versus temperature. A lot of papers inv estigated core losses of spinel ferrites versus \ntemperature but it was mostly for Mn-Zn power ferri tes [5] and not at high frequency. There are \nvery few papers concerning core losses versus tempe rature of Ni-Zn or Ni-Zn-Cu ferrites [6][7], \nmoreover, they only investigated losses of a single  composition measured at low frequency (50 \nkHz). In order to understand how the ferrite compos ition influences the core loss, ferrites with \nthree Ni / Zn ratios were studied (Ni / Zn = 0.43, 1 and 3) with, for each ratio, cobalt substitutions  \nup to 0.035 mol per formula. \n \n \n2 Experimental procedure  \n \n2.1 Sample preparation \n \nFerrites were synthesized using the conventional ce ramic route. The raw materials (Fe 2O3, NiO, \nZnO, CuO) were ball milled for 24 hours in water. C o 3O4 was then added before the calcination \naround 800°C in air for 2 hours. The calcined ferri te powder was then milled by attrition for 30 \nmin. The resulting powder was compacted using axial  pressing.     The sintering was performed at \n935°C for 2 hours in air. Magnetic characterization s were done on ring shaped samples with the \nfollowing dimensions : outer diameter = 6.8 mm; inn er diameter = 3.15 mm; height = 4 mm. \n \n \n2.2 Sample measurements \n \nBulk density was deduced from weight and dimensions . Saturation magnetization was \nmeasured on a magnetic balance. Initial complex per meability was measured versus frequency \nbetween 1 MHz and 1 GHz using an HP 4291impedance-m eter. Static initial permeability (µ s) was \ndefined as µ’ at 1 MHz because for these ferrites µ ’ is constant from very low frequencies to the \nmegahertz range. For the permeability versus temper ature measurements, the rings were wound \nwith a copper wire and placed in an oven going from  –70°C to 150°C. µ s was deduced from the \ninductance measured at 100 kHz by an 4194A impedanc e-meter Agilent. Core losses were \nmeasured at 1.5 MHz and 25 mT using a Clark-Hess 25 8 wattmeter and a 100 W Kalmus HF \namplifier. \n \n \n3 Results and discussion \n \n3.1 Physicochemical characterizations \n \nThree different series of ferrites were studied: \n                                                 \n  3 \n- Ni 0.24 Zn 0.56 Cu 0.20 Co εFe 1.98 O4-γ \n- Ni 0.40 Zn 0.40 Cu 0.20 Co εFe 1.98 O4-γ' \n- Ni 0.60 Zn 0.20 Cu 0.20 CoεFe 1.98 O4-γ'' \n \nFour formulations were done for each series, with c obalt rate of 0, 0.014, 0.028 and 0.035 mol \nper formula. The copper rate was 0.20 mol to allow the densification below 950°C. Three Ni / Zn \nratios were studied in order to investigate the inf luence of cobalt substitution on ferrite with \ndifferent magneto-crystalline anisotropy. Indeed, K 1 of the host crystal increases with nickel \ncontent [8]. \n \nThe samples were sintered at 935°C for 2 hours in a ir. Figure 1 shows the X-ray diffraction \npattern of a Ni 0.40 Zn 0.40 Cu 0.20 Fe 2O4 ferrite sintered at 935°C. Only the spinel structu re can be \nobserved, none of the precursor oxides are present in the sintered materials. \n \nThe bulk densities of the sintered ferrites are sho wn in figure 2. All the ferrites have a low \nporosity ( ρ > 92 % of the theoretical density). The cobalt sub stitution does not seem to have a \nsignificant effect on densification, a slight incre ase of the densification can however be noticed. \nThis phenomenon is probably a consequence of the sl ight increase in the iron default of the ferrite. \n \nMicrostructure of the ferrites were also observed, SEM pictures are shown in figure 3. The \ngrain size is not affected by the Ni / Zn ratio. Ev en if the nickel is known to lead to a more difficu lt \ndensification [9], it is not observed here. The cob alt rate does not change the microstructure. All \nthe ferrites exhibit the same kind of microstructur e, with a low porosity and an average grain size \nbetween 1.5 and 2 µm. The grain size repartition is  relatively homogenous, although one can see a \nfew larger grains with defaults inside. \n \n \n3.2 Magnetic characterisations \n \nSaturation magnetization (M s) at room temperature was measured (see table 1). T his parameter \nmainly depends on the Ni / Zn ratio. M s increases up to a Ni / Zn ratio close to 1, then, it decreases \nwith the zinc content. The behaviour of the develop ed ferrites is coherent with previous studies \nconcerning Ni-Zn [8] and Ni-Zn-Cu [9] ferrites. The  maximum of the saturation magnetization is \nobtained for the Ni 0.40 Zn 0.40 Cu 0.20 Co εFe 1.98 O4-γ' ferrites. Copper slightly lowers M s compared to the \ncorresponding Ni-Zn ferrite [10]. \nCobalt substitution does not have a significant inf luence on the saturation magnetization. \nCobalt introduction must lead to an increase of the  saturation magnetization, due to Co 2+  ions \nmagnetic moment, which is higher than the other div alent ions present in octahedral sites (Ni 2+  and \nCu 2+ ) [11]. The measurement device used for this study is probably not precise enough to discern a \nvariation in M s with such a small amount of Co 2+ .  \n \nFor each ferrite, the static initial permeability a nd the core loss were measured versus \ntemperature. The results are shown in figure 4. \n \nThe initial static permeability versus temperature of the cobalt substituted ferrites is not \nmonotonous ; A local maximum can be observed even i f for high µ s samples, it is flattened. This is \ndue to the contribution of the cobalt ions to the m agneto-crystalline anisotropy (K 1). Several \nstudies were carried out on this phenomenon, which can be useful to increase the permeability \naround a particular temperature [12]. The permeabil ity is due to two contributions : the domain  4wall displacements and the magnetization rotation :  \n \nDKMs\ns ..16 31\n12\n=−µ     for the domain wall motion contribution [13] \n  \n12\n.321KMs\ns∝−µ      for the spin rotation contribution [14] \n \nWith D the average grain size and K 1 the magneto-crystalline anisotropy of the ferrite.  In this \nstudy, we consider that the grain size is similar w hatever the composition, so this parameter does \nnot affect the permeability. The variation of perme ability versus temperature is then only \ndepending on the variations of M s and K 1 versus temperature. \nAs the saturation magnetization of these ferrites i s monotonically decreasing versus \ntemperature, such a change in permeability is there fore due to a change in magnetic anisotropy. K 1 \nof the Ni-Zn-Cu spinels is weak and negative (betwe en –2 and –6,5.10 3 J/m 3, for NiFe 2O4, \nZnFe 2O4 and CuFe 2O4 [8]). The cobalt ferrite has a high and positive K 1 (close to 300.10 3 J/m 3 at \nroom temperature [8]). Addition of a small amount o f cobalt will thus lead to a compensation of \nK1. Looking at the previous relations, if there is a magneto-crystalline anisotropy compensation, \nthe permeability will go through a maximum. This ex plains the apparition of a local maximum in \nµs(T) curves of cobalt substituted ferrites at a comp ensation temperature (T 0) increasing with the \ncobalt content and K 1 of the Ni-Zn-Cu host crystal.  \n \nThe compensation temperature of the three series of  ferrites are shown in table 2, these results \nare established from µ s(T) curves (Figures 4.A1, B1 and C1). For a given N i /Zn ratio, T 0 increases \nwith the cobalt content. For the same cobalt rate, T 0 increases with the nickel content, i.e. with \nmagneto-crystalline anisotropy of the Ni-Zn-Cu host  crystal. At low temperature, the magneto-\ncrystalline anisotropy is positive due to Co 2+  ions contribution. At high temperature, Ni-Zn-Cu- \nhost crystal contribution becomes preponderant and K 1 is negative. The measurements are thus \ncoherent with the single-ion model theory.  \n \nThe cobalt substitutions are also known to pin the domain walls and to lower the domain wall \ndisplacements contribution to the permeability. Thi s results in a decrease of µ s with the increase of \nthe cobalt content, as can be seen in figures 4.A1 and 4.B1. For the third series (Ni / Zn = 3, figure  \n4.C1), the magneto-crystalline compensation is part icularly marked. The permeability of cobalt \nsubstituted ferrites can then become higher than th at of the cobalt-free ferrite (curve B, figure \n4.C1). It seems that for these ferrites, a slight c obalt substitution leads to a smaller K 1 than that of \nthe cobalt-free ferrite. \n \nTemperature dependence of core loss \n \nCore loss variations mainly depend on saturation ma gnetization and effective anisotropy (since \nwe consider that the microstructures are similar fo r all the materials). Indeed, it is known that \nlarger grain size can increase core loss [15]. For the cobalt-free ferrites these two parameters \ndecrease versus temperature. \nThe Ni 0.24 Zn 0.56 Cu 0.20 Fe 1.98 O4 ferrite (curve A of the figure 4.A2) has core loss  decreasing until \n20°C and rapidly increasing above 120°C. To underst and this behaviour, we have to look at the \ntemperature dependence of the permeability. Until 8 0°C, the anisotropy decreases faster than the \nsaturation magnetization, which results in an incre ase of the permeability and a decrease of the \ncore loss. Above 80°C, the initial static permeabil ity reaches 1000, shifting the µ” resonance \nfrequency towards 3 MHz. As the measurements are do ne at 1.5 MHz, relaxation losses become  5therefore predominant and lead to a fast increase o f the core loss at high temperature. \nWhen the nickel content increases, the Curie temper ature rises. For the \nNi 0.40 Zn 0.40 Cu 0.20 Fe 1.98 O4-γ’ ferrite (curve A, figure 4.B2), T c is around 340°C. The core loss \nmeasurements between –50 and 150°C are continuously  decreasing. This is again related to the \nvariation of permeability which constantly increase s while maintaining a resonance frequency far \nenough from 1.5 MHz. \nFor the Ni 0.60 Zn 0.20 Cu 0.20 Fe 1.98 O4 ferrites (curve A, figure 4.C2), one can note that  the \npermeability is almost constant in the studied temp erature range. The core losses are thus nearly \nstable between –50 and 150°C, a slight decrease can  be observed at high temperature. Concerning \nthe core loss values of the Ni-Zn-Cu ferrites, the higher the magneto-crystalline anisotropy, the \nhigher the core loss. The power losses at room temp erature increase from 600 mW/cm 3 to 2800 \nmW/cm 3 when nickel content goes from 0.24 mol to 0.60 mol  per formula. \n \nCobalt substituted ferrites have a completely diffe rent behaviour. The core loss goes through a \nminimum around T 0 (magneto-crystalline compensation temperature) and  their values are also \nhighly reduced near this compensation compared to t he cobalt-free ferrites. This phenomenon \nappears to be similar to the K 1 compensation observed in Mn-Zn spinels [12]. This compensation, \ndue to Fe 2+  ions, also leads to a minimum in the core loss. If  this is well known for Mn-Zn ferrites, \nit is the first time that it is observed in Ni-Zn b ased ferrites. \nSuch low core losses around T 0 are not only due to the K 1 compensation but are also the result \nof the cobalt induced anisotropy (K u). Small amounts of Co 2+  ions in Ni-Zn spinels are known to \npin the domain walls, however, the temperature depe ndence of this induced anisotropy is not well \nknown. In order to study this parameter, Core loss versus magnetic induction of the \nNi 0.40 Zn 0.40 Cu 0.20 Co 0.028 Fe 1.98 O4+γ  ferrite was measured at different temperatures (fi gure 5). The \ncobalt substituted ferrites have not a classical be haviour of core loss versus induction. When \ninduction increases, the core loss increases almost  linearly until a threshold induction \ncorresponding to the required energy to unpin the d omain walls. Above this threshold induction \n(which depends on K 1 and K u), the anisotropy induced by the cobalt is no longe r efficient so the \ncore loss increases fast. This phenomenon appears c learly on figure 5 for the measurement at \n20°C, the core loss increases slightly up to 35 mT and then rises rapidly. When the temperature \nincreases, the threshold induction decreases and, f or the measurement done at 100°C, the core loss \nnearly recovers a classical behaviour with an evolu tion proportional to the square of the magnetic \ninduction. This ferrite has a T 0 close to 10°C, therefore the measurements done on figure 5 tend to \nprove that the cobalt induced anisotropy is more ef ficient around the magneto-crystalline \nanisotropy compensation. \n \nConsequently, around T 0 the magnetic configuration is doubly favourable in  order to have low \npower loss with a weak effective anisotropy and an induced anisotropy that improve the linearity \nof the ferrite. This can then lead to very low core  loss of respectively 150 and 200 mW/cm 3 (at 1.5 \nMHz, 25 mT and 20°C) for Ni / Zn = 0.43 and Ni / Zn  = 1 ferrites with cobalt content of 0.028 \nmol. The third series of ferrites have higher core loss with a minimum of 400 mW/cm 3. It seems \nthat K 1 of the Ni-Zn-Cu host crystal (which increases due to high nickel content) is too high to \nhave very low core loss.  \n  \n3 Conclusion \n \nThe study of the temperature dependence of core los s of cobalt-substituted Ni-Zn-Cu ferrites \nhighlighted that the core losses were minimum aroun d the magneto-crystalline anisotropy \ncompensation. This is extremely interesting for ada pting the ferrite to the operating range \ntemperature for power applications. Today, power el ectronics need materials that can work at high  6frequency and high temperature (> 80°C). The ferrit es studied in this paper can perfectly answer to \nthese issues. Indeed, for the Ni 0.40 Zn 0.40 Cu 0.20 Co εFe 1.98 O4+ γ ferrites (figure 4.B2), core loss at 80°C \ncan be divided by a factor 4 thanks to cobalt subst itution. These materials represent also a real \nimprovement compared to the state of art Ni-Zn ferr ite for radiofrequency, with core loss 4 times \nlower than a commercial ferrite with µ s=120 (figure 6). \n  7References \n \n[1]  T. Y. Byun, S. C. Byeon, K. S. Hong, “Factors affecting initial permeability of Co-substituted \nNi-Zn-Cu ferrites”, IEEE, vol 35, Issue 5, Part 2, pages : 3445-3447 (1999) \n \n[2]  R. Lebourgeois, J. Ageron, H. Vincent and J-P.  Ganne, low losses NiZnCu ferrites (ICF8), \nKyoto and Tokyo, Japan (2000) \n \n[3]  L. Néel, J. Phys. Radium 13, 249 (1952) \n \n[4]  J. G. M. De Lau, A. Broese Van Groenou, J. De Phys. IV 38, C1-17 (1977) \n \n[5] V. T. Zaspalis, E. Antoniadisa, E. Papazogloua,  V. Tsakaloudia, L. Nalbandiana and C. A. \nSikalidisb, “The effect of Nb 2O5 dopant on the structural and magnetic properties o f MnZn \nferrites”, Journal of Magnetism and Magnetic Materi als Volume 250, Pages 98-109, (2002) \n \n[6]  H. Su, H. Zhang, X. Tang, Y. Jing, Y. Liu, “Ef fects of composition and sintering temperature \non properties of NiZn and NiCuZn ferrites” Journal of Magnetism and Magnetic Materials 310, \np17–21, (2007) \n \n[7]  Y. Matsuo, M. Inagaki, T. Tomozawa, and F. Nak ao, IEEE Transactions On Magnetics, Vol. \n37, n°. 4, (2001) \n \n[8]  J. S. Smit and H. P. J. Wijn, Ferrites, Philip s technical library (1961) \n \n[9]  J. Ageron ,Thesis (1999) \n \n[10]  G. K. Joshi, A. Y. Khot and S. R. Sawant, “Ma gnetisation, curie temperature and Y-K angle \nstudies of Cu-substituted and non substituted Ni-Zn  mixed ferrites”, Solid State Communications, \nvol 65 n°12, 1593-1595 (1988) \n \n[11]  E. Rezlescu, L. Sachelarie, P. D. Popa, and N . Rezlescu,“ Effect of Substitution of Divalent \nIons on the Electrical and Magnetic Properties of N i–Zn–Me Ferrites”, IEEE transactions on \nmagnetics, vol 36 n°6 (2000) \n \n[12]  H. Pascard, Basic concepts for high permeabil ity in soft ferrites, J. Phys. IV France, 377-384 \n(1998) \n \n[13]  R. Lebourgeois, C. Le Fur, M. Labeyrie, M. Pa té, J-P. Ganne, “Permeability mechanisms in \nhigh frequency polycrystalline ferrites”, Journal o f Magnetism and Magnetic Materials, Volume \n160, Pages 329-332,(1996) \n \n[14]  J. L. Snoek, Physica 14, 207 (1948) \n \n[15] R. Lebourgeois, S. Duguey, J.-P. Ganne, J.-M. Heintz, “Influence of V 2O5 on the magnetic \nproperties of nickel–zinc–copper ferrites”, Journal  of Magnetism and Magnetic Materials, Volume \n312, Issue 2, Pages 328-330, (2007) \n \n  8 \nFerrites Ms \n(emu/g) \n  Ni/Zn=0.43    Co=0 65.5 \nNi/Zn=0.43    Co=0.014 65.5 \nNi/Zn=0.43    Co=0.028 64.4 \nNi/Zn=0.43    Co=0.035 65.2 \nNi/Zn=1    Co=0 79.8 \nNi/Zn=3    Co=0 72.4 \n \nTable 1 : Saturation magnetization at room temperat ure of materials sintered at 935°C for 2 \nhours in air. \n \n \n \n% Cobalt T0                \nNi /Zn = 0.43  T0                 \nNi /Zn = 1 T0                 \nNi /Zn = 3 \n0.014 -45°C -50°C -28°C \n0.028 -7°C 5°C 30°C \n0.035 3°C 17°C 50°C \n \nTable 2 : Magneto-crystalline anisotropy compensati on of the three series of ferrites, determined \nfrom µ s(T) curves.  9\n              \n \nFigure 1 : X-ray diffraction pattern of Ni 0.40 Zn 0.40 Cu 0.2 Fe 1.98 O4 ferrite (black curve), comparison \nwith JCPDF 00-052-0277 (red peaks). \n \n \n \n \nFigure 2 : Density of the three series of ferrites sintered at 935°C for 2 hours in air. \n \n \n \nNi/Zn=0.43  _  Co=0 \n \nNi/Zn=1  _  Co=0  \n \nNi/Zn=1  _  Co=0.028  \n \nFigure 3 : SEM Micrographics of three materials sin tered at 935°C for 2 hours in air. \n \n  10  \n \nNi 0.24 Zn 0.56 Cu 0.20 Co εε εεFe 1.98 O4+ γγ γγ \n \n(4.A1) \n \n \n(4.A2)  \n \n \nNi 0.40 Zn 0.40 Cu 0.20 Co εε εεFe 1.98 O4+γγ γγ \n \n(4.B1)  \n \n \n(4.B2)  \n \n \nNi 0.60 Zn 0.20 Cu 0.20 Co εε εεFe 1.98 O4+γγ γγ \n \n(4.C1)  \n \n(4.C2) \n \nFigure 4 : Initial static permeability versus tempe rature and core loss at 1.5 MHz and 25 mT versus \ntemperature of Ni 0.24 Zn 0.56 Cu 0.20 Co εFe 1.98 O4+γ, Ni 0.40 Zn 0.40 Cu 0.20 Co εFe 1.98 O4+γ and \nNi 0.60 Zn 0.20 Cu 0.20 Co εFe 1.98 O4+γ ferrites \n \n  11 \n \nFigure 5 : Core loss of the Ni 0.40 Zn 0.40 Cu 0.20 Co 0.028 Fe 1.98 O4+ γ ferrite measured at 1.5 MHz versus magnetic \ninduction for different temperatures.  \n \n \n \nFigure 6 : Core loss versus temperature measured at  1.5 MHz and 25mT. Comparison between a commercial Ni-\nZn ferrite (µ s=120) and the Ni 0.40 Zn 0.40 Cu 0.20 Co 0.028 Fe 1.98 O4+ γ ferrite (µ s=130) "    },    {        "title": "1810.10555v1.Electronic_and_optical_properties_of_spinel_zinc_ferrite___Ab___initio__hybrid_functional_calculations.pdf",        "content": "Electronic and optical properties of spinel zinc ferrite:\nAb initio hybrid functional calculations\nDaniel Fritsch\nDepartment of Chemistry, University of Bath, Claverton Down, Bath BA2 7AY, United Kingdom\n(Dated: July 3, 2021)\nSpinel ferrites in general show a rich interplay of structural, electronic, and magnetic properties.\nHere, we particularly focus on zinc ferrite (ZFO), which has been observed experimentally to crys-\ntallise in the cubic normal spinel structure. However, its magnetic ground state is still under dispute.\nIn addition, some unusual magnetic properties in ZFO thin \flms or nanostructures have been ex-\nplained by a possible partial cation inversion and a di\u000berent magnetic interaction between the two\ncation sublattices of the spinel structure compared to the crystalline bulk material. Here, density\nfunctional theory has been applied to investigate the in\ruence of di\u000berent inversion degrees and\nmagnetic couplings among the cation sublattices on the structural, electronic, magnetic, and optical\nproperties. E\u000bects of exchange and correlation have been modelled using the generalised gradient\napproximation (GGA) together with the Hubbard \\+ U\" parameter, and the more elaborate hybrid\nfunctional PBE0. While the GGA+ Ucalculations yield an antiferromagnetically coupled normal\nspinel structure as the ground state, in the PBE0 calculations the ferromagnetically coupled normal\nspinel is energetically slightly favoured, and the hybrid functional calculations perform much better\nwith respect to structural, electronic and optical properties.\nPACS numbers: 71.15.Mb,71.20.Nr,78.20.Bh\nZinc ferrite or ZnFe 2O4(ZFO) belongs to the min-\neral class of spinel ferrites crystallising in a cubic crys-\ntal structure with the general formula MFe2O4. Com-\nmon to all spinel ferrites is the face-centred cubic (fcc)\narrangement of the oxygen anions. In the so-called nor-\nmalspinel structure the octahedrally coordinated B-sites\n(Oh) are solely populated with Fe3+cations whereas the\ntetrahedrally coordinated A-sites ( Td) accomodate a di-\nvalentd-metal cation, e.g., M2Mn, Fe, Co, Ni, Cu,\nZn, Mg, Cd.1One also \fnds mixed-valence occupancies,\n(M1\u0000\u0015Fe\u0015)[M\u0015Fe2\u0000\u0015]O4, with the inversion parameter \u0015\nranging from \u0015= 0 for the normal to\u0015= 1 for the so-\ncalled inverse spinel. Combining a divalent cation with\na non-integer inversion degree, which can partly be tai-\nlored by experimental growth conditions,2o\u000bers a huge\nvariety of structural, electronic, and magnetic properties\nwithin the mineral class of spinel ferrites.\nAn example for a fully inverse spinel ferrite is NiFe 2O4\n(NFO), whereas CoFe 2O4(CFO) has a partially inverse\nstructure with \u0015\u00190:8.3Both, NFO and CFO, are\nferro(i)magnetic (FM) insulators with high Curie temper-\natures and large saturation magnetisations,4with possi-\nble applications in arti\fcal multiferroic heterostructures\nor spintronics devices.5An example for a normal spinel\nis ZFO. Although experimentally mostly reported to be\nan antiferromagnetic (AFM) insulator, with a compar-\natively low N\u0013 eel temperature of TN\u001910 K,3,6,7there\nis still an ongoing discussion about its magnetic ground\nstate. While very early models suggested various AFM\narrangements of the magnetic moments among the A-\nand B-site sublattices,8{11in recent experiments on high-\nquality single crystals, no AFM order has been observed\nfor temperatures down to 1.5 K, and the AFM order\nis actually thought to originate from defects or inhomo-\ngeneities.12This makes the magnetic structure of spinel ferrites\nparticularly interesting, e.g. the interactions among the\nmagnetic cations distributed over the A- and B-sites, and\npossible phase transitions. A \frst understanding of the\nmagnetic interactions in spinels was provided by the N\u0013 eel\nmodel13which described a collinear arrangement of mag-\nnetic moments among the A- and B-site sublattices. This\nis realised, e.g., in NFO and CFO, where the A- and B-\nsite cations order ferromagnetically among themselves,\nand antiferromagnetically with respect to the other sub-\nlattice. With the d5(Fe3+) andd7(Co2+) \\levels\" of\nd-orbital population the additional choice of high-spin\nvs low-spin complexes arises. However, for the tetrahe-\ndrally coordinated A-sites only high-spin complexes are\nobserved, and within the mentioned materials the B-sites\nhave high-spin complexes as well. Di\u000berent inversion de-\ngrees combined with possible high-spin vs low-spin com-\nplexes add another complication to properly describe the\ncorrect magnetic ground state of spinel ferrites. With\na non-magnetic cation (such as Zn2+in ZFO) populat-\ning the A-sites it becomes more favourable for the Fe3+\ncations to order antiferromagnetically on the B-sites be-\nlow 10 K; contrary to the ferromagnetic coupling in NFO,\nCFO, and ZFO at higher temperatures.3,7\nIn addition, contrary to the normal spinel ground state\nof bulk ZFO, several properties in strained thin \flms\nand nanoparticles, like increased magnetisation, have\nbeen explained based on a partial inversion and/or a\nAFM to FM change in the B-site magnetisation. Ex-\nperimental works so far dealt with single crystals,12epi-\ntaxial thin \flms,2,14{17and nanocrystalline samples,18,19\nrespectively. The optical properties of ZFO have only\nbeen rarely investigated.15,17,20,21Available theoretical\nworks focused on the ground state properties of ZFO\nusing various exchange-correlation functionals, includ-arXiv:1810.10555v1  [cond-mat.mtrl-sci]  24 Oct 20182\ning variants of the generalised gradient approximation\n(GGA),22{24di\u000berent parametrisations of the GGA+ U\nfunctional,16,22,25and a recent GGA plus many-body\ncorrection investigation,21respectively. Although more\nelaborate hybrid functionals have proven to give im-\nproved results for other spinel ferrites,26systematic stud-\nies for ZFO are missing to date.\nHere, we focus on di\u000berences between the collinear\nAFM and FM normal spinel structure, as well as inverse\nstructures with \u0015= 0:5 and\u0015= 1:0. Presented results\ninclude structural (lattice constant a, oxygen parameter\nu, bulk modulus B), electronic (density of states (DOS),\nband structure), magnetic, and optical (dielectric func-\ntions) properties.\nI. COMPUTATIONAL DETAILS\nThe results of the present work have been obtained\nby density functional theory (DFT) calculations employ-\ning the Vienna ab initio Simulation package (VASP)27{29\ntogether with the projector-augmented wave (PAW) for-\nmalism.30Spin-polarisation has been taken into account.\nStandard VASP PAW potentials were used with 14, 12,\nand 6 valence electrons for iron, zinc, and oxygen, re-\nspectively. Approximations to the exchange-correlation\npotential used in the present work are the GGA together\nwith Hubbard \\+ U\" method31based on the Perdew-\nBurke-Ernzerhof (PBE) parametrisation,32and the more\nelaborate hybrid functional PBE0,33where a quarter of\nthe exchange potential is replaced by Hartree-Fock exact-\nexchange. The Hubbard \\+ U\" method has been applied\nin the simpli\fed, rotationally invariant version of Du-\ndarev et al. ,34with a value of Ue\u000b=U\u0000J= 5:25 eV\napplied to the Fe cations. This value of Ue\u000bhas been\nchosen to reproduce the experimental band gap of ZFO\nof around 2.2 eV,35,36is similar to the Ue\u000b= 5:9 eV\napplied to FeO31and is slightly larger than Ue\u000b= 3\neV applied to NFO and CFO.4,37Both functionals, the\nGGA+Uand the hybrid PBE0, are employed to over-\ncome the insu\u000ecient description of electronic exchange\ne\u000bects in standard DFT calculations.31,33\nThe spinel structure crystallises in the cubic space\ngroup Fd \u00163m (No. 227) with the smallest possible unit\ncell accomodating two functional units (f.u.) of ZFO and\n14 atoms. Due to the necessary application of periodic\nboundary conditions, the calculations face an arti\fcial\nsymmetry reduction depending on the chosen inversion\ndegree and magnetic order.4,38,39Similar to our earlier\nworks on spinel ferrites, structural relaxations were per-\nformed for several volumes of the smallest possible unit\ncell within a scalar-relativistic approximation and allow-\ning the internal structure parameter to relax until all\nforces on the atoms were smaller than 0.001 eV \u0017A\u00001.\nDense \u0000-centred kpoint meshes of 8 \u00028\u00028 for the\nBrillouin zone integration and a cuto\u000b energy of 800 eV\nensured converged structural parameters and total ener-\ngies within meV accuracy for the GGA+ Ucalculations.For the more demanding PBE0 calculations, \u0000-centred\nkpoint meshes of 6 \u00026\u00026 and a cuto\u000b energy of 500\neV have been used. Ground-state geometries were de-\ntermined by a cubic spline \ft to the total energies with\nrespect to the unit cell volumes.\nII. RESULTS AND DISCUSSION\nThe obtained structural properties of ZFO are given\nin table I in comparison to available experimental re-\nsults. The GGA+ Ucalculated lattice constants, oxygen\nparameters u, and bulk moduli for the normal spinel are\n8.541 \u0017A, 0.261, and 161.1 GPa, and 8.545 \u0017A, 0.260, and\n162.0 GPa for the AFM and FM arrangement of mag-\nnetic moments among the B-site sublattice, respectively.\nThe respective PBE0 calculated lattice constants, oxy-\ngen parameters u, and bulk moduli are 8.446 \u0017A, 0.261,\nand 173.8 GPa, and 8.452 \u0017A, 0.260, 174.1 GPa, and are\nin much better agreement to the experimental results of\n8.441 \u0017A [40] (8.4599 \u0017A [6]), 0.260 [41], and 175.0 GPa [42],\nrespectively. This is in line with the general trend that\nhybrid functional calculations for oxide semiconductors\nyield better agreement of lattice parameters and bulk\nmoduli with respect to experimental results.43\nThe AFM arrangement of magnetic moments among\nthe B-site sublattice is the ground state for the GGA+ U\ncalculations, in agreement with experiment and other\ntheoretical investigations, whereas the FM arrangement\nis slightly favoured for the hybrid functional PBE0 cal-\nculations.\nAs can be seen from the data in table I, with increasing\ninversion degree \u0015the lattice constants decrease from the\nnormal spinel AFM values for both, GGA+ Uand PBE0\nfunctional calculations, in agreement with experimental\nresults of Wu et al. ,19who report a lattice constant of\n8.377 \u0017A for a partially inverse ZFO sample with \u0015= 0:5.\nBoth, the GGA+ Uand PBE0 calculated bulk moduli\nincrease with increasing inversion degree \u0015.\nThe DOS of normal ZFO with a AFM and FM ar-\nrangement of magnetic moments among the B-site sub-\nlattices, partially inverse ZFO ( \u0015= 0:5) and fully inverse\nZFO (\u0015= 1:0) are shown in \fgure 1. The left and right\npanels depict the results of GGA+ Uand PBE0 calcu-\nlations, and the zero energy is set to the valence band\nmaximum, respectively. In each of the panels the to-\ntal DOS is shown as shaded grey area, and the spin-up\n(spin-down) contributions are given along the positive\n(negative)y-axis. Generally, apart from the larger band\ngap for the PBE0 calculations, the overall shape of the\nDOS looks quite similar for the GGA+ Uand the PBE0\ncalculations.\nFrom the DOS with respect to the inversion degree \u0015it\ncan be seen that an increasing inversion degree shifts the\nvalence band DOS towards higher energies, and exhibits\na growing contribution to the tetrahedral Fe and octahe-\ndral Zn DOS, respectively. From \fgure 1 it can also be\nseen that both, GGA+ Uand PBE0 calculations, yield3\nTABLE I. Ground state structural properties (lattice constant a, oxygen parameter u, and bulk modulus B) of ZFO calculated\nwith the GGA+ Uand PBE0 exchange-correlation functionals and di\u000berent inversion degrees \u0015in comparison to available\nexperimental results. The theoretical data also includes the energy di\u000berence \u0001 Ewith respect to the most stable con\fguration.\nFor the normal spinel (\u0015= 0:0) the \frst (second) data set reports structural properties for the AFM (FM) arrangement of\nmagnetic moments among the B-site sublattice, respectively.\nGGA+U PBE0 Expt.\n\u0015 a u B \u0001E a u B \u0001E a u B\n[\u0017A] [GPa] [eV] [ \u0017A] [GPa] [eV] [ \u0017A] [GPa]\n0.0 8.541 0.261 161.1 0.0 8.446 0.261 173.8 0.030 8.441 [40] 0.260 [41] 175.0 [42]\n8.52 [22] 0.0 [22] 8.4599 [6]\n0.0 8.545 0.260 162.0 0.051 8.452 0.260 174.1 0.0\n8.53 [22] 0.068 [22]\n0.5 8.526 0.259 166.3 0.519 8.424 0.259 192.6 0.175 8.377 [19]\n1.0 8.515 0.250 169.8 0.761 8.407 0.250 203.7 0.213\n8.49 [22] 0.210 [22]\nan insulating behaviour, underlying the important e\u000bect\nof electronic exchange e\u000bects on the electronic properties\ncompared to plain GGA approaches.44\nThe local magnetic moments for Fe cations are found\nto be in the range of 4.1 to 4.3 \u0016B, with the GGA+ U\nvalues being slightly larger than the PBE0 ones, for all\ninversion degrees and arrangements of magnetic moments\namong the B-site sublattices. This is in favourable agree-\nment with the experimental value of 4.22 \u0016B[6] and the\nrange of 4.1 to 4.2 \u0016Breported from other GGA+ Ucal-\nculations,22respectively.\nFor the normal spinel and AFM arrangement of mag-\nnetic moments among the B-site sublattice it is no sur-\nprise that the spin resolved band structure depicted in\nthe upper panels of \fgure 2 shows he same behaviour\nfor the spin-up and spin-down channels, with a band gap\nof 2.21 eV (3.68 eV) for the GGA+ U(PBE0) calcula-\ntions. While the GGA+ Uband gap is close to the ex-\nperimental band gap of around 2.2 eV,35,36the PBE0\nband gap is close to the experimental band gap of 3.31\neV, obtained from spectroscopic ellipsometry measure-\nments.15This changes drastically for the FM arrange-\nment, where the band gap is now between the spin-up\nvalence band and the spin-down conduction band for\nboth, and amounts to 1.68 eV (3.13 eV) for the GGA+ U\n(PBE0) calculations, respectively. Changing the inver-\nsion degree \u0015from 0.5 to 1.0 only has a small in\ruence\non the band gap, which is now around 1.91 eV (3.37 eV)\nfor the GGA+ U(PBE0) calculations.\nBased on the obtained relaxed structural ground states\nfor di\u000berent inversion degrees \u0015and magnetic arrange-\nments on the B-site sublattices the optical properties\nhave been calculated and are depicted in \fgure 3. Gen-\nerally, the onsets of the imaginary parts of the PBE0\ndielectric functions ( \"2) are shifted towards higher ener-\ngies, as to be expected from the larger band gaps already\nseen in the DOS (\fgure 1) and the band structures (\fg-ure 2). Moreover, all the PBE0 dielectric functions are\n\ratter compared to the GGA+ Uones. If one accounts\nfor the slight changes in the observed band gaps, the over-\nall features of the imaginary part of the PBE0 dielectric\nfunction for normal ZFO and a AFM arrangement of the\nmagnetic moment among the B-site cations (upper right\npanel of \fgure 3), is in good agreement with the optical\ndata reported in.15\nIII. SUMMARY AND OUTLOOK\nIn summary, a detailed investigation on the in\ruence\nof the inversion degree \u0015and di\u000berent arrangements of\nthe magnetic moments among the B-site cations on the\nstructural, electronic, magnetic, and optical properties of\nthe spinel ferrite ZFO has been presented. It has been\nshown that the hybrid functional PBE0 performs better\nthan GGA+ U, and yields an overall better agreement\nwith respect to experimental results on structural, elec-\ntronic and optical properties. These hybrid functional\ncalculations provide the basis for future in-depth analyses\nof the optical properties in relation to the band structure\nand DOS, as well as taking into account strain e\u000bects\nfrom underlying substrates.4,39\nACKNOWLEDGMENTS\nThis research has received funding from the European\nUnions Horizon 2020 research and innovation programme\nunder grant agreement No 641864 (INREP). This work\nmade use of the ARCHER UK National Supercomput-\ning Service (http://www.archer.ac.uk) via the member-\nship of the UKs HPC Materials Chemistry Consortium,\nfunded by EPSRC (EP/L000202) and the Balena HPC\nfacility of the University of Bath.\n1J. Smit and H. P. J. Wijn, Ferrites (Cleaver-Hume Press,\nLondon, 1959).2Y. F. Chen, D. Spoddig, and M. Ziese, J. Phys. D: Appl.\nPhys. 41, 205004 (2008).4\n-10-50510DOS [eV-1]Fe (O h)GGA+ U\n-10-50510 λ=0.0 (AFM)PBE0\n-10-50510DOS [eV-1]Zn (T d)\n-10-50510 λ=0.0 (FM)\n-10-50510DOS [eV-1]Fe (T d)\n-10-50510 λ=0.5\n-8 -4 0 4\nE-E Fermi [eV]-10-50510DOS [eV-1]Zn (O h)\n-4 0 4\nE-E Fermi [eV]-10-50510 λ=1.0\nFIG. 1. (Color online) Total and projected DOS per formula\nunit for ZFO calculated with the GGA+ U(left panels) and\nPBE0 (right panels) exchange-correlation functionals for dif-\nferent inversion degrees \u0015and arrangements of the magnetic\nmoments among the B-site sublattice, respectively. The octa-\nhedralOh(tetrahedral Td) states of Fe are shown as straight\n(dashed) green lines, and the tetrahedral Td(octahedral Oh)\nstates of Zn are shown as straight (dashed) red lines, respec-\ntively. The shaded grey area in all panels depicts the total\nDOS. Minority spin projections are shown using negative val-\nues. The zero energy is set to the valence band maximum.\n3V. A. M. 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Fernández  a \na Departamento  de Electrocerámica,  Instituto de Cerámica  y Vidrio (ICV), CSIC, Madrid E-28049,  Spain \nb Departamento  de Física de Materiales,  Universidad  Complutense  de Madrid, Madrid E-28040,  Spain \nc CELLS-ALBA  Synchrotron,  Cerdanyola  del Vallès E-08290,  Spain \nd Instituto de Magnetismo  Aplicado,  UCM-ADIF,  Las Rozas E-28232,  Spain \na r t i c l e i n f o \nArticle history: \nReceived  8 March 2021 \nRevised 11 August 2021 \nAccepted  16 August 2021 \nAvailable  online 20 August 2021 \nKeywords:  \nRare-earth  free permanent  magnets \nHexaferrites  \nCold sintering  process \nCeramic densiﬁcation  \nMagnetic  properties  a b s t r a c t \nThe incessant  technological  pursuit towards  a more sustainable  and green future depends  strongly  on \npermanent  magnets.  At present,  their use is widespread,  making it imperative  to develop  new process-  \ning methods  that generate  highly competitive  magnetic  properties  reducing  the fabrication  temperatures  \nand costs. Herein, a novel strategy  for developing  dense sintered  magnets  based on Sr-hexaferrites  with \nupper functional  characteristics  is presented.  An innovative  cold sintering  approach  using glacial acetic \nacid as novelty,  followed  by a post-annealing  at 1100 °C, achieves  a densiﬁcation  of the ceramic  magnets  \nof 92% with respect to the theoretical  density and allows controlling  the particle  growth.  After the cold \nsintering  process,  a fraction  of amorphous  SrO is identiﬁed,  in addition  to a partial transformation  to \nα-Fe 2 O 3 as secondary  crystalline  phase. 46 wt% of SrFe 12 O 19 remains,  which is mostly recuperated  after \nthe post-thermal  treatment.  These ﬁndings  do not signiﬁcantly  modify the ﬁnal structure  of ferrite mag- \nnets, neither at short- nor long-range  order. The innovative  process has a positive  impact on the magnetic  \nproperties,  yielding  competitive  ferrite magnets  at lower sintering  temperatures  with an energy eﬃciency  \nof at least 25%, which opens up a new horizon  in the ﬁeld of rare-earth  free permanent  magnets  and new \npossibilities  in other applications.  \n©2 0 2 1 The Author(s).  Published  by Elsevier  Ltd on behalf of Acta Materialia  Inc. \nThis is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/  ) \n1. Introduction  \nConstant  technological  requirements  lead to research  into new \nmaterials  and development  of existing  ones with new or improved  \nattractive  functional  properties.  Speciﬁcally,  the M-type  hexafer-  \nrites (MFe 12 O 19 , M = Pb, Sr, Ba) exhibit  remarkable  structural  and \nhard magnetic  properties  [1–3] , and they are widely investigated  \n[ 2 , 4 , 5 ]. Ferrites  have a low cost as compared  to their competitor  \npermanent  magnets  (AlNiCo,  MnBi, MnAl or NdFeB)  and they are \noxides.  Therefore,  they do not present  the corrosion  problems  of \nmetallic  alloys-based  magnets  and their manufacturing  process  is \nmuch easier and cheaper  with a stability  of operation  over time, \ntemperature,  and radiation  [6] . Besides,  the intrinsic  characteristics  \nof hexaferrites  make them a suitable  and green alternative  to rare- \nearth-based  permanent  magnets  in a variety  of current  applica-  \n∗Corresponding  author. \nE-mail address: aida.serrano@icv.csic.es  (A. Serrano).  tions, such as in small electric  motors,  microwave  devices,  record-  \ning media, telecommunication  and electronic  industry,  and also in \nmicro and nano-systems  such as biomarkers,  biodiagnostics  and \nbiosensors  [7–11] . \nThe ceramic  magnets  based on hexaferrites  usually  require  high \nsintering  temperatures  ( > 1200 °C) [12] and long dwell times to \nachieve  density  values > 90% with respect  to the theoretical  den- \nsity. However,  during the solid-state  sintering  using thermal  con- \nventional  methodologies,  a particle  coarsening  process  towards  an \nexaggerated  growth  takes place, which is detrimental  to the mag- \nnetic response  [ 13 , 14 ]. Currently,  many strategies  are being investi-  \ngated to avoid these issues and the results are often unpromising.  \nFor example,  the incorporation  of secondary  phases is required  to \nensure densiﬁcation  at moderate  particle  sizes [12] but the sinter- \ning temperature  are not reduced  < 1200 °C. \nAn interesting  approach  to fabricate  the ceramics  magnets  at \nlower temperatures  and sintering  times, i.e. from a greener  per- \nspective,  may be the cold sintering  process  (CSP). The CSP is a rel- \nhttps://doi.org/10.1016/j.actamat.2021.117262  \n1359-6454/© 2021 The Author(s). Published by Elsevier Ltd on behalf of Acta Materialia Inc. This is an open access article under the CC BY license  \n( http://creativecommons.org/licenses/by/4.0/  ) A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nFig. 1. Representative  scheme of the cold sintering  route using GAA followed  by a \npost-annealing  step at 1100 °C for 2 h for obtaining  dense SFO ceramics,  1 → 5. \natively new sintering  route [15] in which inorganic  powder  com- \npounds  are mixed with an aqueous  solvent  that partially  solubi-  \nlizes, promoting  mass transport  at low temperatures  under an ap- \nplied uniaxial  pressure,  obtaining  a dense material  [16–18]  . This \nsintering  process  has been employed  for the densiﬁcation  of a \nlarge number  of compounds  using aqueous  solvents,  where many \nparameters  can be considered  to tailor the ﬁnal properties  of sin- \ntered materials  [15–18]  . Some works in this matter have even con- \nsidered  the use of pure organic  solvents  during the CSP. Berbano  \net al. employed  ethanol  in the CSP of Li 1.5 Al 0.5 Ge 1.5 (PO 4 ) 3 solid \nelectrolyte  [19] , Ndayishimiye  et al. used TEOS as transient  liquid \nin the preparation  of (1–x ) SiO 2 –x PTFE composites  [20] and Kang \net al. demonstrated  the use of dimethyl  sulfoxide  solutions  for the \ndensiﬁcation  of metal oxides [21] . However,  despite  the great ad- \nvances made in such a short time, there are still many open ques- \ntions and materials  that have not been considered  or sintered  by \nthis innovative  way. No CSP studies  on permanent  magnets  have \nbeen reported  on this matter.  \nHere, a novel route based on CSP employing  a non-aqueous  \nsolvent  is followed  to sinter dense permanent  magnets  based on \nSr-hexaferrites.  The innovative  approach  allows the structural  and \ncompositional  control  of the ceramic  magnets,  reaching  improved  \nmagnetic  properties  at lower sintering  temperatures  and with \ngreater  energy  eﬃciency.  This work also provides  a further  under- \nstanding  of the sintering  of hexaferrite-based  ceramics,  which open \nup a new horizon  in the ﬁeld of rare-earth  free permanent  mag- \nnets and beyond.  \n2. Experimental  methods  \n2.1. Materials  and processing  route \nDense hexaferrite  ceramics  were reached  by CSP using glacial \nacetic acid (GAA) as a pure organic  solvent  plus a subsequent  post- \nannealing.  Fig. 1 illustrates  the sintering  steps followed.  Prior to \nthe CSP, commercial  platelet-shaped  SrFe 12 O 19 (SFO) particles  from \nMax Baermann  GmbH Holding  (Germany)  and GAA from Sigma-  \nAldrich  were mixed in a proportion  of 50/50 wt% in an agate \nmortar  for 10 min until the solvent  is distributed  homogeneously  \naround  the ferrite particles  (Steps 1 and 2). The initial excess of \nGAA ensures  its homogeneous  distribution  and the consequent  par- tial dissolution  of the particle  surfaces.  Subsequently,  the resulting  \nwetted  mixture  of SFO plus GAA was pressed  at 150 MPa (green \npiece, labelled  G) for 5 min at room temperature  (RT) in a cylin- \ndrical die (Step 3). The pieces pressed  into the die were submit-  \nted to CSP (labelled  CSP) heating  with an annealing  rate of 20 °C \nmin −1 at 190 °C and applying  a pressure  of 400 MPa for 2 h in \nair atmosphere  (Step 4). CSP samples  were then extracted  from \nthe die and thermally  treated  at 1100 °C for 2 h in air (Step 5) \nto obtain the ﬁnal magnets  (CSP1100  °C). These sintered  ceramics  \nwere compared  with SFO-based  magnets  obtained  by a conven-  \ntional process,  which were pressed  at 270 MPa and subsequently  \nsintered  at 1100 °C (Cn110 0 °C) and 130 0 °C (Cn130  0 °C) for 4 h in \nair. \n2.2. Characterisation  procedure  \nThe relative  density  of the SFO-based  pieces was evaluated  by \nthe Archimedes  method  and by the mass/dimensions  values. Rel- \native densities  were calculated  considering  the theoretical  density  \nof SFO (5.10 g cm −3 ) [22] , α-Fe 2 O 3 (5.26 g cm −3 ) [23] and SrO \n(5.01 g cm −3 ) [24] , identiﬁed  by X-ray diffraction  (XRD) and X-ray \nabsorption  spectroscopy  (XAS) techniques.  \nThe morphology  of the SFO ceramics  was analyzed  by ﬁeld \nemission  scanning  electron  microscopy  (FESEM),  with an S-4700  \nHitachi  instrument  at 20 kV. Samples  were studied  on fresh frac- \ntured surfaces,  except to those sintered  by conventional  route that \nwere polished  and thermally  etched at a temperature  value of 10% \nbelow the sintering  temperature.  ImageJ software  was employed  to \ngenerate  the particle  size distribution  from FESEM images  and de- \ntermine  the average  particle  size. \nXRD measurements  were carried  out in a D8 Advanced  Bruker \ndiffractometer  using a Lynx Eye detector  and a Cu K αradiation  \nwith a λ= 0.154 nm in the 2 θrange 25–65 deg. Rietveld  reﬁne- \nments of the XRD data were carried  out using the software  FullProf  \n[25] in order to obtain quantitative  information  from the samples.  \nThe propagation  of errors is considered  to calculate  the uncertainty  \nin the reﬁned  parameters.  In the reﬁnements,  a Thompson-Cox-  \nHastings  pseudo-Voigt  function  was used to model the peak-proﬁle  \n[26] . The instrumental-contribution  to the peak-broadening  was \ndetermined  from measurements  of a standard  powder  (NIST LaB6 \nSRM® 660b)  [27] and deconvoluted  from the data. The sample-  \ncontribution  to the broadening  was considered  as purely size- \noriginated.  \nXAS was performed  on the sintered  samples  to evaluate  the ef- \nfect of the sintering  method  in the short-range  structural  modi- \nﬁcations  of the SFO structure.  Powdered  samples  were character-  \nized at the Fe and Sr K-edge  at the BL22 CLÆSS beamline  of the \nALBA synchrotron  facility (Barcelona,  Spain).  XAS measurements  \nwere collected  on pellets prepared  by mixing  proper amount  of \nsample  with cellulose,  at RT in transmission  mode. The monochro-  \nmator used in the experiments  was a double  Si crystal oriented  \nin the (311) direction,  with an incident  energy  resolution  less than \n0.3 eV at Fe K-edge.  Higher harmonic  rejection  was achieved  by \na proper choice of angle and coating  of collimating  and focusing  \nmirrors.  The incident  and transmitted  intensity  were detected  with \ntwo ionization  chambers  ﬁlled with N 2 and inert gases in order to \nhave 10 and 75% absorption,  respectively.  Metal foils were mea- \nsured and used as reference  to calibrate  the energy.  SFO starting  \npowders,  α-Fe 2 O 3 and SrO powder  references  were also measured  \nas reference  from transmitted  photons.  \nFor the theoretical  extended  X-ray absorption  ﬁne structure  \n(EXAFS)  calculations,  Fourier  transform  was performed  in the \nk 3 χ( k ) weighted  EXAFS signal between  2.5 and 13.0 ˚A −1 at the Fe \nK-edge  and between  3 and 11.5 ˚A −1 at the Sr K-edge.  Experimen-  \ntal EXAFS results were ﬁtted in R -space in the range 1.4–3.8  ˚A and \n2.5–4.4  ˚A at the Fe and Sr K-edge,  respectively,  using the FEFFIT \n2 A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nFig. 2. (a) Relative densities  for G (Step 3), CSP (Step 4), CSP1100 °C (Step 5), Cn1100 °C and Cn1300 °C. FESEM images of (b) starting SFO powders  and SFO ceramic processed:  \n(c) Cn1100 °C, (d) Cn1300 °C, (e) CSP and (f) CSP1100 °C. The white bars correspond  to 5 μm. \ncode [ 28 , 29 ]. The ﬁtting was performed  by ﬁxing the shift at the \nedge energy  for each absorption  edge, which were previously  cal- \nculated  from the starting  SFO powders  (used as reference).  There- \nfore, the coordination  number  N , the interatomic  distance  R and \nthe Debye-Waller  (DW) factors for each shell are used as free pa- \nrameters.  Only the most intense  single-scattering  paths were con- \nsidered  for the ﬁtting. At the Fe K-edge  position,  a ﬁrst shell pro- \nduced by the interaction  of a Fe absorbing  atom with six O atoms \nand a second  one constituted  by two shells of six Fe atoms each \nwere considered.  At the Sr K-edge  position,  a three-shell  model \nwas considered  with a ﬁrst and a second  shell produced  each by \nthe interaction  of a Sr absorbing  atom with six O atoms and a third \nshell formed  by the interaction  of a Sr absorbing  atom with ﬁfteen \nFe atoms. \nMagnetic  properties  of the sintered  ferrites  were evaluated  by \na vibrating  sample  magnetometer  (PPMS-VSM  model 60 0 0 con- \ntroller, Quantum  Design system)  and acquired  under a maximum  \napplied  magnetic  ﬁeld of 5 T at RT. Therefore,  here the magnetiza-  \ntion values are given at 5 T (M 5T ) instead  of saturation  magnetiza-  \ntion (M S ) [30] . \n3. Results  and discussion  \n3.1. Morphological  and structural  properties  of hexaferrite  permanent  \nmagnets  \nThe innovative  strategy  by CSP using GAA as a novelty  plus \na subsequent  post-annealing  was followed  to sinter dense hex- \naferrite  ceramics  with improved  magnetic  properties.  During  the \nsintering  process  of the SFO, the relative  density  of the pieces \nwas evaluated  and data are shown in Fig. 2 a. The piece obtained  \nfrom the mixture  of SFO and GAA compacted  in Step 3 (pellet G) presents  a relative  density  of 61%, which is increased  to 85% \nafter the CSP (Step 4). Subsequent  annealing  at 1100 °C for 2 h \n(CSP1100  °C) induces  a remarkable  increase  of the relative  den- \nsity to 92%, which is a satisfactory  ceramic  densiﬁcation  for many \ncurrent  magnetic  applications.  This density  value supposes  a sig- \nniﬁcant  increase  of 20% with respect  to the hexaferrite  magnet  \nsintered  by a conventional  route at same temperature  for 4 h \n(Cn1100  °C). It is required  to pass the temperature  from which the \nferrite particle  size grows signiﬁcantly  (Cn1300  °C) to get the high- \nest relative  density  of 97%, with the negative  consequences  on the \nmagnetic  properties.  \nThe effect of the sintering  process  on the ceramic  morphology,  \nanalyzed  by FESEM,  is shown in Fig. 2 b–f. The starting  SFO pow- \nders show a platelet-like  morphology  as commented  above with \na bimodal  average  size of 1–3 μm and 10 0–50 0 nm, which grow \nas the conventional  route is followed  at 1100 °C to more homo- \ngeneous  particles  with an average  value of 1.7(3) μm. A particle  \ngrowth  with an average  value of 7(2) μm is reached  at 1300 °C. \nIt should be mentioned  that the presence  of triple point junctions  \nhaving 120 °angles with straight  particle  boundaries  and the ex- \nistence  of closed porosity  indicate  that the third stage of sintering  \nis achieved  at 1300 °C. This is not observed  for Cn1100  °C where \nthe particle  morphology  continues  retaining  a platelet  shape with \na slight growth  and an open porosity  that point to the microstruc-  \ntural equilibrium  not being completely  attained.  \nWhen the SFO powders  are processed  by CSP using GAA as or- \nganic solvent,  a particle  reﬁnement  is identiﬁed  with respect  to the \nstarting  powder.  The coalescence  of smaller  particles  in equiaxed  \nparticles  and an average  size of 1.1(4) μm are noted. After the post- \nannealing  process  of the CSP ceramics  at 1100 °C, a change  in the \nparticle  morphology  is identiﬁed  with more rounded  and homoge-  \nneous particles  of around  1.0(2) μm in size that signaled  the inﬂu- \n3 A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nFig. 3. (a) Rietveld reﬁnement  (continuous  lines) of the XRD patterns (grey dots) for \nCSP and CSP1100 °C, indicating  the phase proportion  after the CSP. (b) The crystallite  \nsize (squares)  and the unit cell volume (triangles)  are also represented  along the \nvalues calculated  for the ceramics  sintered by conventional  route and the starting \nSFO powders.  \nence of the sintering  route in the effective  particle  growth  inhibi- \ntion. \nThe identiﬁcation  of phases and long-range  structural  modiﬁ-  \ncations  of the SFO lattice induced  with the sintering  process  have \nbeen investigated  by means of XRD. Fig. 3 a shows the XRD pat- \nterns related  to CSP and CSP1100  °C. For the CSP magnet,  along \nwith the SFO diffraction  pattern  [22] , Bragg’s  peaks related  to α- \nFe 2 O 3 phase [23] are recognized  and a crystalline  fraction  of 34% \nis obtained  by Rietveld  reﬁnement.  Thus, a partial destruction  of \nthe ferrite phase is induced  by the CSP under pressure,  which is \nrecovered  after the annealing  process  at 1100 °C (CSP1100  °C sam- \nple) that also helps to improve  the densiﬁcation.  \nThe model-to-data  agreement  of the XRD patterns  for the rest \nof samples  is shown in Fig. S1 in the supplementary  information  \n(SI). In the ceramics  sintered  by the conventional  methodology,  \nonly the reﬂections  associated  with ferrite are identiﬁed  [22] , and \nno other crystalline  secondary  phases are detected.  In addition,  the \nSFO lattice parameters  (see SI) and unit cell volume  ( Fig. 3 b) cal- \nculated  for the CSP ceramic  are similar  to those obtained  for the \nSFO starting  powders,  while the rest exhibits  an expansion  of the \nlattice. Concerning  the crystallite  size, just the Cn1300  °C ceramic  \nshows a growth  with a value around  117 nm larger than the SFO \npowder  (97 nm). It should be pointed  out that all samples  show Table 1 \nResults of LCF performed  in the XANES range showing the phase \ncomposition  of the permanent  magnets sintered.  Fraction of each \nphase is calculated  considering  the results at both absorption  \nedges. \nSample SFO (%) SrO (%) α-Fe 2 O 3 (%) \nCSP 46.0(5) 28.2(4) 25.8(5) \nCSP1100  °C 92.4(3) 7.6(3) 0.0(1) \nCn1100 °C 96.2(3) 3.8(5) 0.0(1) \nCn1300 °C 99.2(4) 0.8(1) 0.0(1) \nlarge differences  between  the particle  and crystallite  sizes, indicat-  \ning the large number  of crystallites  are forming  the SFO particles.  \nThe crystalline  growth  in the sintered  magnets  during the sintering  \nprocesses  could have been inhibited  by the presence  of secondary  \nand amorphous  phases identiﬁed  by XRD and XAS. \nPossible  non-crystalline  phases,  unidentiﬁed  by XRD, may be \ngiven during the sintering  process,  especially  after the CSP step. In \naddition,  important  modiﬁcations  on the Fe and Sr stoichiometry  \nand on the short-range  SFO structure  could have happened.  This \ninvestigation  was carried  out by XAS experiments,  at the Fe and \nSr K-edge.  Both X-ray absorption  near-edge  structure  (XANES)  and \nEXAFS experiments  were performed.  \nXANES  results for the sintered  ceramic  magnets  along with the \nXANES  proﬁles  of the starting  SFO, α-Fe 2 O 3 and SrO references  are \nshown in Fig. 4 a. From the absorption  edge position  at the XANES  \nregions,  the average  valence  of the absorbing  atoms for all ceram-  \nics is calculated  to be 3 + and 2 + for the Fe and Sr cations,  re- \nspectively  (see Fig. S2 in SI) [ 31 , 32 ]. Besides,  a similar  XANES  pro- \nﬁle of the sintered  ferrite to that of the SFO powders  is identiﬁed  \nin both absorption  edges, with changes  in some resonances  for the \ncase of the CSP ferrite. By a linear combination  ﬁtting (LCF) of the \nXANES  signal, around  a 46% of SFO and a 26% of α-Fe 2 O 3 phase \nare obtained  for the CSP ceramic  ( Fig. 4 b and Table 1 ). These ﬁnd- \nings suppose  a ratio of α-Fe 2 O 3 and SFO phases of 0.56, corrob-  \norating  the XRD results where a α-Fe 2 O 3 /SFO ratio of 0.52 was \nfound. Interestingly,  along the α-Fe 2 O 3 phase, already  identiﬁed  by \nXRD, here a proportion  of a second  secondary  phase is recognized:  \nSrO phase with a 28%, which may be considered  an amorphous  \ncompound  generated  during the CSP step from the Sr cations  lo- \ncated in the Sr ferrite structure.  As the post-annealing  of the CSP \nmagnet  is carried  out, the ferrite phase is almost completely  re- \ncovered,  obtaining  a slight quantity  of amorphous  SrO (8%) and \nwithout  evidences  of the hematite  polymorph.  By comparing  the \nsintering  process  by using GAA with the standard  aqueous  solvent  \nin the CSP step [15] , the aqueous  solvent  inhibits  the densiﬁcation  \nof material  and the recovery  of SFO after the post-annealing  (see \nFigs. S4 and S5 in SI). During  the sintering  process  by CSP using \naqueous  solutions,  more problems  in maintaining  the formulation  \nof the compound  are identiﬁed  [ 21 , 33 ]. Besides,  a low dissolution  \nof particle  surface  during the ﬁrst stages of CSP can be considered  \ntaking into account  the similar  morphological  characteristics  to the \nstarting  SFO particles  and the lower relative  density  of 84% with \nrespect  to the permanent  magnets  sintered  by CSP using pure sol- \nvent (GAA). \nParticularly,  the pre-edge  peak in the XANES  region at the Fe \nK-edge  (related  to 1s → 3d transitions)  [34] shows intensity  mod- \niﬁcations  with the sintering  process,  see Figs. 4 a and S3 in SI. The \nhighest  intensity  is found for the Cn1300  °C sample,  indicating  a \ndistortion  of site symmetry  for Fe 3 + cations  respect  to those of the \nstarting  SFO powders  that show a lower intensity  with a coordina-  \ntion closer to the octahedron.  In addition,  for the Cn1300  °C sam- \nple, the intensity  at the whiteline  is the lowest in agreement  with \nthe largest decrease  of the coordination  number  as the EXAFS re- \nsults conﬁrm  below. With respect  to the position  of the pre-edge  \n4 A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nFig. 4. (a) XANES spectra (continuous  lines) at the Fe and Sr K-edge and LCF curves (dashed lines) for CSP, CSP1100 °C, Cn1100 °C and Cn1300 °C. (b) Fraction of each phase \nobtained  from the LCF of the XANES spectra at the Fe K-edge (left) and the Sr K-edge (right) for all ceramics  sintered.  \nFig. 5. Fourier transform  modulus  of the EXAFS spectra (dots) at the (a) Fe and (b) Sr K-edge and best-ﬁtting  simulations  by a three-shell  model (continuous  lines) of the \nprocessed  magnets and SFO powders,  along with α-Fe 2 O 3 and SrO references.  \npeak, similar  values are identiﬁed  for all samples,  indicating  that \nthe oxidation  state of Fe cations  is the same, 3 + [31] , as calcu- \nlated above. \nShort-range  ordering  analysis  of cations  around  the Fe and Sr \ncations  and the parameters  obtained  by EXAFS are shown in Fig. 5 \nand Table 2 . The most signiﬁcant  alterations  are found in the struc- \nture of the SFO-based  ceramic  after the CSP. Some modiﬁcations  \nat the second  and the third Fe-Fe shells are distinguished  close to \nthe signal of the hematite  structure,  corroborating  the phase trans- \nformation  with the CSP and the XANES  data. With respect  to the \nSr ions, an intensity  reduction  at the three modeled  coordination  \nshells is obtained  after the CSP ( Table 2 ). As the post-annealing  \nis performed,  the SFO structure  is recovered  achieving  similar  EX- \nAFS parameters  than the starting  SFO powders.  For all samples,  no \nmodiﬁcations  in the DW factors are observed  at both absorption  \nedges with respect  to the reference  powders,  neither  with the sin- tering step nor with the sintering  process.  To conclude,  the reduc- \ntion of the coordination  number  at both edges for the sintered  ce- \nramics (mostly  in Cn1300  °C) with respect  to the starting  SFO pow- \nders (see Table 2 ) should be mentioned,  which indicates  the loss \nof some atoms into the structure  inducing  defects  (e.g. vacancies)  \nduring the sintering  process  [14] . \nHence, the selectivity  of XAS analysis  has allowed  to detect \nwhat happens  to the ferrite during the sintering  process  and where \nthe Sr cations  are located  as the ferrite structure  is transformed  \n(i.e. forming  a SrO amorphous  secondary  phase).  Furthermore,  \nthese ﬁndings  can explain  why Sr surplus  is generally  required  to \nproduce  α-Fe 2 O 3 -free SFO as identiﬁed  in some studies  [ 35 , 36 ], \nwhich has represented  an issue in ferrite manufacturing  for a long \ntime. While the stoichiometric  Sr:Fe ratio of the ferrite is 1:12 [3] , \nratios as high as 1:1 have been needed  for its synthesis,  which en- \ntails a great Sr excess resulting  in the formation  of amorphous  SrO. \n5 A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nTable 2 \nResults of the EXAFS ﬁttings by a three-shell  model at the Fe and Sr K-edge of ferrite ceramics.  EXAFS parameters  are compared  with results obtained  from magnets \nprocessed  by the conventional  route and starting SFO, α-Fe 2 O 3 and SrO references.  \nFe K -edge Sr K -edge \nSample shell N R ( ˚A) DW( ˚A 2 ) shell N R ( ˚A) DW( ˚A 2 ) \nStarting  SFO powder Fe-O 6 1.960(4)  0.013(2)  Sr-O 6 2.795(3)  0.007(2)  \nFe-Fe 6 2.970(6)  0.006(2)  Sr-O 6 2.99(1) 0.004(2)  \nFe-Fe 6 3.465(4)  0.009(1)  Sr-Fe 15 3.690(6)  0.015(2)  \nCn1100 °C Fe-O 5.8(2) 1.959(4)  0.013(1)  Sr-O 5.4(4) 2.79(1) 0.007(2)  \nFe-Fe 4.7(1) 2.968(8)  0.005(2)  Sr-O 5.5(3) 2.98(1) 0.004(1)  \nFe-Fe 6.1(2) 3.459(4)  0.009(1)  Sr-Fe 14.3(3) 3.692(7)  0.015(2)  \nCn1300 °C Fe-O 5.3(1) 1.954(4)  0.013(1)  Sr-O 5.6(4) 2.792(9)  0.006(2)  \nFe-Fe 3.9(2) 2.956(8)  0.005(1)  Sr-O 5.9(3) 2.98(1) 0.004(2)  \nFe-Fe 6.1(1) 3.451(4)  0.009(1)  Sr-Fe 14.8(1) 3.688(6)  0.015(1)  \nCSP Fe-O 5.8(1) 1.959(5)  0.012(1)  Sr-O 3.8(3) 2.79(1) 0.006(2)  \nFe-Fe 5.3(1) 2.961(8)  0.003(2)  Sr-O 4.5(3) 2.98(1) 0.005(2)  \nFe-Fe 3.6(2) 3.417(8)  0.005(2)  Sr-Fe 9.5(1) 3.69(1) 0.015(1)  \nCSP1100  °C Fe-O 6.1(2) 1.956(4)  0.012(1)  Sr-O 6.0(4) 2.79(1) 0.006(2)  \nFe-Fe 5.8(2) 2.968(8)  0.006(1)  Sr-O 5.8(3) 2.98(1) 0.004(2)  \nFe-Fe 6.0(2) 3.452(4)  0.008(1)  Sr-Fe 15.0(3) 3.691(6)  0.015(2)  \nα-Fe 2 O 3 powder reference  Fe-O 6 1.96(1) 0.013(1)  \nFe-Fe 4 2.948(7)  0.006(1)  \nFe-Fe 3 3.375(8)  0.006(1)  \nSrO powder reference  Sr-O 6 2.621(3)  0.011(1)  \nSr-Sr 12 3.728(6)  0.021(2)  \nFig. 6. Hysteresis  loops representing  the magnetic  response  for (a) Cn1100 °C and Cn1300 °C and (b) CSP and CSP1100 °C. (c) Dependence  between  the M 5T values and SFO \nwt%. (d) M R /M 5T ratio versus H C obtained  from the magnetic  curves. \n3.2. Magnetic  performance  of hexaferrite  permanent  magnets:  \nimproving  functional  response  \nThe functional  magnetic  response  of the sintered  ferrites  was \nevaluated  and represented  in Fig. 6 . For all ceramic  magnets  a \ncharacteristic  hysteresis  loop of a ferromagnetic  material  is mea- \nsured with singular  properties  depending  on the sintering  pro- \ncess. Cn1100  °C presents  a H C of 2.2 kOe and a M 5T of 72.2 emu \ng −1 . However,  despite  the interesting  magnetic  properties  as hard magnetic  material,  the relative  density  of the magnet  is low and \nthe mechanical  integrity  is poor. By increasing  the sintering  tem- \nperature  to 1300 °C, densiﬁcation  is achieved  as shown above, \nbut reducing  considerably  the H C to 1.0 kOe due to the particle  \ngrowth  (see Fig. 6 a). Conversely,  for CSP magnet  (see Fig. 6 b), who \npresents  85% relative  density,  the partial decomposition  of SFO into \nα-Fe 2 O 3 and SrO compounds  with the sintering  process  is traduced  \nin a reduction  of coercivity  (H C of 1.5 kOe) and magnetization  at \n5 T (M 5T = 49.2 emu g −1 ). These results are in agreement  with \n6 A. Serrano, E. García-Martín,  C. Granados-Miralles  et al. Acta Materialia  219 (2021) 117262 \nthe presence  of non-magnetic  SrO phase and the low saturation  \nmagnetization  of α-Fe 2 O 3 . However,  after the post-annealing  at \n1100 °C (CSP1100  °C), in addition  to the increased  density  to 92%, \na clear improvement  of magnetic  properties  is obtained  with a H C \nof 2.8 kOe and M 5T = 73.7 emu g −1 , as a consequence  of recrystal-  \nlization  of the ferromagnetic  SFO phase. For that, the role of GAA \nin the CSP is crucial allowing  also to reﬁne the particle  size dur- \ning the sintering  process,  which results in an improvement  of the \ncoercivity  value. In all cases, the BH max values are estimated  from \nthe open hysteresis  loops and non-orientated  samples,  obtaining  \nthe highest  merit ﬁgure for the CSP1100  °C magnet  (see Table S3 in \nSI). Besides,  the dependence  between  the M 5T values and the wt% \nof the ferromagnetic  phase (i.e. SFO) is also recognized  and dis- \nplayed in Fig. 6 c. These properties  are perfectly  competitive  when \ncompared  to commercial  ferrite magnets  (Hitachi  metals NMF-7C  \nseries offer H C = 2.8–3.3  kOe and M S = 68 emu g −1 ) [37] . Addi- \ntionally,  given that competent  ferrite magnets  can be sintered  at \n1200–1250  °C by the incorporation  of secondary  phases,  our pro- \ncedure allows reaching  similar  performance  at 8–12% lower tem- \nperatures  which means an energy  eﬃciency  of at least 25%. \nDuring  the CSP, a slight increment  of the degree of magnetic  \nalignment  between  particles  is also attained,  with a M R /M 5T ra- \ntio of 0.54, exceeding  the predicted  value for an assembly  of \nnon-interacting  particles  randomly  oriented  [38] . The M R /M 5T ra- \ntio rises even more up to 0.62 with the post-annealing  for the \nCSP1100  °C magnet,  as Fig. 6 d shows. This magnetic  improvement  \nis likely due to a favourable  arrangement  under pressure  of the \nferrite platelets  facilitated  by the liquid solvent.  \n4. Conclusion  \nHerein a strategy  to sinter ceramic  magnets  based on Sr- \nhexaferrites  with interesting  structural  and magnetic  characteris-  \ntics is developed.  An approach  combining  the CSP as an intermedi-  \nate step, in which GAA is employed  as pure solvent,  plus a sub- \nsequent  post-annealing  has densiﬁed  the ceramic  magnets  to a \nrelative  density  of 92%. Despite  the partial transformation  of the \nmagnetic  phase during the CSP, which is accompanied  by a reduc- \ntion of the magnetic  properties,  after the post-annealing  of the CSP \nceramic  at 1100 °C the magnetic  phase is recovered  with parti- \ncle reﬁnement.  An improvement  in the ﬁnal functional  response  \nof the ceramic  magnets  is achieved,  greatly  surpassing  the per- \nformance  of the hexaferrite  magnets  processed  by conventional  \nroute and with very competitive  response  at lower temperatures.  \nThese investigations  are the ﬁrst step in a very promising  direc- \ntion, demonstrating  a feasible  and green sintering  way towards  the \ndevelopment  of dense rare-earth  free oxide permanent  magnets  for \na large number  of technological  applications.  \nDeclaration  of Competing  Interest  \nThe authors  declare  that they have no known  competing  ﬁnan- \ncial interests  or personal  relationships  that could have appeared  to \ninﬂuence  the work reported  in this paper. \nAcknowledgments  \nThis work has been supported  by the Ministerio  Español  de \nCiencia,  Innovación  y Universidades  (MCIU)  through  the projects  \nMAT2017-86540-C4-1-R  and RTI2018-095303-A-C52  and The Euro- \npean Commission  through  the AMPHIBIAN  Project  (720853).  Part \nof these experiments  was performed  at the CLÆSS beamline  at \nALBA Synchrotron  with the collaboration  of ALBA staff. A.S. ac- \nknowledges  ﬁnancial  support  from Comunidad  de Madrid  for an \n“Atracción  de Talento  Investigador” Contract (2017-t2/IND5395).  C.G.-M.  and A.Q. acknowledge  ﬁnancial  support  from Ministerio  Es- \npañol de Ciencia  e Innovación  (MICINN)  through  the “Juan de la \nCierva” Program  (FJC2018-035532-I)  and the “Ramón  y Cajal” Con-  \ntract (RYC-2017-23320).  \nSupplementary  materials  \nSupplementary  material  associated  with this article can be \nfound, in the online version,  at doi: 10.1016/j.actamat.2021.117262  . \nReferences  \n[1] R.C. Pullar, Hexagonal  ferrites: a review of the synthesis,  properties  and appli- \ncations of hexaferrite  ceramics,  Prog. Mater. Sci. 57 (2012) 1191–1334,  doi: 10. \n1016/j.pmatsci.2012.04.001  . \n[2] S.H. Mahmood,  I. 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A 240 (1948) 599–642,  doi: 10.1098/ \nrsta.1948.0  0 07 . \n8 "    },    {        "title": "1510.07164v2.Graphene_Transverse_Electric_Surface_Plasmon_Detection_using_Nonreciprocity_Modal_Discrimination.pdf",        "content": "arXiv:1510.07164v2  [cond-mat.mes-hall]  21 Jul 2016Graphene Transverse Electric Surface Plasmon Detection\nusing Nonreciprocity Modal Discrimination\nNima Chamanara and Christophe Caloz\n´Ecole Polytechnique de Montr´ eal, Montr´ eal, QC H3T 1J4, Ca nada.\n(Dated: July 10, 2016)\nAbstract\nWe present a magnetically biased graphene-ferrite structu re discriminating the TE and TM plas-\nmonic modes of graphene. In this structure, the graphene TM p lasmons interact reciprocally with\nthe structure. In contrast, the graphene TE plasmons exhibi t nonreciprocity. This nonreciprocity\nis manifested in unidirectional TEpropagation in a frequen cy bandclose to the interband threshold\nfrequency. The proposed structure provides a unique platfo rm for the experimental demonstration\nof the unusual existence of the TE plasmonic mode in graphene .\n1I. INTRODUCTION\nGraphene plasmonics has been an area of extensive research in the past few years1–7.\nGraphene plasmons have enabled photodetection enhancement8, light matter interaction\nenhancement in solar cells9, and novel optical modulators and sensors1,10. The Dirac\nband structure endows graphene with tunability, not easily obtaina ble in other plamonic\nmaterials10. Moreover, the linearity of this band structure leads to the existe nce of unusual\nplasmonic modes that are unique to graphene11. It was shown theoretically in Ref.11,12\nthat, in addition to the conventional transverse magnetic (TM) or longitudinal plasmonic\nmode, graphene also supports an unusual plasmonic mode which is tr ansverse electric (TE).\nCompared to the conventional TM plasmons, the TE mode is more loos ly conﬁned to\ngraphene, has a lower loss and propagates with a faster phase velo city13–18. The TE mode\ncan be excited when the imaginary part of the conductivity acquires a non-Drude sign. This\ncondition is satisﬁed when the interband conductivity of graphene b ecomes dominant over\nits intraband conductivity, which is normally satisﬁed in a frequency w indow close to the\ninterband transition threshold frequency11,13, that can be tuned from the microwave to the\ninfrared frequency bands by adjusting graphene’s chemical pote ntial.\nThe speciﬁc ﬁeld conﬁguration of graphene TE plasmons leads to non reciprocal inter-\naction with a magnetically biased ferrite substrate or superstrate , whereas graphene TM\nplasmons do not nonreciprocally interact with such a structure. Th e magnetic ﬁeld lines of\nthe graphene TE mode and its electric current are shown in Fig. 1, fo r propagation along the\nzdirection. The electric current is transverse to the direction of pr opagation, with sinusoidal\nvariation along z. The magnetic ﬁeld lines loop around current sections, as shown in Fig . 1,\nandtheelectric ﬁeld(notshownintheﬁgure)iscompletely transver se. Such amagneticﬁeld\ngenerally interact nonreciprocally with a properly magnetized ferrit e substrate/superstrate.\nIn order to include nonreciprocal interaction, the ferrite substr ate/superstrate should be bi-\nased by a static magnetic ﬁeld parallel to the plane of graphene and n ormal to the direction\nof propagation, as shown in Fig. 1. The magnetically biased ferrite su bstrate/superstrate ac-\nquires then a tensorial permeability in the yzplane. However, the conductivity of graphene\nis scalar (no cyclotron orbiting) since the magnetic ﬁeld is parallel to it s plane.\nFigure.2shows alongitudinal ( yz)cross-section ofthestructure inFig.1, withthearrows\nrepresenting the magnetic ﬁeld lines. As the magnetic ﬁeld propagat es along graphene, any\npointA/Binside the ferrite substrate/superstrate sees a rotating magne tic ﬁeld in the yz\n2PSfrag replacementsxy\nz\nTE magnetic ﬁeld lines\nTE current density\nferrite substratemagnetic biasferrite superstrate\nFIG. 1. Artistic representation of theTE surfaceplasmons i na graphenesheet sandwiched between\na ferrite substrate and superstrate and propagating in the + zdirection. The electric current is\ntransverse and sinusoidally varying along graphene, with t he magnetic ﬁelds looping around them.\nThe electric ﬁeld (not shown) is completely transverse. The magnetic bias ﬁeld is parallel to the\nplane of graphene and looking in opposite directions in the s ubstrate a superstrate. For clarity\nonly part of the superstrate is shown.\nplane, with clockwise/counterclockwise rotation for forward and c ounterclockwise/clockwise\nrotation for the backward direction. For an antisymmetric permea bility tensor µ=µd(yy+\nzz) +µo(yz−zy), such left and right handed rotating magnetic ﬁelds perceive eﬀec tive\nscalar permeabilities µd+iµoandµd−iµo, respectively. For the magnetic bias conﬁguration\nshowninFig.1, thesubstrate andsuperstrateexhibit oppositesig noﬀ-diagonalpermeability\ncomponents ( µA\no=−µB\no). For a TE wave propagating in the forward direction point A\n(right handed) and B (left handed) perceive µA=µA\nd+iµA\noandµB=µB\nd−iµB\noeﬀective\npermeabilities, respectively. Since µA\no=−µB\nothe eﬀective permeability perceived by both\npoints is µA\nd+iµA\no. Similarly for the backward direction the eﬀective permeability perce ived\nby both points is µA\nd−iµA\no. Therefore, the ferrite structure is eﬀectively seen as diﬀerent\n3media for opposite direction of propagation19, and therefore exhibits nonreciprocity.\nPSfrag replacementsy\nz\nA\nAB\nB\nforward + z backward −zǫr1,¯¯µr1\nǫr2,¯¯µr2\nFIG. 2. Longitudinal ( yz) cross-section of the structure in Fig. 1. Any point A/B perc eives\nclockwise/counterclockwise and counterclockwise/clock wise rotating magnetic ﬁelds, respectively,\nas the wave propagates in the + zdirection and in the the −zdirection, respectively.\nII. ANALYSIS\nForaninﬁnitegraphenesheetbetween twosemi-inﬁnite ferritemed ia, theelectromagnetic\nﬁelds supported by the structure and their dispersion relations ca n be derived analytically.\nThe plasmonic electric ﬁelds in regions 1 (above graphene) and 2 (belo w graphene), are\nexpressed as surface waves propagating along the zdirection with propagation constant\nk, and exponentially decaying in the + yand−ydirections with decay rates α1andα2,\nrespectively ( e+iωtconvention):\n4E1=/bracketleftBig\nEx1, Ey1, Ez1/bracketrightBig\ne−α1y−ikz, (1a)\nE2=/bracketleftBig\nEx2, Ey2, Ez2/bracketrightBig\neα2y−ikz, (1b)\nFor generality of the analysis it is initially assumed that the direction of the magnetic bias\nin regions 1 and 2 are not necessarily opposite and may take arbitrar y values. With the\nrelative permeability tensors\n¯¯µr1=\n1 0 0\n0µrd1−µro1\n0µro1µrd1\n,¯¯µr2=\n1 0 0\n0µrd2−µro2\n0µro2µrd2\n(2)\nof thex-biased superstrate and substrate ferrite material, the magnet ic ﬁelds above and\nbelow graphene are found through Maxwell equations as\nHi=−1\niωµ0¯¯µ−1\nri/squaresmallsolid∇×Ei. (3)\nwhere the subscript i= 1,2 represents the ﬁelds in regions 1 or 2. The eigenmodes and their\ndispersion are then found by the application of the boundary condit ion in the graphene\nplane,\nay×(H1−H2)|y=0=σET|y=0, (4)\nwhere, by continuity of the electric ﬁeld Ex1=Ex2,Ez1=Ez2, so that the tangential electric\nﬁeld simply reads ET=Ex1x+Ez1z. Substituting (1) and (3) in (4) results in\nEx1µ0ωσ/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig\n+\nEx1/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig\n(iα1µrd1−µro1k)+\nEx2/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig\n(iα2µrd2+µro2k) = 0, (5a)\nEy1k−Ey2k+iEz1α1−Ez1µ0ωσ+iEz2α2= 0. (5b)\nThe normal electric ﬁeld components Ey1andEy2are redundant, as they may be expressed\nin terms of Ez1andEz2, respectively, through the divergence relations ∇/squaresmallsolidE1= 0 and\n∇/squaresmallsolidE2= 0, as\nEy1=−iEz1\nα1k, E y2=iEz2\nα2k, (6)\n5leading to\n/bracketleftbig\n−µ0ωσ/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig\n+\n/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig\n(iα2µrd2+µro2k)+\n/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig\n(iα1µrd1−µro1k)/bracketrightbig\nEx1= 0, (7a)\n/parenleftbiggiα1\nµ0ω+iα2\nµ0ω−σ−ik2\nα2µ0ω−ik2\nα1µ0ω/parenrightbigg\nEz1= 0. (7b)\nThe decay rates α1andα2may be expressed interms of thepropagationconstant k, through\nthe electric ﬁeld wave equation in regions 1 and 2,\n∇×¯¯µ−1\nr1/squaresmallsolid∇×E1−ω2µ0ε0εr1E1=0, (8a)\n∇×¯¯µ−1\nr2/squaresmallsolid∇×E2−ω2µ0ε0εr2E2=0, (8b)\nwhich enforce the relations\n\nEx1\nµ2\nrd1+µ2\nro1(−α2\n1µrd1−ǫ0ǫr1µ0µ2\nrd1ω2−ǫ0ǫr1µ0µ2\nro1ω2+µrd1k2)\niEz1\nα1k(α2\n1+ǫ0ǫr1µ0ω2−k2)\nEz1(−α2\n1−ǫ0ǫr1µ0ω2+k2)\n=0, (9)\n\nEx2\nµ2\nrd2+µ2\nro2(−α2\n2µrd2−ǫ0ǫr2µ0µ2\nrd2ω2−ǫ0ǫr2µ0µ2\nro2ω2+µrd2k2)\niEz2\nα2k(−α2\n2−ǫ0ǫr2µ0ω2+k2)\nEz2(−α2\n2−ǫ0ǫr2µ0ω2+k2)\n=0,(10)\nwhereǫr1and¯¯µr1are the relative permittivity and the relative permeability tensor in re gion\n1, respectively, and ǫr2and¯¯µr2are the relative permittivity and the relative permeability\ntensor in region 2, respectively.\nEquations (7), (9) and (10) describe the eigenmodes of the syste m. This set of equations\nadmits two modal solutions. The ﬁrst solution is a TM mode, for which Ex= 0 and\nα1=/radicalbig\n−ǫ0ǫr1µ0ω2+k2, (11a)\nα2=/radicalbig\n−ǫ0ǫr2µ0ω2+k2. (11b)\nThis modehasitsmagneticﬁeldalongtheDCmagneticbias, andtheref oresees theferriteas\nan isotropic medium with scalar permeability µ0. It therefore interacts reciprocally with the\n6ferrite, with identical characteristics in the forward and backwar d directions. The dispersion\nrelation for this mode is given by\nσ+iǫ0ǫr1ω/radicalbig\n−ǫ0ǫr1µ0ω2+k2+iǫ0ǫr2ω/radicalbig\n−ǫ0ǫr2µ0ω2+k2= 0. (12)\nTheoﬀ-diagonalcomponent ofthepermeability tensor hasnocont ribution, conﬁrming recip-\nrocal interaction with the magnetically biased ferrite structure. T he dispersion is a function\nofk2and is therefore reciprocal with respect to the direction of propa gation, as expected.\nThe second solution is a TE surface plasmon mode ( Ez= 0), with decay rates in the\nnormal direction\nα1=/radicalbigg\n−ǫ0ǫr1µ0µrd1ω2−ǫ0ǫr1\nµrd1µ0µ2\nro1ω2+k2, (13a)\nα2=/radicalbigg\n−ǫ0ǫr2µ0µrd2ω2−ǫ0ǫr2\nµrd2µ0µ2\nro2ω2+k2. (13b)\nThismodehasitsmagneticﬁeldperpendiculartotheDCmagneticbias. Asexplainedabove,\nsuch a magnetic ﬁeld perceives diﬀerent eﬀective materials in the for ward and backward\ndirections and is thus nonreciprocal. The dispersion relation for the TE mode is\n−µ0ωσ/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig\n+\n/parenleftbig\nµ2\nrd1+µ2\nro1/parenrightbig/parenleftbigg\niµrd2/radicalbigg\n1\nµrd2(−ǫ0ǫr2µ0µ2\nro2ω2+µrd2(−ǫ0ǫr2µ0µrd2ω2+k2))+µro2k/parenrightbigg\n+\n/parenleftbig\nµ2\nrd2+µ2\nro2/parenrightbig/parenleftbigg\niµrd1/radicalbigg1\nµrd1(−ǫ0ǫr1µ0µ2\nro1ω2+µrd1(−ǫ0ǫr1µ0µrd1ω2+k2))−µro1k/parenrightbigg\n= 0.\n(14)\nThe term µro2kin this relation, which is odd in k, results in a dispersion that is diﬀerent for\npositive and negative k’s, corresponding to nonreciprocity. Note that if the substrate a nd\nsuperstrate ferrites are identical and have the same parallel mag netic bias, this dispersion\nrelation remains symmetric with respect to propagation direction (o pposite signs of k) and\nis thus reciprocal. In this case the nonreciprocity produced by the substrate and superstrate\ncancel out each other. To generate nonreciprocity the magnetic bias should be diﬀerent.\nOppositely directed magnetic ﬁelds generate maximum nonreciprocit y. This nonreciprocity\nis manifested in unidirectional TE plasmon propagation on a frequenc y band close to the\nintrband frequency threshold ( ω= 2µc). Although the magnetic eﬀect produced by the\nferrite is relatively weak at infrared andoptical frequencies, we ne xt show that over a speciﬁc\n7frequencybandwhichistunablebygrapheneandferriteparamete rs, isolationistheoretically\ninﬁnite. Equation (14) does not admit analytic solutions and should be solved numerically.\nSome guidelines regarding numerical solution of dispersion equations are provided in the\nsupplementary material20.\nIII. RESULTS\nConsideragraphenesheetwithchemicalpotential µc= 0.1eV,scatteringtime τ= 0.2ps,\nand temperature T= 300 K. The corresponding interband and intraband conductivities21–23\nare plotted in Fig. 3(a). Close to the interband frequency thresho ld (ω= 2µc), the interband\nconductivity becomes dominant over the intraband conductivity, a nd the imaginary part of\nthe total conductivity ﬂips sign, as shown in Fig. 3(b). This region co rresponds to the\nfrequency band where the TE plasmonic mode of graphene can prop agate along graphene.\nFirst consider a graphene sheet sandwiched between two magnetic ally unbiased media\nwith scalar parameters ǫr1,µr1andǫr2,µr2. The TE dispersion equations for such a structure\nis\nk=/radicalBig\nǫr1µr1k2\n0+α2\n1=/radicalBig\nǫr2µr2k2\n0+α2\n2, (15)\nwherek0is the free space wave number and α1andα2are decay rates normal to graphene\nsurface in regions 1 and 2 with the following relations (details provided in the supplementary\nmaterial20)\nα1/k0=µ1\nµ2(µ2\n1−µ2\n2)/parenleftbig\niµ3\n2ση0±\n/radicalBig\nµ2\n2(−ǫ1µ3\n1+ǫ1µ1µ2\n2+ǫ2µ2\n1µ2−ǫ2µ3\n2−µ2\n1µ2\n2σ2η2\n0)/parenrightbigg\n, (16a)\nα2/k0=−1\nµ2\n1−µ2\n2/parenleftbig\niµ2\n1µ2ση0±\n/radicalBig\n−µ2\n2(ǫ1µ3\n1−ǫ1µ1µ2\n2−ǫ2µ2\n1µ2+ǫ2µ3\n2+µ2\n1µ2\n2σ2η2\n0)/parenrightbigg\n. (16b)\nwhereµi=µri,ǫi=ǫriandη0=/radicalbig\nµ0/ǫ0. Forǫr1=ǫr2andµr1=µr2the solution\nto dispersion equations (15)-(16) for ℑσ >0 lies in the proper Riemann sheet ( ℜαi>\n0), corresponding to a surface wave with exponential decay norm al to the graphene sheet.\nTherefore for identical media on both sides of graphene, the TE mo de propagates at all the\n8frequencies marked by grey color in Fig. 3(b). For ǫr1=ǫr2= 10 and and µr1=µr2= 1\nthe corresponding TE normalized phase constant ℜkz/k0, lossℑkz/k0, propagation length\nζ=−1/ℑkzand conﬁnement factor α/k0are presented in Fig. 4(a) by solid curves. The\ndashed curves correspond to a very small change in the permittivit y and permeability of\nregion 2 by 10−3and 10−4respectively. The dispersion curves change dramatically for such\nasmall contrast inmaterial parameters. ThereforetheTEmodeis highlysensitive tothethe\ncontrast of thematerial parameters onbothsides ofgraphene14. Asthecontrast between the\nmaterialparametersofregions1and2isincreased, thesolutionst odispersionequations(15)-\n(16) quickly moves to the improper Riemann sheet and becomes unph ysical. For ǫr1= 10\nandµr1= 1 the material contrast corresponding to proper surface wave solutions is plotted\nin Fig. 4(b). For the structure of Fig. 1 the substrate and supers trate ferrites should be\nalmost identical, otherwise the TE surface plasmon mode can not pro pagate.\nAssume the ferrites have anti-parallel magnetic bias B0as shown in Fig. 1, saturation\nmagnetization Msand loss factor α. The permeability tensor for such a ferrite substrate has\na resonance at microwave frequencies and its components decrea se as 1/fat higher frequen-\ncies19,24. At infrared and optical frequencies, this magnetic eﬀect become s vanishingly small.\nHowever, due to high sensitivity of the TE mode the resulting nonrec iprocity is signiﬁcant.\nThe normalized conﬁnement factor α/k0is plotted in Fig. 5 for three diﬀerent magnetic\nbias values B0= 0.0035 T corresponding to a ferromagnetic resonance f0= 0.1 GHz in\nred,B0= 0.035 T corresponding to f0= 1.0 GHz in blue and B0= 0.106 T corresponding\ntof0= 3.0 GHz in green. Solid curves represent forward, and dashed curve s backward\npropagation. As expected the TE mode is interacting nonreciproca lly with the magnetically\nbiased structure. In the highlighted frequency bands the TE mode propagates unidirec-\ntionally. This frequency band spans several gigahertz to a few ter ahertz depending on the\nstrength of the magnetic bias. The corresponding phase constan ts and propagation lengths\nare plotted in Fig. 6. The forward and backward plasmons undergo s lightly diﬀerent phases\nand losses as they propagate along the graphene-ferrite struct ure. In the highlighted fre-\nquency bandstheisolationbetween theforwardandbackward mod esistheoreticallyinﬁnite.\nNote that the anti-parallel magnetic bias conﬁguration shown in Fig. 1 may be produced\nthrough a longitudinal DC current. For a 10 µm wide graphene strip and a magnetic bias\nB0= 0.00035 corresponding to a ferromagnetic resonance f0= 0.01 GHz, the required DC\ncurrent is 28 mA. For these parameters the unidirectional propag ation bandwidth is 6 GHz.\nNote that the operation frequency of the structure can be tune d through the chemical po-\n9tential of graphene. For higher amounts of doping the operation f requency is increased and\nfor smaller amounts of doping it is lowered towards the microwave fre quency band. However\nas the chemical potential is reduced, fabrication eﬀects such as in teraction of graphene with\nthe substrate, which may lead to the modiﬁcation of the Dirac band s tructure25,26become\nimportant and should be taken into account17. For undoped graphene these interactions\ngreatly modify the characteristics of graphene plasmons17. The energy scale of such mod-\niﬁcations in the Dirac cones are normally in the order of 5-50 meV25,26. Depending on\nthe fabrication process of the graphene-ferrite structure and the level of chemical potential,\ncalculation of such eﬀects may be necessary. However, such cons iderations are beyond the\nscope of this paper.\nIV. CONCLUSIONS\nWe proposed a graphene ferrite structure that discriminates bet ween the TM and TE\nsurfaceplasmonsofgrapheneusingnonreciprocity, theTEsurfa ceplasmonmode, incontrast\nto its TM counterpart, has a speciﬁc nonreciprocal signature, pr opagating unidirectionally.\nThe TM mode interacts reciprocally. The proposed structure may s erve asa platformfor the\nexperimental demonstration of the existence of currently still elu sive TE plasmonic modes\nin graphene.\n1A. Grigorenko, M. Polini, and K. Novoselov, Nature photonic s6, 749 (2012).\n2A. C. Neto, F. Guinea, N. Peres, K. S. Novoselov, and A. K. Geim , Reviews of modern physics\n81, 109 (2009).\n3A. K. Geim and K. S. Novoselov, Nature materials 6, 183 (2007).\n4A. Politano and G. Chiarello, Nanoscale 6, 10927 (2014).\n5X. Luo, T. Qiu, W. Lu, and Z. Ni, Materials Science and Enginee ring: R: Reports 74, 351\n(2013).\n6N. Chamanara, D. Sounas, and C. Caloz, Optics Express 21, 11248 (2013).\n7N. Chamanara, D. L. Sounas, T. Szkopek, and C. Caloz, Optics E xpress21, 25356 (2013).\n8T. Mueller, F. Xia, and P. Avouris, Nature Photonics 4, 297 (2010).\n9W. L. Barnes, A. Dereux, and T. W. Ebbesen, Nature 424, 824 (2003).\n1010M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F. W ang, and X. Zhang, Nature\n474, 64 (2011).\n11S. Mikhailov and K. Ziegler, Physical Review Letters 99, 016803 (2007).\n12M. Bordag and I. Pirozhenko, Physical Review B 89, 035421 (2014).\n13G. W. Hanson, Journal of Applied Physics 103, 064302 (2008).\n14O. Kotov, M. Kol’chenko, and Y. E. Lozovik, Optics express 21, 13533 (2013).\n15X. Y. He and R. Li, IEEE Journal of Selected Topics in Quantum E lectronics 20, 62 (2014).\n16D. Drosdoﬀ, A. Phan, and L. Woods, Advanced Optical Materials 2, 1038 (2014).\n17J. F. Werra, F. Intravaia, and K. Busch, Journal of Optics 18, 034001 (2016).\n18N. Chamanara and C. Caloz, Forum for Electromagnetic Resear ch Methods and Application\nTechnologies (FERMAT) 10, 1 (2015).\n19B. Lax and K. J. Button, Microwave ferrites and ferrimagnetics (McGraw-Hill, 1962).\n20See Supplemental Material at [URL will be inserted by publis her] for dispersion equations of the\nunbiased graphene-ferrite structure, and guidelines on nu merical solution of the magnetically\nbiased graphene-ferrite structure dispersion.\n21V. Gusynin, S. Sharapov, and J. Carbotte, Journal of Physics : Condensed Matter 19, 026222\n(2007).\n22V. Gusynin and S. Sharapov, Physical Review B 73, 245411 (2006).\n23V. Gusynin, S. Sharapov, and J. Carbotte, New Journal of Phys ics11, 095013 (2009).\n24R. E. Collin, Foundations for microwave engineering (John Wiley & Sons, 2007).\n25G. Giovannetti, P. A. Khomyakov, G. Brocks, P. J. Kelly, and J . van den Brink, Physical\nReview B 76, 073103 (2007).\n26S. Zhou, G.-H. Gweon, A. Fedorov, P. First, W. De Heer, D.-H. L ee, F. Guinea, A. C. Neto,\nand A. Lanzara, Nature materials 6, 770 (2007).\n11101 102 103\nf (THz)−0.20−0.15−0.10−0.050.000.050.10σ (mS)\nℜ(σ) intra\nℑ(σ) intra\nℜ(σ) inter\nℑ(σ) inter\n(a)\n101 102 103\nf (THz)−0.20−0.15−0.10−0.050.000.050.10σ (mS)ℜ(σ)\nℑ(σ)\n(b)\nFIG. 3. Conductivity of a graphene sheet with chemical poten tialµc= 0.1 eV, scattering time τ=\n0.2 ps, and temperature T= 300 K. (a) Intraband and interband conductivities plotted separately.\n(b) Total conductivity. The interband conductivity become s dominant in the highlighted frequency\nrange where the imaginary part of the total conductivity ﬂip s sign.1240 50 60 70 80 90 100\nf (THz)10-910-710-510-310-1101103105107ℜ(kz/k0)\nℑ(kz/k0)\nζ/λ\nℜ(α/k 0)\n(a)\n−0.0010 −0.0005 0.0000 0.0005 0.0010\n∆ǫr−0.00020−0.00015−0.00010−0.000050.000000.000050.000100.000150.00020∆µr\n(b)\nFIG. 4. (a) TE normalized propagation constant, loss, propa gation length and conﬁnement\nfactor for a graphene sheet with chemical potential µc= 0.1 eV, scattering time τ= 0.2 ps, and\ntemperature T= 300 K. The solid lines represent the dispersion for ǫr1= 10.0,µr1= 1,ǫr2= 10.0,\nµr2= 1 and dashed lines for ǫr1= 10.0,µr1= 1,ǫr2= 10.0001,µr2= 1.00001. (b) Sensitivity\nof TE surface plasmons with respect to material contrast. Th e TE mode is supported only for\nmaterial contrasts inside the highlighted strip.1345.5 46.0 46.5 47.0 47.5 48.0\nf (THz)10-610-510-410-3ℜ(α/k 0)\nFIG. 5. Decay rate normal to graphene ( y) for the forward (solid curved) and backward (dashed\ncurves) TE surface plasmons, for a graphene sheet with chemi cal potential µc= 0.1 eV, scattering\ntimeτ= 0.2 ps, and temperature T= 300 K, between two ferrite mediawith anti-parallel magnet ic\nbias as in Fig. 1 and with ferromagnetic resonances f0= 0.1 GHz (red), f0= 1.0 GHz (blue)\nandf0= 3.0 GHz (green). The corresponding magnetic bias ﬁeld is B0= 0.00356 T (red),\nB0= 0.0356 T (blue), B0= 0.106 T (green). A loss factor α= 0.05 is assumed for the ferrites.\n1445.5 46.0 46.5 47.0 47.5 48.0\nf (THz)2.02.53.03.54.0ℜ(kz/k0)\n(a)\n45.5 46.0 46.5 47.0 47.5 48.0\nf (THz)104105106107ζ/λ\n(b)\nFIG. 6. Dispersion curves for the forward (solid curves) and backward (dashed curves) propagat-\ning TE surface plasmons along the graphene-ferrite structu re shown in Fig. 1 with anti-parallel\nmagnetic biases. (a) Normalized propagation constants. (b ) Normalized propagation length. The\nferrites have ferromagnetic resonances f0= 0.1 GHz (red), f0= 1.0 GHz (blue) and f0= 3.0 GHz\n(green). The corresponding magnetic bias ﬁeld is B0= 0.00356 T (red), B0= 0.0356 T (blue),\nB0= 0.106 T (green). A loss factor α= 0.05 is assumed for the ferrites. For clarity in (a) the blue\nand green curves are shifted up by 0.1 and 0.2 units, respecti vely.15"    },    {        "title": "0804.4377v1.The_role_of_carbon_segregation_on_nanocrystallisation_of_pearlitic_steels_processed_by_severe_plastic_deformation.pdf",        "content": " 1The role of carbon segregation on nanocrystallisation of pearlitic steels \nprocessed by severe plastic deformation. \n \n \nX. Sauvage1* and Y. Ivanisenko2 \n \n 1- Groupe de Physique des Matériaux - UMR CNRS 6634, Institut of Material Research, Université de Rouen, 76801 Saint-Etienne-du-Rouvray, France.  2- Institut für Nanotechnologie, Forschungszen trum Karlsruhe, 76021 Karlsruhe, Germany  \n   *corresponding author : Xavier Sauvage  \nxavier.sauvage@univ-rouen.fr  \nTel : + 33 2 32 95 51 42 \nFax : + 33 2 32 95 50 32   \nabstract \nThe nanostructure and the carbon distribution in  a pearlitic steel pr ocessed by torsion under \nhigh pressure was investigated by three-dime nsional atom probe. In  the early stage of \ndeformation (shear strain of 62), off-stoichiometry cementite was analysed close to interphase boundaries and a strong segregation of carbon atoms along dislocation cell boundaries was \nobserved in the ferrite. At a shear strain of 300, only few nanoscaled off-stoichiometry \ncementite particles remain and a nanoscaled e quiaxed grain structure with a grain size of \nabout 20 nm was revealed. 3D-AP data clearly point out a strong segr egation of carbon atoms \nalong grain boundaries. The influence of this ca rbon atom segregation on the nanostructure \nformation is discussed and a scenario accounti ng for the nanocrystall isation during severe \nplastic deformation is proposed.  \nkeywords \nsevere plastic deformation, three-dimensional atom probe, pearlitic steels, nanocrystalline \nmaterial, segregation.        \nPublished in Journal of Ma terial Science 42 (2007) p.1615 \nReceived: 12 May 2006 / Accepted: 2 Augu st 2006 / Published online: 16 December 2006 \n ©Springer Science+Business Media, LLC 2006  \nJ Mater Sci (2007) 42:1615–1621 \nDOI 10.1007/s10853-006-0750-z \n \n  2 \n1. Introduction \n \nProcessing of bulk nanocrystalline materials than ks to severe plastic deformation (SPD) has \nbeen widely investigated in the past two decades. Several SP D techniques have been designed \nto optimise grain refinement like Equal Channe l Angular Pressing (ECAP)  [1], High Pressure \nTorsion (HPT) [1-10] or Accumulated Roll B onding (ARB) [11]. In pure metals, grain sizes \nin a range of 100 to 500 nm are commonly achie ved. Such grain size re duction is usually \nattributed to the formation of dislocation cel l walls which progressively become low angle \ngrain boundaries and finally thr ough higher level of deformation turn into high angle grain \nboundaries [5, 6]. It has been shown however that  in severely deformed pearlitic steels, the \ngrain size could be one order of magnitude smal ler than that in pure iron [12, 13]. Pearlitic \nsteels exhibit a lamellar stru cture made of a mixture of ferrite and of cementite (Fe 3C carbide, \nvolume fraction of about 12 %) [14] and it is now  well admitted  that cementite could be at \nleast partly decomposed during plastic defo rmation [13, 15-20]. Previously published data \nshow indeed that cementite is dissolved in pe arlitic steels processed by HPT and since X-ray \ndiffraction data did not show any significant α-Fe peaks shift, it is though that carbon atoms \nhave segregated along grain bounda ries and dislocations [13]. Such carbon atom segregation \nalong grain boundaries was recently reported fo r ball milled pearlitic steel where the \ncementite phase was also decomposed [21]. This feature could explain why the grain size reduction is more pronounced in such  carbon steel than in pure iron.  \nThe nanostructure formation of  pearlitic steels subjected to  SPD have been extensively \nstudied by TEM, but quantitative measurements of the carbon redistribution following the strain induced decomposition of cementite can only be obtained by a three-dimensional atom \nprobe (3D-AP). The aim of this work was theref ore to carry out some experiments using this \ntechnique to map out the carbon di stribution in 3D for a better understanding of the role of \ndiffusing alloying elements on the nanostructure formation during SPD.    \n2. Experimental \n \nThe investigated material is a carbon stee l UIC 860 : 0.6-0.8 wt. % C, 0.8-1.3 wt. % Mn, 0.1-\n0.5 wt. % Si, 0.04 wt. % P (max), 0.04 wt. % S (max), Fe-balance. This steel was austenitised at 1223 K during 30 minutes and cooled in air to achieve a fully pearliti c microstructure with \nthe thickness of ferrite and cementite lamell ae of 210 and 40 nm, respectively. Then, it was \ndeformed by HPT under a pressure of 7 GPa with  one and five turns of torsion (N) at a \nconstant rate of 1 turn/min (s ee details in reference [13]).  \nThree-dimensional atom probe (3D-AP) and field ion microscopy (FIM) specimens were \nprepared by electropolishi ng [15]. Small rods were cut out fr om HPT discs and needle shaped \nspecimens were prepared so that th e tip was located at a distance of 3 ±0.5 mm from the disc \ncentre (see reference [16] for details ). The corresponding shear strain was γ = 62 ( ± 10 ) and \n300 (± 50 ) for N = 1 and 5, respectively [13]. FIM images were obtained at 80K, using Ne as \nimaging gas. 3D-AP analysis was carrie d out at 80K in UHV (residual pressure 10\n-8 Pa), with \n20% pulse fraction and 1.7 kHz pulse repetition rate. The atom probe was equipped with a \nreflectron device to enhance the mass resoluti on and a CAMECA’s position sensitive detector \n(Energy Compensated Optical Tomographic At om Probe, ECoTAP). During 3D-AP analyses, \ncarbon atoms are collected as ions (C+, C2+) and molecular ions (C 32+, C2+, C42+, C3+) [22]. \n3D-AP reconstructed volumes shown in this paper exhibit the distribution of all these \ndifferent ions. Measurements of carbon c oncentration were performed following the \nprocedure described by Sha and co-authors [22].  3 \n3. Results \n \nExperimental data collected on the undeformed pearlitic steel are not shown in the present \npaper because they are consistent with alrea dy published data [15-18]. The measured carbon \nconcentration is below 0.1 at% in the α-Fe phase and 25 ± 0.5 at.% in cementite lamellae \n(Fe 3C carbides). Moreover, sharp gradients (less than 2 nm) were always detected across α-Fe \n/ Fe 3C interfaces.  \n \nCarbon atom distribution after one tu rn of torsion under high pressure \n \n \nFigure 1 : 3D reconstruction of an analysed volume in the pearlitic steel processed one turn by HPT (a). Only \ncarbon atoms are plotted to exhibit a cementite lamellae.  The carbon concentration profile was computed with \na two nanometers sampling box across th e ferrite/cementite interface (b).  \n \n \nAfter one turn of torsion under high pressure, la rge regions of cementite were detected with \nthe 3D-AP. Such a region is shown in the Fig. 1 (a) where only car bon atoms are exhibited. \nThis is most probably a small part of a former cementite lamellae that might have been \nfragmented and plastically deformed. Th e composition profile computed across the α-\nFe/cementite interface exhi bits a sharp carbon gradient at this  interface (two nanometers wide \ncorresponding to the thickness of  the sampling box). However, in teresting features appear \nboth on the Fe 3C side and in the α-Fe side. In the far left of the profile, the carbon \nconcentration is 25 at.% as expected for cem entite, however along the interface there is a 10 \nnm thick layer with a carbon conten t in a range of 20 to 25 at.% which could be attributed to \noff-stoichiometry cementite. On the ferrite side , it is interesting to note as well that a large \ncarbon gradient appears : there is about 2 at.% carbon in the ferrite clos e to the interface and \nthis value slowly decreases dow n to zero at a distance of about  8 nm from th e interface.   4 \n \nFigure 2: Carbon atom distribution in an analysed vo lume in the pearlitic steel processed one turn by HPT \n(a). A small part of the volume was enlarged to show at  higher magnification carbon (b) and iron distribution \n(c). Carbon atoms are segregated on planar defects sepa rating ferrite zones labelled 1, 2 and 3. Atomic planes \nattributed to (110) α-Fe are clearly exhibited in zone 3. A composition profile was computed with a 1 nm \nsampling box across a carbon enriched planar defect (d). \n \nIn the ferrite, thin bands cont aining a significant amount of car bon were also detected (Fig. 2 \n(a)). Their 3D shape is somewhat complicated : they are curled and twisted. As shown by the \ncomposition profile computed across one of them, their thickness is in a range of 2 to 3 nm \nand their carbon content is about  6 at.% (Fig. 2 (d)). Both th e carbon and iron distribution is \nalso shown at a higher magnifica tion (Fig. 2 (b) and (c)) to poin t out three regions (labelled 1, \n2 and 3) separated by carbon rich layers. The atomic resolution was achieved in the region 3 \nwhere (110) α-Fe atomic planes are clearly exhibited because they were almost perpendicular \nto the analysis directi on. In the other two regions (labelled 1 and 2), these atomic planes are \nnot exhibited. This indicates th at there is a significant miso rientation between these three \nregions and that they are three different ferrite grains. The local  density in the reconstructed \nvolume showing Fe atoms appears slightly highe r at boundaries between ferrite grains (Fig. 2  5(c)). This feature is attributed to local magnification effects resulting from a lower \nevaporation field of these boundary regions which contain a significant amount of carbon. \n \nCarbon atom distribution after five tu rns of torsion under high pressure \n \n \nFigure 3: Field ion microscopy image of the pearlitic steel processed five turns by HPT exhibiting a lamellar \nnanostructure (a). 3D reconstruction of an analysed volume where only carbon atoms are plotted to show two carbon rich lamellae (b). The composition profile was computed with a 2 nm sampling box across the two \ncarbon rich lamellae (c). \n \nAfter five turns of torsion, tw o kinds of nanoscaled structures  were observed by FIM within \nsamples. Some regions exhibit a lamellar-like st ructure with an interl amellar spacing in a \nrange of 10 to 20 nm as shown in the Fig. 3(a)  where three dark lamellae are arrowed. Two of \nthese lamellae were analysed with the 3D-AP, and the volume displayed in the Fig. 3(b) show  6that they contain a significant amount of carbon. Quantitative carbon concentration \nmeasurements were obtained thanks to a compos ition profile (Fig. 3(c)).  They contain about 6 \nat.% carbon, and their th ickness is about 5 nm.  \n \n \nFigure 4: Field ion microscopy image of the pearlitic st eel processed five turns by HPT (a). Grain boundaries \nare arrowed. 3D distribution of carbon atoms within a small analysed volume (b) and carbon concentration \nprofile computed across with a 2 nm sampling box (c). \n \nField ion microscopy observations revealed ot her regions with a very different kind of \nnanostructure. Such a region is shown in th e Fig. 4 (a) where several grain boundaries are \narrowed. The grain size is in a range of 10 to 30 nm, and grains  are equiaxed. The distribution \nof carbon atoms is shown by the small analysed volume displayed in the Fig. 4(b) and the \ncomposition profile computed through this volume (Fig. 4(c)). A nanoscaled cementite particle is located in the middle of the volume.  As shown by the profile, cementite is slightly \noff-stoichiometry and contains between 20 and 25  at. %C. On the left and on the right of the \nparticle, segregation of carbon atoms along two pl anar defects is clearly exhibited. They are \nmost probably the grain boundaries obser ved on the FIM image (Fig. 4(a)). \n  74. Discussion \n Decomposition of cementite  \nThe 3D-AP analyses confirm that cementite is decomposed as it has been previously reported \n[13], but the data shown here provide new in formation about the decomposition mechanism. \nThe cementite particle analysed after one turn of torsion clearly exhibits that its structure is not homogenous. It contains 25 at.% in the core, while in a thin boundary layer it is slightly \noff-stoichiometry (Fig. 1). Moreover, 3D-AP da ta clearly show that even after 5 turns, \ncementite decomposition is not completed (Fig. 4) but cementite particles are much larger at \nlower strains probably because they are less fragmented but also less dissolved. It is \ninteresting to note that afte r five turns of torsion, Fe\n3C particles are so small that they are fully \noff-stoichiometry (Fig. 4). Thus, the decom position of cementite during SPD may proceed in \ntwo steps : first, C-vacancies are introduced in the Fe 3C phase which leads to the formation of \na metastable off-stoichiometry ce mentite [23-25]. Then, in a second step this latter structure is \nfully decomposed. As argued by Gridnev and Ga vrilyuk, dislocations in the ferrite lattice \ncould play an important role in  this process [19]. They propos ed that the transfer of carbon \natoms from cementite to dislocations in the ferrite is energetically favourable since the \ninteraction energy between carbon atoms and di slocations is higher than the bonding energy \nbetween Fe and C atoms in the Fe 3C phase (0.8 eV/atom and 0.5 eV/atom respectively). Thus, \nclose to the interphase bounda ry, carbon atoms may jump from  their lattice site in the \ncementite phase to dislocations located in the ferrite. This mechanism is consistent with the present experimental data showing a thin la yer of off-stoichiometry cementite along the \nFe\n3C/α-Fe interface (Fig.1(b)).  \n \nDiffusion of carbon atom s through the ferrite  \nAfter one turn of torsion, a significant carbon grad ient was exhibited in the ferrite close to the \nFe3C/α-Fe interface with a maximum concentration in a range of 1 to 2 at.% (Fig. 1(b)). Since \nthe solubility of carbon in BCC iron is very low [14], the formation of such a super saturated \nsolid solution is unlikely to happen. More over it has already been  confirmed by X-ray \ndiffraction measurements showing th at the lattice parameter of the α-Fe phase is not affected \nby Fe 3C decomposition even after 5 turn s of torsion [13]. Therefore, this observed gradient is \nnot attributed to downhill diffusion of interstit ial C atoms in the BCC fe rrite lattice but to \ncarbon atom segregation along disloc ations segments pinned by Fe 3C/α-Fe interfaces. In this \ncase the homogenous distribution of carbon obser ved in small 3D-AP reconstructions (Fig. \n1(b)) can be attributed to overlapping Cottrell atmospheres from neighbouring dislocations. \nBroad segregation of carbon atoms around dislocati ons (up to 7nm from the core) have indeed \nbeen reported by Wilde and co-authors [26] . Moreover, TEM observations of severely \ndeformed pearlite always demonstrate the form ation of very high dislocation density in the \nferrite close to the inter-phase  boundary [12,13]. If it is  assumed that this density is about \nρ=1012 cm-2 (typical density in severely deformed meta ls [1]), then the mean distance between \ndislocations is L ~ 1/ √ρ = 10 nm. In such a configurat ion, Cottrell atmospheres from \nneighbouring dislocations could simply overl ap, and a “homogenous” distribution will be \nobserved by 3D-AP. Such a segreg ation of carbon atoms to dislocat ions is consistent with the \nmechanism proposed by Gr idnev and Gavrilyuk [19]. \nOn the concentration profile displayed in the Fig.  1(a), it is interesti ng to note that there are \nmore carbon atoms missing in the Fe 3C than carbon atoms detected in the ferrite. This \nindicates that some of them have diffused ove r long distances. Several mechanisms should be \nconsidered : bulk diffusion, dislocation drag a nd pipe diffusion. However, the deformation \nwas performed at room temperature and no signifi cant increase of the te mperature is expected \nduring the HPT process [13]. In such conditions , the mobility of interstitial carbon atoms is  8very low in the ferrite and as previously repo rted [13], the estimate d diffusion distance of \ncarbon is much lower than 1 nm (during th e deformation time which is about 300 s). \nObviously, from the Fick’s law ( J = -D grad(C) ), such a low diffusivity (D) combined with \nthe low solubility (C) of carbon in the BCC ferrite [14] cannot gi ve rise to a significant carbon \nflux (J) through bulk diffusion. Moreover, this very  low mobility of carbon makes as well the \ndislocation drag mechanism not realistic. Thus, carbon atoms might diffuse through pipe \ndiffusion in dislocation cores. It is indeed well known that bulk diffusion rates of solute \nelements is much lower than their diffusion ra tes along dislocation core s where the activation \nenergy could be lowered by a factor of two [27].  \n \nTransformation of the pearlitic structure during HPT \nThe transmission electron microscopy investiga tion of the microstruc ture evolution of a \npearlitic steel of the same composition during torsion under high pressu re had been carried \nout in [13] and general trends can be summa rized as follows. At the beginning of HPT \ndeformation (e.g. after one turn of torsion), a ve ry high dislocation density is observed in the \nferrite phase in the vicinity of ferrite / cem entite interphase boundary. Then, between one and \nthree turns of torsion, a cellular structure form s in the ferrite phase, pearlite colonies \nprogressively align along the shear directi on and cementite platelets are elongated. Finally, \nhigher shear strains lead to a gradual increase  of misorientations of  cell boundaries in the \nferrite and to their transformation into hi gh angle grain boundaries. This process is \naccomplished after five turns of rotation a nd it gives rise to a very homogenous \nnanocrystalline structure with a grain size in the range of 10 to 20 nm where cementite is \nalmost completely decomposed [13, 28]. This evolution of th e microstructure of pearlitic \nsteels during HPT is consistent with the obser vations reported for othe r processes like wire \ndrawing [17, 18, 29], ball milling [ 21, 30], shot peening [31] and on surfaces of railway rails \n[12, 32].  The present data are consistent with TEM obser vations described above but they reveal quite \nnew features. After five turns of torsion (N=5), th e original lamellar structure of the pearlite is \nstrongly affected by the plastic deformation. Two kinds of nanoscal ed structure were revealed \nby the three-dimensional atom probe analyses. First, some regions still exhibit a lamellar \nstructure which is very similar to the microstr ucture of pearlitic steels processed by cold \ndrawing (steel cords) [17,18]. Carbon rich lamellae containing  less than 10at.% carbon are \nexhibited, the interlamellar spacing is smaller than 20 nm and large carbon gradients appear \n(Fig. 3). Such carbon rich lamellae are former Fe\n3C lamellae that were strongly elongated and \npartly decomposed during the HPT process. Ho wever, only few regions with this kind of \nlamellar structure were revealed (less than 20%  vol.). Other colonies have been completely \ntransformed into an equiaxed structure made of nano-scaled α-Fe grains stabilised by carbon \natoms segregated along grain bound aries (Fig. 4). Such a micros tructure is similar to the \nmicrostructure obtained by Ohsaki  and co-authors by mechanical milling of a pearlitic steel \n[21]. During mechanical milling, im pacts and friction of steel ba lls induce the severe plastic \ndeformation. This process is not continuous a nd the strain rate is much higher than during \nHPT. This could explain why a smaller grai n size was achieved (about 10 nm) and why a \nsignificant amount of carbon was detected in solid solution in the ferrite (up to about 1at.%). \nHowever, it is interesting to note that similar carbon concentr ations along grain boundaries \nwere reported (close to 6at.%).    9\n \nFigure 5: Schematic representation of the nanostructur e formation in the pearlitic steel processed by HPT. \nOriginal microstructure made of stoechiometric Fe 3C and α-Fe phases (a) ; Jump of C atoms (grey dots) from \nFe3C to dislocations in the ferrite and forma tion of C-vacancies (open circles) in the Fe 3C (b) ; Saturation of \ncell boundaries with C atoms, increase of misorientation between cells, further decomposition of Fe 3C and \nformation of new cell boundaries (c) ; Final microstructure made of α-Fe nanograins stabilised by carbon \nsegregation along grain boundaries (d). \n \nOn the base of previous TEM  observations [13] supported by the present 3D AP data we \nsuggest the following mechanism of the forma tion of the equiaxed nanoscaled structure \nduring HPT processing of the pearlitic steel,  sketched out in the Fig. 5. The original \nmicrostructure (N=0) is made of α-Fe with an extremely small amount of carbon in solid \nsolution [14] and stoichiometric Fe 3C lamellae containing 25at.%  carbon Fig. 5(a). At the \nearly stage of deformation (N <1), Fe 3C lamellae are elongated and fragmented. Carbon \natoms in Fe 3C nearby interphase boundaries jump to dislocations in the ferrite leaving \nvacancies and a thin layer of off-stoichiometry Fe 3C (data Fig. 1 and sketch Fig. 5(b)). Then, \nsome of them diffuse along the core of dislocations collected at Fe 3C/α-Fe interfaces. With \nincreasing of strains, dislocation cells devel op in the ferrite and ce ll boundaries trap carbon \natoms (data Fig. 2 and sketch Fig. 5(c)).  At higher level of deformation (1 ≤ N ≤ 5), Fe 3C \nlamellae are further fragmented and decom posed via the mechanism discussed above. \nDislocations previous ly collected at Fe 3C/α-Fe boundaries are satu rated with carbon atoms \nand are locked in the ferrite. Dislocations generated in the course of deformation are \nfrequently pinned with carbon atoms coming fr om cementite decomposition and that diffuse \nin ferrite lamellae along disloca tion cores. These pinned disloca tions initiate the formation of \nnew cell boundaries situated at rather small distances from each other. Simultaneously, an \nincrease of the misorientation between disloca tion cells occurs due to sink of not-pinned \ndislocations in cell boundaries. That way, a smaller cell size th an in pure iron after similar \nlevel of deformation is obtained [5, 6]. Cementite particles are progressively decomposed and after five turns of torsion (N=5), only few Fe\n3C nanoclusters remain (data Fig. 4). Most of  10carbon atoms are located along fo rmer dislocation cell boundarie s that have progressively \nbecome new grain boundaries stab ilised by carbon atoms (data Fi g. 4 and sketch Fig. 5(d)).  \n  \n5. Conclusions \n \ni) After one turn of torsion under high pressu re (shear strain of 62), off-stoichiometry \ncementite with a carbon concentration in a range of  20 to 25 at.% was analysed in the vicinity \nof interphase boundaries. Carbon atoms released by this decomposition of Fe\n3C were detected \nin the ferrite close to the interphase b oundary or along dislocation cell boundaries.  \nii) After five turns of torsion under high pressu re (shear strain of 300), only few nanoscaled \noff-stoichiometry cementite particles remain. Th e microstructure is mostly equiaxed with a \ngrain size of about 20nm but few regions with a lamellar structure stil l exist. Carbon atoms \nare segregated along grain boundaries with a ty pical carbon concentration peak of about 6 \nat.%.  iii) These experimental data confirm the mechanism proposed by Gridnev and Gavrilyuk \ninvolving dislocations [19] : during the plas tic deformation, C-vacancies are created in \ncementite and carbon atoms jump in the core of di slocations located in the ferrite close to \ninterphase boundaries. iv) It is proposed that carbon atoms diffuse along dislocation cores through pipe diffusion. \nDislocations are progressively pinned and serve as origin of new dislocation boundaries. This \nleads to a smaller cell size than in pure  iron. Further deformation induces higher \ndecomposition rate of cementite, saturation of  dislocation cell boundaries with carbon atoms \nand increasing of misorientations. This mechanism is though to promote the nanocrystallisation of pearlite during HPT.   \nAcknowledgements \nProfessor R.Z. Valiev (Institute of Physics of Advanced Materials, Ufa, Russia) is gratefully \nacknowledged for the processing of the samples by torsion under high pressure. \n  \nReferences \n \n[1] R.Z. VALIEV, R.K. ISLAMGALIEV, I. V. ALEXANDROV, Progr ess in Material \nScience 45 (2000) 103.  \n[2] Q. WEI, L. KECSKES, T. JIAO, K. T. HARTWIG, K.T. RAMESH, E. MA, Acta \nMater 52 (2004) 1859. \n[3] KYUNG-TAE PARK, DONG HYUK SHIN , Mater Sci Eng A334 (2002) 79. \n[4] D.H. SHIN, B.C. KIM, Y.S. KIM,  K.T. PARK, Acta Mater 48 (2000) 2247. \n[5] R.Z. VALIEV, Y.V. IVANISENKO, E.F.  RAUCH, B. BAUDELE T, Acta Mater 44 \n(1996) 4705. \n[6] D.A. HUGHES, N. HANS EN, Acta Mater 11 (2000) 2985. \n[7] O.V. MISHIN, V.Y. GERTSMAN, R.Z. VA LIEV, G. GOTTSTEIN, Scripta Mater 35 \n(1996) 873. \n[8] A. VINOGRADOV, H. MIYAMOTO, T. MIMAKI, S. HASHIMOTO, Ann. Chim. \nSci. Mat. 27 (2002) 65. \n[9] R.K. ISLAMGALIEV, W. BUCHGRABER, Y.R. KOLOBOV, N.M. \nAMIRKHANOV, A.V. SERGUEEVA, K. V. IVANOV, G.P. 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TS UCHIYA, Mat Sci Eng A375-377 (2004) 899. \n[31] Z.G. LIU, H.J. FECHT, M. UMEMOTO, Mat Sci Eng A375-377 (2004) 839. [32] H.W. ZHANG, S. OHSAKI, S. MIT AO, M. OHNUMA, K. HONO, Mat Sci Eng A \n421 (2006) 191-199. \n \n "    },    {        "title": "2402.16199v1.High_resolution_numerical_experimental_comparison_of_heterogeneous_slip_activity_in_quasi_2D_ferrite_sheets.pdf",        "content": "High-resolution numerical-experimental comparison of heterogeneous slip\nactivity in quasi-2D ferrite sheets\nJ. Wijnen, T. Vermeij, J.P.M. Hoefnagels, M.G.D. Geers, R.H.J. Peerlings∗\nDepartment of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands\nAbstract\nThe role of heterogeneity in the plastic flow of thin ferrite specimens is investigated in this study.\nThis is done through a recently introduced quasi-2D experimental-numerical framework that allows\nfor a quantitative comparison of the deformation fields of metal microstructures between experi-\nments and simulations at a high level of detail and complexity. The method exploits samples that\nare locally ultra-thin (”2D”) and hence have a practically uniform microstructure through their\nthickness. This allows testing more complex loading conditions compared to uniaxial micromechan-\nical experiments while avoiding the complexity of an unknown subsurface microstructure, which\nlimits comparisons between experiments and simulations in traditional integrated approaches at\nthe level of the polycrystalline microstructure. The present approach enables to study the effect\nof microstructural features such as grain boundaries. To study the role of stochastic fluctuations,\na constitutive model is employed which introduces random heterogeneity into a crystal plasticity\nmodel. A detailed analysis of the simulations is performed at the level of individual slip systems.\nSince both experimental and numerical results are susceptible to stochastic fluctuations, the out-\ncomes of many simulations are compared to the experimentally obtained result. This comparison\nallows us to determine how a single experiment relates to an ensemble of simulations. Additionally,\nresults obtained with a conventional crystal plasticity model are considered. The analysis reveals\nthat the heterogeneity in the plasticity model is essential for accurately capturing the deformation\nmechanisms.\nKeywords: Crystal plasticity, Numerical-experimental, Strain localization, Plastic heterogeneity,\nDislocation sources\n1. Introduction\nPlastic deformation in metals is a complex phenomenon, governed by various length scales.\nContrary to the apparent homogeneous deformation observed at the macroscale, metal plasticity is\nstochastic and heterogeneous by nature, introducing fluctuations at various scales. At the smallest\nscale, plastic deformation of a crystal results from the gliding of dislocations over discrete glide\nplanes. Multiple dislocations move collectively in an avalanche-like motion, resulting in intermit-\ntent plastic deformation in the form of strain bursts [1, 2, 3, 4]. At the scale of single grains,\nheterogeneities in the dislocation substructure, such as the locations and strengths of dislocation\n∗Corresponding author\nEmail address: r.h.j.peerlings@tue.nl (R.H.J. Peerlings )\nPreprint submitted to Journal February 27, 2024arXiv:2402.16199v1  [cond-mat.mtrl-sci]  25 Feb 2024sources, introduce fluctuations and strain localizations in the deformation of metals. Additionally,\na heterogeneous microstructure, e.g. consisting of different phases, promotes strain localizations at\nthe microstructural scale [5, 6]. The inhomogeneity of plastic deformation at a certain length scale\nstrongly affects strain localizations at larger scales, all the way up to macroscopic localization and\nfailure [6, 7, 8]. Therefore, studying fluctuations at different length scales is essential for improving\nour understanding of metal plasticity.\nThis paper presents a numerical-experimental analysis of the plastic heterogeneity at the grain\nscale. The main objective is to demonstrate the role of plastic strain localizations in the defor-\nmation behavior of grains. Key components that facilitate this study are the adopted ’quasi-2D’\nexperimental methodology, recently developed to enable detailed comparisons between experiments\nand simulations at this scale [9], and the employment of a computational model that introduces\nstochastic sub-grain fluctuations of plastic properties that naturally promote strain localizations\n[10]. Additional challenges that obstruct comprehensive experimental-numerical comparisons at\nthis scale are addressed here. The most prominent challenge is how to compare simulations and\nexperiments. Since the precise configuration of the underlying dislocation structure, e.g. the lo-\ncations of dislocation sources, is not known in experiments, obtaining a perfect one-to-one match\nbetween simulated and experimentally obtained strain fields is not possible. Instead, representative\ncharacteristics of a strain field are computed and are used to compare against an ensemble of strain\nfields.\nIn the literature, micro- and nanoscale mechanical testing, commonly performed through the\nloading of miniaturized tension or compression specimens, has been used extensively to investi-\ngate heterogeneous microscale plasticity [11, 12, 13]. These experiments rely on a detailed three-\ndimensional (3D) characterization of the deformation mechanisms, and enable detailed comparisons\nwith simulations, for example, at the level of individual slip systems [14, 15]. Micro- and nanoscale\nexperiments are often accompanied by discrete dislocation (DD) simulations to gain insights into\nthe fundamental deformation mechanisms of the dislocation network, e.g. into the stochastics of\nstrain bursts [16, 17] and dislocation sources [18, 19, 20].\nMost microscale experiments are limited in complexity through the microstructure morphology,\nspecimen shape, and loading path due to the small specimen size in at least two directions and\nthe absence of bulk material in these directions. This impedes the investigation of the interac-\ntion of deformation mechanisms and the influence of microstructure morphology. To advance our\nunderstanding in the direction of the bulk (or sheet) material, integrated experimental-numerical\napproaches with the same level of detail but with a larger degree of microstructural complexity\nare desired. On the other hand, in traditional polycrystalline bulk samples, only the surface mi-\ncrostructure can be characterized. Here, the unknown subsurface structure limits the interpretation\nof the observed deformation and the direct comparison to simulations, since severe two-dimensional\n(2D) simplifications through the thickness are usually made.\nThe recently developed integrated experimental-numerical quasi-2D framework, exploited in\nthis work, aims to bridge the gap between small-scale approaches and traditional integrated ap-\nproaches at the level of a polycrystal [9]. The considered specimens have a thickness of at most a\nfew micrometers in the gauge region, while their in-plane dimensions span multiple grains. This\nresults in pancake-shaped grains, which can be characterized on both the front and rear sides of the\nspecimen. Interpolating these surface microstructures through the (small) thickness is a reasonable\nassumption that results in an accurate 3D representation of the actual microstructure. In this way,\ndirect comparisons between simulations and experiments are possible on specimens that contain\n2several microstructural features, which do not suffer from unknown subsurface effects.\nAt the considered microstructural scale, strain localizations triggered by heterogeneities in the\nunderlying dislocation network are prominent. To analyze the effect of the heterogeneous strain\nfields at this scale, computational material models need to be employed that take this heterogeneity\ninto account. However, most numerical modeling frameworks addressing these inhomogeneities, e.g.\nDD simulations, are computationally too expensive for modeling experiments at real scale in full\ndetail. The discrete slip plane (DSP) model, developed in [10], overcomes this issue by introducing\nfluctuations of the slip resistance into a crystal plasticity model by considering one of the main\ncharacteristics of the underlying dislocation network, i.e. the strength and spatial distribution\nof dislocation sources. A notable advantage of this model is that it can be integrated into a\nconventional crystal plasticity (CP) model, only requiring a pre-processing step to determine the\nlocal material parameters based on the dislocation source stochastics, without directly simulating\ndislocation (source) interactions. It was shown in an earlier study that including this stochastic\nbehavior can explain the diversity in the many active slip systems observed in uniaxial tensile\ntests of single-crystal ferrite [21], contrary to a conventional CP model that does not include\nany heterogeneity at the scale of single crystals. In this paper, we study the role of stochastic\nfluctuations in the deformation behavior of quasi-2D specimens. These thin samples contain more\nmicrostructural complexity than uniaxial tensile tests since only one of the dimensions is small.\nThe combination of the integrated quasi-2D framework and the DSP model allows to study larger\nareas that contain microstructural features such as grain boundaries.\nAs stated before, the stochastic variations in both the experiments and simulations, make it\nimpossible to achieve a perfect match between the two. Additionally, only a single unique quasi-2D\nsample can be fabricated, i.e. an experiment cannot be repeated on the same sample. Therefore,\ninstead of directly comparing deformation fields between experiments and simulations, we aim\nto focus on characteristics of the deformation field that are statistically representative. For this\nreason, experimental strain fields are decomposed into contributions from individual slip systems\nby employing the SSLIP method [22]. Additionally, the degree of heterogeneity in these slip fields\nis determined. By using these characteristic quantities of the strain fields, we aim to determine\nhow a single experimental sample relates to the ensemble of simulations.\nA detailed analysis of two ferrite regions is presented. To investigate the role of plastic hetero-\ngeneity on the deformation of grains, differences between experimental observations, simulations\nwith the DSP models, and simulations with a standard (smeared) CP model are studied. The\nlatter accounts for inhomogeneities between different phases and grains through crystallographic\ntexture but lacks sub-grain fluctuations.\nThis paper has the following structure. In Section 2, the computational material models em-\nployed are briefly summarized. Section 3 presents two novel aspects extending the quasi-2D inte-\ngrated experimental-numerical framework. The set of boundary conditions applied to the regions of\ninterest is one of these aspects. The experimentally measured deformations at the edge of the exam-\nined region are applied in a weak sense, allowing for deviations from the measured displacements\nin the simulations. Furthermore, we introduce some characteristic measures of the deformation\nfield, based on which a detailed statistical comparison at the level of individual slip systems is\nperformed. The above-described sections use a simple single-grain ferrite region for demonstration\npurposes. A region containing a grain boundary is presented as a case study in Section 4. Finally,\nthe conclusions are summarized in Section 5\n32. Material models\nIn this section, the two material models used in simulations throughout this paper are sum-\nmarized. We first briefly review a conventional phenomenological crystal plasticity (CP) model in\nSection 2.1, because it serves as a basis for the more advanced discrete slip plane (DSP) model.\nFurthermore, the standard CP results are used as a classical reference for comparison purposes.\nIn Section 2.2, we discuss the DSP model, which considers the spatial and strength distribution\nof dislocation sources. This results in a heterogeneous slip resistance within a grain and conse-\nquently in heterogeneous slip distributions. Finally, the adopted model parameters are discussed\nin Section 2.3.\n2.1. Crystal plasticity model\nCrystal plasticity (CP) models are commonly used to study polycrystalline materials and mi-\ncrostructures in relation to their macroscopic mechanical properties [23]. In the conventional phe-\nnomenological CP model employed here, the deformation of a material point is the accumulated\n(homogeneous) behavior of slip on many underlying atomic slip planes.\nThe model is formulated in a finite deformation setting, in which the deformation gradient\ntensor, F, is split into an elastic part, Fe, and a plastic part, Fp, via a multiplicative decomposition:\nF=Fe·Fp. (1)\nThe crystallographic orientation of a grain is taken into account in the plastic velocity gradient by\nconsidering the individual contributions of all the slip systems:\nLp=NX\nα=1˙γα⃗ sα\n0⊗⃗ nα\n0, (2)\nwhere ˙ γαis the shear rate on slip system α, which has a slip plane normal ⃗ nα\n0and slip direction\n⃗ sα\n0.Ndenotes the total number of slip systems. The shear rate of a slip system is related to the\nresolved shear stress on that slip system, τα, by a phenomenological power law:\n˙γα= ˙γ0\u0012|τα|\nsα\u00131\nr\nsign ( τα), (3)\nwith slip resistance sα, reference shear rate ˙ γ0and rate-sensitivity exponent r. The slip resistance\nsαhas an initial value s0and its evolution is given by\n˙sα=h0\u0012\n1−sα\ns∞\u0013aNX\nβ=1qαβ|˙γβ|, (4)\nwhere h0andαare hardening parameters and the constants qαβcharacterize the mutual interaction\nbetween slip systems, by latent hardening. qαβtakes a value of 1 when slip systems αandβshare\nthe same slip plane, and a value of qnotherwise.\n42.2. Discrete slip plane plasticity\nAt small scales, the discrete and stochastic nature of plastic slip is more prominent. This is\nparticularly true inside individual grains, where a highly heterogeneous plastic slip activity is gen-\nerally observed. Conventional CP models capture fields that are much smoother than observed\nexperimentally. For this reason, the so-called discrete slip plane (DSP) model has been proposed\nin earlier work, see Wijnen et al. [10]. Based on the assumption that the heterogeneous plastic\ndeformation is predominantly due to the specific configuration of dislocation sources and obstacles\nto dislocation glide, the model introduces a spatial variation of slip system properties probed ran-\ndomly from a stochastic distribution of dislocation sources, which naturally induces more localized\nslip patterns compared to CP.\nIn the DSP model, in its fundamental form, all discrete atomic glide planes of a particular slip\nsystem in the finite size crystal are considered and the amount of slip on them is characterized by\na relative displacement (or disregistry) field v. The initial resistance against slip, i.e against the\nevolution of von an atomic plane is given by:\nsα,i\n0=sα,i\nnuc+sfric+1\n2Gb√ρdis, (5)\nwhere snucdenotes the nucleation stress, sfricthe lattice friction, Gthe shear modulus, bthe length\nof the Burgers vector and ρdisthe initial dislocation density. It is assumed that the nucleation\nstress varies between individual atomic planes due to the presence of one or multiple dislocation\nsources (giving a low snuc) or obstacles (giving a high snuc). For each plane, the nucleation stress is\nrandomly sampled from a probability density function (PDF) based on the physics of dislocation\nsources. Most planes are assigned a high nucleation stress, on the order of the theoretical shear\nstrength of the crystal, representing planes that do not contain a dislocation source. The nucleation\nstress of planes that do contain a dislocation source is based on the strength of a single-arm (SA)\nsource.\nSA sources are considered to be the dominant plastic mechanism in micropillar compression\ntests or microtensile tests [11, 24, 25]. In such geometries, a SA source spirals around its pinning\npoint, resulting in a plastic slip on its plane. The mechanism of a SA source in a thin plate or\nfilm slightly deviates from a SA source in a micropillar. Figure 1a shows a schematic of a thin\nplate with a single plane of a slip system displayed in red. Figure 1b shows the perpendicular\nview of the plane, containing a SA dislocation source. Additionally, Figure 1c shows a schematic\nof the resolved shear stress required during the operation of the source. Initially, at configuration\nt0, the dislocation in Figure 1b is pinned to a point in the center of the plate. The other end of\nthe dislocation is connected to the free surface at the top of the plate. When a resolved shear\nstress is applied to the glide plane of the dislocation, it bows out. The stress that bows out\nthe dislocation increases until it reaches its critical configuration, t1, corresponding to a resolved\nshear stress denoted as τmaxin Figure 1c. In this configuration, the dislocation has the maximum\ncurvature, and, consequently, line tension that needs to be overcome. If the resolved shear stress\nis sufficiently high to overcome this critical configuration, it continues to bow out (configuration\nt2). Finally, part of the dislocation reaches the free surface at the bottom of the plate and the\ndislocation is split in two, as shown in configuration t3. One dislocation segment is still connected\nto the pinning point while its other end is connected to the free surface at the bottom of the plate.\nThe other new dislocation is connected to the free surfaces on both sides of the plate. This process\nrepeats itself on the other side of the pinning point, where it creates another dislocation, shown\nin configuration t4. The pinned dislocation is now back in its initial configuration, t0. Therefore,\n5Figure 1: Schematic of the cross section of a thin plate containing a SA source. (a) Shows the outline of the thin\nplate with the cross section plane depicted in red. (b) Shows a perpendicular view of such a plane containing a SA\nsource. The mechanism of a SA source in a thin plate is depicted by configurations t1through t4. (c) A sketch of\nthe shear stress required during the operation of the SA source.\nin contrast to a SA source in a micropillar, a SA source in a thin plate generates new dislocations\nthat are connected to the free surfaces at both sides of the plate, which means that they cannot\nbe annihilated at the free surface (top or bottom).\nThe required resolved shear stress to activate a SA source is inversely proportional to the length\nover which the dislocation has to bow out in its critical configuration ( t1), i.e. the shortest distance\non the glide plane from the pinning point to one of the free surfaces. Note that in Figure 1c,\nthe two peaks have an equal height since the pinning point in Figure 1b is located exactly in the\nmiddle of the plate. If the pinning point were located closer to the top surface, the height of the\nfirst peak would increase, whereas the second peak would become lower. The opposite effect would\nbe observed if the pinning point was closer to the bottom surface.\nTherefore, the adopted PDF for snucis based on parallelogram-shaped planes for cuboid ge-\nometries, as presented in Reference [21]. One dimension of the cuboid geometry is taken equal to\nthe average thickness of a grain, while the other two dimensions are adopted as the in-plane grain\nsize. Note that since the thickness of most grains in the regions of interest is much smaller than\ntheir in-plane dimensions, the strength of the SA sources, related to the shortest distance of the\npinning point to the free surface, is mainly determined by the thickness.\nThe kinetics of slip over a single atomic plane, i, is described by\n˙vα,i= ˙v0\u0012|τα,i|\nsα,i\u00131\nr\nsign(τα,i), (6)\nwhere ˙ vα,iand ˙v0are velocities instead of the shear rates appearing in the conventional CP coun-\nterpart of this equation, Equation (3). The slip resistance of each plane has an initial value sα,i\n0\nand evolves asymptotically towards sα,i\n∞according to the relationship\n˙sα,i=k0\u0012\n1−sα,i\nsα,i\n∞\u0013a\n|˙vα,i|, (7)\nwhere k0is the initial hardening rate.\n6Table 1: Model parameters used for interstitial free ferrite in the CP and DSP model.\nCP DSP\n˙γ00.001 s−1˙v0 0.1µm/s\nr 0.05 r 0.05\ns015 MPa l 1µm\ns∞250 MPa sfric 20 MPa\nh0120 GPa ρdis 7·1011m−2\na 17.5 s∞ 2s0\nqn1.4 k0 30 GPa/ µm\na 5\nqn 1.4\nModeling all atomic planes of all the slip systems in a three-dimensional volume of the sizes\nconsidered here is computationally untractable. Therefore, atomic slip planes are clustered in\nbands, in which it is assumed that only the weakest slip plane experiences slip. Dividing this\nplastic relative displacement over the width of the band, l, results in a mean shear strain rate\nwithin the band given by\n˙γα=˙v0\nl\u0012|τα|\nsα,imin\u00131\nr\nsign ( τα). (8)\nWe employ this relationship to implement the DSP model as a conventional CP model with band-\nspecific material properties, randomly sampled from a stochastic distribution. Furthermore, the\ninitial slip resistance s0adopted in a band is equal to that of the weakest plane, sα,imin, i.e. the\nplane with the lowest slip resistance. This approach is taken for all ( N) considered slip systems of\nthe crystal, with a separate band for each slip system, implying that a particular material point\n(integration point in a finite element setting) belongs to Nbands, each with its own stochastic\nproperties.\nAs said, the slip resistance in a band is given by the slip resistance of the weakest atomic\nplane that experiences slip. Therefore, Equation (7) of the weakest atomic slip plane in a band is\nrewritten in terms of the mean shear rate in that band, resulting in\n˙sα,imin=k0l\u0012\n1−sα,imin\nsα,imin∞\u0013aNX\nβ=1qαβ|˙γβ|. (9)\nNote that latent hardening is additionally introduced into the above equation through constants\nqαβ.\nA more detailed treatment of the DSP model is presented in Wijnen et al. [10].\n2.3. Model parameters\nThe parameters for both the conventional CP model and the DSP model have been identified\nfrom the same uniaxial microtensile tests on monocrystal ferrite specimens of interstitial free steel,\nin Du et al. [26] and Wijnen et al. [21], respectively. The resulting model parameters, shown in\nTable 1, are adopted in the present study.\n73. Novel aspects of the integrated methodology\nThe DSP model introduced in Section 2.2 enables a detailed analysis of the kinematics of metal-\nlic microstructures. Unlike conventional CP, the stochastic nature of this model entails heteroge-\nneous and partially localized strain distributions with similar mean characteristics, but generally\na different spatial distribution compared to the measured pattern. To deal with this stochastic\nvariability, two additional elements of the quasi-2D integrated experimental-numerical approach of\nVermeij et al. [9] are introduced. First, for the applied boundary conditions the displacements ex-\ntracted from the in-situ SEM-DIC measurements are adopted. However, unlike the earlier method\nby Vermeij et al. [9], fluctuations with respect to the measured displacements are allowed, which\nrelax artificial stress concentrations near the boundary of the region of interest (ROI) and allows for\nstrain localization at other locations than in the experiment. Second, two quantitative indicators\nare introduced to determine how well the experiment matches the ensemble of random realizations\nof the DSP model.\nA simple microstructural ROI, i.e. a single crystal domain without grain boundaries, is used for\ndemonstration purposes throughout this section. The ROI is part of a larger specimen of which the\nthickness in the center is reduced by electrolyte polishing until a small hole is created, as described\nin detail in Reference [9]. Figure 2a shows the electron backscatter diffraction (EBSD) image of an\narea around the hole. The ROI is located in the grain immediately above the hole, as indicated by\nthe yellow rectangle. The thickness profile of the ROI is shown in Figure 2b, and ranges between\n1.5 and 2.1 µm. The discretization that is used in the simulations, containing 124,126 quadratic\ntetrahedral elements and 186,649 nodes, and the slip system orientations of the grain are shown in\nFigures 2c and 2d, respectively.\n3.1. Boundary conditions\nA key characteristic of the employed integrated experimental-numerical framework is that the\ndisplacements measured on the boundary of the region of interest in the experiment are applied\nas boundary conditions in the simulations. Figure 3a shows the displacements in the x-direction\nobtained by SEM-DIC from the experiment. The boundary of the region considered in the simu-\nlations is depicted with a yellow line. In the experiment, only the surface x- and y-displacements\nare measured in the region. However, the simulations are conducted in 3D. Therefore, the mea-\nsured in-plane displacements at the ROI edges are extruded through the thickness, i.e. they are\nprescribed to the full boundary denoted by Γ in the sketch of Figure 3b. This boundary condition\nset is referred to as fully prescribed boundary conditions (FPBCs) in the remainder of this section.\nFPBCs force the ROI to deform in the same way as in the experiment. However, this raises\nseveral concerns. For example, stress concentrations are introduced near the boundary because\nthe material has to comply with the prescribed displacements, which in addition are affected by\nmeasurement noise. This can be observed in Figures 4a and 4d that show the strain and stress field\nfor the case in which the measured displacements on the boundary of the area in Figure 2 are directly\napplied to a simulation with the DSP model. In the equivalent stress field of Figure 4d, the highest\nstress concentrations occur close to the boundary. Measurement noise and errors also add to these\nstress concentrations. Even when these would be reduced by smoothing they may have a significant\nartificial detrimental effect, for example, when identifying material parameters with integrated\ndigital image correlation [27]. Another disadvantage is that these boundary conditions enforce\nstrain localizations at measured locations of plastic slip crossing the boundary in the experimental\nspecimen, while the (random) source distribution is generally different in the simulations and,\n8Figure 2: (a) EBSD map of the area around the hole of the specimen. The small ROI is marked with a yellow\nrectangle. (b) Thickness profile of the ROI. (c) Discretization of the ROI consisting of 124,126 quadratic tetrahedral\nelements and 186,649 nodes. (d) Pole figure showing the orientations of the slip plane normals and slip directions of\nthe considered grain. The blue and orange markers denote the 110 and 112 plane normal, respectively, while the slip\ndirections are denoted by black markers. The plane normal of a slip system is depicted with the same marker as its\nslip direction.\ntherefore, triggers localizations elsewhere on the boundary. Close inspection of the equivalent\nstrain field in Figure 4a shows that many localization bands inside the ROI are constrained close\nto the boundary.\nTo relax the FPBCs, the boundary Γ in the simulation is allowed to deviate from the measured\ndisplacements. This is done by introducing the following boundary condition:\n⃗t=−E∗\nL∗⃗ w∀⃗ x∈Γ, (10)\nwhere E∗is a boundary stiffness, L∗is a characteristic length, and ⃗ wis the fluctuation vector given\nby\n⃗ w=⃗ u−⃗ uM, (11)\n9Figure 3: (a) Displacements in the x-direction measured with SEM-DIC. The yellow border denotes the region\nof interest used in the simulations. (b): Schematic of the region of interest. Ω denotes the volume. Γ denotes\nthe through-thickness boundaries to which the boundary conditions are applied. In reality, these boundaries are\nconnected to surrounding material.\nFigure 4: (a) Equivalent Green-Lagrange strain fields obtained in the ROI using the DSP model with (a) FPBC, (b)\nMKBC, and (c) RBC with E∗= 105MPa. The equivalent Cauchy stress fields for the same simulations are shown\nin (d), (e), and (f), respectively.\n10with ⃗ uthe actual in-plane displacement in the simulation and ⃗ uMthe measured displacement.\nEquation (10) applies a traction to the boundary of the volume element as a linear function of\nthe difference between the actual and measured displacements. It forces the boundaries of the\nsimulated volume element towards the measured displacements. Similar to the FPBCs, the ⃗ uMis\nextruded through the thickness of the boundary, such that ⃗tis applied to the full boundary Γ.\nBy only prescribing the traction at the boundaries, the average deformation in the region can\ndeviate from the prescribed deformation. Therefore, the boundary condition of Equation (10) is\ncomplemented by a constraint that enforces the average deformation gradient in the simulation to\nbe equal to the experimentally measured one. To connect the deformation in the 3D volume of\na simulation to the 2D deformation of the surface area in the experiment, first, a mean in-plane\ndisplacement vector is defined by averaging through the thickness in the model:\n¯⃗ u=1\nLzZz2\nz1⃗ u dz , (12)\nwhere Lz=z2−z1denotes the local thickness of the specimen. The area average of the in-plane\ndeformation gradient calculated with the mean in-plane displacement vector is now enforced to\ntake the same value as its experimental counterpart:\nZ\nΩ2D\u0000\n∇¯⃗ u\u0001T−(∇⃗ uM)TdΩ2D=0, (13)\nwhere Ω2Drepresents the area of the ROI. Using the divergence theorem and consecutively substi-\ntuting Equation (12) and Equation (11) results in\nZ\nΓ1\nLz⃗ w⊗⃗ n dΓ =0, (14)\nwhere ⃗ nis the outward normal vector at the boundary. Since the constraint reduces to a boundary\nintegral, only the displacements measured at the boundary of the simulation domain have to be\nused. Note that the resulting integral is computed on the 3D domain boundary Γ. The term 1 /Lz\naccounts for the non-uniform thickness of the specimen, i.e. it ensures that two opposing faces of\na square plate with non-uniform thickness contribute equally to the constraint. For a constant\nthickness, this term can be taken outside of the integral and does not have an influence on the\nconstraint.\nThe boundary conditions defined by Equations (10) and (14) are termed relaxed boundary\nconditions (RBCs) throughout the remainder of this paper. A similar set of boundary conditions\nwas recently introduced by Wojciechowski [28] in the scope of computational homogenization. The\namount of relaxation is determined by the stiffness E∗/L∗in Equation (10). L∗is adopted here as\nthe length of the geometry in the direction perpendicular to the boundary. Since the considered\nregions have the same xandydimensions, L∗is the same for all boundaries Γ. In this way,\n⃗ w/L∗can be interpreted as the fluctuation in the (linear) average strain tensor over the domain\ncontracted with the boundary normal, and E∗is the stiffness associated with that strain.\nTwo limit cases for E∗can be considered. For E∗→ ∞ , the fluctuation vanishes and the\nFPBCs are recovered. For the other limit case, E∗→0, the constraint of Equation (10) vanishes,\nand only the average deformation is prescribed. This is similar to so-called minimal kinematic\nboundary conditions (MKBCs) used in computational homogenization frameworks, which often\nresult in excessive localization near the weak spots at the boundary, while the rest of the domain\n111011041071010\nBoundary stiffness [MPa]4.04.2Total Strain Energy [nJ]FPBC\nRBC\nMKBCFigure 5: The total strain energy in the volume element as a function of the boundary stiffness E∗in the RBCs.\nThe total strain energies obtained for FPBCs and MKBCs are denoted by dashed lines.\nstays undeformed [29, 30]. Figures 4b and 4e show the same simulation as performed in Figure 4,\nbut with MKBCs. Localization is indeed more severe, e.g. in the bottom right corner, compared\nto the simulation with FPBCs, especially in the lower-right corner of the region.\nFigure 5 shows the total strain energy of the considered volume element for a range of traction\nstiffnesses for the simulation as done with the FPBCs and MKBCs of Figure 4. The total strain\nenergies obtained for the limit cases of MKBCs and FPBCs are also shown in the figure. Clearly, the\ntotal strain energy with RBCs changes from the value obtained with MKBCs to the value obtained\nwith FPBCs with increasing E∗. In this analysis, we adopt an intermediate traction stiffness of\nE∗= 105MPa, which results in a total strain energy approximately halfway between the two\nlimit cases. The equivalent strain and equivalent stress maps for this set of boundary conditions\nare shown in Figures 4c and 4f. The deformations are similar to the results with FPBCs, but\nlocalizations near the boundary are less suppressed. Yet the unphysical strain peak in the bottom\nright corner of Figure 4b is now absent. Significant differences also appear in the stress field, where\nhigh values are no longer visible close to the boundary, in contrast to Figure 4d.\n3.2. Slip activity characteristics for comparison of experiments and simulations\nThe experimentally determined strain and slip fields are highly heterogeneous at the sub-grain\nlevel. The DSP model, unlike conventional CP, replicates this heterogeneity. However, the precise\nlocations of slip bands and other features of the heterogeneity depend on the random properties of\nthe slip system bands, i.e. simulated by the variability of the nucleation stress at the integration\npoints, which represents the distribution of dislocation sources. This implies that each realization of\nthe random microstructure will trigger different strain and slip activity patterns, none of which will\nbe exactly identical to the experimental pattern. In addition, the width of the slip bands and other\nfeatures in the modeling approach are set by the band width lintroduced in the implementation of\nthe DSP model, which, due to computational efficiency reasons, is typically larger than the width\nof the slip bands observed in the DIC strain fields. Note that the experimental strain patterns\ndepend on the facet size used in the DIC analysis, which generally also broadens the true width\nof slip bands. These observations raise the question of how to objectively assess and compare the\n12Figure 6: The 2D equivalent Green-Lagrange strain field (a) in the experiment, (b) obtained with the CP model and\nobtained for the (c) ”best” and (d) ”worst” realizations with the DSP model.\nsimilarity (or dissimilarity) of both strain patterns with different length scales and configurations,\nwhich nevertheless may look similar to the observer’s direct view.\nBefore addressing this question, we first illustrate the problem for the single-grain ferrite re-\ngion considered above. In both the CP simulation and the DSP simulations, 12 {110}⟨111⟩and\n12{112}⟨111⟩slip systems are taken into account. Since the out-of-plane deformation in the ex-\nperiment is unknown, a 2D equivalent strain measure based on the in-plane components of the\nGreen-Lagrange strain tensor is used for plotting the deformation in both the experiment and\nthe simulations [5]: Eeq=√\n2\n3q\n(Exx−Eyy)2+E2xx+E2yy+ 6E2xy. The resulting strain fields are\nshown in Figure 6, where Figure 6a displays the experimental equivalent strain field and Figures 6c\nand 6d the computational strain fields for two realizations of the random source distribution. These\ntwo realizations are termed ”best” and ”worst”, which will be clarified later in this section. The\nresult obtained with conventional CP is also shown, for reference, in Figure 6b. In the latter, the\ndeformation is nearly homogeneous. Only the strains around the bottom center of the region are\nsomewhat higher due to the thickness profile and the boundary conditions. The experimental data,\nin Figure 6a, on the other hand, shows a fine-scale pattern of intense localization bands, in which\nthe strain locally exceeds the mean value by a factor of 10 or more. Clearly, the maximum strain\nin the region is much higher in the experiment than in the CP result.\nThe equivalent strain fields for the two different realizations of the DSP model reveal fluctua-\ntions that are at least qualitatively similar to those observed in the experiment. In particular, the\n13predicted bands have similar orientations to those in the experimental result, and their amplitude\npeaks in the same region near the bottom of the considered ROI. However, as expected, neither the\nwidth of the bands nor their precise location matches the experiment. Similar observations may be\nmade for the slip distributions associated with the individual slip systems. For the modeling, this\ndata is readily obtained from the simulations. For the experiments, it may be extracted from the\nmeasured deformation fields by the SSLIP method, introduced in Vermeij et al. [22]. This method\nlocally decomposes the deformation gradient field, measured on the specimen surface, into con-\ntributions from the theoretical slip systems. This is achieved by solving an optimization problem\nin which the measured kinematics optimally captures a combination of slip systems, each with a\nto-be-determined amount of slip. As an example, the slip activity of the most active slip system\nin the experiment denoted as SS1 in the pole figure of Figure 2d is shown in Figure 7a. The slip\nactivity of the same slip system in the DSP simulation of Figure 6c can be seen in Figure 7b.\nObtaining a perfect match between the slip activity and, consequently, the equivalent strain\nmap in a simulation and in the experiment is not possible. This is because, at the considered scale,\nstochastic fluctuations have a prominent effect on the experiment. These stochastic fluctuations\nare also present in the DSP model. Therefore, a comparison is made based on the amount of\nslip on the individual slip systems and the degree of localization in the slip fields. However, the\nspatial localizations observed in both the DSP model as well as the experiments are limited due to\nnumerical and experimental resolutions. Accordingly, a measure is introduced that represents the\namount of localization, given a certain length scale.\nA typical characteristic of strong localizations is a high gradient in the strain field. Localization\nof a specific slip system usually follows its slip plane trace. Therefore, the gradient of the slip\nactivity field perpendicular to the slip plane trace is a measure of fluctuations in that slip system\naround a particular point. However, the fluctuations in the deformation field obtained with the\nDSP model are dependent on the band width l. Preferably, the adopted band width is as small\nas possible. It is, however, limited due to the computational cost of a simulation. Therefore, a\nlocalization quantity, ψ, is introduced that approximates the gradient in the slip activity field by\nthe difference in slip activity of two points separated by a vector ⃗dψ(similar to a central difference\napproximation). This vector is perpendicular to the in-plane trace of the considered slip plane\nand has a length, ⃗dψ, that is equal to the adopted band width in the DSP model, projected on\nthexy-plane, since this is the smallest length over which fluctuations in the simulations can be\nresolved. Note that the smallest length over which fluctuations can take place in the experimental\ndata is also limited, for example, by the subset size used in DIC. However, for comparing the two\ndata sets the largest of these length scales should be used. i.e. the numerical band width.\nThe localization quantity is given by\nψ(⃗ x) =||γ(⃗ x−1\n2⃗dψ)| − |γ(⃗ x+1\n2⃗dψ)||\n|max ⃗ x(γ)|, (15)\nwhere γis the slip activity. It is normalized by the maximum value of the slip activity in the\nspecimen. In this way, ψtakes a value of 1 when two points separated by ⃗dψhave slip activity\nequal to 0 and max ⃗ x(γ).\nThe localization quantity of the slip activity field shown in Figure 7a is calculated by Equation\n15. This is schematically depicted by the red line. The localization value at the midpoint of this\nline, i.e. the red square, is calculated by the slip activity at the end of this line, i.e. the red dots.\nA localization quantity field is obtained by moving the midpoint of this red line over all the pixels\n14Figure 7: The slip activity fields of the most active slip system, i.e. slip system (110)[ ¯111] marked by SS1 in the pole\nfigure of Figure 2d, for (a) the experiment of Figure 6a and (b) the DSP simulation of Figure 6c. The localization\nquantity field of SS1 in (c) the experiment and (d) the DSP simulation.\nof the slip activity field. The result is shown in Figure 7c. Note that the length of the red line in\nFigure 7a is not the actual length of ⃗dψthat is used in the analysis, but is magnified for clarification.\nThe localization quantity field calculated from the slip activity field of Figure 7b is displayed in\nFigure 7d.\nThe slip activity and localization fields are defined locally. However, in order to examine many\nresults together, both measures were averaged over the region. Figure 8 displays the resulting\nvalues for the experiment, the CP model, and 100 realizations of the DSP model for the three most\nactive slip systems in the experiment. The horizontal and vertical axes represent the average slip\nactivity, ¯ γ, and the average localization quantity, ¯ψ, respectively.\nFor the most active slip system in the experiment, i.e. the (110)[ ¯111] slip system denoted by\nSS1, shown in Figure 8a, the experimental data point lies in the center of the point cloud of the\nDSP results, close to the mean. This indicates that the experiment, at least in terms of the two\ncharacteristics employed here, may be captured as a particular realization of the ensemble of DSP\nmodels represented by the blue point cloud. The CP simulation has a higher average slip activity\nthan the experiment and DSP simulations. Furthermore, the localization quantity for the CP\nsimulation is much lower, which can be expected based on the strain field in Figure 6b.\nThe results for the second most active slip system in the experiment, i.e. the ( ¯110)[ ¯1¯11] slip\n150.00 0.01 0.02 0.03 0.04\n̄γ [-]0.00.20.40.60.81.0̄ψ [-]\nDSP\nEXP\nCP\nBest\nWorst(a) SS1 (110)[ ¯111]\n0.00 0.01 0.02 0.03 0.04\n̄γ [-]0.00.20.40.60.81.0̄ψ [-]\nDSP\nEXP\nCP\nBest\nWorst (b) SS2 (¯110)[ ¯1¯11]\n0.00 0.01 0.02 0.03 0.04\n̄γ [-]0.00.20.40.60.81.0̄ψ [-]\nDSP\nEXP\nCP\nBest\nWorst (c) SS3 (12¯1)[¯111]Figure 8: The average slip activity and localization value of the region of Figure 2 for the experiment, 100 DSP\nsimulations, and the CP simulation, for three different slip systems. The kernel density estimates of the DSP\nsimulations is plotted on both axes.\nsystem denoted by SS2, are shown in Figure 8b. The amount of scatter in the DSP results,\nparticularly in the average slip activity ¯ γ, is larger than in Figure 8a. Nevertheless, the average\nslip activity in the experiment is lower than in all the simulations. The average slip activities for\nthe DSP simulations are located around the average slip activity obtained with the CP model. The\nlocalization value of the experiment is similar to a few DSP results, but most simulations have a\nlower value.\nFigure 8c shows the results for the third most active slip system in the experiments (SS3), i.e.\nthe (12 ¯1)[¯111] slip system denoted by SS3. The experimental data point lies in the point cloud of\nthe DSP results. Contrary to the most active slip system, the slip activity in the CP simulation is\nlower than in the experiment and the DSP simulations.\nFor most slip system specific quantities, the experiment falls within the range of the DSP\nsimulations. Only SS2 is significantly more active in most DSP simulations. Interestingly, there is\nno slip system compensating for this higher activity, i.e. there is no slip system that is significantly\nmore active in the experiment than in the simulations. A possible explanation is that, in contrast to\nthe in-plane deformation, the average out-of-plane deformation in the simulations is not enforced\nto be equal to that of the experiment. The difference in average slip activity between the CP\nsimulation and the DSP simulations for SS1 and SS2 shows that introducing fluctuations in the\nslip resistance not only results in a more localized strain field but also influences the relative activity\nof the slip systems. In general, a better match between experiment and simulation is obtained with\nthe DSP model compared to the CP model, for both the slip fluctuations as well as the slip activity.\nOur previous study [21] showed that the DSP model gives a more accurate slip system activity\ncompared to the CP model for uniaxial tensile tests. The results presented in this section show\nthat this conclusion can be extended to other geometries with more complex loading conditions.\nIt is instructive to consider, in the results of Figure 8, those DSP simulations that match\nthe experiment the best and worst, solely in terms of ¯ γandψ. The absolute difference between\nthe experiment and all the simulations is calculated, for both the slip activity and localization\nvalue. The differences are normalized by the maximum difference between the experiment and a\nsimulation, such that for both properties there is a single data point with a value of 1 for every\n16slip system. Next, the distance between the experiment and a simulation for a single slip system\nis calculated by taking the Euclidean norm of the vector that contains the normalized difference in\nslip activity and localization value. Finally, the distances of the three slip systems are added up,\nresulting in a single distance per DSP simulation. The simulation that matches ”best” (shortest\ndistance) and ”worst” (largest distance) with the experiment are both marked in the plots of\nFigure 8. It can be observed that the best simulation has a localization value that is similar to\nthe experiment for all three slip systems. The slip magnitude for SS2 of the best simulation is\nrelatively low, closer to the experiment than most other DSP simulations. To compensate for the\nlow slip magnitude of SS2, the slip magnitudes of SS1 and SS3 are relatively high.\nThe equivalent strain fields of the best and worst matching simulations are shown in Figures 6c\nand 6d, respectively. These figures show that the worst matching simulation has less pronounced\nstrain fluctuations than the best matching simulation. The latter contains more small areas with\nan equivalent strain close to zero and higher equivalent strain values in general. The smoother\nstrain field of the worst agreeing simulation can also be recognized in the data of Figure 8, where\nit reveals a low localization value for all three slip systems.\n4. Case study of ROI containing a grain boundary\nSo far, the framework has been applied to a relatively simple ROI, i.e. a single-phase region\nwithout grain boundaries. In this section, a ferrite region containing two grains is examined.\nThe ROI is taken from the same specimen as the ROI considered in Section 3. Its location is\nmarked in Figure 9a. The thickness profile of the specimen, together with the grain boundaries\non the front surface (black line) and rear surface (gray line), are depicted in Figure 9b. The\nregion is approximately split in half by a curved grain boundary. To perform simulations, the\n3D geometry was discretized with 64,366 quadratic tetrahedral elements and 103,136 nodes, as\ndepicted in Figure 9c. The pole figure in Figure 9d shows the orientation of the slip systems in\ngrain 1, denoted in Figure 9b.\n4.1. Equivalent strain fields\nThe 2D equivalent strain fields of the experiment are presented in Figure 10a. At some posi-\ntions, predominantly near the grain boundary, DIC data was not correctly correlated due to large\ndeformations. Therefore, interpolation was performed on the displacement fields to recover the\ncomplete deformation fields, as discussed in Vermeij et al. [9]. The equivalent strain field shows a\nstrong localization close to the boundary. Yet, a detailed inspection of the secondary electron (SE)\nand backscatter electron (BSE) images did not reveal clear signs of any damage. Furthermore,\ndue to the high resolution and the precise alignment of the images, the strong localization was\nidentified to be located in grain 1, i.e. at the right side of the grain boundary. No indication of\ngrain boundary sliding was observed. Away from the boundary, grain 1 shows many fluctuations\nin the strain fields. Grain 2, i.e. the left grain, shows only small strain localizations coming in at\nthe top of the region, propagating through approximately one-third of the grain.\nFigure 10b shows the equivalent strain field obtained with the conventional CP model. It reveals\nhigh strains in grain 1 near the grain boundary, similar to the experiment. The strain gradually\ndecreases further away from the boundary. Clearly, the strain fields are again much smoother\ncompared to the experiment.\nThe equivalent strain fields for the ”worst” and ”best” realizations obtained with the DSP\nmodel are shown in Figures 10c and 10d, respectively. The simulations show more fluctuations in\n17Figure 9: (a) EBSD map of the indicated area with a grain boundary around the hole of the specimen. The smaller\nROI is marked with a yellow rectangle. (b) The thickness profile of the ROI, where the grain boundary edges are\nshown as an overlay with a black line for the front surface and a gray line for the rear surface. (c) Discretization\nof the ROI consisting of 64,366 quadratic tetrahedral elements and 103,136 nodes. (d) The orientations of the slip\nplane normals and slip directions of the grain 1. The blue and orange markers denote the 110 and 112 plane normal,\nrespectively, while the slip directions are denoted by black markers. The plane normal of a slip system is depicted\nwith the same marker as its slip direction.\nthe strain field than the conventional CP simulations. However, in both simulations, strain bands\ntend to line up with the grain boundary in grain 1. This indicates that the influence of fluctuations\non the slip resistance is less pronounced close to the grain boundary. Instead, the morphology of\nthe grain boundary dictates the degree of plasticity here. In grain 2, similar strain localizations as\nin the experiments are observed, especially for the simulation of Figure 10c. Compared to the CP\nresult, the strains in grain 2 obtained with the DSP model propagate further downwards.\n4.2. Statistical slip system analysis\nDespite the lack of fluctuations in the strain field, the conventional CP model appears to describe\nthe deformation in the experiment reasonably well in an average sense. However, a more detailed\n18Figure 10: 2D equivalent Green-Lagrange strain fields of the ROI shown in Figure 9 for (a) the experiment, (b) the\nCP simulation, (c) the ”best” DSP simulation and (d) the ”worst” DSP simulation. The orange dotted line and the\neye in (b) show the location of the cross-section and camera angle that will be used later in Figure 13.\nanalysis with a focus on slip system activities and degree of localization, similar to Section 3.2,\nwas performed here as well. Since the majority of the deformation took place in grain 1, the\nanalysis is restricted to this grain. The strain fields are again decomposed into the contributions of\nthe individual slip systems, both experimentally and numerically, and the localization quantity is\ncalculated through Equation (15). The averages of the slip activity and the localization quantity\nin grain 1 in the experiment, the conventional CP simulation, and 100 DSP simulations are shown\nin Figure 11, for the three most active slip systems in the experiment. These are the (1 ¯21)[¯1¯11],\n(110)[ ¯1¯11] and (12 ¯1)[¯111] slip systems, denoted by SS1, SS2 and SS3, respectively. They are ordered\nbased on their average slip activity in the experiment. Their orientations are indicated in the pole\nfigure of Figure 9d.\nSurprisingly, the most active slip system in the experiment (SS1) is not at all active in the\nconventional CP simulation. Instead, most slip in the CP simulation emerges from SS3 (Fig-\nure 11b), where the slip magnitude is significantly larger than in the experiment. Although the\nROI is deformed via a complex loading path, its proximity the the hole in the sample renders the\ndeformation in the x-direction dominant. The Schmid factor of SS3 for uniaxial tension in the\nx-direction is close to 0.5, which means that this slip system is likely to activate. However, due\nto the microstructure of this region, strain bands aligned with the grain boundary tend to form.\n190.000 0.025 0.050 0.075 0.100\n̄γ [-]0.00.20.40.60.81.0ψ [-]\nDSP\nEXP\nCP\nBest\nWorst(a) SS1 (1¯2¯1)[¯1¯11]\n0.000 0.025 0.050 0.075 0.100\n̄γ [-]0.00.20.40.60.81.0ψ [-]\nDSP\nEXP\nCP\nBest\nWorst (b) SS2 (¯110)[ ¯1¯11]\n0.000 0.025 0.050 0.075 0.100\n̄γ [-]0.00.20.40.60.81.0ψ [-]\nDSP\nEXP\nCP\nBest\nWorst (c) SS3 (12¯1)[¯111]Figure 11: The average slip magnitude and localization value in grain 1 of Figure 9 for the experiment, 100 DSP\nsimulations, and the CP simulation, for three different slip systems. The kernel density estimates of the DSP\nsimulations is plotted on both axes.\nIn the conventional CP simulation, this strain band is almost fully accommodated through slip on\nSS3, even when its slip plane is not fully aligned with the grain boundary, by a combination of slip\nand kinking. The kink mechanism is a localization mode inherent to the constitutive behavior of\ncrystal plasticity, in which the localization band is oriented perpendicular to the slip plane [31, 32].\nPhysically, kinking requires a large number of dislocation sources that are aligned perpendicular to\nthe slip plane [33]. Furthermore, it introduces a high geometrically necessary dislocation (GND)\ndensity which is energetically unfavorable [34]. Therefore, in the experiment, deformation modes\nparallel to the slip systems are preferred and SS3 is not activated. Instead, the strain band follow-\ning the grain boundary is a result of slip on both SS1 and SS3. Because these slip systems have\nthe same slip direction, cross-slip can occur between them, as confirmed by the experimental strain\nmap in which the localization band is curving along the grain boundary (Figure 10a).\nIn the DSP simulations, slip does take place on SS1, as well as on SS3. For most simulations, the\naverage slip activity on SS1 is lower than in the experiment, but several realizations have a similar\naverage slip activity. Also, the slip activities of SS2 and SS3 in the DSP simulations are similar\nto those in the experiment. However, the amount of slip on SS3 is slightly overpredicted in most\nrealizations. Because the slip resistance of a slip system in the DSP model varies in the direction\nperpendicular to the slip plane due to the limited number of dislocation sources, the slip mechanism\nparallel to the slip planes is much more favorable compared to the kink mechanism, as is the case\nin the experiment. As a result, the activation of SS3 in the strain band near the grain boundary is\nsuppressed. Figures 10c and 10d reveal that the strain band following the grain boundary consists\nof short slip bands on alternating slip systems, resulting in a zig-zag pattern. This alternation\nhappens mainly between SS1 and SS2. Although no cross-slip mechanism is incorporated explicitly\ninto the model, the two slip systems having the same slip direction are most likely preferred due\nto compatibility. Note that a combination of the slip planes of SS1 and SS3 also results in a\ndirection approximately parallel to the grain boundary, but that out-of-plane components of these\nslip directions are significantly different, entailing deformation incompatibilities.\nThe average localization quantity in grain 1 of the experiment lies in the point cloud of the\nDSP simulations for all three slip systems. The ”best” and ”worst” matching DSP simulations\n20with the experiment were again determined based on the average quantities, as described before.\nThe equivalent strain fields of the ”best” and ”worst” matching simulations (green and red dots\nin Figure 11, respectively) are shown in Figures 10c and 10d, respectively. When only considering\nthese equivalent strain fields, Figure 10c does not seem to be much closer to the experiment than\nFigure 10d. However, Figure 11 reveals that the ratios of slip system activities of the ”best”\nmatching DSP simulation are much closer to the experiment than the ”worst” matching DSP\nsimulation.\n4.3. Slip system fields\nThe ”best” matching DSP simulation is most similar to the experiment in terms of the average\ncharacteristics of the slip fields. However, this provides no information on the spatial agreement\nbetween slip fields. Figure 12 presents the slip activity fields of the three most active slip systems\nin grain 1 for the experiment in (a-c) and for the ”best” realization of the DSP model in (e-f). Near\nthe grain boundary, all three slip systems reveal a significant amount of slip in the experiment.\nFurthermore, SS1 is active throughout the entire grain. The activity of SS2 is most pronounced in\nthe region close to the grain boundary. Slip on SS3 is also observed in the upper-right region of\nthe grain.\nIn the DSP simulation, SS1 is most active near the grain boundary, while slip is observed\nthroughout almost the entire grain, similar to the experiment. Localization bands of SS2 are\nvisible close to the grain boundary. The amount of slip in these localization bands fades out\nfurther away from the grain boundary. SS3 has slip bands visible in the right part of the grain,\nprimarily in the upper-right region. Furthermore, a small slip band is observed in the lower-left\nregion, against the grain boundary. So it can be concluded that regions of the grain in which\nthe particular slip systems are active, are in adequate agreement between the experiment and the\n”best” DSP simulation. Note again that a perfect match is not feasible since both the experiment\nand the DSP simulation are significantly influenced by the stochastic fluctuations. Furthermore,\nthis DSP realization closest resembles the experiment based on the average characteristics of the\nslip fields, and not on the exact position of the localizations or the width of the slip bands.\n4.4. 3D deformation\nContrary to the experimental data, the numerical data is fully 3D. This allows us to inspect\nthe 3D through-thickness deformation of the considered region. Figure 13 shows the deformed\ncross-sections of the conventional CP simulation in subfigure (a) and of the ”best” DSP simulation\nin subfigure (b), from a perspective view. The cross-sections were taken from the upper-left corner\nto the lower-right corner of the regions while the viewpoint is from approximately the lower-left\ncorner, looking at the front surface, as depicted by the orange dotted line and eye symbol in\nFigure 9b.\nDespite the 2D character of the ultra-thin geometry, the equivalent strain fields clearly reveal\nthe 3D orientation of the slip systems. In the DSP simulation, a strain band is observed that\ntouches the grain boundary on the rear. The strain band follows the orientation of the slip plane of\nSS2 (see Figure 9d). As a result, the same strain band observed on the front is located at a small\ndistance from the grain boundary. This shift of the strain localization with respect to the front and\nrear is much less pronounced in the CP model. Here, the strain band is the result of slip on SS3.\nThe slip plane of this slip system makes an angle of approximately 45◦with the xy-plane, i.e. the\nplane of the geometry, however, the strain localization is roughly straight through the thickness\nof the geometry. This again demonstrates that the conventional CP model has no restrictions in\n21Figure 12: (a-c) Slip activity fields of the three most active slip systems in the experiment and (d-f) the slip activity\nfields of the corresponding slip systems in the ”best” matching DSP realization.\nforming strain bands that are not aligned to the slip plane of the active slip system, in contrast to\ncommonly observed slip system mechanics.\n5. Summary and conclusions\nIn this study, a detailed numerical-experimental analysis was performed on thin regions of\na ferrite polycrystalline sheet. The quasi-2D nature of the microstructure enabled us to fully\ncharacterize the specimen’s morphology and deformation. This enables direct comparisons between\nexperiments and simulations. The richness of the available experimental data allows to assess and\npossibly improve current materials models to include more relevant physical phenomena.\nThe framework was used to study the deformation kinematics of two ferrite regions. This was\ndone at the level of individual slip systems. Two types of models were employed, namely, a con-\nventional crystal plasticity model, in which the average behavior of many underlying atomic planes\nis accounted for, and the recently introduced discrete slip plane (DSP) model, which considers\nfluctuations in slip resistances on atomic planes due to stochastic variability in the location and\nstrength of dislocation sources.\n22Figure 13: 3D cross-sections of the deformed geometries for (a) the conventional CP model and (b) the ”best”\nrealization with the DSP model. The deformation is on the true scale. The colormap shows the (3D) equivalent\nGreen-Lagrange strain. The cross-sections are taken from the upper-left corner to the lower-right corner of the region.\nThe viewpoint is approximately from the lower-right corner, looking at the front surface.\nA detailed analysis of 100 DSP realizations revealed a remarkable difference in slip system\nactivity between conventional CP simulations on the one hand and DSP simulations and the\nexperiment on the other hand, especially for the ferrite region considered in the case study of\nSection 4. Although the deformation fields obtained with the conventional homogeneous CP model\nmay appear in reasonable agreement with the experiment at first sight, the conventional CP model\ncompletely failed to predict the correct active slip systems. Almost all deformation took place\non one favorably oriented slip system, following a strain path that was not perpendicular to the\nslip system. In contrast, the fluctuations introduced by the DSP model through the stochastics\nof dislocation sources facilitate strain bands parallel to the slip systems. As a result, the grain\nboundary in the considered region had a significant effect on the slip system activity in the DSP\nsimulations, similar to what was observed experimentally.\nThe amounts of slip and the amounts of localization on the most active slip system were used\nto ascertain how an experiment relates to the ensemble of simulations. For both regions, the\nexperimental data fell within the spread of the values obtained with the DSP model. The adequate\n23agreement demonstrates that at the scale of the performed experiments, the inherent discrete and\nstochastic nature of plastic deformation needs to be taken into account to capture the correct\ndeformation mechanisms.\nDeclaration of Competing Interest\nThe authors declare that they have no known competing financial interests or personal rela-\ntionships that could have appeared to influence the work reported in this paper.\nAcknowledgments\nThis research was carried out under project number S17012 in the framework of the Partnership\nProgram of the Materials innovation institute M2i (www.m2i.nl) and the Netherlands Organization\nfor Scientific Research (www.nwo.nl) (NWO project number 16348).\nReferences\n[1] M.-C. Miguel, A. Vespignani, S. Zapperi, J. Weiss, J.-R. Grasso, Intermittent dislocation flow in viscoplastic\ndeformation, Nature 410 (2001) 667–671. doi:10.1103/PhysRevE.63.041201 .\n[2] O. Kapetanou, V. Koutsos, E. Theotokoglou, D. Weygand, M. 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Kamenetskii\n1 ,  \n \n1\nMicrowave Magnet ic Laboratory, Department of Electrical and Computer Engineering,   \nBen Gurion University of the Negev, Beer Sheva, Israel  \n2\n Department of Electrical and Electronics Engineering, Shamoon College of Engineering,  \nBeer Sheva, Israel  \n \nDecember  10, 201 5 \n \nAbstract  \nThe near fields originated from a small quasi -two-dimensional ferrite disk with magnetic -dipolar -\nmode (MDM) oscillations are the fields with broken dual (electric -magnetic) symmetry. \nNumerical studies show that such fields – called the magnetoelectric (ME) fields – are \ndistinguished by the  power -flow vortices and helicity parameters  [E. O. Kamenetskii, R. Joffe, \nand R. Shavit, Phys. Rev. E 87, 023201 (2013)]. These numerical studies can well explain recent \nexperimental results w ith MDM ferrite disks. In the present paper , we obtain analytically \ntopological characteristics  of the ME -field modes. For this purpose, we use a method of \nsuccessive approximations. In the  second  approximation we take into account the influence of the \nedge regions of an open ferrite disk, which are excluded in the first -approximation solving of the \nmagnetostatic (MS) spectral problem. Based on t he analytical method , we obtain  a “pure”  \nstructure of the electric and magnetic fields outside the MDM ferrite di sk. The analytical studies \ncan display some fundamental features that are non -observable in the numerical results. While in \nnumerical investigations, one cannot separate the ME fields from the external electromagnetic \n(EM)  radiation, the present theoretica l analysis allows clearly distinguish the eigen topological \nstructure of the ME fields.  Importantly , this ME-field structure gives evidence for certain \nphenomena that can be related to  the Tellegen and bianisotropic coupling effects.  We discuss the \nquestio n whether the MDM ferrite disk can exhibit  properties of the cross magnetoelectric \npolarizabilities.  \n  \nPACS number(s): 41.20.Jb; 42.50.Tx; 76.50.+g  \n  \nI. INTRODUCTION  \n \nThe electric displacement current in Maxwell equations allows prediction of wave propagat ion of \nelectromagnetic fields. In a small ferrite sample (with sizes much less than the free -space \nelectromagnetic wavelength),  at ferromagnetic -resonance frequencies,  one has negligibly sma ll \nvariation of electric energy. In this case, a dynamical process  in a sample is described by three \ndifferential equations , without the electric displacement current , [1 – 4]  \n \n                                                                      \n0B \n ,                                                                (1) \n                                                                     \nBEt \n ,                                                          (2) \n                                                                      \n0H \n .                                                              (3) \n  2 A formal use of a set of differential Eqs. (1) – (3), does not allow consideration of any retardation \neffects. However , without Eq. (2), based  on Eqs. (1) , (3) and the constituti ve relation  \n \n                                                                         \nBH\n ,                                                             (4)  \n \nwhere \n\n  is the permeability tensor, one can obtai n solutions for propagating waves inside a \nferrite sample. Taking into account the temporal -dispersion properties of a ferrite material at a \nferromagnetic resonance, one obtains the Walker equation for magnetostatic -potential (MS -\npotential) wave function \n  (introduced by a relation \nH\n ) [5]. The o scillations in small \nferrite particles , described by the Walker equation,  are called magnetostatic -wave (MS -wave) or \nmagnetic -dipolar -mode (MDM) oscillations [ 1 – 5].  \n    Interaction of small ferrite particles with electromagnetic radiation is not a trivial problem. \nMDM oscillations in ferrite spheres excited by external microwave fields were experimentally \nobserved, for the first time, by White and Solt in 1956 [ 6]. Afterwards, experiments with disk -\nform ferrite specimens revealed unique spectra of oscillations. While in a case of a ferrite sphere \none observed only a few wide absorption peaks of MDM oscillations [ 6], for a ferrite disk there \nwas a multiresonance (at omic -like) spectrum with very sharp resonance peaks [ 7 – 9]. \nAnalytically, it was shown [ 10 – 12] that , contrary to spherical geometry of a ferrite particle \nanalyzed in Ref. [ 5], the Walker equation (together with the homogeneous boundary conditions \nfor fu nction \n  and its derivatives) for quasi -2D geometry of a ferrite disk gives the Hilbert -space \nenergy -state selection rules for MDM spectra. There are so called G-mode spectral solutions  [10 \n– 15].  \n    When we aim to obtain the MD M spectral solutions taking into account also the electric fields \nin a ferrite disk [ to obtain the solutions taking into account Eq. (2)], we have to consider the \nboundary conditions for a magnetic flux density, \nB \n . Analytically, i t was shown that in \nthis case (because of specific boundary conditions for a magnetic flux density on a lateral surface \nof a ferrite disk), one has the helical -mode resonances and the spectral solutions are described by \ndouble -valued functions [12, 13]. Th ere are so called L-mode spectral solutions. For L modes, the \nelectric field in a vacuum region near a ferrite disk has two parts: \ncp E E E\n , where \ncE\n  is the \ncurl-field component and \npE\n  is the potential -field component [15]. While the curl electric field \ncE\n in vacuum we define from the Maxwell equation \n0 cHEt \n , the potential electric \nfield \npE\n  in vacuum is calculated by integr ation over the ferrite -disk region, where the sources \n(magnetic currents \n()m mjt\n ) are given. Here \nm\n  is dynamical magnetization in a ferrite disk. It \nwas shown that in vacuum near a ferrite disk, the regions wit h non -zero scalar product \n pcEE \n can exist [15].  This scalar product is called the field helicity . For time -harmonic \nfields,  the time -averaged helicity factor is expressed as [ 15] \n \n                                                    \n*0Im4pcEE F \n .                                                      (5) \n \n    MDM oscillations in a quasi -2D ferrite disk are macroscopically coherent quantum states, \nwhich experience broken mirror symmetry and also broken time -reversal symmetry  [12, 13]. \nFree-space microwave fields, emerging from magnetization dynamics in quasi -2D ferrite disk ,  3 carry orbital angular momentums and are characterized by power -flow vortices and non -zero \nhelicity. Symmetry properties of these fields – called magnetoe lectric (ME) fields – are different \nfrom symmetry properties of free -space electromagnetic (EM) fields. At the MDM frequency \nMDM\n, we have for magnetic induction  \n\nMDMiBE \n . The ME -field helicity density is \nnonzero only at the resonance frequencies of MDMs and is expressed as  \n \n                         \n* * * 00\n2Im Re Re4 4 4MDM MDM MDMiE B E B E H Fc               \n ,               (6)      \n \nwhere \n00 1c . In this equation, both the electric and magnetic fields are potential fields. \nThe parameter  defined by Eq. (6) is different from the time -averaged optical (electromagnetic) \nchirality density, which is obtained for both, the electric and magnetic, curl fields and is \nexpressed as [ 16 – 19] \n \n                                                       \n* 0Im2EB\n .                                                              (7) \n \nThe ME-field helicity density F was analyzed in numerical studies [15]. A numerical analysis \nshows  also that distribution of the real power -flow densit y \n* 1\n2ReEH\n  of a ME field \nconstitutes the vortex topological structure  in vacuum  [15, 20]. Also, in Ref. [ 21] it was shown \nnumerically that together with the real power -flow density, a ME field is characterized by the \nimaginary \n* 1\n2ImEH\n  power -flow density.  \n    ME-coupling properties, observed in the near -field structure, are originated from \nmagnetization dynamics of MDMs in a quasi -2D ferrite disk. In general, ME -coupling effects \nmanifest in numerous macroscopic phenomen a in solids. Physics underlying these phenomena \nbecomes evident through a symmetry analysis. In isolating crystal materials, in which both spatial \ninversion and time -reversal symmetries are broken, a magnetic field can induce electric \npolarization and, con versely, an electric field can induce magnetization [ 1, 22]. Without \nrequirements of a special kind of a crystal lattice, a ME -coupling term appears in magnetic \nsystems with topological structures of magnetization. In particular, there can be chiral, toroi dal, \nand vortex structures of magnetization [2 3, 24]. Other examples on a role of magnetization \ntopology in the ME -coupling effects concern orbital magnetization. As it was discussed in Refs. \n[25, 26], an adequate description of magnetism in magnetic mater ials should not only include the \nspin contribution, but also should account for effects originating in the orbital magnetism. It was \nshown that in the two -dimensional case, orbital magnetization is exhibited due to exceeding of \nchiral -edge circulations in one direction over chiral -edge circulations in opposite direction [2 6]. \nRecently, it was shown that ME coupling can occur also in isotropic dielectrics due to an effect of \norbital ME polarizability – topological ME -coupling effect [ 27 – 29]. In such a case , one has the \ncontribution of orbital currents to the ME coupling. The orbital ME polarizability is due to the \npseudoscalar part of the ME coupling and is equivalent to the addition of a term to the \nelectromagnetic Lagrangian – the axion electrodynamics te rm [30]. That is why the orbital ME \nresponse in isotropic dielectrics is referred as the axion orbital ME polarizability [ 27, 28].  \n    In this paper, we develop a theoretical analysis of the near fields originated from a MDM \nferrite disk. The electric and  magnetic fields in vacuum are obtained based on the magnetization \ndistributions of MDMs inside a ferrite disk . With use of  this model, w e study  analytically  4 topological characteristics of the ME -field modes. We show  that in a vacuum region very near to \na ferrite sample there is  a good correspondence between the analytical and numerical results  for \nthe field structure. For an incident electromagnetic field, the MDM ferrite disk looks as a trap \nwith focusing to a ring, rather than a point.  While in numerical  investigations, one cannot \nseparate the ME fields from the external EM radiation, the theoretical analysis allows clearly \ndistinguish the eigen topological structure of ME fields. This may concern , in particular,  an \nimportant  question whether the MDM ferr ite disk can exhibit properties of the magnetoelectric \n(cross) polarizabilities. In the paper, we examine the analytically obtained ME -field structure in a \nview of the Tellegen and bianisotropic coupling effects. Based on the analytical study , we \ninvestiga te the active and reactive components of the complex power -flow density of the ME \nfields. The topological structure of ME fields demonstrated by such power flow distributions can \nexplain stability of the eigenstates of MDM oscillations.   \n \n II. THE MODEL AND ANA LYTICAL RESULTS FOR THE ME -FIELD  STRUCTURES  \n \nWe use a method of successive approximations. In the first -approximation solving of the MDM \nspectral problem, we obtain analytically the RF magnetization inside a ferrite disk. The known \nmagnetization and t he magnetic current derived from this magnetization are considered as \nsources for the magnetic and electric fields outside a ferrite disk. Based on the second \napproximation, we find the magnetic and electric components of the ME field and analyze \ntopology of the ME fields in vacuum. In the theoretical analysis, we use the same disk parameters \nas in Refs. [ 14, 15 ]: the yttrium iron garnet (YIG) disk has a diameter of \n23 mm and the disk \nthickness is \n0.05d mm; the disk is n ormally magnetized by a bias magnetic field \n 49000H\nOe; the saturation magnetization of the ferrite is \n 1880  4sM G. We assume that \nmagnetic losses in a ferrite disk are negligibly small.  \n \nA. The first -approximation solution s for MS-potential wave functions  \n \nIn the first -approximation solution , we ignore the inf luence of the edge regions outside a quasi -\n2D ferrite disk  [10] (see Fig. 1) .  \n \nFig. 1. An open quasi -2D ferrite disk.  \n \nBased on this model, we can use separation of varia bles. Analytically, there are two spectral \nmodels for the MDM oscillations. These models give so -called G- and L-modes.  The MS -\npotential wave function for L-mode is written as [ 10, 12 – 15] \n  5                                                             \n, , , , ( ) ( , )n n n nC z r      \n ,                                                (8)  \n \nwhere a dimensionless effective membrane function \n,( , )nr\n  is defined by the Bessel -function \norders \n1, 2, 3,...  and the numbers of zeros of the Bessel func tions corresponding to different \nradial variations \n1, 2, 3,...n  In Eq. (8), \n,()nz  is a dimensionless function of the MS -potential \ndistribution along z axis. \n,,nnC QC  is an amplitude coefficient  where \nQ  is a dimensional unit \ncoefficient of 1A and \n,nC  is a normalized dimensionl ess amplitude . Inside a ferrite disk \n(\n,   2 2 r d z d    ) the MS -potential wave function is represented as  \n \n                             \n, 1, , , cos  sin  i i t\nnrr z t C J z z e e \n   \n  \n       ,                       ( 9)                                              \n \nwhere \n  is a propagation constant for MS waves along z axis  and \n  is a diagonal component of \nthe permeability tens or. Solutions in a form of Eq. ( 9) show the azimuthally -propagating -wave \nbehavior for MS -potential membrane functions.   \n    To define the  normalized dimension less amplitudes \n,nC  we will use t he Hilbert -space energy -\neigen state G-mode spe ctral solutions .  Assuming that the scalar -wave membrane function for the \nG-mode spectral solution is represented as  [12 – 15] \n \n                                                                      \n,,\n,nn\nnC\n\n ,                                                      (10) \nwe can write  \n \n                                                                 \n2\n2*\n,,  nn\nSC dS\n .                                                  (11)  \n \nHere integration is made over an entire open -disk region in the \n,r  plane.  Normalization of \nmembrane function \n\n  is expressed as  \n \n                                                                    \n2\n,\n,1n\nnC\n .                                                           (12) \n \nFig. 2 shows the calculated coefficients \n,nC  for the 1st-order Bessel -function (\n1 ) and the \nnumbers of radial variations (\n1, 2, 3,...n ). \n  6 \n \n \nFig. 2. Normalized amplitude coefficients for MS -potent ial wave functions. The MDMs are with the \n1st-order Bessel -function (\n1 ) and the numbers of radial variations (\n1, 2, 3,...n ). \n \n    Based on solution (9) for every MDM, we can find the first -approximation solutions for the \nmagnetic field both inside and outside a ferrite disk  \n \n                                                                      \nMDM MDMH \n .                                                (1 3)  \n \nAlso, we can find the spectral magnetization distr ibution inside a ferrite disk:  \n \n                                                                        \nMDM MDMm \n ,                                          (1 4) \n \nwhere \n\n  is the magnetic susceptibility tensor [4].  For a normally magnetized ferrite disk, we \nhave only the radial and azimuthal components of magnetization. These components are \nexpressed as  \n \n          \n,1cos sin ( , , , )i a\nrit\nnmm z z J J errr z t C e\n\n \n \n   \n    \n              ,      (1 5) \n  7           \n,1cos sin ( , , , )i a it\nnmm i z z J J errr z t C e\n  \n \n \n   \n    \n              ,  (16) \n \nwhere  \n, \na  are diagonal and off -diagonal  components of the magnetic susceptibility tensor, \nrespectively  [4], and \nJ  is a derivative (with respect to the argument) of the Bessel function of \norder \n . \n    Figs. 3 and 4 show MS -potential wave functions for the 1st and 2nd MDMs, respectively.  \n \n \n \n \n \nFig. 3. MS -potential wave functions (the 1st approximation) for the 1st MDM. The MDM is described \nwith the 1st-order Bessel -function (\n1 ) and the radial variation \n1n . The variation along z axis, \n()z\n, corresponds to the main thickness mode. ( a) The function \n()z ; (b) the radial distribution of \nthe MS -poten tial wave function; ( c) the picture of intensity of membrane MS -potential wave \nfunction \n( , )r\n , which rotates in the plane perpendicular to the disk axis z. \n  8 \n \nFig. 4. MS -potential wave functions (the 1st approximation) for the 2nd MDM. Th e MDM is \ndescribed with the 1st-order Bessel -function (\n1 ) and the radial variation \n2n . The variation \nalong z axis, \n()z , corresponds to the main thickness mode. ( a) The function \n()z ; (b) the radial \ndistribution of the MS -potential wave function; ( c) the picture of intensity of membrane MS -\npotential wave function \n( , )r\n , which rotates in the plane perpendicular to the disk axis z. \n \nThese modes are described  by the 1st-order Bessel -functions (\n1 ). The 1st and 2nd MDMs have  \nthe numbers of radial variations n = 1 and n = 2, respectively .  The variation along z axis, \ndescribed by the function\n()z , corresponds to  the main t hickness mode.  Based on the known \nMS-potential wave functions, radial and azimuthal components of magnetization  were calculated. \nThe in -plane distributions of the magnetization  (at the plane \n2 zd ) are shown in Fig.  5 for the \n1st and 2nd MDMs.   9 \n \nFig. 5. The magnetization distribution in a quasi -2D ferrite disk. ( a), (b) The 1st and 2nd MDMs, \nrespectively, for the phase \n0t\n ; (c), (d) the 1st and 2nd MDMs, respectively, for the phase \n90t\n. Every elementary magnetic dipole rotates in the xy plane. Because of the azimuthally -\npropagating -wave behavior for MS -potential membrane functions, the entire pictures of the \nmagnetization rotate as well. Direction of the magnetization rotation, clockwise or counterclockwise, \ndepends on a direction of a bias magnetic field. The arrows are unit vectors.  \n \nEvery elementary magnetic dipole rotates in the xy plane. Because of the azimuthally -\npropagating -wave behavior for MS -potential membrane functions, the enti re pictures of the \nmagnetization rotate as well.  Direction of the magnetization rotation, clockwise or \ncounterclockwise, depends on the direction of the bias magnetic field.      \n \nB. The second -approximation solution s \n \nThe known, from Eqs. (15), (16) , MDM magnetization distribution s inside a ferrite disk  allow  \nmaking an analysis by taking into consideration  the edge regions, shown in Fig 1. As a result,  one \ncan obtain  an entire structure of the MS -potential wave function and the fields outside a ferrite \ndisk. There are the second -approximation solutions.  \n    Using  Eqs. (1), (1 3) and taking into ac count that \n0() B H m\n , we obtain the following \nPoisson’s equation  \n  \n                                                                     \n2m \n ,                                                            (17) \n  10 where the right -hand -side term is considered as an effective magnetic charge. If a ferrite disk has \na volume V and surface S, we specify \nm\n  inside V as magnetization of a certain MDM and assume \nthat it falls suddenly to zero at the surface S. For a given MDM, the solution of Eq. ( 17) for the \nMS-potential wave function outside a ferrite disk  is [31]   \n \n                                         \n1 ( ') ( ')( )  4 ' 'm x n m xx dV dSx x x x    \n ,                                    (18) \n \nwhere \nn\n  is the outwardly directed normal to surface S. The quantity \nnm\n  can be considered as \nan effective surface magnetic charge density. Because of non-uniform magnetization distributions \nthroughout the volume V, both integrals in the right -hand -side of Eq. ( 18) contribute to the MS -\npotential solution. Since there are only the radial and azimuthal components of the MDM \nmagnetization, surface S in Eq. (18) is a lateral surface of a ferrite disk. Based on Eq . (18), we \nfind the second -approximation solutions for the potential magnetic field outside the ferrite disk:  \n \n                             \n\n33( ') ' ( ') ' 1( )  4 ''m x x x n m x x xH x dV dS\nx x x x       \n .                      (19)  \n \nWe can also obtain the second -approximation solutions for the magnetic flux density outside the \nferrite disk:  \n \n                                                                 \n0 ( ) ( )B x H x\n .                                                            (20) \n \nFig. 6 shows the  normalized magnetic field distributions  \nˆ / | | H H H\n  outside the ferrite disk \n(on the xz cross -sectional plane) for the 1st and 2nd MDMs.   11 \n \nFig. 6. Normalized magnetic field distributions  \n ˆ/ | | H H H\n  outside a ferrite disk (on the xz cross -\nsectional plane) shown for a certain phase, \n0t\n . (a), (b) The 1st and 2nd MDMs, respectively, for \nthe 1st approximation; ( c), (d) the 1st and 2nd MDMs, respectively, for the 2nd approximation. Th e \narrows are unit vectors.  \n \nOne can compare the magnetic fields calculated based on the 1st approximation [Figs. 6 ( a) and \n(b)] with the fields calculated based on the 2nd approximation [Figs. 6 ( c) and ( d)].      \n    Inside a ferrite disk the Faraday equa tion (2) is written as  \n \n                                                                   \n()\n0mHj Ei  \n ,                                           (21) \n \nwhere we introduced a magnetic current \n()mj\n , expressed as  \n \n                                                                          \n()\n0mj i m\n .                                                    (22) \n \nOutside a ferrite disk, in vacuum, we consider an electric field as composed by two components: \ncp E E E\n. The curl electric field \ncE\n is defined as  \n \n                                                                     \n0 c H Ei \n .                                                 (23)                        \n  12 Here \nH\n  is the potential magnetic field in vacuum, found based on Eq. (1 9). The potential electric \nfield \npE\n  in vacuum is originated from a source region: a MDM ferrite disk with a magnetic \ncurrent \n()mj\n . Based on an evident duality with  the classical -electrodynamics  problem of the \nmagnetostatic magnetic field originated from the electric -current source region  [31], in Ref. [15] \nit was shown that the potential electric field \npE\n  outside the ferrite disk is defined  as \n \n                                                \n()\n31( )  4m\npj x x xE x dV\nxx \n\n .                                          (2 4) \n \nFor every  MDM, the potential electric field \npE\n  outside  the ferrite disk is obtained based on Eq. \n(24). A magnetic current \n()mj\n is calculated based on  Eqs. (15), (16), (22).   \n    Fig. 7 shows the  normalized  electric field distributions  \nˆ / | | E E E\n  outside the ferrite disk \n(on the xz cross -sectional plane) for the 1st and 2nd MDMs.  \n \nFig. 7. Distributions of the  normalized  potential electric field  \nˆ/ | | E E E\n  outside a ferrite disk (on \nthe xz cross -sectional plane) shown for a certain phase, \n0t\n . (a) The 1st MDM; ( b) the 2nd MDM. \nThe arrows are unit vectors.  \n \nIn Fig. 8 one can see the calculated distributions of the  magnetic and electric fields for the 1st and \n2nd MDMs on a vacuum xy plane 20 um above the ferrite disk.   13 \n \nFig. 8. The f ield distributions on a vacuum xy plane 20 um above a ferrite disk shown for a certain \nphase, \n0t\n . (a), (b) The magnetic and electric fields, respectively, for the 1st MDM; (c), (d) the \nmagnetic and electric fields, respectively, fo r the 2nd MDM. The arrows are unit vectors.  \n \nIn these figures we present only the distributions of the potential fields \npE\n  for the 1st and 2nd \nMDMs. A comparative analysis of the analytically derived distribution of the potential f ields \npE\n  \nand numerically obtained distribution of the total field \npc E E E\n , shown below in Figs. 15 and \n16, gives evidence for a negligibly small role of the curl electric field \ncE\n  in the  field topology. \nTheoretically, the potential electric field \npE\n  in a vacuum region near a ferrite disk is found by \nintegration over a ferrite -disk volume for a known magnetic current \n()mj\n  [Eq. (24)]. To obtai n the \ncurl electric field \ncE\n , one has to solve the differential equation (23) taking into account that the \nmagnetic field in a vacuum region is the evanescent -wave field. For geometry of an open ferrite \ndisk [with taking into consi deration the edge regions (see Fig. 1)], analytical solutions of Eq. (23) \nis associated with substantial difficulties. Because of a negligibly small role of the curl electric \nfield \ncE\n  in the near -field topology, an analysis of such  a component of the electric field is beyond \nthe frames of this work.  \n \n \n \n  14 C. Topological characteristics of ME fields  \n     \nBased on the known analytical solutions for the MS -potential wave function and the fields \noutside the ferrite disk, we can obtain topolo gical characteristics of the ME fields. The \ndistributions  of the helicity density and the power -flow density comprise  these characteristics . \nThe ME -field helicity density, expressed by Eq. (6), was calculated based on Eqs. (19) and (24) \nfor potential magne tic and electric fields.  Fig. 9 shows the calculated ME-field helicity density \ndistributions for the 1st and 2nd MDM s. One can see that a dimension of the ME -field helicity \ndensity F is Joule/meter4. For a potential field, this is a gradient of energy dens ity which defines \nfield strength. So, in our case, factor F is a strength of a ME field. For potential electric and \nmagnetic fields, magnitudes and directions of the field strengths are determined by forces acting \non a test electric charge or an electric c urrent element. The question what kind of a test element \nshould be introduced to determine the strength of the ME field is open. This question is beyond \nan analysis in the present paper.   \n \n \nFig. 9. T he ME -field helicity density analytically calculated bas ed on Eq. (6). ( a) and ( b) The \ndistributions on a vacuum xy plane 20 um above a ferrite disk for the 1st and 2nd MDMs, \nrespectively; (c) and (d) the distributions on a cross -sectional xz plane for the 1st and 2nd MDMs, \nrespectively.  \n \nIn an analysis of topological properties of ME fields , an important question arises what is a \ndistribution of  an angle between the potential electric and magnetic fields in the near -field space. \nWhile for regular EM -field problems, the vectors of electric and magnetic fields  in vacuum are \nmutually perpendicular in space, in our case , a space angle between the electric and magnetic  15 fields in vacuum is, definitely, not \n90\n . An angle between the potential electric and magnetic \nfields we will characterize by the parameter called t he normalized ME -field helicity:  \n \n                                                              \n*Re\ncosp\np\npEH\nEH\n\n .                                                  (25)  \n \nHere subscript p for an angle \n  means “potential”. The distribution s of \ncosp  on a cross -\nsectional xz plane are shown in Fig. 10 for the 1st and 2nd MDMs.   \n \nFig. 10. T he normalized ME -field helicity (\ncosp ) distributions on a cross -section al xz plane. ( a) \nFor the 1st MDM; ( b) for the 2nd MDM. The insertions illustrate mutual orientations of the electric \nand magnetic fields in different points on the z axis above a ferrite disk.  \n \nIn fact, this parameter represents  mutual orientations of the electric - and magnetic -field vectors \nshown in Figs. 6 (c, d), 7, and 8 .\n  \n    In an analysis of the complex power -flow density of the ME fields, we will use the following \nconsideration.  A simple analysis of the energy balance equa tion for monochromatic MS waves in \na magnetic medium with small losses [12] shows that the  real power flow density for a MDM is \nexpressed as  \n \n                                                       \n* * *Re2s s s s s s sip i B B B    \n .                               (26) \n \nHere the mode number s includes both the Bessel -function orders \n  and the numbers of zeros of \nthe Bessel functions corresponding to different radial variations n. With use of the first -\napproximation solution for the MS -potential  wave function, expressed by Eq. (9), we can \ncalculate the quantity \nsp\n . It is easy  to show [ 32] that inside a ferrite disk \n0ssrzpp . The \nonly non -zero component of the real power flow density inside a ferrite dis k is the azimuth \ncomponent, which is expressed as [ 21, 32 ]: \n  16                \n\n**\n2 2 * *\n2\n2 2 21( , ) ( )2\n( , ) 1                  ( ) ( , ) ,2s s s s\ns s s s a s s\ns\ns s s aip r z C z ir r r\nrC z rrr          \n                          \n    \n          (27)       \n \nwhere \n  and \na  are, respectively, the diagonal and off -diagonal components of the ten sor \n\n . \nThe quantities \n ( , )sp r z , circulating around a circle\n2r (where \nr ), are the MDM power -\nflow-density vortices with cores at the disk center. At a vortex center the amplit ude of \nsp  is \nequal to zero. In a vacuum region, outside a ferrite disk we have from Eq. ( 27) \n \n                                             \n 2 2 21( , )  ( ) ( , )s s s sp r z C z rr    \n .                                    (28)                                        \n \nIt is clear that \n0sp \n , both inside and outside a ferrite disk.   \n    In Ref. [21], an analysis of the energy relations for MDMs was extended by an introduction \nalso the imaginary power flow density for a MDM. For a mode  s, this imaginary power flow \ndensity is expressed as  \n \n                                                   \n * * *Im2s s s s s s sq i B B B    \n .                                     (2 9) \n \nSimilarly to the above calculation of the imaginary power flow density \nsp\n , we can calculate the \nimaginary power flow density \nsq\n  based on  the first -approximation solution (9). In this case, we \ncan show that  inside a ferrite that \n0sq , while \n0srq  and \n0szq . For \nsrq , we obtain:  \n \n        \n\n**\n2 2 * *\n2\n2 2 21( ) ( )2\n() 1            ( ) ( ) .2s s s s\ns s s s a s s r\ns\ns a sq z C z ir r r\nrC z rrr          \n                        \n  \n                      (30)                     \n \nOutside the ferrite disk we have  \n \n                                             \n2\n2 2 ()( , ) ( )2s\nss rrq z C zr \n .                                           ( 31) \n \nFor \nszq , we obtain  \n                                                  \n2 2 ()( ) ( , ) s\ns s s szzq C z rz   \n .                                        (32) \n \n    The above results for the complex po wer-flow density of the ME fields , obtained based on the \nfirst-approximation solutions , can be improved with use of the second -approximation solutions  17 for MS -potential wave functions, in which an entire structure of the fields outside a ferrite disk is \ntaken into account. For the second -approximation solutions , quantities of the real power flow \ndensity \nsp\n  and the imaginary power flow density \nsq\n  outside a ferrite disk are calculated based on \nEqs. (2 6) and (2 9) using  Eqs. (18) and (20) for the MS -potential wave function and the magnetic \nflux density, respectively . Figs. 11  and 1 2 show the ME -field active and reactive power -flow \ndistributions outside a ferrite disk for the 1st and 2nd MDMs  calculated based on Eq s. (26) and \n(29).  \n \nFig. 11. T he ME -field active -power -flow distribution outside a ferrite disk calculated based on Eq. \n(26). ( a) and ( b) The distributions on a vacuum xy plane 20 um above a ferrite disk for the 1st and 2nd \nMDMs, respectively; (c) and (d) the distributions on a cross -sectional xz plane for the 1st and 2nd \nMDMs, respectively. The arrows are unit vectors.   18 \n \nFig. 12. T he ME -field re active -power -flow distribution outside a ferrite disk calculated based on Eq. \n(29). ( a) and ( b) The distributions  on a vacuum xy plane 20 um above a ferrite disk for the 1st and 2nd \nMDMs, respectively; (c) and (d) the distributions on a cross -sectional xz plane for the 1st and 2nd \nMDMs, respectively. The arrows are unit vectors.  \n \n    The real and imaginary power flow  densities, defined by Eqs. (2 6) and (2 9), are related to the \ntime averaged real and imaginary part s of a vector product of the curl electric and potential \nmagnetic fields . A simple manipulation ( taking into account that \n0B \n  and \nH\n ) shows \nthat \n \n                                    \n* * * *()ccE H H E i B i B             \n .                            (33) \n \nFollowing Ref. [ 33], one can conclude that this equation gives  \n \n                                                              \n**\ncE H i B\n .                                                          (34)                                                           \n \nFor mode s, we have  \n \n                                             \n**Re Res s s c ssp i B E H    \n                                        (35) \n \nand \n  19                                              \n**Im Ims s s c ssq i B E H    \n .                                       (3 6) \n \nImportantly, despite the fact that such expression s as \n*ReEH\n  and \n*ImEH\n looks like the \nreal and imaginary Poynting vector s, the MS -wave power flow densit ies cannot be basically \nrelated to  the EM -wave power flow densities . The Poynting vector is obtained for EM radiation \nwhich is described by the two curl operator  Maxwell equations for the electric and magnetic \nfields  [31]. This is not the case  described by Eq. (33) , where , for the MS waves,  we have \npotential magnetic  and curl electric  fields.  \n    While Eqs. (3 5) and (3 6),  are relevant only for a curl electric field \ncE\n  in a vacuum region near \na ferrite disk , the question about a role of the potential electric field \npE\n  in th e vector -product \nquadratic relations arises  as well . Formally, w e can extend our analysis of the active power flow \nalso to calculations of the quantity \n*RepEH\n . This quantity, for the 1st and 2nd MDMs, is shown \nin Fig. 1 3. Similar to th e results shown in Fig. 11, in the picture in Fig. 13 we can see that there \nare the power -flow-density vortices with cores at the disk center. At a vortex center the amplitude \nis equal to zero.  \n \nFig. 13. The quantity \n*RepEH\n  outside a ferrite disk. (a) and ( b) The distributions on a vacuum \nxy plane 20 um above a ferrite disk for the 1st and 2nd MDMs, respectively; (c) and (d) the \ndistributions on a cross -sectional xz plane for the 1st and 2nd MDMs, respectively. The arrows are \nunit vect ors. \n  20     In vacuum , near the central  region of the MDM  ferrite disk, one has in -plane rotating vectors \nof the potential electric and magnetic fields [15]. It was shown [21] that for these spinning \nelectric - and magnetic -field vectors, a time averaged imag inary part of a vector product of th e \nelectric and magnetic fields is related to the reactive power flow density.  Based on Eq. (19)  and \n(24), we can find analytically the distribution of the quantity \n*ImpEH\n  outside the ferrite disk. \nFig. 14 shows the calculated quantity of the reactive power flow \n*ImpEH\n  outside the ferrite \ndisk for the 1st and 2nd MDMs.  Contrary to \n*RepEH\n , the quantity \n*ImpEH\n  has a maximal \namplitude at the d isk center.  \n \nFig. 14. The quantity of the reactive power flow \n*ImpEH\n  outside a ferrite disk. (a) and ( b) The \ndistributions on a vacuum xy plane 20 um above a ferrite disk for the 1st and 2nd MDMs, \nrespectively; (c) and (d) the distri butions on a cross -sectional xz plane for the 1st and 2nd MDMs, \nrespectively. The arrows are unit vectors.  \n \n    In Section IV of the paper we discuss more in details the obtained analytical results on the ME -\nfield topology.  \n \nIII. VERIFICATION OF THE ANALY TICAL RESULTS BY NUMERICAL STUDIES  \n \nOur analytical results of the ME -field structure are well verified by numerical studies based on \nthe HFSS simulation program.  Comparison between the analytical and numerical studies of the \nfield components, shown in Fig s. 15 and 1 6 for the 1st and 2nd MDMs, gives good \ncorrespondence between these results. In these figures, the analytically derived electric fields are \npotential fields calculated based on Eq. (24).   21 \n \nFig. 15. Comparison between the analytical and numerica l results of the field components for the 1st \nMDM. The analytically derived electric fields are potential fields calculated based on Eq. (24). The \nfield components are shown as the quantities normalized to the modulus of the field vectors. ( a) The \nelectric -field components; ( b) the magnetic -field components. A comparative analysis is made for \nthe field components on a vacuum xy plane 50 um above a ferrite disk.  \n  22 \n \nFig. 16. Comparison between the analytical and numerical results of the field components for t he 2nd \nMDM. The analytically derived electric fields are potential fields calculated based on Eq. (24). The \nfield components are shown as the quantities normalized to the modulus of the field vectors. ( a) The \nelectric -field components; ( b) the magnetic -field components. A comparative analysis is made for \nthe field components on a vacuum xy plane 50 um above a ferrite disk.  \n \nThe field components are shown as the quantities normalized to the modulus of the field vectors. \nAs an illustrative example of the ME -field topology obtained based on both the theoretical and \nnumerical analyses, we show in Fig. 1 7 the normalized  ME-field helicity density distributions  \nnumerically calculated  for the 1st and 2nd MDMs.    23 \n \nFig. 17. The ME -field helicity density distributions calculated numerically. ( a) and ( b) The \ndistributions on a vacuum xy plane 20 um above a ferrite disk for the 1st and 2nd MDMs, \nrespectively; ( c) and ( d) the distributions on a cross -sectional xz plane for the 1st and 2nd MDMs, \nrespectively.  \n \nOne can compa re these distributions with the analytically calculated ME -field helicity densi ty \ndistributions shown in Fig. 9 . Evidently, for such a topological parameter, there is good \ncorrespondence between the two types of analyses.  We restrict now further verificati on of the \nobtained analytical results by proper references of our previous numerical studies. The active -\npower -flow vortices shown in Figs. 1 1 and 1 3 are well verified by such vortices obtained \nnumerically in Refs. [14, 15, 20, 21, 32]. The reactive power flows  \n*ImpEH\n  shown in Fig. 1 4 \nare in good correspondence with the numerical results of the reactive  power flows in Ref. [21]. \nThe analyt ical studies, however, display s ome more important features that are non -observable in  \nthe numerica l results. We discuss this in the next Section of the paper.  \n \nIV. DISCUSSION  \n \nThe question of the active power flows of ME fields is not comprehensively clear. The real power \nflow density for MDMs expressed  by Eq. (26), is obtained based on energy balance equation for \nmonochromatic MS waves in a magnetic medium with small losses [12]. Following a formal \nanalysis made in Ref. [33], one can conclude that such real power flow density for MDMs is \nrelated to a real part of the vector product of the curl electric  field and potential magnetic field. At \nthe same time , as we analyzed above, such an expression is not related to the electromagnetic  24 Poynting vector. Our analytical results reveal also some more interesting features. When one \ncompares the active power flo w distributions in Figs. 1 1 and 13, one finds evident similarity \nbetween these pictures. It means that there is similarity between the quantities  \n*RecEH\n  and \n*RepEH\n. So, the roles of the curl electric field and pot ential electric field in the active power \nflow distributions are almost indistinguishable. Certainly, in the numerical studies we are dealing \nwith the total electric fields, without any separation to the curl and potential parts.  \n    Another interesting q uestion concerns the reactive power flows of ME fields.  As we have \nshow n above, there are two types of reactive power flows. The first one, defined by Eq. (29), is \nshown in Fig. 12, while the second one, defined as \n*ImpEH\n , is shown in Fig. 1 4. One can see \nthat these  two types of the reactive power flows are localized at different parts of the ferrite disk  \nand have different directions  with respect to the disk . While in the picture shown in Fig. 12 we \nhave the drain -type vector orientati ons, for the picture shown in Fig. 14 there are the source -type \nvector orientations. At the same time, i t is worth noting that the magnitude of the reactive power \nflow in Fig. 1 2 is sufficiently smaller than such a magnitude in Fig. 14 . This fact can expla in why \nin the numerical analysi s in Ref. [21] we observe only the reactive power flow defined as \n*ImpEH\n.   \n    A very important issue becomes evident when one analyses the analytical results of the \nnormalized ME -field helicity , \ncosp , shown in Fig. 10 . In fact, there are the distributions of a n \nangle between the potential electric and magnetic fields . From Fig. 10 (a) one can see that for the \n1st MDM , the potential electric and magnetic fields in an entire vacuum r egion are mutually \nparallel  (above a ferrite disk)  or mutually anti -parallel (below a ferrite disk).  For the 2nd MDM \n[see Fig. 10 (b)], above a ferrite disk there are the regions both with  mutually parallel  and anti -\nparallel  electric and magnetic fields . Between these two regions, there is an intermediate area  of \nmutual orientations of the field vectors . Similar distributions one has below a ferrite disk. Now \nthe question arises: Whether such distributions of the potential electric and magnetic fields can b e \nconsidered as the fields originated from a Tellegen particle?  Tellegen considered an assembly of \nelectric -magnetic dipole twins, all of them lined up in the same fashion (either parallel or anti -\nparallel) [ 34]. Since 1948, when Tellegen suggested such \"g lued pairs\" as structural elements for \ncomposite materials, electromagnetic properties of these complex media was a subject of serious \ntheoretical studies (see, e.g. Refs. [3 5 – 37]). Till now, however, the problem of creation of the \nTellegen medium is a s ubject of strong discussions. The question , whether  the Tellegen particles \nreally exist in electromagnetics, is still open. The electric polarization is parity -odd and time -\nreversal -even. At the same time, the magnetization is parity -even and time -reversal -odd [31]. \nThese symmetry relationships make questionable an idea that a simple combination of two \n(electric and magnetic) small dipoles can give  the local cross -polarization (magnetoelectric) \neffect . In our case of the ME fields we do not have properties of the cross (magnetoelectric ) \npolarizabilities. The ME-coupling properties are originated from magnetization dynamics of \nMDM oscillations  in a quasi -2D ferrite disk.  These  oscillations are macroscopically coherent \nquantum states, which experience broken m irror symmetry and also broken time -reversal \nsymmetry [12, 13].  The potential electric and magnetic field of the ME -field structure are the \nfields originated from the topological properties of magnetization  [15].   \n    It is also worth noting here that t he normalized ME -field helicity calculated based on Eq. (25)  \nand shown in Fig. 10 , is different from such a parameter shown  in the numerical results [38 – 40]. \nIn the numerical investigations, one cannot separate the ME fields from the external EM \nradiation. Also, one cannot separate the potential and curl fields. It means that in the numerical  25 studies, t he normalized ME -field helicity is calculated as \n*Re\ncospEH\nEH\n\n , where the fields \nE\n and \nH\n  contain both the potential and curl components  of the external -radiation EM and \neigen -oscillation ME fields . The analytical and numerical results on the normalized ME -field \nhelicity are essentially  different in vacuum regions, where a magnitude  of the potenti al electric \nfield is reduced (the vacuum regions sufficiently far from a ferrite disk and the vacuum regions \nwhich are peripheral with respect to the disk center) .        \n \nV. CONCLUSION  \n \nSince MDM oscillations are energetically orthogonal ( G modes), one ha s the same positions of \nthe spectral peaks (with respect to a signal frequency at a constant bias magnetic field or with \nrespect to a bias magnetic field at a constant signal frequency ) in different microwave structures \nwith an embedded quasi -2D ferrite di sk. At the same time, in such different structures there are \ndifferent numerically observed topological characteristics of the microwave fields. While in \nnumerical investigations, one cannot separate the ME fields from the external EM radiation, the \ntheore tical analysis allows clearly distinguish the eigen topological structure of ME fields.  In the \npresent paper, we obtained analytically topological characteristics of the ME -field modes. For \nthis purpose, we used a method of successive approximations. Base d on the analytical method, \nwe have shown  a “pure” structure of the electric and magnetic fields outside a MDM ferrite disk. \nWe analyzed theoretically the fundamental topological characteristics, which are  not observed in \nthe numerical results. The analyti cal studies of topological properties of the ME fields can be \nuseful for novel near - and far -field microwave applications. Strongly localized ME fields o pen \nunique perspective for sensitive microwave probing of structural characteristics of chemical and \nbiological objects. The presence of a biological sample with chiral properties will necessarily \nalter the near -field topology  which in turn will change the spectral characteristics of the MDM \nferrite disk.  \n \nReferences  \n \n[1] L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media , 2nd  ed (Pergamon, \nOxford, 1984).  \n[2] D. C. Mattis, The Theory of Magnetism  (Harper & Row Publishers, New York, 1965).  \n[3] A. I. Akhiezer, V. G. 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Phys. 12, 053032 (20 10).  \n[29] S. Coh, D. Vanderbilt, A. Malashevich, and I. Souza, Phys. Rev. B 83, 085108 (2011).  \n[30] F. Wilczek, Phys. Rev. Lett. 58, 1799 (1987).  \n[31] J. D. Jackson, Classical Electrodynamics , 2nd ed (Wiley, New York, 1975).  \n[32] M. Sigalov, E. O. Kamenet skii, and R. Shavit, J. Phys.: Condens. Matter 21, 016003 (2009).  \n[33] D. D. Stancil, Theory of Magnetostatic Waves  (Springer -Verlag, New York, 1992).  \n[34] B. D. H. Tellegen, Philips. Res. Rep. 3, 81 (1948).  \n[35] E. J. Post, Formal Structure of Electromag netics  (Amsterdam, North -Holland, 1962).  \n[36] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov and A J Viitanen, Electromagnetic Waves in \nChiral and Bi -Isotropic Media (Boston, MA, Artech House, 1994).  \n[37] A. Lakhtakia, Beltrami Fields in Chiral Media  (Singa pore, World Scientific, 1994).  \n[38] M. Berezin, E. O. Kamenetskii, and R. Shavit, J. Opt. 14, 125602 (2012).  \n[39] R. Joffe, E. O. Kamenetskii and R. Shavit, J. Appl. Phys. 113, 063912 (2013).  \n[40] M. Berezin, E. O. Kamenetskii, and R. Shavit, Phys. Rev. E 89, 023207 (2014).  \n  \n \n "    },    {        "title": "2301.11373v1.Magnetic__Optoelectronic__and_Rietveld_refined_structural_properties_of_Al3__substituted_nanocrystalline_Ni_Cu_spinel_ferrites__An_experimental_and_DFT_based_study.pdf",        "content": "1 \n Magnetic, Optoelectronic, and Rietveld Refined Structural Properties of Al3+ Substituted \nNanocrystalline Ni-Cu Spinel Ferrites: An Experimental and DFT Based Study. \nN. Hasana*, S. S. Nishatb, S. Sadmanc, M. R. Shaownd, M. A. Hoquee, M. Arifuzzamanf g, A. Kabirc \naDepartment of Electrical and Computer Engineering, North South University, Dhaka-1229, Bangladesh \nbMaterials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180, United States \ncDepartment of Physics, University of Dhaka, Dhaka 1000, Bangladesh \ndDepartment of Industrial and Production Engineering, Bangladesh University of Textiles, Dhaka-1208, \nBangladesh \neBangladesh Council of Scientific and Industrial Research, Dhaka-1205, Bangladesh \nfDepartment of Electronic Materials Engineering, Research School of Physics, The Australian National \nUniversity, Canberra ACT 2600, Australia \ngDepartment of Mathematics and Physics, North South University, Dhaka-1229, Bangladesh \ncDepartment of Physics, University of Dhaka, Dhaka 1000, Bangladesh \nKeywords : Ni-Cu ferrites, Sol-gel route,, surface morphology, VSM, TEM, Rietveld \nrefinement, density functional theory,  optoelectronic properties. \n*Corresponding author: alamgir.kabir@du.ac.bd , md.arifuzzaman01@northsouth.edu   \nanazmul.hasan05@northsouth.edu \nbshahriyar007@gmail.com  \ncsarker.md.sadman@gmail.com  \ndmrshaownbut44@gmail.com  \ne azizphy0043bcsir@gmail.com   \nfmd.arifuzzaman01@northsouth.edu  \ncalamgir.kabir@du.ac.bd  \n 2 \n Magnetic, Optoelectronic, and Rietveld Refined Structural Properties of Al3+ Substituted \nNanocrystalline Ni-Cu Spinel Ferrites: An Experimental and Dft Based Study. \nAbstract \nThe nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4 (x=0.00: 0.02: 0.10) are prepared through the sol-gel auto \ncombustion route. The structural, surface morphology, magnetic and optoelectronic properties of Al3+ \nsubstituted Ni-Cu spinel ferrites have been reported. The crystallinity, phase structure, and structural \nparameters of the synthesized nanoparticles (NPs) have been determined through x-ray diffraction \n(XRD) and further refined by maneuvering the Rietveld refinement approach. Both XRD and Rietveld \nconfirm the single phase cubic spinel structure of the investigated materials. Microstructural surface \nmorphology study also confirms the formation of NPs in the highly crystalline state with a narrow size \ndistribution. The Rietveld-refined average crystallite size of the Al3+ doped Ni-Cu ferrite nanoparticles \nfalls in the range (61 – 71 nm), and the average grain size is found to vary from 59 to 65 nm. All other \nstructural parameters refined by the Rietveld refinement analysis are corroborated to single-phase cubic \nspinel formation of the NPs. Leveraging a vibrating sample magnetometer (VSM), the consequence of \nAl3+ substitution on the magnetic parameters is studied. The saturation magnetization (M S) and Bohr \nmagneton are found to decrease with Al3+ substitution. The Remanence ratio and coercivity (H C)  are \nobserved to be very low,  suggesting the materials are soft ferromagnetic. First-principle calculations \nwere carried out using the density functional theory (DFT) to demonstrate the optoelectronic behavior \nof the materials. The electronic bandgap is found low as E g=2.99eV for the explored materials with \nobserving  defect states at 0.62eV. The optoelectronic properties of Al3+ substituted Ni-Cu ferrite NPs \nhave been characterized through the DFT simulation for the first time, demonstrating their potentiality \nfor optoelectronic device applications. The materials' optical anisotropy is observed along the x-axis, \nwhich manifests their tunability through light-matter interaction. \n1. Introduction \nFerrites are classified as magnetic materials with astounding electromagnetic, dielectric, and other \nfunctional properties. Notably, the size and shape-dependent tunable properties of nanocrystalline \nferrites (NCFs) make them promising for multi-purpose high-frequency applications. Their potential 3 \n applications have been reflected through their tunable dielectric and electromagnetic properties over the \nlast decade, which considered them suitable for a range of electronic and biomedical applications such \nas multi-layer chip inductors, magnetic sensors, high-density magnetic storage devices, isolators, \nmicrowave devices, wireless power transfer, hyperthermia, drug delivery, magnetic resonance imaging, \ngene therapy and delivery, DNA and RNA separation, and ferrofluids and so on [1–5]. As reported in \n[6], spinel ferrites (metal oxide semiconductors) are proven as the superior class of magnetic materials \npossessing the properties of high sensitivity, fast response, protracted stability, and low cost. Moreover, \nnanocrystalline spinel ferrite thin films have been focused in recent years on understanding the magneto-\noptical behaviors of annealing at comparatively low temperatures [7]. Furthermore, nanocrystalline \nferrite materials are latterly being used in the production of thinner EM wave absorbers with a broader \nabsorption bandwidth to sustain a safe and stable environment for both devices and lives by minimizing \nelectromagnetic wave interference damage [8,9].  \nIn [A][B] 2[O]4 nano-spinel ferrites, the hetero-structure nature is highly essential for application \npurposes, as cation distributions over tetrahedral [A] and octahedral [B] lattice sites, as well as their \ninter-site exchange interactions eventually modify their magnetic characteristics  [8,10]. Among various \nNCFs, nickel ferrites are magnetically soft materials in nature, which offer a range of practical \napplications due to their high magnetic permeability, moderate saturation magnetization, low eddy \ncurrent loss, high resistivity, low dielectric tangent, suitable optical band gap, and high mechanical \nstrength  [9,11–13]. During the synthesis of ferrite nanoparticles, some distinct characteristics (i.e., high \nsurface-to-volume ratio, small-size effect, and quantum tunneling effect) are observed due to bulk-to-\nnano-scale transition that leads to the exalted physical, magnetic, and optoelectronic properties of the \nmaterials. Consequently, ferrites' structural and magnetic properti are influenced by a range of factors \nsuch as synthesis methods, doping, processing time, sintering temperature, grain size, purity, and \nsintering resources [14–19]. \nIt is noteworthy to mention that selecting a suitable substituent in forming a compatible ferrite sample \nis vital for fine-tuning the materials' physical properties and ameliorating the applications at a broad \nrange [20,21]. Here, Al3+ is chosen as the substituent since it undergoes a phase transition via a reduction 4 \n in the symmetry of ferrite crystals as the effect of the Jahn-Teller effect, which in turn improves the \nmaterials' electromagnetic properties [22,23]. Moreover, Al3+ incorporation in ferrites has a standing of \nimproving the crystallinity with maintaining the homogeneity of magnetic nanoparticles (MNPs) [24–\n28]. To synthesize MNPs, the sol-gel approach is widely used among other approaches because of its \nimproved control over powder morphology, homogeneity, and elemental composition, providing a \nnarrow particle size distribution at relatively low temperatures. It also provides a uniformly nano-sized \nmetal cluster, which is crucial for improving the properties of nanoparticles for high-frequency (HF) \ndevice applications [29].  \nSeveral attempts have been made sporadically to examine the structural, electrical, morphological,  \nmagnetic and dielectric characteristics of Ni-based ferrite NPs. Research in spinel ferrites is still \nadvancing, substituting several atoms as the dopant in A and B-sites to improve their physical, dielectric, \nand magnetic properties. V. A. Bharati et al.  [30] investigated the effect of Al3+ and Cr3+ parallel doping \non the structural, morphological, and magnetic properties of Ni ferrite NPs. In [31], K. Bashir et al.  \nstudied electrical and dielectric properties of Cr3+ doped Ni-Cu ferrite N', demonstrating thematerials' \npotential HF applications and photocatalytic activity. Le-Zhong Li et al.  [32] studied Al3+ substituted \nNi-Zn-Co ferrites and found that saturation magnetization dropped dramatically and dc resistivity \nincreased for Al3+ substitution with x>0.10. The morphological and magneto-optical parameters of Ni \nferrite NPs were investigated in [33], where the authors reported the bandgap, E g = 1.5 eV, and the \ndecreasing trend was observed in the variation of saturation magnetization and Tc with Al3+ content. N. \nJahan et al. [34] assessed the consequences of diamagnetic aluminum (Al3+) substitution on the \nmorphological and magnetic properties of Ni-Zn-Co spinel ferrites fabricated using the conventional \nceramic technique. They reported a steady decrease in lattice constant with increasing Al3+ content and \nnoted the maximum saturation magnetization (M s) of 93.06 emu/g at x = 0.12. The structural, optical, \nmagnetic, and photocatalytic behavior of Al3+ substituted nickel-ferrites were reported in [35], which \nrevealed n excellent photocatalytic activity showing the optical bandgap ranges between 1.60 and 1.89 \neV. Q. Khan et al.  [36] scrutinized the influence of inserting Al3+ on the structural and dielectric \nbehavior for Ni-Cu spinel ferrites, divulging a maximum dielectric loss of 0.4 at 2.5 GHz. Consequently, \nvarious groups explored the impact of Al doping on spinel ferrite nanoparticles' characteristics . Mn-Ni-5 \n Zn ferrite [37], Ni-Co ferrite [38], Mn–Zn ferrite [39], Co-Zn ferrites [40], Ni-Mn-Co [41], and Ni-Zn \nferrite [42,43]. According to the DFT study [44], mixed spinel ferrites exhibited half-metallic properties, \nwhereas semiconducting behavior showed pure compositions. Substitution of transitional atom content \nin spinel ferrite enhances the lattice parameter linearly, whereas a decrement in magnetization was \nobserved with the weakening of the super-exchange effect in A, and B sites, as examined through DFT \nstudy in [45]. Another study employed first-principle GGA+U energy calculations for NiFe 2O4 to \nexamine the sensitivity of the cation distribution with strain modulation [46]. However, tunability in \noptical properties by bandgap modulation within spinel ferrite as studied through DFT calculations can \noffer potential for storage and photovoltaics, multifunctional materials and devices applications [47]. \nMoreover, DFT investigations are gradually increasing to tune the optoelectronic properties of various \nspinel ferrite structures, viz. MgFe 2O4 [48], ZnFe 2O4 [49], CoFe 2O4 [50], VFe 2O4 [51]. \nIn [52], we performed on structural, dielectric, and electrical transport properties for Al3+ substitution \n(x=0.00 to 0.10, in the step of 0.02) of nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4. However, the synthesized \nnano spinel ferrites' Rietveld-refined structural characteristics and magnetic properties, as well as the \nDFT-based optoelectronic properties for such a mixed spinel ferrite structure, have not yet been \nreported. Therefore, this study aims to investigate how Al3+ incorporation affects the structural (Rietveld \nrefinement) and magnetic properties of sol-gel produced Ni-Cu ferrite NPs. The optoelectronic \nperformances are also analyzed using the first-principle density functional theory (DFT) simulations for \nNi0.7Cu0.3AlxFe2-xO4 (x=0.06) spinel ferrite structure. \n2. Experimental and Computation Details \n2.1   Materials preparation \nThe nanocrystalline powder samples of Ni 0.7Cu0.3AlxFe2-xO4, with x varying as 0:0.02:0.1, were \nsynthesized via the sol-gel route. Analytical grade of nickel (II) nitrate (Ni(NO 3)2), copper (II) nitrate \n(Cu(NO 3)2), iron (III) nitrate hexahydrate (Fe(NO 3)3.9H2O), and aluminum (V) nitrate Al(NO 3)3.9H2O \nwere used as the raw materials. These metal nitrates were dissolved in de-ionized water with adding a \nfew drops of ethanol in a 1:2 molar ratio to obtain the initial solution with keeping its pH value at 7. The \ndry gel was obtained by vigorously swirling metal nitrates at 70ºC in a thermostatic water bath, then \ndried for 5 hours in a 200ºC electric oven. Following the process, the resultant compositions were burned 6 \n and grounded before sintering at specific temperatures, and the self-ignition process progressively \ntransformed it into fluffy-loose powder. The yielded fluffy-loose powder was annealed at 700ºC for an \nadditional 5 hours to obtain the highly crystalline materials without impurity. The powder was \nhomogenized further by hand-milling in a mortar to assemble disk-shaped samples. Afterward, the \nnanocrystalline powder was condensed into disk-like forms using a 65 MPa hydraulic press for 2 \nminutes. The processed samples had a diameter of 12.02 mm, and a thickness of 2.3 mm. Structure and \nmagnetic properties were measured using the annealed powder samples. \n2.2   Characterizations and properties measurements  \nCharacterization of the synthesized spinel ferrite nanoparticles has been performed using a variety of \ntechniques such as x-ray diffraction (XRD), field emission scanning electron microscopy (FESEM), \nenergy dispersive x-ray analysis (EDX), transmission electron microscopy (TEM), vibrating sample \nmagnetometer (VSM) and UV-Vis spectroscopy. Detail analysis for structural, electrical, and \nmorphology using XRD, FESEM, EDX are presented in our previous study [52]. The structural \nproperties of the prepared ferrites powder (i.e., lattice parameters (a), the crystallite size (D), \ndisplacement density, and so on) were investigated employing an x-ray diffractometer with Cu-K ( 𝜆= \n1.5418 Å) radiation. Synthesized nanoparticles' magnetic properties were measured using a Vibrating \nSample Magnetometer (VSM; Micro Sense, EV9). The following relationship has been used to \ninvestigate the net magnetic moment [53]: \n                              𝜇 =భ∗ೄ\nହହ଼ହ               (1)   \nwhere M 1 indicates the molecular weight of the samples, and M s denotes the magnetic saturation. \nMagnetic coercive force (coercivity) is a material's resistance to demagnetization since it allows us to \nstudy the material's magnetic properties than magnetization resistance, and it follows the relationship as \n[54]: \n                                                                 𝐻=.ଽ\nெೞ                    (2)   \n, where 𝐾=ୌୡ∗ୱ\nଽ is used in Eq. (2) to determine the anisotropic constant.   \n2.3.  Computational method 7 \n In order to learn about the mixed spinel ferrite ' 'material's light-matter interaction potentials, its \noptoelectronic properties were investigated with the spin-polarized Density Function Theory (DFT) [55] \nusing VASP 6.1 (Vienna Ab Initio  Simulation Package) [56–60]. Using the Local Spin Density \nApproximation (LSDA) and the Generalized Gradient Approximation (GGA) with Perdew-Burke-\nErnzerhof (PBE) exchange correlation functions, the electronic charge density was optimized [61]. \nPseudopotentials from the VASP library were used to describe each atom; these potentials were built on \nplane-wave basis sets obtained using the projector-augmented wave (PAW) approach and were \nparameterized for each formalism (LSDA and GGA) [62,63]. It is well-established that these \npseudopotentials are more accurate for magnetic systemshan the classical ultra-soft pseudopotentials \n(USPPs) [60]. Taking into account on-site Coulomb interactions with the Dudarev method (LSDA+U) \n[63] yielded a band gap that is more precise and in good agreement with experimental data. The \nelectronic self-consistence force convergence threshold was considered 1×10-7 eV [64].  The Brillouin \nZone was integrated using Γ-centered k-points mesh 4×4×2 generated with Monkhorst-Pack Scheme \n[65] and the kinetic energy cut-off was chosen 400eV. For each orbital of all atoms, partial occupancy \nwas chosen by Gaussian smearing with a smearing width of 0.05eV to integrate the Brillouin Zone. The \nband structure was obtained using Wannier90 [66] interpolation to get a more accurate band structure. \nThe number of interpolated bands was equal to the number of bands obtained from  the plane wave basis \ncode (VASP). \n \n \n3. Results and discussion \n3.1   Structural Investigations \nX-ray diffraction \nThe x-ray diffractometer (XRD) spectra of the prepared Al3+ substituted Ni-Cu ferrite nanoparticles are \nportrayed in Fig. 1, where the vital peaks are reflected from the planes of (1 1 1), (2 2 0), (3 1 1), (2 2 \n2), (4 0 0), (4 2 2), (5 1 1) and (4 4 0). XRD peaks confirm the single-phase spinel structure of the \nsynthesized magnetic nanomaterials in cubic shape with no other phase pesence. The crystallite size of 8 \n Ni-Cu ferrite nano samples was found to vary with Al3+ substitutions estimated by the highest diffraction  \n(3 1 1) plane utilizing Scherrer formula. In this study, variations in the structural parameters have been \nobserved for 2-10% doping of Al3+ in the lattice of Ni-Cu, which comes out with no significant \nalteration. The evaluated structural parameters utilizing XRD  are listed in Table 1 in [52].  \n \nFig. 1. XRD spectra profiles for the Ni 0.70Cu0.30AlxFe2-xO4 (x = 0: 0.02: 0.1) nanoparticles. \nLattice Spacing \nBragg's law was employed to calculate the distance between atom centers (lattice spacing) d that depends \nupon the direction in the lattice as following: \n𝑑=ఒ\nଶௌ          (3) \nwhere n signifies the order of diffraction is taken as 1 as well as  𝜆, 𝜃 is the x-ray wavelength and Bragg's \nangle, respectively.  \nLattice constants 10 20 30 40 50 60 70\n  Intensity (a.u.)\n2q ()(111)\n(220)\n(311)\n(222)\n(422)\n(511)(400)\n(440)\nx=0.00x=0.02x=0.04x=0.06x=0.08x=0.109 \n The miller indices values, i.e., (h k l) = (3 1 1), were used to calculate the lattice constants employing \nEq. (4) as follows [67]: \n                                                                𝑎=𝑑√ℎଶ+𝑘ଶ+𝑙ଶ          (4) \nBesides, the theoretical lattice constant (a th) was estimated employing the following relation:  \n                                       𝑎௧=଼\nଷ√ଷൣ(𝑟+𝑅)+√3(𝑟+𝑅)൧        (5) \nwhere R0 stands for the oxygen ion's radius (1.32Å) and the radius of A- and B-sites atoms denoted by \n𝑟 and 𝑟 are evaluated utilizing the given formula [68]: \n                                                       𝑟=𝑎√3(𝑢−0.25)−𝑅         (6) \n                                                       𝑟=𝑎ቀହ\n଼−𝑢ቁ−𝑅                 (7)  \nwhere u is the oxygen position parameter considered as 3/8 for an ideal FCC crystal. Moreover, the \ncomputed lattice parameters for each crystal-plane, displayed against the Nelson–Riley function, \nfollowing: \n                                               𝐹(𝜃)=ଵ\nଶቂమఏ\nௌఏ+௦మఏ\nఏቃ                 (8) \nwhere a diffracted line is obtained for each sample employing the diffraction angle 𝜃. \n \nCrystallite Size \nThe average crystallite size (D) for the prepared Ni-Cu nanoparticles were estimated following Debye-\nScherrer's equation from the most rising peak plane (3 1 1) [69]: \n                                                                     𝐷=ఒ\nఉ௦ఏ             (9) \nwhere the structural shape factor (Scherrer's constant) k = 0.94 is used for small cubic crystal, the full \nwidth at half maximum (FWHM) is denoted by β, 𝜆 symbolizes the wavelength of incident x-rays and \n𝜃 is the diffraction angle known as Bragg's angle.  10 \n Dislocation density \nDislocation density is the parameter used to analyze the strength and ductility of the crystal arrangement \nand is varied by the sample annealing. In a crystal structure, the overall dislocation length per unit \nvolume is projected by the number of etch pits per unit area on the etched surface, as determined by \nequation [70]: \n                                                                          𝛿=ଵ\nమ           (10) \nwhere D represents the crystallite size. Dislocation density and particle size have followed an inverse \nrelationship for the synthesized nanoparticles as reported in [52]. \nLattice Strain \nThe unit length deforms when an object is subjected to pressure, reflecting the sample's strain. Due to \nthe formation of defects in crystal structure and flaws in the crystal structure, the atoms' typical lattice \norientations exhibit slight changes. Lattice strain quantifies the distribution of lattice constants induced \nby crystal defects and imperfections, such as interstitial and/or impurity atoms and lattice dislocations \n[71]. The following relation (Eq. 11) was used to evaluate the lattice strain for the synthesized spinel \nferrites. \n                                                                      𝜀௦=ఉ\nସ୲ୟ୬ఏ               (11) \nwhere 𝛽, and 𝜃 represent the full-width at half maximum (FWHM) of the diffraction peak and Bragg's \nangle, respectively. \nMicro-strain \nThe most prevalent sources of deformation are dislocations, plastic deformation, point defects in the \ncrystal structure, and abnormalities in domain boundaries,which happen in one part per million (10-6) of \nthe material [72]. Hence, peak broadening is a crucial aspect of micro-strain, as defined by the following \nequation: \n𝜀௦=ఉ௦ఏ\nସ                (12) 11 \n Stacking Faults  \nDuring crystal formation, point defects can condense into stacking faults (SF), which are distortions \nfrom the normal lattice structure induced by the layer arrangement or faults that arise in the atomic \nplanes of the crystal [73]. The stacking fault was calculated and reported that SF values varied inversely \nonly with the tangent of the diffraction angle 𝜃 [52]. \n \n Fig. 2: Variation in lattice constant and cell volumes of Ni 0.7Cu0.3AlxFe2-xO4 with Al3+ content.  \nEmploying Eq. (9), the average crystallite size of the samples is estimated on the (h k l) = (3 1 1) plane \nvalues which are found to be in range (60-71 nm), with projecting uncertainties listed in Table 1 in [52]. \nThe change in lattice constants and associated cell volume with increasing Al3+ content is depicted in \nFig. 2. Both the lattice parameter values and the cell volumes are observed to vary harmoniously with \nAl3+ content, except for x = 0.02 and 0.08, decrease linearly with Al3+ concentration as illustrated in Fig. \n2. In variation of lattice parameters, a decreasing trend with Al3+ content is noticed, which is justified \nas a larger ionic radius of Fe3+ (0.63 Å) is replaced by Al3+ (0.53 Å) having a smaller ionic radius \nfollowing Vegard's law [74]. The literature reveals similar trends in the variation of lattice parameters \n[17,36,74]. The depicted downward shift of crystallite size with Al3+ content after x=0.04 in Fig. 3 is \nalso highlighted as high Al3+ concentrations result in Al2+ ions, compelling them to exchange lattice \npositions. This phenomenon has been appeared by a random distribution of Al3+ and Fe3+ ions, resulting \nin the increase of stress and strain in the samples. Hence, Al3+ reduces to Al2+ migrating from B-site to 0.00 0.02 0.04 0.06 0.08 0.100.833000.833250.833500.833750.834000.834250.834500.834750.83500\n Lattice Parameter\n Cell volume\nAl3+ content (x)Lattice parameter, a(nm)\n0.57800.57850.57900.57950.58000.58050.58100.58150.5820\n Cell volume, V(nm3)12 \n A-site after a certain amount of Al incorporation. As consequence, it is observed from Fig. 3 that the \naverage grain sizes for the samples are increased up to x=0.04 Al3+ incorporation and then the values \nwere decreased. \n \nFig. 3: Crystallite size and grain sizes variations with Al3+ content. \n Strain parameters help to understand the correlation between strain-induced magnetism and non-\nmagnetic Al3+ concentration in the ferrite nanoparticles. However, the variation in both lattice and micro \nstrain values are portrayed in Fig. 4 to understand a clear project of the lattice distortion behavior with \nthe Al3+ concentrations. As depicted in Fig. 4, both of the parameters are varied in the same manner \nexcept for x=0.10, a slight splitting appears. FESEM extracted surface morphology of the investigated \nmaterials exhibited that the grains were mostly about spherical in shape and are distributed uniformly \nand evenly due to the separating grain boundaries as described in [52]. FESEM micrographs, on either \nhand, revealed multi-grain phenomena composed of grains and grain boundaries, with some \nagglomerations appearing due to the dipole-dipole interaction within magnetic nanoparticles and the \nhigh surface to volume ratio. The mean particle size of the samples was found to be in the nano-size \nrange (59 - 65 nm). The intensity peaks in the Energy-dispersive X-ray spectroscopy (EDX) spectra are \nexacerbated by the energy gap between two electronic states generated by the cannonade of composites \nby the electron beams of the SEM. As reported in [52], there are no impurity peaks, reconfirming the \nmaterials' single-phase structure observed after ascertaining the proper compositional proportions of \nelements contained in the synthesized nanocrystalline ferrite samples.  0.00 0.02 0.04 0.06 0.08 0.105560657075\n Average crystallite size\n Average grain size\nAl3+ content (x)Average crystallite size, D (nm)\n5860626466\n Average grain size, G (nm)13 \n  \nFig. 4: Variation in lattice strain and micro strain of nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4. \nRietveld refinement \nThe refinement of XRD extracted data  is carried out through the Rietveld refinement analysis using the \nFullProf software as illustrated in Fig. 5. In this refining process, a nonlinear least-squares fitting method \nwith a Pseudo-Voigt profile was utilized. Throughout the fitting procedure, instrumental and \nbackground parameters were carefully considered to refine the structural parameters and this process \nwas repeated until a minimal residual (difference between calculated and observed intensities) was \nfound along with achieving a good value of fitting parameter, χ2.  \nThe Rietveld fitting parameters are presented in Table 1, with  R-factors and χ2 values. For each sample's \nfitting, Fig. 5 displays the observed intensity (Y obs), the estimated intensity (Y cal), and the residual, which \nare obtained from the refinement process. Also, Table 1 displays the refined crystal parameters (i.e., \nlattice parameter, unit cell volume, and average crystallite size). 0.00 0.02 0.04 0.06 0.08 0.101.61.71.81.92.02.1\n Lattice Starin\n Micro strain\nAl3+ content (x)Lattice strain (×10-3)\n4.85.05.25.45.65.86.06.26.4\n Micro strain (×10-3) (line-2/m-4)14 \n  \nFig. 5: Rietveld refinement of various samples in nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4 annealed at 700 ºC. \nThe overlapping between the calculated and observed patterns indicate that well-fitted patterns, found \nin all cases in these analyses of Fig. 5. Similar to the correlation profiles, the difference between the \ncalculated and observed patterns is almost linear with just slight fluctuations, denoting that the data is \noptimally-fitted. For the Al3+ concentration in the nano-spinel ferrite compounds at different values of \nx (0.00, 0.02, 0.04, 0.06, 0.08 and 0.10), the goodness-of-fit values for χ2 observed are 0.7292, 0.5632, \n0.7437, 0.6862, 0.5074, and 0.7072. The value of χ2 = 2.00 is considered as a better fitting indicator in \n15 \n the Rietveld refinement analysis [72] and thus the observed values of χ2 have accorded well with the \nobserved fitting patterns.  \nTable 1. Rietveld refined fitting parameters, lattice parameters, cell volume, and average crystallite \nsize for the Al3+ doped Ni-Cu spinel nanoferrites. \nAl3+ content \n(x) Rp R wp Rexp 𝝌𝟐 DW-stat aexp(Å) V(Å)3 Average Crystallite Size \nD (nm) \n0.00 \n0.02 \n0.04 \n0.06 \n0.08 \n0.10 12.5 \n12.1 \n13.4 \n12.7 \n11.6 \n15.5 8.05 \n7.04 \n7.89 \n7.81 \n6.52 \n8.05 9.43 \n9.38 \n9.15 \n9.43 \n9.15 \n9.56 0.73 \n0.56 \n0.74 \n0.67 \n0.51 \n0.71 0.166 \n0.108 \n0.125 \n0.295 \n0.154 \n0.175 8.346 \n8.347 \n8.342 \n8.336 \n8.338 \n8.332 581.40 \n581.75 \n580.53 \n579.35 \n579.73 \n578.43 71.32 \n61.43 \n68.17 \n61.81 \n65.03 \n64.59 \n \nHopping Lengths \nThe hopping length refers to the average distance traveled by an ion from one neighboring lattice-site \nto another. We used the following relations to determine the hopping distance between A-sites, B-sites, \nand shared sites, respectively [75]: \n𝐿ି=√ଷ\nସ     (17) \n𝐿ି=\nଶ√ଶ       (18) \n𝐿ି=√ଵଵ\n଼    (19) \nTo study the hopping mechanism and further cation distributions over the lattice sites, the hopping \nlengths for both tetrahedral and octahedral sites are evaluated following Eq. (17-19). The estimated \nhopping lengths for A-sites (L A), B-sites (L B), and shared A-B sites (L A-B) are tabulated in Table 2 The \nvariations in both L A and L B with Al3+ substitution follow a similar trend as shown in Fig. 6.    \nAs the grain size shifts, the distance between magnetic ions shifts, which in turn affects the L A and L B \nvalues [76]. Fig. 6 clearly reveals that L A>LB, suggesting that the probability of electron hopping \nbetween ions in tetrahedral A and octahedral B sites is lower than that between octahedral B-B sites. \nMoreover, hoping lengths also gives the insight to understand the electrical conduction characteristics \nsince the effectiveness of the scattering process of hopping electrons directly affects the conductivity \n[77]. The other structural parameters as listed in Table 2- bond lengths (d A× and dB×), tetrahedral edge 16 \n (dA×E), shared (d B×E) and unshared (d B×Eu) octahedral edges for the synthesized Ni 0.7Cu0.3AlxFe2-xO4 \nsamples are evaluated using lattice parameter value a and the oxygen positional parameter u = 0.381 Å \nutilizing the following equations [78]:  \n𝑑×=𝑎√3ቀ𝑢−ଵ\nସቁ               (20) \n𝑑×=𝑎ට3𝑢ଶ−ଵଵ\nସ𝑢+ସଷ\nସ   (21) \n𝑑×ா=𝑎√2ቀ2𝑢−ଵ\nଶቁ                    (22) \n𝑑×ா=𝑎√2(1−2𝑢)               (23) \n𝑑×ா௨=𝑎ට4𝑢ଶ−3𝑢+ଵଵ\nଵ   (24) \n \nFig. 6: Variation in hoping lengths (L A and LB) with Al3+ content in nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4.                          \nTable 2. Hopping lengths (L A, LB, and L A-B), tetrahedral and octahedral bond lengths (d A× and d B×), \ntetrahedral edge (d A×E), shared (d B×E) and unshared (d B×EU) octahedral edge for the synthesized Ni-\nCu ferrite nanoparticles \nAl3+ \ncontent (x)  LA(Å) L B(Å) L A-B(Å) dA×(Å) d B×(Å) d A×E(Å) d B×E(Å) d B×EU(Å) \n0.00 \n0.02 \n0.04 \n0.06 \n0.08 \n0.10 3.605 \n3.592 \n3.601 \n3.610 \n3.591 \n3.596 2.943 \n2.933 \n2.940 \n2.948 \n2.932 \n2.936 3.451 \n3.440 \n3.448 \n3.456 \n3.438 \n3.443 1.894 \n1.894 \n1.893 \n1.891 \n1.892 \n1.890 2.038 \n2.038 \n2.037 \n2.035 \n2.036 \n2.034 3.787 \n3.093 \n3.091 \n3.089 \n3.089 \n3.087 2.809 \n2.810 \n2.808 \n2.806 \n2.807 \n2.806 2.953 \n2.953 \n2.951 \n2.949 \n2.950 \n2.947 \n \nTransmission electron microscopy (TEM) is used to validate the structural morphology of \nnanocrystalline nature of the synthesized ferrites. TEM micrographs of Ni 0.7Cu0.3Fe2O4 ferrite \nnanoparticles annealed at 700ºC are shown with different magnifications in Fig. 7 (A & B), where the 0.00 0.02 0.04 0.06 0.08 0.103.563.583.603.623.64  LA\n LB\nAl3+ content (x)LA(Å)\n2.912.922.932.942.952.962.972.98\n LB(Å)17 \n uniformity and nano-spherical shape of the particles is evident. When the plane of the atoms is in the \nsame direction and linear, high crystallinity is achieved, and the average d-spacing is 0.458 nm which \nare exemplifies perfectly through the HTEM image in Fig. 7(C). In Fig. 7(D), the selected area electron \ndiffraction (SAED) pattern is shown to be well matched with the x-ray diffractogram results. The SAED \npattern confirms that the spotty diffraction rings correspond to planes (1 1 1), (2 2 0), (3 1 1), (2 2 2), (4 \n0 0), (4 2 2), (5 1 1), and (4 4 0), reiterating the single spinel phase of the synthesized nanocrystalline \nferrites. \n \nFig. 7: TEM [(A) & (B)], HTEM (C) morphological micrographs, and (D) SAED patterns of \nnanocrystalline Ni 0.70Cu0.30AlxFe2-xO4. \n3.2 Magnetic Properties \nThe magnetic properties of ferrites are affected by the composition of metal ions and their distribution \nin the spinel lattice. The variation in cation distribution over tetrahedral (A-site) and octahedral (B-site) \nC \nB \nA \nD 18 \n lattice sites causes the variation in magnetic properties. At room temperature, the magnetic parameters \nof Al3+ incorporated Ni-Cu ferrite nanoparticles are determined using the VSM method with varying \napplied magnetic fields (H). The M-H curves of the synthesized  materials are represented in Fig. 9. \nFrom the VSM loops, the magnetic parameters (i.e., Ms, Mr, Hc, K, and so on) are calculated and they \nare listed in Table 4. \nM-H Hysteresis loop   \nFig. 8 depicts the magnetization variation of the investigated ferrite nanoparticles with reference to the \napplied magnetic field (H) (known as the M-H loop) at room temperature, which is one of the crucial \nparameters in determining the plausible application of magnetic materials. In Fig. 8, it is observed that \nmagnetization (M) for all the compositions of the synthesized ferrites increases up to 0.5T, thereafter it \nbecomes steady to rise and then appears with the saturation magnetization value for the applied field to \n2T. The magnetization variation trend indicates the soft ferrimagnetic nature of the investigated Ni-Cu \nspinel ferrite nanoparticles. The magnetization curve was extrapolated to µ 0H = 0 to evaluate the M s for \nall the studied compositions. However, from the curves extracted from VSM, estimated values of \nmagnetic parameters as- M s : saturation magnetization, M r : remanence magnetization, H c : coercivity, \nMr/Ms : remanence ratio, η exp(µB) : experimental magnetic moment, and K : magnetic anisotropy \nconstant are listed in Table 3.  \n                              -40-30-20-10010203040\n0.00 0.02 0.04 0.06 0.08 0.10303234363840Ms (emu/g)\nAl content (x)M (emu/g)\nMagnetic field strength , µ0H (T) x = 0.10\n x = 0.08\n x = 0.06\n x = 0.04\n x = 0.02\n x = 0.00\n0.0 0.5 1.0 1.5 -0.5 -1.0 -1.519 \n Fig. 8: Variation in magnetization with applied field strength of nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4 at room \ntemperature. \nFig. 9(A) demonstrates that M s steadily decreased with the gradual increase of Al3+ substitution in \ncomposition. In the examined Ni-Cu ferrites, the cations are distributed over A- and B-sites according \nto Neel's two sub lattice model, where the magnetic moments of various cations are Ni2+ (2.3 μB), Cu2+ \n(1 μB), Al3+ (0 μB), and Fe3+ (5 μB), respectively. The insertion of Al3+ ions might well have occupied \nthe B sublattice, and the probable A sub lattice occupying states were very small. As a result, the number \nof magnetic moments on the B-site was expected to decrease as the Al3+ concentration in the samples \nchanged. The total magnetization, on the other hand, is equal to the difference between B-site and A-\nsite magnetization. As a result, increasing the content of Al3+ ions in the compositions reduces the value \nof Ms. However, as B-site is occupied by a nonmagnetic Al3+ leads to no change in magnetization. The \nearlier studies [15,17] suggest that with further Al3+ incorporation in the ferrites, exchange interaction \nby the crystal symmetry reduction in lattice occurs, therefore, Cu2+ partially migrates its site from A to \nB, Fe3+ shifts from B to A and Al3+ alleviates to Al2+ thus relocating to site A in the manner can be \ndescribed as Cu2+⟷Cu3++e-, Fe3+ +e-⟷ Fe2+ and Al3+ + e− ⟷ Al2+. Facts of cation distribution \naccording to the Neel's lattice model and the exchange interactions among lattice sites (J AA, JAB, and \nJBB) where each is influenced by oxygen ions enable it to be comprehended the trend seen in Fig 9(A) \non the variation of magnetic saturation values and magnetic moment with Al3+ concentrations [79]. \nFurthermore, one unpaired electron contains in Al3+ with a magnetic moment of √3, whereas Al2+ has \nno valence electrons that are unpaired. However, Oxygen seems to have two unpaired electrons in its \nvalence state which compensate for neutralizing the charge of the samples. Hence, the lattice constant \nchanges influencing the J AB interaction leads to altering the magnetic performance due to cationic \nredistribution between A and B sites and lattice distortion in the crystal symmetry of the ferrites [80,81]. \nThe saturation magnetization (M S= MB-MA) and consequently the experimental magnetic moment (η exp) \nare found to decrease with the increase of Al3+ content as depicted in Fig. 9(A) for destabilization of the \nA-B inter-site interaction. However, both parameters are shown maximum for the pristine sample with 20 \n no Al3+ content.  Similar variations in such magnetic behaviors for the ferrite nanoparticles have been \nreported in the literature [17,28,82].  \n \nFig. 9: Variations in (A) saturation magnetization and magnetic moment, (B) remanence magnetization \nand coercivity with Al3+ content.  \nUtilizing the M-H loops evaluated values of M r and H C and listed in Table 3 as well as the variation of \nboth parameters is presented in Fig. 9(B). As shown in Fig. 9(B), both M r and H C demonstrate a similar \nbehavior with the incorporation of Al3+ content. The resultant low coercivity values for the examined \nferrite nanomaterials classify them as soft magnetic materials. Fig. 10 portrayed the projection of \nchanges between the M r/Ms (remanence ratio) and K (magnetic anisotropy constant) for \nNi0.7Cu0.3AlxFe2-xO4 samples with Al3+ substitution changes. Several factors contribute to the variation \nof HC and K values with Al3+ concentration, in which magnetic domain walls and corresponding \nmagnetic moments are significant [53]. The K values are found to be increased with Al3+ concentration, \nwhich is attributed to the site exchange interactions among magnetic nanoparticles. The remanence ratio \nof the synthesized ferrites is found to be very low in a range of (0.000 – 0.094) [Table 3], indicating the \npresence of magnetic nanoparticles with a multi-domain nature in the samples as also reported earlier \n[53,83,84].  \n \n \nA \n B \n0.00 0.02 0.04 0.06 0.08 0.10020406080100  Hc\n Mr\nAl3+ content (x)Hc (Oe)\n-0.50.00.51.01.52.02.53.03.5\nMr (emu/g)\n0.00 0.02 0.04 0.06 0.08 0.10303234363840  Saturaion magnetization\n Bohr Magneton\nAl3+ content (x)Ms (emu/g)\n1.31.41.51.61.7\n Bohr Magneton ηexp(µB)21 \n  \n \nTable 3. Measured magnetic parameters for the synthesized Ni 0.7Cu0.3Fe2-xAlxO4 samples. \nAl3+ content (x) Ms \n(emu/g) Mr \n(emu/g) Hc (Oe) Mr/Ms  ηexp(µ𝑩) K (erg/Oe)  \n0.00 \n0.02 \n0.04 \n0.06 \n0.08 \n0.10 39.98 \n38.84 \n37.06 \n33.64 \n32.97 \n31.01 0.00 \n0.78 \n3.22 \n2.58 \n2.68 \n2.92 0.00 \n13.74 \n93.57 \n57.76 \n58.31 \n70.94 0.000 \n0.020 \n0.087 \n0.077 \n0.081 \n0.094 1.6887 \n1.6361 \n1.5573 \n1.4101 \n1.3786 \n1.2934 0.00 \n555.90 \n3612.19 \n2024.01 \n2002.58 \n2291.51 \n \n                       \nFig. 10: Changes in the remanence ratio and the magnetic anisotropy constant at room temperature \nwith varying Al3+ concentration.  \nUV-Vis Analysis: \nOptical absorption spectrum (UV–Vis) study is one of the suitable methods for understanding the optical \nproperty of the materials. Fig. 11 shows the UV-vis absorption spectra of the prepared Ni-Cu nano spinel \nferrites that exhibit absorption spanned in a wide wavelength range of 200-800 nm. It is seen from Fig. \n11(A) that there is a tendency to show different slopes at different wave lengths, which can be ascribed \nto the electron transitions between the oxygen ions and cations. To estimate the direct bandgap energies \n(Eg), the UV–Vis spectra were plotted; (αhν)2 vs  photon energy (E=hν) using the formula: E bg (eV) ≤ \n1240/𝜆 [85,86], and represented in Fig. 11(B). Optical absorption spectra of Al3+ substituted \nNi0.7Cu0.3Fe2O4 nanoparticles resulted the direct band gap energy (E bg) 2.60 eV, 3.00 eV, 3.24 eV, 3.07 \neV, 3.37 eV, and 2.80 eV respectively for x=0.00,0.02,0.04,0.06,0.08 and 0.10 concentration. It can also 0.00 0.02 0.04 0.06 0.08 0.100.000.020.040.060.080.10  Mr/Ms     \n K \nAl3+ content (x)Mr/Ms\n01000200030004000\n Anisotropy constant, K (erg/Oe)22 \n be seen that the UV–Vis absorption spectra exhibit a slight spline shape indicating to the localized (by \nmeans of structural defects) electronic levels above the valence band gap. Additionally, it has been found \nthat in wide bandgap semiconductors, shallow defect levels near the conduction band or valence band \ndo not play an effective role as a recombination site. However, the deepest level within the forbidden \nband is where the most effective recombination takes place [87]. It appears that the defects at the \noctahedral location with Al and Fe in the prepared Ni-Cu spinel ferrites could kick off recombination. \nHowever, the optical properties since associated with the crystal formation, hence lattice and micro \nstrain along with the hopping lengths effectively play role to alter the properties.  \n \nFig. 11: Variations in (A) Absorptions and (B) Band gap values for Ni-Cu nano spinel ferrites with \nvarying Al3+ content. \nDensity Functional Theory analysis \nTo mimic the Ni 0.7Cu0.3Fe1.94Al0.06O4 sample, standard DFT simulation was performed, and the \nelectronic and optical properties were investigated. A 2×1×1  supercell was constructed from the \nconventional Ni 0.7Cu0.3Fe2O4 unit cell of Fd 3തm space group in the Cubic crystal system. Since previous \nexperiments reported Fe, Ni and Cu with radii 0.77Å,0.69Å,0.71Å  respectively [88][89]. The Al3+ ions \nwith ionic radius ~0.48Å [88], the Cu atoms occupied the tetrahedral sites. They occupied the positions \nof Ni atoms, and our experiment suggests the Al doping into the structure there it occupied the \ntetrahedral site of Fe atoms. Therefore, five Ni atoms were then replaced with five Cu atoms to design \na 30% Cu doping in the Ni occupied A-site, after that two Fe atoms were replaced with two Al atoms \n(B) \n (A) \n200 300 400 500 600 700 8000.050.100.150.200.250.300.35Absoprtion (a.u.)\nWavelength (nm) x=0.00\n x=0.02\n x=0.04\n x=0.06\n x=0.08\n x=0.10Ni0.7Cu0.3Fe2-xAlxO4\n1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5036912(ahv)2(eVcm-1)2\nEnergy (eV) x=0.00\n x=0.02\n x=0.04\n x=0.06\n x=0.08\n x=0.10Ni0.7Cu0.3Fe2-xAlxO423 \n for the sake of Al (x=0.06) doping in the B-site and the DFT simulations were performed. Spin polarized \ncalculations with GGA-PBE functionals were performed to relaxed the structure.Table 4 shows the \nstructural parameters from DFT calculations. It is noticed that the lattice parameter ( 𝑎), cell volume (V), \nand inter bonds are well consistent with the experimental data.  \n \nFig. 12: Optimized atomic structure of the Ni 0.7Cu0.3Fe1.94Al0.06O4 spinel ferrite nanoparticle (a) 001 \nplane, (b) 111 plane.  \nTable 4. DFT extracted structural optimized parameters for Ni 0.7Cu0.3Fe1.94Al0.06O4.  \nNi0.7Cu0.3 \nFe1.94Al0.06\nO4 Lattice \nParameter \n𝑎 (Å) Cell Volume \nV (Å3)  LA-B (Å) d A (Å) dB (Å) \nalpha 8.29     569.72 \n Ni-Fe: 3.38 \n \nNi-Al: 3.51 \n \nCu-Fe: 3.55 \n \nCu-Al: 3.51 Ni-O: 1.96 \n \nCu-O:  2.02 \n Fe-O: 2.043 \n \nAl-O: 1.928 \n \n \nElectronic Properties                                                                      \nAs we know, the standard DFT packages underestimate the band gap [90], so we used the DFT+U \nformalism in the Dudarev approach. The U eff was applied to the 3d orbitals of Fe, Ni and Cu atoms by \nthe amount 4eV, 6.4eV and 4eV correspondingly [91] and found the electronic band gap was consistent \nwith the experimentally observed value. The equilibrium structure obtained from the calculation was \nTriclinic in phase, and the lattice parameters were a=b=8.288 Å, and c=17.078 Å does have a good \nCu \nNi \nFe \nAl \nO \n(b) \n (a) 24 \n agreement with the experimentally observed data. As referred, the material to be a magnetic structure, \nthe magnetic nature of Fe, Ni, O, Cu, and Al was introduced, and found the equilibrium magnetic \nmoments of corresponding elements as 4.257 𝜇, -1.796𝜇, 0.259𝜇 and -0.608 𝜇. However, as the \nmaterial is magnetic in nature, so the spin polarized calculations were performed. Hence the Spin-UP \nbands (represented in Fig. 13(a)) and Spin-Down (represented in Fig. 13(b)) bands were obtained. The \nWannier90 interpolated band structure is presented in Fig. 13. From the figure, it is clearly evident that \nthe doped material has a direct band gap at the Γ-point in both cases.  \n  \nFig. 13:  Electronic Band Structure of Ni 0.7Cu0.3 Fe1.94Al0.06O4 nanocrystalline ferrite. \nTherefore, there can be a direct transition of electrons from the valance band to the conduction band \nwithout phonon interaction. The Spin-UP case has a band gap of 2.99eV, and the Spin-Down bands \nhave a band gap of 1.66eV. The DFT calculated optical band gap nature and value is found in tune with \nthe experimental data as presented in Fig. 11. The direct band gap makes the material a potential \ncandidate for optoelectronic device applications.  However, a defect band state evident in the Spin-UP \nband structure in the conduction band region. That defect states located 0.62eV above the valence band \nmaxima. \nTo investigate the defect region, we performed the Density of States (DOS) and Partial Density of States \n(PDOS) calculations presented in the Fig. 14. The DOS shows that the material has a defect state near \nfermi region in the conduction band. This occurred due to the strong hybridization between Cu-3d orbital \nand O-2p orbital; a small contribution exists due to the Fe-3d orbital but no significant contribution of \nAl-3(s,p) orbitals can exist be observed in the defect region. Therefore, it can be concluded that this \n25 \n material has an opportunity to perform as a photocatalyst in color degradation. Moreover, the above \nexperiment reports that the material is magnetic in nature, which also can be observed in the non-\nsymmetric density of states represented in Fig. 14. From the PDOS it can be easily inferred that the \nmagnetism is mostly driven by the Fe-3d orbital electrons and the 3d orbitals of Ni and Cu \nsimultaneously.  \n \nFig. 14: Density of States of Ni 0.7Cu0.3 Fe1.94Al0.06O4 nanoparticle. \nOptical Properties \nAs previously mentioned, the above material has a potential optoelectronic device application. \nTherefore, the optical profile is crucial for this purpose. So, we did an investigation in its optical \nproperties with the GGA-PBE+U formalism. Where the dielectric function plays the critical role, and \nabsorption coefficient 𝛼, reflectivity R, dielectric energy loss function L, refractive index 𝜂, optical \nconductivity 𝜎 like information concealed. To perceive these optical properties of this material at-first \nwe extracted the dielectric function and its corresponding real and imaginary parts from the Kramer-\nKronig relation - 𝜖(𝜔)= 𝜖(𝜔)+𝑖𝜖 (𝜔) [92]. Hence, the light-matter interaction opens \ntremendous opportunities in case of magnetic matter, we considered polarization along three orthogonal \naxes-x,y,z which leads us to the optical anisotropy of the concerned material along the x-direction. The \noptical anisotropy of the doped material constituted in Fig. 15(a, b), which evokes the anisotropy along \nx-axis. In the static limit, 𝜔 → 0, 𝜀_௫(0)=6.05eV when 𝜀_௬= 𝜀_௭= 5.84eV, where 𝜀_ is \n26 \n the real part of dielectric function along 𝑖(𝑥,𝑦,𝑧)-direction. But, the imaginary part of the dielectric \nfunction at the static limit vanishes in all three cases i.e. 𝜀_௫=𝜀_௬=𝜀_௭= 0 eV. Therefore, \nthe average of 𝜀 is 5.91eV and the maximum of 𝜀 lies at 2.86eV along the x-direction polarization \nand 2.92eV for polarization along y,z-directions. The absorption spectra presented in the Fig. 15(c) \ninfers that there is no absorption below the band gap energy nor any band transition appeared.  \n \nFig. 15: DFT extracted (a) Real part of Dielectric Function, (b) Imaginary part of Dielectric Function, (c) \nAbsorption spectra, (d) Dielectric Energy Loss, (e) Real pat of Refractive Index,  \n(f) Extinction coefficient for Ni 0.7Cu0.3 Fe1.94Al0.06O4 nanoparticle.   \nThe dielectric energy loss plotted in Fig. 15(d), showing results that there is a sharp gain starting around \n11.63eV which diminishes around 21.35eV. The sharp peak around 15.6-16.0eV corresponds to the \nplasmon frequency. Refractive Index of the Al3+ (6%) doped Ni 0.7Cu0.3Fe2O4 has been studied along \nthree orthogonal polarization directions 𝑥,𝑦,𝑧 those have been extracted from the complex dielectric 0.0 4.0 8.0 12.0 16.0 20.0-1.00.01.02.03.04.05.06.07.08.0Re( e(w) ) xx\n yy\n zz\nEenergy (eV)(a)\n0.0 5.0 10.0 15.0 20.00.01.02.03.04.05.06.07.0(b)\nImag (e(w))\nEnergy (eV) xx\n yy\n zz\n0.0 5.0 10.0 15.0 20.0 25.00.02.0x1054.0x1056.0x1058.0x1051.0x1061.2x106(c)Absorption (A.U.)\nEnergy (eV) a\n0.0 5.0 10.0 15.0 20.0 25.00.01.02.03.04.0(d)\nEnergy Loss\nEnergy (eV) xx\n yy\n zz\n0.0 5.0 10.0 15.00.00.51.01.52.02.53.0(e)\nhreal\nEnergy (eV) xx\n yy\n zz\n0.0 5.0 10.0 15.0 20.0 25.00.00.20.40.60.81.01.21.41.61.8(f)\nK\nEnergy (eV) xx\n yy\n zz27 \n function, that yields the refractive index of the respective material in a similar fashion. Fig. 15(e,f), \nshows that the above material has a homogeneous refractive index along the polarization direction 𝑦 & 𝑧, \nbut it poses different over the 𝑥-direction, therefore this material showed up with two different indices \n2.37 (along- 𝑥) and 2.31(along- 𝑦,𝑧), hence it encoded itself with the optical birefringence. At limit 𝜔→\n0, the average refractive index of the material 2.34. The extinction coefficient K, from Fig. 15(f), \nvanishes below the band gap, nearly the band gap region it has a small peak and later it progresses \naround 5eV region and thereafter it decreases towards the 20eV with a single peak, and finally it \ndiminishes. Maximum attenuation occurs at 5.16eV( 𝑥-direction) and 4.96eV( 𝑦,𝑧-direction). \nReflectivity was extracted from the refractive index and extinction coefficients, represented in Fig. 16, \nat static limit 𝜔→ 0, there was a 16% reflectivity (in case of 𝑥-direction polarization) and about 15% \nreflectivity for the other cases. Maximum reflectivity was achieved at 5.14eV for the 𝑥-orientation, \nwhich was about 32% , whether the other two cases had maximum reflectivity in the same energy level \nwhich of amount  31%. \n \nFig. 16: Reflectivity of Ni 0.7Cu0.3Fe1.94Al0.06O4 nanoparticle.  \n4. Conclusion \nUsing the sol-gel auto-combustion method, a series of nanocrystalline Ni 0.7Cu0.3AlxFe2-xO4 has been \nproduced with varying Al3+ concentrations (0.00≤x≤0.10, in the step of 0.02) and annealed at 700ºC. X-\nray diffraction patterns confirm that all investigated nanoparticles possess the same cubic single-phase 0.0 5.0 10.0 15.0 20.0 25.0 30.00.000.050.100.150.200.250.300.35R (%)\nEnergy (eV) xx\n yy\n zz28 \n structure with no impurities. The Rietveld refined lattice parameters of the annealed nanoparticles fall \nin the range (0.833-0.835 nm). It shows steady decrease in lattice parameter size with the increment of \nAl3+ except for x=0.02 and 0.08. The average crystallite size of the investigated nanomaterials has been \nmeasured using Scherrer's formula considering the most prevalent XRD peaks (3 1 1), and the values \nare found in the range (61–71 nm) after the refinement. The VSM technique is employed to carry out \nthe magnetic parameters. The Ni-Cu ferrite nanoparticles under this study are shown to be soft \nferrimagnetic against an applied magnetic field. A decreasing trend is observed for shifts in saturation \nmagnetization and Bohr magneton with Al3+ incorporation. The saturation magnetization is observed as \nmaximum for the sample with Al3+ content (x=0.00). Moreover, the remanence ratio (0.00-0.094), \ncoercivity (0.00-93.57 Oe), and the magnetic anisotropy constant (0.00-3612.19 erg/Oe) are varied with \nAl3+ content. 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T homson Avenue, Cambridge, CB3 0HE, United Kingdom \n2Centro Brasileiro de Pesquisas Físicas, Rua Dr Xavi er Sigaud 150, Rio de Janeiro, 22290-180, Brazil \n3Quantum Materials Laboratory, Cambridge, CB3 9NF, U nited Kingdom \n4Physics Department, Missouri University of Science and Technology, Missouri, 65409, United States of A merica \n5Physics Department, Lancaster University, Lancaster , LA1 4YB , United Kingdom \n6School of Chemistry, St. Andrews University, St. An drews, Fife, KY16 9ST, United Kingdom  \n7IMEM-CNR, Parco Area delle Scienze 37/A, 43124 Parm a, Italy  \n8Schools of Chemistry and of Physics and Astronomy, St. Andrews University, St. Andrews, Fife, KY16 9ST , United \nKingdom \n \n \nHexagonal ferrites do not only have enormous commer cial impact (£2 billion/year in sales) \ndue to applications that include ultra-high density  memories, credit card stripes, magnetic \nbar codes, small motors and low-loss microwave devi ces, they also have fascinating magnetic \nand ferroelectric quantum properties at low tempera tures.  Here we report the results of \ntuning the magnetic ordering temperature in PbFe 12-xGa xO19  to zero by chemical substitution \nx. The phase transition boundary is found to vary as  \u0001\u0002~\u0004\u0005 − \u0007 \u0007 \b⁄ \n\u000b/\r with xc very close to \nthe calculated spin percolation threshold which we determine by Monte Carlo simulations, \nindicating that the zero-temperature phase transiti on is geometrically driven. We find that \nthis produces a form of compositionally-tuned, insu lating, ferrimagnetic quantum criticality.  \nClose to the zero temperature phase transition we o bserve the emergence of an electric-dipole \nglass induced by magneto-electric coupling.  The st rong frequency behaviour of the glass \nfreezing temperature Tm has a Vogel-Fulcher dependence with Tm finite, or suppressed below \nzero in the zero frequency limit, depending on comp osition x.  These quantum-mechanical 2 of 17  \n properties, along with the multiplicity of low-lyin g modes near to the zero-temperature phase \ntransition, are likely to greatly extend applicatio ns of hexaferrites into the realm of quantum \nand cryogenic technologies. \n \nM-type hexagonal ferrites (hexaferrites) including BaFe 12 O19 , SrFe 12 O19  and PbFe 12 O19  are popular \nmagnetic materials for their use in a wide range of  applications [1, 2]. Moreover, they also have \ninteresting magnetic and ferroelectric properties a t low temperatures [3, 4].  Here we study the \neffects of tuning the magnetic ordering (Néel) temp erature all the way to zero resulting in a \ngeometrically driven zero-temperature phase transit ion of the underlying spin system and the \nemergence of an electric-dipole glass.  These prope rties are expected to be important for a wide \nrange of advanced quantum and cryogenic application s including, for example, electro-caloric and \nmagneto-caloric refrigeration,  and quantum memory devices, as the materials can be  readily \ncontrolled by magnetic fields and voltage gates. \n \nBaFe 12 O19 , SrFe 12 O19  and PbFe 12 O19  crystalise in the magnetoplumbite structure and ar e Lieb-\nMattis [5] ferrimagnets with Néel temperatures of a pproximately TN ≈ 720K and saturated \nmagnetisations in the low temperature limit of 20 µB per double formula unit [6].  The crystal \nstructure can be seen in the right inset to Fig. 1a  which shows a double unit-cell.  The underlying \nspin structure comprises collinear anti-ferromagnet ic order below TN with a total of 16 spins \npointing up and 8 spins pointing down located on Fe3+  sites per double unit-cell resulting in \nferrimagnetism.  The Fe 3+  ions, each in the high S = 5/2 spin state, are located on five sub-lattices \nas follows: six spin-up on octahedral sub-lattice ( k), one spin-up on octahedral sub-lattice (2a), one  \nspin-up on pseudo-hexahedral sub-lattice (2b), two spin-down on tetrahedral sub-lattice (4f IV ) and \ntwo spin-down on octahedral sub-lattice (4f VI ) [6].  The M-type hexaferrites are n-type \nsemiconductors [7] with bandgaps of Eg ≈ 0.63 eV and rather heavy electrons and holes: m(light e) \n= 5.4 me; m(heavy e) = 15.9 me; m(light h) = 10.2 me; m(heavy h) = 36.2 me and highly anisotropic \nconductivity.  For electric fields applied normal t o the c-axis, the electrical conductivity is circa \nfifty times greater than along c.  An example of the hexahedral (bi-pyramid) sites is shown in the \nright inset to Fig. 1a where its faces have been sh aded in grey.  All of the bi-pyramids comprise a 3 of 17  \n single Fe 3+  ion at the centre surrounded by five O 2- ions at the corners.  The vibration of these \npositive ions within their negatively charged oxyge n enclosures is along the c direction and \ngenerates polar transverse-optic A 2u -symmetry phonon modes.  The lowest of these is obs erved \nto drop in frequency at long wavelengths ( q = 0) to 42 cm -1 (5.2meV) as T approaches zero resulting \nin an incipient ferroelectric state [8, 9].  This s tate has been investigated in recent work with \nevidence of uniaxial ferroelectric quantum critical  behaviour along the c direction [3] and anti-\nferroelectric frustration on the triangular lattice  of dipoles in the a-b plane [4]. \n \nIn this paper we study a different part of the phas e diagram close to the insulating magnetic zero \ntemperature critical point, achieved by suppressing  TN to zero by randomly substituting Ga ions for \nthe Fe ions in PbFe 12-xGa xO19 . As x increases, the lattice of spins is diluted as the non-magnetic \ngallium ions act as quenched spinless impurities. T his results in a drop in TN as determined by \nMössbauer and magnetic measurements [6].  The Mössb auer data also indicate that the Ga ions  \ndistribute themselves with nearly equal probability  in all the available sublattices, at least for not  \ntoo large x. As shown in Fig. 1, by extrapolating the trend to  T = 0K we find that the critical value \nof x for which TN goes to zero is x = xc ≈ 8.6.   \n \nThe zero temperature transition between the ferrima gnetic and nonmagnetic ground states as a \nfunction of iron concentration can either be geomet rically driven or driven by quantum fluctuations.  \nIn the first scenario, the transition is a percolat ion transition.  It occurs when the iron concentrat ion \nfalls below the percolation threshold pc of the lattice of iron sites where p is the probability of a \nsite containing an iron atom [related to x by x = 12(1-p) ].  Long-range magnetic order is then \nimpossible because the iron atoms form disconnected  finite-size clusters.  In the second scenario, \nthe zero-temperature phase transition occurs before  the iron concentration falls below pc because \nthe magnetic order is destroyed by quantum fluctuat ions of the iron spins. \n \nTo help distinguishing the two scenarios, we have d etermined the percolation threshold of the \nlattice of iron atoms in the M-type hexagonal ferri tes by means of computer simulations.  This 4 of 17  \n requires knowledge about the connectivity of the ir on atoms, i.e., about the exchange interactions \nbetween the iron atoms on the five different sub-la ttices. The exchange interactions in BaFe 12 O19  \nwere determined from phenomenological fits of exper imental sub-lattice magnetization data in \nRefs. [10] and [11].  More recently, these interact ions were also calculated from first principles \n[12]. Even though the exact values of the interacti ons differ between these papers, they agree on \nthe basic structure:  The four dominating interacti ons are between the following sub-lattices: 2a-\n4f IV , 2b-4f VI , 12k-4f IV , and 12k-4f VI  (see Fig 1a). These interactions are antiferromagn etic, and they \nare not frustrated because each couples a spin-up a nd a spin-down sub-lattice.  All other \ninteractions are significantly weaker, and they are  frustrated because they couple spins in the same \nsub-lattice or in different sub-lattices with the s ame spin direction.  The exchange interactions in \nall the M-type hexagonal ferrites, Pb, Sr and Ba ar e expected to be very similar.  In our percolation \nsimulations we have therefore only included the bon ds corresponding to the four dominating \n(unfrustrated) interactions.  The weaker frustrated  interactions may become important at dilutions \nclose to the percolation threshold and at low tempe ratures.  Because they are frustrated, they are \nexpected to suppress the ferrimagnetic order compar ed to a scenario that includes only the leading \nunfrustrated ones.  We have further assumed that al l iron sites have the same occupation \nprobability (i.e., the gallium doping is completely  random). \n \nTo find the percolation threshold for the thus defi ned lattice of iron atoms, we have implemented a \nversion of the fast Monte Carlo algorithm due to Ne wman and Ziff [13].  We have studied systems \nwith sizes of up to 200x200x200 double unit cells ( 192 million Fe sites), averaging over several \nthousand disorder realizations for each size.  The percolation threshold is determined from the \nonset of a spanning cluster.  Extrapolating the res ults to infinite system size, yields pc = 0.2628(5) \nwhere the number in brackets is the error of the la st digit (estimated from the very small statistical  \nerror of the data and the robustness of the extrapo lation).  In the material PbFe 12-xGa xO19 , this \ncorresponds to a gallium concentration  x = 8.846(6).  We note that the percolation threshol d for \nour realistic model of the magnetic interactions in  the hexaferrites is also very close to the thresho ld \nfor a simple three-dimensional hexagonal stacked st ructure [14, 15].  Fig. 1b shows a projection 5 of 17  \n of the relevant Fe ions into the a-b plane for the parent compound (left) and a percolating magnetic \ncluster (right) for x close to xc. \n \nThe experimentally observed value of xc = 8.6 is very close to that determined above from our \nmodel calculations suggesting that the zero tempera ture phase transition is predominantly \ngeometrically driven by the percolation of magnetic  ions through the crystal.  The closeness of \nthe measured and calculated values of xc is further evidence, along with the results of Mös sbauer \nexperiments [6] mentioned above, that Ga ions are s ubstituted randomly onto the Fe sites of the \nparent compound. For x > xc, magnetic long-range order is impossible. This is confirmed by the \nmeasured heat capacity of a sample with x = 9, shown in Fig. 2a, which does not feature a ph ase \ntransition down to the lowest measured temperatures . However, due to statistical fluctuations in the \ndistribution of the Ga ions, we expect disconnected  Fe-rich clusters of magnetic order to exist \nwithin a background rich in the non-magnetic Ga ion s, resulting in a paramagnetic Griffiths phase \n[16, 17].  The weak hysteresis observed in the magn etisation-field curve of the x=9 sample, shown \nin Fig. 2b, supports this picture as it can be attr ibuted to the contribution to the uniform \nmagnetisation from the ferrimagnetic clusters. The experimental confirmation of the paramagnetic \nGriffiths phase will require further study.   At th e percolation threshold, and for x < xc, there is \nan “infinite percolation cluster” that spans the en tire crystal, resulting in a finite value of the gl obal \nTN and long-range magnetic order.  The point T = 0K and x = xc can be understood as a multi-\ncritical point (MCP) because it combines the geomet rical criticality of the zero-temperature \npercolation transition (characterised by the percol ation critical exponents [14]) and the thermal \ncriticality of the finite-temperature phase boundar y (characterised by the usual thermodynamic \ncritical exponents).  Despite extensive theoretical  work on classical and quantum magnetic \npercolation phase transitions, there are relatively  few experimental examples.  Notable examples \ninclude work on the square lattice two-dimensional La 2Cu 12-z(Zn,Mg) zO4 system [18] and on \ntransition metal halides [19].  PbFe 12-xGa xO19  is unique in that it is a three-dimensional hexago nal \nmagneto-electric system that can be successfully tu ned up to and beyond the percolation threshold \nwith a novel phase transition boundary TN (x) as discussed below. The fact that the measured va lue 6 of 17  \n of xc is a little less than that determined from calcula tions could be due to a small degree of Ga \nclustering or alternatively due to effects of quant um fluctuations arising from the sub-dominant \nfrustrated magnetic interactions referred to above.  \n \nAs shown in Fig. 1 we find that the shape of the ph ase transition boundary follows a striking \n\u000e\u000f~\u00041 − \u0011/\u0011 \u0012\n\u0013/\u0014 dependence over the entire concentration range fro m x = 0 to x = xc. Where \ndoes this power law come from? As the phase boundar y starts at high temperatures ( \u000e\u000f\u00040\n=\n720\u0019), one might expect \u000e\u000f\u0004\u0011\n to follow the form predicted by classical percolat ion theory, \n\u001a\u0004\u000e\u000f\n~ \u0004\u0011\u0012− \u0011\n\u001b where \u001a\u0004\u000e\n is the appropriate spin Hamiltonian temperature sc aling function.  \n\u001a\u0004\u000e\n~\u001c\u001d\u0013\u001e \u001f  ! ⁄ for a system with Ising symmetry and \u001a\u0004\u000e\n~\u000e for a system with continuous (e.g. \nHeisenberg) symmetry and \" is a crossover exponent usually defined as the rat io of percolation \nand thermal correlation length critical exponents \" = # $/#! [14, 19-21].  Over the range of \ntemperatures and chemical compositions tested so fa r, our measured \u000e\u000f\u0004\u0011\n curve is quite different \nfrom the predictions of these classical models in w hich usually \" ≥ 1. \n \nAlternatively, the shape of the phase boundary may be governed the zero-temperature quantum \nphase transition occurring as the composition x is tuned through xc. Quantum phase transitions are \nsubtly different from the more familiar classical p hase transitions occurring as a function of \ntemperature at high temperatures. In the present ca se, the zero entropy state at T = 0 K (or at \nsufficiently low temperatures for the third law of thermodynamics to apply) of long-range \nferrimagnetic order for x < xc, is transformed into a paramagnetic state with no conventional \nmagnetic ordering for x > xc.  This zero temperature state still has zero entro py (assuming a non-\ndegenerate ground state). Magnetic quantum phase tr ansitions are a highly active area of research.  \nDepending on specific material details (precise lat tice geometry) and dimensionality, they can lead \nto a rich tapestry of exotic phases such as dimer s tates, valence bond solids, spin glasses, quantum \nspin liquids and topological entities, such as spin  spiral states and others [22-26]. In the presence \nof quenched disorder, quantum phase transitions can  give rise to smeared phase transitions as well \nas to the above-mentioned Griffiths phases [16] tha t are characterized by singular low-temperature 7 of 17  \n thermodynamic functions with gapless excitations ov er a range of tuning parameter variables [17, \n25, 27, 28] . \n \nIn ‘clean’ quantum critical systems, where the inte ractions are tuned, for example, by lattice density  \nor magnetic field, the effects of quantum criticali ty can often be felt over a wide range of \ntemperatures and tuning parameters below a temperat ure scale T*  set for example by the spectrum \nof magnetic excitations ( \u000e∗= ℏ(∗/)*). In contrast to classical critical points, thermo dynamic \nproperties near quantum critical points are affecte d by fluctuations of the order-parameter field in \nspace and time. This implies that thermodynamic qua ntities are functions of the dynamical \nexponent z characterizing the spectrum Ω\u0004,\n~,- of modes close to the critical point where q is \nthe wavevector.  For insulators where the modes are  typically propagating (heavily under-damped) \nΩ\u0004,\n is a frequency-wavevector dispersion of the normal  modes, whereas for metals where the \nmodes are typically dissipative (heavily over-dampe d) Ω\u0004,\n is a relaxation rate spectrum.  If \n. + 0 ≥ 4  (where  d is the dimension of space, thermodynamic quantitie s in the quantum critical \nregime may be calculated by the one-loop (Hartree) approximation used in renormalization group \nmodels and self-consistent-field models of quantum criticality[23, 29-41].  Close to the quantum \ncritical point, this yields the magnetic susceptibi lity 2~1/\u000e34 where the (thermal) critical \nexponent 5!= \u0004. + 0 − 2\n 0 ⁄.  The critical temperature is found to vary as \u000e\u0012~\u00041 − 6 6 \u0012⁄ \n7/3 4 \nwhere g is the (non-disorder inducing) quantum tuning para meter.  We note that the value 2/3 of \nthe phase transition boundary exponent found in Fig . 1 would be consistent with a dynamical \nexponent z = 2 and d = 3.  Such an exponent z in a magnetic insulator may arise for example from  \nthe dynamics of spin precession [23, 25, 42].   \n \nHowever, the situation is different in the presence  of strong disorder as introduced, for example by \nthe dilution of the magnetic lattice. According to the Harris criterion [43], disorder is typically a \nrelevant perturbation at a quantum phase transition  and therefore destabilizes the clean critical \nbehavior. For the specific case of magnetic percola tion quantum phase transitions, theories predict \nthat thermodynamic functions either depend on a new  dynamical exponent 0′ defined in terms of 8 of 17  \n the fractal dimension Df of the percolation transition [14] and the dynamic al exponent z of the clean \nquantum phase transition [27, 28] or they show even  more unconventional activated scaling \nbehavior 20 . However, in both cases, the phase boundary is pre dicted to follow the classical behavior \ndiscussed above, in disagreement with our observati on.  \n \nThe origin of the unusual phase transition boundary  with a 2/3 power law may be due to the \ninteraction of magnetic and ferroelectric degrees o f freedom, or the combined effects of quantum \nfluctuations (arising from the frustrated magnetic interactions referred to above) and those of the \ngeometrically-driven percolation transition.  In a future study the relative importance of these \neffects may be separated by employing further tunin g parameter such as pressure or field in addition \nto chemical composition. \n \nWe now turn to the results of measurements of the d ielectric susceptibility for samples with x close \nto xc.  The dielectric susceptibility ε probes the electrical dipole response of the syste m and typical \nresults are shown in Fig. 3.  We find that for samp les measured close to TN the dielectric function \nexhibits a frequency dependent peak at a temperatur e Tm(f) in the real part ε’(T), which arises from \nmagneto-electric coupling, presumably via striction , plus a peak in the imaginary part ε”(T) at \nslightly lower temperatures.  Spin-phonon coupling in BaFe 12 O19  has been reported previously \nfrom Raman spectroscopy [44-46], and detailed dynam ics given by Fontcuberta’s group [47-49].  \nA model for dielectric loss at Néel temperatures ha s been given by Pir č et al [50], and their graph \nof ε’(T) and ε”(T) for low-frequency probes is given in Fig. 6 of Re f. [50] for realistic parameters, \nassuming a magneto-electric interaction through str iction. The data indicate that polarized clusters \nform around TN with glassy dynamics which freeze at Tm(f).  Such a state is known as a \nferroelectric relaxor or electric-dipole glass [51,  52].  Relaxor dynamics are characterized by a \nbroad distribution of relaxation times, and the fre ezing process at Tm(f) is associated with the \ndivergence of the longest relaxation time.  The pre sent data satisfy a Vogel-Fulcher relationship \nas in Fig. 3c with frequencies f from 100 Hz to 1 MHz.  The glass freezing temperat ure Tf = \nTm(f→0) is found to be finite for samples with TN > 0 and suppressed below zero for samples with 9 of 17  \n paramagnetic ground states x > xc.  The electrically polarized clusters which form t he glass state \nare likely supported within the magnetic Fe rich cl usters.  These magnetic clusters are of \ndiminishing size as the Ga concentration is increas ed suppressing Tm(f).  Since the glass freezing \ntemperature can be tuned through zero with frequenc y and composition, future studies may involve \nmodels of a quantum dipole glass (quantum relaxor) analogous to those studied in spin systems.  \nBoth the ferroelectric-glass and the magnetic clust ers will contribute to the heat capacity which is \nlikely to be the origin of the non-cubic temperatur e dependence as observed over any abscissa \nrange in Fig. 2a. \n \nIn summary, randomly substituting nonmagnetic Ga io ns for magnetic Fe ions in the ferrimagnetic \nhexagonal ferrite PbFe 12-xGa xO19  suppresses the Néel temperature to zero at a criti cal composition \nxc close to the magnetic ion percolation threshold as  calculated for the hexaferrite structure. The \nphase transition boundary features a \u000e\u000f~\u00041 − \u0011/\u0011 \u0012\n\u0013/\u0014 dependence over a wide range of T and \nx. This has not been observed experimentally in other percolation systems, and the origin of this \nbehavior is currently unexplained by theory. Close to xc the system develops magnetic clusters and \nan electric-dipole glass with Vogel-Fulcher behavio ur.  The magneto-electric effect raises the \nexciting possibility of manipulating the low temper ature magnetic phases by electric fields (voltage \ngates) and the electric-dipole glass by magnetic fi elds.  Future experimental and theoretical \nstudies are likely to be key in elucidating the exo tic spin and electric-dipole states and their \napplications expected to arise close to zero temper ature phase transitions in hexaferrites. \n \nMETHODS  \n \nM-type hexaferrite samples were prepared by the flu x method. The raw powders of PbCO 3, Fe 2O3, \nGa 2O3 and fluxing agent Na 2CO 3 were weighed in the correct molar ratio and mixed well. The \nmixed raw powder was put in a platinum crucible and  heated to 1250°C for 24 hours in air, then \ncooled down to 1100°C at a rate of 3°C/min and fina lly quenched to room temperature. The samples \n(ca. 2 mm across) were characterized by x-ray diffr action at room temperature using a Rigaku X-10  of 17  \n ray diffractometer.  Heat capacity was measured as a function of temperature using the relaxation \ntechnique on a 5mg sample.  The low temperature DC magnetisation was measured using a \nSQUID magnetometer up to fields as high as 5 T.  Th e dielectric measurements were carried out \nin a liquid-cryogen free cryostat at temperatures a s low as 6 K. Silver paste was painted on the \nsurfaces of a thin plate of each crystal and an And een-Hagerling, Agilent 4980A and QuadTech \nLCR instruments were used to measure the dielectric  susceptibility at frequencies in the range 100 \nHz to 1 MHz. \n \nACKNOWLEDGEMENTS \n \nWe would like to thank M. Continentino and G. G. Lo nzarich for useful help and discussions.  \n \nCONTRIBUTIONS \n \nS.E.R. and J.F.S. designed the project.  S.E.R., A. T.J., W.G., J.O., F.D.M., N.L., E.B.S., B.E.W., \nT.V. and J.F.S. collected the data and interpreted the results.  B.E.W. prepared the samples. T.V. \nperformed the numerical calculations.  S.E.R., T.V.  and J.F.S. analysed the data and wrote the \npaper. \n \nFUNDING  \n \nS.E.R. and E.B.S. acknowledge support from a CONFAP  Newton grant. T.V. acknowledges support \nfrom the NSF under Grant No. DMR-1506152. \n \n \n \n \n 11  of 17  \n FIGURES  \n \n \n \n \nFigure 1 –Magnetic phase diagram and crystal struct ure of PbFe 12-xGa xO19 .  The Néel \ntemperature, TN ≈ 718K, in PbFe 12 O19  separating paramagnetic and ferrimagnetic phases i s \nsuppressed via non-magnetic Ga substitution and tun ed through a geometrically driven percolation \nphase transition located at T = 0K and x = x c ≈ 8.6 as shown in (a).  x = 8.6 is close to the calculated \npercolation threshold x = 8.85 referred to in the main text for the hexafer rite structure.  The right \ninset shows a double unit-cell of PbFe 12 O19  as explained more fully in the main text with the \ncrystallographic c direction indicated by the arrow.  Values of TN were determined by Mössbauer \n(Néel Order)\n(C)FerrimagneticParamagnetic\nParamagnetic\n(Magnetic clusters / \nshort-range order)\nb) 12  of 17  \n and magnetisation measurements [6].  The value of TN as a function of Ga x in the related materials \nBaFe 12-xGa xO19  and SrFe 12-xGa xO19  differ from those shown above for PbFe 12-xGa xO19  by only a \nfew per cent.  The main figure shows \u000e\u000f\u0014/\u0013 (blue dots) plotted against x and the straight line is a \nbest fit to the data with an equation of the form \u000e\u000f\u0004718 K\n⁄ = \u00041 − \u0011/\u0011 \u0012\n\u001b with critical Ga \nconcentration xc = 8.56, and the power-law exponent determined as φ = 0.67 ± 0.02, i.e. 2/3.  The \nregion labelled (A) is where uniaxial quantum criti cal ferroelectric fluctuations have recently been \nreported in BaFe 12 O19 and SrFe 12 O19  [3, 9].  The regions labelled (B) and (C) are wher e an \nelectric-dipole glass state (ferroelectric relaxor)  is observed, induced by magneto-electric coupling \nas explained in the main text and later figures.  T he dashed line separates the classical \nparamagnetic phase and the paramagnetic phase compo sed of disconnected clusters of \nferrimagnetic order.  The region labelled (C) is wh ere one might expect to search for exotic spin \nand thermodynamic states [27, 28].  In (b) the left  image shows a projection into the a-b plane of \nthe relevant magnetic ions (one spinel block shown)  used in the percolation calculation explained \nin the main text for the un-doped parent compound P bFe 12 O19 .  The right image shows a \npercolating magnetic cluster under the conditions o f magnetic dilution in PbFe 12-xGa xO19  with x \nclose to xc. \n \n \n \n \n \n 13  of 17  \n  \nFigure 2 – Thermal and magnetization measurements i n PbFe 3Ga 9O19  (x = 9).  The heat \ncapacity as a function of temperature in (a) demons trates along with Mössbauer experiments [6] \nthe absence of a bulk phase transition and thus no long-range order in a sample with x > xc.  The \nweak hysteresis measured at 2K in (b) indicates the  contribution to the uniform magnetization from \n‘rare regions’ - small disconnected ferrimagnetic c lusters - in the paramagnetic phase at low \ntemperatures.  Close to the zero temperature percol ation phase transition the magnetic clusters are \n14  of 17  \n randomly distributed in space but perfectly correla ted in time leading to the possibility of singular \nthermodynamic functions over a range of tuning para meter variables [20, 28]. \n \n15  of 17  \n  \n \nFigure 3 - Real and imaginary parts of the dielectr ic constant (a, b) and Vogel-Fulcher plots \n(c), for PbFe 12-xGa xO19  showing dipole-glass behaviour.  Figures (a) and (b) show the real ε and \nimaginary parts ε’’  of the dielectric constant measured at different f requencies plotted against \ntemperature T for samples of PbFe 12-xGa xO19  with x = 8.4 and x = 9.6 respecitvely.  A Vogel-\nFulcher fit to the data of the peak temperature Tm versus measurement frequency f is shown for the \nsame two samples in (c). 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B 76, 020101(R) (2007). \n \n "    },    {        "title": "1702.06033v1.Effect_of_annealing_on_the_magnetic_properties_of_zinc_ferrite_thin_films.pdf",        "content": "arXiv:1702.06033v1  [cond-mat.mtrl-sci]  20 Feb 2017Eﬀect of annealing on the magnetic properties of zinc ferrite t hin\nﬁlms\nYogesh Kumara,b, Israel Loritea, Michael Lorenza, Pablo Esquinazia, Marius Grundmanna\naFelix-Bloch Institute for Solid State Physics, Fakult¨ at f ¨ ur Physik und Geowissenschaften, Linn´ estrasse 5, 04103\nLeipzig, Germany\nbSolid State Physics Division, Bhabha Atomic Research Centr e, Mumbai 400 085, India\nAbstract\nWe report on the magnetic properties of zinc ferrite thin ﬁlm deposited on SrTiO 3single crystal\nusing pulsed laser deposition. X-ray di ﬀraction result indicates the highly oriented single phase\ngrowth of the ﬁlm along with the presence of the strain. In com parison to the bulk antiferro-\nmagnetic order, the as-deposited ﬁlm has been found to exhib it ferrimagnetic ordering with a\ncoercive ﬁeld of 1140 Oe at 5 K. A broad maximum, at ≈105 K, observed in zero-ﬁeld cooled\nmagnetization curve indicates the wide grain size distribu tion for the as-deposited ﬁlm. Reduc-\ntion in magnetization and blocking temperature has been obs erved after annealing in both argon\nas well as oxygen atmospheres, where the variation was found to be dependent on the annealing\ntemperature.\nKeywords: Annealing, Magnetization, Thin ﬁlms, Epitaxial growth, Ox ygen vacancies\n1. Introduction\nSpinel ferrites, AB 2O4, generally exhibit cubic spinel structure, where oxygen an ions re-\nside at fcc lattice sites and cations occupy the tetrahedral ly and octahedrally coordinated in-\nterstitial sites forming A and B sublattices [1]. These mate rials can have normal, inverse and\nmixed spinel structures and possess di ﬀerent kind of magnetic characters (ferrimagnetic, antifer -\nromagnetic and paramagnetic) depending on the nature of cat ions and their distribution among\ndiﬀerent sites[1, 2]. The zinc ferrite (ZnFe 2O4) has been proposed to be a candidate for spin-\ntronic applications, and various studies have been carried out on its magnetic and electrical\nproperties[2, 3, 4, 5, 6, 7, 8, 9]. Bulk zinc ferrite, in perfe ct oxygen stoichiometry, is known\nto exhibit normal spinel structure with all Zn2+and Fe3+ions occupying the tetrahedral and oc-\ntahedral sites, respectively and exhibits antiferromagne tic ordering below 10.5 K [2]. However,\nnanoparticles [4, 5] and thin ﬁlms of ZnFe 2O4[6, 7, 8, 9] are reported to have the ferrimagnetic\norder. This is normally attributed to the placement of iron a nd zinc ions at both the sites, altering\nthe spinel structure from normal to mixed state and inducing strong negative JABinteractions\nbetween iron ions [10, 11, 12]. Apart from this redistributi on of cations, the oxygen vacancy\nconcentration is also believed to play a crucial role in cont rolling the magnetic properties of zinc\nEmail address: lucky1708@gmail.com (Yogesh Kumar)\nPreprint submitted to Materials Letters September 22, 2018ferrite. Greater magnetic response has been observed for th e samples prepared under low oxygen\npartial pressures [6, 13]. In fact, it is still not establish ed whether the observed ferrimagnetism is\nonly related to cation (partial) inversion and /or oxygen vacancies. In this paper, we present mag-\nnetic studies of ZnFe 2O4thin ﬁlm deposited on SrTiO 3(100) substrate, as zinc ferrite is known\nto grow epitaxially on these substrates [3, 9]. Also, SrTiO 3single crystals do not have mag-\nnetic impurities and annealing at the studied temperatures is not known to aﬀect their magnetic\nproperties. Hence, SrTiO 3is a good choice as a substrate to study the changes in the magn etic\nproperties of the deposited ﬁlms.\n2. Experimental\nPulsed laser deposition (KrF excimer laser: λ=248 nm) was used to grow a thin ﬁlm ( ∼20 nm\nthick) of zinc ferrite on SrTiO 3(100) substrate kept at 773 K and under a relatively low oxyge n\npartial pressure of 6 ×10−5Torr. Prior to deposition, the base pressure of the chamber w as re-\nduced to∼7.5×10−8Torr. The crystalline structure of the deposited ﬁlm was stu died by x-ray\ndiﬀraction (XRD)θ−2θscan, performed using the Phillips X’Pert Bragg-Brentano d iﬀractome-\nter with Cu Kαradiations. Temperature and ﬁeld dependent magnetization measurements were\ncarried out using a MPMS-7 superconducting quantum interfe rence device magnetometer from\nQuantum Design. Zero-ﬁeld cooled (ZFC) and ﬁeld cooled (FC) curves were recorded with an\napplied ﬁeld of 1000 Oe. To study the e ﬀect of annealing environment and temperature on mag-\nnetic properties, the as-deposited ﬁlm was ﬁrst annealed in argon at 773, 823, and 873 K and\nsubsequently in oxygen at similar temperatures. The magnet ization measurements were carried\nout after each step.\n3. Results and Discussions\nX-ray diﬀractionθ−2θpattern of the pristine zinc ferrite ﬁlm is shown in Fig. 1, wh ich\nindicates the epitaxial growth of the ﬁlm on SrTiO 3without presence of any kind of secondary\nphase. The out of plane lattice parameter has been calculate d to be∼8.47 Å. It should be noted\nthat the lattice parameter is higher than that of bulk value ( 8.44 Å)[14], which may be due to\nthe substrate induced strain during the thin ﬁlm growth. Ave rage crystallite size in the deposited\nﬁlm is∼20 nm, as calculated using the Scherrer’s formula. No change has been observed in the\nlattice parameter and crystallinity of the ﬁlm, within the r esolution limit of the instrument, even\nafter the ﬁnal annealing in the oxygen (not shown here).\nField dependent magnetization at 5 K and 300 K, and ZFC-FC cur ves obtained for the pris-\ntine sample are shown in Fig. 2. The data has been presented he re after subtracting the dia-\nmagnetic contribution from the substrate. In comparison to the antiferromangetic order found\nin bulk zinc ferrite grown under optimal conditions, the obs ervation of S-shaped M-H curves in\nthe as-deposited ﬁlm indicate the existence of ferrimagnet ic order. Such kind of magnetic be-\nhaviour has also been observed previously for the thin ﬁlms o f zinc ferrite [6, 7, 8, 9]. Saturation\nmagnetization ( Ms) at 5 K has been found to be around 20.5 emu /g, equivalent to a magnetic\nmoment of∼0.44µb/Fe, with a coercive ﬁeld of ≈1140 Oe. At 300 K, along with negligible\ncoercive ﬁeld, Msis around 12.1 emu /g corresponding to a magnetic moment of ∼0.26µb/Fe\n(see Fig. 2(b)). In comparison to 300 K, ﬁlm is magnetically m uch harder at low temperature,\nindicating the presence of some sort of blocking mechanism, which is also supported by the ir-\nreversibility in the ZFC-FC curves at lower temperatures. A part from this, a broad maximum\n2/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s42/s83/s84/s79 /s32/s40/s49/s48/s48/s41\n/s83/s84/s79 /s32/s40/s52/s48/s48/s41/s90/s110/s70/s101\n/s50/s79 \n/s52/s32/s40/s56/s48/s48/s41/s83/s84/s79 /s32/s40/s51/s48/s48/s41\n/s32/s32\n/s32/s32/s112/s114/s105/s115/s116/s105/s110/s101/s32/s102/s105/s108/s109/s83/s84/s79 /s32/s40/s50/s48/s48/s41/s90/s110/s70/s101\n/s50/s79 \n/s52/s32/s40/s52/s48/s48/s41/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s114/s97/s114/s121/s32/s117/s110/s105/s116/s115/s41\n/s50 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s42\nFigure 1: X-ray di ﬀraction scans of pristine zinc ferrite ﬁlm grown under an oxy gen partial pressure of 6 ×10−5Torr.\nPeaks marked with ”*” correspond to the tungsten and related compounds from xrd tube.\nis present in the ZFC curve. Here, the magnetization ﬁrst inc reases with the temperature and\nafter∼105 K, known as blocking temperature ( TB), it reduces suggesting a wide particle size\ndistribution for the as-deposited ﬁlm.\nIt has previously been observed that the oxygen vacancies pl ay a major role in the mag-\nnetic properties of zinc ferrite, with larger magnetic repo nse for ﬁlms grown under lower oxygen\npartial pressures [6]. Hence, at ﬁrst glance, it seems that t he observed magnetization can be\nexplained on the basis of oxygen vacancies, as the ﬁlm is grow n at quite low oxygen partial pres-\nsure of 6×10−5Torr. To verify this assumption and to avoid any removal of ox ygen vacancies,\nwe ﬁrst annealed the ﬁlm at 773, 823 and 873 K in the argon gas. A fter these treatments, we\nperformed annealing in oxygen environment to remove the oxy gen vacancies. Magnetization\nmeasurements were carried out after each treatment. It is im portant to mention here that in each\ncase the annealing temperature /atmosphere was changed only after getting the saturation e ﬀect\nat the current temperature, which was studied by the magneti zation measurements with multiple\ntreatments under similar conditions. In most cases, satura tion eﬀect was achieved after 4 hours\nof annealing. Interestingly, with annealing in the argon at mosphere, magnetization of the sample\nhas been found to be reduced with the rise in the temperature. The saturation magnetization ( Ms)\nis only aﬀected at 300 K, while at 5 K remanence ( Mr) reduces after annealing (see Fig. 3(a)).\nZFC-FC curves still exhibit the irreversibility at low temp eratures; however, the blocking tem-\nperature decreases after each treatment (shown separately in Fig. 3(d)). Such e ﬀect can not be\nexplained on the basis of oxygen vacancies as the argon-anne aling is not supposed to remove\n3these vacancies. However, some kind of structural changes c an also take place during annealing\nat higher temperatures. In fact, in one of our recent article , we have shown that the substrate\ntemperature plays a major role on occupancy of tetrahedral s ites by iron ions thereby a ﬀecting\nthe magnetic properties [9]. It was found that with the rise i n the substrate temperature, zinc\nferrite tends to have normal spinel structure with reduced m agnetization. Hence, under argon-\nannealing, reduction in magnetization for studied sample w ith rise in temperature can also be\nexplained on the basis of stabilization of normal spinel str ucture.\nAfter argon-annealing, the ﬁlm was annealed in oxygen and ob tained results are plotted in\nFig. 3. Magnetization was reduced with annealing in oxygen a s well, again showing the major\nchanges in the saturation values at 300 K. Also, the blocking temperature decreases further with\ndependence on temperature (see Fig. 3(d)), along with the ap pearance of irreversibility in ZFC-\nFC curves even after the ﬁnal annealing step. As the annealin g under oxygen was performed\nat temperatures not higher than that used for argon-anneali ng, hence, we can ignore the possi-\nbility of further redistribution of iron ions at di ﬀerent sites. Thus, these changes can be easily\nexplained on the basis of removal of oxygen vacancies, which is consistent with the reports on\nzinc ferrite ﬁlms where less magnetization has been observe d for the samples prepared under\nhigher oxygen partial pressures [6]. Hence, on the basis of o ur observations, we can say that the\nmagnetic properties of zinc ferrite ﬁlms are a ﬀected both by structural changes through cation\ninversion as well as oxygen vacancies, and can be controlled by the annealing temperature and\nthe atmosphere. It is important to mention here that similar kind of magnetization study was also\nperformed on other comparable zinc ferrite ﬁlms, which exhi bited similar results. However, for\nthe sake of brevity this data is not presented here.\n4. Conclusions\nIn summary, we have studied the magnetic properties of highl y oriented zinc ferrite thin ﬁlm\ngrown on SrTiO 3single crystals and their dependence on annealing in the tem perature range\nof 773-873 K and under di ﬀerent atmospheres. Pristine ﬁlm exhibits ferrimagnetic or der and a\nbroad zero ﬁeld cooled magnetization curve. Magnetization has been found to be reduced after\nannealing in argon and oxygen with larger changes observed f or measurements at 300 K. Our\nobservations demonstrate that the magnetic properties of z inc ferrite ﬁlms can be controlled by\nvarying the annealing temperature under di ﬀerent environments.\nAcknowledgments\nThis work was supported by the DFG within the Collaborative R esearch Center (SFB 762)\n“Functionality of Oxide Interfaces”. One of the authors (YK ) would like to thank Department of\nScience and Technology (DST), India for the ﬁnancial suppor t. We also acknowledge Annette\nSetzer for the technical support.\nReferences\n[1] V . Brabers, Chapter 3 progress in spinel ferrite researc h, in: Handbook of Magnetic Materials, Elsevier BV, 1995,\npp. 189–324.\n[2] W. Schiessl, W. Potzel, H. Karzel, M. Steiner, G. M. Kalvi us, A. Martin, M. K. Krause, I. Halevy, J. Gal, W. Schfer,\nG. Will, M. Hillberg, R. Wppling, Magnetic properties of the ZnFe 2O4spinel, Phys. Rev. B 53 (14) (1996) 9143–\n9152.\n4[3] M. Lorenz, M. Brandt, K. Mexner, K. Brachwitz, M. Ziese, P . Esquinazi, H. Hochmuth, M. Grundmann,\nFerrimagnetic ZnFe 2O4thin ﬁlms on SrTiO 3single crystals with highly tunable electrical conductivi ty, Phys. Sta-\ntus Solidi RRL 5 (12) (2011) 438–440.\n[4] S. J. Stewart, S. J. A. Figueroa, J. M. R. L´ opez, S. G. Marc hetti, J. F. Bengoa, R. J. Prado, F. G. Requejo,\nCationic exchange in nanosized ZnFe 2O4spinel revealed by experimental and simulated near-edge ab sorption structure,\nPhys. Rev. B 75 (7).\n[5] S. A. Oliver, H. H. Hamdeh, J. C. Ho, Localized spin cantin g in partially inverted ZnFe 2O4ﬁne powders, Phys.\nRev. B 60 (5) (1999) 3400–3405.\n[6] Y . F. Chen, D. Spoddig, M. Ziese, Epitaxial thin ﬁlm ZnFe 2O4: a semi-transparent magnetic semiconductor with high curi e temperature,\nJournal of Physics D: Applied Physics 41 (20) (2008) 205004.\n[7] M. Sultan, R. Singh, Magnetic and optical properties of r f-sputtered zinc ferrite thin ﬁlms, J. Appl. Phys. 105 (7)\n(2009) 07A512.\n[8] S. Nakashima, K. Fujita, K. Tanaka, K. Hirao, T. Yamamoto , I. Tanaka,\nFirst-principles XANES simulations of spinel zinc ferrite with a disordered cation distribution, Phys. Rev. B\n75 (17).\n[9] V . Zviagin, Y . Kumar, I. Lorite, P. Esquinazi, M. Grundma nn, R. Schmidt-Grund,\nEllipsometric investigation of ZnFe 2O4thin ﬁlms in relation to magnetic properties, Appl. Phys. Le tt. 108 (13)\n(2016) 131901.\n[10] F. Lotgering, The inﬂuence of fe3+ions at tetrahedral sites on the magnetic properties of ZnFe 2O4, Journal of\nPhysics and Chemistry of Solids 27 (1) (1966) 139–145.\n[11] T. Kamiyama, K. Haneda, T. Sato, S. Ikeda, H. Asano, Cati on distribution in ZnFe 2O4ﬁne particles studied by neutron powder di ﬀraction,\nSolid State Communications 81 (7) (1992) 563–566.\n[12] J. L. Dormann, M. Nogues, Magnetic structures in substi tuted ferrites, Journal of Physics: Condensed Matter 2 (5)\n(1990) 1223–1237.\n[13] C. E. R. Torres, G. A. Pasquevich, P. M. Z´ elis, F. Golmar , S. P. Heluani, S. K.\nNayak, W. A. Adeagbo, W. Hergert, M. Ho ﬀmann, A. Ernst, P. Esquinazi, S. J. Stewart,\nOxygen-vacancy-induced local ferromagnetism as a driving mechanism in enhancing the magnetic response of ferrites,\nPhys. Rev. B 89 (10).\n[14] C. D. Spencer, D. Schroeer, Mssbauer study of several co balt spinels using Co57and Fe57, Phys. Rev. B 9 (9)\n(1974) 3658–3665.\n5/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s51/s54/s57/s49/s50/s49/s53\n/s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s45/s49/s52/s45/s55/s48/s55/s49/s52\n/s45/s52 /s45/s50 /s48 /s50 /s52/s45/s50/s49/s45/s49/s52/s45/s55/s48/s55/s49/s52/s50/s49\n/s40/s98/s41/s40/s97/s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s112/s114/s105/s115/s116/s105/s110/s101/s32/s102/s105/s108/s109/s49/s48/s53/s32/s75/s77/s32/s40/s101/s109 /s117/s47/s103/s41\n/s51/s48/s48/s32/s75/s77/s32/s40/s101/s109 /s117/s47/s103/s41\n/s65 /s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41/s53/s32/s75/s77/s32/s40/s101 /s109/s117 /s47/s103 /s41\n/s65/s112/s112/s108/s105/s101/s100/s32/s70/s105/s101/s108/s100/s32/s40/s84/s41\nFigure 2: Variation in mass magnetization (a) with temperat ure (ZFC-FC) while applying a ﬁeld of 1000 Oe and (b) with\nﬁeld at 300 K for the as-deposited zinc ferrite ﬁlm. Inset: ma ss magnetization vs ﬁeld curve at 5 K.6/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48\n/s45/s50 /s45/s49 /s48 /s49 /s50/s45/s49/s50/s45/s56/s45/s52/s48/s52/s56/s49/s50\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s51/s54/s57/s49/s50/s49/s53/s40/s97/s41/s53/s32/s75/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s70/s105/s101/s108/s100/s32/s40/s84/s41/s32/s80/s114/s105/s115/s116/s105/s110/s101\n/s32/s65/s114/s55/s55/s51\n/s32/s65/s114/s56/s50/s51\n/s32/s65/s114/s56/s55/s51\n/s32/s79/s55/s55/s51\n/s32/s79/s56/s50/s51\n/s32/s79/s56/s55/s51/s40/s98/s41\n/s32/s80/s114/s105/s115/s116/s105/s110/s101\n/s32/s65/s114/s55/s55/s51\n/s32/s65/s114/s56/s50/s51\n/s32/s65/s114/s56/s55/s51\n/s32/s79/s55/s55/s51\n/s32/s79/s56/s50/s51\n/s32/s79/s56/s55/s51\n/s32/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s70/s105/s101/s108/s100/s32/s40/s84/s41/s51/s48/s48/s32/s75\n/s32/s79/s55/s55/s51\n/s32/s79/s56/s50/s51\n/s32/s79/s56/s55/s51/s32/s112/s114/s105/s115/s116/s105/s110/s101\n/s32/s65/s114/s55/s55/s51\n/s32/s65/s114/s56/s50/s51\n/s32/s65/s114/s56/s55/s51/s40/s99/s41\n/s32/s32/s77/s40/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41\n/s80/s114/s105/s115/s116/s105/s110/s101\n/s65/s114/s55/s55/s51 /s65/s114/s56/s50/s51 /s65/s114/s56/s55/s51\n/s79/s55/s55/s51 /s79/s56/s50/s51 /s79/s56/s55/s51/s54/s56/s49/s48/s49/s50/s40/s100/s41\n/s32/s77\n/s114/s40/s53/s32/s75/s41\n/s32/s77\n/s83/s40/s51/s48/s48/s32/s75/s41/s32/s77/s32/s40/s101/s109/s117/s47/s103/s41\n/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s32/s84\n/s66\n/s84\n/s66/s32/s40/s75/s41\nFigure 3: Variation in mass magnetization with ﬁeld at (a) 5 K and (b) 300 K ; (c) ZFC-FC curves recorded with an\napplied ﬁeld of 1000 Oe and, (d) extracted residual magnetiz ation ( Mr) at 5 K, spontaneous magnetization ( Ms) at 300\nK and blocking temperature( TB) for pristine zinc ferrite ﬁlm and after annealing under var ious conditions.\n7"    },    {        "title": "1402.1950v1.Synthesis_of_Fe3O4_CoFe2O4_MnFe2O4_trimagnetic_core_shell_shell_nanoparticles.pdf",        "content": "Synthesis of Fe 3O4@CoFe 2O4@MnFe 2O4trimagnetic core/shell/shell nanoparticles\nV\u0013 eronica Gavrilov-Isaac, Sophie Neveu, Vincent Dupuis, Delphine Talbot, and Val\u0013 erie Cabuil\nSorbonne Universit\u0013 es, UPMC Univ Paris 06, UMR 8234, PHENIX,\nF-75005 Paris, France CNRS, UMR 8234, PHENIX, F-75005 Paris, France\u0003\nPACS numbers:\nMagnetic nanoparticles with spinel structure MFe 2O4\n(M = Fe, Co, Mn, Zn, Ni, Cu...) have been\nextensively studied for their various magnetic ap-\nplications ranging from magnetic energy storage to\nbiomedical applications.[1];[2] Di\u000berent synthesis meth-\nods, such as co-precipitation[3];[4] ;forced hydroly-\nsis in a polyol medium[5];[6] ;micro-emulsions[7] ;hy-\ndrothermal synthesis[8];[9] ;micro\ruidic process[10] ;or\nthermal decomposition[11];[12] ;have been used to con-\ntrol size, shape and composition of these nanoma-\nterials. Thermal decomposition of metal precursors\nhas been demonstrated to be a very e\u000bective method\nto prepare monodisperse nanoparticles with controlled\nmorphology[13] . To develop original magnetic properties\nbimagnetic core/shell nanostructured particles have been\nsynthesized and characterized.[14] These particles are a\ncombination of a magnetic hard phase (e.g. CoFe 2O4)\nand a magnetic soft phase (e. g. MnFe 2O4, ZnFe 2O4\nor Fe 3O4), and possess unique magnetic properties.[15].\nThey are expected to have a good e\u000eciency for magnetic\nhyperthermia.[16]\nWe report here the synthesis and characteriza-\ntion of what we call trimagnetic core/shell/shell\nFe3O4@CoFe 2O4@MnFe 2O4nanoparticles. These par-\nticles are a combination of a hard phase (CoFe 2O4)\nand two soft phases (Fe 3O4and MnFe 2O4), and\nhave unique magnetic characteristics. The Fe 3O4\ncore particles were synthesized according to the proce-\ndure described by Sun and all[13] by high-temperature\ndecomposition ( \u0018280/uni2103) of a mixture of Fe(acac) 3,\noleic acid, oleylamine, 1,2-hexadecanediol and benzyl\nether. To synthesize Fe 3O4@CoFe 2O4core/shell and\nFe3O4@CoFe 2O4@MnFe 2O4core/shell/shell nanoparti-\ncles, a seed-mediated growth at high temperature method\nwas used. The Fe 3O4nanoparticles seeds (1.5 mmol)\ndispersed in heptane were mixed under a \row of nitro-\ngen with a mixture of Fe(acac) 3(1 mmol), Co(acac) 2\n(0.5 mmol), oleic acid (6 mmol), oleylamine (6 mmol),\n1,2-hexadecanediol (10 mmol), benzyl ether (20 mL).\nThe solution was \frst heated to 100 /uni2103for 30 min to re-\nmove heptane, then to re\rux ( \u0018300/uni2103) for 1h. The \fnal\nmixture was cooled down to room temperature, washed\nwith ethanol and a black precipitate was collected af-\nter magnetic precipitation. The separated nanoparti-\ncles were re-dispersed in heptane, and a black ferro\ruid\n\u0003Electronic address: veronica.gavrilov-isaac@upmc.frcomposed of Fe 3O4@CoFe 2O4bimagnetic core@shell\nnanoparticles was produced. Under the same conditions,\nFe3O4@CoFe 2O4@MnFe 2O4core/shell/shell nanoparti-\ncles dispersed in heptane, were obtained by mixing the\nFe3O4@CoFe 2O4bi-magnetic seeds (1.5 mmol) with a\nmixture made of 1 mmol of Fe(acac) 3and 0.5 mmol of\nMn(acac) 2.\n50\t\r nm\t\r (a) \n50\t\r nm\t\r (c) \n50\t\r nm\t\r (b) 05101520250.00.10.20.30.4\nDiameter [nm]Probability densityd0 = 5.7 nmσ = 0.21\n05101520250.000.050.100.150.200.25\nDiameter [nm]Probability densityd0 = 8.2 nmσ = 0.23\n05101520250.000.050.100.15\nDiameter [nm]Probability densityd0 = 11.9 nmσ = 0.19\nFIG. 1: TEM images and size distribution histograms\nof (a) 6 nm Fe 3O4core nanoparticles, (b) 8 nm\nFe3O4@CoFe 2O4core/shell nanoparticles and (c) 12 nm\nFe3O4@CoFe 2O4@MnFe 2O4core/shell/shell nanoparticles\nobtained with a JEOL 100CX (x93000).\nFigure 1 shows the transmission electron mi-\ncroscopy (TEM) images of 6 nm Fe 3O4core,\n8 nm Fe 3O4@CoFe 2O4core/shell, and 12 nm\nFe3O4@CoFe 2O4@MnFe 2O4core/shell/shell nanopar-\nticles. TEM size analysis indicates that particles are\nmonodisperse with narrow size distributions. Histograms\nof core, core/shell, and core/shell/shell nanoparticles\nprovide a nice illustration of the progressive increase of\nparticles size as soon as a new magnetic shell is added.\nWe compare the magnetic properties (blocking tem-\nperature and coercivity) of the core, core/shell andarXiv:1402.1950v1  [cond-mat.mtrl-sci]  9 Feb 20142\nTemperature [K]M/Mmax [a.u.]05010015020025030035000.20.40.60.81 (a) FeFe@Co Fe@Co@Mn  \nMagnetic field [Oe]Magnetization [emu/g]-50000-30000-10000100003000050000-80-60-40-20020406080 (b) \nFeFe@Co Fe@Co@Mn  \nFIG. 2: (a) Blocking temperature and (b) coercivity at 5K\nof Fe 3O4core nanoparticles (Fe), Fe 3O4@CoFe 2O4core/shell\nnanoparticles (Fe@Co) and Fe 3O4@CoFe 2O4@MnFe 2O4\ncore/shell/shell nanoparticles (Fe@Co@Mn).\ncore/shell/shell particles. Figure 2a shows the zero-\n\feld cooled (ZFC) temperature dependence of magne-\ntization under a 50 Oe \feld. The blocking tempera-\nture (T B) increases when comparing core, core/shell,\nand core/shell/shell structures. Fe 3O4nanoparticles dis-\nplay a blocking temperature at 25 K, although this\nof Fe 3O4@CoFe 2O4is around 210 K. The increase be-\ntween the blocking temperatures of the core/shell andcore/shell/shell structures (T B= 305 K) is lower, indi-\ncating that the magnetic hard phase shell (CoFe 2O4) has\na more important impact on the blocking temperature\ncompared to magnetic soft phase shell (MnFe 2O4).\nMagnetization as a function of the magnetic \feld ac-\nquired at 5K, is displayed in Figure 2b. The tem-\nperature is lower that the blocking temperature and a\nhysteresis look is obtained for each sample. This re-\nsult is quite di\u000berent of the two phase magnetic be-\nhavior that would have been obtained with physically\nmixed CoFe 2O4and MnFe 2O4nanocrystals[15]. This\ncon\frmes the core/shell and core/shell/shell structures\nof the synthesized particles. Coercivity H Cis signi\f-\ncantly di\u000berent in bimagnetic core/shell and trimagnetic\ncore/shell/shell nanoparticles compared to magnetic core\nnanoparticles. Hysteresis measurements show that coer-\ncivity increases when the magnetic soft phase Fe 3O4core\nis coated with a magnetic hard phase CoFe 2O4shell.\nIt changes from 0.2 kOe for Fe 3O4nanoparticles to 9\nkOe for Fe 3O4@CoFe 2O4nanoparticles. These results\nregarding the core/shell particles are in good accordance\nwith those of Song and Zhang[15] who have evidenced\na coercivity increase for MnFe 2O4particles coated with\na CoFe 2O4shell and a decrease for CoFe 2O4particles\ncoated by a MnFe 2O4shell. In their paper, the au-\nthors discussed their observations in terms of a simple\nmodel in which coercitivity is ruled by the proportion\nof hard and soft phases within a particle. Our results,\nfor a Fe 3O4@CoFe 2O4core/shell particles coated with\nan additional shell made of a magnetic soft phase (here\nMnFe 2O4but similar results were obtained for a second\nshell made of Fe 3O4), show that contrary to expecta-\ntions from this simple model, the coercivity is increased\n(Hc= 17 kOe). This shows that the physics governing\nthe magnetic properties of trimagnetic core/shell/shell\nnanoparticles is certainly more complex than anticipated\nfrom the results on bimagnetic core/shell nanoparticles\nand should be investigated more thoroughly by numerical\nsimulations and on the experimental side by varying the\nshell thicknesses and the nature of materials. In the same\ntime, it provides new opportunities towards a \fne tuning\nof the magnetic anisotropy of magnetic nanoparticles.\nAcknowledgments\nThe author thanks Aude Michel for the technical as-\nsistance, and P. Beaunier for the access to the TEM plat-\nform.\n[1] N. A. Frey, S. Peng, K. Cheng, and S. Sun, Chem. Soc.\nRev.38, 2532 (2009).\n[2] A.-H. Lu, E. L. Salabas, and F. Sch uth, Angew. Chem.\nInt. Ed. 46, 1222 (2007).\n[3] F. A. Tourinho and R. Franck, J. Mater. Sci. 25, 3249(1990).\n[4] S. Neveu, A. Bee, M. Robineau, and D. Talbot, J. Colloid\nInterface Sci. 255, 293 (2002).\n[5] S. Ammar, A. Helfen, N. Jouini, F. Fi\u0013 evet, I. Rosenman,\nF. Villain, P. Molini\u0013 e, and M. Danot, J. Mater. Chem.3\n11, 186 (2001).\n[6] D. Caruntu, Y. Remond, N. H. Chou, M.-J. Jun,\nG. Caruntu, J. He, G. Goloverda, C. OConnor, and\nV. Kolesnichenko, Inorg. Chem. 41, 6137 (2002).\n[7] N. Moumen and M. P. Pileni, Chem. Mater. 8, 1128\n(1996).\n[8] T. J. Daou, G. Pourroy, S. B\u0013 egin-Colin, J. M. Gren\u0012 eche,\nC. Ulhaq-Bouillet, P. Legar\u0013 e, P. Bernhardt, C. Leuvrey,\nand G. Rogez, Chem. Mater. 18, 4399 (2006).\n[9] O. Horner, S. Neveu, S. de Montredon, J.-M. Siaugue,\nand V. Cabuil, J. Nanopart. Res. 11, 1247 (2009).\n[10] A. Abou-Hassan, S. Neveu, V. Dupuis, and V. Cabuil,\nRSC Adv. 2, 11263 (2012).\n[11] L. P\u0013 erez-Mirabet, E. Solano, F. Mart\u0013 \u0010nez-Juli\u0013 an,\nR. Guzm\u0013 an, J. Arbiol, T. Puig, X. Obradors, A. Pomar,R. Y\u0013 a~ nez, J. Ros, et al., Mater. Res. Bull. 48, 966 (2013).\n[12] Q. Song and Z. J. Zhang, J. Am. Chem. Soc. 126, 6164\n(2004).\n[13] S. Sun, H. Zeng, D. B. Robinson, S. Raoux, P. M. Rice,\nS. X. Wang, and G. Li, J. Am. Chem. Soc. 126, 273\n(2004).\n[14] O. Masala, D. Ho\u000bman, N. Sundaram, K. Page, T. Prof-\nfen, G. Lawes, and R. Seshadri, Solid State Sci. 8, 1015\n(2006).\n[15] Q. Song and Z. J. Zhang, J. Am. Chem. Soc. 134, 10182\n(2012).\n[16] J.-H. Lee, J.-T. Jang, J.-S. Choi, S. H. Moon, S.-H.\nNoh, J.-W. Kim, J.-G. Kim, I.-S. Kim, K. I. Park, and\nJ. Cheon, Nature Nanotechnol. 6, 418 (2011)."    },    {        "title": "0812.0484v2.Phase_transitions_in_multiferroic_BiFeO3_crystals__thin_layers__and_ceramics__Enduring_potential_for_a_single_phase__room_temperature_magnetoelectric__holy_grail_.pdf",        "content": " 1Phase transitions in multiferroic BiFeO 3 crystals, thin-layers, and ceramics:  \nEnduring potential for a single phase, room -temperature magnetoele ctric ‘holy grail’ \n \nA. M. Kadomtseva1, Yu.F. Popov1, A.P. Pyatakov1,2* G.P. Vorob’ev1, А.К. Zvezdin2,  \nand D. Viehland3 \n1 M. V. Lomonosov Moscow State University, Leninskie gori, MSU, Physics department, Moscow 119992, Russia \n2Institute of General Physics Russian Academy of Sc ience, Vavilova st., 38, Moscow 119991, Russia \n \n3Dept. of Materials Science and Engineer ing, Virginia Tech, Blacksburg, VA 24061  \n \n Magnetic phase transitions in multiferroic bismuth ferrite (BiFeO3) induced by magnetic field, epitaxial \nstrain, and composition modifica tion are considered. These transitions from a spatially modulated spin spiral state to \na homogenous antiferromagnetic one are accompanied by  the release of latent magnetization and a linear \nmagnetoelectric effect that makes BiFeO3-based material s efficient room-temperature single phase multiferroics.  \n \nSince the beginning of the multiferroic era (in the early 1960s), afte r Soviet scientists \ndiscovered a new class of materials that was cal led “ferroelectromagnets” [1], to our present \ntime, bismuth ferrite BiFeO 3 has remained the prototypical example of a multiferroic: having a \nrelatively simple structure and at the same ti me quite diversified and uncommon properties.  \nOn the one hand due to its simple chemical and crystal structure, BiFeO 3 is a model \nsystem for fundamental and theoretical studies of  multiferroics [2]. However, on the other hand, \nits unusual magnetic symmetry properties (space and time symmetry violation both in its crystal \nand magnetic structures) results in  a variety of nontrivial conse quences, including: (i) the unique \ncoexistence of weak ferromagne tism and linear magnetoelectricity [3,4]; (ii) a to roidal moment, \ni.e., a special magnetic type of ordering [5-8];  (iii) the existence of an incommensurately \nmodulated spin structure [9], previously only obs erved in magnetic metals; and (iv) magnetically \ninduced optical second harmonic generation, observed fo r the first time in this  particular material \n[10].  \n Furthermore, BiFeO 3 is the material with unique high ferroelectric Curie (T C=1083 K)  \n[11] and antiferromagnetic Neel (T N=643K) [12] temperatures. However, its potential has yet to \nbe realized, and might never be fully explo ited. Difficulties persist as the magnetoelectric \nexchange and weak ferromagnetism are locked with in a spin cycloid. A fundamental problem is \nthat electronic configurations that favor magnetism are antagonistic to those that favor \npolarization [2] – compromise is necessary. R ecently, investigations have shown that the \nmultiferroic properties of BiFeO 3 can be dramatically increased by (i) epitaxial constraint [13], \nand/or (ii) rare earth substituents. These findings  coupled with those in ME two-phase (nano and \nmacro) composites of piezoelectric  and magnetostrictive materials ha ve served as triggers for a \n“magnetoelectric renaissance”: the revival of hope  to find a room temperature magnetoelectric \nmaterial with significant coupling of polar and magnetic subsystems [14-16]. As a consequence, \nmultiferroics are now being considered as prom ising materials for spintronics [17,18], magnetic \nmemory systems, sensors, and tunable microw ave devices [19]: offering the potential to \nrevolutionize electromagnetic  material’s applications.   \n* Author to whom correspondence should be addressed: alexander.pyatakov@gmail.com   2Part I. BiFeO 3 single crystals: unrealized potential. \nI.1 Early Studies: Existence of both locally antiferromagnetic and long-range cycloid order \nThe crystal structure of bismuth ferrite is a rhombohedrally-distort ed perovskite that \nbelongs to the space group R3c. In Figure 1(a), the unit cell of BiFeO 3 is shown in its hexagonal \nrepresentation ([001] hex, [100]  hex , [110]  hex, [010] hex are hexagonal axis). A lternatively, in some \ncases, a pseudocubic representation has been used, where [111] c is equivalent to [001] hex. \nOxygen atoms (not shown) occupy face-centere d sites of the Bi cubic framework.   \nBiFeO 3 has been shown to be ferroelectric w ith its polarization oriented along the \nrhombohedral c-axis (i.e., [111] c) due to the displacement of Bi, and Fe, O relative to one another \n[20]; and shortly thereafter, neutron diffract ion studies revealed antiferromagnetic (AFM) \nordering along [111] c [21]. Spins in neighboring positions are antiparallel with each other, \nresulting in an AFM ordering of  the G-type, as illustrated  in Fig. 1(b).  \n \n       \nС-axis \n[001] hex Fe\nFeFe\nFe\nFeFeFe\n \n(a)        (b) \n \nFig. 1 Crystallographic (a) and magnetic (b) structure of BiFeO 3 \n \nIn spite of a high Curie temperature [11] and a large polar displacement of ions [22], the \nmeasured value of the spontaneous pola rization has been re ported to be 6.1 μC/cm2 [11], which is \nsurprisingly small compared to protot ypical ferroelectrics such as PbTiO 3 (~100μC/cm2). \nInterestingly, the magnetic symmetry of BiFeO 3 allows the linear magn etoelectric effect and \nweak ferromagnetism given in a thermodynamic potential as kjiHLE  and k jiMLP terms. For the \ncase of cubic symmetry these magnetoelectric terms are expressed in simple form:  \n[ ] ( )H×⋅⋅−=− lEαinduced MEF ;    (1a) \n[ ] ( )SM×⋅ ⋅−=− l PS βs spontaneou MEF ;   (1b) \nwhere α, β are magnetoelectric constants, Ps is the spontaneous polarization, l = L1/2M0 the unit \nantiferromagnetic vector, M0  the magnitude of the magnetizati on vector of the sublattices,  Ms the \nin-plane spontaneous magnetization, and E and  H the electric and magnetic fields. It is \nnoteworthy that in contrast to conventiona l weak ferromagnetism that is expressed in \nDzyaloshinskii-Moriya tems jiML  [23,24], weak ferromagnetism is possible only in the presence \nof spontaneous polarization Ps: which is why k jiMLP  terms (1b) are often referred to as  3Dzyaloshinskii-Moriya-like . Due to small in-plane anisotropy, the spontaneous magnetization in \nBiFeO 3 was expected to be easily controlled by a magnetic field of ~1 Oe; furthermore, as weak \nferromagnetism in BiFeO 3 is magnetoelectric (ME) in orig in, a unique coexistence of weak \nferromagnetism and linear magnetoelectricity – fo rbidden under conventional circumstances [3] \n– was expected. However, neither weak ferro magnetism nor linear magne toelectric couplings \nwere observed. Unfortunately, this fact ha s undermined the great potential of BiFeO 3 as an \nunusual ferroelectromagnetic material with high magnetoelectric coupling at high temperatures: \nwhich has been the lingering goal of scientists working in the field since the 1960s. Only since \n2000 has single phase BiFeO 3 materials been developed with enhanced multiferroic properties in \nepitaxial thin-layer form [13] (see Part II ), i.e., there persists promise. \nThe reason for the lack of weak-ferromagnetism and a linear  magnetoelectric effect was \nfound in the 1980’s by precise time-of-flight neut ron measurements [9]. This investigation \nrevealed that the G-type AFM structure is not a complete description of the spin structure in \nBiFeO 3: rather, in addition, it  has a long-wavelength modulation th at forms a spin cycloid with a \nwave-vector that is oriented along [110] hex, which is perpendi cular to the [111] c, as shown in \nFig.2. This modulation results in a zero valu e for the volume-averag ed ME effect and \nspontaneous magnetization. Thus, only a quadratic magnetoelectric effect whose value averaged \nover cycloid period is not zero was reported [25].   \n \nFig. 2 Long range magnetic order: incommensurate spin cycloid. The arrows correspon d to the antiferromagnetic \nvector L that changes its orientation in space  ()xθθ= . \n \n The position of satellite peaks observed in time-of-flight neutron studies along the \n[110] hex (see Fig.3a) enabled determination of the period of the spin cycloid to be o\nA620~.  \nThese results demonstrated that the periodicity of the cycloid was incommensurate with that of \nthe lattice parameters. The presen ce of only a single diffraction peak along the (003) shows that \nthere is no modul ation along the [111] c. It is noteworthy to me ntion that before these \nmeasurements spatially modulated  structures were only observed in magnetic metals: so, the \nobservation of a spin cycloid in magnetic dielectric BiFeO 3 was rather surp rising. Additional \nexperimental evidence of a spin cycloid wa s provided by nuclear magnetic measurements \n(NMR) in BiFeO 3 ceramic doped with 57Fe [26]. Instead of a si ngle peak corresponding to a \nhomogeneous structure, the NMR line was more complicated, exhibiting two maxima \ncorresponding to the spin orientations  perpendicular and parallel to [111] c (see Fig.3b). The \nNMR lineshape was asymmetri cal at low temperature ( T=4.2 K) and became increasingly  4symmetric at higher temperatures . This behavior was explained in  terms of anharmonic cycloid: \nover most of the cycloid’s period, spins are at a small angle ( θ) with respect to [111] c, as can be \nseen from the stronger intensity of the high fre quency peak, and with increasing temperature the \nanharmonicity of the spin profile decrease s resulting in nearly linear dependence of θ on \ncoordinate along spin cycloid dir ection at room temperatures [26] . However in a recent report on \na high resolution neutron powder diffraction study [ 27] it was stated that the character of the \nmodulated cycloidal ordering of the Fe3+ magnetic moments remains the same from 4 K up to the \nNeel temperature of 640 K. Thus the temperat ure behavior of the spin cycloid remains an \nintriguing question.  \n \nFig. 3 Experimental evidences for sp in cycloid existence: (a) Time-of-flight neutron difraction measurements [9]; \n(b) Nuclear magnetic resonance measurements [26]. \n \n The existence of a spin cycloid in BiFeO 3 has been explained in terms of a relativistic \nLifshitz-like  invariant lkji L LP∇  [28-30]: named by analogy to the Lifshitz invariant \nxllxlli j j i ∂∂−∂∂ . For cubic symmetry, it takes the simple form of  \n()( ) [ ]L L LL ∇⋅−∇⋅⋅−=− P Flike Lifshitz γ ;   (2) \nwhere γ is the coefficient of the inhomogenous ma gnetoelectric interaction. The omitted terms \ncan be written as the total derivative ()Lf∇ , and do not contribute to the equation for the spin \ncycloid. Like the Dzyaloshinskii–Moriya-like ex change (1a), this inho mogenous interaction is \nmagnetoelectric in origin and a spin cycloid is  only possible in the presence of spontaneous \npolarization. It is remarkable  that inhomogenous magnetoelectr ic interaction (2) can explain \nmagnetoelectric effect in magnetic domain walls [31-33] and ferroelectrici ty in spiral magnets \n[34] under the assumption of polarization  induced by magnetic inhomogeneity. Furthermore, \nthere is a profound analogy between spatially modul ated spin structures in multiferroics, and \nspatially modulated structures in nematic liquid crystals [35,36]. The peri odic vector director \nstructures in a nematic liquid crysta l arise in an external electric fi eld (i.e., a flexoelectric effect), \nand can be tuned under an  applied electric field.  \nThus, the inhomogeneous magnetoel ectric interaction of  equation (2) that gives rise to the \ncycloid prevents the observation of the homogene ous interaction of equa tion (1), which would \nmanifest itself in a weak ferromagnetism (1a) a nd a linear magnetoelectric effect (1b). The \nnecessary condition for observation of  these two effects is the suppr ession of the spin modulated \nstructure. Due to the magnetoelectric nature of the spin cycloid, it can be controlled by a \nmagnetic and an electric field, and even de stroyed under high field c onditions that induce a \nhomogeneous spin state (see Section  I.2). Alternatively, the spatia lly modulated structure of \nBiFeO 3 can be disrupted by epitaxial c onstraint in thin-layers (see Section II.1 ) and/or by rare-\nearth substitutions (see Section II.2 ). Here, in the remaining parts of Section I.1 , the theoretical \nFrequency  5framework for magnetoelectric in teractions and spatially modulat ed spin structures in BiFeO 3 \nwill be discussed.   \nSymmetry and magnetoelectric interactions  \nMagnetic symmetry is an elegant and effi cient tool for understanding the physical \nproperties of crystals with complex magnetic st ructures, notably antiferromagnets. On the basis \nof this approach, the Dzyaloshinski–Moriya in teraction and weak ferro magnetism [23,24, 37,38], \nthe linear magnetoelectric effect [39-41], piezo magnetism [42,43], linear magnetostriction [44], \nand a variety of unusual optical e ffects associated with the AFM vector (e.g., quadratic Faraday \neffect [45] or linear bire fringence [46,47]) have been  studied over many years. \nThe occurrence of one or the other of th ese effects in an AFM crystal with a \ncentrosymmetric crystallographic structure depends on the space par ity of its magnetic structure. \nFor example, weak ferromagnetism arises in crysta ls with an even antiferromagnetic structure, \nwhereas the linear magnetoelectric effect is  forbidden in them. Conversely, a linear \nmagnetoelectric effect is allowed and weak ferr omagnetism forbidden in crystals with an odd \nantiferromagnetic structure. In this respect, cr ystals with trigonal symmetry – rhombohedral \nMnCo 3, FeBO 3, and α-Fe 2O3 antiferromagnets with even magnetic structure, and the \nrhombhedral Cr 2O3 antiferromagnet with an odd magnetic structure – have been studied most \nthoroughly. The crystal st ructures of all these materials pos sess an inversion center (space group \nсR3 ). Here, in this subsection, the symmetry properties of BiFeO 3 are considered: it is akin to \nthe aforementioned rhombohedral antiferromagnets, but yet different from them in that an \ninversion center is abse nt in both its crystal and magnetic structures. It will be shown that \nBiFeO 3 has unusual properties due to precisely this fact. \nWe choose the space group сR3 as a “parent” symmetry for the R3c symmetry under \nstudy. In reality, the phase transition in BiFeO 3 at the Curie point TC differs from a сR cR 3 3→   \ntransition. However, this differen ce is not of importance in dete rmining the adequate invariants \nthat are responsible for the magnetoelectr ic properties of the system. The space group сR3 \ndiffers from R3c only by the presence of a polar vector P= (0, 0, Ps). Indeed, the parent symmetry \ncan be used to develop the perturbation theory  for determining the thermodynamic potential and \nother physical quantities of the system under the assumption of the smallness of P; i.e., we \nexpand with respect to P. Correspondingly, aa/Δ=ξ  is a small parameter, where a is the lattice \nconstant, and aΔ is the characteristic atomic deviati on from the symmetric positions about the \nspace inversion in R3c.  \nThe magnetic exchange structure (i.e., the mutu al directions of ma gnetic moments in the \ncrystal) is determined by the following code using Turov’s nomenclature [3]: I -, 3z+, 2x+ where I \nis the space-inversion element; 3 z (aligned with the c-axis) the threefold axis and 2 x the twofold \naxis are the group genera tors; and the indices ± of these elements that specify their parity about \nthe transposition of magnetic sublattices (i.e., “+” indicates that the symmetry element transposes \nthe ions within the same magnetic sublattice of an  antiferromagnet, and “–” that the sublattice is \ntransposed into one with opposite spin direc tion). Using these symmet ry operators, the AFM \nvector can be shown to obey th e following transformation rules: \n±IL=±L, x x x L L ±=±2 , )( )( 2zy zy x L Lm=±. \nFor the other vectors, the action of elemen ts with different indices is the same: \n±Im=m , x x x m m=±2 , )( )( 2zy zy x m m −=±, ±IP=  – P, x x x P P=±2 , )( )( 2zy zy x P P −=±. \nThe layout of the group elements relative to the magnetic ions in  bismuth ferrite is shown in \nFig.4. It is interesting to compare the code of BiFeO 3 with the codes of other antiferromagnetic \ncompounds belonging to the same space group: for hematite (32OFe−α ) I+, 3x+, 2x-, and for  6chromite (Cr 2O3) I-, 3Z+, 2x-, whose exchange structures are illustrated in Figs.4b and 4c, \nrespectively1. \n \nFig. 4 . Exchange structures of (a) bismuth ferrite BiFeO 3, L1 = M1 – M2; (b) hematite L2 = M1 – M2 – M3 + M4; \nand (c) chromite Cr 2O3, L3 = M1 – M2 + M3 – M4. \nAs crystal and magnetic cells are identical in BiFeO 3 the reduced group ~\nG can be used \nwhere all translation on cell period are consider ed as unit operators. The space group contains \neight irreducible representa tions: four one-dimensional ( Г1, Г2, Г4, Г5) and two two-dimensional \n(Г3, Г6) ones (see Table I). Their matrix representa tions are given in the columns corresponding \nto the generating symmetry elements. The vector components of electric field E, magnetic field \nH, electric polarization P, magnetization m, and antiferromagnetic vectors L1, L2, and L3 for the \nexchange structures of bismuth ferrite, hematit e, and chromite, respectively (see Fig. 4), are \ngiven in the table according to their transformati on properties. For instance, it follows from this \ntable that the ( Lz)1 component changes sign and the vector L⊥ = (Lx, Ly)1 transforms into ( Lx, –\nLy)1 under the symmetry operation 2x+. The transformation properties of the products miLi, HiEi, \nLiEi for bismuth ferrite are th en given in Table II.  \nTable I \n E+ I+/- 3z+ 2x+/- Ei ; P i H i; m i L i \nГ1 1 1 1 1   ()2zL \nГ2 1 1 1 -1  Hz; m z  \nГ3 \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001  ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001 R \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−1 001  \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nyx\nHH\n;⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nyx\nmm\n \n2⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−xy\nLL \nГ4 1 -1 1 1   ()3zL  \nГ5 1 -1 1 -1 \nzE;zP ; z∇  ()1zL \nГ6 \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−\n1001 R \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−1 001 x\nyE\nE⎛⎞\n⎜⎟\n⎝⎠; x\nyP\nP⎛⎞\n⎜⎟\n⎝⎠;\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∇∇\nyx  \n1⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nyx\nLL;\n3⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−xy\nLL; \nTable of irreducible representations of the reduced сR3 space group (translation on cell period are considered as \nunit operators). Symbol R in table stands for operator of rotation at an angle of 120 ˚ \n                                                 \n1 It is worth noticing that the exchange structure of BiFeO 3 still remains an open question. In the paper of Claude \nEderer and Nicola A. Spaldin [Phys. Rev. B 71, 06040 1(R) (2005)] the position of inversion symmetry center is \nchosen at Fe atoms so th e inversion element becomes I+. The possible solution of this inconsistency can be found in \nthe paper of R. de Suosa and Joel E. Moore arXiv:0806.2142 (this comment is made in 2008).  7Table II \n E+ I- 3z+ 2x+ miL i; H iLi; \nHiEi;LiEi; \nГ1 1 1 1 1 ( )yy xx LE LE +  \nГ2 1 1 1 -1 ( )xy yx LE LE −  \nГ3 \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001 R \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−1 001  \nГ4 1 -1 1 1 ( )yy xx LmLm +  \nГ5 1 -1 1 -1 ( )yx xy LmLm − ;  \n( )yx xy LHLH − ; \n( )yx xy EH EH − ; \nГ6 \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n1001 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−−\n1001 R \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n−1 001 ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n+−\nxy yxxx yy\nLHLHLHLH \nTable of irreducible representations of the reduced сR3 space group (translation on cell period are considered as \nunit operators).  R in table stands for operator of rotation at an angle of 120 ˚ \nOne can readily see in Table I that the ( )xy yx lHlH −  combination in hematite, where l = \nL2/2M0 is the unit antiferromagnetic vector and M0  is the magnitude of the magnetization vector \nof the sublattices, corresponds to the first irreducible representati on, i.e., its invariant. This \ninvariant is responsible for the formation of a weak magnetization ( )( )x y y x ll mm −,~ , ; i.e., the \nlatter is a weak ferromagnet w ith the magnetization vector perpe ndicular to the antiferromagnetic \none. It can also be shown from this table that  a spontaneous magnetization is forbidden in Cr 2O3. \nThe combinations kjlm (where l=L3/2M0) are not invariant. At the same time, the symmetry of \nCr2O3 allows a linear magnetoelect ric effect because of the ( )zy y x x lHE HE + , zz zlHE , \n( )yy xx z lElEH + , and ( )yy xx z lHlHE +  invariants. \nOne can also easily find the following invariant for bismuth ferrite, using Tables I and II:  \n( )... ... + − −=yx xy z lmlmP F ;   (3) \nwhere Pz is the component of spontaneous polarization ( )zP,0,0=P  along the c-axis, \nm=(M1+M2)/2M0 and l = L1/2M0 the unit magnetization and antiferromagnetic vectors, \nrespectively, and M0 the magnitude of the magnetization vector of the sublattices. This \ninteraction is Dzyaloshinski–Moriya-like (1), and gives rise to a w eak ferromagnetism with a \nmagnetization of \n( )0, , ~xz yz lP lP − m .   (4) \nThis result is by no means contradictory to  the well-known theorem in the theory of \nantiferromagnetism about the im possibility of weak ferroma gnetism coexisting with a \nmagnetoelectric effect [3]. Rather, this theorem is related to those antiferromagnets whose space \ngroup contains a space-inversion element (i.e., a crystal structure even about space inversion). \nFor BiFeO 3, we deal with ferroelectromagnets where this requirement is not fulfilled. It is worth  8noting that the physical nature of weak ferro magnetism in a ferroelectromagnet is basically \ndifferent from that of the Dzyaloshinski–Mo riya case. A weak ferromagnetic moment in \nferroelectromagnets results from magnetoelectric interaction: in other words, this magnetic \nmoment arises due to the intern al effective electric field. The expression for the free-energy \ndensity that includes the magnetoelectric te rms proportional to the invariants of the HiEili type \ncan be given as \n( ) ( ) [ ]\n() ( ) ( ) ......\n4 3 21\n+ − − − − − −+ + − −=\nyx xy z yx xy z xy yx zxy yx y xx yy x\nEH EHla lHlHEa lElEHalHlHE lHlHEa f\n  (5)  \nThe tensor relating the magnetic-field-induced pola rization to the magnetic vector for the linear \nmagnetoelectric effect is \n03 32 1 4 12 1 4 1\nx yx x z yy y z x\nij\nla lala la lalala lala la\n−− −+ −\n=α .   (6)  \nThe coexistence of weak ferromagnetism (4) and magnetoelectric effect (5) not only remarkable \nitself but also may have an interesting conseq uence such as magnetoelectric effect, magnetic \nsusceptibility, and electric polarizability enhancem ent as well as a renormalization of the values \nof the spontaneous electric polari zation and magnetization [48].  \nApart from the magnetoelectric (ME) effect and spontaneous magnetization, the magnetic \nsymmetry of bismuth ferrite also allows for magne tic ordering of a special toroidal type [5,6]. \nThe toroidal moment is a va lue conjugate to the product [ ]HE× , and results from the [ ] ( )HET×  \nterm in the free energy expansion. It follows that  the vector components of the toroidal moment \nare determined by the antisymmetric part of the linear ME tensor, given as:  \njk ijk iT αε= ,        ( 7 a )  \n32 23 1 α α− =T , 13 31 2 α α− =T , 21 12 3 α α− =T  ;  (7b)  \nwhere ijkεis the Kronecker tensor. By us ing (7b) and (6), one can read ily verify that the vector \ncomponents of the toroidal moment in BiFeO 3 are proportional to the components of the \nantiferromagnetic vector: \nz z\nyx\nyxl Tll\nTT\n~; ~   ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛.     (8) \nThe analysis of the irreducible representations of  crystal space groups allows one to predict not \nonly the tensor components of magnetic materials,  but also their microscopic magnetic structure. \nTaking into account that the differential operator ∇ transforms as a polar vector, we can include \nthis new term in the expression for the free energy density (Table I), rewriting the free energy as:  \n( )yy z xx z zy y zx xz L lllll lllP F ∇− ∇− ∇+ ∇ ⋅=γ .  (9) \nThis form has a Lifshitz-lik e invariant (2), which sets up the thermodynamic conditions \nresponsible for the spin cycloid.  \nTherefore, there are two types of ME intera ctions that must be considered in BiFeO 3: a \nhomogeneous interaction characterized by the te nsor of the linear ME effect (6), and an \ninhomogeneous interaction characterized by the γ constant in (9). A di stinctive feature of \nbismuth ferrite is the presence of  a spontaneous polari zation. Thus, a space- inversion element is \nabsent in its crystallographic space group, and ta ken together with the sp ace-parity violation in  9its magnetic structure results in the coexiste nce of weak ferromagnetis m (4), magnetoelectric \neffect (6), a toroidal moment (8) and an inhomogeneous ME interaction (9). \n \n \nSpatially modulated spin structure  \nThe total expression for the free- energy density has the form  \nan L exch F F FF + + = ,      ( 1 0 )  \nwhere     \n() () () ( ) ∑\n=∇ + ∇ = ∇ =\nzyxii exch A l A F\n,,2 2 2 2sin ϕθ θ    (11) \nis the exchange energy, A the constant of inhomogeneous excha nge (or exchange stiffness), and \nθ and φ are the polar and azimuthal angles of the unit antife rromagnetic vector \n() θ ϕ θ ϕ θ cos, sin sin, cos sin=l  in the spherical coordinate syst em with the polar axis aligned \nwith the principal axis c, and where \nθ2cosu an K F −=       ( 1 2 )  \nis the anisotropy energy, and Ku the anisotropy constant.  \nMinimization of the fr ee-energy functional Ff d V=∫⋅  by the Lagrange–Euler method \nin the approximation ignoring anisot ropy [30] gives for the functions (, ,)xyz θ  and (, ,)xyz ϕ   \n0 ()y\nxqconst arctgqϕ== ; yqxqy x+ =0θ ;   (13) \nwhere q is the wave vector of the cycloid. Eq uation (13) describes a cycloid whose plane is \nperpendicular to the basal plane and oriented along the propagation direction of the modulation \nwavevector. \nThe exact solution that takes into account an isotropy gives the following expressions for \nthe spin distribution and cycloid period [49,50]:  \nθθ 2cos 1mmAK\ndxdu−⋅=       (14a) \n()\nuKmAmK⋅=14λ ;      (14b) \nwhere ()∫−=2\n021cos 1π\nθθ\nmdmK  is an elliptical integral of the first kind, and m the modulus \nparameter of the elliptical integral that is found by minimization procedure of the free-energy \n(10) [50]. For an anisotr opy constant much smaller than the exchange energy 2\nuKA q<< , the \nmodulus parameter m tends to zero, and solution (14a) becomes harmonic with a linear \ndependence of θ on coordinates (13). By substituting (13)  into (10), one can obtain the volume-\naveraged free-energy density for a harmonic cycloid approximation, as \n()22 u\nsKqP Aq F − − = γ .       ( 1 5 )  \nThe wave-vector corresponding to the energy minimum is then  10APqs\n22\n0⋅= =γ\nλπ.        ( 1 6 )  \nKnowing that λ=620Аo\n [9], and assuming the polarization of BiFeO 3 to be \n2 6/ 106 cmC Pz−⋅=  [11] and the excha nge constant to be cmerg1037−⋅=A  [12], one can estimate \nthe inhomogeneous ME coefficient to be γ=105 erg/C=10-2 V. \n \nI.2 High magnetic field studies. Fi eld-induced transition: cycloidal Æhomogenous state \nThe magnetoelectric origin of the spin cycloid offers a means to control the \nmagnetization state of BiFeO 3 by magnetic and/or elec tric fields. It is r easonable to suppose that \nan applied external magnetic and/or  electric field above that of a critical va lue would result in the \nspin cycloid becoming energetically unfavorable. A bove this critical point,  a transition will occur \nto a homogenous antiferromagnetic spin state: c onsequently, all of the latent properties of \nbismuth ferrite hidden by the cycloid – such as linear magnetoelectri c effect, spontaneous \npolarization, and toroidal moment – should beco me apparent. We will first show experimental \nevidence from a series  of magnetic, magnetoelectric, magn etic resonance st udies under high \nmagnetic fields [4,7,28,29,51] that demonstrate such a spin transformation. Then, we will provide theoretical grounds fo r understanding the transition.  \nIn Fig.5, we show the magneti zation and polarization of BiFeO\n3 induced under high \nmagnetic fields.  Measurements were performed at low temperature (T=10K) in order to suppress noise: as stated above similar magnetic a nd magnetoelectric proper ties should persist to \nroom temperature. The experimental magnetiza tion curves at low and high fields are well \ndescribed by the linear dependences \n⎩⎨⎧\n> + =< ⋅=\n⊥ (17b) ;   ,a) (17     100   ,\n]001[ cspontH HH M MkOe HH M\nχχ    \nwhere⊥ = χ χ 6/5 ; and 5105−\n⊥ ⋅=χ  is the magnetic susceptibility  in the direction perpendicular \nto the AFM vector  l. However, abrupt changes can be seen in Fig.5 for both the magnetization \nand polarization near a critical field of Hc≈200kOe. We ascribe these changes to an induced \ntransition between spatially-modulated  and homogeneous spin states.  \n \nFigure 5. (a) Magnetization of bismuth ferrite as a function of magnetic field [4]. Dots are the experimental data \nobtained in a field oriented along the [001]c direction, and solid line is the theoretical dependence (Eq. (29)); and \n(b) Dependence of the longitudinal electric polarization on  the magnetic field for crystallographic direction along \nBi atom cube edges [28].  \n  11In Fig. 5a, weak ferromagnetism appears in th e offset of the linear dependence of M on H \nin the field range of H> Hc (17b). The value of spontaneous  magnetization in this weak \nferromagnetic state can be found by extrapolation of  the high-field linear slope of the M-H curve \nto the ordinate axis: ≈spontM]001[ 0.25emu/g (see Fig.5a). Taking into  account the orie ntation of the \ncrystal (the magnetizati on was measured along [001] c). We can arrive at a value for the in-plane \nspontaneous magnetization of G g emu g emu Ms 5 / 6.0 / 25.023≈ = =  [4]. The changes in the \nmagnetization curve in the vicinity of H c reflect the cycloidal Æhomogeneous transformation, \nwhich we will treat in more theoretical detail below in this section.  \nIn Fig. 5b, a linear magnetoel ectric effect can be seen from  the linear dependence of P on \nH in the field range of H>H c. For H<H C the electric polarization has a near-quadratic dependence \non H; whereupon, near H c, it undergoes an abrupt jump that  is accompanied by the onset of a \nlinear dependence on H. The value of the linear ME  effect can be determined from the slope of \nthe P-H curve at H>H c, and can be found as ~10-10 C/(m2Oe) [28]. \nTo determine the toroidal moment in BiFeO 3, the antisymmetric part of the ME tensor (6) \nwas examined at fields exceeding H C for orientations at an angle of  45° to the a- and b-axes of \nthe basal plane. Using such a field orientati on, it is possible to measure both polarization \ncomponents P a(H) and P b(H). A nonzero antisymmetr ic component in the magnetoelectric tensor \n(6) appears for H>H c, evidencing a toroidal moment T z~(α12– α21) upon inducing the \nhomogeneous spin state [4,7]. \n \nFigure 6 (a) Antiferromagnetic resonance frequencies as functions of magnetic field H [51]; (b) Magnetic hysteresis \nof the absorption peak. Solid line is for the increasing field and dashed one is for the decreasing field [51]. \n \n All of the abovementioned high-field measur ements [4,7,28,29] of macroscopic property \nchanges associated with the induced transition were performed using  pulsed fields. However, \nlocal probes have also been used  to investigate changes in elect ron spin resonance (ESR) spectra \nof BiFeO 3 [51] under static magnetic fields. M easurements were done at liquid helium \ntemperatures for the same reasons as those made  under pulsed field (i.e ., reduce noise) at the \nNational High Magetic Fiel d Laboratory (Tallahass ee, FL). Figure 6a show s the frequency of the \nESR signal as a function of H for 50<H<250kOe. Strong changes in the resonance frequency \nwere observed near a critical field of Hc =180kOe. A Landau-Gi nzburg phenomenological \ntheory was developed to theoretically model the electron spin resonance spectra in the \nhomogenous spin state (H>H c) by taking into account the Dzyaloshinski-Moriya-like \nmagnetoelectric interaction (4), represented in Fig. 6a by a dashed line. Fitting of the data to the \nphenomenological model yielded a value of Ms=6G  for the spontaneous magnetization, in \nagreement with the direct measurements of th e magnetization changes under a pulsed field [4]. \nAlso, please note that the induced phase transformation was accompanied by appreciable  12hysteresis in the resonance spectra between incr easing and decreasing fiel d strengths, as shown \nin Fig. 6b: evidencing a 1st order type transition.   \n The combinations of the data in Figures 5 and 6 de monstrate agreement between \nmacroscopic property changes and those made us ing a local probe: both revealing an induced \nphase transition near a critical magnetic field. The results are in  agreement with respect to the \ncritical field levels required to induce the transformation, and induced magnetization and \npolarization changes. In what follows, we will discuss the thermodynamic framework which explains these observations based on an i nduced transformation between cyloidal and \nhomogeneous spin states.  \n \nTheory of magnetic and electric field-induced phase transitions \n Application of high magnetic and electric fi elds will result in changes in the effective \nanisotropy of BiFeO 3, i.e., ),(HE Keff . This tunability of the uniaxia l anisotropy constant K eff by \nE or H offers the possibility of a spin cycloid transformation, in accordan ce with (14). Consider \nthe case of  H||E||c. It is convenient to use dimensionless units of electric and magnetic fields:  \nsz\nPEe||κ= ,      ( 1 8  a )  \n2\n02AqHhz⊥=χ;     (18 b) \nwhere ||κ is the electric susceptibility in the direction perpendicular to the c-axis, ⊥χthe \nmagnetic susceptibility  in the direct ion perpendicular to the AFM vector  l, q0 (16) the value of \nthe wavevector of the cycloid corresponding to the minimum of the free energy (10) in the \nabsence of external fields and neglecting anisot ropy. The electric polarizat ion of the medium and \nthe magnetization of the magnetoelectric origin in the linear approximation can be represented in \nthe form:  \n        ()e P Ps z + = 1      ( 1 9  a )  \n()e M Ms+ = 1 ;     (19 b) \nwhere s s DP M⊥=χ  is the spontaneous magnetization due  to a Dzyaloshin skii-Moriya-like \ninteraction (1a), and D is a homogenous magnetoelec tric interaction constant. \nThe free energy density can be conv eniently written as the sum  \nan L exch f f ff + + = ,     ( 2 0 )  \nwhere the energies of exchange, inhomogeneous magnetoelectric intera ction, and effective \nanisotropy are normalized to the exchange energy 2\n0Aq of the harmonic cycloid in the absence of \napplied fields: that is, \n2\n2\n02\n01⎟⎠⎞⎜⎝⎛= =dxd\nq AqFfexch\nexchθ    ( 2 1 )  \ndxd\nqe\nAqFfL\nLθ⋅+−= =\n02\n012     ( 2 2 )  \n() θ2cos,hek fan−= ;       ( 2 3 )   13where () () ()2 2\n2\n01 , e h k\nAqKhekueff+⋅− − = = β  is the dimensionless eff ective anisotropy constant, \nwhich accounts for the effects of electric e and magnetic h fields. For the homogeneous state \nwith an AFM unit vector cl⊥ (2πθ=) we have   \n0=⊥f .        ( 2 4 )  \nAs for the homogeneous state  l || c (θ=0), it was shown in [50] that the parallel phase is only \npossible under electric fields applied antiparallel to P s, which are much larger than the \nferroelectric coercive field of BiFeO 3; and thus unlikely to exist for this particular material.  \nIn the spin cycloid state, we can obtain the expression for the total free energy (20) \naveraged over the period ()∫=2\n04π\nλθθθλdddxf f  by taking into account (14a):  \n() ()\n()()emhek\nK mKmK\nmhekf + −⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− −= 1,21,\n1 12 π\nλ;  (25) \nwhere ()∫−=2\n021cos 1π\nθθ\nmdmK , () θθπ\nd m mK ⋅ − =∫2\n02\n2 cos 1  are elliptical integrals of the \nfirst and second kind, respectively. To each pair of field strengths ( e, h) there corresponds a \nmodulus m of the elliptic integral for which the energy is minimum. Physically, this means that \nthe cycloid’s profile changes under applied fi elds. Under strong fields, its shape differs \nsignificantly from that of a ha rmonic profile, becoming sim ilar to a function describing a \nperiodic structure of domains separated by walls (solitons) w hose widths are considerably \nsmaller than the domain width. It follows from  (24) that, upon the transformation to the phase \nwith the AFM unit vector l ⊥c, the energy of the domain walls λfchanges sign and the \nspatially modulated spin state becomes energe tically unfavorable. The zero-order approximation \nthen corresponds to a harmonic cycloid with a linear spin distribution (13), which upon taking \ninto account (20-23), has the simplified form \n()2),(12 heke f − +−=λ.   (26) \nThen, by using (24) and (26), we can obtain an alytical expressions co rresponding to the phase \nboundary between the spatially modulated  and the homogeneous states as \n() ⎟\n⎠⎞⎜\n⎝⎛− + + =21 122 2 β\nz u e k h .       (27)  \nUsing (27), the phase diagram can be constr ucted according to the results of numerical \ncalculations, as shown in  Figure 7. The energy of the spat ially modulated structure (25) was \nminimized with respect to the modulus m at each point ( e, h). The calculated energy was \ncompared with the energy of th e homogeneous state ( 24). In the calculations, the anisotropy \nparameter, the magnetoelectric interaction para meter, and the coercive force were taken as \n2=uk , 2 ,1=β , and 04 ,0=ce , respectively. The material constants for BiFeO 3 corresponding \nto these values are: 5106⋅=uK erg/cm3, 7103−⋅=A erg/cm, 6\n010=q cm-1,5105−\n⊥ ⋅=χ , \n6=sM Gauss, 06 ,0=sP C/m2, 50 =cE kV/cm, and 50 1|| ≈−=ε κ  [50].  \nIn Figure 7, the regions I and II are separated by a boundary s hown as a solid line. This \nboundary corresponds to that be tween the homogeneous (with l ⊥c) and the spatia lly modulated \nspin states. The limit of the spatially-modulat ed spin phase in the harmonic approximation  14according to expression (27) is i ndicated by a boundary represented as a dashed line. It can be \nseen that, owing to the change in the shape of the cycloid (charact erized by the modulus m of the \nelliptic integrals), a solitonic cycloid can exist in the regions in which the harmonic cycloid is \nenergetically unfavorable (the region between da shed and solid lines). The critical magnetic \nfields chfor numerical and analytical (27) solutions differ by ~10% at zero electric field; and, \nthis discrepancy increases slightly with increasi ng electric field. In the absence of an applied \nelectric field, the phase transition occurs at a point h=1.8, which corresponds to a magnetic field \nof ~190kOe. This is in good agre ement with the experi mental data reported in [4,7,28,29,51]. At \nthe critical magnetic field, the phase tran sition can be shifted by h=0.1 (10kOe) upon \nsimultaneous application of an electric field of  0.08 (100kV/cm). Figure 7 also illustrates the \nspin cycloid transformation under dually-applied el ectric and magnetic fields: the period of the \ncycloid decreases with increasing electric field; and at the field region between dashed and solid \nlines, the cycloid is domain-like and described by a soliton (14a) with −∞→m at Ch h→ .  \n \n \nFig 7 Electric field-magnetic field phase diagram: e dimensionless electric field: 0.1 correspond to =125kV/cm; h \ndimensionless magnetic field: h=1 correspond to 110kOe ; (1-6) fragment x-projectio n of antiferromagnetic vector \non coordinate along direction of cycloid propagation (1-4) quasiharminic cycloid, (5,6) anharm onic (domain-like) \ncycloid. \n \nThe anharmonic character of the cycloid in hi gh magnetic fields becomes apparent in the \nnonlinear behaviour of the magnetization curv e (see Fig.5a) for 150<H<200kOe [4]. This \nobserved experimental dependence (17) can be understood in terms of the sum of spontaneous  15magnetization Mspont averaged over the cycloid’s period, and the magnetization MH  induced by \nexternal field dependence on external field H:  \n)( )( )( H H HH spontM M M + = .    (28)  \nExperimental configurations some what more complex than that in the case of the phase diagram \nof Fig.7 can be treated: where th e field is aligned with the [001] c, or ( )3 ,32,0 H H=H  in \nhexagonal representation. The total magnetization along [001] c in this case is \n() ()\n444344421321\nс]001[с]001[с2\n]001[ с]001[ sin31\n32sin )(\nzHyHM Mspontx H H x M H M\nλ λθ χ χ θ⊥ ⊥+ + = ; (29) \nwhere θ is the angle between the spontaneous polarization Pz (c axis) and the antiferromagnetic \nvector; and с]001[yHM  and с]001[zHM  are the projections of the y-  and z-  components of the \ninduced magnetization HM onto the [001] c, respectively. Equation (29)  can then be used to \nexplain the observed dependences of th e spin profile: in the field range of ()cH H ,0∈ , the spin \nprofile changes from quasi-harmonic to that of a single domain antiferromagnetic state, as shown \nin Figure 8. Thus, averaged over an entire period θθ λπ\ndddxxf∫2\n0)(1, the values of sine ()λθy sin  \nand the square of sine ()\nλθy2sin  vary within intervals of (0 , 1) and (1/2, 1), respectively \n(fig.8). In turn, the dependence of (29) turns to that of (17a) at 0→H , and to that of (17b) at \ncH H→ . \n \nFig 8. The transformation of the cycloid in th e high magnetic field directed along [001] c axis (i.e. magnetic field has \nnonzero projection on the basal (001) hex plane). \n \nSummary  \n In spite of many investigations spanning nearly 50 years, multiferroic properties in \nBiFeO 3 crystals have not been realized: except under  extremely high fields. In a nutshell, this \ndisappointment is due to the trapping of th e magnetic vector and linear magnetoelectric \ninteractions within a spin cycloid: which can be destabilized at high fields. It is fundamental that \nelectronic configurations that favor magnetism ar e antagonistic to those that favor polarization \n[2] – compromise is necessary. But, how? Low field \n() 0 sin =λθy  \n()21sin2=\nλθy  \nHigh field \n() 1 sin →λθy  \n() 1 sin2→\nλθy  \n 16Part II. Epitiaxial films, and rare- earth substitutents: Lingering potential  \nAs we learned above in Part 1 , the necessary condition to manifest multiferroic \nproperties in bismuth ferrite is the suppression of the spin cycloi d. Apart from application of an \nextremely high magnetic field of ~200kOe, other means exist of suppressing the spatially \nmodulated structure in BiFeO 3. Recently, the mentioned suppression of the spin cycloid – \naccompanying by weak ferromagnetism and linear ma gnetoelectricity – has been observed in \nbismuth ferrite epitaxial thin-layers, and by rare- earth substituted crystalline solutions; as we will \ndiscuss in more detail below. These approaches offer hope that the special properties of BiFeO 3 \nmay still be realized. \n \nII.1  Epitaxial thin-layers: Release of late nt magnetization and observation of large \npolarization.  \nEpitaxial thin-films of BiFeO 3 have been grown on (001) c SrTiO 3 [13]. The P s of (001) c \nBiFeO 3 thin films (50-500nm) is ~60 C/m2 – which is ~20x larger than that of a bulk crystal \nprojected onto the same orientation. Hetero-epi taxy induces significant an d important structural \nchanges [52-54]. Synchrotron studies [54] on BiFeO 3 thin films of have shown that the \nmonoclinic M a phase is stable for (001) c epitaxial films, with lattice parameters of \n);,2,2( βmm m cb a= (3.907, 3.973, 3.997Å; 89.2o), which are distinctly different from the \nrhombohedral ones of bulk single crystals and ceramics of (a r=3.96 Å). However, the studies \nhave also shown that the rhombo hedral phase is stable for (111) c epitaxial films [52], whose \nlattice parameters are identical to  those of the bulk crystal. Cl early, the structure of BFO films \ncan be engineered by epitaxial constraint. \nCorrespondingly, the crystallographic orientation of the film state ha s significant effect \non the physical properties of BiFeO 3. Remanent polarizations P r of ~55 μC/cm2 for (001) c films, \n~80 μC/cm2 for (110) c films, and ~100 μC/cm2 for (111) c films have been reported [52]. Pulsed \npolarization studies have confirmed that this large value is due to a true dielectric displacement, \nwithout notable contributions fr om conduction. Fig. 9a shows √3·P(001)c, √2·P(110)c, and P (111)c as a \nfunction of E for the variously oriented films. In this figure, the values of the projected \npolarizations can be seen to be nearly equivalent. This reveals that the direction of spontaneous polarization lies close to (111)\nc, and that the values  measured along (110) c and (001) c are simply \nprojections onto these orientations. Clearly, similar to bulk crystals and ceramics, the spontaneous polarization is oriented close to (111)\nc, but yet is dramatically increased relative to \nthe reported bulk values. The question still remain s whether this enhancement for epitaxial thin \nlayers is due to the constraint (as was theoretically  justified in [55]), or rather simply due to the \nlower leakage currents and higher dielectri c breakdown strengths which can allow for the \nobservation of polarization switching in samples with large coercive fields. The studies of electrical and conductive properties of bism uth ferrite films [56-58] have shown that  oxygen \nvacancies are the main cause for the high leakage current in BiFeO\n3, while doping of higher \nvalence ions such as Ti4+ [57] or substitution Bi with Tb ions [58] reduces the dc conductivity by \nmore than three order of magnitude down to 10−11 Ω-1 cm−1.  \nThe M-H properties of the variously oriented films have also recently been characterized \n[53], using a superconducting quantum interfer ence device (SQUID). The magnetization of the \nfilms at low magnetic field levels has been found to be dramatically increased, relative to that of bulk crystals. This is illustrated in Fig. 9b for a (111)\nc film, where it can be seen that the induced \nmagnetization is on an order of 0.6 emu/g at H ac=1Tesla – more than one order of magnitude \nhigher than that of bulk crystals under the same field.  \n \n  17\n \nFigure 9. (a) P-E response for BiFeO 3 epitaxial thin layers grown on (001) c, (110) c, and (111) c SrTiO 3 substrates by \npulsed laser deposition; and (b) M-H response for a BiFeO 3 epitaxial thin layer grown on (111) c SrTiO 3, plotted \nnext to that of a (111) c oriented BiFeO 3 bulk single crystal [53]. \nIt should be noted that (i ) the value of the saturation magnetization for the (111) c film is \nclose to the spontaneous magnetization observed in bulk crystals at the cycloidal to homogenous \nantiferromagnetic spin transition field of H ≅18 Tesla [4, 51]; and (ii) the structure of (111) c \nBiFeO 3 thin films is identical  to that of bulk crystals – no st ructural phase cha nges were found to \naccompany changes in magnetization and polariz ation. The influence of epitaxy on the LG \nformalism can then be accounted for by adding a rhombic perturbation of K pert to the uniaxial \nmagnetic anisotropy K u. A phase transition from the cycloidal to the homogeneous \nantiferromagnetic spin states will occur at a critical value of the perturbationc\npertK , when the \nenergy of the cycloidal state is equal to that of  the homogeneous one (see Section I.2). This will \noccur when the anisotropy constant fulfills  the critical perturbation condition of \n()\n362 2\n1022 2 cmerg MKAPKs\nus c\npert ×≈ − + >\n⊥χγ.   (30) \nAnalysis has shown that the in-pla ne epitaxial constraint in (111) c BiFeO 3 thin films is sufficient \nto break the cycloida l spin order [53]. \nNotably higher magnetizations of 10emu/g have been reported in (001) c epitaxial layers \n[13]. We know for sure that the latent magnetizat ion of the AFM state can be released: it is much \nlower than reported in [13], but still exceeds by mo re than one order the value of that induced by \nmagnetic field of several Tesla in bulk single crystals. The possibili ty exists that ME films with \nmuch higher magnetizations might be engineered by crystal-chemical design [59]: however, this \nremains controversial [60, 61]. Interestingly, it has recently been shown [62] that La substitution (5 at %) can increase the magnetiz ation by a factor of two in BiFeO\n3; and up to 3.5emu/g for \n(BiFeO 3)1−x–(PbTiO 3)x films [63]. The substitution of Bi with Tb results in ferromagnetic \nmagnetization of 600 G independent of the film thickness [64]. The influence of rare earth \nsubstitution on magnetic properties of BiFeO 3 is discussed in more detail in Section II.2. \nThe enhanced multiferroic properties of BiFeO 3 appear not only as large electric \npolarization and magnetization but al so in enhanced magnetoelectric  effect of 3V/(cm·Oe) [13]. \nThere were several models to explain this phenomenon: an enha ncement due to existence of \nk jiMLP interaction that renormalizes the magnetoelect ric effect [48], and epitaxial constraint \n[53,55]: but it is still remains an open question. \nIn addition, as BiFeO 3 is a rhombohedral perovskite, it crys tallizes in the same structure as \nseveral known half-metallic ferromagnets: such as La 2/3Sr1/3MnO 3, La 2/3Ca1/3MnO 3, and \nSr2FeMoO 6. This makes it possible to combine BiFeO 3 with these other perovskites in \nmultifunctional epitaxial heterostructures. Such hete rostructures could be used in spintronics as \n 18magnetic and ferroelectric tunnel junctions, controllable by el ectric and magnetic fields [ 18]. \nRecently, epitaxial bilayers integrating a BiFeO 3 layer on a La 2/3Sr1/3MnO 3 bottom electrode and \na SrTiO 3 substrate  were successfully grown [65] . Structural, electrical and magnetic properties of \nsaid heterorostructures were studied; and it wa s shown that the magnetic properties of \nLa2/3Sr1/3MnO 3 are preserved and that the BiFeO 3 layers are insulating and ferroelectric down to \nthicknesses of 5 nm. Clearly, BiFeO 3 ultra-thin layers have the potential to fulfill some important \ncriteria for use as ferroelectric tunnel barriers. Moreover, the antiferromagnetic domain \nswitching induced by electric field ha s been demonstrated in BiFeO 3 thin films very recently \n[66] that provide the new possi bility for tunnel switching devices2. \n \nII.2  Effect of rare earth substituents. \nCompounds with the formula RFeO 3 (i.e., rare-earth orthoferrite s) also have a perovskite \nstructure; though, orthorhombically distorted. The introduction of ra re-earth substitu ents can thus \nchange the anisotropy constant, so that the pres ence of spatially modulated structures would be \nenergetically unfavorable. The firs t experiments of this type were  done in the late 1980s - early \n1990s on crystals of Bi 1-x RxFeO 3 (R = La, Gd, Dy) for 0.4< x<0.5 [67-69], which revealed the \npresence of a linear ME effect up to liquid nitr ogen temperatures, as shown in Figure 10a. The \nhigh magnetic field measurements showed that  the presence of lanthanum additives x~0.1 \nreduces the transition field from the spatially  modulated state to hom ogenous [70,71]. Direct \nevidence of cycloid suppression in La-substituted bismuth ferrite ceramics was provided by nuclear magnetic resonance [72]. The NMR spectra for Bi\n1-xLaxFeO 3 for x=0.1, 0.2, 0.3 are \nshown in Figure 10b. It is clearly  seen that the doubled NMR line (corresponding to that of the \ncycloid) transforms into a single broad one  (corresponding to that of the homogenous \nantiferromagnetic state) with increasing La content.  \n  \n \nFigure 10. (a) Magnetoelectric effect in the orthorhombic crystals Bi 1-xRxFeO 3 (plane 001) H ⊥(001) (1) \nBi0.45La0.55FeO 3 T=4.2K (2) Bi 0.55Gd 0.45FeO 3 T=4.2K (3) Bi 0.55Dy0.55FeO 3 T=4.2K (4) Bi 0.45Dy0.55FeO 3 T=77K; [69] \n(b) Spin cycloid suppression in Bi 1-x LaxFeO 3 with increasing of La content: (1) x=0 (2) x=0.1 (3) x=0.2 [72]. \n  \nIn search of BiFeO 3-based materials with enhanced multiferroic properties [73-76], \nattention has also be drawn to solid solutions of BiFeO 3–xPbTiO 3 ceramics substituted with La \non the Bi sites. For Bi 0.8La0.2FeO 3-43%PbTiO 3 [75], an anomalously high polarization (see \n                                                 \n2 This text is dated 2006. Nowdays the electric field control of magnetization is implemented in exchanged coupled \nstructures of Co 0.9Fe0.1 ferromagnetic layer on the BiFeO 3 antiferromagnetic substrate, see Y.H. Chu et al  Nature \nMaterials, v. 7, p. 478 (this comment is made in 2008).  19Fig.11a) and the appearance of a small spontaneous magnetization (see Fig.11b) have been \nreported in the vicinity of a morphotropic phase boundary (MPB) between rhombohedral and \ntetragonal ferroelectric phases, about which numer ous monoclinic and an orthorhombic bridging \nphases have also been reported in relate d ferroelectric perovskites [77-81].  \n \n    ( a )       ( b )  \nFigure 11.  Polarization and magnetization behaviors for Bi 0.8La0.2FeO 3-43%PbTiO 3  ceramics:  (a) P-E response; \nand (b) M-H response. Data for a [111] c oriented BiFeO 3 single crystal is also shown for comparison [75]. \nIt is interesting to note that  MPB solid solutions of BiFeO 3–xPbTiO 3 were initially \nreported in 1962, at the time of the earliest multiferroic studies [77]. However, enhanced \nmultiferroic properties were not reported until recently. Reminding us that persistence is \nimportant in the search for high perf ormance single-phase ME materials! \n   \nPart III. Lesson(s) learne d, and new horizons: ‘ Zhong young zhi dao’,  \n The multiferroic properties of BiFeO 3 result from ‘compromises’. The weak \nantiferromagnetism of the homogeneous spin st ate only arises due to magnetoelectric \ninteractions with an internal effective electr ic field created by the pol arization. This weak \nmagnetization is not expressed in  the zero field conditi on in bulk crystals, ra ther prefers to ‘ sleep \nquietly ’ within a long-period incommensurately modulated  spin structure. It is worth noting that, \nin general, incommensuration itself results from ‘ compromises ’ between competing interactions \n[30]. This homogeneous spin state can be ‘ awakened ’ by ordering fields  that perturb the \nsymmetry of the magnetocrystallin e anisotropy, upon: (i) application of high magnetic fields, in \nbulk crystals; (ii) imposing epitax ial constraints, in thin layers; and (iii) local orthorhombic \ndistortions (or random field-like in teractions) induced by rare-earth  substituents. Thus, the lesson \nthat we should have learned is that of ‘ Zhong young zhi dao’ : the ancient Chinese ‘doctrine of \nthe mean or middle grounds’.   \n We can cite laminate ME composites [82] and self-assembling nano-composites as \nexamples of new horizons offered by understanding this ‘ middle ground’ . Composites assembled \n(at length scales ranging from nm to mm) from  two phases of entirely different symmetries and \nelectronic configurations: allowing polariz ation and magnetization to each have ‘ their best \nspace ’. Thus, systems which reveal the coexis tence of large polarizations and large \nmagnetizations can be fabricated. In the ca se of laminated composites, magnetoelectric \nsusceptibilities as high as αme=5x10-7s/m [or 50V/cm-Oe] have b een reported near the \nelectromechanical resonance frequency, which is nearly five orders of magnitude higher than \nthat of the value for classical magnetoelectric Cr 2O3. In the mentioned (nano- or macro-) \ncomposite systems, the magnetoelectric propert ies are not restricted by the transformation \ncharacteristics of Table II, but rather are pr oduct tensor properties created by the elastic ‘ co-\n 20operation ’ between the magnetostriction of the magne tic phase and the electrostriction of the \npolar one.  \n We conceptually illustrate this approach in Figure 12, following work of Zheng et. al. \n[82], which shows a self-assemble d nanocomposite consisting of CoFe 2O4 nano-rods quasi-\nperiodically dispersed in a BaTiO 3 matrix. This epitaxial film was grown on SrRuO 3/SrTiO 3 \nelectrode/substrate by pulsed laser deposition from an initially two-phase ceramic target. The structures have different lattice parameters, and thus this on-growth morphology is naturally \ncreated to minimize the elastic energy. The le ft-hand figure shows the polarization hysteresis \nloop for the BaTiO\n3 matrix, and the right-hand one show s the magnetization hysteresis loop for \nthe CoFe 2O4 nano-rods3. Clearly, large polarizations and magn etizations co-exist on an intimate \nlength scale in this artificially architectured film . The bottom part of the figure clearly shows that \nthe spontaneous magnetization experiences a cha nge on going through the ferroelectric Curie \ntemperature: demonstrating exchange between the polarization and magnetic subsystems.   \n   \n \nFigure 12.  Illustration of the morphology and properties of self-assembled eptiaxial CoFe 2O4 nano-rods embedded \nin a BaTiO 3 matrix, both deposited epitaxially on top of a SrRuO 3/SrTiO 3 electrode/substrate. Top figure: bright \nfield  image of the nano-composite morpholo gy; left-hand figure: P-E response of BaTiO 3 matrix; right-hand figure: \nM-H response of CoFe 2O4 nano-rods; and bottom figure: magnetization as a function of temperature, illustrating a \nchange near the ferroelectric Curie temperature[82]. \n \n Clearly, composites offer a ‘middle grounds ’ approach to room temperature ME \nmaterials. The system is not single phase; al though the morphological re gularity of Fig.12 has \nsome limited characteristics of a lattice arrangement  of phases. New horizons lay in this nano-\nscale, and not just thin-layered ones that naturally asse mble under epitaxial co nstraint; but, rather \ntruly architecturally-engineered systems are concei vable, if and when they can be made. Using \nsaid approach, we could envisi on bulk materials in which ferr omagnetic and ferroelectric phases \nare intimately mixed on fine scales, in a mann er allowing for continuity of both flux lines, \ncreating an effective medium that appears to be  single phase to electr omagnetic radiation.  \n Finally, we illustrate several simple architecturally-engineered nano-structured BiFeO 3 \nlayers. First, by varying the oxygen pressure during deposition, the dominant phase in a BiFeO 3 \n                                                 \n3 It is worth noticing that self assembled BiFeO 3/CoFe 2O4 columnar nanostructures demonstrate not only \nferroelectric and ferromagnetic hysteresis but also electric field-induced magnetization switching, see F. Zavaliche \net al, Nano Letters,  5, 1793 (2005)  21film has been shown to continuously change from BiFeO 3 to \na mixture of α-Fe 2O3 and γ-Fe 2O3. 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XRD confirms spinel formation with  \nparticle size  in the 15-200 nm  range , with increased size with annealing temperature . \nMicrowave magnetoabsorption data of annealed lithium ferr ite nanoparticles,  obtained \nwith a broadband system based on a network analyzer operating  up to  8.5 GHz  are \npresented . At fields up to 200 mT we can observe a broad absorption peak that shifts to \nhigher frequencies with magnetic field according to ferromagnetic resonance theory . \nThe amplitude of absorption, up to 85 %, together with the frequency width of about 4.5 \nGHz makes this material suitable as wave absorber . Samples annealed at higher \ntemperatures show a behaviour  similar to  polycrystalline samples, thus suggesting their \nmultidomain character . Keywords  A. Magnetic materials , A. Oxides , B. Sol -gel chemistry,  B. Magnetic \nproperties  \n \n1. Introduction  \nFerrimagnetic materials  are widely used for electrotechnical equipment since their \ndiscovery in the forties.  In particular, spinel ferrites are very good choices  for magnetic \nrecording, transformer cores and rod antennas  due to their very high electrical \nresist ivity, low eddy current losses , the possibility of  tailoring the ir magnetic properties  \nby changing composition , easy synthesis and cost production [1]. Lithium ferrite \n(Li0.5Fe2.5O4) exhibit s high Curie temperature, square hysteresis loop, moderate \nsaturation magnetization and good thermal stability of their magnetic properties [2]. \nDue to the absence of divalent iron,  it can also be employed in microwave devices , such \nas circulators, isolators, magnetostatic resonators, filters, switches, limiters and tunable \nelectroptic modulators , replacing YIG for lower mass production costs [3]. Lithium \nferrites are also good candidates in the field of rechargeable Li -ion batteries [ 4]. \n \nLithium ferrite is a soft ferrimagnetic material  with bcc cubic crystal  lattice  and \nmagnetic ordering of  inverse spinel structure , in which Li cation s occupy octahedral B \npositions.  In order to optimize their magnetic properties to the different  microwave \ndevices,  appropriate chemical substitutions can be carried out, especially to reduce  the \nlosses. For this purpose  it is known that  small amounts of Mn  added to Li ferrites \ndecrease the dielectric loss, improve the remanence, and reduce the stress sensitivi ty of \nthe remanence  [5]. Moreover, Mn doping  reduces  porosity, grain size,  \nmagnetocrystalline anisotropy , magnetostriction, and also modifies the coercive field \ndue to either displacement of Li cation to tetrahedral A sites or by reduction of some ferrous iron, this effect being especially remarkable in polycrystalline ferrites  [6]. There \nare several studies that deal  mainly  with structural and dielectric properties of Mn \nsubstituted Li ferrites [7 -10], employing different fabrication techniques different than  \nconventional ceramic technique, because sintering at high temperatures promotes \nevaporation of lithium that made this material  technologically difficult to prepare  in this \nway. The sol -gel method provides an easy alternative for the preparation of  nano sized  \nlithium ferrites  at low er annealing  temperatures . \n \nAt present, due to exponential growth in microwave communication through mobile and \nsatellite communications , there is strong interest in materials that absorb radio \nfrequency energy . For wireless telecommunications , electronic measuring equipment , \nand to avoid interference noise in the frequency range up to 8 GHz, inexpensive, \nlightweight absorbers of electromagnetic  radiation  are needed.  For broadband operation \nmagnetic nano particles  can be used  [2]. Magnetic field induced microwave absorption \nin nanoscale ferrites is a recent and active area of research  useful in this context [11].  In \nthe present work , broadband  microwave magnetoabsorption data of Mn substituted  \nlithium ferrite nanoparticles are presented . \n \n2. Experimental  \n2.1 Sample preparation  \n \nMn-doped lithium ferrite nanoparticles were prepared by sol -gel technique by means of \nPechini method . Starting materials were LiNO 3 (Fluka), Fe(NO 3)3·9H 2O (Merck), \nMnSO 4·H2O ( Merck ), citric acid ( Sigma Aldrich) and etilenglycol ( Fluka). Molar \nratio among  LiNO 3, MnSO 4·H2O and Fe(NO 3)3·9H 2O was kept in 1 :0.01:5 , so that spinel ferrite formation is optimized [6] with 1% mol Mn doping . Molar ratio for  citric \nacid:  metal was 3:1, and citric acid: etilenglycol was 1:2 . In this way,  nitrates and \nsulphates were  solved in distilled water together with citric acid . Solutions were mixed \nwith magnetic stirring for 30 min at room temperature. Then they were put together, \nmixed with etilenglycol and stirred for 1 h. to complete esterification reaction . Gel \nprocessing was achieved by drying at 90º  C during  12h for water releasing , then  heated \nat 150º C during  12h, and finally  at 250º C for 1h. Gel volume grows indicating NO 2, \nO2 and CO 2 releasing . Powders thus obtained were annealed at different temperatures in \nthe 400-1000º  C range . In all cases, the annealing procedure was carried out  with 5 \nºC/min heating rate  and keeping the samples  4h at annealing temperature . This route of \npreparation has revealed to be one efficient and cheap technique to obtain high quality \nnanosized ferrite powder.  \n \n2.2 Measurement setup  \n \nX-ray diffractograms were obtained o n a Siemens D5000  apparatus employing Cu -K \nradiation  (=1.54056 Å) at 40 kV and 30 mA with a curved graphite monochromator, \nan automatic divergence slit (irradiated length  20.00 mm), a progressive receiving slit \n(height 0.05 mm), and a flat plane sample holder in a Bragg -Brentano parafocusing \noptics configuration. Intensity data were collected by the step -counting method (step \n0.02 º/s).  \n \nMagnetic field induced microwave absorption of nickel ferrite nanoparticles has been \nobtained with the help of an automatic measuring system based on a network analyzer \nAgilent model E5071C working from 0.1 MHz  to 8.5 GHz . The sample holder  is placed into the polar pieces of an electromagnet which produc e magnetic fields up to 600 mT \nwith a bipolar DC power supply Kepco B OP 50 -8M. The magnetic field in the sample \nis measured with a gaussmeter  FWBell 6010 with a calibrated perpendicular Hall probe. \nAll the system is controlled with a PC with an appropriate Agilent VEE control \nprogram.  Non-magnetic  sample holder  is placed at the end of a copper shorted semi -\nrigid  coaxial line. The powdered sample is pressed into a toroidal shape that completely \nfills the space between the inner and outer conductors, which are short circuited at the \nend plane of the sample, ensuring that the sample is located in an area with minimum rf \nelectric field and maximum rf magnetic field. Microwave absorption is obtained with \nthe reflected rf signal by means of S 11 parameter, after translating the measurement \nplane to the sample position, and subtracting  the signal obtained with the empty sample \nholder , so that we get only the absorption produced in the sample in the whole \nfrequency range analyzed.  This setup allow the broadband measurement of microwave \nabsorption  and hence the ferromagnetic resonance (FMR)  with varying continuously \nboth the operating frequency and DC magnetic field  [12]. In addition, this measurement \nsetup also allow s the measurement of permittivity and permeability of small amounts of \nmagnetic materials in the above mention ed frequency range  with only the S11 parameter  \n[13]. \n \n3. Results and Discussion  \n \nIn the figure 1 we can see that single phase spinel ferrite  (JCPDS card 17 -115) is \nobtained for all the annealing temperatures analyzed , except  for the as -prepared powder . \nThe average grain diameter has been obtained from them by using the Scherrer’s \nformula for the [311] diffraction peak . Sample particle size  increase s with annealing temperature  in good agreement with the increase in the sharpness of diffraction lines, \nrelated with the effect of annealing temperature on the higher crystallinity of the sample . \nFigures obtained with Scherrer’s formula are the following: 15 nm  for as prepared \npowders at 250º  C, and 25 nm, 40 nm, 100 nm and  205 nm for samples annealed at 400º  \nC, 600º  C, 800º  C and 1000º  C resp . These values are similar to the findings by other \nauthors [ 9, 14]. As expected, the sizes are smaller than Li ferrite samples  with identical \nfabrication but  without Mn doping  (due to the fact that similar sizes are expected for \nthe as prepared samples after identical fabrication route, the size for the as \nprepared sample of Li ferrite is presented here as a reference, because XRD for th e \nMn doped sample cannot produce a representative value of particle size ) [15].  With \nthese average sizes superparamagnetic behaviour is not expected at room temperature . \nHigher annealing temperatures have not been analyzed , due to the  formation of a \nsecondary phase with  lithium ferrate  caused by the  volatility of lithium [ 4]. \n \nIn the figure 2, magnetic permeability data obtained with an LCR at 1 kHz are \npresented. We can see three different behaviour s depending on the annealing \ntemperature: samples annealed at 400º  C- 800º C behave in a similar  way, whereas \npowders obtained at 250º  C without annealing are non -magnetic  due to incomplete \nformation of ferrimagnetic spinel ferrite . Finally,  the sample at 1000º  C has a higher \nrelative ma gnetic permeability  regarding the rest of samples. As we will discuss below , \nit is probably related to the existence of magnetic domain s in this sample . The absence \nof local maxima in these curves suggests that the ferrimagnetic characte r of the samples \nremains in the  temperature range  80 K - 420 K .  \n As a reference of the broadband nature of experimental results, we show in the figure 3a \na 3D plot of the microwave absorption. Similar information  can be displayed  in colour \n2D plots of absorption vs magnetic field o r absorption vs frequency.  In the figure 3b we \ncan see the linear FMR behaviour of the sample annealed at 400º C.  Additionally the \nusual FMR curves of derivative of absorption vs magnetic field  at a fixed frequency  \ncould also be prepared.  \n \nIn the figure 4 we present the results of microwave magnetoabsorption curves of the \ndifferent thermally annealed Mn doped lithium ferrites  (the as -prepared sample at 250º \nC is not shown as it doesn't exhibit microwave absorption) . In each graph the frequency \nbehaviour of the measured reflected rf signal for some selected values of the applied DC \nmagnetic field is shown. In all the curves we observe a single peak of maximum \nabsorption that shifts to higher frequencies with increasing the applied magnetic field. \nIn the frequency range available by our measurement setup, we can observe the \nabsorption  peaks at applied field s up to 200 mT. Higher fields shift the peak to \nfrequencies beyond the capability of our network analyzer. These peaks are very broad \nin frequency (up to 4 GHz half width  in sample sintered at 800º C), so that they could \nbe good candidates to microwave absorbers. In addition, the sample annealed at 1000º  C \nalso exhibits a secondary process near the higher frequency range available by our \nsystem. The behaviour of this sample is rather similar to the results obtained in a sample \nof polycrystalline Li ferrite prepared in similar conditions , which  is also presented  for \ncomparison. It is also noteworthy that the amount of sample needed to fill the sample  \nholder is similar in these last two samples , about 100  mg, and four times  higher than  the \nrest, pointing to a higher densification  of the sample annealed at 1000º C regarding the \nother  samples  annealed at lower temperatures .  \nConcerning the strength  of microwave absorption, we can establish  two different \nbehaviour s: samples sintered up to 800º C have similar figures of about 40 % absorption  \n(about 2% per gram) . On the other hand, in the sample annealed at 1000º C t he \nabsorption increases up to 85 % (i.e. 0.85% per gram) , with a qualitatively different \nbehaviour, because the maximum absorption is obtained without applied field, whereas \nin the rest of the samples the absorption increases with the applied field. The figure of \nmaximum absorption is also closely related to the result obtained with bulk Li ferrite.  \nThe fact that the results of magnetoabsorption qualit atively and quantitavely match,  \npoint to the possibility that sample annealed at 1000º C contains multiple magnetic \ndomains. This fact is supported by the average crystallite size (205 nm) , as observed in \nother magnetic systems [ 12], the higher density of the obtained compound,  comparable \nto polycrystalline ferrite, and the higher magnetic permeability  values  as shown in \nFigure 1.  \n \nThese results resemble to previous ones obtained by us and other authors [ 12, 16], and \ncan be ascribed to ferromagnetic resonance of the spinel ferrite. In order to obtain the \ncharacteristic parameters of FMR, the resonance frequency as a function of the \nmagnetic applied field is represented in the figure 5.  Data for samples annealed at 1000º  \nC and bulk ferrite are difficult to extract due to overlapping of two phenomena:  thermal \nannealing enhance crystallite growth and promotes multidomain structure which  allow \nspin as well as domain wall resonances, that broaden and distort the resonance line  at \nlower frequencies  (this effect can also be present to some extent in the sample annealed \nat 800º C). In addition, the  secondary process near our top frequency range  also affect s the resonance data. With this in mind, we have tried to fit the data to the Kittel \nexpression for FMR in spheres:  \n()A r r B Bgf += ·2·\n\n        (1) \nwith fr the resonance frequency, Br the applied magnetic f ield at resonance , g is the \nspectroscopic splitting factor,  is the gyromagnetic ratio, and B A the effective \nanisotropy field . It is noteworthy that in this frequency range, the anisotropy field \ncannot  be ignored, a s it is similar, or even higher  than the applied magnetic field . The \nresults are not conclusive , but we can extract g -values around 2.18 , similar to the  2.139 -\n2.158  range  found in  literature data  for bulk ferrites  [17]. Anisotropy fields lie in the \nrange from 0.9 kOe for sample annealed at 400º C to 3.30-3.70 kOe for the rest of the \nsamples  (values have been converted to  H in cgs units to allow comparison with \nliterature data) , similar to the figures obtained previously for other nanoparticulate  \nspinel ferrite  system s [11, 12], but one order of magnitude higher than bulk Li ferrites \n[18]. \n \nIn order to obtain additional parameters  to ascertain the validity of our results,  we have \nmeasured  the magnetic parameters  of the sample annealed at 1000º C, obtaining a \nsaturation magnetization  of 57 emu/g  and a coercive field of 250 Oe. We observe that \nthe coercive field  is fairly lower  than the calculated anisotropy field , as it happens in \nmultidomain compounds . MS is lower than bulk magnetization, thus suggesting spin \ndisordering in the surface. We can make a rough estimation of the shell thickness with \nthe expression  \n()Dt\nSBulk S 61· M)D(M − =\n        (2) \n  obtaining a thickness around 5 nm, i.e. 6 lattice steps of surface layer, so that we can \nmodel our samples as a core -shell assembly of interacting particles . The fact that  MS is \n10 % higher than the non-doped  Li ferrite is not surprising, because  the Mn doping  in Li \nferrite polycrystals  promote a diminution of ferrous iron content that appears in the \nannealing process [ 6]. In this case we can also consider that  Mn cations enter into the  \noctahedral sites in the  spinel structure  [8, 19], displacing Li atoms to tetrahedral sit es as \nwell as to the chemically disorder ed surface layer , thus promoting the observed \nmagnetization increase  regarding undoped Li ferrites.  \n \nWith the measured M S and the obtained anisotropy field after FMR fitting, we can \ncalculate the value of the effective anisotropy constant, to get 3.75·104 ±0.55·104 J/m3 \nfor the sample annealed at 1000º C . This value is four times higher than reported for \nbulk lithium ferrites . It is noteworthy that f or magnetic nanoparticles  the value of the \neffective anisotropy constant is determined not only by the contribution of the bulk \nmagnetocrystalline anisotropy, but also by the surface, strain and shape anisotropy, as \nwell as the anisotropy arising from interparticle interactions , and in our case the value \nobtained is very close to the  value of 4·104 J/m3 obtained with a different measurement \napproach  by Wang in a very similar  ferrite  system  [14] assuming a core -shell \nconfiguration of the nanoparticles  annealed at lower temperatures , and multidomain \nbehaviour in the higher size particles , in a similar way as our findings . \n \nWith this in mind, we  can consider the effect of Mn  on the microwave absorption of \nnanoparticle lithium ferrites. It is well known that in polycrystals, Mn substitution  \nprevent the grain growth and reduce porosity. Our results confirm that average \nnanoparticle size is lower than undoped samples annealed at the sa me temperature  [15]. In addition, Mn cations can impede the existence of ferrous cations by appropriate \nvalence change, and then  Mn2+ and some  amount of  Mn3+ are present. Mn2+ has low \ninfluence in anisotropy, but Mn3+ is a Jahn Teller cation that can alter the local crystal \nfields, and hence the anisotropy constant  [6]. It is also known that magnetostriction \nconstants are strongly lowered by Mn addition in Li ferrites [5]. The increase in surface \nto volume ratio produces  a non negligible shell in which th e broken bonds produce a \ndifferent chemical and exchang e effect than in the core. In this ferrite system some Li \ndisplacement  takes place from the core to the shell  [19]. The overall effect  of the above \nmentioned factors  is an increase in surface anis otropy  than can exceed the bulk \nanisotropy up to two orders of magnitude . The increased surface anisotropy and the \ninhomogeneities in the shell broaden  the resonance  [6]. Preliminary measurements show \nthat the  resonance  linewidth is 10% higher in Mn doped Li ferrite nanoparticles than the \nnon doped sample s prepared with the same annealing temperature  [15]. Finally, t he \nobserved variation in the microwave absorption results of the  analyzed samples  \nannealed at different temperatures arise  mainly  from the different particle sizes , because \nthe Mn content do not change , together with the previously mentioned multidomain \nformation at higher annealing temperatures . For lower annealing temperature s, the \nlower particle size promotes a higher anisotropy and surface to volume ratio that causes \na broader and higher absorption.  \n \n4. Conclusions  \n \nMagnetic field induced microwave absorption at frequencies up to 8.5 GHz have been \nmeasured in different thermally annealed Li ferrite nanoparticles obtained with sol-gel \nPechini method.  The results agree well with the theory of ferromagnetic resonance. The wide micro wave absorption observed reveal  that this material can be used as a bsorber in \nthis frequency band.  Nanoparticle Li ferrites have higher effective anisotropy fields than \nbulk ferrites , caused by the increased contribution to the effective anisotropy due to the \nsurface inhomogeneity . \n \nAcknowledgements  \nFunding: This work was supported by the Spanish Ministerio de E conomía, Industria y \nCompetitividad, Agencia Estatal de Investigación  with FEDER , project id. MAT2016 -\n80784 -P \n \nReferences  \n[1] J. Smit, H. P. J. Wijn, Ferrites, Philips Technical Library, 1959 . \n[2] J. Nicolas, Ferromagnetic materials vol. 2, ch. 4, ed. E. P. Wolfarth, North Holland \nPubl. 1980 . \n[3] M. Pardavi -Horvath , Microwave applications of soft ferrites , J. Magn . Magn . Mater . \n215-216 (2000) 171 -183. \n[4] S. S. Teixeira, M. P. F. Graça, L. C. Costa, Dielectric and structural properties of \nlithium ferrites, Spectrosc . Lett. 47 (2014) 356 -362. \n[5] P. Baba , G. Argentina , W. Courtney , G. Dionne , D. Temme , Lithium ferrites for \nmicrowave devices , IEEE Trans Magn. 8(1) (1972) 83 -94. \n[6] G. O. White, C. E. Patton, Magnetic properties of lithium ferrite microwave \nmaterials, J. Magn. Magn. Mater. 9 (1978) 299 -317. \n[7] P. P. Hankare, R. P. Patil, U. B. Sankpal, S. D. Ja dhav, I. S. Mulla, K. M. Jadhav,  B. \nK. Chougule, Magnetic and dielectric properties of nanophase manganese -substituted \nlithium ferrite, J. Magn . Magn . Mater . 321(19) (2009 ) 2977 -3372 . [8] E. Wolska, P. Piszora, W. Nowicki, J. Darul, Vibrational spectra of lithium ferrites: \ninfrared spectroscopic studies of Mn -substituted LiFe 5O8, Int. J. Inorg . Mater . 3 (2001) \n503-507. \n[9] Yen-Pei Fu, Chin -Shang Hsu, Li 0.5Fe2.5-xMn xO4 ferrite sintered from microwave -\ninduced combustion, Solid State Commun. 134 (2005) 201 -206. \n[10] M. J. Iqbal, M. I. Haider, Impact of transition metal doping on Mössbauer, \nelectrical and dielectric parameters of structurally modified lithium ferrite \nnanomaterials , Mater . Chem . Phys . 140(1)  (2013) 42-48. \n[11] G.V. Kurlyandskaya, S.M. Bhagat, C. Luna and M. Vazquez , Microwave \nabsorption of nanoscale CoNi powders , J. Appl. Phys.  99 (2006) 104308 . \n[12] P. Hernández -Gómez,J. M. Muñ oz, M. A. Valente ,  Field -Induced Microwave \nAbsorption in Ni Ferrite Nanoparticles , IEEE Trans Magn. 46(2) (2010) 475 -478. \n[13] D. González -Herrero, J. M. Muñoz, C. Torres, P. Hernández -Gómez, Ó. Alejos, C. \nde Francisco, A new method to measure permittivity and permeability in nanopowder \nmaterials in microwave range , Appl . Phys . A 112(3)  (2013) 719-725. \n[14] H. Yang, Z. Wang, L. Song, M. Zhao, J. Wang and H. Luo. A study on the \ncoercivity and the magnetic anisotropy of the lithium ferrite nanocrystallite , J. Phys. D: \nAppl. Phys. 29 (1996) 2574 -2578 . \n[15] P. Hernández -Gómez, M. A. Valente, M.P.F. Graça, J. M. Muñoz , Synthesis, \nstructural characterization and b roadband ferromagnetic resonance in Li ferrite \nnanoparticles, J. All oys Comp d. 765 (2018) 186 -192. \n[16] S. E. Lofland, H. García -Miquel, M. Vazquez, S. M. Bhagat, Microwave \nmagnetoabsorption in glass -coated amorphous microwires with radii close to skin depth , \nJ. Appl. Phys. 92(4)  (2002)  2058 -2063 . \n[17] W. H. von Aulock, Handbook of Microwave Ferrite Materials , ed. Academic  Press , New York, 1965, p. 407  \n[18] C. J. Brower and  C. E. Patton , Determinat ion of anisotropy field in polycrystalline \nlithium ferrites from FMR linewidths , J. Appl . Phys . 53 (1982)  2104 . \n[19] N. G. Jović, A. S. Masadeh, A. S. Kremenović, B. V. Antić, J. L. Blanuša, N. D. \nCvjetičanin, G. F. Goya, M. V. Antisari and E. S. Božin, Ef fects of Thermal Annealing \non Structural and Magnetic Properties of Lithium Ferrite Nanoparticles, J. Phys. Chem. \nC, , 113 (48)  (2009) 20559 . Figure Captions  \nFigure 1 X ray diffractograms of the as prepared and annealed Mn doped Li ferrite \nsamples.  \n \nFigure 2. Magnetic permeability of the annealed Mn doped Li ferrite nanoparticles . \n \nFigure 3 a) 3D plot of magnetoabsorption vs applied magnetic field and frequency \ncorresponding to sample annealed at 400º  C. b) 2D colour  map of absorption at different \napplied magnetic field vs frequency.  \n \nFigure 4. Microwave absorption as a function of frequency, at different applied \nmagnetic fields (curves shift with increasing applied field from left to right in each \ngraph), corresponding to Mn doped Li Fe2.5O4 nanoparticle samples annealed at 400 º C, \n800º C, and 1000 º C, as well as bulk ferrite sample.  \n \nFigure 5. Resonance frequency as a function of applied magnetic field, corresponding to \nMn doped Li ferrite  nanoparticles annealed at 400 -1000º C, together with bulk ferrite.  \n  \n \nFigure 1  X ray diffractograms of the as prepared and annealed Mn doped Li ferrite \nsamples.   \nFigure 2 . Magnetic permeability of the annealed Mn doped Li ferrite nanoparticles .  \n \nFigure 3  a) 3D plot of magnetoabsorption vs applied magnetic field and frequency \ncorresponding to sample annealed at 400º  C. b) 2D colour  map of absorption at different \napplied magnetic field vs frequency.   Figure 4. Microwave absorption as a function of frequen cy, at different applied \nmagnetic fields (curves shift with increasing applied field from left to right in each \ngraph), corresponding to Mn doped Li Fe2.5O4 nanoparticle samples annealed at 400º  C, \n800º C, and 1000º  C, as well as bulk ferrite sample.   \n \nFigure 5. Resonance frequency as a function of applied magnetic field, corresponding to \nMn doped Li ferrite  nanoparticles annealed at 400 -1000º C, together with bulk ferrite.  \n "    },    {        "title": "2008.13659v3.Power_Module__PM__core_specific_parameters_for_a_detailed_design_oriented_inductor_model.pdf",        "content": "arXiv:2008.13659v3  [physics.app-ph]  20 Oct 2020Power Module (PM) core-speciﬁc parameters for a\ndetailed design-oriented inductor model\n1stAndr´ es Vazquez Sieber\n* Departamento de Electr ´onica\nFacultad de Ciencias Exactas, Ingenier ´ıa y Agrimensura\nUniversidad Nacional de Rosario\n** Grupo Simulaci ´on y Control de Sistemas F ´ısicos\nCIFASIS-CONICET-UNR\nRosario, Argentina\navazquez@fceia.unr.edu.ar2ndM´ onica Romero\n* Departamento de Electr ´onica\nFacultad de Ciencias Exactas, Ingenier ´ıa y Agrimensura\nUniversidad Nacional de Rosario\n** Grupo Simulaci ´on y Control de Sistemas F ´ısicos\nCIFASIS-CONICET-UNR\nRosario, Argentina\nmromero@fceia.unr.edu.ar\nAbstract —This paper obtains shape-related parameters and\nfunctions of a Power Module ferrite core for a design-orient ed\ninductor model, which is a fundamental tool to design any\nelectronic power converter and its control policy. To impro ve\naccuracy, some particular modiﬁcations have been introduc ed\ninto the standardized method of obtaining characteristics core\nareas and lengths. Also, a novel approach is taken to obtain t he\nair gap reluctance as a function of air gap length for that spe ciﬁc\ncore shape.\nIndex Terms —power module ferrite core, ungapped core\nmodel, air gap reluctance model, air gap length computation ,\ncoil former\nI. I NTRODUCTION\nFErrite-core based inductors are commonly found in the\nLC output ﬁlter of voltage source inverters (VSI) [1],\n[2], as energy storage devices in DC/DC converters [3], [4]\nand as line-input ﬁlters in PFC converters [5], [6], among\nmany other power conversion applications. Due to the ferrit e\nmaterial properties, these inductors have to deal with rela tively\nhigh-frequency currents, sometimes being superimposed on\nrelatively large-amplitude low-frequency currents. It is of\nparamount importance to design these inductors in a way that\na minimum inductance value is always ensured which allows\nthe accurate control and the safe operation of the electroni c\npower converter. In order to efﬁciently design these induct ors,\na method to ﬁnd the required minimum number of turns Nmin\nand the optimum air gap length goptto obtain a speciﬁed\ninductance at a certain current level is needed. This method has\nto be based upon an accurate inductor model which needs to\nbe parametrized, among other things, according to the speci ﬁc\ncore shape, size and the air gap arrangement employed.\nIt is a standard practice to use the effective core area Aeand\nlengthlespeciﬁed in the core datasheet to compute the core\nreluctance Rc, which assumes homogeneous core temperature\nand magnetic induction [7], [8]. A detailed inductor model\nwould require the determination of a core cross-sectional a rea\nvectorAcand a core length vector Lc, to describe how an\nungapped core is divided aiming at a better modelling of Rc\nand how the ferrite permeability dependence with magnetic\ndh2\nrh2d2d3\nh3\nd1dh1g\nFig. 1. Typical PM ferrite core\ninduction and temperature impacts on that reluctance. Furt her-\nmore, the necessary air gap length gto be manufactured has\nto be obtained as a function of the required air gap reluctanc e\nRgg, denoted as g(Rgg). This mandates to ﬁrstly develop\nthe inverse model, i.e. reluctance as a function of air gap\nlength,Rgg(g), which requires the input of many mechanicalAc1,Ac6Ac5,Ac10 Ac9 Ac2Ac4 Ac7γ(r) αϕr\nAc8,Ac3Ac8 Ac3\nAc3Ac8Ac10Ac5\nAc10 Ac5Ac6Ac1\nFig. 2. Cross-sectional areas ( Acz) corresponding to the zones in which the PM core is divided\ndimensions very speciﬁc to the ferrite core shape. The usual ly\nsimpliﬁed models frequently require an experimental adjus t-\nment of the gap length [9], [10] and hence a more complex\nRgg(g)is needed for a higher accuracy, especially at relatively\nlarge air gaps. Finally, some dimensions of the correspondi ng\ncoil former are used to determine the winding utilization fa ctor\nkuand the coil DC-resistance RDC.\nIn this paper, all those parameters and functions are obtain ed\nfor a speciﬁc shape of ferrite core: the Power Module (PM).\nA dimensional drawing of a PM core extracted from a typical\ndatasheet is shown in Figure 1. The core depicted has only one\nair gap placed in the central leg where the two core halves are\nfaced (here referred to as qg= 1). Two more gaps similarly\nlocated but on each of the external legs can be introduced if\na spacer is placed between two equal ungapped ferrite pieces\n(here referred to as qg= 2). This paper is organized as follows.\nIn section II, vectors AcandLcfor an ungapped core are\nobtained. An accurate function Rgg(g)is obtained in section\nIII. A simple method to obtain g(Rgg)is presented in section\nIV. Section V determines the winding height wh, winding\nwidthwwand average turn length ltbased on the dimensions\nof the coil former. Finally, conclusions are presented in se ction\nVI.\nII. U NGAPPED CORE REGIONS\nA comprehensive inductor model should divide the fer-\nrite core into a number of Zparts, where Ac=\n[Ac1... Acz... AcZ]andLc= [lc1... lcz... lcZ]. The PM\ncore is then sectorized in Z= 10 regions in full compliance\nwith [11] and so z= 1...10. The location of the magnetic\npathLc, cross-sectional areas Aczand core lengths lcz, which\nare shown in Figure 2 and Figure 3, are also determined\naccording to [11], with the exceptions that are detailed nex t.\nAll the needed dimensions to apply [11] come from Figure 1\nsupposing an ungapped core, i.e. g= 0, using the average\nFig. 3. Magnetic path length divisions ( lcz) corresponding to the zones in\nwhich the PM core is divided\nvalues detailed in the corresponding core datasheet, like f or\nexample [13].\nWe have observed in several numerical examples that\nAc3=Ac8calculated following [11] yield smaller values than\nthe correspondingly obtained for Ac1. This seems to be in\ncontradiction with [12] which states that the minimum cross -sectional area ( Amin) should coincide with Ac1in all PM\ncores models. To save this discrepancy, we propose ﬁrstly th at\nlc3andlc8are equal to the distance between the central and\nthe external legs, shown in Figure 3. Secondly, Ac3andAc8\nare then adjusted in such a way that\nlc3\nAc3=lc8\nAc8=1\nh2−h1/integraldisplayd2\n2\nd1\n2dr\nrγ(r)(1)\nwhererandγ(r)are shown in Figure 2 for regions z= 3and\nz= 8. The deﬁnite integral in (1) has to be numerically solved.\nThe values of Ac3=Ac8in this way obtained comply with\n[12]. Other slight divergence exists in the calculation of lc4\nandlc9. This takes into account the alteration of the magnetic\npath situation along the external legs, due to the notches in\nAc5andAc10, introducing the correction factor sx, which is\nneglected in [11].\nFinally, the equations for each lczandAczare\n•lc1=lc6=h1\n2\n•lc2=lc7=π\n16(h2−h1+2sy)\n•lc3=lc8=d2−d1\n2\n•lc4=lc9=π\n16(h2−h1+2sx)\n•lc5=lc10=h1\n2\n•Ac1=Ac6=π\n4(d2\n1−d2\nh1)\n•Ac2=Ac7=π\n4/bracketleftBig\nd2\n1−d2\nh1\n2+d1(h2−h1)/bracketrightBig\n•Ac3=Ac8=(h2−h1)(d2−d1)\n2/integraltextd2\n2\nd1\n2dr\nrγ(r)\n•Ac4=Ac9= (d2\n3−d2\n2)π\n2−α\n4−π\n4dh2rh2+π\n2−α\n2d2(h2−h1)\n•Ac5=Ac10= (d2\n3−d2\n2)π\n2−α\n2−π\n2dh2rh2\nwhere the correction factors are\nsy=d1−/radicalbigg\nd2\n1+d2\nh1\n2\nsx=/radicalBigg\nd2\n3+d2\n2\n2−2π\nπ\n2−αdh2rh2−d2\nand the required angular variables are\nα= arcsin/parenleftbiggh3\nd2/parenrightbigg\nγ(r) =π−2 arcsin\n\nsin(π−ϕ)\nr\nd1\n2cos(π−ϕ)\n+/radicalBigg\nr2−d12\n4[1−cos2(π−ϕ)]\n\n\n\nϕ= arctan/parenleftbiggh3\nd2cos(α)−d1/parenrightbigg\nNote that regions located at same relative places in their\nrespective core pieces, actually have same Aczandlcz. In\nfact, [11] considers the core divided into only ﬁve zones,\nFig. 4. Half external gap in PM core. For R′\nbxe:l=g\n2,w= (π−2α)d3+d2\n4\nandh=h2\n2. ForR′\nbyie:l=g\n2,w=w(x)andh=h1\n2. ForR′\nbyee:l=g\n2,\nw=w(x)andh=h2\n2.\ngrouping the pairs with same Aczinto a single area with\nits length doubled, to calculate core effective area Aeand\neffective length le. However, taking Z= 10 does not add an\nextra calculation effort and allows to consider a different core\ntemperature in each section with equal cross-sectional are a, by\nusing a core temperature vector Tc= [Tc1... Tcz... TcZ].\nIII. A IR GAP RELUCTANCE Rgg\nIn a gapped PM core having qg= 2,Rggis composed by\nthe air gap reluctance placed on the central leg, Rggcplus\nthe combined air gap reluctances located in the external leg s,\nRgge. Assuming an equal distribution of the magnetic ﬂux in\nboth external legs\nRgg=Rggc+Rgge (2)\nIfqg= 1, thenRgge= 0.\nA. External reluctance Rgge\nThe fundamental idea used to determine Rgge, shown in\nFigure 4a, is to transform each half of the curved surfaces Ac5\nandAc10into rectangular ones having the same cross-sectional\narea, maintaining the same gap length g. In this way, the 3D\nreluctance of this equivalent air gap is addressed using the\napproach of [14]. The computation starts considering a basi c\nair gap disposition where the 2D reluctance per unit of lengt h\nis known. This basic 2D reluctance is referred to as R′\nbasic,\nR′\nbasic=1\nµo/bracketleftbigw\n2l+2\nπ/parenleftbig\n1+lnπh\n4l/parenrightbig/bracketrightbigand is depicted in Figure 5. R′\nbasic considers the fringing\neffects in the magnetic ﬂux near the air gap in the x-zplane\nbut it assumes an inﬁnite length in the ydimension, i.e.\nthere is no fringing effects in this direction. For the core\ndimensions shown in Figure 4b ( x-zplane), the actual air gap\nis decomposed into a parallel/series structure comprised o f\nbasic gap dispositions each having a reluctance R′\nbxeto obtain\nthe 2D reluctance per unit of length, considering no fringin g\neffects towards the center of the core, i.e. in the ydimension.\nThis total reluctance obtained is referred to as R′\nxe. Next, the\nfringing factor σxeis obtained as the relationship between R′\nxe\nand the 2D reluctance per unit of length neglecting the fring ing\neffects. Accordingly, considering the fringing effects in thex-\ndirection, the basic 2D reluctance per-unit-of-length R′\nbxe, the\ntotal 2D reluctance per-unit-of-length in the x-zplaneR′\nxeand\nthe fringing factor σxewould be\nR′\nbxe=1\nµo/bracketleftBig\n(π−2α)(d2+d3)\n4g+2\nπ/parenleftBig\n1+lnπh2\n4g/parenrightBig/bracketrightBig\nR′\nxe=2R′\nbxe\n2=R′\nbxe\nσxe=R′\nxe\n4g\nµo(π−2α)(d2+d3)\nFor the y-direction, the y-zplane shown in Figure 4c is\nconsidered, noting that now there is no fringing ﬂux in the\nx-dimension. Here there are two basic reluctances R′\nbyee and\nR′\nbyie since the distance from the gap to the next corner of\nthe core in the external side is different from the internal s ide\nof the core. Moreover, due to the small notch of radio rh2\non the outer side of the core, the air gap has no uniform\nwidth. In this situation, the gap width w(x), the inner and outer\nbasic reluctances per unit of length, R′\nbyie(x)andR′\nbyee(x)\nrespectively, are used to ﬁnd the fringing factor σye(x)all\nalong the gap length. Accordingly,\nR′\nbyee(x) =1\nµo/bracketleftBig\nw(x)\ng+2\nπ/parenleftBig\n1+lnπh2\n4g/parenrightBig/bracketrightBig\nR′\nbyie(x) =1\nµo/bracketleftBig\nw(x)\ng+2\nπ/parenleftBig\n1+lnπh1\n4g/parenrightBig/bracketrightBig\nwhere\nw(x) =\n\nd3−d2\n2ifa >|x|> b\nd3−d2\n2−rh2/radicalbigg\n1−4/parenleftBig\nx\ndh2/parenrightBig2\nifb≥ |x|\n\n\na=(π−2α)(d2+d3)\n8b=dh2\n2\nTo obtain σye(x), the same reasoning used before for the\nx-zplane is now applied to the y-zplane, leading to\nR′\nye(x) = 2R′\nbyee(x)//2R′\nbyie(x) =2R′\nbyie(x)R′\nbyee(x)\nR′\nbyie(x)+R′\nbyee(x)\nσye(x) =R′\nye(x)\ng\nµow(x)\nFig. 5.R′\nbasic\nFig. 6. Central gap in PM core. For R′\nbyic:l=g\n2,w=d1−dh1\n2and\nh=h1\n2−e. ForR′\nbyec:l=g\n2,w=d1−dh1\n2andh=h2\n2−e.\nTo take into account the overall effect of the non homogeneou s\ngap width, an average fringing factor σyis obtained as\nσye=4\n(π−2α)(d2+d3)/integraldisplay(π−2α)(d2+d3)\n8\n−(π−2α)(d2+d3)\n8σye(x)dx\nAfter some manipulation and ﬁnally applying symbolic inte-\ngration tools, σyeturns to be\nσye=a\na+c+8\n(π−2α)(d3+d2)\n\ndh2\n2/parenleftbigg\n1−a\na+c/parenrightbigg\n+c\n2rh2\ndh2/braceleftBigg\nπ\n2−a+c/radicalbig\n(a+c)2−rh22/bracketleftBigg\nπ\n2\n+arctan/parenleftBigg\nrh2/radicalbig\n(a+c)2−rh22/parenrightBigg/bracketrightBigg/bracerightBigg\n\n\na=d3−d2\n2\nc=2g\nπ/parenleftbigg\n1+lnπ√h1h2\n4g/parenrightbiggFig. 7. Typical coil former for a PM core\nThe reluctance of an air gap lacking of fringing ﬂux is given\nby\nRggo=g\nµoAcg\nwhereAcg=Ac5is the core area in contact with the\ngap. The fringing factors σxeandσyetake into account the\nfringing ﬂuxes by introducing the effective air gap area Agee\nthat virtually increases Acgin bothxandydirections, thus\nreducing the reluctance. Consequently, the actual Rggecan be\ncalculated as\nRgge=σxeσyeRggo=g\nµoAgee(3)\nAgee=Ac5\nσxeσye\nB. Central reluctance Rggc\nThe center gap reluctance Rggcis obtained by considering\nthe rounded central leg with its centered hole as an unfolded\nannulus as shown in Figure 6a, transforming Ac1andAc6\ninto equivalent rectangular surfaces. In the x-zplane shown in\nFigure 6b, it is assumed that no fringing ﬂux exists in the x-\ndirection since there is an endless closed ring. On the contr ary,\nin the y-zplane shown in Figure 6c, there is one side in contact\nwith the windings and the other side in contact with the small\ncentral hole. Note that the actual fringing ﬂux in this side w ill\nbe more constrained to ﬂow than the existing over the opposit e\nside. Thus in this case, the formulae used before cannot be\napplied in a straightforward way. As a workaround, we propos e\nthe use of a basic 2D reluctance per-unit-of-length that is a\ngeometric average ( R′\nbyec) between two extremes. On the one\nhand, the case with no central hole and thus no fringing ﬂux\n(R′\nbyenf ) and in the other hand, the case where the hole radius\nis large enough to impose no restrictions over the fringing ﬂ ux\nto circulate ( R′\nbyef), like at the other side of the central leg.\nConsequently, in the x-direction\nσxc= 1and in the y-direction we get\nR′\nbyef(e) =1\nµo/bracketleftbigg\nd1−dh1\n2g+2\nπ/parenleftbigg\n1+lnπ(h2\n2−e)\n2g/parenrightbigg/bracketrightbigg\nR′\nbyenf=2g\nµo(d1−dh1)\nR′\nbyec(e) =/radicalBig\nR′\nbyef(e)R′\nbyenf(4)\nR′\nbyic(e) =1\nµo/bracketleftbigg\nd1−dh1\n2g+2\nπ/parenleftbigg\n1+lnπ(h1\n2−e)\n2g/parenrightbigg/bracketrightbigg\nR′\nbyc(e) =R′\nbyec(e)R′\nbyic(e)\nR′\nbyec(e)+R′\nbyic(e)\nR′\nyc=/braceleftBigg\nR′\nbyc(e=g)+R′\nbyc(e= 0) ifqg= 1\n2R′\nbyc(e= 0) ifqg= 2/bracerightBigg\nσyc=R′\nyc\n2g\nµ0(d1−dh1)(5)\nBeing now Acg=Ac1, the actual Rggcturns to be\nRggc=σxcσycRggo=g\nµoAgec(6)\nAgec=Ac1\nσxcσyc\nIV. A IR GAP LENGTH g\nAccording to (2), to obtain gas a function of Rgg, the\nimplicit equation\nRgg=/braceleftbiggRggc(g) ifqg= 1\nRggc(g)+Rgge(g)ifqg= 2/bracerightbigg\n(7)\nhas to be numerically solved. As it is shown in [14], Rgg=\n0⇔g= 0 andRgg(g)is increasing with g. This makes\neasy to invert Rgg(g)by ﬁnding the solution of (7) within the\nclosed interval [g∗\nmin,g∗\nmax]deﬁned next, employing a simple\nroot-solver like Matlab’s fzero function. An adequate g∗\nmin\nwould be\ng∗\nmin=µ0AcgN2\nLi\nAcg=/braceleftbiggAc1 ifqg= 1\nAc1Ac5\nAc1+Ac5ifqg= 2/bracerightbigg\nwhich corresponds to the ideal case of no existing fringing ﬂ ux\nand initial permeability µi→ ∞ ;Liis the initial inductance\nandNis the number of winding turns of the current inductor\nbeing designed. If qg= 2,g∗\nmax should be selected as\n3g∗\nmin≤g∗\nmax≤10g∗\nmin\nin most design cases, though there is not a deﬁnite limit\nbecause the spacers can be made as thick as desired. However,\nifqg= 1then we are limited to g∗\nmax=h1\n2which is the height\nof a core piece central leg. In this case, if the required gwere\ngreater than g∗\nmax, thenqg= 2 or a larger core should be\ninitially chosen.Fig. 8. Alternative dispositions of the winding inside the c oil former\nV. W INDING HEIGHT ,WINDING WIDTH AND AVERAGE\nTURN LENGTH\nThe drawing of a typical coil former for PM cores is\nshown in Figure 7. The minimum winding width wwand the\nmaximum winding height hware\nww=h1cf−2h2cf\nhw=d2cf−d1cf\n2\nThe winding is arranged inside the coil former according to\none of the alternatives depicted in Figure 8. The average len gth\nof a winding turn ltas a function of the winding thickness tw\nis then\nlt(tw) =π/parenleftbig\nd1cf+2dwb+tw/parenrightbig\n(8)\nwhere the distance between the winding and the bobbin dwb\nis\ndwb=\n\n0 inward alignment (cf. Figure 8.a)\nhw−tw\n2centered alignment (cf. Figure 8.b)\nhw−twoutward alignment (cf. Figure 8.c)\n\n\nVI. C ONCLUSION\nThis paper thoroughly develops the required core-speciﬁc\nparameters and functions exclusively related to the shape o f\nPower Module (PM) cores, which are needed by a comprehen-\nsive design-oriented inductor model. This is a fundamental tool\nto properly design and control any type of electronic power\nconverter. The same procedures followed in this paper can be\neasily adapted to obtain the required data for other shapes\nof ferrite cores. Further work will be devoted in developing\nthe exact expression of (4), the external part of the central\ngap basic reluctance in the y-z plane R′\nbyec, by extending the\nreasoning of [14] to this particular structure. Experiment al\nmeasurements of the initial inductance of several PM-core\nbased inductors are in process, showing so far very good\nagreement with theoretical predictions.ACKNOWLEDGMENT\nThe ﬁrst author wants to thank Dr. Hernan Haimovich for\nhis guidance and constructive suggestions.\nREFERENCES\n[1] E. Gurpinar and A. Castellazzi, Tradeoff Study of Heat Sink and Output\nFilter Volume in a GaN HEMT Based Single-Phase Inverter , in IEEE\nTransactions on Power Electronics, vol. 33, no. 6, pp. 5226- 5239, June\n2018.\n[2] R. Otero-De-Leon, L. Liu, S. Bala and G. Manchia, Hybrid active power\nﬁlter with GaN power stage for 5kW single phase inverter , 2018 IEEE\nApplied Power Electronics Conference and Exposition (APEC ), San\nAntonio, TX, 2018, pp. 692-697\n[3] B. Ray, H. Kosai, J. D. Scoﬁeld and B. Jordan, 200°C Operation of a\nDC-DC Converter with SiC Power Devices , APEC 07 - Twenty-Second\nAnnual IEEE Applied Power Electronics Conference and Expos ition,\nAnaheim, CA, USA, 2007, pp. 998-1002.\n[4] L. Schrittwieser, J. W. Kolar and T. B. Soeiro, 99% Efﬁcient three-phase\nbuck-type SiC MOSFET PFC rectiﬁer minimizing life cycle cos t in DC\ndata centers , CPSS Transactions on Power Electronics and Applications,\nvol. 2, no. 1, pp. 47-58, 2017.\n[5] A. Leon-Masich, H. Valderrama-Blavi, J. M. Bosque-Monc us´ ı and L.\nMart´ ınez-Salamero, A High-Voltage SiC-Based Boost PFC for LED\nApplications , IEEE Transactions on Power Electronics, vol. 31, no. 2,\npp. 1633-1642, Feb. 2016.\n[6] Q. Li, H. Y . Zhao and J. C. Song, Design and loss analysis of the\nhigh frequency PFC converter , 2015 IEEE International Conference\non Applied Superconductivity and Electromagnetic Devices (ASEMD),\nShanghai, 2015, pp. 124-125.\n[7] C. Wm. T. McLyman, Transformer and Inductor Design Handbook ,\n3rd ed. New York, USA: Marcel-Dekker, 2004.\n[8] M. K. Kazimierczuk, High-Frequency Magnetic Components , 2nd ed.\nWest Sussex, England: John Wiley & Sons, 2014.\n[9] N. Mohan, T. M. Undeland and W. P. Robbins, Power Electronics.\nConverters, Applications, and Design , 3rd ed. New York, USA: John\nWiley & Sons, 2003.\n[10] A. Van den Bossche and V . C. Valchev, Inductors and Transformers for\nPower Electronics , 1st ed. Boca Raton, USA: Taylor & Francis, 2005.\n[11] Calculation of the effective parameters of magnetic piece p arts, IEC\n60205 edition 3.1, International Electrotechnical Commis sion, Geneva,\nSwitzerland, Aug. 2009.\n[12] PM-cores made of magnetic oxides and associated parts - Dime nsions ,\nIEC 61247 edition 1.0b., International Electrotechnical C ommission,\nGeneva, Switzerland, 1995.\n[13] EPCOS AG, PM 62/49. Core and accessories , Ferrite\nand Accessories, May 2017. [Online]. Available://www.tdk -\nelectronics.tdk.com/inf/80/db/fer/pm 6249.pdf.\n[14] J. M¨ uhlethaler, J. W. Kolar and A. Ecklebe, A Novel Approach for 3D Air\nGap Reluctance Calculations , in Proc. of the International Conference\non Power Electronics - ECCE Asia, pp. 446-452, Jun. 2011."    },    {        "title": "1902.08797v1.Tailoring_the_crack_tip_microstructure__A_novel_approach.pdf",        "content": "1 \n Tailoring the crack -tip microstructure: A novel approach  \nKhilesh Kumar Bhandari a,*, Arka Mandal a, Md. Basiruddin Sk a, Arghya Deb b, \nDebalay Chakrabarti a \na Department of Metallurgical and Materials E ngineering, Indian Institute of Technology, \nKharagpur,  Kharagpur -721302, India  \nb Department of Civil Engineering, Indian Institute of Technology, Kharagpur, Kharagpur -\n721302, India  \n \n* Corresponding author (Last name, First name):  \nBhandari, Khilesh Kumar  \nEmail id: khileshbha009@gmail.com , khileshkrbhandari@iitkgp.ac.in  \nAddress: Department of Metallurgical and Materials Engineering, IIT Kharagpur, \nKharagpur – 721 302, West Bengal, India . \nContact details: (+91) 9547976523  \n \nCo-authors (Last name, First name):  \n \nMandal, Arka  \nEmail id: arkametbesu@gmail.com  \n \nSk, Md. Basiruddin  \nEmail id: basiruddin.sk@gmail.com  \n \nDeb, Arghya  \nEmail id: arghya@civil.iitkgp.ac.in  \n \nChakrabarti, Debalay  \nEmail id: debalay@metal.iitkgp.ernet.in  \n \n 2 \n Tailoring the crack -tip microstructure: A novel approach  \nKhilesh Kumar Bhandari a,*, Arka Mandal a, Md. Basiruddin Sk a, Arghya Deb b, \nDebalay Chakrabarti a \na Department of Metallurgical and Materials E ngineering, Indian Institute of Technology, \nKharagpur, Kharagpur -721302, India  \nb Department of Civil E ngineeri ng, Indian Institute of Technology, Kharagpur, Kharagpur -\n721302, India  \n \nAbstract  \nThis investigation demonstrates  a way to innovatively modify the ferritic mi crostructure at a \nlocal scale, particularly at the failure prone area such as Charpy V -notch ( CVN ) root. Tensile \npre-strain (PS) up to 6 percent and 12 percent  were employed before annealing  (An)  the \nsamples at 650°C for 15 minute s. Ferrite grain size increased sharply and gradually (along the \ndistance ahead of  the notch root) within the microstructural ly modified region in 6 -12 percent \npre-strained and annealed  samples , respectively. Critical strain which promotes strain induc ed \nboundary migration (SIBM), was  found to be 0.1 which resulted in abnormally coarse ferrite \ngrains.  \nKeywords:  Tensile pr e-strain; Annealing; Recrystallized microst ructure; Plastic deformation; \nGrain boundary migration . \nIntroduction  \nFailure of a  structural component takes place  when a crack propagates under application of an \nexternal load. The b rittle failure is more catastrophic i n nature than the ductile failure. Since, \nbrittle failure occurs via propagation of nucleated micro -cracks  [1], the role of \nmicrostructural parameters need s to be carefully reviewed . So far, the studies  [2–6] mostly \ninvestigated the effect of several micr ostructural parameters, such as grain size, second -phase \nparticle size, fraction of different microstructural constituents  and crystallographic orientation  \non the resistance to brittle (cleavage) crack propagation.  An overall fine grain or bim odal \ngrain structure is reported  to possess b etter mechanical pr operties  in ter ms of the strength -\ntoughness combination  [7,8]. In a bimodal grain structure, the fine grains provide high \nstrength and resistance to crack propagation, whi le the coarse grains contribute  to strain 3 \n hardening ability and the overall ductilit y [9,10] . Nevertheless, producing such a functionally \ngraded microstructure over a lar ge structural section/component is a huge challenge . \n Several studies have bee n carried out to understand the effect of pre -strain on various \nmechanical properties and the as sociated failure mechanisms [11–17]. Pre -straining at room \ntemperature can alter dislocation substructure by acute dislocation i nteraction and \nmultiplication [11], resulting in an increased  yield and tensile strength [12,13] . However , \nductility presumably follows an inverse relationship with the degree of pre-straining [12–14]. \nFractu re toughne ss has been reported to be decreased continuously as a function of pre -\nstraining [16,17] , while some investigators also found that it decreased  after a certain small \npre-strain level  [15]. Besides, pre -straining makes the material  susceptible to brittleness and \nmaterial can f ail at a smaller strain  on subsequent reloading , as reported  for low carbon steel \n[14]. \n Unlike above mentioned studies where homogeneous pre -straining is practised, \nloading/pre -straining of notched/pre -cracked specimen inevit ably influences the stress and \nstrain distribution by forming plastic zon e ahead of crack  tip [18]. The concept of stress \nsingularity as estimated by elastic stress field equations is no t effective at crack tip, due to the \npresence of the plastic zone [19]. By using the elastic stress f ield equation for σ yy in plane θ = \n0, the crack tip plastic zone size (rp*) can be quantitatively estimated by \nequation\n , where \n  is the stress intensity factor in crack opening mode and \n is the yield stress. Later, the  crack was considered to be longer than its actual size (i.e. a eff \n= a act + δ), by Irwin,  in order to eliminate the ambiguity in plastic zone size estimation and it \nwas found to be twice of the r p* [20]. Howeve r, the actual estimation of the plastic zone \nshape can be obtained by considering the entire range of theta θ (± 90°), instead of a circular \nshape.  \n When a material deform s, inhomogeneity is induced in the form of dislocation s and \npoint defects.  To bring the material in its lower energy configuration,  subsequent annealing \nresults in partial recovery by the annih ilation and rearrangement of those defects. Recovery is \na series of events where the initially tangled dislocations fir st lead to cell formation [21], then \nsub-grain develo pment and its growth. These phenomena are strongly dependent on factors \nlike amount of prior strain, deformation and annealing temperature.  In addition, \nrecrystallization is another energy minimization process (governed by internal stored energy \nin cold wo rked metal) where new strain -free grains form and grow subsequently.  4 \n Recrystallization is discontinuous and generally involves large strain relaxation as compared \nto recovery.  The recrystallization kinetics is very similar to that of phase transformation  [21] \nand can get affected by processing parameters (amount of prior strain, strain rate, \ndeformation temperature, mode of deformation, annealing time and temperature) and by \nmicrostructural parameter s (initial grain size, second -phase particle, grain orientation)  [22–\n24]. \n Recrystallized grains having high surface energy , which makes the structure unstable, \ncan further grow . Grain growth decreases the grain boundary area, resulting in reduction of \nthe total surface energy. This process can be classified either as normal or abnormal grain \ngrowth  [25]. Former is a continuous process where all grains constituting the material grow \nuniformly.  Latter occurs when normal grain growth is inhibited, except for some grain s \nhaving favourable condition to grow at the expense of neighb ouring recrystallized grains \n[26]. Abnormal grain growth is more pronounced in alloy s having second -phase particles, \ncausing Zener pinning [27–29]. Other factors, e.g. crystallographic texture and surface effect \ncan also  contribute to this process [26,30] . Abnormal grain growth can also happen in a \npartially recrystallized microstructure, where the boundary of the strain -free recrystallized \ngrain moves to consume t he neighbouring strain matrix containing deformed/recovered \ngrains. This phenomenon is known as  SIBM [31,32] . \n Considering the difficulty to generate the bimodal grain structure  in large component, \nthe present study ai ms to develop suitable microstructure at a local scale, particularly at the \nfailure prone area (such as the crack front ) in a single phase ferritic  steel by tensile pre-\nstraining and subsequent annealing.  \nMaterial s and Methods  \nStandard sub -size Charpy V -notched (5x10x55 mm) specimens were extracted from a fully \nferritic steel with the average composition of 0.03 C, 0.25 Mn, 0.01 S, 0.02 P, 0.008 Si, 0.07 \nAl, 0.02 Cr, 0.02 Ni, and 0.0036 Ni (wt. %) . Investigated material consisted of equiaxed \nferrite grains with average grain size of 37 ± 7 µm. T ensile pre -strain up to 6 percent  (6%PS) \nand 12% (12%PS) were employed before annealing  (An)  at 650°C for 15 minutes under inert \natmosphere  (to prevent any damage at the notch tip) . A sample without pre -straining (0%P S) \nwas also annealed at the same condition for comparison. An elastic -plastic finite element \n(FE) simulation was  performed in  ABAQUS /explicit  software to quantitatively estimate the 5 \n stress -strain distribution ahead of notch root  with root radius of 0.25 mm  after pre -straining . \nMicrostructural characterization was performed using optical microscopy, electron back -\nscatted diffraction (EBSD) and nanoindentation after conventional preparation of the \nsamples.  \nResult s \nBasic power hardening law, \n  was considered for \nFE simulation in ABAQUS, where , elastic modulus, E = 210 GPa and Poisson’s ration, ν = \n0.27. Plastic strain conto ur maps revealing the plastic zone in figure 1(a, b)  display that the  \nfarthest boundary from  notch tip  is located at an angle of ~58.5 ° and ~48° in 6%PS and \n12%PS sample s, respectively, in comparison  to 69 °, as reported by Tuba [33]. The increase \nin pre -strain significantly increases the plastic zone siz e as in figure 1(a, b) . The shape of the \nplastic zone obtained from FE simulation in this investigation is in good agreement with the \nfindings rep orted by Tuba [33] and Hahn and Rosenfield [34]. Figure 1(c, d ) shows the stress \nand strain distribution as a function of distance from the notch tip in 6  percent and 12 percent \npre-strained samples . An increase in stress  and strain values is observed as the degree of pre -\nstrain increased , while the distribution followed same pattern . The maxi mum normal stress, \nσyy, for both pre -strained samples has values more than 2.5  times the ge neral yield stress, \nfigure 1(c) , which is close to the theoretical values of 3 [19]. On the other hand, the plastic \nstrain, ε yy, attains its maxima  at the notch tip and decreases sharply moving away from the \nnotch tip  (figure 1(d) ). \n Ferrite grain size remained unchanged , however, a minimal directionality  in grai n \nmorphology just ahead of the notch tip  was observed in the pre-strained samples . Anneal ing \nof 0%PS sample did  not show considerable microstructural alteration in figure 2(a ). \nMoreover, Figure  2(b, c ) clearly display the regions over which microstructura l modification \nhave occurred after annealing in accordance wit h the plastic zone size of t he 6-12% pre-\nstrained sample . The modified region revealed substantial gradient in grain size; from finer \nferrite grains (~25 µm) at notch root to abnormally coarse  grains (~200 -300 µm) at  the end  of \nthe modified region. Interestingly, 12%PS -An sample exhi bited a gradual increase in grain \nsize with the coarsest grains (~200 -250 µm) located towards the end of the modified region \n(3 mm away from notch root), figure 2(c ). In contrast , 6%PS -An sample displayed steep \ngrain -size gradient with the coarsest grains (~270 -300 µm) situated at a distance of just 1 mm \naway from the notch tip, figure 2(b ) (refer grain size distribution  (right)  in figure 2(b, c) ). 6 \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 2 . IPF ND maps revealing enclosed modified region (left), optical microstructure showing modified \nferrite grains (centre), and corresponding ferrite grain size distribution with distance from notch tip (right ) \nin pre -strained and annealed (PS -An) samples: (a) 0%PS -An, (b) 6%PS -An, and (c) 12%PS -An. \n 7 \n  \n \n \n \n \n \n \n \n \n \n \n \n Grain orientation spread (GOS) map in figure 3  is an effective measur e of the strain \nin terms of misorientation, particularly within a grain.  Average GOS value is calculated by \naveraging deviation between the orientation of each da ta point within a grain and average \norientation of that grain. A small GOS value (0.56) repres ents the characteristic of an \nannealed structure in 0%PS sample, figure 3(a) . The pre -straining of CVN samples induced \nplastic deformation of material ahead of notch root, forming plastic zone  and resulted in \nhigher GOS value in 6%PS (2.26) and 12%PS (2.80 ) samples, figure 3(b, c) , ahead of the \nnotch tip . Supposedly, GOS value did not change in 0%PS -An sample, figure 3(d) , since \nthere was no prior strain which could be released during annealing.  The modified region \nobtained after annealing the 6%PS and 12%P S samples, figure 3(e, f) , divulged strain -free \ngrains as a result of strain relaxation . Additionally, the unmodified region surrounding this \nmodified region still possessed deformed structure  as revealed in the corresponding GOS \nmaps.   \nAverage in -grain mi sorientation (avg. GOS distribution ) ahead of the notch root is \ncalculated by creating square subsets at distances and plotted as a function of distance from \nthe notch tip, figure 4(a) . GOS values follow  an increasi ng trend with distance in 6%PS -An \nFigure 3 . Grain orientation spread (GOS) maps, obtained by EBSD analysis to qualitatively display \nstrains in the investigated samples: (a) 0%PS, (b) 6%PS, (c) 12% PS, (d) 0%PS -An, (e) 6%PS -An, and \n(f) 12%PS -An. \n 8 \n and 12% PS-An samples, starting from the lowest (0.28) at the notch tip. Presence of a \ndeformed region ahead of the modified region, which remained unaltered after annealing, was \nconfirmed by high GOS values (1.4 in 6%PS -An and 1.8 in 12%PS -An). On the contrary, \npre-strained CVN samples (6%PS and 12%PS) demonstrate an opposite i.e. decre asing trend \nin their GOS values in  figure 4(a) , since strain a ccumulation is  the maximum at the notch \nroot and decreases with distance. A uniform distribution  of GOS value  as the function of \ndistance  in as -received (0%PS) and subsequent annealed (0%PS -An) samples is also shown \nfor comparison.  \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Dislocation density (DD) obtained via nanoindentation technique following Nix and \nGao model [35,36]  confirms the presence of highly deformed grains (DD ~ 2.48x1014 ± \n1.1x1013) just ahead  the notch tip  in pre -strained samples, figure 4(b) . DD decreases on \nmoving away from the notch tip, as expected. An opposite, increasing trend with the increase \nFigure 4 . (a) Average GOS distribution, and (b) Dislocation density distribution ahead of notch root. (c) \nPlot predicting critical strain for strain -induced boundary migration (SIBM) using grain size distribution \n(normalised w.r.t. 0%PS grain size) and notch tip s train distribution curve.  \n 9 \n in distance from notch tip is observed in the annealed specimens having lowest DD \n(~1.95*1014 ± 3.1x1012) at the notch tip , figure 4(b) .  \nDiscussion  \nThe present work is performed in single phase ferritic steels  just to curtail microstructural \nvariables other than the grain size. Pre -straining promoted plastic flow resulting in increase in \nthe grain aspect ratio from 1.1 to 1.3. Subsequent annealing of the deformed structure \nreleases the accumulated strain by mec hanisms like recovery, recrystallization and grain \ngrowth. It is well recognized that the restoration mechanism such as recrystallization and \nSIBM depends primarily on amount of prior deformation, subsequent annealing temperature \nand time. Since, the annea ling temperature and time was kept constant, prior strain was the \nsole parameter controlling the final ferrite grain size and the grain size distribution. Presence \nof fine, equiaxed and strain free ferrite grains ahead of the notch root in figure 2(b, c) i s due \nto complete recrystallization which occurs when the local strain exceeds a critical strain. \nHowever, strain below this critical value could not yield complete recrystallization and ended \nup in structural restoration by grain coarsening following SIBM  toward the end of the \nmodified region over a distance of ~ 1.20 mm, away from the notch root, figure 2(b, c). \nAbnormal grain growth by SIBM was reported earlier in several studies [31,37,38] , wher e \nlocally accumulated strain could not reach the critical strain (i.e. driving force) required for \ncomplete recrystallization.  \n GOS distribution plot shown in figure 4(a) confirmed complete recrystallization at the \nnotch tip and a deformed region just ahea d of the modified region in the annealed samples. \nAlthough, the mechanistic calculations clearly define the plastic zone size, its boundary may \nnot necessarily be a sharp one, rather a diffused boundary spread over a distance. The trends \nfound in GOS distr ibution plots are in good agreement with the dislocation density profile as \na function of distance ahead of a crack -tip for all the samples, figure 4(a, b). Therefore, GOS \nmaps can effectively be used for qualitative estimation of the local strain in both pre-strained \nand annealed condition.  \n Since, the critical strain for recrystallization cannot be estimated solely by optical \nmicrographs, an attempt is made in order to find out the critical strain for SIBM by analysing \nmodified ferrite grain size (of anne aled samples) and plastic strain (of pre -strained samples) \ndistribution plots together, figure 4(c). It shows the variation in ferrite grain size (normalised 10 \n with respect to the grain size of the as -received sample) as a function of plastic strain ahead \nof the notch tip within the modified region. The plastic strain corresponding to the coarsest \nferrite grains (at a distance of 1 mm and 3 mm in 6%PS -An and 12%PS -An samples, \nrespectively, from the notch tip) is found to be 0.1 in both 6%PS and 12%PS samples.  The \npresence of these coarse grains in annealed samples is a consequence of SIBM which occurs \nat a strain lower than that necessitated for complete recrystallization. With this it can be \nconcluded that there exists a critical strain, 0.1 in this investiga tion, which promotes SIBM \nwhen annealed. The microstructural modifications extended up to a distance decided by the \nlocal strain level (down to ε = 0.1), beyond which the local strain is insufficient (but not \nnecessarily zero) to drive any microstructural restoration. As the local strain exceeds 0.1, \ncomplete recrystallization and associated grain refinement governed the microstructural \nrestoration.  \n The positive effect of functionally graded microstructure (finer or bimodal type) on \nvarious properties are stated in many studies before [5–8]. Release of residual strain and grain \nstructure modification are expected to influence the fracture toughness positively by resisting \nthe crack propagation through crack b lunting. Future investigation aims to look into the effect \nof pre -straining and subsequent annealing on impact transition behaviour and other \nmechanical properties. Devising a model to predict critical strain with ferrite grain size as an \ninput parameter c an be another future aspect.  \nConclusions  \n In summary, this investigation unveils a new methodology of tailoring microstructure \nat a local scale, especially at failure prone area via pre -straining and subsequent annealing, \nsince developing a suitable micros tructure over a large structural component is challenging.  \nA method to predict critical strain, using grain size (after annealing) and plastic strain (after \nFE simulation) distribution, promoting SIBM is also presented in this work.  The critical \nstrain for  SIBM has been found out to be 0.1 and it decreased gradually hereafter, forming a \ndiffused boundary of the plastic zone.  It is noteworthy to mention that if a pre -existing crack \ncan be detected within a structural component by non -destructive testing, sui table heat \ntreatment can be employed following the current approach to modify the local microstructure \nand enhance the life of the whole component.  \n 11 \n Acknowledgement s: \n Authors acknowledge support from Dr B. Syed, TATA steel, Jamshedpur for \nproviding m aterial for this investigation.  Authors gratefully concede the thoughtful \ndiscussion with late Prof. John F. Knott, Birmingham, UK and Dr. S. V. Kamat, DS & DG – \nNS & M, Defence Metallurgical Research Laboratory (DMRL) Hyderabad, India.   \nReferences:  \n[1] A.W. Thompson, J.F. Knott, Metall. Trans. A 24 (1993) pp.523–534. \n[2] D.A. Curry, J.F. Knott, Met. Sci. 12 (1978) pp.511–514. \n[3] D.A. Curry, J.F. Knott, Met. Sci. 13 (1979) pp.341–345. \n[4] A. Ghosh, A. Ray, D. Chakrabarti, C.L. Davis, Mater. Sci. Eng. A 561 (2013) pp.126–\n135. \n[5] N.J. 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Charact. 107 (2015) pp.174–181. \n  "    },    {        "title": "1201.1154v1.Ferrite_cavities.pdf",        "content": "Ferrite cavities\nH. Klingbeil\nGSI Helmholtz Centre for Heavy Ion Research, Darmstadt, Germany\nAbstract\nFerrite cavities are used in synchrotrons and storage rings if the maximum\nRF frequency is in the order of a few MHz. We present a simple model for\ndescribing ferrite cavities. The most important parameters are deﬁned, and the\nmaterial properties are discussed. Several practical aspects are summarized,\nand the GSI SIS18 ferrite cavity is presented as an example.\n1Introduction\nThe revolution frequency of charged particles in synchrotrons or storage rings is usually lower than\n10MHz. Even if we consider comparatively small synchrotrons (e.g., HIT/HICAT in Heidelberg, Ger-\nmany, or CNAO in Pavia, Italy, of about 20–25m diameter, both used for tumour therapy), the revolution\ntime will be greater than 200 ns since the particles cannot reach the speed of light. Since, according to\nfRF=h\u0001frev;\nthe RF frequency is an integer multiple of the revolution frequency, the RF frequency will typically be\nlower than 10MHz if only small harmonic numbers hare desired. For such an operating frequency, the\nspatial dimensions of a conventional RF resonator would be far too large to be used in a synchrotron. One\nway to solve this problem is to reduce the wavelength by ﬁlling the cavity with magnetic material. This\nis the basic idea of ferrite-loaded cavities [1]. Furthermore, this type of cavity offers a simple means to\nmodify the resonant frequency in a wide range (typically up to a factor of 10) and in a comparatively short\ntime (typically at least 10ms cycle time). Therefore, ferrite cavities are suitable for ramped operation in\na synchrotron.\nOwing to the low operating frequencies, the transit-time factor of traditional ferrite-loaded cavities\nis almost 1and therefore not of interest.\n2Permeability of magnetic materials\nIn this article, all calculations are based on permeability quantities \u0016for which\n\u0016=\u0016r\u00160\nholds. In material speciﬁcations, the relative permeability \u0016ris given which means that we have to\nmultiply with \u00160to obtain\u0016. This comment is also valid for the incremental/differential permeability\nintroduced in the following.\nIn RF cavities, only so-called soft magnetic materials which have a narrow hysteresis loop are of\ninterest since their losses are comparatively low (in contrast to hard magnetic materials which are used\nfor permanent magnets1).\nFigure 1 shows the hysteresis loop of a ferromagnetic material. It is well known that the hysteresis\nloop leads to a residual induction Brif no magnetizing ﬁeld His present and that some coercive\nmagnetizing ﬁeld Hcis needed to set the induction Bto zero.\nLet us now assume that some cycles of the large hysteresis loop have already passed and that\nHis currently increasing. We now stop increasing the magnetizing ﬁeld Hin the upper right part of\n1No strict separation exists between hard and soft magnetic materials.arXiv:1201.1154v1  [physics.acc-ph]  5 Jan 2012H\nHcBrB\n2 B/c215/c682 H/c215/c68\nHc\nBrFig. 1: Hysteresis loop\nthe diagram. Then, His decreased by a much smaller amount 2\u0001\u0001H, afterwards increased again by\nthat amount 2\u0001\u0001H, and so forth2. As the diagram shows, this procedure will lead to a much smaller\nhysteresis loop where Bchanges by 2\u0001\u0001B. We may therefore deﬁne a differential orincremental\npermeability3\n\u0016\u0001=\u0001B\n\u0001H\nwhich describes the slope of the local hysteresis loop. It is this quantity \u0016\u0001which is relevant for RF\napplications. One can see that \u0016\u0001can be decreased by increasing the DC component of H. SinceHis\ngenerated by currents, one speaks of a bias current that is applied in order to shift the operating point to\nhigher inductions Bleading to a lower differential permeability \u0016\u0001.\nIf no biasing is applied, the maximum \u0016\u0001is obtained which is typically in the order of a few\nhundred or a few thousand times \u00160.\nThe hysteresis loop and the AC permeability of ferromagnetic materials can be described in a phe-\nnomenological way by the so-called Preisach model which is explained in the literature (cf. [2]). Un-\nfortunately, the material properties are even more complicated since they are also frequency-dependent.\nOne usually uses the complex permeability\n\u0016=\u00160\ns\u0000j\u001600\ns (1)\nin order to describe losses (hysteresis loss, eddy current loss and residual loss). The parameters \u00160\nsand\n\u001600\nsare frequency-dependent. In the following, we will assume that the complex permeability \u0016describes\nthe material behaviour in rapidly alternating ﬁelds as does the above-mentioned real quantity \u0016\u0001when a\nbiasing ﬁeld Hbiasis present. However, we will omit the index \u0001for the sake of simplicity.\n2The factor of 2was assumed in order to have the same total change of 2\u0001\u0001Has in the equation HAC(t) = \u0001Hcos!t\nwhich is usually used for harmonic oscillations.\n3In a strict sense, the differential permeability is the limit \u0016\u0001=dB\ndHfor\u0001H;\u0001B!0.\n23Magnetostatic analysis of a ferrite cavity\nFigure 2 shows the main elements of a ferrite-loaded cavity. The beam pipe is interrupted by a ceramic\ngap. This gap ensures that the beam pipe may still be evacuated but it allows a voltage Vgapto be induced\nin longitudinal direction. Several magnetic ring cores are mounted in a concentric way around the beam\nand beam pipe (ﬁve ring cores are drawn here as an example). The whole cavity is surrounded by a\nmetallic housing which is connected to the beam pipe.\nFig. 2: Simpliﬁed 3D sketch of a ferrite-loaded cavity\nFigure 3 shows a cross-section through the cavity. The dotted line represents the beam which\nis located in the middle of a metallic beam pipe (for analysing the inﬂuence of the beam current, this\ndotted line is regarded as a part of a circuit that closes outside the cavity, but this is not relevant for\nunderstanding the basic operation principle). The ceramic gap has a parasitic capacitance, but additional\nlumped-element capacitors are usually connected in parallel — leading to the overall capacitance C.\nStarting at the generator port located at the bottom of the ﬁgure, an inductive coupling loop surrounds\nthe ring core stack. This loop was not shown in Fig. 2.\nNote that due to the cross-section approach, we get a wire model of the cavity with two wires\nrepresenting the cavity housing. This is sufﬁcient for the practical analysis, but one should remember\nthat the currents are distributed in reality.\nAll voltages, currents, ﬁeld, and ﬂux quantities used in the following are phasors, i.e., complex\namplitudes/peak values for a given frequency f=!=2\u0019.\nLet us consider a contour which surrounds the lower left ring core stack. Based on Maxwell’s\nsecond equation in the time domain (Faraday’s law)\nI\n@S~E\u0001d~l=\u0000Z\nS_~B\u0001d~S\nwe ﬁnd\nVgen= +j!\btot (2)\nin the frequency domain. If we now use the complete lower cavity half as integration path, one obtains\nVgap= +j!\btot:\nHence we ﬁnd\nVgap=Vgen: (3)\n3Vrt\nrV\nII\nBeam□Pipe/c70\n/B67/B65/B6e/B69/B6f/B62/B65/B61/B6d\n/B67/B65/B6e/B62/B65/B61/B6d\nVC\n/B67/B61/B70I /B43Fig. 3: Simpliﬁed model of a ferrite cavity\nHere we assumed that the stray ﬁeld Bin the air region is negligible in comparison with the ﬁeld inside\nthe ring cores (due to their high permeability). Finally, we consider the beam current contour:\nVbeam= +j!\btot=Vgap:\nFor negligible displacement current we have Maxwell’s ﬁrst equation (Ampère’s law)\nI\n@S~H\u0001d~l=Z\nS~J\u0001d~S:\nWe use a concentric circle with radius raround the beam as integration path:\nH2\u0019r=Itot: (4)\nThis leads to\nB=\u0016Itot\n2\u0019r(5)\nwith\nItot=Igen\u0000IC\u0000Ibeam: (6)\n4For the ﬂux through one single ring core we get\n\b1=Z\n~B\u0001d~S=tZro\nriB dr =t\u0016Itot\n2\u0019lnro\nri:\nWith the complex permeability\n\u0016=\u00160\ns\u0000j\u001600\ns\nand assuming that Nring cores are present, one ﬁnds\nVgap=j!\btot=j!N \b1=j!Nt(\u00160\ns\u0000j\u001600\ns)Itot\n2\u0019lnro\nri:\nTherefore we obtain\nVgap=Itot(j!Ls+Rs) =ItotZs; (7)\nif\nZs=1\nYs=j!Ls+Rs; (8)\nLs=Nt\u00160\ns\n2\u0019lnro\nri;\nRs=!Nt\u001600\ns\n2\u0019lnro\nri=!\u001600\ns\n\u00160sLs=!Ls\nQ(9)\nare deﬁned. Here,\nQ=\u00160\ns\n\u001600s=1\ntan\u000e\u0016(10)\nis the quality factor (orQ factor ) of the ring core material. Using Eq. (6) we ﬁnd\nVgapYs=Itot=Igen\u0000Ibeam\u0000Vgapj!C\n)Vgap=Igen\u0000Ibeam\nYs+j!C=Ztot(Igen\u0000Ibeam): (11)\nThis equation corresponds to the equivalent circuit shown in Fig. 4. In the last step we deﬁned\nYtot=1\nZtot=Ys+j!C:\nIn the literature one often ﬁnds a different version of Eq. (11) where Ibeamhas the same sign as Igen. This\ncorresponds to both currents having the same direction (ﬂowing into the circuits in Figs. 4 and 5). In any\ncase, one has to make sure that the correct phase between beam current and gap voltage is established.\n4Parallel and series lumped element circuit\nIn the vicinity of the resonant frequency, it is possible to convert the lumped element circuit shown in\nFig. 4 into a parallel circuit as shown in Fig. 5. The admittance of both circuits shall be equal:\nYtot=j!C +1\nRs+j!Ls=j!C +1\nRp+1\nj!Lp\n)Rs\u0000j!Ls\nR2s+ (!Ls)2=1\nRp+1\nj!Lp:\n5R/B73\nV/B67/B61/B70I/B62/B65/B61/B6d I/B67/B65/B6e\nC\nL/B73Fig. 4: Series equivalent circuit\nR/B70V/B67/B61/B70I/B62/B65/B61/B6d I/B67/B65/B6e\nCL/B70 Fig. 5: Parallel equivalent circuit\nA comparison of real and imaginary part yields:\nRp=R2\ns+ (!Ls)2\nRs(12)\n!Lp=R2\ns+ (!Ls)2\n!Ls: (13)\nFor the inverse relation, we modify the ﬁrst equation according to\n(!Ls)2=Rs(Rp\u0000Rs)\nand use this result in the second equation:\n!Lpq\nRs(Rp\u0000Rs) =RsRp\n)(!Lp)2(Rp\u0000Rs) =RsR2\np\n)Rs=(!Lp)2\nR2p+ (!Lp)2Rp:\nEquations (12) and (13) directly provide\nRpRs= (!Lp)(!Ls) (14)\nwhich leads to\n!Ls=Rp\n!LpRs=R2\np\nR2p+ (!Lp)2!Lp:\nSince it is suitable to use both types of lumped element circuit, it is also convenient to deﬁne the complex\n\u0016in a parallel form:\n1\n\u0016=1\n\u00160p+j1\n\u001600p: (15)\nThis is an alternative representation for the series form shown in Eq. (1) which leads to\n1\n\u0016=\u00160\ns+j\u001600\ns\n\u00160s2+\u001600s2:\nComparing the real and imaginary parts of the last two equations, we ﬁnd:\n\u00160\np=\u00160\ns2+\u001600\ns2\n\u00160s; (16)\n6\u001600\np=\u00160\ns2+\u001600\ns2\n\u001600s: (17)\nThese two equations lead to\n\u00160\np\u00160\ns=\u001600\np\u001600\ns:\nTogether with Eqs. (9), (10), and (14) we conclude:\nQ=\u00160\ns\n\u001600s=!Ls\nRs=Rp\n!Lp=\u001600\np\n\u00160p: (18)\nWith these expressions, we may write Eqs. (16) and (17) in the form\n\u00160\np=\u00160\ns\u0012\n1 +1\nQ2\u0013\n(19)\n\u001600\np=\u001600\ns\u0000\n1 +Q2\u0001\n: (20)\nIf we use Eq. (18)\nQ=!Ls\nRs;\nwe may rewrite Eqs. (12) and (13) in the form\nRp=Rs(1 +Q2) (21)\nLp=Ls\u0012\n1 +1\nQ2\u0013\n: (22)\nBy combining Eqs. (21) and (9) we ﬁnd\nRp= (1 +Q2)!Nt\u001600\ns\n2\u0019lnro\nri:\nWith the help of Eqs. (18) and (19) one gets\n\u001600\ns=\u00160\ns\nQ=\u00160\np\nQ+1\nQ=\u00160\npQ\n1 +Q2:\nThe last two equations lead to\nRp=!Nt\u00160\npQ\n2\u0019lnro\nri=Nt\u00160\npQf lnro\nri:\nThis shows that Rpis proportional to the product \u00160\npQfwhich is a material property. The other param-\neters refer to the geometry. Therefore, the manufacturers of ferrite cores sometimes specify the \u0016rQf\nproduct (for the sake of simplicity, we deﬁne \u0016r:=\u00160\np;r).\nForQ\u00155, we may use the approximations\nRp\u0019RsQ2; Lp\u0019Ls; \u00160\np\u0019\u00160\ns; \u001600\np\u0019\u001600\nsQ2\nwhich then have an error of less than 4%.\n75Frequency dependence of material properties\nAs an example, the frequency dependence of the permeability is shown in Figs. 6 and 7 for the special\nferrite material Ferroxcube 4 assuming small magnetic RF ﬁelds without biasing. All the data presented\nfor this material are taken from Ref. [3]. It is obvious that the behaviour depends signiﬁcantly on the\nchoice of the material. Without biasing, a constant \u00160\ns\u0019\u00160\npmay only be assumed up to a certain\nfrequency (see Fig. 6). Increasing the frequency from 0upwards, the Q factor will decrease (compare\nFigs. 6 and 7). Figure 8 shows the resulting frequency dependence of the \u0016rQfproduct.\nIf the magnetic RF ﬁeld is increased, both Qand\u0016rQfwill decrease in comparison with the\ndiagrams in Figs. 6 to 8. The effective incremental permeability \u0016rwill increase for rising magnetic RF\nﬁelds as one can see by interpreting Fig. 1. Therefore, it is important to consider the material properties\nunder realistic operating conditions (the maximum RF B-ﬁeld is usually in the order of 10–20mT).\n/B31/B30/B30/B30/B30\n/B31/B30/B30/B30\nf/MHz/c109 /c61 /c109s,rRe{ }//c1090\n/B31/B30/B30\n/B31/B30\n/B31/B30 /B31/B30/B30 /B31 /B30/B2e/B31/B31\n/B32\n/B33’\nFig. 6:\u00160\ns;rversus frequency for three different types\nof ferrite material (1: Ferroxcube 4A, 2: Ferroxcube\n4C, 3: Ferroxcube 4E). Data adopted from Ref. [3].\n/B31/B30/B30/B30\n/B31/B30/B30\nf/MHz/B31/B30\n/B31\n/B31/B30 /B31/B30/B30 /B31 /B30/B2e/B31/B31\n/B32\n/B33\n/c109 /c61 /c109s,rIm{ }//c1090\n’’Fig. 7:\u001600\ns;rversus frequency for three different types\nof ferrite material (1: Ferroxcube 4A, 2: Ferroxcube\n4C, 3: Ferroxcube 4E). Data adopted from Ref. [3].\nf/MHz/c109rQf/GHz\n/B31/B31/B30/B31/B30/B30\n/B31 /B32 /B30/B2e/B32 /B35 /B30/B2e/B35 /B31/B30 /B32/B30 /B35/B30/B31/B32/B33\nFig. 8:\u00160\ns;rQfproduct versus frequency for three different types of ferrite material (1: Ferroxcube 4A, 2: Ferrox-\ncube 4C, 3: Ferroxcube 4E). No bias ﬁeld is present, and small magnetic RF ﬁeld amplitudes are assumed. Data\nadopted from Ref. [3].\nIf biasing is applied, the \u0016rQfcurve shown in Fig. 8 will be shifted to the lower right side; this\neffect may approximately compensate the increase of \u0016rQfwith frequency [3]. Therefore, the \u0016rQf\n8product may sometimes approximately be regarded as a constant if biasing is used to keep the cavity at\nresonance for all frequencies under consideration.\n6Quality factor of the cavity\nThe quality factor of the equivalent circuit shown in Fig. 5 is obtained if the resonant (angular) fre-\nquency\n!0= 2\u0019f0=1p\nLpC\nis inserted into Eq. (18):\nQ0=Rps\nC\nLp:\nIn general, all parameters \u00160\ns,\u001600\ns,\u00160\np,\u001600\np,Rs,Ls,Rp,Lp,QandQ0are frequency-dependent. It depends\non the material whether the parallel or the series lumped element circuit is the better representation in\nthe sense that its parameters may be regarded as approximately constant in the relevant operating range.\nIn the following, we will use the parallel representation.\nWe brieﬂy show that Q0is in fact the quality factor deﬁned by\nQ0=!Wtot\nPloss\nwhereWtotis the stored energy and Plossis the power loss (both time-averaged):\nPloss=jVgapj2\n2Rp(23)\nWel=1\n4CjVgapj2\nWmagn =1\n4LpjILj2=1\n4LpjVgapj2\n!2L2p=jVgapj2\n4!2Lp\nAt resonance, we have Wel=Wmagn which leads to\nQ0= 2!Wel\nPloss= 2!RpC\n2=Rps\nC\nLp\nas expected. The parallel resistor Rpdeﬁned by Eq. (23) is often called shunt impedance .\n7Impedance of the cavity\nThe impedance of the cavity\nZtot=1\n1\nRp+j\u0010\n!C\u00001\n!Lp\u0011=q\nLp\nC\n1\nRpq\nLp\nC+j\u0012\n!p\nLpC\u00001\n!p\nLpC\u0013\nmay be written as\nZtot=Rp\nQ0\n1\nQ0+j\u0010\n!\n!0\u0000!0\n!\u0011\n9)Ztot=Rp\n1 +j Q0\u0010\n!\n!0\u0000!0\n!\u0011:\nThe Laplace transformation yields\nZtot(s) =Rp\n1 +sQ0\n!0+Q0!0\ns=Rp!0\nQ0s\ns!0\nQ0+s2+!2\n0;\nwhich may be found in the literature in the form\nZtot(s) =2Rp\u001bs\ns2+ 2\u001bs+!2\n0\nif\n\u001b=!0\n2Q0\nis deﬁned.\n8Length of the cavity\nIn the previous sections, we assumed that the ferrite ring cores can be regarded as lumped-element in-\nductors and resistors. This is of course only true if the cavity is short in comparison with the wavelength.\nAs an alternative to the transformer model introduced above, one may therefore use a coaxial\ntransmission line model. For example, the section of the cavity that is located on the left side of the\nceramic gap in Fig. 3 may be interpreted as a coaxial line that is homogeneous in longitudinal direction\nand that has a short-circuit at the left end. The cross section consists of the magnetic material of the\nring cores, air between the ring cores and the beam pipe, and air between the ring cores and the cavity\nhousing. This is of course an idealization since cooling disks, conductors and other air regions are\nneglected. Taking the SIS18 cavity at GSI as an example, the ring cores have \u0016r= 28 at an operating\nfrequency of 2:5MHz. The ring cores have a relative dielectric constant of 10–15, but this is reduced\nto an effective value of \u000fr;eff = 1:8since the ring cores do not ﬁll the full cavity cross section. These\nvalues lead to a wavelength of \u0015= 16:9m. Since 64 ring cores with a thickness of 25mm are used, the\neffective length of the magnetic material is 1:6m= 0:095\u0015(which corresponds to a phase of 34\u000e). In\nthis case, the transmission line model leads to deviations of less than 10% with respect to the lumped-\nelement model. The transmission line model also shows that the above-mentioned estimation for the\nwavelength is too pessimistic; it leads to \u0015= 24 m which corresponds to a cavity length of only 24\u000e.\nThis type of model makes it understandable why the ferrite cavity is sometimes referred to as a\nshortened quarter-wavelength resonator .\nOf course, one may also use more detailed models where subsections of the cavity are mod-\neled as lumped elements. In this case, computer simulations can be performed to calculate the overall\nimpedance. In case one is interested in resonances which may occur at higher operating frequencies, one\nshould perform full electromagnetic simulations.\nIn any case, one should always remember that some parameters are difﬁcult to determine, espe-\ncially the permeability of the ring core material under different operating conditions. This uncertainty\nmay lead to larger errors than simpliﬁcations of the model. Measurements of full-size ring cores in the\nrequested operating range are inevitable when a new cavity is developed. Also parameter tolerances due\nto the manufacturing process have to be taken into account.\nIn general, one should note that the total length and the dimensions of the cross-section of the\nferrite cavity are not determined by the wavelength as for a conventional RF cavity. For example, the\nSIS18 ferrite cavity has a length of 3m ﬂange-to-ﬂange although only 1:6m are ﬁlled with magnetic\n10material. This provides space for the ceramic gap, the cooling disks, and further devices like the bias\ncurrent bars. In order to avoid resonances at higher frequencies, one should not waste too much space,\nbut there is no exact size of the cavity housing that results from the electromagnetic analysis.\n9Cavity ﬁlling time\nThe equivalent circuit shown in Fig. 5 was derived in the frequency domain. As long as no parasitic\nresonances occur, this equivalent circuit may be generalized. Therefore, we may also analyse it in the\ntime domain (allowing slow changes of Lpwith time):\nIC=C\u0001dVgap\ndt,Vgap=Lp\u0001dIL\ndt,Vgap= (Igen\u0000IL\u0000IC\u0000Ibeam)Rp\n)IL=\u0000Vgap\nRp+Igen\u0000IC\u0000Ibeam (24)\n)Vgap=Lp\u0012\n\u00001\nRpdVgap\ndt+d\ndt(Igen\u0000Ibeam)\u0000Cd2Vgap\ndt2\u0013\n)LpCVgap+Lp\nRp_Vgap+Vgap=Lpd\ndt(Igen\u0000Ibeam)\n)Vgap+2\n\u001c_Vgap+!2\n0Vgap=1\nCd\ndt(Igen\u0000Ibeam): (25)\nHere we used the deﬁnition\n\u001c= 2RpC:\nThe product RpCis also present in the expression for the quality factor:\nQ0=Rps\nC\nLp=RpCp\nLpC=1\n2\u001c!0\n)\u001c=2Q0\n!0=Q0\n\u0019f0:\nUnder the assumption !0>1\n\u001c(Q0>1\n2), the approach Vgap=V0e\u000bt(with a complex constant \u000b) for the\nhomogeneous solution of Eq. (25) actually leads to\n\u000b=\u00001\n\u001c\u0006j!x\nwith the exponential decay time \u001cand the oscillation frequency\n!x=!0s\n1\u00001\n(\u001c!0)2=!0s\n1\u00001\n4Q2\n0:\nThis leads to !x\u0019!0even for moderately high Q> 2(error less than 4%).\nThe time\u001cis the time constant for the cavity which also determines the cavity ﬁlling time . Fur-\nthermore, the time constant \u001cis relevant for phase jumps of the cavity (see, e.g., Ref. [4]).\n1110Power ampliﬁer\nUp to now, we only dealt with the so-called ‘unloaded Q factor’ Q0of the cavity. An RF power\nampliﬁer that feeds the cavity may often be represented by a voltage-controlled current source (e.g., in\nthe case of a tetrode ampliﬁer). The impedance of this current source will be connected in parallel to the\nequivalent circuit thereby reducing the ohmic part Rp. Therefore, the loaded Q factor will be reduced in\ncomparison with the unloaded Q factor. Also the cavity ﬁlling time will be reduced due to the impedance\nof the power ampliﬁer.\nIt has to be emphasized that for ferrite cavities 50 \n impedance matching is not necessarily used\nin general. The cavity impedance is usually in the order of a few hundred ohms or a few kilo-ohms.\nTherefore, it is often more suitable to directly connect the tetrode ampliﬁer to the cavity. Impedance\nmatching is not mandatory if the ampliﬁer is located close to the cavity. Short cables have to be used\nsince they contribute to the overall impedance/capacitance. Cavity and RF power ampliﬁer must be\nconsidered as one unit; they cannot be developed individually in the sense that the impedance curves of\nthe cavity and the power ampliﬁer inﬂuence each other.\n11Cooling\nBoth the power ampliﬁer and the ferrite ring cores need active cooling. Of course, the Curie temperature\nof the ferrite material (typically >100\u000eC) must never be reached. Depending on the operating conditions\n(e.g., CW or pulsed operation), forced air cooling may be sufﬁcient or water cooling may be required.\nCooling disks in-between the ferrite cores may be used. In this case, one has to make sure that the\nthermal contact between cooling disks and ferrites is good.\n12Cavity tuning\nWe already mentioned in Section 2 that a DC bias current may be used to decrease \u0016\u0001which results in\na higher resonant frequency. This is one possible way to realize cavity tuning . Strictly speaking, one\ndeals with a quasi-DC bias current since the resonant frequency must be modiﬁed during a synchrotron\nmachine cycle if it is equal to the variable RF frequency. Such a tuning of the resonant frequency f0to\nthe RF frequency fRFis usually desirable since the large Ztotallows one to generate large voltages with\nmoderate RF power consumption.\nSometimes, the operating frequency range is small enough in comparison with the bandwidth of\nthe cavity that no tuning is required.\nIf tuning is required, one has at least two possibilities to realize it:\n1. Bias current tuning\n2. Capacitive tuning\nThe latter may be realized by a variable capacitor (see, e.g., Refs. [5,6]) whose capacitance may be varied\nby a stepping motor. This mechanical adjustment, however, is only possible if the resonant frequency is\nnot changed from machine cycle to machine cycle or even within one machine cycle.\nIn the case of bias current tuning, one has two different choices, namely perpendicular biasing\n(also denoted as transverse biasing ) and parallel biasing (also denoted as longitudinal biasing ). The\nterms parallel and perpendicular refer to the orientation of the DC ﬁeld Hbiasin comparison with the RF\nﬁeldH.\nParallel biasing is simple to realize. One adds bias current loops which may in principle be located\nin the same way as the inductive coupling loop shown in Fig. 3. If only a few loops are present, current\nbars with large cross sections are needed to withstand the bias current of several hundred amperes. The\nrequired DC current may of course be reduced if the number Nbiasof loops is increased accordingly\n(keeping the ampere-turns constant). This increase of the number of bias current windings may be\n12limited by resonances. On the other hand, a minimum number of current loops is usually applied to\nguarantee a certain amount of symmetry which leads to a more homogeneous ﬂux in the ring cores.\nPerpendicular biasing is more complicated to realize; it requires more space between the ring-\ncores, and the permeability range is smaller than for parallel biasing. The main reason for using perpen-\ndicular biasing is that lower losses can be reached (see, e.g., Ref. [7]). One can also avoid the so-called\nQ-loss effect orhigh loss effect . The Q-loss effect often occurs when parallel biasing is applied and if\nthe bias current is constant or varying only slowly. After a few milliseconds, one observes that the in-\nduced voltage breaks down by a certain amount even though the same amount of RF power is still applied\n(see, e.g., Refs. [8, 9]). For perpendicular biasing, the Q-loss effect was not observed. The Q-loss effect\nis not fully understood yet. However, there are strong indications that it may be caused by mechanical\nresonances of the ring cores induced by magnetostriction effects [10]. It was possible to suppress the\nQ-loss effect by mechanical damping. For example, in some types of ferrite cavities, the ring cores are\nembedded in a sealing compound [11] which should damp mechanical oscillations. Not only the Q-loss\neffect but also further anomalous loss effects have been observed [8].\nWhen the inﬂuence of biasing is described, one usually deﬁnes an average bias ﬁeld Hbiasfor the\nring cores. For this purpose, one may use the magnetic ﬁeld\nHbias=NbiasIbias\n2\u0019\u0016r\nlocated at the mean radius\n\u0016r=priro:\nOf course, this choice is somewhat arbitrary from a theoretical point of view, but it is based on practical\nexperience.\nA combination of bias current tuning and capacitive tuning has also been applied to extend the\nfrequency range [12].\n13Further complications\nWe already mentioned that the effective differential permeability depends on the hysteresis behaviour of\nthe material, i.e., on the history of bias and RF currents. It was also mentioned that, owing to the spatial\ndimensions of the cavity, we have to deal with ranges between lumped-element circuits and distributed\nelements. The anomalous loss effects are a third complication. There are further points which make the\nsituation even more complicated in practice:\n– If no biasing is applied, the maximum of the magnetic ﬁeld is present at the inner radius ri. One\nhas to make sure that the maximum ratings of the material are not exceeded.\n– Bias currents lead to an r\u00001dependency of the induced magnetic ﬁeld Hbias. Therefore, biasing is\nmore effective in the inner parts of the ring cores than in the outer parts resulting in a \u0016\u0001which\nincreases with r. According to Eq. (5), this will modify the r\u00001dependency of the magnetic RF\nﬁeld. As a result, the dependence on rmay be much weaker than without bias ﬁeld.\n– The permeability depends not only on the frequency, on the magnetic RF ﬁeld, and on the biasing.\nIt is also temperature-dependent.\n– Depending on the thickness of the ferrite cores, on the conductivity of the ferrite, on the material\nlosses and on the operating frequency, the magnetic ﬁeld may decay from the surface to the inner\nregions reducing the effective volume.\n– At higher operating frequencies with strong bias currents, the differential permeability will be\nrather low. This means that the magnetic ﬂux will not be perfectly guided by the ring cores any-\nmore. The fringe ﬁelds in the air regions will be more important, and resonances may occur.\n1314Cavity conﬁgurations\nA comparison of different types of ferrite cavities can be found in Refs. [13–15]. We just summarize a\nfew aspects here that lead to different solutions.\n– Instead of using only one stack of ferrite ring cores and only one ceramic gap as shown in Fig. 3,\none may also use more sections with ferrites (e.g., one gap with half the ring cores on the left\nside and the other half on the right side of the gap — for reasons of symmetry) or more gaps.\nSometimes, the ceramic gaps belong to different independent cavity cells which may be coupled\nby copper bars (e.g., by connecting them in parallel). Connections of this type must be short to\nallow operation at high frequencies.\n– One conﬁguration that is often used is a cavity consisting of only one ceramic gap and two fer-\nrite stacks on both sides. Figure-of-eight windings surround these two ferrite stacks (see, e.g.,\nRef. [16]). With respect to the magnetic RF ﬁeld, this leads to the same magnetic ﬂux in both\nstacks. In this way, an RF power ampliﬁer that feeds only one of the two cavity halves will indi-\nrectly supply the other cavity half as well. This corresponds to a 1:2 transformation ratio. Hence,\nthe beam will see four times the impedance compared with the ampliﬁer load. Therefore, the\nsame RF input power will lead to higher gap voltages (but also to a higher beam impedance). The\ntransformation law may be derived by an analysis that is similar to the one in Section 3.\n– Instead of the inductive coupling shown in Fig. 3, one may also use capacitive coupling if the\npower ampliﬁer is connected to the gap via capacitors. If a tetrode power ampliﬁer is used, one\nstill has to provide it with a high anode voltage. Therefore, an external inductor (choke coil)\nis necessary which allows the DC anode current but which blocks the RF current from the DC\npower supply. Often a combination of capacitive and inductive coupling is used (e.g., to inﬂuence\nparasitic resonances). The coupling elements will contribute to the equivalent circuit.\n– Another possibility is inductive coupling of individual ring cores. This leads to lower impedances\nwhich ideally allow a 50 \n impedance matching to a standard solid-state RF power ampliﬁer (see,\ne.g., Ref. [17]).\n– In case a small relative tuning range is required, it is not necessary to use biasing for the ferrite\nring cores inside the cavity. One may use external tuners (see, e.g., Refs. [18, 19]) which can\nbe connected to the gap. For external tuners, both parallel and perpendicular biasing may be\napplied [20].\nNo general strategy can be deﬁned as to how a new cavity is designed. Many compromises have to\nbe found. A certain minimum capacitance is given by the gap capacitance and the parasitic capacitances.\nIn order to reach the upper limit of the frequency range, a certain minimum inductance has to be realized.\nIf biasing is used, this minimum inductance must be reached using the maximum bias current but the\neffective permeability should still be high enough to reduce stray ﬁelds. Also the lower frequency limit\nshould be reachable with a minimum but non-zero bias current. There is a maximum RF ﬁeld BRF;max\n(about 15mT) which should not be exceeded for the ring cores. This imposes a lower limit for the\nnumber of ring cores. The required tuning range in combination with the overall capacitance will also\nrestrict the number of ring cores. As mentioned above, the ampliﬁer design should be taken into account\nfrom the very beginning, especially with respect to the impedance. The maximum beam impedance that\nis tolerable is deﬁned by beam dynamics considerations. This impedance budget also deﬁnes the power\nthat is required. If more ring cores can be used, the impedance of the cavity will increase, and the power\nloss will decrease for a given gap voltage.\n15The GSI ferrite cavities in SIS18\nAs an example for a ferrite cavity, we summarize the main facts about GSI’s SIS18 ferrite cavities (see\nFigs. 9 and 10). Two identical ferrite cavities are located in the synchrotron SIS18.\n14Fig. 9: SIS18 ferrite cavity\nFig. 10: Gap area of the SIS18 ferrite cavity\n15The material Ferroxcube FXC 8C12m is used for the ferrite ring cores. In total, N= 64 ring cores\nare used per cavity. Each core has the following dimensions:\ndo= 2ro= 498 mm ,di= 2ri= 270 mm ,t= 25 mm\n\u0016r=priro= 183 mm:\nFor biasing,\nNbias= 6\nﬁgure-of-eight copper windings are present. The total capacitance amounts to\nC= 740 pF;\nincluding the gap, the gap capacitors, the cooling disks, and other parasitic capacitances. The maximum\nvoltage that is reached under normal operating conditions is Vgap= 16 kV.\nTable 1: Equivalent circuit parameters for SIS18 ferrite cavities (without inﬂuence of tetrode ampliﬁers)\nResonant frequency f0 620kHz 2:5MHz 5MHz\nRelative permeability \u00160\np;r 450 28 7\nMagnetic bias ﬁeld at mean radius Hbias 25A=m 700A=m 2750 A=m\nBias current Ibias 4:8A 135A 528A\n\u00160\np;rQfproduct 4:2\u0001109s\u000013:7\u0001109s\u000013:3\u0001109s\u00001\nQ-factorQ 15 53 94\nLs 88:2\u0016H 5:49\u0016H 1:37\u0016H\nLp 88:5\u0016H 5:49\u0016H 1:37\u0016H\nRs 22:8 \n 1:63 \n 0:46 \nRp 5200 \n 4600 \n 4100 \nCavity time constant \u001c 7:7\u0016s 6:7\u0016s 6:0\u0016s\nTable 1 shows the main parameters for three different frequencies. All these values are consis-\ntent with the formulas presented in the paper at hand. It is obvious that both \u00160\np;rQfandRpdo not\nvary strongly with frequency justifying the parallel equivalent circuit. This compensation effect was\nmentioned at the end of Section 5.\nAll the parameters mentioned here refer to the beam side of the cavity. The cavity is driven by an\nRF ampliﬁer which is coupled to only one of two ferrite core stacks (consisting of 32 ring cores each).\nThe two ring core stacks are coupled by the bias windings. Therefore, a transformation ratio of 1:2 is\npresent from ampliﬁer to beam. This means that the ampliﬁer has to drive a load of about Rp=4 = 1:1 k\n .\nFor a full amplitude of Ugap= 16 kV atf= 5MHz the power loss in the cavity amounts to 31kW.\nThe SIS18 cavity is supplied by a single-ended tetrode power ampliﬁer using a combination of\ninductive and capacitive coupling.\nIt has to be emphasized that the values in Table 1 do not contain the ampliﬁer inﬂuence. Depending\non the working point of the tetrode, Rpwill be reduced signiﬁcantly, and all related parameters vary\naccordingly.\n16Further practical considerations\nFor measuring the gap voltage, one needs a gap voltage divider in order to decrease the high-voltage RF\nto a safer level. This can be done by capacitive voltage dividers. Gap relays are used to short-circuit the\ngap if the cavity is temporarily unused. This reduces the impedance seen by the beam which may be\nharmful for beam stability. If cycle-by-cycle switching is needed, semiconductor switches may be used\n16instead of vacuum relays. Another possibility to temporarily reduce the beam impedance is to detune the\ncavity.\nThe capacitance/impedance of the gap periphery devices must be considered when the overall\ncapacitance Cand the other elements in the equivalent circuit are calculated. Also further parasitic\nelements may be present.\nOn the one hand, the cavity dimensions should be as small as possible since space in synchrotrons\nand storage rings is valuable and since undesired resonances may be avoided. On the other hand certain\nminimum distances have to be kept in order to prevent high-voltage spark-overs. For EMC reasons, RF\nseals are often used between conducting metal parts of the cavity housing to reduce electromagnetic\nemission.\nIn order to fulﬁl high vacuum requirements, it may be necessary to allow a bakeout of the vacuum\nchamber. This can be realized by integrating a heating jacket that surrounds the beam pipe. One has to\nguarantee that the ring cores are not damaged by heating and that safety distances (for RF purposes and\nhigh-voltage requirements) are kept.\nIn case the cavity is used in a radiation environment, the radiation hardness of all materials is an\nimportant topic.\n17Magnetic materials\nA large variety of magnetic materials is available. Nickel-Zinc (NiZn) ferrites may be regarded as the\ntraditional standard material for ferrite-loaded cavities. The following material properties are of interest\nfor the material selection and may differ signiﬁcantly for different types of material:\n– permeability\n– magnetic losses\n– saturation induction (typically 200–300mT for NiZn ferrites)\n– maximum RF inductions (typically 10–20mT for NiZn ferrites)\n– relative dielectric constant (in the order of 10–15for NiZn ferrites but very high for MnZn ferrites,\nfor example) and dielectric losses (usually negligible for typical NiZn applications)\n– maximum operating temperature, thermal conductivity, and temperature dependence in general\n– magnetostriction\n– speciﬁc resistance (very high for NiZn ferrites, very low for MnZn ferrites).\nIn order to determine the RF properties under realistic operating conditions (large magnetic ﬂux, biasing),\nthorough reproducible measurements in a ﬁxed test setup are inevitable.\nAmorphous and nanocrystalline magnetic alloy (MA) materials have been used to build very\ncompact cavities that are based on similar principles to those of classical ferrite cavities (see, e.g.,\nRefs. [6, 15, 21–23]). These materials allow a higher induction and have a very high permeability. This\nmeans that a smaller number of ring cores is needed for the same inductance. MA materials typically\nhave lower Q factors in comparison with ferrite materials. Low Q factors have the advantage that fre-\nquency tuning is often not necessary and that it is possible to generate signal forms including higher\nharmonics instead of pure sine signals. MA cavities are especially of interest for pulsed operation at high\nﬁeld gradients. In case a low Q-factor is not desired, it is also possible to increase it by cutting the MA\nring cores.\nMicrowave garnet ferrites have been used at frequencies in the range 40–60MHz in connection\nwith perpendicular biasing since they provide comparatively low losses (see, e.g., Refs. [24–26]).\n17Acknowledgements\nThe author would like to thank all the GSI colleagues with whom he discussed several RF cavity issues\nduring the past years, especially Priv.-Doz. Dr. Peter Hülsmann, Dr. Hans Günter König, Dr. Ulrich Laier,\nand Dr. Gerald Schreiber. He is also grateful to the former staff members of the ring RF group, especially\nDr. Klaus Blasche, Dipl.-Phys. Martin Emmerling, and Dr. Klaus Kaspar for transferring their RF cavity\nknow-how to their successors. Last but not least, the author thanks Dr. Rolf Stassen (FZ Jülich) for\nreviewing the manuscript.\nIt is impossible to provide a complete list of references. The following list cites only a few refer-\nences regarding the most important aspects. Many other important publications exist.\nReferences\n[1] J. M. Brennan, inHandbook of Accelerator Physics and Engineering, Eds. A. W. Chao and\nM. Tigner (World Scientiﬁc, Singapore, 1999), pp. 570–2.\n[2] I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, 2nd ed., e-book\n(Elsevier, San Diego, 2003).\n[3] F. G. Brockman, H. van der Heide, and M. W. Louwerse, Ferroxcube für Protonensynchrotrons,\nPhilips Technische Rundschau , 30 (1969/70) 323–342.\n[4] S. Papureanu, Ch. Hamm, A. Schnase, and H. Meuth, Performance test of a ferrite-loaded cavity\nunder operation conditions, in15th IEEE Particle Accelerator Conference, Washington, DC, USA,\n1993 (IEEE, Piscataway, NJ, 1994), pp. 962–4.\n[5] M. Morvillo, R. Garoby, D. Grier, M. Haase, A. Krusche, P. Maesen, M. Paoluzzi, and C. Rossi,\nThe PS 13.3–20 MHz RF system for LHC, in20th IEEE Particle Accelerator Conference, Portland,\nOR, USA, 2003, Ed. J. Chew (JACoW, Geneva, 2003), pp. 1724–6.\n[6] P. Hülsmann, G. Hutter, and W. Vinzenz, The bunch compressor system for SIS18 at GSI, inEPAC,\nLuzern, 2004, pp. 1165–7.\n[7] W. R. Smythe and T. G. Brophy, RF cavities with transversely biased ferrite tuning, IEEE Trans.\nNucl. Sci. NS-32 (1985) 2951–3.\n[8] J. E. Grifﬁn and G. Nicholls, A review of some dynamic loss properties of Ni-Zn accelerator RF\nsystem ferrite, IEEE Trans, Nucl. Sci. 26 (1979) 3965–7.\n[9] K. Kaspar, H. G. König, and T. Winnefeld, Studies on maximum RF voltages in ferrite-tuned ac-\ncelerating cavities, inEPAC, Luzern, 2004, pp. 985–7.\n[10] H. G. König and S. Schäfer, Reduction of Q-loss effects in ferrite-loaded cavities, inEPAC, Genoa,\nItaly, 2008, pp. 985–7.\n[11] V . S. Arbuzov et al., Accelerating RF station for HIRFL-CSR, Lanzhou, China, inRuPAC XIX,\nDubna, 2004, pp. 332–4.\n[12] X. Pei, S. Anderson, D. Jenner, D. McCammon, and T. Sloan, A wide tuning range RF cavity with\nexternal ferrite biasing, in15th IEEE Particle Accelerator Conference, Washington, DC, USA, 1993\n(IEEE, Piscataway, NJ, 1994), pp. 1421–3.\n[13] A. Susini, Low frequency ferrite cavities, inEPAC, Rome, 1988, Ed. S. Tazzari (World Scientiﬁc,\nSingapore, 1989), pp. 1416–7.\n[14] I. S. K. Gardner, Ferrite dominated cavities, inCAS - CERN Accelerator School: RF Engineering\nfor Particle Accelerators, Oxford, 1991, S. Turner (Ed.) CERN 92-3, pp. 349–374.\n[15] A. Schnase, Cavities with a swing, inCAS - CERN Accelerator School: Radio Frequency Engi-\nneering, Seeheim, 2000, J. Miles (Ed.) CERN 2005-3, pp. 236–272.\n[16] A. Krusche and M. Paoluzzi, The new low frequency accelerating systems for the CERN PS\nBooster, inEPAC, Stockholm, 1998 (IOP, Bristol, 1998), pp. 1782–3.\n18[17] J. Dey, I. Kourbanis, and D. Wildman, A new RF system for bunch coalescing in the Fermilab Main\nRing, in16th IEEE Particle Accelerator Conference, Dallas, TX, USA, 1995 (IEEE, Piscataway,\nNJ, 1995), pp. 1672–4.\n[18] R. M. Hutcheon, A perpendicular-biased ferrite tuner for the 52 MHz PETRA II cavities, in12th\nIEEE Particle Accelerator Conference, Washington, DC, USA, 1987 (IEEE, New York, 1987),\npp. 1543–5.\n[19] C. C. Friedrichs, R. D. Carlini, G. Spalek, and W. R. Smythe, Test results of the Los Alamos ferrite-\ntuned cavity, in12th IEEE Particle Accelerator Conference, Washington, DC, USA, 1987 (IEEE,\nNew York, 1987), pp. 1896–7.\n[20] R. L. Poirier, Perpendicular biased ferrite-tuned cavities, in15th IEEE Particle Accelerator Confer-\nence, Washington, DC, USA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 753–7.\n[21] C. Fougeron, P. Ausset, D. de Menezes, J. Peyromaure, and G. Charruau, Very wide range and short\naccelerating cavity for MIMAS, in15th IEEE Particle Accelerator Conference, Washington, DC,\nUSA, 1993 (IEEE, Piscataway, NJ, 1994), pp. 858–61.\n[22] K. Saito, K. Matsuda, H. Nishiuchi, M. Umezawa, K. Hiramoto, and R. Shinagawa, RF accelerating\nsystem for a compact ion synchrotron, in19th IEEE Particle Accelerator Conference, Chicago, IL,\nUSA, 2001 (IEEE, New York, 2002), pp. 966–8.\n[23] R. Stassen, K. Bongardt, F. J. Etzkorn, H. Stockhorst, S. Papureanu, and A. Schnase, The HESR\nRF system and tests in COSY , inEPAC, Genoa, 2008 (JACoW, Geneva, 2008), pp. 361–3.\n[24] L. M. Earley, H. A. Thiessen, R. Carlini, and J. Potter, A high-Q ferrite-tuned cavity, IEEE Trans.\nNucl. Sci. NS-30 (1983) 3460–2.\n[25] K. Kaspar, Design of Ferrite-Tuned Accelerator Cavities Using Perpendicular-Biased High-Q Fer-\nrites, Technical report, Los Alamos National Laboratory, New Mexico, USA, November 1984,\nLA-10277-MS.\n[26] G. Schaffer, Improved ferrite biasing scheme for Booster RF cavities, inEPAC, Berlin, 1992 (Édi-\ntions Frontières, Gif-sur-Yvette, 1992), pp. 1234–6.\n19"    },    {        "title": "0810.2770v1.Magnetic_susceptibility_and_magnetization_fluctuation_measurements_of_mixed_Gadolinium_Yttrium_Iron_Garnets.pdf",        "content": "arXiv:0810.2770v1  [cond-mat.mtrl-sci]  15 Oct 2008Magnetic susceptibility and magnetization ﬂuctuation mea surements of mixed\nGadolinium-Yttrium Iron Garnets\nS. Eckel,∗A. O. Sushkov, and S. K. Lamoreaux\nYale University, Department of Physics, P.O. Box 208120, Ne w Haven, CT 06520-8120\n(Dated: October 31, 2018)\nWe study the magnetic properties of Gadolinium-Yttrium Iro n Garnet (Gd xY3−xFe5012,x=\n3,1.8) ferrite ceramics. The complex initial permeability is me asured in the temperature range 2 K\nto 295 K at frequency of 1 kHz, and in the frequency range 100 Hz to 200 MHz at temperatures 4 K,\n77 K, and 295 K. The magnetic viscosity-induced imaginary pa rt of the permeability is observed at\nlow frequencies. Measurements of the magnetization noise a re made at 4 K. Using the ﬂuctuation-\ndissipation theorem, we ﬁnd that the observed magnetizatio n ﬂuctuations are consistent with our\nmeasurements of the low-frequency imaginary part of permea bility. We discuss some implications\nfor proposed precision measurements as well as other possib le applications.\nI. INTRODUCTION\nRare-earth garnets have generated sustained interest\nin various soft-ferrite applications, and, more recently, in\nmore fundamental research. These materials combine at-\ntractive magnetic properties (large permeability, low loss\nangle) with high resistivity, which leads to their exten-\nsive use in microwave applications [1, 2]. Most of their\napplications up to now have been at room temperature,\nalthough there are some more recent suggestions for low-\ntemperature applications also [3]. Gadolinium iron gar-\nnet (GdIG), for example, has found use in adiabatic de-\nmagnetization refrigeration(ADR) in satellites [4]. From\ntheperspectiveofmorefundamentalmeasurements,rare-\nearth garnets have generated great interest as attrac-\ntive materials for experiments searching for the parity\nand time-reversalinvariance-violatingpermanent electric\ndipole moment of the electron [5, 6, 7, 8, 9], as well as\nfor measurements of nuclear anapole moments [10], and\npossibly measurements of the Weinberg angle [11].\nThe ﬁrst evidence that GdIG retains its large initial\npermeability when it is cooled to cryogenic temperatures\ncame from the study by Pascard [12] of the real part of\nthe magnetic susceptibility versus temperature, spanning\nthe range from 4 K to 564 K (the Curie temperature of\nGdIG). The complex permeability µ=µ′−iµ′′of GdIG\nwas measured eariler at radio-frequencies at three tem-\nperatures: 300 K, 195K, and 77 K [13]. Other properties\nhave also been studied at low temperature, including the\nspontaneousandsaturationmagnetizations[14, 15], crys-\ntal anisotropy [16], and ferrimagnetic order [17]. How-\never, to our knowledge, there have been no studies of\nthefull complexinitialpermeabilityofGd-containingfer-\nrites in the temperature range between 4 K and 300 K,\nand at frequencies up to 1 GHz. Such a study is neces-\nsary to evaluate the suitability of these materials for low-\ntemperature applications and precision measurements.\nAn important emerging application of soft ferrites\n∗Electronic address: stephen.eckel@yale.eduis magnetic shielding. Sensitive magnetic ﬁeld mea-\nsurements have to be performed inside a set of per-\nmeable shields, and are often limited by the mag-\nnetic ﬁeld noise generated by Johnson currents in the\nshields themselves [18]. Constructing the shields out of\nhigh-resistivity soft ferrite materials greatly reduces this\nsource of magnetic noise at frequencies above approxi-\nmately 50 Hz [19]. At lower frequencies, however, ﬁnite\nimaginary part of the permeability generates extra mag-\nnetic noise with 1/f power spectrum, in agreement with\nthe ﬂuctuation-dissipation theorem [19, 20].\nThis magnetic noise is related to magnetic viscosity,\nwhich arises when the magnetization of a sample is de-\nlayed following application of a magnetic ﬁeld [21]. Al-\nthough initially seen in ferrous materials, this eﬀect has\nrecently been found in nanowires and other nanostuc-\ntures [22, 23]. Anomalous magnetic viscosity eﬀects have\nalso been seen in various ferrite materials [24]. Magnetic\nviscosity is typically measured by looking at the repsonse\nofthe magnetizationafter asudden changein the applied\nﬁeld; however, it is also characterized by a small µ′′that\nis independent of frequency for a given magnetic ﬁeld\nstrength and temperature [25, 26, 27]. As shown in the\nabove references, the temperature and ﬁeld dependences\nof this eﬀect can be quite complex.\nIn thisworkwestudy the magneticpropertiesofmixed\nGadolinium and Yttrium iron garnets with the chemical\nformula Gd xY3−xFe5012. Here,xquantiﬁes the ratio of\nGadolinium to Ytrrium present within the garnet. We\npresent measurements of both real and imaginary parts\nof the complex initial susceptibility in the temperature\nrange of 4 K to 300 K, and in the frequency range of\n100 Hz to 200 MHz. We also measure the magnetization\nnoise of these materials, and conﬁrm the consistency of\nour results with the ﬂuctuation-dissipation theorem.\nII. EXPERIMENTAL SETUP\nThe inductance of a tightly wound coil on a toroidal\ncore is proportional to the permeability µof the core\nmaterial. The presence of a complex permeabililty µ=2\nZTR=ωL′′\nL=L′ZTEquivalent Circuit: SQUID circuit:\nMsq\nLin =\nSQUID\nFIG. 1: Equivalent circuit for an inductor with a dissipa-\ntive core (see text) and circuit schematic for the SQUID ﬂux\npickup loop, described in Section IV.\nµ′−iµ′′creates a complex inductance L=L′−iL′′. The\nimpedance for such an toroidal inductor is ZT=iωL′+\nωL′′, which is equivalent to a resistance of R=ωL′′\nin series with an inductor L′, as shown in Fig. 1. We\ndeduce the real and imaginary parts of the permeability\nfrom measurements of the complex impedance of such a\ntoroid.\nTwo toroidal samples: one pure GdIG (Gd 3Fe5012),\nand one mixed GdYIG (Gd 1.8Y1.2Fe5012), were pur-\nchased from Paciﬁc Ceramics. These were manufactured\nusingtheir standardprocess. Powderwasgranulatedand\nscreened, using 99.99% pure Gd, 99.999% pure Y, and\n99.6% pure Fe oxides. The typical grain size was 5-10\nmicrons. The samples were pressed and ﬁred at approxi-\nmately 1400◦C for 15 hours. The toroids were made with\nsquare cross sections, with inner radius of 0.5 cm, outer\nradius of 1.5 cm, and a height of 1 cm.\nTo measure the real part of the permeability, µ′, at\nfrequencies less than 1 MHz, the toroidal samples were\nwrapped with approximately 30 turns of wire, and the\nresulting self-inductance was measured with a QuadTech\nSeries 1200 LCR meter. This method does not give reli-\nable measurements for the imaginary part of the perme-\nability, however, since the ohmic resistance of the wind-\ning dominates the loss due to µ′′. For measurements of\nµ′′at frequencies less than 1 MHz, the toroidal samples\nwere wrapped with a primary winding of approximately\n35 turns, and a secondary winding of approximately 80\nturns. Litz wire was used to reduce skin eﬀects. The\ncomplex mutual inductance ofthe two windings was then\nmeasured by a four-point measurement with the LCR\nmeter. Since the input impedance of the voltage inputs\non the meter is large ( >10 MΩ), the current ﬂowing\nin the secondary winding is very small, thus the ohmic\nresistance of the secondary does not aﬀect the measure-\nment of the imaginary part of the susceptibility (on the\nlevel of a fraction of one percent). The inter-winding ca-\npacitance, however, limits this technique to frequencies\nbelow100kHz, athigherfrequenciesthesmallimpedance\nof the shunt capacitor results in a substantial current in\nthe secondary winding.\nA Hewlett-Packard series 4100A impedance analyzer\nwas used for measurements in the range of 1 MHz to200 MHz. The sample was placed in a toroidal-shaped\ncopper housing to reduce the number of turns around\nthe toroid to one. This minimized the electrical length of\nthe wrapping such that it was always less than the wave-\nlength. However, it introduced capacitive eﬀects which\nprevented reliable measurements at frequencies above\n200 MHz. The self inductance of the single turn induc-\ntor was measured with the impedance analyzer. Ohmic\nresistance eﬀects lead to errors on the order of 1%, since\nthe series impedance due to µ′′dominates at such high\nfrequencies.\nFor all the inductance measurements, the strength\nof the applied magnetic ﬁeld was no more than H0∼\n3 A/m, estimated from the current ﬂowing in the toroid\nwindings. This is much smaller than the ferrite’s satu-\nration ﬁeld ( H0∼105A/m) [17], which ensures that we\nalways measure the initial susceptibility of the material.\nLow-temperature measurements were performed with\nthe samples mounted in a G-10 holder inside a Janis\nmodel 10CNDT cryostat. Temperature was measured\nwith a LakeShore model DT-670C-SD silicon diode tem-\nperature sensor, read out by a LakeShore Model 325\ntemperature controller, with accuracy of 0.1 K. At 4 K,\ntemperature-dependent eﬀects on the calibration of the\nQuadTech LCR meter were corrected by making a mea-\nsurement of tan δ=µ′′/µ′using a SQUID magnetome-\nter. Multiple-turn coils were wrapped around toroidal\nnon-magnetic (G10) and GdIG samples and the coils\nwere connected in series. A single-turn superconduct-\ning pickup loop was wrapped about each sample and\neach pickup loop was connected to a SQUID. An sinu-\nsoidalcurrentwaspassedthroughthecoils,andthephase\nshift between the non-mangetic and GdIG samples was\nrecorded. The tan δrecorded by the SQUIDs was then\nused to correct the phase-shift error in the calibration of\nthe QuadTech LCR meter at 4 K. More details on the\nSQUID setup can be found in Sec. IV and Ref. [28]. For\ntemperature dependent measurments, temperature vari-\nation was accomplished by removing the cryogens and\nletting the cryostat warm up, while a control computer\nread out the temperature controller and the LCR me-\nter. Temperature dependent measurements were typi-\ncally separated by 0.1 K, and the total warming time\nwas approximately 12 hours.\nIII. INITIAL SUSCEPTIBILITY\nThe measured real part of permeability, µ′, as a func-\ntion of temperature is shown in Fig. 2 for GdIG and\nGdYIG ferrites. It is evident that both these ferrites\nretain their large permeability down to 4 K. The mag-\nnetic properties of mixed Gadolinium-Yttrium iron gar-\nnets are well described by a model with three magnetic\nsublattices: the c-sites containing Gd3+ions, the a-sites\ncontaining Fe3+ions, and the d-sites containing Fe3+\nions [29]. The Y3+ions are not magnetic, they substitute\nfor Gd3+. At the compensation temperature, the magne-3\n0 50 100 150 200 250020406080100120µ'\nTemperature(K)x= 3.0 exp. (GdIG)\nx= 1.8 exp. (GdYIG)\nx= 3.0 theory (GdIG)\nFIG. 2: The real part of the initial permeability of GdIG\nand GdYIG versus temperature. The Globus model (Eq. 1)\ndescribed in the text is shown in ﬁne dashed red as the best ﬁt\nto the data between 65 K and 150 K. All data were obtained\nat a frequency of 1 kHz and H≈3 A/m applied.\ntizations of the three sublattices compensate each other,\nso that the net magnetization vanishes, and the perme-\nability approaches unity [30]. As evident from Fig. 2,\nthe compensation temperature for Gd 1.8Y1.2Fe5012is\n155K,ingoodagreementwithmeasurementsinRef.[17].\nThe compensation temperature for pure GdIG is 290\nK [14, 17, 31], which is not visible in the plot.\nAs claimed in Ref. [12], the model of domain wall\nbulging due to Globus [32, 33] provides a satisfactory ex-\nplanation of the temperature dependence of µ′for GdIG\n(x= 3.0). In this model\nµ′−1 =3πM2\ns\n4l|K1|Dm, (1)\nwhereMsisthe saturationmagnetization, K1isthemag-\nnetocrystalline anisotropy, Dmis the mean grain size,\nandlis a constant, with dimensions of length, that de-\npends on the properties of the material. Taking the data\nforMsandK1from Refs. [14, 16] and ﬁtting in the\nleast squares sense the constant lyields the theoretical\nﬁt in Fig. 2. There is good agreement with our data be-\ntween 65 K and 200 K. Lack of agreement below 50 K,\nhowever, suggests that the single-crystal samples used\nin Refs. [14, 16] do not have the same low-temperature\nproperties as the pressed ceramic used for the present\nwork.\nOur measurements indicate the maximum permeabil-\nity of 112 at 47 K for GdIG. Susceptibility data from\nRef. [12], however, shows a peak in the GdIG susceptibil-\nity ofµ′≈70at 56K. This discrepancycan be accounted\nfor within the model (1) by the diﬀerence in grain size,\nDm, ofthe ceramicsamples: the samplesused inRef. [12]\nhaveDm∼4µm, whereasour sampleshave Dm∼7µm.1 2 3 4 5 60.81.01.21.41.61.82.02.22.4\n777879808182 µ''Data\nµ''Linear Fit\nµ'Data\nµ'Linear Fitµ''\nAverageH(A/m)\nµ'\nFIG. 3: The real and imaginary parts of permeability of GdIG\nversus applied ﬁeld H, averaged over the volume of the sam-\nple. This data was obtained at a frequency of 1 kHz and\ntemperature of 4 K.\nAt ﬁxed temperature and frequency, the complex sus-\nceptibility depends on the applied ﬁeld H. The mea-\nsurements of µ′andµ′′for various applied ﬁelds Hare\nshown in Fig. 3. We always observed linear dependence\nof the magnetic susceptibility on H, implying that the\nmagnetization is given by M=χH+1\n2αH2. This non-\nlinear dependence of magnetization on applied ﬁeld leads\nto the low-ﬁeld Rayleigh hysteresis loop [34]. Our four-\npoint measurements of µ′′at low frequency ( f <100\nkHz) were always made at several applied ﬁelds Hand\nlinearly extrapolated to zero, as shown in Fig. 3.\nThe measurements of the complex susceptibility as a\nfunction of frequency at 4 K, 77 K, and 295 K are shown\nin Fig. 4. No data are shown for pure GdIG at 295 K,\nbecause this is close to its compensation temperature, so\nthe sample was very nearly non-magnetic at this temper-\nature. It is evident from the data that near 20 MHz the\npermeability of both samples exhibits a resonance. This\n“natural resonance” occurs at the frequency ωr=γHA,\nwhereγis the gyromagnetic ratio, and HA∝K1/Msis\nthe crystalline anisotropy ﬁeld [34, 35]. Using Eq. (1) we\ncan re-write the resonance frequency as:\nωr∝Ms\nµ′−1. (2)\nUsing the data for saturation magnetization in Ref. [14],\nand our data for µ′−1, it is apparent that as the tem-\nperature decreases, Msgrows faster than µ′−1, thus the\nnatural resonance should shift to higher frequency with\nlower temperature, which is indeed the case as can be\nseen from the data in Fig. 4. A similar trend is observed\nin Ref. [13].\nIn spite of the shift of the natural resonance to higher\nfrequency, the low-frequency imaginary part of perme-4\n1021031041051061071080.1110\n  µ''\nFrequency50100\n  \n µ '\nx=3.0 (GdIG)\n4K\n77K0.1110\n  \n µ ''10100\n  \n µ'\nx=1.8 (GdYIG)\n4 K\n77 K\n295K\nFIG. 4: The real and imaginary parts of permeability ofGdIG\nand GdYIG versus frequency. Since µ′∼1 for GdIG at room\ntemperature, no data is shown. All data were obtained with\nan applied ﬁeld of H≈3 A/m. Low frequency ( f <100\nkHz)µ′′data were obtained by a four point measurement and\nextrapolated to zero applied ﬁeld, as described in the text.\nability increases as the temperature is lowered. This, as\nwell as inspection of the data in Fig. 4, shows that the\nfrequency dependence of the permeability is not as sim-\nple as the models suggested in Ref. [36], for example. It\nappears that at low frequencies ( f <100 kHz) the mag-\nnitude of µ′′is dominated not by the wing of the natural\nresonance, but by some frequency-independent dissipa-\ntion mechanism. This is known as magnetic viscosity.\nThe presence of a non-zero µ′′is indicative of a phase\nshift ofmagnetizationwith respecttothe appliedﬁeld H.\nThe phase lag, or loss angle, δis given by tan δ=µ′′/µ′.\nFor a given temperature, and at frequencies much lower\nthan the natural resonance, the loss angle is empirically\nparameterized by the lag equation [37]:\n2πtanδ\nµ′=c+aH+ef, (3)\nwherec,a, andeare parameters, and fis the frequency\nofthe appliedﬁeld H, which ismuchlessthanthe satura-\ntionﬁeld. Thelasttermaccountsforeddycurrenteﬀects.\nHowever, in the present case, eddy current lossesare neg-\nligible since our ferrite samples are very good insulators.TABLE I: Parameters of the loss equation (3) for GdIG and\nmixed GdYIG at various temperatures. Values were compiled\nby least squares ﬁt to all available data less than 10 kHz.\nError bars are extracted from the ﬁt (1 σ).\nx T c a (A/m)−1\n3.0 77 K (3 .51±0.40)×10−4(2.73±0.13)×10−4\n4 K (1 .14±0.14)×10−3(2.30±0.48)×10−4\n1.8 295 K (5 .75±0.44)×10−4(3.68±0.18)×10−4\n77 K (2 .93±0.12)×10−3(6.75±0.31)×10−4\n4 K (2 .05±0.30)×10−3(6.73±1.13)×10−4\nThe second term describes hysteresis eﬀects described\nabove, thus the parameter ais a measure of non-linearity\nin the system. The parameter cis a constant allowing for\nfrequency-independent loss at zero applied ﬁeld, this is\nthemagneticviscosity. Usingthelow-frequencymeasure-\nments of µ′′versus frequency and applied ﬁeld H, values\nofaandcwere extracted at 4 K, 77 K, and 295 K. These\nvalues are shown in Table I. Once again, we do not give\nany results for pure GdIG ( x= 3) at 295 K, since this is\nvery close to its compensation temperature.\nMagnetic viscosity is characterized by time dependent\nchange in the magnetization after a change in the ap-\nplied magnetic ﬁeld. When a small magnetic ﬁeld is ap-\nplied to a ferromagnet, magnetization changes primarily\ndue to domain wall movement. As it moves, the domain\nwall traverses a complex potential energy landscape [34].\nThe time scale of the thermally-driven movement across\na potential energy barrier of height Edis given by an\nArrhenius-type equation: τ=τ0exp(Ed/kBT) [35]. In\na macroscopic sample, there are multiple domain walls,\neach seeing some distribution of such activation energies.\nIf this distribution is ﬂat over a wide range of potential\nbarrier heights, then µ′′is independent of frequency, and\nthe magnetic viscosity term cappears in the expression\n(3) for the lag angle [25, 26, 27, 38].\nThe temperature dependence of magnetic viscosity of\nourGdYIG samplescanbeinferredfromFig.5, where µ′′\nis plotted as a function of temperature. There are several\ncompeting eﬀects that leadto a complicated temperature\ndependence [25]. On the one hand, at low temperature,\nthe domain wall relaxation times grow longer, increasing\nthe magnitude of µ′′. On the other hand, as the temper-\nature is lowered even further, some of the domain walls\nbecome pinned, no longer contributing to the initial sus-\nceptibility, decreasing the magnitudes of both µ′andµ′′,\nas seen in the x= 3.0 curve in Figs. 5 and 2.\nIV. MAGNETIZATION NOISE\nThe magnitude of the magnetization noise is the key\nproperty of the GdYIG garnet ferrites that determines\nthefeasibilityoftheirapplicationinmagneticshieldingas\nwell as in experiments searching for parity and/or time-\nreversal symmetry violations. We measured the magne-5\n0 50 100 150 200 2500.00.51.01.52.02.53.03.54.0\n  µ''\nTemperature(K)x= 3.0 (GdIG)\nx= 1.8 (GdYIG)\nFIG. 5: Measurement of µ′′versus temperature for GdIG and\nGdYIG. All data were obtained at a frequency of 1 kHz and\nH∼3 A/m applied. Data below 25 K was unreliable due to\nsigniﬁcant temperature dependent eﬀects on the QuadTech\nLCR meter’s calibration.\ntization noise of our samples and compared the results\nwith our data for the complex initial permeability using\nthe ﬂuctuation-dissipation theorem.\nThe noise measurements were made at 4 K inside the\ncryogenic dewar described in Section II. A Quantum De-\nsign model 50 superconducting quantum interference de-\nvice (SQUID) magnetometer was connected to the sam-\nple under study by a niobium wire pickup loop. The one-\nturn loop was wound around the square cross-section of\nthe toroidal sample, so that the total ﬂux pickup area\nwas 1 cm2. The SQUID control and readout was per-\nformed by the control unit (model 5000), connected to\nthe SQUID via a 4-meter MicroPREAMP cable. The\nSQUID and the sample were mounted on G-10 holders,\nand enclosed in superconducting shielding, made of 0.1-\nmm thick Pb foil, glued to the inner surfaces of two G-\n10 cylinders. The magnetic ﬁeld shielding factor of our\nsetup is greaterthan 109. Some more details on our noise\nmeasurement setup are given in Ref. [28].\nThe measured magnetization noise amplitude spec-\ntrum for a pure GdIG sample is shown in Fig. 6. The\nnoise data for the mixed GdYIG sample is within 40% of\nthe pure GdIG noise; we do not show it, since the two\ncurves would lie on top of each other, given the plot’s\nlogarithmic scale. The measurement bandwidth of 1 kHz\nis set by the output analog ﬁlter to prevent aliasing with\nour sampling rate of 12 kHz. The SQUID magnetometer\nintrinsic sensitivity in this frequency range is 7 fT/√\nHz;\nwithin our bandwidth, the magnetization noise is always\nabove the magnetometer sensitivity.\nAccording to the ﬂuctuation-dissipation theorem, any\ndissipativematerialexhibits thermalnoise. Wemeasured\nthe dissipation of our ferrite samples in Section III, this\nis given by the imaginary part µ′′of the complex perme-0.01 0.1 1 10 100 100010-710-610-510-4\nGd3Fe5O12magnetization noise at 4K\nNyquist theorem predictionMagnetizationNoise(A/(mHz1/2))\nFrequency\nFIG. 6: Magnetization noise of GdIG at 4.2 K. The blue\nline is the result of using the ﬂuctuation-dissipation theo rem,\nEq. (5), with the value for the imaginary part of permeabilit y\nµ′′= 1.1, taken from measurements shown in Fig 4. The\nSQUID white noise level corresponds to 10 fT /√\nHz on this\nplot.\nability. We can therefore check that our measurements\nof the magnetization noise and the dissipation are consis-\ntent using the ﬂuctuation-dissipation theorem. Consider\nthe circuitshowninFig. 1, wherethe compleximpedance\nZTof the pickup loop around the permeable sample is in\nseries with the input inductance Linof the SQUID. The\nreal part of ZT,ωL′′, is a source of voltage noise, whose\nspectral density is given by the Nyquist theorem:\n(δV)ω=/radicalbig\n4kBTωL′′, (4)\nwherekBis Boltzmann’s constant, Tis temperature,\nω= 2πfis angular frequency, and L′′is the imaginary\npart of the pickup loop inductance. The current noise\nin the circuit can be obtained by dividing Eq. (4) by the\ntotal impedance, ZT+ωLin≈ω(L′+Lin). This current\ncouples to the SQUID through the mutual inductance\nMsq. The resulting ﬂux through the SQUID is the same\nas would be created by a lossless pickup loop around the\nsample with magnetization noise given by:\n(δM)ω=1\nµ0A/radicalBigg\nkBTL′′\n2πf, (5)\nwhereA= 1.0 cm2is the area of the pickup loop and\nµ0is the permeability of free space. The dependence of\nthe single turn pickup loop’s inductance on χ=µ−1\nwas found empirically and is described well by a linear\nexpansion, e.g. L=L0+χL1, where L0andL1are\nthe expansion coeﬀecients. Since these coﬀecients are\nreal, the imaginary part of the inductance is given by\nL′′=χ′′L1=µ′′L.\nUsing Eq. (5) and our empirical determination of L′′,\nwe can compare our data for µ′′of GdIG at 4 K with6\nour measurement of its magnetization noise. From the\nGdIG data shown in Fig. 4 we can deduce, with good ac-\ncuracy, a constant µ′′= 1.1 for frequencies below 1 kHz.\nWith this value, Eq. (5) gives a prediction for the mag-\nnetization noise, shown as the blue line in Fig. 6. Note\nthe 1/√fdependence characteristic of magnetic viscos-\nity. The agreement with experimental noise measure-\nments is remarkable, given that µ′′was measured with\nan entirely diﬀerent method.\nV. CONCLUSIONS\nWehavemeasuredthefullcomplexpermeabilityoftwo\nGdYIG ceramics in the temperature range 2 K to 295 K\nand conﬁrmed that the real part of the permeability re-\nmains large (78 and 52 respectively) at 4 K. The Globus\nmagnetization model ﬁts the results well between 65 K\nand 200 K. We also measured the full complex perme-\nability as a function of frequency in the range 100 Hz to\n200 MHz at temperatures of 4 K, 77 K, and 295 K. We\nobserved the natural resonances near 20 MHz, and de-\ntected the frequency-independent imaginary permeabil-\nity due to magnetic viscosity eﬀects for frequencies less\nthan 100 kHz: µ′′= 1.1 for GdIG and µ′′= 0.8 for\nGdYIG at 4 K. To our knowledge, such eﬀects have not\nbeenstudiedforrare-earthferritesinsuchawidetemper-\nature range. Using the ﬂuctuation-dissipation theorem,\nthese results were compared with direct measurements ofthe magnetization noise, with very good agreement.\nOur noise measurements at 4 K show that the intrinsic\n1/√fmagnetization noise in GdIG and GdYIG is larger\nthan the SQUID magnetometer noise for frequencies be-\nlow approximately 100 kHz. This implies that precision\nmeasurements involving these materials either have to be\ncarried out at higher frequencies, to avoid this noise, or\nhave to make use of the geometry-dependent demagne-\ntizing ﬁelds to reduce the coupling of the noise to the\nmeasurement device [28].\nA possible application of the materials studied in the\npresent work is to magnetic shielding of high-T cSQUIDs\nat 77 K. For a cylindrical shield of radius 5 cm and thick-\nness 1 cm (the dimensions in Ref. [19]), made of GdIG,\nthe shielding factor is 6. The magnetic ﬁeld noise on the\naxis can be predicted from our measurement of µ′′= 0.5\nat 77 K, resulting in δB= 10 fT/√\nHz at 1 Hz. This is at\nthe level of a high-T cSQUID’s sensitivity. Such a shield\ncan be used inside a set of µ-metal shields to screen ex-\nternal magnetic ﬁelds and the Johnson noise due to the\ninnermost layer of the µ-metal shields, without generat-\ning excessive Johnson noise itself.\nAcknowledgments\nThe authors would like to acknowledge useful discus-\nsions with Woo-Joong Kim and Dmitry Budker. This\nwork was supported by Yale University.\n[1] M. Pardavi-Horvath, Journal of Magnetism and Mag-\nnetic Materials 215-216 , 171 (2000).\n[2] I. Zaquine, H. Benazizi, and J. C. Mage, Journal of Ap-\nplied Physics 64, 5822 (1988).\n[3] G. F. Dionne, The 41st annual conference on magnetism\nand magnetic materials 81, 5064 (1997).\n[4] T. T. King, B. A. Rowlett, R. A. Ramirez, P. J. Shirron,\nE. R. Canavan, M. J. DiPirro, J. S. Panek, J. G. Tut-\ntle, R. D. 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Section A 62, 562 (1949)."    },    {        "title": "2401.11488v2.HARDCORE__H_field_and_power_loss_estimation_for_arbitrary_waveforms_with_residual__dilated_convolutional_neural_networks_in_ferrite_cores.pdf",        "content": "1\nHARDCORE: H-field and power loss estimation for\narbitrary waveforms with residual, dilated\nconvolutional neural networks in ferrite cores\nWilhelm Kirchg ¨assner∗, Nikolas F ¨orster∗, Till Piepenbrock∗, Oliver Schweins∗, Oliver Wallscheid∗\n∗Department of Power Electronics and Electrical Drives\nPaderborn University, 33095 Paderborn, Germany\nAbstract —The MagNet Challenge 2023 calls upon competi-\ntors to develop data-driven models for the material-specific,\nwaveform-agnostic estimation of steady-state power losses in\ntoroidal ferrite cores. The following HARDCORE (H-field and\npower loss estimation for Arbitrary waveforms with Residual, Di-\nlated convolutional neural networks in ferrite COREs) approach\nshows that a residual convolutional neural network with physics-\ninformed extensions can serve this task efficiently when trained\non observational data beforehand. One key solution element is an\nintermediate model layer which first reconstructs the bhcurve\nand then estimates the power losses based on the curve’s area\nrendering the proposed topology physically interpretable. In ad-\ndition, emphasis was placed on expert-based feature engineering\nand information-rich inputs in order to enable a lean model\narchitecture. A model is trained from scratch for each material,\nwhile the topology remains the same. A Pareto-style trade-off\nbetween model size and estimation accuracy is demonstrated,\nwhich yields an optimum at as low as 1755 parameters and\ndown to below 8 % for the 95-th percentile of the relative error\nfor the worst-case material with sufficient samples.\nIndex Terms —Magnetics, machine learning, residual model\nI. I NTRODUCTION\nThe MagNet Challenge 2023 is tackled with a material-\nagnostic residual convolutional neural network (CNN) topol-\nogy with physics-informed extensions in order to leverage\ndomain knowledge. Topological design decisions are dictated\nby peculiarities found in the data sets and by the overall goal\nof maximum estimation accuracy at minimum model sizes.\nThe topology’s central idea is the calculation of the area\nwithin the bhpolygon based on a preceding hsequence\nestimate, see Fig. 3. The area within the polygon formed by the\nsequences b,h∈R1024can be calculated using the shoelace\nformula or surveyor’s area formula [1]. The shoelace method\nassigns a trapezoid to each edge of the polgyon as depicted\nin Fig. 1. The area of these trapezoids is defined according to\nshoelace either with a positive or negative sign, according to\nthe hysteresis direction. The negative areas compensate for the\nparts of positive trapezoids that extend beyond the boundaries\nof the polygon. Provided that the polygon is shifted into the\nfirst quadrant by some offsets hosandbos, the power loss in\nW m−3caused by magnetic hysteresis effects can be computed\nwith the frequency f,M= 1024 , and circular padding by\nˆphyst=f·1\n2M−1X\ni=0bi(hi−1−hi+1). (1)\nhosbos\nˆphyst\nbin T\nhin A/m\nFig. 1. Visualization of the shoelace formula applied to a bhpolygon.\n101103Emp.\nprob.3C94 78 N30 3E6 3F4\n101103Emp.\nprob.N87 3C90 N49 N27 77\n−505\nRel. error\nin percent101103Emp.\nprob.A\n−505\nRel. error\nin percentB\n−505\nRel. error\nin percentC\n−505\nRel. error\nin percentD\n−505\nRel. error\nin percentE\nFig. 2. Relative error (ˆphyst−p)/phistogram between provided scalar pand\nˆphystcalculated from the likewise provided bhpolygon area.\nWhen applying (1) on the given sequences b,h∈RM\nwithM= 1024 , it becomes evident that the calculated area\ndoes not equal the provided loss measurements exactly. Fig. 2\nshows the discrepancy with respect to the provided scalar loss\npfor all materials. The relative error ranges up to over 7 %\nfor certain materials (e.g. 3F4, N49, D, E). Consequently, if\nmerely an h-predicting model was to be identified, the lower\nbound on the rel. error would be significantly elevated by this\ncircumstance alone.\nSince the power losses calculated from neither the ground\ntruth bhcurve area (assuming ideal knowledge on the h\nsequence) nor the estimated area ( ˆhreconstructed via a CNN)\ndo perfectly match the provided loss measurement values\n(targets), an additional residual correction mechanism is addedarXiv:2401.11488v2  [eess.SY]  23 Jan 20242\nfeature\nengineering #1bh curve\nestimationbh curve-based\npower loss est.data-driven\nloss correction\ntarget\nestimate\nfeature \nengineering #2input data\nFig. 3. Overview of the physics-inspired HARDCORE modeling toolchain.\nto compensate for this. A high-level view on the proposed\nresidual, physics-inspired modeling toolchain is depicted in\nFig. 3, which is coined the HARDCORE approach (H-field and\npower loss estimation for Arbitrary waveforms with Residual,\nDilated convolutional neural networks in ferrite COREs), and\nits details are discussed in the following.\nII. M ODEL DESCRIPTION\nA residual CNN with physics-informed extensions is uti-\nlized for all materials. Such a CNN is trained for each material\nfrom scratch. Yet, the topology is unaltered across materials,\nsignal waveforms, or other input data particularities.\nA. One-dimensional CNNs for h-estimation\nA 1D CNN is the fundamental building block in this\ncontribution, which consists of multiple trainable kernels or\nfilters per layer slided over the multi-dimensional input se-\nquence in order to produce an activation on the following\nlayer [2]. These activations denote the convolution (more\nprecisely, the cross-correlation) between the learnable kernels\nand the previous layer’s activation (or input sequence). In this\nstateless architecture, circular padding ensures that subsequent\nactivation maps are of equal size. Circular padding can be\nutilized here instead of the common zero-padding as sequences\ndenote complete periods of the bandhcurve during steady\nstate. Moreover, a kernel does not need to read strictly adjacent\nsamples in a sequence at each point in time, but might use a\ndilated view, where samples with several samples in between\nare used. The dilated, temporal CNN update equation for the\ni-th filter’s activation a(l)\ni[k]at time kand layer lwith the\nlearnable coefficients Wi∈RA×κapplied on Aprevious\nlayer’s filters, an uneven kernel size of κ∈ {2x+1 : x∈N0},\nand the dilation factor δreads\na(l)\ni[k] =A−1X\np=0(κ−1)/2X\nj=−(κ−1)/2Wi;(p,j)·a(l−1)\np[k+jδ].(2)\nSince the task at hand does not require causality of CNN\nestimates along the time domain (losses are to be estimated\nfrom single bsequences), the sliding operation can be effi-\nciently parallelized, and sequential processing happens merely\nalong the CNN’s depth. All 1D CNN layers are accompanied\nby weight normalization [3]. A conceptual representation of\nthe 1D CNN for estimating ˆhis visualized in Fig. 6 (left part).B. Feature engineering\nThe term feature engineering encompasses all preprocess-\ning, normalization, and derivation of additional features in an\nobservational data set. The input data contains the frequency\nf, the temperature T, the measured losses pas well as the\n1024 sample points for the bandhwaveforms. Especially the\ncreation of new features that correlate as much as possible\nwith the target variable (here, the hcurve or the scalar power\nlossp) is an important part of most machine learning (ML)\nframeworks [4].\n1) Normalization: As is typical in neural network training,\nall input and target features have to be normalized beforehand.\nAll scalar and time series features are divided by their max-\nimum absolute value that occurs in the material-specific data\nset, with the exception of the temperature and the frequency,\nwhich will be divided by 75◦Cand150 kHz , respectively,\nregardless the material. Moreover, for an accurate hestimate, it\nwas found to be of paramount importance to normalize each b\nandhcurve again on a per-profile base in dependence not only\non the ℓ∞norm of |b|, but also on the maximum absolute band\nhappearing in the entire material-specific data set. The latter\ntwo values are denoted blimandhlim, and can be understood as\nmaterial-specific scaling constants. In particular, the per-profile\nnormalized bandhcurves for a certain sample read\nbn=b\nmax k|b[k]|,hn=h\nhlim·blim\nmax k|b[k]|, (3)\nwithhlim= max i,k|hi[k]|,blim= max i,k|bi[k]|, and ibeing\nthe sample index in the entire material-specific data set. Then,\nbnis added to the set of input time series features, and hnis\nthe target variable for the hestimation task.\nFig. 4. Exemplary samples of the normalized bandhcurves.\nThebnoverhncurves are displayed in Fig. 4, which\nunderlines how the polygon area becomes roughly unified (no\nlarge area difference between samples). In the following, all\nfeatures that get in touch with the model are normalized values\nwithout any further notational indication.\n2) Time series features (feature engineering #1): As dis-\ncussed in Sec. II-A, 1D CNNs build the core of the imple-\nmented model. The inputs to the CNNs are the (per-profile)\nnormalized magnetic flux density bnand the corresponding\nfirst and second order derivatives ( ˙bnand¨bn) as time series. In\na macroscopic measurement circuit context, ˙bcorresponds to3\n0.1\n0.00.1b in Vs / m²\n~magnetic flux\n0200db/dt in V/m²\n~voltage\n0 200 400 600 800 1000\ndatapoints10\n010d2b/dt2 in V/(m²µs)\n~voltage slew rate\nFig. 5. Magnetic flux density examples and their first and second order\nderivatives for a sinusoidal, triangular and one unclassified waveform with\na circuit-based interpretation in terms of their proportionality to magnetic\nflux, voltage and the voltage slew rate.\nthe applied magnetizing voltage throughout the measurement\nprocess of the data. Accordingly, ¨brepresents the voltage slew\nrate during the commutation of the switches in the test setup.\nConsequently, the second derivative allows to detect switching\nevents and to characterize them according to their maximum\nslew rate. Fig. 5 shows, that the sinusoidal waveform (green) is\ngenerated without any fast transient switching behaviour, prob-\nably with a linear signal source. The nonsinusoidal examples\nshow typical switching behaviour with different voltage slew\nrates during the single transitions and voltage overshoots as\nwell as ringing. The second derivative of binforms the ML\nmodel about switching transition events and how fast changes\nin time are.\n3) Scalar features (feature engineering #2): Although\nsequence-based CNNs take up the main share of the ML model\nsize, scalar environmental variables also have a considerable\nimpact on handp. While the temperature Tis passed to the\nmodel unaltered (but normalized), the frequency is presented\nby its logarithm ln(f). The sample time 1/fis passed directly\nto the model. Furthermore, some b-derived scalar features\nare also passed to the model to feed in a priori knowledge.\nFor example, the peak-to-peak magnetic flux ∆bas well as\nthe mean absolute time derivative |˙b|are directly fed into\nthe network. Each waveform is automatically classified into\n”sine”, ”triangular”, ”trapezoidal”, and ”other” by consulting\nthe form and crest factors, as well as some Fourier coefficients.\nThe waveform classification is presented to the model by one\nhot encoding (OHE). A summary of all expert-driven input\nfeatures is presented in Tab. I.\nC. Residual correction and overall topology\nThe model topology comprises multiple branches that end\nin the scalar power loss estimate ˆp. An overview is sketched\nin Fig. 6. Two main branches can be identified: an h-predictorTABLE I\nUTILIZED INPUT FEATURES .\nTime series features Scalar features\nmag. flux density b temperature T\nper-profile norm. bn sample time 1/f\n1st derivative ˙bn log-frequency ln(f)\n2nd derivative ¨bn peak2peak ∆b\ntan-tan-b tan(0 .9·tan(bn)) log peak2peak ln(∆b)\nmean abs dbdt |˙b|\nlog mean abs dbdt ln(|˙b|)\nwaveform (OHE)\nand a wrapping p-predictor. The h-predictor utilizes both\ntime series and scalar features with CNNs and multilayer\nperceptrons (MLPs), and estimates the full hsequence. The\np-predictor predicts p, on the other hand, and leverages the\npredicted magnetic field strength ˆhwith the shoelace formula\nand a scaling factor. The latter accounts for the losses inex-\nplicable by the bh-curve (recall Fig. 2), and is predicted by a\nMLP that utilizes the scalar feature set only.\nTheh-predictor merges time series and scalar feature in-\nformation by the broadcasted addition of its MLP output to a\npart of the first CNN layer output. This effectively considers\nthe MLP-transformed scalar features as bias term to the time-\nseries-based CNN structure.\nOn the merged feature set, two further 1D CNN layers\nfollow that end the transformation in a 1024-element sequence.\nThe per-profile scaled bnsequence from the set of input\ntime series is element-wise added to this newly obtained\nestimation (residual connection). This results in the CNN\nmodel to merely learn the difference between hnandbn[5].\nEventually, this sequence becomes the hestimation ˆhnwhen\nthe sequence’s average along the time domain, that is, across\nall 1024 elements, is subtracted from each element. This is\na physics-informed intervention in order to ensure a bias-\nfreehestimate ˆhafter denormalization. Note that all such\noperations are still end-to-end differentiable with an automatic\ndifferentation framework such as PyTorch [6], [7].\nSince the resulting ˆhcan only be trained to be as close as\npossible to the provided hsequence, which is not leading to\nthe correct pground truth (cf. Fig. 2), another MLP is branched\noff the scalar input feature set, and denotes the start of the p-\npredictor. This MLP inherits two hidden layers and concludes\nwith a single output neuron. This neuron’s activation, however,\nis not ˆpbut rather an area scaling factor s∈[−1,1]to be\nembedded in the shoelace formula (1) with\nˆp=f·\u0000\n0.5 + (0 .1·s)\u0001M−1X\ni=0bi(ˆhi−1−ˆhi+1). (4)\nConsequently, the p-predictor branch can alter the shoelace\nformula result by up to 10 % in positive and negative direction.\nThep-predictor can be justified physically, when referring\nto Fig. 2 again. As the comparison shows, the hysteresis loss\nrepresents the total loss within a variation of −10 % to+5 % .\nThe positive deviation ( ˆphyst> p) indicates some measurement\ndiscrepancy between the measured loss and the given band\nhcurves. For parts of the negative deviations, a physical\nexplanation can be found in eddy current losses, related to4\n1D CNN\n1024 samplesMLPMLP\nAddAddArea\nscaling\nfactor\nTime series \nfeaturesScalar featuresSubtract mean along \ntime domainPolygon\narea calculation\nFig. 6. The residual 1D CNN topology is shown while applied on time series and scalar features, which also contain engineered features from Tab. I.\na high dielectric constant and non-zero conductivity. Due to\nthe small thickness of the used toroidal cores and the limited\nexcitation frequency, eddy current losses are assumed to be of\nminor effect.\nThe physical interpretability of the intermediate estimate\nˆhis a key advantage of the HARDCORE approach: First,\nit enables utilizing full htime series simulation frameworks\n(e.g., time domain FEM solvers). Secondly, for future designs\nof magnetic components with arbitrary shapes it becomes\nindispensable to accurately take into account also geometrical\nparameters of the core. This is only possible by distinguishing\nbetween the magnetic hysteresis and the (di-)electric losses.\nD. Training cost functions\nThe training process involves two cost functions for a\ntraining data set with size N: First, the hestimation accuracy,\nwhich is assessed with the mean squared error (MSE) as\nLMSE,H =1\nNMN−1X\nn=0M−1X\ni=0(ˆhi,n−hi,n)2. (5)\nSecond, the power loss estimation accuracy is to be gauged.\nDespite the relative error being the competition’s evaluation\nmetric, the mean squared logarithmic error (MSLE) is selected\nLMSLE,P =1\nNN−1X\nn=0(ln ˆpn−lnpn)2(6)\nin order to not overemphasize samples with a relatively low\npower loss [8]. As LMSLE,P also depends on ˆhthrough (4), the\nquestion arises, how both cost functions are to be weighted.\nIn this contribution, a scheduled weighting is applied with\nLtotal=αLMSLE,P + (1−α)LMSE,H, (7)\n0 1000 2000 3000 4000 5000\nEpochs103\n102\n101\nTraining or validation loss\ntrain_h\ntrain_p\nval_h\nval_pFig. 7. Exemplary training and validation loss curve for material A, seed 0\nand fold 3.\nwhere α= (β·iepoch)/K epoch with β∈[0,1]denoting a\nhyperparameter scaling factor, Kepoch being the number of\ntraining epochs, and iepoch∈ {0,1, . . . K epoch−1}representing\nthe current epoch index. The scheduled weighting ensures that\nthe model focuses on ˆhin the beginning of the training, where\nmore information is available. Later though, the model shall\ndraw most of its attention to the power loss estimate, possibly\nat the expense of the hestimation accuracy. A training example\nfor material A is depicted in Fig. 7.\nIII. H YPERPARAMETERS , PARETO FRONT AND RESULTS\nThe proposed topology features several degrees of freedom\nin form of hyperparameters. An important aspect is the model\nsize, which is defined by the number of hidden layers and\nneurons in each layer. A simple trial-and-error investigation5\n1135175522314711595110943\nModel size102\n101\n100Relative loss error\nA\n1135175522314711595110943\nModel size\nB\n1135175522314711595110943\nModel size\nC\n1135175522314711595110943\nModel size\nD\n1135175522314711595110943\nModel size\nEQuantile\nAverage 95th 99th\nFig. 8. Pareto front for the evaluation materials (A, B, C, D and E) showing\nmodel size (amount of parameters) vs. relative error of the power loss\nestimation.\n2 µs\n100\n50\n050100b in mT\n50\n 0 50h in A/m2 µs1646 kW/m³1651 kW/m³\ng. truth\nestimate\nFig. 9. Exemplary ground truth vs. estimated bhcurve comparison.\ncan provide fast insights into the performance degradation\nthat comes with fewer model parameters. In Fig. 8, several\nparticularly selected model topologies are illustrated against\ntheir achieved relative error versus the inherited model size.\nThe scatter in each quantile is due to different random number\ngenerator seeds and folds during a stratified 4-fold cross-\nvalidation. Topology variations are denoted by the amount\nof neurons in certain hidden layers. In addition, the largest\ntopology has an increased kernel size with κ= 17 , and the\nsmallest topology has the second CNN hidden layer removed\nentirely (the green layer in Fig. 6).\nA slight degradation gradient is evident as of 5 k parameters\nfor materials A and C, whereas for the other materials the trend\nis visible only when removing the second hidden layer. Over-\nall, the material performance scales strictly with the amount\nof training data available. Since fewer model parameters are\na critical aspect, the chosen final model has 1755 parameters,\nwhich is at an optimal trade-off point on the Pareto front.\nIn Tab. II, the model size of a corresponding PyTorch model\nfile dumped to disk as just-in-time (jit) compilation is reported.\nThe exemplary bh-curve and h-curve estimation is shown in\nFig. 9. Reported error rates come from the best seed out of five\nduring a four-fold cross-validation ( β= 1, Kepochs = 5000 ,\nNesterov Adam optimizer). It shows effectively that any ma-TABLE II\nFINAL MODEL DELIVERY OVERVIEW\nRelative error\nMaterial Parameters Training Model size Average 95-th\ndata quantile\nA 1755 2432 43.13 kB 2.34 % 6.20 %\nB 1755 7400 43.13 kB 1.10 % 2.68 %\nC 1755 5357 43.13 kB 1.46 % 3.70 %\nD 1755 580 43.13 kB 7.03 % 25.76 %\nE 1755 2013 43.13 kB 2.51 % 7.10 %\nterial can be modeled with the same topology at high accuracy\nas long as a critical training data set size is available (which\nis not the case for material D, see available training data in\nTab. II). The final model is already a trade-off between model\nsize and accuracy, such that in case one of the two criteria can\nbe softened, the other can be further improved.\nThe final model delivery is trained on all training data sam-\nples (no repetitions with different seeds), and with Kepoch=\n10000 , δ= 4, κ= 9. This final topology features a CNN\nwith12(TanH) →8(TanH) →1(linear) kernels, a MLP with\n11(TanH) neurons, and a p-predictor MLP with 8(TanH) →\n1(TanH) neurons.\nIV. C ONCLUSION\nA material-agnostic CNN topology for efficient steady-state\npower loss estimation in ferrite cores is presented. Since the\ntopology remains unaltered across materials and waveforms at\na steadily high accuracy, the proposed model can be considered\nuniversally applicable to plenty of materials. As long as\nsufficient samples of a material are available (roughly, 2000),\nthe relative error on the 95-th quantile remains below 8 %.\nThus, the contributed method is proposed to become a standard\nway of training data-driven models for power magnetics.\nREFERENCES\n[1] B. Braden, “The surveyor’s area formula,” The College Mathematics\nJournal , vol. 17, no. 4, pp. 326–337, 1986.\n[2] A. Krizhevsky, I. Sulskever, and G. E. Hinton, “ImageNet Classification\nwith Deep Convolutional Neural Networks,” in Advances in Neural\nInformation and Processing Systems (NIPS) , vol. 60, no. 6, 2012, pp.\n84–90. [Online]. Available: https://doi.org/10.1145/3065386\n[3] T. Salimans and D. P. Kingma, “Weight Normalization: A Simple\nReparameterization to Accelerate Training of Deep Neural Networks,”\nArXiv e-prints: 1602.07868 [cs.LG], 2016. [Online]. Available: http:\n//arxiv.org/abs/1602.07868\n[4] P. Domingos, “A few useful things to know about machine learning,”\nCommunications of the ACM , vol. 55, no. 10, p. 78, 2012. [Online].\nAvailable: http://dl.acm.org/citation.cfm?doid=2347736.2347755\n[5] S. Bai, J. Z. Kolter, and V . Koltun, “An Empirical Evaluation of Generic\nConvolutional and Recurrent Networks for Sequence Modeling,” 2018.\n[Online]. Available: http://arxiv.org/abs/1803.01271\n[6] A. Paszke, S. Gross et al. , “PyTorch: An Imperative Style, High-\nPerformance Deep Learning Library,” in Advances in Neural Information\nProcessing Systems 32 . Curran Associates, Inc., 2019, pp. 8024–8035.\n[7] A. G. Baydin, B. A. Pearlmutter et al. , “Automatic differentiation\nin machine learning: A survey,” Journal of Machine Learning\nResearch , vol. 18, pp. 1–43, 2018. [Online]. Available: https:\n//arxiv.org/abs/1502.05767\n[8] C. Tofallis, “A better measure of relative prediction accuracy for model\nselection and model estimation,” Journal of the Operational Research\nSociety , vol. 66, pp. 1352–1362, 2015."    },    {        "title": "0902.1418v2.Induced_Violation_of_Time_Reversal_Invariance_in_the_Regime_of_Weakly_Overlapping_Resonances.pdf",        "content": "arXiv:0902.1418v2  [nlin.CD]  30 Jul 2009Induced Violation of Time-Reversal Invariance in the Regim e of\nWeakly Overlapping Resonances\nB. Dietz,1T. Friedrich,1,2H. L. Harney,3M. Miski-Oglu,1\nA. Richter,1,4,∗F. Sch¨ afer,1J. Verbaarschot,5and H. A. Weidenm¨ uller3\n1Institut f¨ ur Kernphysik, Technische Universit¨ at Darmst adt, D-64289 Darmstadt, Germany\n2GSI Helmholtzzentrum f¨ ur Schwerionenforschung GmbH, D-6 4291 Darmstadt, Germany\n3Max-Planck-Institut f¨ ur Kernphysik, D-69029 Heidelberg , Germany\n4ECT∗, Villa Tambosi, I-38100 Villazzano (Trento), Italy\n5Department of Physics and Astronomy,\nSUNY at Stony Brook, NY 11794, USA\n(Dated: November 16, 2018)\nAbstract\nWe measure the complex scattering amplitudes of a ﬂat microw ave cavity (a “chaotic billiard”).\nTime-reversal ( T) invariance is partially broken by a magnetized ferrite pla ced within the cavity.\nWe extend the random-matrix approach to Tviolation in scattering, determine the parameters\nfrom some properties of the scattering amplitudes, and then successfully predict others. Our work\nconstitutes the most precise test of the theoretical approa ch toTviolation within the framework\nof random-matrix theory so far available.\nPACS numbers: 24.60.Ky, 05.45.Mt, 11.30.Er, 85.70.Ge\n1We measure the eﬀect of partial violation of time-reversal ( T) invariance on the exci-\ntation functions of a ﬂat microwave cavity induced by a magnetized f errite placed within\nthe cavity. The classical dynamics of a point particle moving within the cavity and elas-\ntically reﬂected by the walls, is chaotic. The statistical properties o f the eigenvalues and\neigenfunctions of the analogous quantum system are, therefore , expected to follow random–\nmatrix predictions [1]. Random-matrix theory (RMT) provides a unive rsal description of\ngeneric properties of chaotic quantum systems. In particular, RM T yields analytical expres-\nsions for correlation functions of scattering amplitudes [2] that ca n be generalized to include\nTviolation. Althoughwidelyused(todiscover signaturesof Tviolationincompound-nucleus\nreactions [3] in the Ericson regime [4], to describe electron transpor t through mesoscopic\nsamples in the presence of a magnetic ﬁeld [5], and in ultrasound trans mission in rotational\nﬂows[6]), that generic model for Tviolation has, to the best of our knowledge, never been\nexposed to a detailed experimental test. With our data we perform such a test.\nOur aim is not a detailed dynamical modeling of the properties of the ca vity. With the\nexception of the average level density we determine the paramete rs of the RMT expressions\nfrom ﬁts to some of the data. We then test the RMT approach by us ing it to predict\nother data, and by subjecting our ﬁts to a thorough statistical t est. All of this is in the\nspirit of a generic RMT approach since a dynamical calculation of the r elevant parameters\nis not possible for many systems. Such a calculation works only for sp ecial chaotic quantum\nsystems like some cavities where the semiclassical approximation can be used [7, 8]. We use\nthat approximation only to determine the average level density, an d to estimate the range\nof validity of RMT in terms of the shortest periodic orbit.\nMicrowave cavities have been used before to study the eﬀect of T-invariance violation\non the eigenvalues [9, 10, 11] and on the eigenfunctions [12]. Here we study ﬂuctuations of\nthe scattering amplitudes versus microwave frequency. For our c avity the average resonance\nspacingdis of the order of the resonance width Γ, and we work in the regime of weakly\noverlapping resonances.\nExperiment. The ﬂat copper microwave resonator has the shape of a tilted stad ium [13]\n(see Fig. 1) and a height of 5 mm. The excitation frequency franges from 1 to 25 GHz. In\nthat range, only one vertical mode of the electric ﬁeld strength is e xcited. The Helmholtz\nequationforthetiltedstadiumisthenmathematicallyequivalenttoth eSchr¨ odingerequation\nof a two-dimensional chaotic quantum billiard [14]. An Agilent PNA-L N52 30A vector\n2network analyzer (VNA) coupled rf energy via one of two antennas labeled 1 and 2 into the\nresonator and determined magnitude and phase of the transmitte d (reﬂected) signal at the\nother(same)antennainrelationtotheinputsignaland, thus, the elements Sab(f)witha,b=\n1,2 of the complex-valued 2 ×2 scattering matrix S(f). Distorting eﬀects of the connecting\ncoaxial cables were removed by calibration. We measured the elemen ts ofS(f) in the\nfrequency range 1–25 GHz at a resolution of 100 kHz. To improve th e statistical signiﬁcance\nof the data set, an additional scatterer (an iron disc of 20 mm diame ter) was placed within\nthecavity. Itcouldbefreelymovedandallowedthemeasurement of statisticallyindependent\nspectra, so-called “realizations”.\nTime-reversal invariance is violated [15] by a ferrite cylinder (4 πMS= 1859 Oe, ∆ H=\n17.5 Oe, courtesy of AFT Materials GmbH, Backnang, Germany) of 4 mm diameter and\n5 mm height. The cylinder was placed inside the resonator and magnet ized by an external\nmagnetic ﬁeld B. The ﬁeld was provided by two NdFeB magnets (cylindrical shape, 20 mm\ndiameter and 10 mm height) attached from the outside to the billiard. Field strengths of\nup to 360 mT could be attained. Here we focus on the results at B= 190 mT as there the\neﬀects are most clearly visible. The spins within the ferrite precess c ollectively with their\nLarmor frequency about the external ﬁeld. The rf magnetic ﬁelds of the resonator modes\nare, in general, elliptically polarized and couple to the spins of the ferr ite. The coupling\ndepends on the rotational direction of the rf ﬁeld. An interchange of input and output\nchannels changes the rotational direction and thus the coupling of the resonator modes to\nthe ferrite. Figure 2 demonstrates that reciprocity, deﬁned by S12(f) =S21(f) and implied\nbyTinvariance, is violated.\nAs a measure of the strength of T-invariance violation, we deﬁne the cross-correlation\nFIG. 1: The tilted stadium billiard (schematic). The two ant ennas 1, 2 connect the resonator to\nthe VNA. The ferrite is ﬁxed, the scatterer can be moved freel y.\n3coeﬃcient Ccross(ǫ= 0) where\nCcross(ǫ) =Re(/angbracketleftS12(f)S∗\n21(f+ǫ)/angbracketright)/radicalbig\n/angbracketleft|S12(f)|2/angbracketright/angbracketleft|S21(f)|2/angbracketright. (1)\nIfTinvariance holds, we have Ccross(0) = 1 while for complete breaking of Tinvariance S12\nandS21are uncorrelated and thus Ccross(0) = 0. The average /angbracketleft·/angbracketrightover the data is taken in\nfrequency windows of width 1 GHz and over 6 realizations, i.e., position s of the additional\nscatterer. The upper panel of Figure 3 shows Ccross(0) for the diﬀerent frequency windows.\nThe cross-correlation coeﬃcient is seen to depend strongly on falthough complete violation\nofTinvariance is never attained. At 5–7 GHz the Larmor frequency of t he ferrite matches\nthe rf frequency, and the ferromagnetic resonance directly res ults inCcross(0)≈0.8. Around\n15 GHz the eﬀects of T-invariance violation are strongest, Ccross(0)≈0.5. A third minimum\nis observed at about 24 GHz. The connection of the latter two minima to the properties of\nthe ferrite is not clear.\nAnalysis. We analyze the data with a scattering approach developed in the con text of\ncompound-nucleus reactions [16]. The scattering matrix for the sc attering from antenna b\nto antenna awitha,b= 1,2 is written as\nSab(f) =δab−2πi/parenleftbig\nW†(f−Heﬀ)−1W/parenrightbig\nab. (2)\nThe matrix Wµais rectangular and describes the coupling of the Nresonant states µin the\ncavity withtheantennas a= 1,2. Weassumethat T-invarianceviolationisduetotheferrite\nonly. Then Wµais real. The resonances in the cavity are modeled by Heﬀ=H−iπ˜W˜W†.\nHereHis the Hamiltonian of the closed resonator. The elements of the real matrix˜Wµcare\n0.20.5|Sab|\n-π0+π\n16.0 16.5 17.0\nFrequency (GHz)Arg(Sab)\nFIG. 2: Transmission spectra for B= 190 mT in the range 16–17 GHz. The amplitudes and phases\nofS12(solid) and S21(dashed) are seen to diﬀer.\n40.51.0Ccross(0)\n0.10.3\n0 5 10 15 20 25\nFrequency (GHz)ξ\nFIG.3: Experimentallydeterminedvaluesof Ccross(0)(upperpanel)fromEq.(1)andtheparameter\nξforT-invariance violation deduced from these (lower panel) wit h the help of Eq. (3). The error\nbars indicate the r.m.s. variation of Ccross(0) over the 6 realizations.\nequal to those of Wµcforc= 1,2. As done successfully before [17, 18], Ohmic absorption\nof the microwaves in the walls of the cavity and the ferrite is mimicked [ 19] by additional\nﬁctitious weakly coupled channels c. The classical dynamics of a point particle within the\ntilted stadium billiard is chaotic. Therefore [1], we model Hby an ensemble of random\nmatrices. The N-dimensional Hamiltonian matrix Hof the system (the cavity) is written\nas the sum of two parts [20, 21, 22], H=Hs+i(πξ/√\nN)Ha. The real, symmetric, and\nT-invariant matrix Hsis taken from the Gaussian Orthogonal Ensemble (GOE) while the\nreal, antisymmetric matrix Hawith Gaussian-distributed matrix elements models the T-\ninvariance breaking part of H. Forπξ/√\nN= 1 the Hamiltonian Hbelongs to the Gaussian\nUnitary Ensemble (GUE) describing systems with complete Tbreaking. However, for N→\n∞,Tinvariance is signiﬁcantly broken already when the dimensionless para meterξis close\nto unity [23]. In the same limit Ccross(0) in Eq. (1) can be expressed analytically in terms of\na threefold integral involving the parameter ξ. For the derivation we extended the method\nof Ref. [24] where the ensemble average of |Sab|2was computed as function of the parameter\nξ. The cross-correlation coeﬃcient Ccross(0) is obtained by setting ǫ= 0,σ=−1, and\n5a,b= 1,2 in the function\nFσ\nab(ǫ|Ta,Tb,τabs,ξ) =1\n8/integraldisplay∞\n0dx1/integraldisplay∞\n0dx2/integraldisplay1\n0dx\n×µ(x,x1,x2)\nF·exp/parenleftbigg\n−iπǫ\nd(x1+x2+2x)/parenrightbigg\n×/productdisplay\nc1−Tcx/radicalbig\n(1+Tcx1)(1+Tcx2)/bracketleftBig/braceleftBig\nJab(x,x1,x2)\n×/bracketleftbig\nFE++(λ2\n2−λ2\n1)E−+4tR(λ2\n2E−+F(E+−1))/bracketrightbig\n+σ·2(1−δab)TaTb/bracketleftbig\nE−Kab(λ,λ1,λ2|Ta,Tb,ξ)\n+/parenleftbigg\nE+−E−\ntF/parenrightbigg\nLab(λ,λ1,λ2|Ta,Tb,ξ)/bracketrightbig/bracerightBig\n·exp(−2tG−)\n+ (λ1↔λ2)/bracketrightBig\n, (3)\nwith the notations\nt=π2ξ2,R= 4(x+x1)(x+x2),\nU= 2/radicalbig\nx1(1+x1)x2(1+x2),F= 4x(1−x),\nE±= 1±exp(−2tF), λ= 1−2x,\nλi=/radicalbig\n(1+x1)(1+x2)+x1x2−(−1)iU,\nGi=λ2\ni−1,i= 1,2. (4)\nThe integration measure µ(x,x1,x2) and the function Jab(x,x1,x2) are given explicitly in\nRef. [2], while the functions Kab(λ,λ1,λ2|Ta,Tb,ξ) andLab(λ,λ1,λ2|Ta,Tb,ξ) can be read oﬀ\nEq. (2) of Ref. [25]. We checked our analytic results by numerical RM T simulations. The\nparameters of Eq. (3) for ǫ= 0 areξ, the transmission coeﬃcients Ta= 1−|/angbracketleftSaa/angbracketright|2fora=\n1,2, and the sum τabsof 300 transmission coeﬃcients that model the Ohmic losses [18, 19].\nForatypical set T1,T2,τabs, Fig.4 shows Ccross(0)versus ξ. Withinthefrequency range1–\n25GHz,Ccross(0) depends very weakly on T1,T2,τabs, andFig.4 canbetaken to beuniversal.\nFor each data point shown in the upper panel of Fig. 3 the correspo nding value of ξwas\nread oﬀ Fig. 4 and the result is shown as function of fin the lower panel of Fig. 3. Figure 3\nshows that in the interval from 1 to 25 GHz, the ratio of the averag e resonance width Γ to\nthe average resonance spacing dvaries from Γ /d≈0.01 to Γ/d≈1.2 while the strength\nξofTbreaking varies from zero to 0 .3. Numerical calculations show that for ξ= 0.3 the\nspectral ﬂuctuations of the Hamiltonian Hfor the closed resonator deﬁned below Eq. (2)\n6almost coincide with those of the GUE [26]. We also found that for ξ= 0.4 they do not\ndiﬀer signiﬁcantly from those presented in Ref. [9], where the conclu sion was drawn, that\ncomplete Tbreaking is achieved. However, even for ξ= 0.4 the value of Ccross(0) is still far\nfrom zero. This shows that Ccross(0) is a particularly suitable measure of the strength ξof\nTviolation.\nAutocorrelation function. SinceCcross(0) depends only weakly on the values of T1,T2and\nτabs, we used the autocorrelation function Cab(ǫ) for a more precise determination of these\nparameters, especially of τabs. The function\nCab(ǫ) =/angbracketleftSab(f)S∗\nab(f+ǫ)/angbracketright−|/angbracketleftSab(f)/angbracketright|2(5)\nwas calculated analytically with the method of Ref. [24] as a function o fT1,T2,τabs,ξ, andd\nand is obtained from Eq. (3) by setting σ= +1. It interpolates between the well-known re-\nsultsfororthogonalsymmetry[2](full Tinvariance)andforunitarysymmetry[27](complete\nviolation of Tinvariance). The mean level spacing dwas computed from the Weyl formula.\nThe Fourier transform of the function Cab(ǫ) was then ﬁtted to the data as in Ref. [18]. As\nstarting points we used the values of T1andT2obtained from the measured values of Saa(f)\nand ofξdetermined from Ccross(0). For each of the 6 realizations the spectra of Sab(f) were\ndivided into intervals ∆ fof 1 GHz length. In each interval the Fourier transform ˜Cab(tk)\nof the autocorrelation function (5) was calculated for values of tkbetween 5 ns and 200 ns.\nThe lower limit is determined by the length of the shortest periodic orb it in the classical\nbilliard; for smaller values of tkthe Fourier coeﬃcients are nongeneric [8]. At tk≈200 ns the\nvalues of ˜Cab(tk) have decayed over more than three orders of magnitude, and no ise limits\nthe analysis. The time resolution was 1 /∆f= 1 ns. We measured four excitation functions\n0.00.51.0\n0.0 0.5 1.0Ccross\nξ\nFIG. 4: Dependence of the cross-correlation coeﬃcient Ccross(0) on the parameter ξas predicted\nby the random-matrix model for partial violation of Tinvariance. Also shown is how Ccross(0) =\n0.49(3) translates into ξ= 0.29(2).\n7Sab(f) taking ( a,b) = (1,1),(1,2),(2,1),(2,2 yielding a total of 4800 Fourier coeﬃcients\nfor each interval. For f >10 GHz the ﬁtted values for T1andT2diﬀer by not more than\n7 % from the initial ones. (For smaller fthe intervals of 1 GHz width comprise only few\nresonances). The spread of the data is large, see the left panel o f Fig. 5. Going to the time\ndomain is useful since the Sab(f) are correlated for neighboring fwhereas the correlations\nare removed in the ratios of the experimental and the ﬁtted values for˜Cab(tk). The latter\nare stationary and ﬂuctuate about unity. Thus the statistical an alysis is much simpliﬁed.\n12131415\n 0  100  200log[C~12(t)]\nTime (ns)(a)\n0.020.030.04\n 0  5  10C12(ε)\nε (MHz)(b)\nFIG. 5: Autocorrelation function for S12in the range of 16–17 GHz and at B= 190 mT. In the\ntime domain (a) the data (dots) scatter around the theoretic al ﬁt (solid) for T1= 0.37,T2= 0.41,\nτabs= 2.9 andξ= 0.25. Transforming the results back into frequency domain (b) conﬁrms the\ngood agreement between data and theory. We observe that neig hboring data points in (b) are\ncorrelated, whereas those in (a) are not.\nFor each realization the parameters τabsandξwere obtained by ﬁtting the analytical\nexpression for ˜Cab(tk) to the experimental results. The values of ξdetermined from these\nﬁts agree with the ones found from the cross-correlation coeﬃcie nt. To reduce the spread\nwe combined the data from all realizations within a ﬁxed frequency int erval. The result\nwas analyzed with the help of a goodness-of-ﬁt (GOF) test (see Re f. [18]) that distinguishes\nbetween full, partial, and no violation of Tinvariance. We deﬁned a conﬁdence limit such\nthat the GOF test erroneously rejects a valid theoretical descrip tion of the data with a\nprobability of 10 %. With this conﬁdence limit the test rejects the ﬁtt ed expressions for\n˜Cab(tk) in only 1 out of the 24 available frequency windows or in 4.2 % of the tes ts. Thus,\nthe RMT model correctly describes the ﬂuctuations of the S-matrix for partial violation of\nTinvariance in the regimes of isolated and weakly overlapping resonanc es.\nElastic enhancement factor. As a second test of the theory we use the values of ξob-\n8tained from the cross-correlation coeﬃcients (see Fig. 3) and the parameters Ta,Tb,τabs\nresulting from the ﬁt of ˜Cab(tk) to predict the values of the elastic enhancement factor\nW=/parenleftbigg\n/angbracketleft|Sﬂ\n11|2/angbracketright /angbracketleft|Sﬂ\n22|2/angbracketright/parenrightbigg1/2\n//angbracketleft|Sﬂ\n12|2/angbracketrightwithSﬂ\nab=Sab−/angbracketleftSab/angbracketrightas a function of f. We use that\nW=/radicalbig\nC11(0)C22(0)/C12(0), see Eq. (5). For T-invariant systems, the elastic enhance-\nment factor decreases from W= 3 for isolated resonances with many weakly coupled open\nchannels to W= 2 for strongly overlapping resonances (Γ ≫d). The corresponding values\nfor complete violation of Tinvariance are W= 2 and W= 1, respectively [28]. Figure 6\ncompares the analytic results for the enhancement factor W(ﬁlled circles) to the data (open\ncircles). For small f(where Γ /d≪1 andξ≈0) the experimental results diﬀer from the\nprediction W= 3. Here only few resonances contribute and the errors of the ex perimental\nvalues for Ware large. Moreover Wis determined from only a single value Cab(0) of the\nmeasured autocorrelation function while the analytic result is based on a ﬁt of the complete\nautocorrelation function. As fincreases so does Γ /d, andWtakes values well below 3.\nAt frequencies where ξis largest Wdrops below the value 2 as predicted, a situation that\ncannot arise for T-invariant systems. The overall agreement between both data se ts above\n≈10 GHz corroborates the conﬁdence in the values of ξdeduced from the cross-correlation\ncoeﬃcients.\n 1 2 3\n 0  5  10  15  20  25Enhancement Factor\nFrequency (GHz)\nFIG. 6: Comparison of elastic enhancement factors. For the e valuation of Wthe autocorrela-\ntion coeﬃcients Cab(0) were determined either directly from the data (open circ les) or from the\nanalytic result for partial violation of Tinvariance (ﬁlled circles) with ξdetermined from the cross-\ncorrelation coeﬃcient. The error bars indicate the variati ons within the 6 realizations. The dashed\nhorizontal lines indicate the limits of WforT-invariant systems in the regime of isolated ( W= 3)\nor overlapping ( W= 2) resonances.\nSummary. We have investigated partial violation of Tinvariance with the help of a\n9magnetized ferrite placed inside a ﬂat microwave resonator (a chao tic billiard) with two\nantennas. We measured reﬂection and transmission amplitudes in th e regime of isolated\nand weakly overlapping resonances in the frequency range from 1 t o 25 GHz and determined\nthe cross-correlation function, the autocorrelation functions, and the elastic enhancement\nfactor from the data. The results were used as a test of random- matrix theory for scattering\nprocesses with partial Tviolation. That theory yields analytic expressions for all three\nobservables. The parameters of the theory ( T1,T2,τabsand the parameter ξforTviolation)\nwere partly obtained directly from the data but improved values res ulted from ﬁts to the\nautocorrelation function. We ﬁnd that 0 ≤ξ≤0.3. The validity of the theory was tested\nin two ways. (i) A goodness-of-ﬁt test of the Fourier coeﬃcients o f the scattering matrix in\nfrequency intervalsof1GHzwidthyielded excellent agreement. (ii)T heelasticenhancement\nfactor predicted from the ﬁtted values of the parameters shows overall agreement with the\ndataforfrequenciesabove10GHzwheretheexperimental error saresmall. Weconcludethat\nthe random-matrix description of S-matrix ﬂuctuations with partially broken Tinvariance\nis in excellent agreement with the data.\nF. S. is grateful for the ﬁnancial support from the Deutsche Tele kom Foundation. This\nwork was supported by the DFG within SFB 634.\n∗Electronic address: richter@ikp.tu-darmstadt.de\n[1] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev. Lett .52, 1 (1984).\n[2] J. J. M. Verbaarschot, H. A. Weidenm¨ uller, and M. R. Zirn bauer, Phys. Rep. 129, 367 (1985).\n[3] W. von Witsch, A. Richter, and P. von Brentano, Phys. Rev. Lett.19, 524 (1967); E. Blanke\net al., ibid.51, 355 (1983).\n[4] T. Ericson, Phys. Rev. Lett. 5, 430 (1960).\n[5] G. Bergman, Phys. Rep. 107, 1 (1984).\n[6] J. Rosny et al.Phys. Lett. 95, 074301 (2005).\n[7] R. Bl¨ umel and U. Smilansky, Phys. Rev. Lett. 60, 477 (1988).\n[8] R. Bl¨ umel and U. Smilansky, Phys. Rev. Lett. 64, 241 (1990).\n[9] P. So, S. M. Anlage, E. Ott, and R. N. Oerter, Phys. Rev. Let t.74, 2662 (1995).\n[10] U. Stoﬀregen et al., Phys. Rev. Lett. 74, 2666 (1995).\n10[11] O. Hul et al., Phys. Rev. E 69, 056205 (2004).\n[12] D. H. Wu, J. S. A. Bridgewater, A. Gokirmak, and S. M. Anla ge, Phys. Rev. Lett. 81, 2890\n(1998).\n[13] H. Primack and U. Smilansky, J. Phys. A 27, 4439 (1994).\n[14] H.-J. St¨ ockmann and J. Stein, Phys. Rev. Lett. 64, 2215 (1990).\n[15] B. Dietz et al., Phys. Rev. Lett. 98, 074103 (2007).\n[16] C. Mahaux and H. A. Weidenm¨ uller, Shell-Model Approach to Nuclear Reactions (North-\nHolland Publ. Co., Amsterdam, 1969).\n[17] C. H. Lewenkopf and A. M¨ uller, Phys. Rev. A 45, 2635; R. Sch¨ afer, T. Gorin, T. H. Seligman,\nand H.-J- St¨ ockmann, J. Phys. A 36, 3289 (2003).\n[18] B. Dietz et al., Phys. Rev. E 78, 055204(R) (2008).\n[19] P. W. Brouwer and C. W. J. Beenakker, Phys. Rev. B 55, 4695 (1997).\n[20] A. Pandey, Ann. Phys. (N.Y.) 134, 110 (1981).\n[21] A. Pandey and M. L. Mehta, Comm. Math. Phys. 87, 449 (1983).\n[22] A. Altland, S. Iida, and K. B. Efetov, J. Phys. A 26, 3545 (1993).\n[23] Time-reversal invariance is signiﬁcantly broken for πξ/√\nN≃d/v. Here,d=vπ/√\nNdenotes\nthe average level spacing and v2the variance of the oﬀ-diagonal matrix elements of Hs,Ha.\n[24] Z. Pluhaˇ r et al., Ann. Phys. 243, 1 (1995).\n[25] U. Gerland and H. A. Weidenm¨ uller, Europhys. Lett. 35, 701 (1996).\n[26] O. Bohigas, M. J. Giannoni, A. M. Ozorio de Almeida, and C . Schmit, Nonlinearity 8, 203\n(1995).\n[27] Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers, J. Phys. A : Math. Gen. 38, 10731 (2005).\n[28] D. V. Savin, Y. V. Fyodorov, and H.-J. Sommers, Acta Phys . Pol. A109, 53 (2006).\n11"    },    {        "title": "1203.6220v1.Geometric_and_disorder____type_magnetic_frustration_in_ferrimagnetic__114__Ferrites__Role_of_diamagnetic_Li__and_Zn2__cation_substitution.pdf",        "content": " 1 Geometric and disorder – type magnetic frustration in ferrimagnetic “114” \nFerrites: Role of diamagnetic Li+ and Zn2+ cation substitution.  \n \nTapati  Sarkar, V. Caignaert, V. Pralong and B. Raveau * \n \n \nLaboratoire CRISMAT, UMR 6508 CNRS ENSICAEN,  \n6 bd Maréchal J uin, 14050 CAEN, France  \n \nDedicated to Professor Jacques Friedel on the Occasion of His 90th Birthday.  \n \nAbstract  \n \nThe comparative study of the substitution of zinc and lithium for iron in the “114” ferrites, \nYBaFe 4O7 and CaBaFe 4O7, shows that these diamagne tic cations play a major role in tuning \nthe competition between ferrimagnetism and magnetic frustration in these oxides. The \nsubstitution of Li or Zn for Fe in the cubic phase YBaFe 4O7 leads to a structural transition to a \nhexagonal phase YBaFe 4-xMxO7, for M = Li (0.30 \n  x \n 0.75) and for M = Zn (0.40 \n  x \n \n1.50). It is seen that for low doping values i.e. x = 0.30 (for Li) and x = 0.40 (for Zn), these \ndiamagnetic cations induce a strong ferrimagnetic component in the samples, in contrast to \nthe spin glass behaviour of the cubic phase. In all the hexagonal phases, YBaFe 4-xMxO7 and \nCaBaFe 4-xMxO7 with M = Li and Zn, it is seen  that in the low doping regime (x ~ 0.3 to 0.5), \nthe competition between ferrimagnetism and 2 D magnetic frustration is dominated by the  \naverage valency of iron. In contrast, in the high doping regime (x ~ 1.5), the emergence of a \nspin glass is controlled  by the high degree of cationic disorder, irrespective of the iron \nvalency.  \n \n \n \nKeywords : “114” Ferrites, ferrimagnetism and magnetic frus tration.    \n \n \n \n \n* Corresponding author: Prof. B. Raveau  \ne-mail:  bernard.raveau @ensicaen.fr  \nFax: +33 2 31 95 16 00   \nTel:  +33 2 31 45 26 32   2 1. Introduction   \n \n Strongly  correlated electron systems, involving transition metal oxides with a “square” \ncrystal lattice, namely perovskites, have been the subject of numerous investigations during the \nlast thirty years, for their superconducting properties in cuprates as well as their magnetic \nproperties in CMR mangani tes. Oxides with a triangular lattice have also been studied , as \nshown for the spinel family , [1 – 3] that exhibits strong ferrimagnetism and unique magnetic \ntransitions, as for example in Fe 3O4, and for the pyrochlore family and for spinels with a \npyrochl ore sublattice , [4 – 6] which have been investigated for magnetic frustration. \nNevertheless, the number of oxides with a triangular lattice is much more limited, and the \nrecent synthesis of the “114” cobaltites and ferrites [7 – 11] with original structure s closely \nrelated to the spinel offers a new playground for the investigation of the competition between \nmagnetic ordering and geometric frustration in this class of materials [12 – 14]. The \nconsideration of the structure of the “114” ferrites, cubic LnBaF e4O7 (Fig. 1 a), and hexagonal \nCaBaFe 4O7 (Fig. 1 b), shows that these oxides, both consist of similar layers of FeO 4 \ntetrahedra, called triangular (T) and kagom é  (K), and that their “Fe 4O7” frameworks can be \ndeduced from each other by a translation of one  triangular layer out of two. Therefore, the \ncorresponding iron sublattice is very different in the two structural families: it consists of a \npure tetrahedral framework of corner sharing “Fe 4” tetrahedra  in the cubic LnBaFe 4O7 (Fig. 2 \na), whereas it is bui lt up of rows of corner – sharing “Fe 5” bipyramids running along \n , \ninterconnected through “Fe3” triangles in the (001) plane for hexagonal CaBaFe 4O7 (Fig. 2 b ). \nIt is this geometry of the iron framework which is at the origin of two different kinds of \ngeometric frustration. The pure tetrahedral iron sublattice of LnBaFe 4O7 oxides is similar to \nthat of pyrochlore [4] and consequently generates a 3 D magnetic frustration. In contrast, the \nmixed “ bipyramidal – triangular” sublattice of CaBaFe 4O7, allows a com petition between a 1 D \nmagnetic ordering along \n  and a 2 D magnetic frustration in the (001) plane.  \n Recently, we have shown the possibility of stabilizing the iron “bipyramidal – \ntriangular” lattice at the cost of the tetrahedral lattice by substitution of zinc for iron in the \ncubic LnBaFe 4O7 oxides [15]. Surprisingly, it was observed that this doping with a \ndiamagnetic cation destroys the spin glass behaviour of LnBaFe 4O7 and induces \nferrimagnetism. However, with progressive increase of the Zn concentrat ion, the ferrimagnetic \ninteraction starts to weaken, and we get a spin glass for very high doping concentration. In \norder to understand the role of the different factors which govern the magnetic properties of  3 these ferrites, we have studied the substituti on of lithium and zinc , two diamagnetic cations \nwith different valencies,  for iron, in the ferrites YBaFe 4O7 and CaBaFe 4O7. We discuss, herein, \nthe relative influence of valence effects and cationic disordering upon the competition between \nmagnetic orderin g and frustration in these systems.  We will specifically attempt to decouple \nthe role of the Fe valency, and that of the disorder on the Fe sites in order to determine how the \ntwo separately affect the magnetic ground state. This will allow us the possibil ity to tune and \ncustomize the magnetic properties of these oxides by understanding the role of the two \ngoverning factors – the average Fe valency, and the degree of disorder on the Fe sites.  \n \n2. Experimental  \n \n All the samples used in this study were prepa red by standard solid state reaction \ntechnique. The details of the synthesis procedure can be found in our earlier publications [ 15, \n16]. The samples were chemically monophasic, and the phase purity was checked from X -ray \ndiffraction patterns registered wi th a Panalytical X’Pert Pro diffractometer. The d. c. \nmagnetization measurements were performed using a superconducting quantum interference \ndevice (SQUID) magnetometer with variable temperature cryostat (Quantum Design, San \nDiego, USA). The a.c. susceptib ility, \n ac(T) was measured with a PPMS from Quantum Design \nwith the frequency ranging from 10 Hz to 10 kHz (H dc = 0 Oe and H ac = 10 Oe). All the \nmagnetic properties were registered on dense ceramic bars of dimensions ~ 4 \n  2 \n 2 mm3. \n \n3. Results and discus sion \n \n3.1. Zn substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In our previous work [15], we had synthesized the oxide series YBaFe 4-xZnxO7 with \nthe hexagonal symmetry , for x ranging between 0.4 – 1.5. For the sake of relevant \ncomparison, we have prepared CaBaFe 4-xZnxO7 samples with x = 0.5 and 1.5. We have \nspecifically chosen these two values of x so that we can investigate how the two factors, i.e. \nthe average Fe valency and the cationic disorder , affect the magnetic properties in two \nseparate regimes: the low do ping regime and the high doping regime.  The XRPD patterns of \nthese two samples clearly show that they are monophasic, keeping the hexagonal symmetry  of \nCaBaFe 4O7, and with cell parameters close to those of the virgin oxide i.e. a = 6.3527 (1)  Å \nand c = 10.3274 (2)  Å for x = 0.5, and a = 6.3668 (1)  Å and c = 10.29 75 (1)  Å for x = 1.5.   4  \n3.1.1.  Low doping regime: YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 \n \n YBaFe 3.5Zn0.5O7 and CaBaFe 3.5Zn0.5O7 fall in the low doping regime. The degree of \ndisorder (measured in terms of the % of substituent cation) is the same and relatively small in \nthese two samples. We show the d. c. M vs T for the two samples in Fig. 3 . Both samples \nshow a ferrimagnetic transition (seen as a sharp rise in the M vs T curves) , similar to that \nobserve d for CaBaFe 4O7, [9] in accordance to the fact that the samples have been stabilized in \nthe hexagonal symmetry . However, their ordering temperatures , and importantly, their \nmagnetic moments are much smaller than those observed for CaBaFe 4O7 (TC = 270 K and  \nMFC(5K)  = 2.6 µ B/f.u.). Moreover,  the magnetic moments and the ordering temperatures  for the \ntwo samples  are very different , though they exhibit the same degree of disorder.  The \nCaBaFe 3.5Zn0.5O7 sample exhibits a much higher magnetic moment ( MFC(5K)  = 1.7 µB/f.u.) and \ntransition temperature (T C ~ 203.0 K) compared to the YBaFe 3.5Zn0.5O7 sample for which \nMFC(5K)  = 0.54 µ B/f.u. and T C ~ 119.5 K . Bearing in mind that both these oxides exhibit the \nsame “bipyramidal – triangular” iron sublattice ( Fig. 2 b ), the se results show that the \ncompetition between the 1 D ferrimagnetism that appears along \n  and the 2 D frustration in \nthe (001) plane of the hexagonal structure is strongly affected by the average valence of iron. \nIndeed, for the same degree of disorder (12. 5 % Zn), T C decreases and the magnetic \nfrustration increases significantly as the average value of Fe decreases from 2.57 for \nCaBaFe 3.5Zn0.5O7 to 2.29 for YBaFe 3.5Zn0.5O7. \nThe effect of the higher value of the average Fe valency of CaBaFe 3.5Zn0.5O7 is also  \nseen in  the M vs H loops of the two samples  (shown in the inset of Fig. 3 ). Not only is the \ncoercivity and remanence magnetization higher for CaBaFe 3.5Zn0.5O7, the shape of the M-H \nloops are also very different. CaBaFe 3.5Zn0.5O7 has a square loop, reminis cent of hard \nferrimagnets, while YBaFe 3.5Zn0.5O7 has a much softer M -H loop  signifying a weakening of \nthe ferrimagnetic interaction in YBaFe 3.5Zn0.5O7 compared to that in CaBaFe 3.5Zn0.5O7. Thus, \nin the low doping regime, the average Fe valency  of the ferri te is clearly the governing factor \nthat controls  the magnetic state and the strength of the magnetic interaction.  The degree of \ndisorder here plays a relatively minor role.  \n \n \n \n  5 3.1.2.  High  doping regime: YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 \n \nAn increase of  the doping concentration of the diamagnetic substituent subsequently \nleads to the appearance of spin glass behaviour in both YBaFe 4-xZnxO7 as well as CaBaFe 4-\nxZnxO7. This can be seen in Fig. 4 , where we have shown the d. c. M vs T  curves  for \nYBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7. In contrast to the sharp rise in the M(T) curves below \nthe ordering temperatures seen in the low doped samples (x = 0.5), the M(T) curves of the \nhigher doped samples (x = 1.5) show a more gradual rise in the magnetization with the \ndecrease in temperature terminating in a cusp -like behaviour at low temperature. The \ntemperature at which the ZFC M(T) curves of the two samples show cusps are T cusp = 35.5 K \nand 40.5 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively.  A. C. susceptib ility \nmeasurements \n '(T) of the two samples measured using different frequencies in the range 10 \nHz – 10 kHz  (Fig. 5 ) show that both samples show similar frequency dependent peaks with  Tg \n= 45 K and 50 K for YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectivel y. The two samples \nalso have very similar narrow S – shaped loops with almost the same values of the coercivity \nand remanence magnetization  (see inset of Fig. 4). Moreover , the important point to note here \nis that the average Fe valency of these two compou nds is very different (Fe val = 2.40 and 2.80 \nfor YBaFe 2.5Zn1.5O7 and CaBaFe 2.5Zn1.5O7 respectively).  In spite of such a large difference in \nthe Fe valency, the two compounds behave strikingly similar to each other. We explain this \nresult as the dominating role of the cationic disorder in samples where the degree of disorder \nis large. Thus, for higher doped samples, the degree of disorder plays the deciding factor in \nstabilizing the magnetic ground state, and the Fe valency plays only a minor role.  \n Remarkab ly, the magnetic behaviour of these highly doped hexagonal phases is very \nsimilar to that of the virgin cubic sample YBaFe 4O7, which was shown to be a spin glass, with \na rather similar T g ≈ 50 K [10]. Thus, a high degree of cation disordering in the hexago nal \nphase has an effect similar to the pure geometric frustration of the tetrahedral iron sublattice \n(Fig. 2 a ) of the cubic phase i.e. it allows a complete magnetic frustration to be reached.  \n \n3.2. Li substitution in YBaFe 4O7 and CaBaFe 4O7 \n \n In the previo us sections, we have shown that the magnetic properties of hexagonal Zn \nsubstituted “114” ferrites  YBaFe 4O7 and CaBaFe 4O7 can be tuned on the basis of two doping \nregions – the low doping regime and the high doping regime. While the average Fe valency \nplays  a crucial factor in determining the magnetic state of the oxide in the low doping regime,  6 the degree of cationic disorder becomes the dominant factor controlling the magnetic state in \nthe regime of high doping.  In order to confirm that this effect is not specific to Zn, but is, in \nfact, a rather general phenomenon, we carry out similar studies on YBaFe 4O7 with a second \ndiamagnetic substituent, Li+, and we compare the magnetic behaviour of YBaFe 4-xLixO7 with \nthat previously observed for CaBaFe 4-xLixO7 [16].  \n Due to its univalent character, lithium has the advantage of inducing an average iron \nvalency which is different from that of the zinc compounds for the same substitution rate, \nthereby allowing the relative effects of  the Fe valence and the cationic dis order to be \ncompared further. The size of Li+, which is similar to that of Zn2+, and its ability to adopt the \ntetrahedral coordination , are favourable to such a substitution. In contrast to the case of zinc \nsubstitution, the maximum amount of lithium that was substituted was limited by the \nexperimental conditions of synthesis. Indeed, working in sealed tubes, and using the \nprecursors Y 2O3, BaFe 2O4, LiFeO 2, Fe 2O3 and Fe in order to avoid any reaction with the \nsupport and any Li 2O volatization, only the compo sitions YBaFe 4-xLixO7 with x \n  0.75 could  \nbe prepared , keeping the oxygen and lithium stoichiometry intact.  \n The first important point deals with the fact that lithium substitution, like zinc, \nstabilizes the hexagonal symmetry at the cost of the cubic phas e. Quite remarkably, a smaller \nlithium content, x = 0.30 only, is sufficient to stabilize the hexagonal form, instead of x = 0.40 \nfor Zn . In any case, the cell parameters of the Li substituted yttrium phase vary only slightly \nwith composition from a = 6.30 74 (1)  Å, c = 10.35 86 (2)  Å for x = 0.30 to a = 6.293 2 (1)  Å, c \n= 10.3 104 (1)  Å for x = 0.75.  \n The magnetic study of the compounds YBaFe 4-xLixO7 clearly show s that for the \nmaximum substitution rate i.e. x = 0.75, the complete spin glass behaviour cannot be  reached. \nThus, for the sake of comparison with other substituted phases, we discuss, in this section, the \nresults obtained in the low doping regime.  \n  \n3.2.1.  YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7 \n \n Li+ substitution in hexagonal CaBaFe 4O7 has been studied b y us before [16]. For the \npurpose of comparison with Li+ in YBaFe 4O7, we choose the samples with x = 0.3 from the \ntwo series. This is because, as we have stated before, x = 0.3 is the minimum amount of Li+ \nrequired to stabilize monophasic YBaFe 4-xLixO7 with the hexagonal symmetry.  \n The d. c. magnetization results  of YBaFe 3.7Li0.3O7 have been shown in Fig. 6 . In \naccordance with its hexagonal symmetry, YBaFe 3.7Li0.3O7 is ferrimagnetic. However,  7 considering the fact that the average Fe valency in YBaFe 3.7Li0.3O7 (Fe val = 2.35) is much less \nthan that in Ca BaFe 3.7Li0.3O7 (Fe val = 2.62), the ferrimagnetic interaction in YBaFe 3.7Li0.3O7 \nshould be weaker than that in Ca BaFe 3.7Li0.3O7. This is indeed the case as can be seen from  \nTable 1 , where we have compared the va lues of T C, M FC(T=5K)  and H C(T=5K)  for the two \nsamples.  The values for Ca BaFe 3.7Li0.3O7 in Table 1  have been quoted from reference 16.  \nCaBaFe 3.7Li0.3O7 has higher values of T C, M FC(T=5K)  and H C(T=5K)  than those of \nYBaFe 3.7Li0.3O7 , showing that the ferrim agnetic interaction is stronger and the magnetic \nfrustration is weaker in CaBaFe 3.7Li0.3O7.  \n \n3.2.2.  CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 \n \n In all the above cases, we have compared samples with different average Fe valenc ies, \nand shown that in the regime of low doping, the Fe valency controls the magnetic properties \nof the oxide, while in the regime of high doping, the degree of cationic diso rder is the \ndeciding factor.  We have seen that in the low doping regime, samples with the same degree of \ndisorder, b ut with different Fe valency behave differently vis – à – vis their magnetic \nproperties. In our final section, we investigate what happens in the reverse case i.e. for \nsamples with different degree of disorder (but well within the low doping regime), but  having  \nthe same average Fe valency. For this purpose , we compare the  two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7 which fall in the low doping regime, and have almost the same average \nFe valency (Fe val = 2.57 and 2.58 for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7 respectively).   \nThe d. c. magnetization results of CaBaFe 3.5Zn0.5O7 have been shown in Fig. 7. In accordance \nwith its hexagonal symmetry, CaBaFe 3.5Zn0.5O7 shows a sharp ferrimagnetic transition, and a \nlarge square hysteresis loop. However, what is more striking is the almost exact one – to – one \ncorrespondence of the magnetic parameters obtained for the two samples CaBaFe 3.8Li0.2O7 \nand CaBaFe 3.5Zn0.5O7. These are listed in Table 2 .  The values for CaBaFe 3.8Li0.2O7 have \nbeen quoted from reference 16.  This remarkable correspondence in the magnetic properties of \nthe two samples with the same average Fe valency proves unambiguously that in the low \ndoping regime the Fe valency is indeed the main factor controlling the magnetic state of the \noxide.   \n \n \n \n  8 4. Conclusion  \n \n This study shows the great impact of the substitution of diamagnetic cations such as \nzinc or lithium for iron upon the competition between ferrimagnetism and magnetic frustration \nin “114’ ferrites. The first effect is structural – it is seen t hat the substitution of these cations \nfor iron in the cubic phase, YBaFe 4O7, leads to a hexagonal symmetry, and consequently \ndestroys the 3 D geometric frustration at the benefit of a competition between a 2 D geometric \nfrustra tion and 1 D magnetic orderin g, thereby inducing ferrimagnetism. The second effect, \nobserved in the hexagonal phases such as CaBaFe 4O7, modifies the competition between the 2 \nD frustration and the 1 D magnetic ordering in two different ways, depending on the \nsubstituent concentration.  For low doping values, the modification of the average iron valency \nthat is induced by this substitution dominates the magnetism of these compounds leading to a \ndecrease of ferrimagnetism at the benefit of 2 D magnetic frustration, whereas in the high \ndoping regime, the disordering of the cations dominates, inducing a complete magnetic \nfrustration, irrespective of the iron valency. The nature of the ferrimagnetism of these ferrites, \ntill date, has not been completely elucidated, and in particular, it is st ill not known whether the \niron spins lie in plane or out of the triangular planes, so that a vast field is still open for the \ninvestigation and understanding of this new type of magnetic frustration.  \n \n5. Acknowledgements  \n \nWe acknowledge the CNRS and the Co nseil Regional of Basse Normandie for financial \nsupport in the frame of Emergence Program  and N°10P01391 . V. P. acknowledges support by \nthe ANR -09-JCJC -0017 -01 (Ref: JC09_442369).  \n \n \n \n \n \n \n \n \n \n \n  9 6. References  \n  \n      [1]    Walz , F.: J. Phys. Condens. Matter 14, R285  (2002)   \n      [2]    Verwey , E. J. W.,  Haayman, P. W.: Physica  8, 979  (1941 ) \n      [3]    Goodenough,  J. B. Magnetism and the Chemical Bond, Interscience Monographs on  \n               Chemistry , Vol. 1 5 Huntington, New York , 1976 )  \n      [4]    Greedan,  J. E.:  J. Alloys and Compounds  408 – 412, 444  (2006 ) \n      [5]    Muraoka,  Y., Tabata , H., Kawai,  T.: J. Appl. Phys. 88, 7223  (2000)  \n      [6]    Delgado,  G. E.,  Sagredo , V., Bolzoni,  F.: Cryst. Res. Technol . 43, 141  (2008)  \n      [7]    Valldor , M., Andersson,  M.: Solid State Sciences   4, 923  (2002 ) \n      [8]    Valldor,  M.: J. Phys.: Condens. Matter . 16, 9209  (2004)  \n      [9]    Raveau, B., Caignaert,  V., Pralong,  V., Pelloquin  D., Maignan,  A. : Chem. Mater. 20,  \n              6295  (2008)  \n      [10]   Caignaert,  V., Abakumov,  A. M., Pelloquin, D., Pralong,  V., Maignan,  A., Van   \n                Tendeloo , G., Raveau,  B.: Chem. Mater . 21, 1116  (2009)  \n      [11]   Pralong, V., Caignaert,  V.,  Maignan , A., Raveau,  B. : J. Mater. Chem . 19, 8335  \n                (2009)   \n      [12]   Chapon, L. C., Radaelli, P. G., Zheng , H., Mitchell, J. F.: Phys. Rev. B  74, 17240  \n               (2006)  \n      [13]   Manuel, P., Chapon,  L. C.,  Radaelli, P. G., Zheng , H., Mitchell, J. F.:  Phys. Rev.  \n                Lett. 103, 037202  (2009 ) \n      [14]   Caignaert, V., Pralong,  V., Hardy,  V., Ritter C., Raveau,  B.: Phys. Rev. B  81, 094417  \n                (2010 ) \n      [15]    Sarkar, T., Pralong,  V., Caignaert , V., Raveau,  B. : Chem. Mater . 22, 2885 (2010)  \n      [16]    Vijayanand hini, K., Simon, Ch., Pralong,  V., Caignaert V., Raveau,  B.: Phys. Rev. B  \n                79, 224407  (2009 ) \n \n \n \n \n \n \n \n  10 Figure Captions  \n \nFigure 1 : Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two \nsorts of layers of FeO 4 tetrahedra  called triangular (T) and kagomé (K).  \nFigure 2 : Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \nFigure 3 : MZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at \nH = 0.3 T. The ins et shows the magnetization as a function of magnetic field at T = 5 K for \nthe two samples.  \nFigure 4 : MZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at \nH = 0.3 T. The inset shows the magnetization as a function of magnetic field  at T = 5 K for \nthe two samples.  \nFigure 5 : Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \nFigure 6 : MZFC(T) and M FC(T) curves  for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets \n(a) show the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation of T C.  \nFigure 7 : MZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0. 3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C. \n \nTabl e Captions  \n \nTable 1 : TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \nTable 2 : T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \n \n \n \n \n  11 \n \n \nFig. 1. Structure of (a) cubic LnBaFe 4O7 and (b) hexagonal CaBaFe 4O7, built up of two sorts \nof layers of FeO 4 tetrahedra called triangular (T) and kagomé (K).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  12 \n  \nFig. 2 . Schematic representation of the iron sublattice in (a) cubic LnBaFe 4O7 and (b) \nhexagonal CaBaFe 4O7. \n \n \n \n \n  13 \n \n \nFig. 3. M ZFC(T) and M FC(T) curves for YBaFe 3.5Zn0.5O7 and Ca BaFe 3.5Zn0.5O7 measured at H \n= 0.3 T. The inset shows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  14 \n \n \nFig. 4. M ZFC(T) and M FC(T) curves for YBaFe 2.5Zn1.5O7 and Ca BaFe 2.5Zn1.5O7 measured at H \n= 0.3 T. The inset s hows the magnetization as a function of magnetic field at T = 5 K for the \ntwo samples.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  15 \n \n \nFig. 5. Real (in -phase) component of a.c. susceptibilities  for (a) YBaFe 2.5Zn1.5O7 and (b) \nCaBaFe 2.5Zn1.5O7 as a function of temperature measured using a frequency range 10 Hz – 10 \nkHz.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  16 \n \n \nFig. 6. M ZFC(T) and M FC(T) curves for YBaFe 3.7Li0.3O7 measured at H = 0.3 T. The insets (a) \nshow the magnetization as a function of magnetic field at T = 5 K and (b) dM/dT as a \nfunction of T for estimation  of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  17 \n \n \nFig. 7. M ZFC(T) and M FC(T) curves for CaBaFe 3.5Zn0.5O7 measured at H = 0.3 T. The insets \n(a) shows the plot of d. c. magnetic susceptibility as a function of temperature along with the \nCurie – Weiss fit, (b) the magnetization as a  function of magnetic field at T = 5 K and (c) \ndM/dT as a function of T for estimation of T C.  \n \n \n \n \n \n \n \n \n \n \n \n \n  18 Table 1. TC, M FC(T=5K)  and coercive field (H C) for YBaFe 3.7Li0.3O7 and CaBaFe 3.7Li0.3O7. \n \n \nSample  \n  \nTC (K) \n  \nMFC(T=5K)   (µB/f.u.)  \n HC (at T = 5 K) (T)  \nYBaFe 3.7Li0.3O7  115.7  0.34 0.87 \nCaBaFe 3.7Li0.3O7 150.7  0.50 0.98 \n \nTable 2. T C, Curie -Weiss temperature (θ CW), effective paramagnetic moment (µ eff) and \ncoercive field (H C) for CaBaFe 3.8Li0.2O7 and CaBaFe 3.5Zn0.5O7. \n \nSample  \n TC (K) \n θCW (K) \n µeff (µB/f.u.)  \n HC (at T = 5 K) \n(T) \nCaBaFe 3.8Li0.2O7 191.6  211.9  4.37 1.17 \nCaBaFe 3.5Zn0.5O7 203.0  218.8  4.05 1.12 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n "    },    {        "title": "1304.1278v2.Surface_parameters_of_ferritic_iron_rich_Fe_Cr_alloy.pdf",        "content": "arXiv:1304.1278v2  [cond-mat.mtrl-sci]  20 Jun 2013Surface parameters of ferritic iron-rich Fe-Cr alloy\nS Sch¨ onecker1, S K Kwon2, B Johansson1,3and L Vitos1,3,4\n1Applied Materials Physics, Department of Materials Science and Engin eering,\nRoyal Institute of Technology, Stockholm SE-10044, Sweden\n2Graduate Institute of Ferrous Technology, Pohang University of Science and\nTechnology, Pohang 790-784, Korea\n3Department of Physics and Astronomy, Division of Materials Theory , Uppsala\nUniversity, Box 516, SE-75120 Uppsala, Sweden\n4Research Institute for Solid State Physics and Optics, Wigner Rese arch Center for\nPhysics, Budapest H-1525, P.O. Box 49, Hungary\nE-mail:stesch@kth.se\nAbstract. Using ﬁrst-principles density functional theory in the implementatio n of\nthe exact muﬃn-tin orbitals method and the coherent potential ap proximation, we\nstudied the surface energy and the surface stress of the therm odynamically most stable\nsurface facet (100) of the homogeneous disordered body-cent red cubic iron-chromium\nsystem in the concentration interval up to 20 at.% Cr. For the low-in dex surface facets\nof Fe and Cr, the surface energy of Cr is slightly larger than the one of Fe, while the\nsurface stress of Cr is considerably smaller than the one of Fe. We ﬁ nd that Cr addition\nto Fe generally increases the surface energy of the Fe-Cr alloy, ho wever, an increase of\nthe bulk amount of Cr also increases the surface stress. As a resu lt of this unexpected\ntrend, the (100) surface of Fe-Cr becomes more stable against r econstruction with\nincreasing Cr concentration. We show that the observed trends a re of magnetic origin.\nIn addition to the homogeneous alloy case, we also investigated the im pact of surface\nsegregation on both surface parameters.\nPACS numbers: 75.70.-i,75.50.Bb,68.35.bd,73.20.QtSurface parameters of ferritic iron-rich Fe-Cr alloy 2\nSubmitted to: J. Phys.: Condens. Matter\n1. Introduction\nIn recent years, iron rich Fe-Cr alloys, as the basis of ferritic (bod y-centred cubic (bcc)-\nbased) stainless steel, have attracted much scientiﬁc attention f or their potential use\nin the next generation ﬁssion and prospective fusion reactors [1]. E mployed as ﬁrst\nwall and blanket material, or fuel cladding, this steel must withstan d neutron-induced\nradiation damage [2], e.g., swelling and void formation. Moderate Cr add ition to bcc\nFe in the order of 10% most beneﬁcially improves its swelling and irradiat ion creep\nbehaviour [3,4]. Chromium substitution is further known to inﬂuence other mechanical\nproperties such as the ductile-brittle transition temperature and radiation-induced\nhardening [5]. The experimental corrosion resistance of ferritic st eel in an oxidising\nenvironment improves drastically if the bulk alloy concentration exce eds 9-13 weight\npercent Cr and this self-healing protection attributed to the form ation of a chromium\noxide scale [6,7]. There is, however, a limit to the amount of Cr that ma y be added\nto steels as the beneﬁcial low corrosion rate is shadowed by the enh anced precipitation\nof intermetallic phases which often degrade the mechanical proper ties of ferritic steel.\nMoreover, it has also been recognised that alloying with additional ele ments further\nincreases the pitting and crevice corrosion resistance in certain ag gressive environments,\ne.g., molybdenum in chloride environments [8].\nThe surface of the bcc Fe-Cr system has often been used as the p rototype reference\nsystem to study the behaviour of surfaces of ferritic stainless st eels. There is a\ngreat deal of phenomenological modelling on the passivity of stainles s steels [9–11],\nwhile ﬁrst principles investigations of surfaces focused on the atom ic level behaviour\ninvolving surface segregation and the surface magnetic structur e for the technologically\nrelevant Fe-rich Fe-Cr alloys [12–20]. Understanding the aforeme ntioned threshold\nbehaviour of the corrosion resistance and the particular role of Cr in the segregation\nprocess has been a primary target for modelling. Experimental evid ence indicates Cr\nenrichment at the surface at high temperatures and bulk Cr conce ntrations larger than\n13at.% [21–24]. Recent ab initio calculations indeed predicted a sharp t ransition from\nCr-free to Cr enriched surfaces at around 9at.% Cr in the bulk alloy [ 12–14], attributed\nto complex magnetic interactions between ferromagnetically intera cting Fe and anti-\nferromagnetically interacting Cr species, which are likely to govern m any essential\ncharacteristics of the Fe-Cr phase diagram below the Curie temper ature [25].\nNot much is known about the ferritic Fe-Cr system concerning two e ssential\nmacroscopic parameters that describe the thermodynamic prope rties of its crystalline\nsurface: surface energy and surface stress. That is surprising since the signiﬁcance of\nstress and strain eﬀects on surface physics has been widely discus sed [26–29]. The\nequilibrium shape of mesoscopic crystals is the one that minimises its su rface free\nenergy [30]. The surface energy is further of eminent relevance in c onnection to faceting,\nroughening, crystal growth phenomena at surfaces and has bee n discussed in relationSurface parameters of ferritic iron-rich Fe-Cr alloy 3\nto segregation. There has been an increasing experimental and th eoretical activity\nto understand the importance of stress on many physical proper ties associated with\nsurface relaxation and reconstruction [31,32], segregation [33], s urface adsorption [34],\nand its role in bottom-up self-organisation and surface melting [26]. O n the other\nhand, a theoretical study of surface parameters of crystalline s urfaces of the individual\nelements, Fe and Cr, was subject to a number of publications [35–39 ]. According to\nthe general expectation, surface stress of clean surfaces is te nsile due to the increased\nelectron density within the surface layer. However, it was demonst rated recently [36,37]\nthat magnetism can overwrite this picture leading to exceptional, co mpressive surface\nstresses as predicted for the thermodynamically stable surfaces of bcc Cr and cubic Mn,\nthough not in the case of bcc Fe. It is hence worthwhile to investigat e the surface of\nthe bcc Fe-Cr system to gain information on its essential surface p arameters keeping\nin mind the presence of complex magnetic interactions in this alloy syst em that may\nalter our expectation on their behaviours. In this context, it is impo rtant to mention\nthe particular role of ab initio calculations in the determination of surf ace parameters\nsince experimental methods to determine their absolute value ofte n lack reliability and\naccuracy [27,28,40–45].\nThis work deals with an ab initio determination of surface parameters for the\nthermodynamically stable surface of bcc Fe-Cr ((100) facet) in th e concentration range\nof 0-20at.% Cr. The paper is organised a follows: In Section 2 we brieﬂ y overview\nthe theory of surface energy and surface stress. Numerical de tails of our computation\nare presented in Section 3. We discuss our results in Section 4 for tw o diﬀerent surface\nchemistries: aperfectlytruncatedbulksystemwithoutspacialco ncentrationdependence\nand a system involving surface segregation. The reason for the lat ter is to account for\nthe observed transition from Cr-free to Cr enriched surfaces ar ound the aforementioned\nthreshold bulk Cr concentration.\n2. Surface parameters\nSurface energy and surface stress are two fundamental quant ities to characterise the\nmacroscopic properties of surfaces. Qualitatively, the scalar sur face free energy, γ, was\nintroduced by Gibbs as the reversible work per unit area to create a surface [27,28]. The\ntensorial surface stress, τij,i,j={x,y}, is the reversible work per unit area to stretch\na surface elastically in the surface plane which is here assumed to lie in t hex-y-plane.\nIfγ <0 for a particular surface of a solid, then this surface is unstable an d the crystal\nfragments spontaneously. Hence γis positive for stable bulk systems. The components\nofτmaybebothpositive (tensilesurfacestress) ornegative(compre ssive surfacestress).\nTensile surface stress on a surface favours smaller in-plane lattice constants than the\nbulk value while a surface with compressive surface stress favours a larger one.\nIn ab initio total energy calculations, the surface energy is usually c omputedSurface parameters of ferritic iron-rich Fe-Cr alloy 4\nas [26–28],\nγ=Esurf−Ebulk\nA, (1)\nwhereEsurfandEbulkspecify the energy of two semi-inﬁnite bulk systems and the\ninﬁnite bulk system, respectively, normalised to the unit area A. Surface energies are\nconveniently extracted from slab calculations and diﬀerent proced ures were proposed\nto yield convergent numbers with the slab size [46–48]. In the presen t work, we model\nthe Fe-Cr system by considering two distinct subsystems: one tha t includes the surface\n(surface subsystem) and one without (bulk subsystem). Due to t he periodic boundary\nconditions parallel to the surface, the size of the slab has to be con verged with respect to\nthe thickness of the slab only. Here, we follow essentially [46] and der ive both Esurfand\nEbulkfrom slabs with the same thickness characterised by the total num ber of layers,\nn, taken as a multiple of the bulk equilibrium lattice parameter oriented n ormal to the\nsurface plane. In case of the surface subsystem, the slab consis ts of an atomic part with\nthickness nmand vacuum which is needed to decouple the two surfaces of the slab from\nanother (across the vacuum). This surface-surface distance is denoted by nv. Since the\nslab representing the surface subsystem contains two equal sur faces, a factor of one half\nis added to (1) to yield the surface energy of one surface, viz.\nγ=En\nsurf−nm\nnEn\nbulk\n2A, (2)\nwheren=nm+nv, and both En\nsurfandEn\nbulkrefer to the total energy of the n-layer\nslab.En\nbulkis scaled by a factor of nm/nto the correct number of atoms. Only geometry\nrelaxation in the direction perpendicular to the surface is allowed and may be included\ninEn\nsurfas appropriate.\nThesurfacestress tensorcanbedeﬁnedasthestrainderivative ofthesurfaceenergy\nin the Lagrangian coordinate system (surface area Ais standard state of strain) [26,49],\nτij=∂γ\n∂ǫij/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nǫij=0, (3)\nwhereǫijdenotes the components of the strain tensor specifying an elastic in-plane\ndeformation of the surface. The τij’s are evaluated at the unstrained state ( ǫij= 0).\nFor a high-symmetry surface facet such as the bcc (100) facet, τxx=τyy,τxy=τyx= 0,\nfurther assuming an isotropic distortion, ǫxx=ǫyy=ǫ(zero otherwise), and using the\nsurface energy from (2), we arrive at\nτ≡τii=1\n4A∂(En\nsurf(ǫ)−nm\nnEn\nbulk(ǫ))\n∂ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nǫ=0. (4)\nA factor of 1 /2 appears due to the applied isotropic strain. The previous equation is\nconveniently [36,37,50–52] used to compute the surface stress from the elastic energies\nof a surface subsystem and a bulk reference system employing slab s.\nNumerically, the elastic energies in (4) are ﬁtted to second order po lynomial\nfunctions with ﬁt coeﬃcients canddas a function of strain,\nEn\nsurf(ǫ)−En\nsurf =csurfǫ+dsurfǫ2(5a)Surface parameters of ferritic iron-rich Fe-Cr alloy 5\nnm\nn(En\nbulk(ǫ)−En\nbulk) =cbulkǫ+dbulkǫ2. (5b)\nThe reference energies, En\nsurf/bulk, are the energies of the unstrained surface and\nbulk states. Assuming the previous ﬁt functions, we may express τby\nτ=csurf−cbulk\n4A. (6)\nAlthough in theory cbulk≡0 for bulk in equilibrium, it may be ﬁnite in calculations due\nto numerical errors [37].\nWe conclude this section by mentioning the excess surface stress [2 7,28,53], |τ−γ|.\nA larger value of the excess surface stress, |τ−γ|, indicates a higher tendency of a\nsurface towards reconstruction [53,54].\n3. Electronic structure calculations\nTotal energy calculations within the framework of density function al theory (DFT)\nwere done by means of the exact muﬃn tin orbitals (EMTO) method [55 –57], which\nis a screened Korringa-Kohn-Rostoker type of method [58–60] a nd solves the Kohn-\nSham equations in a Green’s function formalism. This enables us to com pute the\nelectronic structure of substitutionally disordered alloys using the coherent-potential\napproximation (CPA) [61–63]. The one-electron potential is repres ented by optimised\noverlapping muﬃn-tin potential spheres describing more accurate ly the exact crystal-\npotential compared to conventional muﬃn-tins or non-overlappin g spherical symmetry\npotentials [55, 64]. Combined with the full-charge density technique [ 65–68] for\ntotal energy calculations, [55, 56] the EMTO method has proven to yield reliable\ntotal energies and electronic structure in practise including the Fe -Cr system under\nconsideration[13,14,69–75]. TheCPA,beingasingle-citeapproxima tiontotheimpurity\nproblem, is a standard technique for electronic structure calculat ions in totally random\nalloys, suited for the case of alloy components having similar sizes. Du e to its single-\nsite nature, screening corrections to the potential and the tota l energy must be taken\ninto account which was done within the screened impurity model of Ko rzhavyiet al\n[76–78]. The CPA cannot directly treat atomic short-range order ( ASRO), which was\nobserved for the nearest neighbour coordination shell for the Fe -Cr system by means of\nneutron-diﬀusive scattering [79,80]. Accordingly, there is a tende ncy to form an ordered\ncompound for Cr concentrations smaller than 11at.% while larger Cr c ontents incline\nshort-range clustering. Recent studies by M¨ ossbauer spectro scopy reported that the\nASRO inversion occurs at 6at.% Cr [81,82]. Inclusion of ASRO is, howev er, beyond\nthe scope of this work. We note in this context that the surface en ergy and the surface\nstress are deﬁned as excess quantities, i.e., contributions from th e bulk part of the\nsurface subsystem and the bulk subsystem are expected to canc el each other out (e.g.,\nASRO) and only surface eﬀects survive. To the best of our knowled ge, there is neither\nexperimental nor theoretical evidence on ASRO at the surface of Fe-Cr.\nTotal energies in EMTO were obtained using the Perdew-Burke-Ern zerhof (PBE)\nparametrisation [83, 84] of the exchange-correlation energy fun ctional, while self-Surface parameters of ferritic iron-rich Fe-Cr alloy 6\nconsistent charge densities were computed in the local-density app roximation in the\nparametrisation of [85]. Switching oﬀ gradient corrections to the ex change-correlation\npotential is justiﬁed since functionals in the generalised-gradient a pproximation are\nknown to overestimate the magnetic moment of iron. This perturba tive approach gives\naccurate total energies which was also tested in the case of Fe-Cr [75,86]. The EMTO\npartial waves were expanded into s,p,d, andforbitals, and the core states were\nrecalculated at each iteration step. Integration over the Brillouin z one was done on a\nk-point grid of 15 ×15×1points in case of the surface subsystem. A single k-point along\nthe short reciprocal lattice vector (corresponding to the direct lattice vector parallel to\nz) is suﬃcient since bands are dispersionless in this direction (ensured by the converged\nthickness of vacuum). For the bulk reference system, the numbe r ofk-points in this\ndirection was converged to the value of two. Increasing the k-point grid of the surface\nsubsystem (bulksubsystem) to20 ×20×1 (20×20×3) showed thatthe totalenergies are\nconverged at a level of 1meV/atom. The Green’s function was evalu ated for 16 complex\npoints distributed exponentially on a semicircle including states below t he Fermi level.\nFor the particular case of Fe we also carried out DFT calculations with the\nfull-potential local-orbital scheme, FPLO-9.01-35 [87] and PBE. T he convergence\nof numerical parameters and the basis was carefully checked. Line ar-tetrahedron\nintegrations with Bl¨ ochl corrections on a 12 ×12×1 (12×12×3) mesh in the full\nBrillouin zone for the surface subsystem (bulk subsystem) were su ﬃcient to converge\nthe total energy at a level of 1meV/atom compared to a 18 ×18×1 (18×18×5) mesh.\nThe valence basis of Fe comprised 3 spd, 4spd, and 5sstates.\nFerritic Fe-Cr crystallises in the bcc crystal structure, and the ( 100) surface was\nreportedtobethemoststablefacet[13]. IncaseofEMTO,wemode lledthissurfacebya\nslab with a converged thickness of 13 atomic layers (approximately 1 8˚A) and separated\nby vacuum with a thickness of nv= 7 (approximately 10 ˚A). A somewhat larger surface\nsubsystem with nm= 19 and nv= 7 was required to yield converged surface parameters\nin case of FPLO. The symmetry of the slabs includes a mirror plane par allel to the\n(identical) surfaces.\nSurface geometries may diﬀer from ideally truncated bulk crystals s ince relaxation\nand reconstruction may occur. Surface reconstruction on the c lose-packed surfaces of\nthe transition metals is rather uncommon [88], however relaxation o f the surface layer\nand of subsurface layers is frequently observed. Results of [89] a nd references therein\nindicate that layer relaxations of the most stable (110) and the sec ond most stable\n(100) surface facets of Fe are minor and change the correspond ing surface energies in\nthe order of 1% (cf.literature values in table 2). Punkkinen et alrecently compared\nthe surface energy and the surface stress of the most stable su rfaces facets of Fe and Cr\nfor non-relaxed geometry with values of fully relaxed surface geom etries [36]. Due to\nthe enhanced surface magnetism in both systems, relaxations hav e only a minor eﬀect\nand they were found not to markedly alter the surface energy and the surface stress. We\nexpect for the same reasons that the surface geometry of the b cc (100) surface of Fe-Cr\nremains close to the truncated bulk one. To further support this p oint, we computed theSurface parameters of ferritic iron-rich Fe-Cr alloy 7\nsurface-layer relaxation of the bcc (100) surface of Fe and of Fe -Cr (up to 20 at.% Cr)\nwith EMTO. We found a relative change in length of the interlayer dista nce of−2.2%\nfor Fe (inward relaxation) and an accompanied reduction of the sur face energy by 1.5%.\nThe available experimental data for Fe as obtained from low-energy electron diﬀraction\non the top-layer relaxation of the bcc (100) surface of Fe is ambigu ous with values of\n+0.5% and−1.4±3%, both from [90], and −5±2% from [91]. For Fe-Cr, the interlayer\ndistances reduce between 2.6% to 2.7% and the surface energies de crease the most for\nFe80Cr20by 1.4% and the least for Fe 95Cr05by 1.2%. Based on these arguments, we\nconclude that relaxation is a minor eﬀect in Fe and Fe-Cr and we held ﬁx ed all atomic\npositions to the ideal bcc lattice sites for the results presented in s ection 4.\nThe total energies as a function of strain, cf.(5 a) and (5b), were computed in a\nstrain interval of |ǫ|= 0.02.\nTable 1. Inﬂuence of the choice of the size of the surface subsystem, cha racterised\nbynm, and the strain interval for elastic energy ﬁts, |ǫ|, for the surface parameters of\nthe (100) surface facet of Fe. The surface parameters of the r eference system, given\nbynm= 13 and |ǫ|= 0.02, are highlighted in boldface. The absolute diﬀerence ∆ as\nwell as the relative change in % in parenthesis are speciﬁed, and stat ed at the end of\neach row and at the end of each column for a decrease of |ǫ|from 0.02 to 0.01 and\nan increase of nmfrom 13 to 15, respectively. γ,τ, and the absolute diﬀerence are in\nunits of J ·m−2. In all cases nv= 7.\nnmsurface energy γ surface stress τ ∆\n|ǫ|= 0.02|ǫ|= 0.01\n13 2.615 0.57 0.50 -0.07 (-12)\n15 2.626 0.51 0.56 0.05 (10)\n∆ 0.011 (0.4) -0.06 (-11) 0.06 (12) -\nWe conclude this section by establishing the precision of the calculate d surface\nparameters in EMTO with respect to the selected size of the surfac e subsystem\n(nm= 13) and the strain interval ( |ǫ|= 0.02) using the example of Fe. The previous\nset of numerical parameters deﬁne the reference system, which we consider to yield\nconverged surface related quantities in this work. The surface en ergy and the surface\nstressofFewerecomputedforalargersurfacesubsystem( nm= 15)ontheonehand, and\nthe surface stress was ﬁtted to total energies from a narrower strain interval ( |ǫ|= 0.01)\non the other hand. Table 1 lists both the absolute values of the surf ace parameters\nand the change (∆) with respect to the reference set of surface parameters. Apart from\npure Fe, we assessed the precision for several alloy concentratio ns of Fe-Cr in the same\nway, and noted that the tabulated ∆’s in table 1 represent charact eristic values for all\nconcentrations tested. ∆ may be used to deﬁne the precision of ou r calculation with\nrespect to the choices of nmand of|ǫ|, which is hence of the order of 0.01-0.02J ·m−2\nfor surface energies, and approximately 0.04-0.08J ·m−2for surface stresses. The energy\nscale for surface stress calculations is roughly one order of magnit ude smaller than the\none for surface energy calculations which explains the diﬀerence in t he correspondingSurface parameters of ferritic iron-rich Fe-Cr alloy 8\n∆’s.\nDue to the neglect of ASRO in this work, we model a chemically homogen eous bulk\nalloy, i.e., there is no spatial probability (composition) dependence of the distribution\nof the alloys components. First we consider an ideally truncated bulk system, i.e., a\nsystem with surface, for which no atomic redistribution (segregat ion) occurs. Hence the\nsurface composition is identical to the one in the bulk for all composit ions. Second, we\nallow for a change of the surface chemistry.\n4. Results and discussion\n4.1. Surface parameters of Fe\nFor the theoretical equilibrium lattice parameter of ferromagnetic (FM) bcc iron, we\nobtained 2.837 ˚A and for the spin moment a value of 2 .21µB. This is to be compared\nto experimental values, 2.867 ˚A [92] and 2 .21µBfor the total magnetic moment [93].\nTable 2 lists the surface energy and the surface stress of the bcc (100) surface of\nFM iron as calculated in this work and compared to available data from t he literature.\nThe multitude of comparable ab initio data from full-potential (FP) an d projector-\naugmented-wave (PAW)methodsallowstodrawconclusions ontypic al scatter ofsurface\nenergy and surface stress calculations, as well as a critical evalua tion of the present\nresults. Concerning the surface energy of Fe, the only outlier see ms to be the value\nobtained with FPLO, since the remaining surface energies scatter in the range of\napproximately 2.3-2.6J ·m−2. The particular choice of the gradient corrected density\nfunctional may have an eﬀect on the surface energy as all PBE valu es are larger than\nthe values obtained with the parametrisation of Perdew et al[94,95]. Our EMTO\nvalue ofγ= 2.62J·m−2is in close agreement to the PAW and the FP linear augmented\nplane wave (FP-LAPW) results of Punkkinen et al. The too high surface energy\nfrom FPLO may be related to a too strongly contracted wave funct ion at the bulk-\nvacuum interface, cf., e.g., the analysis in [96]. Concerning the surfa ce stress of bcc\nFe, the FP and PAW values scatters in the range of approximately 1.1 -1.4J·m−2.\nThe EMTO value, τ= 0.57J·m−2, is comparatively small and may thus indicate a\nsystematic underestimation of the surface stress in Fe and the Fe -based system. This\nunderestimation may be ascribed to the muﬃn-tin approximation to t he one-electron\npotential.\nThe experimental value of the surface energy of Fe from [97], γ= 2.41J·m−2,\nis estimated from surface stress measurements of the liquid-vacu um interface at the\nmelting temperature and surface stress measurements of the liqu id-solid interface, and\nextrapolated to T= 0K.\nBecause of the reduced coordination number, the magnetic momen t of Fe at the\nsurface is enhanced compared to the bulk. We obtained a surface m agnetic moment of\n2.97µBinvery closeagreement with arecently reportedvalue, 2 .96µB[12]. Calculations\nof Punkkinen et al[36,37] suggested analmost linear relationship between themagnet icSurface parameters of ferritic iron-rich Fe-Cr alloy 9\nTable 2. Surface parameters of the bcc (100) surface facet of FM Fe. All methods\nemployed gradient corrected density functionals.\nmethod surface energy γsurface stress τreferences\n(J·m−2) (J ·m−2)\nEMTO, PBE 2.62 0.57 this work\nFPLO, PBE 3.09, 3.07a1.15 this work\nPAW, PBE 2.55b, 2.50a1.39a[37]\nFP-LAPW, PBE 2.6b- [37]\nPAW, GGA [94,95] 2.48, 2.47a- [89]\nPAW, GGA [94,95] 2.32, 2.29a- [98]\nexperiment 2.41c- [97]\nasurface layer relaxation included\nbestimated from ﬁgure\ncestimated at T= 0 K\nmoment enhancement, ∆ m2, and the magnetic surface stress, τmag, on the basis of their\ncomputed surface stresses for the most stable surfaces of mag netically ordered Cr, Mn,\nFe, Co, and Ni, i.e.,\nτmag∝∆m2=m2\nsurf−m2\nbulk, (7)\nwheremsurfandmbulkare the magnetic moments at the surface and in the bulk,\nrespectively. The magnetic contribution to τ,τmag, is deﬁned as the diﬀerence\nbetween the nonmagnetic and the magnetic values of τ, that is evaluated without spin-\npolarisation ( τnsp) and with spin-polarisation ( τsp) for identical surface geometry, viz.\nτmag=τnsp−τsp. The geometry of the spin-polarised system is the reference stat e if not\nstated otherwise. The above proportionality was veriﬁed for eleme nts with FM order\n(Fe, Co, and Ni) and anti-ferromagnetic (AFM) order (Cr and Mn). The present EMTO\nvalues for FM Fe are ∆ m2= 3.91µ2\nBandτmag= 3.14J·m−2, respectively, which ﬁt very\nwell to the correlation established by Punkkinen et al(see ﬁgure 3 from [36]).\n4.2. Chemically homogeneous Fe-Cr alloy\nFirst we consider the case of chemically homogeneous surface alloys , i.e., it is assumed\nthat the chemical composition at the surface is identical to the bulk composition.\n4.2.1. Lattice constants and surface parameters The theoretical equilibrium lattice\nparameters of ferritic Fe-rich Fe-Cr alloys (0-20at.%Cr) were pre viously calculated with\nEMTO-CPA and discussed in detail in [71–75]. Since our computed lattic e parameters\npractically reproduce these earlier results, we refer the reader t o those references. It is,\nhowever, important to point out the non-linear behaviour of the lat tice parameter of\nthe Fe-Cr system.\nIn the atomic concentration range of 0-20at.% Cr, we identify a clea r trend of the\nconcentration dependence of all surface parameters in Fe-Cr allo ys, see ﬁgure 1. TheSurface parameters of ferritic iron-rich Fe-Cr alloy 10\nsurfaceenergyincreasesmonotonicallyby0 .33J·m−2foranincreaseoftheconcentration\nof Cr from 0 to 20at .%. This trend is not entirely unexpected since the surface energy\nof the non-relaxed bcc (100) surface facet of Cr was found to be larger by 0.5-0.8J ·m−2\nin theory than the one of Fe. [37] That is, the surface energy of th e disordered alloy\nwith low Cr content (0-20at.% Cr) follows a monotonic trend (rule of m ixing) given by\nthe boundary values of pure Fe and pure Cr (the exact PAW values f rom [37] for the\nrelaxed surface are 2 .50J·m−2for Fe and 3 .06J·m−2for Cr).\nIn previous theoretical considerations for the most stable surfa ces [36,37], the\nsurface stress of Cr was reported to be 1 .9J·m−2smaller than the corresponding value\nof Fe for relaxed surface geometries and likewise was the diﬀerence for non-relaxed\ngeometries. As further stated, this diﬀerence amounts to 1 .7J·m−2if speciﬁcally the\nrelaxed bcc (100) surface facet is considered (the exact PAW valu es from [37] for the\nrelaxed surface are1 .39J·m−2for Feand −0.32J·m−2for Cr). Since relaxationseems to\nhave a similar eﬀect on τfor both elements for the most stable surface, it is reasonable\nto assume that the eﬀect of relaxation on τfor the (100) surfaces of Fe and of Cr are\nalso similar. Hence, the surface stress of the non-relaxed bcc (10 0) surface of Cr is\npresumably still considerable smaller than the one of Fe (roughly by 1 -2J·m−2). On\nthis ground, we expect an overall decrease of the surface stres s of Fe-Cr with increasing\nCr content. Our ﬁndings in the dilute Cr concentration range are ho wever contrary\nto this expectation (ﬁgure 1). In the range up to 10at .% Cr in the iron matrix, the\nsurface stress strongly increases to a maximum value of 1 .78J·m−2being approximately\n1.25J·m−2larger than the corresponding value of Fe. τlevels oﬀ for concentrations\nhigher than 10at .% Cr.\nThe third surface-characteristic quantity depicted in ﬁgure 1 is th e excess surface\nstress,|τ−γ|. It evidently decreases strongly in the concentration range up to 10at.%\nCr relative to the value of pure Fe, which is mainly due to the accompan ied increase\nofτ. For Cr concentrations above 10at .%,|τ−γ|remains almost unchanged. The\nsurfacereconstructionispredictedtooccurwhentheexcess su rfacestressbecomeslarger\nthan the characteristic surface strain energy associated with th e reconstruction [53,54].\nThe latter may be expressed in terms of the shear modulus and the B urgers vector.\nNow, taking into account that the elastic moduli of Fe-Cr alloys show a rather weak\ncomposition dependence for the present concentration interval [74], one may conclude,\nthat the (100) surface of Fe-rich Fe-Cr alloys is considerable more stable against\nreconstruction than the surface of pure Fe.\n4.2.2. Magnetism and magnetic surface stress The magnetic structure of ferritic Fe-\nCr is governed by interactions between Fe atoms, that prefer to a lign their magnetic\nmoments in parallel, andCr atoms, that favour ananti-parallel alignm ent. Itsenergetics\nis rather well described within collinear magnetism of ﬁxed Ising spins [2 5,73,75]. In the\niron-richferriticFe-Cralloys, themomentatCrsitesarecoupledan ti-paralleltotheones\nof Fe necessarily implying that they are aligned in unfavourable paralle l orientation with\nrespect to other Cr atoms. In the dilute limit, however, the averag e Cr-Cr distance isSurface parameters of ferritic iron-rich Fe-Cr alloy 11\n0 5 10 15 20\n2.62.83.03.23.4surface energy γ [J/m2]\nspin-polarized\nnon-spin-polarized\n01234surface stress τ [J/m2]\n0 5 10 15 20\nat.% Cr x0.00.51.01.52.0excess surface stress | τ−γ| [J/m2]τmag\nFigure 1. Surface parameters (in units of J ·m−2) of chemically homogeneous Fe-rich\nFe-Cr alloys (0-20 at.% Cr): (top panel) surface energy γ, (middle panel) surface stress\nτ, (bottom panel) excess surface stress |τ−γ|. Lines are a guide to the eye.Surface parameters of ferritic iron-rich Fe-Cr alloy 12\nlarge and their mutual interaction energy small [14,15]. The EMTO- CPA spin moment\nat Cr is−1.62µBon the impurity level. This calculation was done with 0 .05at.% Cr. A\nnegative sign of the magnetic moment indicates an antiparallel alignme nt with respect\nto the moment of Fe which was deﬁned to possess a positive sign. Klav eret alobtained\na slightly larger spin moment of Cr inthe dilute limit, −1.8µB, calculated with thePAW\nmethod and PBE for a super cell with an eﬀective Cr concentration o f 0.19at.% [15].\nThus, a Cr atom in a surrounding Fe matrix at very low Cr concentrat ions is much\nstronger polarised than in AFM ordered pure Cr (the experimental spin moment of the\nlong wave spin-density ground state of Cr is approximately 0 .59µB).\nWithin the random solid solution description provided by EMTO-CPA, inc reasing\nthe Cr concentration beyond dilute levels results in a gradual loss of the modulus of the\nmagnetic moment at Cr sites while the Fe spin moment hardly changes, see ﬁgure 2.\nThese ﬁndings are in line with previously published theoretical assess ments within the\nCPA [70,73]. This concentration dependent eﬀect on the Cr moment s seems to be\nwell understood on the basis of an increased number of unfavoura ble Cr-Cr interactions\n(frustration) with increasing number of Cr atoms in the Fe matrix. T he total net\nmagnetic moment of the alloys decreases in the same concentration interval (ﬁgure 2).\nAs reported in [73], the total net magnetic moment of Fe-Cr obtaine d in the CPA is in\nclose agreement with the measured net magnetic moments in the FM p hase of Fe-Cr.\nKlaveret aland Korzhavyi et alshowed by means of the super cell technique, that\nclustering of Cr atoms in Fe-Cr in the concentration range ≤20at.% Cr leads to a\nreduction of the absolute value of the Cr magnetic moments in compa rison to dispersed\nCr atoms due to frustration [15,75]. As mentioned above, the onse t of clustering of\nCr atoms can be connected to the experimentally determined invers ion of the ASRO\nparameter at approximately 6at .% Cr [81,82] or at approximately 11at .% Cr [79,80].\nBoth the magnetic moments of Fe atoms and of Cr atoms located at t he surface\nare enhanced with respect to their bulk values. The EMTO-CPA magn etic moment\nof Cr located at the surface in the dilute limit amounts to −3.11µBbeing thus even\nlarger in absolute value than the corresponding value of Fe (2 .96µB). We realise from\nﬁgure 2 that Fe moments in the surface layer and in the bulk change lit tle as a function\nof concentration in contrast to the magnetism at the Cr sites. The modulus of the spin\nmoment of a Cr atom localised at the surface undergoes a slight decr ease in the range of\nincreasing Cr content from0to20at.%, whichisinfactsimilar tothere duction oftheFe\nsurface moment. The diﬀerent behaviours of the Cr-Cr interactio n is due to the diﬀerent\naverage distance between two Cr atoms at the surface and in the b ulk (for the same\nconcentration it is larger at the surface) and the interaction ener gy which scales with\nthe number of atoms in nearest neighbour shells (which is larger in the bulk) [14]. As\ndiscussed above, the absolute value of the bulk Cr moment drops co nsiderably resulting\nin a drastically higher moment enhancement of Cr at the surface. Th is strong moment\nenhancement is in fact a propensity of atoms in the surface layer on ly; Fe and Cr\nmagnetic moments in subsurface layers of the bcc (100) surface p ossess almost bulk\nvalues [12,14].Surface parameters of ferritic iron-rich Fe-Cr alloy 13\n0 5 10 15 20\n-3-2-10123site magn. moment [ µB]\nFe bulk\nFe surface\nCr bulk\nCr surface\n0 5 10 15 20\nat.% Cr x0.00.51.01.52.02.53.0conc. averaged magn. moment [ µB]\nbulk\nsurface\ndifference\nFigure 2. Magnetism in Fe-Cr alloys for the chemically homogeneous surface an d\nbulk reference systems. Top panel: site resolved spin magnetic mom ents in the bulk\nand at the surface; bottom panel: concentration averaged spin m agnetic moments in\nthe bulk and at the surface, and the diﬀerence between surface a nd bulk magnetic\nmoments. EMTO-CPA calculations on the impurity level were done for 0.05 at.% Cr.\nLines are a guide to the eye.Surface parameters of ferritic iron-rich Fe-Cr alloy 14\n01234conc. weighted ∆m2 [µ2\nΒ]Fe\nCr\n0 5 10 15 20\nat.% Cr x01234\nτmag [J/m2]\nFigure 3. Contributions of Fe and Cr to the surface moment enhancement, ∆ m2,\nweighted by their atomic concentration (left-hand ordinate) and m agnetic surface stress\nτmag(right-hand ordinate) for the chemically homogeneous reference systems. Note\nthatτmagequals the indicated τmagfrom ﬁgure 1. Lines are a guide to the eye.\nAs a consequence of the distinct moment behaviours of the individua l alloys\ncomponents, the total net surface moment ofthe alloy diminishes m ore pronounced than\nthe total net bulk moment (ﬁgure 2). Their diﬀerence drops to zer o at approximately\n20at.% Cr, i.e., the net surface moments equals the net bulk moment. We su ggest that\nthe changes in the magnetic structure determine the trends of γandτas we argue\nbelow.\nTo understand the contribution of magnetism to the noticed behav iour of surface\nparameters we return to ﬁgure 1 where we included data of non-sp in-polarised\ncalculationsaswell. Theseweredoneforexactlythesamegeometry asthespin-polarised\ncalculations. The magnetic contribution to the surface energy, γmag, is likewise deﬁned\ntoτmagas the diﬀerence between the nonmagnetic and the magnetic values ofγ. In\nagreement with previous investigations for 3 dtransition metals [35,36], we ﬁnd that\nmagnetism generally reduces surface energies and surface stres ses in the case of Fe-Cr.\nThe magnitude of the magnetic contribution to the stress is clearly la rger than its eﬀect\non the surface energy. Furthermore, τmagreduces strongly by approximately 1.3J ·m−2\nin the concentration range from 0 to 10at.% Cr while it is approximately constant\nfor higher concentrations. γmagexhibits the same behaviour, however less pronounced.\nThese behaviours conﬁrm that changes in the magnetic structure drive the observed\ntrends of γandτ.Surface parameters of ferritic iron-rich Fe-Cr alloy 15\nBoth Fe and Cr are polarised in Fe-Cr, hence both species contribut e toτmag. To\nbetter understand their individual contributions, we explore the c oncentration weighted\nsurface moment enhancement, that is, we evaluate (7) separate ly for Fe and Cr and\nweight the results by (1 −x) andx, respectively. The resulting data in ﬁgure 3 signals\na correlation between (1 −x)∆m2(Fe) and τmagin the concentration range x≤20at.%\nCr. The eﬀect of Cr on τmagis strongly diminished since the monotonically increasing\nweighted surface moment enhancement of Cr, x∆m2(Cr), is not reﬂected in the trend\nofτmag. Chromium may, however, be associated with the levelling-oﬀ of the m agnetic\nsurface stress for Cr concentrations in the range 10% ≤x≤20%. Thus, in Fe-rich\nFe-Cr alloys the trend of τmagas a function of concentration seems to be dominated by\nthe magnetism of Fe.\n4.2.3. Magnetic pressure Starting from the deﬁnition of the magnetic surface stress\n(τmag=τnsp−τsp) and using (6), we regroup all appearing terms according to\nτmag=cnsp\nsurf−cnsp\nbulk\n4A−csp\nsurf−csp\nbulk\n4A\n=cnsp\nsurf−csp\nsurf\n4A−cnsp\nbulk−csp\nbulk\n4A\n≡τsurf\nmag−τbulk\nmag. (8)\nIn the previous line, we deﬁned the magnetic stress of the surface reference system, τsurf\nmag,\nand the magnetic stress of the bulk reference system, τbulk\nmag. Notice that the deﬁnition of\nτbulk\nmagdoesnotincludeanyparameterrelatedtosurfacesanymore. Itq uantiﬁeshowmuch\nmagnetism contributes to the bulk stress and it is thus closely relate d to the magnetic\npressure known since the advent of band structure theory [99,1 00]. Magnetic pressure is\nforexample associatedwithincreased atomicvolumes inFMtransition metalscompared\nto what their volumes would be in the absence of spin-polarisation.\nThe magnetic pressure of the bulk reference system per layer and per particle,\nτbulk\nmag/nm, as a function of concentration for the Fe-Cr system as plotted in ﬁgure 4\nis non-linear and tensile, indicating that without magnetism the lattice constant, or\nequivalently, the Wigner-Seitz radius would be smaller. The order of τbulk\nmagmay be\nconnected to the diﬀerence in the Wigner-Seitz radius, rWS, between the FM Fe-Cr\nsystem and the nonmagnetic Fe-Cr system. Our data shows that t he equilibrium\nWigner-Seitz radius of the nonmagnetic Fe-Cr system follows a linear concentration\ndependence, i.e, Vegard’s rule ( rWS(Fe1−xCrx)∼(1−x)rWS(Fe) +xrWS(Cr)), with\nrWS(Cr)> rWS(Fe). As mentioned in the beginning of Sec. 4.2.1, the equilibrium\nWigner-Seitz radius of the FM Fe-Cr system changes non-linearly as a function of the\nCrconcentration. Thediﬀerence intheequilibriumWigner-Seitzradiu sbetween theFM\nFe-Cr alloy and the NM Fe-Cr alloy, ∆ rWS, is plotted in ﬁgure 4. We ﬁnd that τbulk\nmagand\n∆rWSare strongly correlated: τbulk\nmagas a quantitative measure for the magnetic pressure\nin the bulk Fe-Cr system correlates with the deviation of the equilibriu m Wigner-Seitz\nradius from Vegard’s rule.Surface parameters of ferritic iron-rich Fe-Cr alloy 16\n0 5 10 15 20\nat.% Cr x1.11.21.31.41.51.6τbulk\\surf\nmag/nm [J/m2]bulk reference\nsurface reference\n0.060.070.08\n∆rWS [Bohr]\nFigure 4. Correlation between the magnetic stress of the chemically homogen eous\nbulk and surface reference systems per layer and per particle (lef t-hand ordinate) and\nthe diﬀerence in the equilibrium Wigner-Seitz radius between the FM Fe -Cr system\nand the non-polarised Fe-Cr system (right-hand ordinate). Lines are a guide to the\neye.\nThe magnetic pressure of the surface reference system ( τsurf\nmag) tells us about the\neﬀect of magnetism at the surface on τcompared to a magnetism-free surface. The\nsurface reference systems, however, include bulk-like contribut ions, as eﬀects due to\nthe presence of a surface decay towards the interior of the surf ace reference systems\nwhich thus become gradually more bulk-like as the distance to the sur face is increased.\nTherefore τsurf\nmagand ∆rWSare also correlated in the same way as τbulk\nmagand ∆rWSare, see\nthe plot of τsurf\nmag/nmin ﬁgure 4. The diﬀerences between τsurf\nmagandτbulk\nmag, i.e., both the\nabsolute value and the trend as a function of x, are ascribed to both the spin-polarised\nand the non-spin-polarised surfaces. It is interesting to note tha t the maximum of τsurf\nmag\nseems to be well below 5at.% Cr while it is above 5at.% Cr for τbulk\nmag.\nMagnetism is the driving force for the enlarged lattice parameters o f the FM Fe-Cr\nsystem compared to the non-polarised model system. Magnetism le ads on the other\nhand to increased magnetic stresses of the bulk and the surface s ystems. Magnetic\nstresses are compressive, i.e., have a tendency to expand the latt ice. According to\nﬁgure 4, τsurf\nmagis larger than τbulk\nmag, i.e., the magnetic contributions to τfavour a larger\nsurface lattice parameter than in the bulk.Surface parameters of ferritic iron-rich Fe-Cr alloy 17\n4.3. Chemically inhomogeneous Fe-Cr alloy\nThere is experimental and theoretical evidence for segregation o n Fe-Cr surfaces.\nAccording to ﬁrst-principles calculations of Ropo et alfor the equilibrium segregation\nproﬁle of the (100) surface at various temperatures, the surfa ce chemistry seems to be\ndetermined by the bulk conﬁguration which leads to the peculiar thre shold behaviour at\napproximately10at.%Crbulkcontent[13,14]. Belowthisvalue, theth ermodynamically\nmost stable (100) surface was found to be essentially Fe-terminat ed in agreement with\nﬁrst-principles surface segregation calculations of Cr in Fe-rich Fe -Cr [16], while this\nsurface facet is enriched in Cr to even higher than bulk concentrat ions above this\nthreshold up to approximately 17at.% Cr in the bulk. Owning the comple xity of these\ncalculations only the surface layer concentration was variable.\nSince one nevertheless may expect that the (magnetic) contribut ions to both excess\nquantities originate from the topmost layers on every surface, we have good reasons\nto believe that changing only the Cr concentration of the surface la yer captures the\ndominant eﬀect on the trends of both surface energy and surfac e stress with surface\nalloying. It is then of course of interest to track how these surfac e parameters behave\nand accordingly how the stability of the surface is aﬀected.\nThe surface energies for chemically inhomogeneous concentration proﬁles{xα}of\nbinary A 1−xBxalloys are obtained according to [17,101],\nγ({xα}) =En\nsurf({xα})−nm\nnEn\nbulk(x)\n2A\n−∆µbulk(x)/summationtextnm\nα=1(xα−x)\n2A, (9)\nwhere{xα}=x1,x2,...,x nmdenotes the concentration of the B element within the\nlayerαperpendicular to the surface, x1andxnmbeing the concentrations of the two\nsurface layers and xi=xnm−i+1due to the symmetry of the slab. The bulk eﬀective\nchemical potential, ∆ µbulk(x), equals the diﬀerence of the chemical potentials of the two\nalloy components in the bulk and is derived from the bulk energy (per a tom), viz.\n∆µbulk(x) =dEbulk(x)\ndx. (10)\nThe surface stress of the bcc (100) surface facet for a chemica lly inhomogeneous\nconcentration proﬁle is then obtained by, using (4) for the n-layer slab,\nτ({xα}) =1\n4A/parenleftbigg∂(En\nsurf−nm\nnEn\nbulk)\n∂ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nǫ=0(11)\n−∂(∆µbulk(x)/summationtextnm\nα=1(xα−x))\n∂ǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nǫ=0/parenrightbigg\n.\nIn this work, we are mostly interested in identifying trends that aris e if the\nhomogeneous concentration proﬁle is altered towards a chemically in homogeneous\nproﬁle. Here we were guided by the aforementioned computed equilib rium proﬁle at\nT= 0K for the bcc (100) surface of Fe-Cr as reported by Ropo et al[13,14], i.e., the\nconcentration proﬁles are variable in the surface layer concentra tion,x1(x1=xnm),Surface parameters of ferritic iron-rich Fe-Cr alloy 18\n0 2 5\nsurface at.% Cr x12.6602.6652.6702.675surface energy γ [J/m2](a)\nFe95Cr05\n1.241.261.281.30\nsurface stress τ [J/m2]\n5 10 15\nsurface at.% Cr x12.712.722.732.74surface energy γ [J/m2]\n1.771.781.79\nsurface stress τ [J/m2](b)\nFe90Cr10\nFigure 5. Surface parameters of Fe-rich Fe-Cr alloys with chemically inhomoge neous\nconcentration proﬁles for two distinct bulk alloy systems, (a) Fe 95Cr05and (b)\nFe90Cr10, as a function of the concentration of Cr in the surface layer, x1. Lines\nare a guide to the eye.Surface parameters of ferritic iron-rich Fe-Cr alloy 19\nand all other concentrations are ﬁxed to the bulk value, x2=x3=...=xnm−1=x.\nTwo distinct bulk alloy systems, Fe 95Cr05and Fe 90Cr10, were selected. The former has a\npredicted equilibrium surface concentration of 0at.% Cr, i.e., a surfa ce Cr concentration\nlower than the bulk one, and the latter possesses a predicted equilib rium surface\nconcentration of 15at.% Cr, i.e., a surface Cr concentration larger than the bulk one.\nFor the Fe 95Cr05bulk alloy system, we thus monitored γandτwith gradual reduction\nof the Cr surface concentration from 5 to 0at.%. For the Fe 90Cr10bulk alloy we varied\nthe Cr amount at the surface between 5 and 15at.%.\nThe eﬀective chemical potential depends on the strain (11). Apply ing a strain of\nthe sizeǫ=±0.02 led to a relative change of |∆µbulk(x)|of approximately 10−5. Since/summationtextnm\nα=1(xα−x) = 2(x1−x) is at most 0.1 in the present work, the second term in (11)\ncontaining the eﬀective chemical potential is approximately by a fac tor of 10−3smaller\nthan the ﬁrst onefor the current system. Thus, we assume that the strain dependence of\ntheeﬀective chemical potentialcanbeneglected, and τisagainobtainedfrom(6). Thus,\nthe coeﬃcient csurfis the one for the chemically homogeneous system with ﬁxed bulk\nconcentration x. This seems reasonable since the surface chemistry should not aﬀe ct\nthe bulk contributions to the surface stress. We recall that csurfshould be identically\nzero at bulk equilibrium.\nThe surface parameters for the two diﬀerent bulk alloy systems ar e compiled in\nﬁgure 5. For the Fe 95Cr05bulk alloy, we identify the following trend for the spin-\npolarised calculations: less surface Cr reduces both the surface e nergy and the surface\nstress, i.e., the Cr free surface possesses the lowest surface en ergy and the lowest surface\nstress. The excess surface stress (not shown) increases slight ly by 0.02J ·m−2when the\nsurface Cr concentration is reduced from 5at.% to 0at.%. The bulk a lloys containing\n10at.% Cr show diﬀerent behaviours: the richer the surface in Cr is t he smaller is the\nsurface energy and the larger is the surface stress. As a result o f these trends, the excess\nsurface stress of the surface with x1=15at.% Cr is by 0.03J ·m−2more stable than the\nlow-Cr surface.\nOur data show that an Fe-rich surface has the lowest surface ene rgy for a bulk\nconcentration below the anticipated threshold concentration, wh ile above this threshold\na Cr-enriched surface possesses the lowest surface energy. Th is ﬁnding is in qualitative\nagreement with calculations for the surface energy of the bcc (10 0) facet of Fe-\nCr from [14] done for ﬁxed x1={0,10}at.% Cr and variable bulk concentration.\nRecalling that the global trend of γas a function of the bulk Cr concentration shows a\nhomogeneously increasing tendency in x(ﬁgure 1), we realise that changing the surface\ncomposition may alter this picture, as in the case of the Fe 90Cr10alloy.\nWe ﬁnd for the investigated inhomogeneous surfaces, that a large r amount of Cr\nin the bulk and at the surface increases the surface stress τand decreases the magnetic\nsurface stress τmag(data for the inhomogeneous surface is not shown). That is, Cr\naddition drives the tendency towards smaller in-plane lattice consta nts at the surface\ncompared to the bulk, and Cr addition has a tendency to reduce the compressive\nmagnetic contribution to the total surface stress.Surface parameters of ferritic iron-rich Fe-Cr alloy 20\n5. Conclusion\nUsing the EMTO method and the CPA we computed the surface energ y, the surface\nstress, and the excess surface stress of the thermodynamically most stable surface facet\n(100) of the homogeneous disordered bcc Fe-Cr system in the con centration interval\nup to 20 at.% Cr. We found that the surface energy increases mono tonically with Cr\naddition thereby following the rule of mixing. An increase of the bulk am ount of Cr\nalso increases the surface stress, which is unexpected, since the surface stress of Cr is\nconsiderably smaller than the one of Fe. As a result of this surprising trend, the excess\nsurface stress reduces with increasing Cr concentration meaning that the (100) surface\nof Fe-Cr becomes more stable against reconstruction than the sa me surface of Fe.\nThe reduction of the compressive magnetic contribution to the tot al surface stress\n(magnetic surface stress) was identiﬁed to dominate this increase of the surface stress.\nWe showed further that mainly the magnetic moment enhancement o f Fe is correlated\nwith the behaviour of the magnetic surface stress. Thus, we conc lude that mainly the\nmagnetism of Fe in Fe-Cr up to 20 at.% Cr is responsible for the unexpe cted trend of the\nsurfacestress. SincethesurfacestressofpureCrlow-indexsu rfacefacetsismuchsmaller\nthantheoneofpureFe(which waspreviously showntobeduetomag netism), weexpect\nthat Cr replaces Fe to dominate the magnetic surface stress for la rger concentrations\nthan 20 at.% Cr.\nWe also investigated the impact of surface segregationon thesurf ace parameters for\nthe Fe 95Cr05and Fe 90Cr10alloys. The former was previously shown to be Fe-terminated\nwhile the latter was shown to be enriched in Cr in vacuum [13,14]. Varyin g only the\nconcentration of the surface layer, we established the following tr ends: a larger amount\nof surface Cr increases the surface stress for both systems, w hile Cr addition raises\n(lowers) the surface energy for the bulk Fe 95Cr05(Fe90Cr10) alloy.\nFor all investigated chemically homogeneous and inhomogeneous diso rdered surface\nproﬁles, a larger amount of Cr in the alloy favours a smaller in-plane lat tice constants\nat the surface than in the bulk, and the addition of Cr shows a tende ncy to reduce the\ncompressive magnetic contribution to the total surface stress.\nAcknowledgments\nThe Swedish Research Council, the Swedish Steel Producers’ Asso ciation, the European\nResearch Council, and the Hungarian Scientiﬁc Research Fund (res earch project OTKA\n84078) are acknowledged for ﬁnancial support. S. S. gratefully a cknowledges the\nCarl Tryggers Stiftelse f¨ or Vetenskaplig Forskning and the Olle Er ikssons Stiftelse f¨ or\nMaterialteknik.\nReferences\n[1] I. 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Lett. , 90:026105, 2003."    },    {        "title": "1111.4359v1.Microwave_magnetoelectric_fields.pdf",        "content": "Microwave magnetoelectric fields  \n \nE.O. Kamenetskii \n \nDepartment of Electrical and Computer Engineering, \nBen Gurion University of the Negev, Beer Sheva, Israel \n \nNovember 16, 2011 \n \nAbstract \n \nWe show that in a source-free subwavelength regi on of microwave fields there can exist the field \nstructures with local coupling between the tim e-varying electric and ma gnetic fields differing \nfrom the electric-magnetic coupling in regular-p ropagating free-space electromagnetic waves. To \ndistinguish such fi eld structures from regular electromagne tic (EM) field structures, we term \nthem as magnetoelectric (ME) fields. We study a structure and conservation laws of microwave \nME near fields. We show that there exist so urces of microwave ME near fields – the ME \nparticles. These particles are represented by small quasi-2D ferrite disks with the magnetic-\ndipolar-oscillation spectra. The ne ar fields originated from such particles are characterized by \ntopologically distinctive power-flow vortices, non- zero helicity, and a tors ion degree of freedom. \nOur studies of the microwave ME near fields ar e combined in two successive papers. In this \npaper we give a theoretical bac kground of properties of the electr ic and magnetic fields inside \nand outside of a ferrite particle with magnetic-d ipolar-oscillation spectra resulting in appearance \nof the microwave ME near fields. Based on the obt ained structures of th e ME near fields, we \ndiscuss effects of so-called ME interactions obs erved in artificial electromagnetic materials. In \nthe next paper, we represent numerical and ex perimental studies of the microwave ME near \nfields and their interactions with matter.   PACS number(s): 41.20.Jb ; 76.50.+g; 84.40.-x; 81.05.Xj \n \nI. Introduction \n \nThe concept of ME near fields arises from an  idea of local coupling be tween the time-varying \nelectric and magnetic fields. The \"local\" means an assumption that a free-space region, inside \nwhich one can observe such a field coupling, is much less than the free-space plane-wave EM \nwavelength. In such an EM subwavelength regi on – the ME-field region – the nature of the \nelectric-magnetic coupling is diffe rent from the nature of the electric-magnetic coupling in a \nregular-propagating free-space EM wave. It m eans that in the ME-field region, symmetry \nbetween the time-varying electric and magnetic fi elds (the \"electric-magnetic democracy\" – a \nnice term coined by Berry [1]) should be in a distinctive form compared to a symmetry \nrelationship between the time-va rying electric and magnetic fiel ds in a free-space EM plane \nwave.      The problem of ME near fields incorporat es a number of fundamental  physical aspects. There \nare, for example, such questions as topological pha ses, chirality and helicity of the fields, and a \ntorsion degree of freedom of the fi elds. In a point ME particle – a source of the ME fields – the \ncoupling between the electric and magnetic fields  is due to specific topological (geometric) \nphases. For a regular-wave EM fi eld, the ME-field region behave s as a topological singularity. It \nis well known that topological phase singularities can be constr ucted based on an interference \nprocess of regular EM waves in free space. The connection between topological phase \nsingularities in 3D free space and the regular-wav e decomposition is complicated as the position  2of nodes depends nonlinearly on the amplitudes and directions of the regular waves. In \nparticular, free-space optical vort ices are created by interference of  three [2] or more [3] plane \nwaves, as well as by interference of spherical waves [4]. A combination of multiple free-space regular EM waves may result in other very speci fic singular properties. Recently, a superposition \nof multiple free-space plane waves was analyzed on the subject of near-field experiments with chiral molecule structures [5 – 7]. As one of  parameters characterizing such a non-plane-wave \nEM field, it was used a so-called the optical chirality density (OCD). This quadratic-form quantity, appeared, for the first time, in the Li pkin's analysis [8], does not have an evident \nphysical meaning in a view of some general disc ussions on conservation laws in the free-space \nMaxwell electrodynamics [8 – 11]. Using Gaussi an units, we represent here the OCD as: \n \n                                                    \n1\n8CE E H H       \n.                                                (1) \n \nHere E\n and H\n are, respectively, the real  electric and magnetic fields . As it was discussed in \nRefs. [5, 6], the EM fields with non-zero OC D, can effectively interact with particles \ncharacterizing by the cross polarizati on effects. This assumes interaction of the electric field with \nelectric dipole moments induced by the magnetic field and interaction of the magnetic field with \nmagnetic dipole moment induced by the electric fiel d. In tiny regions of space, the authors of \nRef. [5] say, the “superchiral” light  – the light with non-zero parameter C – would twist around \nat rates hundreds of times higher than ordinary circ ularly polarized light. As it was shown in Ref. \n[7], the near fields of plasmon-resonance planar-c hiral-metamaterial structural elements [12, 13] \nhave non-zero chiral-density parameter C. Based on numerical results in Ref. [7], one can \nobserve separate free-space regions with posi tive and negative quantities of parameter C. This \nallowed effective spectroscopic characterization of  chiral biomolecules [7]. Nevertheless, from \ngeneral electrodynamics aspects, a physical meani ng of the light twistness proposed in Refs. [5, \n6] remains unclear. As it was discussed in Ref. [11], the chirality density (1) is directly related to the polarization helicity of light in the momentum  (free-space plane-wave) representations. This \nis not, however, the case of studies in Ref. [7] where the evanescent modes are taken into \nconsideration. For near fields , the question, how parameter \nC is related to helicity of light, is \nopen.      Formally, the terms in Eq. (1) correspond to  general definitions of helicity in hydrodynamics \nand in plasma physics. The density of  hydrodynamical helicity is defined as \nuu, where u is \nthe fluid velocity. In plasma physics, the density of magnetic he licity is defined as A A\n, \nwhere A\n is the vector potential of the magnetic fiel d strength. As it was discussed in Ref. [14], \nthe properties of helicity can be observed also  for two circularly polarized transverse \nelectromagnetic waves propagating in vacuum opposite each other (in such a way that their \nPoynting vectors are cancelled out). Th e authors of Ref. [14] showed that in this case one has an \nEM field with ||EH\n and the magnetic helic ity characterizing by non-zero quantity of AA\n. \nThis statement, however, was disproved in Ref. [15], where it was shown that the field energy \nfor the Chu and Ohkawa [14] soluti on diverges and the time average of EH\nis zero. Also in \nRef. [16], based on symmetry and topological ar guments, it was shown that the claim by Chu \nand Ohkawa [14], that a general class of transverse EM waves in vacuum with the ||EH\n field \nstructure exists, is false. Lee [16] stated, in pa rticular, that the possible existence of waves with \nthe ||EH\n field structure may require that the bound ary conditions for the physical space be \nrather more complex and nontrivial than us ed in the Chu and Ohkawa analysis.   \n    Which term should we properly use for a de scription of the light tw istness in free space: \n\"chirality\" or \"helicity\"? Discussing this probl em, it is relevant  to concern some fundamental  3physical aspects on the relation be tween chirality and helicity. Ma thematically, helicity is the \nsign of the projection of the spin  vector onto the momentum v ector. In elemen tary particle \nphysics, helicity represents the projection of the pa rticle spin at the dire ction of motion. Helicity \nis conserved for both massive and massless particles. Chirality (the property related to handedness) is the same as the helicity only when the particle mass is zero or it can be neglected. \nIn condensed matter physics, chirality is to be associated with en antiomorphic pairs which \ninduce optical activity. The existenc e of enantiomorphic pairs, in a crystalline sense, requires the \nlack of a center of symmetry. This is a necessary condition for optical activity. At the same time, the wave helicity, related to a Faraday effect, doe s not require a lack of a center of symmetry. \nWhile the rotation of the plane of polarization by optical activity is a reciprocal phenomenon, \nrotation of the plane of polariz ation by the Faraday effect is  a non-reciprocal phenomenon. \nRegarding the results in Ref. [7],  the observed effect of free-space optical chirality is related to \nthe handedness property: the results are obtai ned with enantiomorphic pairs of plasmon-\nresonance planar-chiral-metamaterial structural elements [12, 13]. The question, however arises: \nWhether the observed effect of the free-space light twistness is really a near-field (or \nsubwavelength) effect? Optical plasmonic oscillat ions in metallic particles are electrostatic \nresonances. For electrically sma ll particles, these resonances  are not accompanied with any \nelectromagnetic retardation effect s. One has the free space-retard ation electromagnetic effects, \nwhen sizes of the plasmonic-resonance partic le become comparable with the free-space \nelectromagnetic wavelength. As we can see from Ref. [7], an in-plane size of the metallic gammadion structure used for creation of superchira l fields is minimum a half of the free-space \nelectromagnetic wavelength. So the effect of optical  chirality in Ref. [7] cannot be considered as \na pure subwavelength effect.     With respect to electrostatic (plasmonic) resonances in small metallic particles at optical \nfrequency, one can consider, to a certain exte nt, magnetostatic (MS) [or magnetic-dipolar-mode \n(MDM)] resonances in small ferrite samples as a dual effect at microwave frequencies [17, 18]. \nThere is, however, a fundamental difference betw een these two effects. In a case of metal \nparticles with electrostatic (plasmonic) resonances one has strong localization of \nelectric  fields in \na subwavelength region [19, 20]. The description is based on the assumption that the velocities \ninvolved are sufficien tly low so that the magnetic field can be neglected. It is  evident that in \nexperiments of Ref. [7], a role of electrostatic (plasmonic) resona nces is to enhance an electric \nenergy of EM fields. In contrast , in small ferrite part icles with MDM oscillations one has strong \nlocalization of both magnetic and electric  fields in a subwavelengt h region. This results in \nappearance of subwavelength power-flow-density vortices [17, 18, 21 – 24]. The phase \ncoherence in magnetic dipole-dipole interactions in a qusi-2D ferrite disk is in the heart of the \nexplanation of many interesting phenomena observed both inside a ferrite and in an exterior \n(vacuum) region. Because of dynamics of th e magnetization motion in a ferrite disk, \ncharacterizing by azimuth symmetry breaking, small ferrite particles with MDM spectra originate free-space microwave fields with uni que symmetry properties [17, 18]. We consider \nsmall quasi-2D ferrite disks with the MDM spectra as point sources of peculiar microwave fields \n– the ME fields. As we will show in this paper, a structure of such a ME field is characterized by \nhelicity properties and a tors ion degree of freedom.   \n    Due to the ANSOFT HFSS program, one can we ll observe some of the ME-field properties in \na microwave subwavelength region of regular EM waves. The HFSS program, in fact, composes \nthe field structures from interferences of multi ple plane EM waves inside  and outside a ferrite \nparticle. We showed that such a very co mplicated EM-wave process in a numerical \nrepresentation can be well modeled analytically with use of a so-called magnetostatic-potential \n(MS-potential) wave function \n. A boundary-value-problem soluti on for a scalar wave function \n in a quasi-2D ferrite disk shows a multiresonance MDM spectrum and unique topological \nvortex structures of the mode fields. There is a very good corre spondence between the results of  4numerical (based on the HFSS program ) and analytical (based on the -function spectra) studies \n[17, 18, 22, 23]. It is worth noting that the MS-potential wave function  is not just a formal \nmathematical representation. This function bears a clear physical meaning. Based on the - \nfunction spectral solutions in a quasi-2D ferrite disk, one obt ains such physically observable \nquantities as energy eigenstates, ei gen electric moments, eigen-mode  power flow densities.      \n    ME fields are parity-violation and time- varying fields with specific space-time geometry. \nInterestingly, the properties of su ch fields can be related to a so -called torsion degree of freedom \n– a subject of heightened interest in modern literature of the field structures. Torsion of \nspacetime – coupling the time and the angular coordi nates of the field – might be connected with \nthe intrinsic angular momentum of matter. The torsion-structure fields  can be created by \nferromagnet structures with its intrinsic ordered spin motion. In the case of a ferromagnet, the \nspin motion originates from fermions (\"spinor matte r\"). It is not possible to eliminate this motion \nthrough transition to a suitable rotating frame of reference. The spin angular momentum can be \nconsidered as the source of the fields which are inseparably coupled to the geometry of \nspacetime [25 – 27]. However, the effects of torsi on in a gravitational cont ext are far negligible \nexperimentally [26, 27]. At the same time, it is  shown that condensed ma tter systems can provide \nuseful laboratories for the study of torsion. Solid and liquid crysta ls with topological defects in \nthe continuum limit can also be described by a ma nifold where the curvature and torsion fields \nare proportional to the topological charge  densities of the defects [28, 29].  In uniform plasmas \none can observe a torsional Alfvén  mode. There is a twisting of magnetic field lines forming a \nconcentric flux shell [30]. One of important aspect s of the torsion degree of freedom concerns a \ntorsion contribution to helicity [31]. Such topologi cal interpretation of helic ity allows describing \nunique symmetry properties of the ME fields originated from a MDM ferrite-disk particle. Due \nto the intrinsic angular momentum of \"spinor  matter\", a quasy-2D ferrite disk with MDM \noscillations (characterized by the long-rang e phase coherence in magnetic dipole-dipole \ninteractions between a pair of spins) can behave  as a torsional defect for propagating-wave EM \nfields. Existence of a torsion de gree of freedom can be considered  as one of the most important \ndistinctive features of the ME fields created by MDM particles. \n    Our studies of microwave ME fields origin ated from MDM ferrite particles are combined in \ntwo successive papers. In this paper we give theoretical background of properties of microwave \nME near fields. In our forthcoming paper [32], we  give the results of numerical and experimental \nstudies of microwave ME near fields and their interactions with matter.  \n \nII. On the electrostatic and magnetos tatic resonances in small particles   \n \nIt is worth starting our studies fr om general aspects of quasistatic  oscillations and a comparative \nanalysis of microwave magnetic-di polar (magnetostatic) resonances in small ferrite samples with \noptical plasmonic (electrostatic) resonances in small metallic particles. Based on this analysis, \nwe show why small particles with magnetostatic  resonances can exhibi t ME properties (coupling \nbetween the time-varying electric and magnetic fi elds in a subwavelength region) and no such \nproperties can be observed in a case of sma ll particles with electrostatic resonances. \n    It is well known that in a general case of  small (compared to the free-space electromagnetic-\nwave wavelength) samples made of media with  strong temporal disp ersion, the role of \ndisplacement currents in Maxwell equations can be negligibly small, so os cillating fields are the \nquasistationary fields [33]. For a case of plasm onic (electrostatic) resonances in small metallic \nparticles, one neglects a magnetic displacement cu rrent and has quasistati onary electric fields. A \ndual situation is demonstrated fo r magnetic-dipolar (magnetostatic ) resonances in small ferrite \nparticles, where one neglects an electric displa cement current. As an appropriate approach for \ndescription of quasistatic oscillations in small pa rticles, one can use a classical formalism where \nthe material linear response at frequency  is described by a local bul k dielectric function – the  5permittivity tensor () – or by a local bulk magnetic f unction – the permeability tensor (). \nWith such an approach (and in neglect of a di splacement current) one ca n introduce a notion of a \nscalar potential: an electrostatic potential  for electrostatic resona nces and a magnetostatic \npotential  for magnetostatic resonances. It is eviden t that these potentials do not have the same \nphysical meaning as in the problems of \"pure\" (n on-time-varying) electrostatic and magnetostatic \nfields [33, 34]. Because of the resonant behavior s of small dielectric or small magnetic objects \n[confinement phenomena plus temporal -dispersion conditions of tensors () or  ()], one \nhas scalar  wave functions: an electrostatic-potential wave function ( , ) rt and a magnetostatic-\npotential wave function ( , ) rt, respectively. The main note is that since we are on a level of the \ncontinuum description of media [based on tensors ( )  or ( )], the boundary conditions for \nquasistatic oscillations should be im posed on scalar wave functions ( , ) rt or ( , ) rt and their \nderivatives, but not on the RF f unctions of polarization  (plasmons) or magnetization (magnons). \nOne has to keep in mind that in phenomenologi cal models based on the effective-medium [the \n()- or  ( )- continuum] description, no electr on-motion equations and boundary conditions \ncorresponding to these equations are used.  \n    Fundamentally, subwavelength sizes eliminat e effects of the electr omagnetic retardation. \nWhen one neglects displacement currents (magnetic  or electric) and cons iders scalar functions \n(,)rt or ( , ) rt as the wave functions, one becomes f aced with important questions, whether \nthere could be the propagation behaviors  for the quasistatic wave proc esses and, if any, what is \nthe nature of these retardation effects. In a cas e of electrostatic resonances, the Ampere-Maxwell \nlaw gives the presence of a curl magnetic field.  With this magnetic field, however, one cannot \ndefine the power-flow density of propagating elect rostatic-resonance waves. Certainly, from a \nclassical electrodynamics point of  view [34], one does not have a physical mechanism describing \nthe effects of transformation of a curl magnetic fiel d to a potential electric field. In like manner, \none can see that in a case of magnetostatic resonances, the Fara day law gives the presence of a \ncurl electric field. Analogously , with this electric fi eld one cannot define the power-flow density \nof propagating magnetostatic-resonance waves sin ce, from a classical electrodynamics point of \nview, one does not have a physical mechanism desc ribing the effects of transformation of a curl \nelectric field to a potential magnetic field [34]. Formally, from Maxwell equations it can be \nshown that in a case of electrostatic resonances , characterizing by a sc alar wave function ( , ) rt, \nthe time-varying electric fields cannot at all be accompanied with the RF magnetic fields. \nSimilarly, one can show that in a case of magne tostatic resonances, char acterizing by scalar wave \nfunction ( , ) rt, the time-varying magnetic fields cannot  at all be accompanied with the RF \nelectric field. This fact can be perceived, in  particular, from the remarks made by McDonald \n[35]. In frames of the quasielectrostatic appr oximation, we introduce electrostatic-potential \nfunction (,)rt neglecting the magnetic displacement current: 0B\nt\n. At the same time, from \nthe Maxwell equation (the Ampere-Maxwell law), 1DHct \n, we write \n \n                                                            2\n21\n HD\ntc t \n.                                                           (2)                       \n \nIf a sample does not posses any magnetic anisotropy, we have   6                                                                     2\n20D\nt\n.                                                                  (3)                       \n \nSimilarly, in frames of the quasimagnetostat ic approximation, we introduce magnetostatic-\npotential function (,)rt neglecting the electri c displacement current: 0 D\nt\n. From Maxwell \nequation (the Faraday law), 1BEct \n, we obtain \n \n                                                             2\n21\n EB\ntc t  \n.                                                        (4)                       \n \n If a sample does not posses any dielectric anisotropy, we have  \n                                                                     \n2\n20B\nt\n.                                                                  (5)                       \n \nAs it follows from Eqs. (3) and (5 ), the electric field in small res onant dielectric objects as well \nas the magnetic field in small resonant magnetic objects vary linearly with time. This leads, however, to arbitrary large fields  at early and late times, and is excluded on physical grounds. An \nevident conclusion suggests itself at once: the el ectric (for electrostatic resonances) and magnetic \n(for magnetostatic resonances) fields are consta nt quantities. Such a co nclusion contradicts the \nfact of temporally disp ersive media and thus any resonant  conditions. Another conclusion is \nmore unexpected: for a case of el ectrostatic resonances the Ampere-Maxwell law is not valid and \nfor a case of magnetostatic resonances the Faraday law is not valid. The above  analysis definitely \nmeans that from classical electrodynamic s, the spectral problem formulated \nexceptionally  for the \nelectrostatic-potential function ( , ) rt do not presume use of magnetic  fields and, similarly, the \nspectral problem formulated exceptionally  for the magnetostatic-potential function ( , ) rt  do \nnot presume use of electric fields. This statem ent lives open a question of the propagation-wave \nbehaviors for the quasista tic-resonance processes. \n    The eigenvalue problem for electrostatic resonances in nanopartic les occurs at optical \nfrequencies when an isotropic di electric medium exhibits strong te mporal dispersion and its real \npart assumes a negative value. The resonant wavelengths are determined by shapes of nanonostructures and dielectric responses of c onstituents [36, 37]. When the material linear \nresponse is described by a bul k dielectric scalar function ( )\n, the electrostatic resonances can \nbe found as solutions of the equation [38]: \n \n                                                                       0 r .                                                  (6)  \n \nFor homogeneous negative permittivity particles ( 0p) in a uniform transparent immersion \nmedium  ( 0s) and with use of conventional Diri chlet-Neumann boundary conditions for \nelectrostatic-potential function, this equation ac quires a form of a linear generalized eigenvalue \nproblem:   7                                                                     2rs   ,                                             (7) \n \nwhere r equals 1 inside the particle and zero outside the particle, and 11ps s  . The \neigenmodes (surface plasmons) are orthogonal an d are assumed to be normalized as [38, 39] \n \n                                                               *2 3\n, qq q qrr d r  .                                            (8) \n \nIt was pointed out that for electrostatic reso nances in nanoparticles one has a non-Hermitian \neigenvalue problem with bi-ort hogonal (instead of regular-o rthogonal) eigenfunctions [40]. \nElectrostatic (plasmonic) resonance excitations, ex isting for particle sizes much smaller than the \nfree-space electromagnetic wave length, are described by the evanescent-wave  electrostatic-\npotential functions (,)rt. No retardation effects are presumed in such a description. In optics, \nthe above electrostatic theory applies only to nanopa rticles, when retardation effects are \nnegligible.    For a spherical nanoparticle of arbitrary radius provided that the latter is much smaller than the \nfree-space wavelength of incident radiation, the resonance permittivity values are consistent with \nthe classical Mie theory [20]. In an analysis of  scattered electromagnetic fields, a small metal \nparticle with ES oscillations can be treated as a point electric dipole precisely oriented in space \n[41, 42]. Importantly, a role of the magnetic fiel d in plasmonic oscillations becomes appreciable \nwhen one deviates from the electrostatic approxi mation to the full-Maxwell-equation description. \nThe retardation effects appear when particle  sizes are comparable with the free-space \nelectromagnetic wavelength. Corrections to electr ostatic resonance modes due to retardation can \nbe found by using series expansions of the so lutions to time harmonic Maxwell equations with \nrespect to the small ratio of the object size to the free-space wavelength. There is the electromagnetic-wave process with a coupling between the electric and magnetic fields [43]. It \nwas shown recently that anomalous  light scattering with quite un usual scattering diagrams and \nenhanced scattering cross sections near plasmon (polariton) resonance frequencies is non-Rayleigh scattering. The observed power-flow patte rns cannot be understood within the frame of \na dipole approximation and the terms of high er orders with respect to size parameter \n2qa  \nshould be taken into account [44 – 46].  \n    The eigenvalue problem for magnetostatic re sonances in small ferrite particles looks quite \ndifferent. At microwave frequencies, in a region  of a ferromagnetic resonance, ferrodielectric \nmaterials (ferrites) are characterized by strong temporal disper sion of the magnetic susceptibility \n[33]. A notion of the magnetic susceptibility has a physical meaning if a size of a ferrite sample l \nis much greater than atomic scales. When a sample size l is also much greater than a \ncharacteristic scale of the exchan ge interaction processes, one can neglect the spatial dispersion \nof the magnetic susceptibility. Imposing now the upper bound for las                      \nlc , one obtains a behavior of quasistationary fields. There are esse ntially magnetostatic \nfields, but because of the temporal dispersion of the magnetic susceptibility, the fields are time-\ndependent and are functions of the coordinates. For small magnetic samples with strong temporal \ndispersion, one has so-called non-uniform ferromagnetic (o r magnetostatic) resonances \ncharacterizing by resonance values  of the magnetic susceptibility [33, 47 – 50]. In such ferrite \nsamples, the electromagnetic boundary problem cannot be formally reduced to the full-Maxwell-\nequation description and the spectral properties are analyzed based on the Walker equation for \nMS-potential wave function (,)rt [50]: \n                                                                    8                                                                       0                                                      (9)  \n \nOutside a ferrite this equation becomes the La place equation. A distinctive feature of MS \nresonances in ferrite samples (in comparison with electrostatic resonances in metal nanoparticles) is the fact that because of the bi as-field induced anisotropy in a ferrite one may \nobtain the \nreal-eigenvalue spectra for scalar wave functions. Such regular multiresonance \nspectra are clearly observed in microwave experi ments with quasi-2D ferrite disks [51 – 55]. In \nsolving the spectral problem, the homogeneous  boundary conditions are imposed on the MS-\npotential function and a normal component of th e magnetic flux density. In such a case one \nobtains a quasi-Hermitian eigenvalue problem for propagating-wave  scalar functions ( , ) rt. \nThis presumes non-electromagnetic  retardation effects in small ferrite samples. A formulation of \nquasi-Hermitian eigenvalue problem and analytical  spectral solutions were shown recently for \nMS modes in a thin-film ferrite disk [21, 24, 56, 57].       In solving a spectral problem for MS oscillati ons, special aspects concern properties of the RF \nelectric fields. It is very important that a role  of the electric fields in MDM ferrite particles \nbecomes evident when one does not deviate fr om the MS approximation to the full-Maxwell-\nequation description. The problem regarding electric fields in MS  resonances acquires a peculiar \nmeaning in a view of fundamental discussions in literature on the sources of the curl electric \nfields in macroscopic electrodynamics of media w ith magnetization oscillations. It is well known \nthat in frames of classical electrodynamics one can assume the existence of a macroscopic \nmagnetic-current density together with a magneti c displacement current [34]. On the other hand, \nit is known that in a macroscopic electrodynamics , there are two models for a magnetic dipole: \nthe amperian-current (electric current loop) model and the magnetic-charge model [58]. The choice between these models in macroscopic electro dynamics is not so evident. In our case of \nMDM oscillations, a choice betw een these two models acquire s a special meaning.   \n    Recent studies of interac tions between EM fields and sm all ferrite part icles with MDM \noscillations reveal strong localization of elect romagnetic energy in microwaves. It was shown \nthat small ferrite-disk particles with MDM spectra are characteri zed by the vortex behaviors [22, \n23]. A quasi-2D MDM ferrite disk can be modeled as a small region of media rotating with very \nlow group velocities [17, 18]. Scattering of the EM fields from the MDM-vortex particle is purely topological. For incident EM waves, such vortex topological singularities act as traps, \nproviding strong subwavelength co nfinement of the microwave fields  [17, 18]. It appears that a \nvortex may turn out to generate a \"radius of no return\", beyond which the incident EM fields falls inevitably towards the vortex singularity. In such a case, the MDM vortex becomes an EM \n\"black hole\" in microwaves. An EM \"black holes\"  in optics are well known [59]. In such optical \n\"black holes\", a vortex flow imprints a long -ranging topological effect  on incident light. \nAccording to Fermat's principle [60] light rays follow the shortest optical paths in media. EM \nfields near the vortex particle behave as the fields in the empty but curved space-time region \n[59]. For external EM fields, a small MDM-vortex pa rticle is a singular point and so is of a great \nparadox. What actually occurs at such an infini tesimally-defined point? When the curvature of \nspace-time at the singularity is in finite, the Maxwell theory does not describe analytically the \nphysical conditions at this point. All this may presume a very interesting problem for \nmathematical mapping of desired distortions of space-time. Presently, the idea of studying a \ngravitational system by replicati ng certain of its aspects in a laboratory environment through \nother, analogous, means has gained wide popularity.     We should come back now to the McDonald 's statement [35] that, formally, no RF magnetic \nfields are available in a case of electrostatic reso nances and no RF electric fields are available in \na case of magnetostatic resonances. As we discusse d in this section, in particles with plasmonic \noscillations one has a non-Hermitian eigenvalue pr oblem and the retardation effects appear when  9particle sizes are comparable with the free-spac e electromagnetic wavelength. So a role of the \nmagnetic field in plasmonic oscillations become s appreciable only when one deviates from the \nelectrostatic approximation to the full-Maxwell-equa tion description. In a ca se of MS resonances \nin small ferrite particles, situation is complete ly different. In these particles one has specific \nretardation effects. The electr ic fields arise from the MS-wave spectral problem solutions \ncharacterizing by symmetry breakings . This results in appearance of peculiar fields – the ME \nfields. The main point is that these ME fields are eigen-mode fields of the MDM spectra. \n \nIII. Formal structure of electr ic fields in ferrite particle s with magnetostatic resonances \n \nOne of the main questions of the field structur es of MS resonances is the question of the \nrelationship between the electric and magnetic fields. As we will show, the \"electric-magnetic \ndemocracy\" in MDM ferrite-disk particles appear s with a very specific form of the field \nsymmetry. Such fields we will characterize as the ME fields.      Differential equations for the fields in side a small non-conducting magnetic sample with \nstrong temporal dispersion can be  obtained from a full system of Maxwell equations with a \nformal assumption that the permittivity of a medium is equal to zero. In a region of electromagnetic transparency of a dielectric medium with tem poral dispersion of the magnetic \nsusceptibility, an averaged density of the electro magnetic energy for harmonic fields is expressed \nas [33]  \n                                        \n 1\n16UE E H H\n \n  \n   ,                                               (10)            \n \nwhere  is the medium permittivity and  are the components of the permeability tensor . \nIn an assumption of negligibly small variation of electric energy in a small sample of a medium \nwith strong temporal dispersion of the magne tic susceptibility, one obtains three differential \nequations instead of the four-Maxwell-equation description of electromagnetic fields [33, 47, \n48]: \n \n                                                       0B \n,                                                                             (11) \n \n                                                       1BEct \n,                                                                   (12)  \n \n                                                       0H \n.                                                                           (13) \n \nTaking into account a constitutive relation  \n                                                               \n4 BHm,                                                             (14) \n \nwhere m is the magnetization, one obtains from Eq. (11): \n \n                                                           4 Hm    .                                                          (15)                       \n \nAt the same time, based on Eqs. (12) (13), and (14) one has for harmonic fields (with the ite \nfactor):  10 \n                                                         4Ei mc     .                                                (16) \n \n    Following the Helmholtz decomposition of any vector field into and potential and curl parts \n[61], we have  \n                                                                 \npot curl mm m,                                                        (17) \n \nwhere  \n                                                    0\npotm        and      0curlm.                                     (18) \n \nIn \"pure\" magnetostatics (non-time-varying fields ) of ferromagnets, the potential and curl parts \nof the magnetization are consider ed as physical notions  defining, respectively, the solutions for \nthe magnetic scalar potential (M H\n) and the vector potential ( B A\n) [34]. From our \nstudies of MS oscillations, it follows that the magnetization field potm is related to the potential \nmagnetic field, while the magnetization field curlm is related to the curl electric field. Formally, \nequations for the potential magnetic field and th e curl electric field can be considered as \ncompletely separate differential equations. It tu rns out, however, that the magnetic and electric \nfields are united because of the spectral properties  of MS oscillations in a ferrite sample. In \nframes of the MS description ( H\n), eigen MS-potential wave functions (,)rt have no \nsolutions related separately to the potential and curl parts of RF magnetization. A total \nmagnetization field is expressed as   \n                                                                \nm ,                                                             (19) \n \nwhere  is the susceptibility tensor [48].  \n    In a MS-resonance ferrite sample, condition 0m presumes the presence of the magnetic \ncharge density:  \n \n                                                              ()mm .                                                                (20) \n \nFor time varying fields, one can suppose that there exists the macroscopic magnetic current \ndensity (magnetization current) introduced analogously to th e electric-polarization current \ndensity [58]:  \n                                                                \n()m mjt.                                                                   (21) \n \nThe magnetic current density and the magnetic charge density should satisfy the conservation law:  \n                                                           \n()\n()m\nmjt .                                                             (22) \n  11    Along with the above condition 0m  for the magnetic curren t density, the condition \n0m  presumes the presence of the electric current density in macroscopic Maxwell's \nequations [33, 34, 58]. In macroscopic electrodyn amics, in general, a question about the models \nfor a magnetic dipole (the amperian-current mode l with an electric current density component \nm, and/or the model, when there is a magnetic current density component tm) is not so \nevident. It was discussed, in particular, that a choice between these models depends on the way \nof measurement of the force on a magnetic dipole which, in nonstationary cases, is different for two models [62]. The controversies about torque  and force on a magnetic dipole [58, 62 – 65] \nraise important questions about properties of magnetization dynamics. In a series of recent \npublications, the problem about magnetic currents and electric fields induced by such currents in magnetic structures appears to be a subject for nu merous discussions. It was shown in Refs. [66, \n67] that moving magnetic dipoles in ferromagnetic metals can induce an electric field. In paper [68], authors raised the question: Can a \"pure ma gnetic current\" induce an electric field? They \nconsidered a situation of existence of a spin cu rrent without a charge current: Spin-up electrons \nmove to one direction while spin-down electrons  move to the opposite direction. An electric \nfield produced by a steady state spin curren t (\"magnetic-charge current\") is described by the \n\"Biot-Savart law\" [69]. It was shown that in the ring geometry there exist persistent spin \ncurrents. There are different mechanisms that can sustain a pure persistent spin current. In the absence of a conventional electromagnetic flux through the ring, the system with \ninhomogeneous magnetic field can support persiste nt spin and charge currents. The Berry-phase \ncurrents were calculated with decoupled the orb ital and spin degrees of freedom [69]. At the \nsame time, it was shown that in a non-magnetic semiconductor ring, a sp in-orbit inte raction can \nsustain a pure spin current in the absence of th e external magnetic field or a magnetic flux [70, \n71]. Recently, the persistent spin current (magne tization current) carried by bosonic excitations \nhas also been predicted in a ferromagnetic He isenberg ring with the inhomogeneous magnetic \nfield [72]. The persistent spin currents are exhibi ted as topological properties of a system. It was \nsupposed that these currents can generate electric fields which ar e described by the \"Biot-Savart \nlaw\" [71, 72]. The electric field originated from a persistent magnetic current in ring geometry is \nan observable quantity due to a Berry phase. A st andard Berry phase is a circuit integral of the \ndifferential phase in a parameter space [73]. The Berry's connection plays the same role as the \nordinary vector potential in  the theory of the Aharonov -Bohm effect. The appropriate \ngeneralization of Stoke's theorem transforms a linear integral of the Berry's connection to a \nsurface integral of the curl of the Berry's connection. The Berry' s connection is gauge dependent \nand nonobservable, while the integrand (the Be rry's curvature) of the surface integral is \nobservable [74]. The above small survey shows a very nontrivial character of our problem of \ncurl electric fields created by a small ferrite disk particle with microwave magnetization oscillations.     In our analysis we will distinguish two types of electric fields, which we conventionally \ndenote as the \n\n and \n fields. For harmonic fields (with the ite factor) the electric field \n \ninside a ferrite is described by the Faraday law: \n \n                                                   4 iiB Hi mcc c       .                                     (23) \n \nAssuming that a ferrite disk is characterized by is otropic dielectric properties, so that inside a \nferrite   \n                                                                        \n0\n ,                                                            (24) \n  12and taking into account Eq. (16) , one obtains the Poisson equa tion for the electric field \n [22]: \n \n                                                                  2( )\n24 e\neffj\nc \n .                                                       (25) \n \nHere we denoted  \n                                                                     \n()e\neffj ic m                                                         (26) \n \nas an effective electric current density. From Eq . (23) it evidently follows that outside a ferrite \nsample (in a dielectric or in vacuum) one has   \n                                                                  iHc   \n ,                                                      (27) \n \nAt the same time, from Eq. (25) one has for the outside electric field   \n                                                                         \n20\n .                                                            (28) \n \nContrarily to a case of the full-Maxwell-equa tion description (givi ng the wave equation \n2\n2\n20 EE\nc  ), we obtained, for time-varying fields, the Laplace-type equation for the vector \nfield \n. Evidently, the curl electric field \n does not \"recognize\" any electr ic charges. It is also \nevident that, by virtue of the Faraday-law equati on (27), outside a ferr ite the electric field \n is \nperpendicular to the magnetostatic-mode field H\n.     \n    Experiments show that the electric field of MDM oscillations in a ferrite  disk is an observable \nquantity. One can excite MDM oscillations by exte rnal quasistatic RF elec tric fields [53] and \ndistinguish (by the MDM spectrum transformation)  the dielectric properties of a surrounding \nmedium [55]. Experimental result s shown in Ref. [55] are conf irmed by numerical studies [32]. \nFrom numerical studies [17, 18,  32] it follows also that for MDMs, the electric and magnetic \nfields in the near-field region above or below a fe rrite disk are not mutually perpendicular. This, \nbeing a violation of Eq. (27), presumes unusual prope rties of the near fields. As we will show in \nthis paper, the electric near field in vacuum can  be strongly different fr om the Faraday-law field \ndefined by Eq. (27). This is due to  the fact that both the electric and magnetic fields in the near-\nfield region arise from specific magnetization dynamics of MDMs. \n    We introduce now another type of a curl  electric field, which we designate as the \n field. \nThis field is described by the differential equation:  \n \n                                                                 () 4mjc   ,                                                     (29) \n \nwhere, for harmonic variables, ()mj im. Formally, the \n field can be considered as a part of \nthe \n field inside a ferrite region [see Eq. (23)]. Inside a ferrite sample with MS oscillations, \nthe \n and \n electric fields may be very slightly dist inguishable quantities. Really, with use of \nthe relation mH in Eq. (23), we have 4 iI Hc      ,  where I\n is the unit  13matrix. In a normally magnetized ferrite disk, the multiresonance MDM spectra are observed in \na region of a ferromagnetic resonance when for  (a diagonal component of tensor ) there is a \nrelation 41 (or for a diagonal component of the permeability tensor there is  1) [51 \n– 57]. It means that for MDM resonan ces one has inside a ferrite sample: \n \n                                                  () 44mim jcc         .                                   (30) \n \n    While different current sources (the currents ()e\neffj ic m  and ()mj im) originate slightly \ndifferent quantities of curl  electric fields (the \n and \n fields, respectively) inside a ferrite, \nthese fields have very different physical nature. From Eq. (29), one sees that there is an evident \nduality with definition of the electric field \n and the macroscopic magnetic field in \nmagnetostatic problems of classical electrodynam ics [34]. Formally, we can complete this \nanalogy by adding an equation for divergence of an electric flux density. We denote this flux \ndensity as \nD. When one supposes that the electric field \n is accompanied with a certain electric \nflux density, one can assume the presence of certa in electric charges. Since there are no real \nelectric free charges in the MDM-problem solutions, we have   \n                                                                        \n0\nD .                                                            (31) \n \nIt is worth also noting that in troducing the electric flux density \nD in our problem does not \npresume the presence of the el ectric displacement current t\nD . In frames of the MS \ndescription of a small ferrite sample, we still ne glect time variations of  the electric energy in \ncomparison with time variations of the magnetic  energy, both inside and outside a ferrite. \n    Eqs. (29) and (31) are completely analogous  to their magnetostatic counterparts in classical \nelectrodynamics [34]. To complete our description, there must be a constitutive relation between \n and \nD. For a region inside a ferrite, we write  this constitutive relation in a form \n \n                                                                     ()4e\nm\nD   ,                                                   (32) \n \nwhere we denoted ()e\nm\n as effective electric polarization originated from magnetization motion.  \n    It is well known that ther e exits a relativ istic effect when a moving magnetization has an \nassociated electric polarization (see, e.g. [34] ). On the other hand, from general symmetry \narguments it can be shown that in some ma gnetic structures with  symmetry breaking of \nmagnetization distribution there could be the phenomenological couplin g mechanisms between \nthe electric polarization and magnetization. When, in particular, the magnetization breaks chiral \nsymmetry, the system can sustain a macrosc opic electric polarizat ion [75 – 77]. For MDM \noscillations, the last mechanism of coupling be tween the electric polarization and magnetization \nis the most appropriate. In Ref. [23] it was shown that in quasi-2D ferrite disks, the MDMs are characterized by symmetry breaking of the ma gnetization structures resulting in induced \nelectric-polarization properties. This occurs since the magnetization of the MDM is spiraling \nalong the disk axis. Based on Eqs. (29)  and (32), one has inside a ferrite \n  \n                                                       \n() ( ) 44me\nm jc       D .                                          (33) \n  14    From Eq. (29), we can see that  outside a ferrite disk we have  \n \n                                                                        0\n                                                            (34) \n \nand so the electric field \n is a potential field: \n \n                                                                       ()e\n .                                                        (35) \n \nHere ()e is introduced as an effectiv e electric scalar pot ential. For an outside region we have \n0   \nD=   and so \n \n                                                                       2( )0e .                                                          (36) \n \nWhile for a region outside a ferrite sample, an electric field \n is a curl field, an electric field \n \nis a potential field. There is no evidence  that in vacuum th e electric near field \n should be \nperpendicular to a magnetic near field H\n.  \n    In our analysis, one can consider two limit cases: (a)  () ( ) 1\n4me\nm jc    and (b) \n() ( ) 1\n4me\nm jc  . In a limit case (a), we have from Eq . (33) for a region inside a ferrite  \n \n                                                              () 4mjc  D .                                                     (37) \n \nWhen we represent the \nD field as a curl of certain vector potentials  \n \n                                                                ()mA \nD,                                                           (38) \n \nwe have the following relation between the vector potential ()mA\n and the magnetic-current \nsource (see Appendix A): \n \n                                                             2( ) ( ) 4 mmAjc  \n .                                                        (39) \n \nFor a limit case (b), we have inside a ferrite   \n                                                         \n()40e\nm    \nD   .                                             (40) \n \nThis gives an electrostati c-type Poisson equation  \n \n                                                               2( ) ( )4ee   ,                                                        (41) \n \nwhere ()e is an effective electric charge density defined as \n \n                                                                  () ()ee\nm \n.                                                        (42)  15 \nIt is worth noting that since MS resonances  are observed only in bo unded ferrites [48], the \nelectric field \n should be found as a result of a solution of  an integro-differential problem. This \npresumes very complicate forms of constitutive relation between \n and \nD. For a case (a), \nhowever, we have a simple situation when a constitutive parameter is equal to unit. This gives \nthe boundary conditions on an interface between a fe rrite and vacuum: continuity of normal and \ntangential components of the electric field \n.      \n    Together with the question on the physical obs ervability of the fields  originated from a MDM \nferrite disk, the quadratic cons ervation laws, characterizing phys ical properties of the fields, \nshould be physically observable. The above analysis of the electr ic fields in MS oscillations \nraises the problem of the ambiguou s definition of the power-flow de nsity, as well as the linear \nand angular momentum densities. In clas sical electrodynamics, the Poynting vector  \n \n                                                       *Re  8cp EH ,                                                           (43) \n \nobtained from the full-Maxwell-eq uation representation, characteri zes the power flow density for \npropagating waves with a mutual transformati on between the curl electric and curl magnetic \nfields. For MS waves, describe d by Eqs. (11) – (13), there are no evident mechanisms for a \nmutual transformation between the potential magnetic  and curl electric fields. This leaves open \nthe question of physical relevance of Eq. (43) fo r MS modes. As it was discussed in Refs. [21, \n22], a relevant equation for anal ytical studies of the power fl ow density of MS waves is  \n \n                                                                **\n16ip BB .                                              (44) \n \nAt the same time, in the HFSS numerical analysis  (which composes the field structures from \ninterferences of multiple plane EM waves inside and outside a ferrite particle) use of Eq. (43) gives proper solutions for the power flow density of MS modes in quasi-2D ferrite disks. Based \non numerical studies, one can see the power-flow- density vortices both inside a MDM ferrite \ndisk and in the near-field region outside the particle [17, 18]. It was shown also [22, 23] that the \nnumerically obtained [with use of Eq. (43)] t opological structures of the power-flow-density \nvortices are in a very good corresp ondence with the vortex structur es derived analytically from \nEq. (44).       The presence of the power-flow-density vortices should presume an angular momentum of \nthe fields. It is well known from Maxwell's theo ry that electromagneti c radiation carries both \nenergy and momentum [34]. The momentum may have both linear and angular contributions. In \nfree space, the angular momentum density is calculated as  \n \n                                                           \n* 1Re8rE Hc                                                    (45) \n \nand is related in a simple wa y to the Poynting vector as  \n \n                                                                 21rpc.                                                             (46)  \n  16In an electromagnetically dense medium, the correct classical-electrodynamics definition of \nelectromagnetic flux has long been controversial with the main competition between the Abraham and Minkowski forms [34]. While a time- averaged quantity of Abraham's density of \nmomentum in an electromagnetically dense medium is expressed as  \n                                                              \n*Re81HEcgA  ,                                                 (47) \n \nfor Minkowski's density of mo mentum in an electromagnetic ally dense isotropic and \nhomogeneous medium with the scalar material parameters  and , one has \n \n                                                             *Re8HEcgM  .                                                 (48) \n \n    In a case of a dispersive anisotropic de nse medium, the question: \"What happens to the \nmomentum of a photon when it  enters a medium\" becomes mu ch more complicated. In free \nspace, the physical quantities characterizing elec tromagnetic fields – energy density, momentum \ndensity, and angular momentum density – are cons erved.  The conservati on can be expressed as \na continuity equation relating a density and a fl ux density of the conserved quantity. One has the \nenergy balance equation (Poynting's theorem) fo r the energy density and the energy flux density, \nthe continuity equation for the momentum density  and the momentum flux density (the Maxwell \nstress tensor), the continuity equation for the angular momentum density and the angular-\nmomentum flux density. In free space, the field has an entire structure which substantiates \nobservance of all of these conservation laws [ 34]. This is not the case for a ferrite medium. \nInside a ferrite medium, local circularly rotating electromagnetic fields and local rotating \nmagnetization are mutually stipulated and the momentum densities found from the conservation \nlaws contain contributions from both the elect romagnetic field and from the matter. Generally, \nthe terms of mechanical variable s in matter can be included in th e constitutive relations for the \nmaterial response. A phenomenological nature of  the constitutive relations leaves little room, \nhowever, for definite physical interpretation of the above conservation laws. For a small \nconfined ferrite structure, physical picture for momentum properties starts to be more \nunpredictable. Propagating electromagnetic fields  in free space carry momentum and angular \nmomentum parallel to each othe r due to relativity requirements fo r transverse waves propagating \nat the speed of light. The situation is different in the sub-wavelength vicinity of electromagnetic \nfield sources, i.e. in the near-field regime. Un der the influence of the material environment the \nelectromagnetic near fields are spatially non-h omogeneous and so no conservation of quantities \nunder parallel displacement and rotation can be a priory assumed.      Magnetic-dipolar fields in small saturated ferrite samples are neither \"pure\" electromagnetic fields nor \"pure\" magnetization fi elds. This presumes the fact th at a total angular momentum of \na ferrite disk should be composed with the \"fie ld part\" and the \"mechani cs part\". For stable \nMDM states, these two parts should compensate one  another, so that a total angular momentum \nshould be equal to zero. The \"field part\" of an angular momentum is due to the presence of phase \ntopology defects in the magnetic-dipolar wave pro cess above. The existence of \"mechanics part\" \nis of pure magnetization-dynamics origin. The only  non-zero \"field part\" of angular momentum \nwhich we can see in the vacuum near-field re gion is not an evidence for the presence of the \nangular momentum of an entire system. The spect ral analytical and nume rical studies of MDMs \nin a normally magnetized quasi-2D MDM ferrite disk gives evidence for discrete states of eigen \noscillations with the vortex stru ctures. There are azimuthally rota ting fields with the vortices of \nthe power flow density in a subw avelength region. For a given dire ction of a bias magnetic field,  17one can see the same directions of the vortex rota tions in three regions: (a) a vacuum near-field \nregion below a ferrite disk, (b) a region inside a ferrite disk, and (c) a vacuum near-field region \nabove a ferrite disk [17, 18, 22, 23]. It is evident that  for every of the MD M eigen states, there \nshould be a conserved angular moment um of an entire structure.  \n    The discussed above quadratic conservati on law concerning power-flow density vortices and \nthe angular momentum density in a MDM ferri te disk are related to another quadratic \nconservation law – the law characterizing the fi eld helicity. The presence of the vortex core \nshould definitely be asso ciated with specific symmetry propertie s of the fields. Formally, we can \nintroduce a quantity corresponding to the parameter of the field helicity based on Eq. (1) and \nwith taking into account proper ties of the MS description ( 0H\n). For a real electric field, \nwe have: \n \n                                                              1\n8FE E\n .                                                      (49) \n \nThis parameter we call as the helicity density of the MDM fields. Evidently, for MDM oscillations, an equation for the he licity density does not have a symmetrical form of the optical \nchirality density expressed by Eq. (1 ). With use of the Faraday law 1BEct   \n for \nmonochromatic fields, one obtains that 0 EB\n, when 0F. The helicity characterizes the \nway in which the field lines curl themselves. As  we showed above, an electric field in vacuum, \noutside a MDM ferrite disk, is a co mposition of a curl electric field \n and a potential field \n. \nThis composition shows unique helicity properties of the near fields originated from a MDM \nferrite disk – the microwave ME near fields. For a vacuum near-field regi on, we can rewrite Eq. \n(49) as  \n                                                                 1\n8F\n.                                                     (50)  \n \nAs we will show below, non-zero parameter F defined by Eq. (50) pres umes the presence of a \ncertain geometrical-phase factor  giving an additional (with resp ect to a dynamical phase) phase \nshift between the electric and magn etic fields. This property of the microwave ME near fields \nappears as a very important topological charact eristic of the microwave ME near fields, \nespecially in a view of recent interest in topology of electromagnetic fields [83 – 86].   \n    Following the above general consideration of the electric fields in fe rrite-disk particles with \nMS oscillations, we clarify now the problem based on the spectral solutions for the MS-wave \nfunction . All the unique features of the electri c fields are observable only at the MS \nresonances. In an assumption of separation of variables, a magnetostatic-potential (MS-\npotential) wave function in a ferrite disk is  represented in cylindrical coordinates , , zr as \n \n                                                         ,, ( ) ,rzC z r   ,                                                  (51) \n \nwhere  is a dimensionless membrane MS-potential wave function, ( ) z  is a dimensionless \namplitude factor,  and C is a dimensional coefficient. For a membrane MS-potential wave \nfunction , the boundary condition of continuity of a radial component of the magnetic flux \ndensity on a lateral surface of  a ferrite disk of radius  is expressed as [21, 57]:  \n  18                                                 a\nrr rirr                  ,                               (52) \n \nwhere  and a are, respectively, diagonal and off- diagonal components of the permeability \ntensor . The term in the right-hand  side of Eq. (52) has the off-diagonal component of the \npermeability tensor, a, in the first degree. There is also the first-order derivative of function  \nwith respect to the azimuth coordi nate. It means that for the MS- potential wave solutions one can \ndistinguish the time direction (given by the direc tion of the magnetization precession and \ncorrelated with a sign of a) and the azimuth rotation di rection (given by a sign of  ). For \na given sign of a parameter a, there are different MS-potential wave functions, () and (), \ncorresponding to the positive and negative directions  of the phase variations with respect to a \ngiven direction of azimuth coordinates, when 02 . There is an evidence for path \ndependence of the problem solutions . To bring a system to its ini tial state one should involve the \ntime reversal operations [18]. Wh en, however, only one direction of  a normal bias magnetic field \nis given, the MS wave rotati ng in a certain azimuth  direction (either counterclockwise or \nclockwise) should make two rotations around a disk axis to come back to its initial state. It \nmeans that for a given direction of a bi as magnetic field, a membrane function , describing \nMS-wave oscillations in a quasi-2D ferrite disk, behaves as a doubl e-valued function.  \n    To make the MDM spectral problem analytic ally integrable, two approaches were suggested. \nThese approaches, distinguishin g by differential operators a nd boundary conditions used for \nsolving the spectral problem, give two types of the MDM oscillation spectra in a quasi-2D ferrite disk. These two approaches are named as the G- and L-modes in the magnetic -dipolar spectra \n[23, 24]. The MS-poten tial wave function \n manifests itself in diffe rent manners for every of \nthese types of spectra . In a case of the G-mode spectrum, where the physically observable \nquantities are energy eigenstates a nd eigen electric moments of a MDM ferrite disk  [51 – 53], the \nMS-potential wave function  appears as a Hilbert-space scalar  wave function [21, 56, 57]. In a \ncase of the L modes, the MS-potential wave function  is considered as a generating function \nfor the vector harmonics of the magnetic fiel ds [18, 22, 23, 24]. For the classical-like L-mode \nspectrum, the physically observable quantities ar e the eigen-mode fields  and the power flow \ndensities. Two types of the MDM spectrum (the G and L modes) appear from the fact that the \npermeability tensor depends both on the frequency and on a bias magnetic field: 0 (, )H  . \nThe G modes, describing MS oscillations with respect to a bias magnetic field 0H\n at a constant \nfrequency , can be considered as the MS-magnon modes . At the same time, the L-mode, \ndescribing MS oscillations with respect to frequency  at a constant bias magnetic field 0H\n, are  \nthe MS-photon modes . There is a univocal co rrespondence between the G-mode and L-mode \nspectrum solutions. The effect of connection between the G- and L-mode spectrum solutions \n(which we can call conventionally as the MS-m agnon – MS-photon interaction) arises from the \ntopological phase factor  on a lateral surface of a ferrite disk.  \n    While the G-mode and L-mode spectrum solutions represen t two ways of an analytical \nintegration of the problem, the HFSS-program solu tions appear as the results of the frequency-\ndomain numerical integration. These numeri cal solutions include both the dynamical and \ntopological (geometrical-phase) e ffects [17, 18, 22, 23]. For a proper interpretation of the \nnumerical-integration spectra and topological properties of the micr owave ME fields, we have to \ndwell on the main aspects of the L-mode and G-mode spectrum solutions. \n \n  19IV. Electric fields in the L-mode solutions \n \nThe L-mode solutions appear from an analysis of MS  resonances in a helical coordinate system. \nAs it was shown in Ref. [24], in a case of a quasi-2D ferrite disk, one has double-helix resonance \nsolutions which can be reduced to solutions in a cylindrical coordinate system. For a ferrite disk \nof radius  and thickness d placed in a z-directed bias magnetic field, the L-mode solutions in \ncylindrical coordinates are represented as [22, 23]:   \n \n                                                       ,,, ( )ii t rrz tC z J e e\n\n\n                                (53) \n \ninside a ferrite disk ( ,  2 2 rd z d    ) and \n  \n                                                        ,,, ( )  ii trz tC z K r e e\n                                     (54) \n \noutside a ferrite disk (for , 2 2 rd z d    ). In these equations,  is a wavenumber of a \nMS wave propagating in a ferrite along z axis,  is a positive integer azimuth number, J and \nK are the Bessel functions of order  for real and imaginary arguments. The function ( ) z  is \ndefined in a general form as () c o s s i nzA z B z  . Eqs. (53), (54) show that the modes in a \nferrite disk are MS waves standing along z axis and propagating along an  azimuth coordinate in a \ncertain (given by a vector of a normal bi as magnetic field) azimuth direction.  \n    One can consider a MS-potential function  as a generating function for the vector harmonics \nH\n. Based on such a MS-potential func tion one defines the magnetic field ( H\n) inside a \nferrite disk as  \n \n                                          ,,, ( )ii t\nrrHr z t Cz J e e\n\n   ,                               (55) \n   \n                                          ,,, ( )ii t rHr z t i C z J e er\n\n\n ,                                  (56)    \n \n                                          ,,,  ( )ii t\nzrHr z t C z J e e\n \n .                                    (57) \n \nTaking into account that mH, where the magnetic susceptibi lity is expressed as [48] \n  \n                                                              0\n0\n00 0a\nai\ni\n  \n  \n   ,                                                  (58)                       \n \none obtains the components of magnetization.  With known magnetization distributions for \nMDMs one can find eigen electric polarization ()e\nm\n and eigen magnetic current ()mj. The  20electric polarization in a MDM ferrite disk, originated from the chiral-order magnetization and \ndefined by the relationship ()e\nm mm    , was analyzed in Ref. [23]. In the present study, \nwe will dwell, however, on a case of a prevailing role of a magnetic current \n() ( ) 1\n4me\nm jc    .  As we will show, such a case well describes the numerical and \nexperimental results of MDM osci llations in a 2D ferrite disk. \n    The radial and azimuth components of a magnetic current ()mj im are the following:  \n \n                               ()    ()m ii t\nrarrj iC z J J e er\n    \n            ,                (59) \n \n                               ()    ()m ii t\narrj Cz J J e er\n     \n           .                 (60) \n \nLet us consider circular co mponents of a magnetic current: \n \n                                                                    () () ()mmm\nr jj i j  .                                                    (61) \n \nBased on Eqs. (59) and (60), we obtain  \n                             \n()    ()m ii t\narrj iC z J J e er\n    \n \n        .           (62) \n \nTaking into account th e known relations for Bessel functions: 1 () () ()Jx Jx J xx \n   and \n1 () () ()Jx Jx J xx \n , we obtain from Eq. (62) \n \n                                          ()  \n1 ()mi i t\narj iC z J e e\n  \n\n\n                           (63) \n \nand  \n                                          \n()  \n1 ()mi i t\narj iC z J e e\n  \n\n\n   .                      (64) \n \nThe quantity a has strict frequency-resonance dependence, while for the quantity a \nno resonance occurs [48]. In a frequency regi on of the MDM oscillati on spectra, there is \naa    . When we take 1 and restrict our analysis by a central region of a disk \n()r  [57], we have, evidently, () ()mmj j . In this case \n  21                                         ()  \n0 ()mi i t\narj iC z J e e   \n\n\n   .                           (65) \n \nThis equation describes a vector characterizi ng by a circular polari zation and a right-hand \nrotation (with respect to a bias magnetic field 0H\n directed along a disk axis z). In neglect of \ncurrent ()mj (assuming that()0mj), we have () ()mm\nrj ij . This shows that a central region of a \ndisk can be considered as a domain with a homogeneously precessing magnetic current.   \n    Now, we write Eq. (29) for components of the electric field \n: \n \n                                               () 14m z\nrrjrz c\n      , \n \n                                               () 4m rzjzr c      ,                                            (66) \n \n                                                110r\nzr\nrr r\n       . \n \nEvidently, homogeneous precession of a magnetic current (with only the  and r components) in \na central region of a quas-2D ferrite disk cause s homogeneous rotation of  the electric field \n in \na disk plane. Because of such  a homogeneous rotation of the \n field in a centr al region of a \nferrite disk, we can also assume that in vac uum (above and below a ferrite and closely to the \nplane surfaces of a disk) there are only the  and r components of the field. So, for the field \n\n in a central region of a disk, both inside a disk and outside near a disk, we can write \n \n                                                       () 4m\nrrjzc \n , \n                                                                                                                                                    (67) \n                                                       () 4m rjzc \n  \n \nThe homogeneously rotating electric field \n is shown in Fig. 1. \n    We consider now circular com ponents of an in-plane magnetic field: \n \n                                                                      r HH i H  .                                                     (68)                       \n \nSimilar to the above analysis of circular components of a magnetic current, we can show that for \n1 and r , there is HH . Under this condition, we get \n \n                                                0 ()ii t rHC z J e e \n\n\n  .                                      (69)                \n  22As we can see, the in-plane components of a magnetic field H\n are perpendicular to the \ncomponents of a magnetic current ()mj. At the same time, from Eqs.  (67) it follows that the in-\nplane components of an electric field \n are also perpendicular to the components of a magnetic \ncurrent ()mj.  So, in the vacuum near-f ield region, we have for th e in-plane fields that || H\n. \nThis allows writing that for in-plane components there is  ||\n | .  \n    Based on the above equations, we can determ ine the helicity parameter for the time-harmonic \nnear fields as   \n                                                           \n* 1Im16F  \n | .                                             (70) \n \nWe can also calculate an angle between vectors \n and \n: \n \n                                                            *Im\ncos \n\n\n\n\n.                                             (71) \n \nWhen one moves away from a disk plane along a disk axis, one expects reduction of the helicity \nparameter of the near field. Such reduction shou ld be not only due to attenuation of amplitudes \nof vectors \n and \n, but also due to variation of sin. This property should give us \nevidence of a torsion structure of the MDM near fields. Based on numerical studies, in Ref. [32] \nit is shown that there are really non-zero helicity parameters F in the near fields above and below \na ferrite disk. It is also shown that when one  moves away from a disk plane along a disk axis, \nthere is reduction of parameter F. \n    For the main thickness mode in a ferrite disk, the function ( ) z  is a symmetrical function \nwith respect to z axis [57]. This gives the symmetri cal-function distribu tion of the field \n \nwith respect to z axis, while the field \n is characterized by the antisymmetrical distribution with \nrespect to z axis. As a result, one has opposite  signs for the helicity density F above and below a \nferrite disk. These signs will change when one reorients the direction of  a normal bias field. \nSuch a statement is clear from the following. At a bias magnetic field oriented along \nzdirection, there is the 90 phase advance of the current ()mj with respect to the magnetic \nfield H. When one reorients a bias magnetic field along zdirection, there will be the \nmagnetic field H and the magnetic current ()mj. Also, there will be the 90 phase delay of the \ncurrent ()mj with respect to the magnetic field H. Thus, at reorientation of a bias magnetic \nfield there is the 180 phase difference between the currents ()mj and ()mj. From Eq. (64), one \nsees that in this case the field \n oppositely changes its direction.  It gives an opposite sign for \nthe helicity density F. Fig. 2 represents qualitative di stributions of the helicity density F for the \nfields above and below a ferrite disk for different  orientations of a bias magnetic field. This \ndistributions show that the near-field structure of the MDM electric fiel d is characterized by the \nspace and time symmetry breakings. However, for su ch near fields, there are evident properties \nfor the  invariance [18]. When one ma kes successively the parity (  ) and time-reversal (  ) \ntransformations, one restores the field structure.    23    In analytical studies of the near-field helicity density F, there are no difficulties in finding \n\n. When a spectral problem for L modes is solved, one easily obtains the magnetic field H\n \nand thus the vector \n of MDMs in a near-f ield region. At the same time, a problem in \nfinding the field \n is not so trivial. It can be supposed that when the magnetic current is known, \nthere exists a direct way to find the \n fields based on solutions of Eqs. (29) and (39). At \nresonance frequencies of MDMs (res ), one formally represents the solutions for the electric \nfield \n as \n \n                                                      ()\n() 3 1()  m\nm res\nresjx\nAx d xcx x \n                                         (72)                        \n \nand   \n                                 \n  ()\n() 3\n3()1() ()  m\nm res\nres resjx x x\nx Ax d xc xx  \n   \n   .                  (73)                      \n \nIn these solutions, however, we have to take into  account the retardation e ffects arising from the \nfact that the magnetic-current sources are azimuth ally running waves. For this reason, one has \nthe causal Green function which means that the source-point time is always earlier than the \nobservation-point time.  \n    To be able to find proper solutions for the \n fields, let us consider now a problem in views of \ntwo observers. The first one is placed in a labora tory frame. This observer identifies the phase \nover-running of 2during a time period of the oscillation T but is unable to differentiate the \n\"orbital\" and \"spin\" polarization angle changes for the \n-field vector. The second observer is \nplaced on a frame rotating with an angular velocity 2frame T  . Contrarily to the first \nobserver, this observer can distinguish \"s pin\" polarization angle changes for the \n-field vector. \nHis view corresponds to Fig. 1 in condition that the time phase tis frozen. It is evident that the \n field vectors are mutually para llel around a closed loop in a disk  plane. The failure of parallel \ntransport around a closed loop, measured by Berry's  phase, is a hallmark of intrinsic curvature. \nSuch intrinsic curvature, in our  case, is due to the double-helix loops of the MDM oscillations in \na quasi-2D ferrite disk [24]. \n    As a starting point in the studies, we will exclude the retardation effect in solutions for the \n \nfields. For this purpose we will make an analysis in a rotating frame. In this case, one has to do \ntransformation of a magnetic susceptibility tensor from a laboratory frame to a rotating reference \nframe. When a frame rotates at a re sonance frequency of MDM resonances (frame res ), \nEqs. (59) and (60) should be rewritten as: \n \n          ()   () s i n ( )m\nrr o t a res rot rot res resrrjC z J Jr     \n               ,    (74) \n \n         ()   () c o s ( )m\nar o trot res rot res resrrjC z J Jr      \n                ,    (75)   \n  24where rot res  and arot res are the diagonal and off- diagonal components of the \nsusceptibility tensors in a rotating reference fram e at the frequencies of MDM resonances (see \nAppendix B). For known distributions of ()m\nrrot resjand ()m\nrot resj inside a ferrite disk, one \ncan obtain solutions for ()()m\nrot resAx\n  and ()\nrot resx     in a rotating reference frame. An \ninversion from the rotating frame to the laborat ory frame will give us the required quantities \n()()m\nresAx\n  and  ()\nresx . The \n field acquires geometric (topological) phase by the MDM \ncarrying an orbital angular moment um. Since the solutions for the \n fields can appear only due \nto integration over geometrical (topological) phases, we can conclude that the electric \n fields \nin the MDM solutions in a ferrite disk are ex clusively the topological  fields. We can also \nconclude that the helicity density F of the MDMs appears because of the presence of topological \nfields \n. When we exclude the retarda tion effects, we can consider  the near-field space above \nand below a ferrite disk as being s liced into the plates with the same in-plane distributions of the \n-field vectors. In this case, one observe s only attenuation of amplitudes of vectors \n without \nany change of an angle between vectors \n and \n. When, however, the retardation effects \nare taken into account, one will observe also  variation of an a ngle between vectors \n and \n \nin the near-field region. So, when the retardation effects in solutions for the \n fields are taken \ninto consideration, one can observe analytical ly certain torsion properties of the MDM near \nfields.    \nV. Electric fields in the G-mode solutions \n \nIn the L-mode representation, the MS-potential  functions serve as generating functions and \nthe observables are the fields of MDMs. In case of G-modes, where the observables are energy \neigenstates of MDMs, the MS-potential wave functions behave as orthogonal quantum-like \nscalar wave functions.      The G-mode solutions are characterized by the singlevalued MS-potential membrane \nfunctions. However, to satisfy the boundary co nditions for magnetic flux density on a lateral \nsurface of a disk, one has to impose a geometrical phase factor. This phase factor appears due to \nsingular edge wave functions [21]. For a G-mode membrane wave function \n~, the boundary \ncondition on a lateral surface of a ferrite disk is the following:  \n \n                                                          0\nrrrr\n      .                                            (76) \n \nOn a lateral border of a ferrite disk, the co rrespondence between a double-valued membrane \nwave function  and a singlevalued function ~ is expressed as:  rr   , where \niqfe\n  is a double-valued edge wave function on contour 2 . The azimuth number \nq is equal to 1\n2l , where l is an odd quantity ( l = 1, 3, 5, …). For amplitudes we have \n f f  and f = 1. Function  changes its sign when  is rotated by 2 so that \n21iqe . As a result, one has the energy-eigenstat e spectrum of MS-mode oscillations with \ntopological phases accumulated by the edge wave function . On a lateral surface of a quasi-2D  25ferrite disk, one can distingui sh two different functions , which are the counterclockwise and \nclockwise rotating-wave edge functions with respect to a membrane function ~. A line integral \naround a singular contour : 2\n**\n01()  ()\nrid i d\n \n\n         \n  is an observable \nquantity. It follows from the fact that b ecause of such a quantity one can restore \nsinglevaluedness (and, therefore, Hermicity) of the G-mode spectral problem. Because of the \nexisting the geometrical phase factor on a lateral boundary of a ferrite disk, G-modes are \ncharacterized by a pseudo-electric field (the gaug e field) [21]. We will denote here this pseudo-\nelectric field by the letter €\n. \n    The geometrical phase factor in the G-mode solution is not single-valued under continuation \naround a contour  and can be correlated with  a certain vector potential ()m\n€\n. We define a \ngeometrical phase for a MDM as [21] \n \n                                        2\n*( )\n0[( )( ) ] 2m\nr€ id K d q\n       \n .                        (77) \n \nwhere 1\nre \n\n  and eis a unit vector along an azimuth coordinate. In Eq. (77), K is \na normalization coefficient. The physical meaning of coefficient K we will discuss below. Here, \nit is necessary to note that in Refs. [21, 55, 90], the coefficient K was conventionally taken as \nequal to unit. In Eq. (77) we inserted a connect ion which is an analogue of the Berry phase. In \nour case, the Berry's phase is generated from the broken dynamical symmetry. The confinement \neffect for magnetic-dipolar osc illations requires proper phase re lationships to guarantee single-\nvaluedness of the wave functions. To compensate  for sign ambiguities and thus to make wave \nfunctions single valued we adde d a vector-potential-type term ()m\n€\n (the Berry connection) to the \nMS-potential Hamiltonian. On a singular contour 2 , the vector potential ()m\n€\n is related \nto double-valued functions. It can  be observable only via the ci rculation integral over contour , \nnot pointwise. The pseudo-electric field €\n can be found as  \n \n                                                                      ()m\n€ € \n.                                               (78) \n \nThe field €\n is the Berry curvature. In c ontrast to the Berry connection ()m\n€\n, which is physical \nonly after integrating around a clos ed path, the Berry curvature €\n is a gauge-invariant local \nmanifestation of the geometric properties of the MS-potential  wavefunctions. The corresponding \nflux of the gauge field €\n through a circle of radius  is obtained as: \n \n                                        () ( )2me\n€\nSK€ d S K d K q        \n ,                          (79) \n \nwhere ()e\n  are quantized fluxes of pseudo-electric fields. There are the positive and negative \neigenfluxes. These different-sign fluxes should be inequivalent to  avoid the cancellation. It is  26evident that while integration of the Berr y curvature over the re gular-coordinate angle  is \nquantized in units of 2, integration over the spin-coordinate angle  1\n2 is quantized \nin units of . The physical meaning of coefficient K in Eqs. (77), (79) co ncerns the property of \na flux of a pseudo-electric field. It should, definitely, be rela ted to the notion of a magnetic \ncurrent in the G-mode analysis. As we will show, in a case of G modes, magnetic currents \nappear due to \"surface magnetic conductance\" . It differs from the situation with L modes where \na magnetic current is the magnetization current.  \n    It is worth noting that the pse udo-electric field (the Berry curvature) €\n can be characterized \nas the density of the pseudo-electric flux and so the quantity €\nt\n\n can be considered as the \ndensity of the pseudo-electric displacement curren t. However, in frames of the magnetostatic \ndescription (when there is a small sample of a medium with strong tem poral dispersion of the \nmagnetic susceptibility), this pseudo-electric displacement current does not define locally the \nmagnetic field of magnetic-dipolar modes. It al so follows that in our case of magnetostatic \ndescription (and contrary to an analysis  of the magnetic monopole and magnetic fluxon \ndynamics based on of the full-Maxwell-equa tion description [83, 84]), the term ()m\n€\nt\n\n does not \ndefine a magnetic field of the MDM. Follo wing Eq. (77), one sees that on contour 2  the \nvector potential ()m\n€\n can be represented as a gradient of a certain scalar function [85]. It means \nthat on a border contour there is ()0m\n€  \n. So the pseudo-electric field €\n is equal to zero \non contour . However, at any points exte rior to a singular contour , the curl the vector \npotential ()m\n€\n (and, therefore, the field €\n) is not equal to zero. To have fields €\n as observable \nquantities in every point of a square S, a singular contour 2  should be excluded from a \nsquare S. This is possible when one assumes that 2S , where   with  . \n    In further analysis of the  G-mode solutions we will stick to a slightly different model than in \nRefs. [21, 55, 90]. Solutions for G modes are determined by the essential boundary conditions, \nwhile for L-modes there are the natural boundary c onditions. The difference between the \nessential boundary conditions and the natura l boundary conditions is defined by the term \na riH , where rH is an annular magnetic field [ 21, 57]. This singular border term, \nwhich expresses the discontinui ty of a radial component of magnetic flux density for G-modes, \ncan be represented as the effective surface magnetic charge density: \n \n                                                             ()4m\nas riH  .                                                 (80) \n \n    For an annular magnetic field at a give n coordinate z, one obtains  \n  \n                                        () () ()r rrHz Cz Cz              .             (81) \n \n Generally, the magnetostatic description presumes  that any close-loop line integral of a magnetic \nfield is zero. Since, however, a magnetic field,  expressed by Eq. (81), is described by a double-\nvalued function, a close-loop integral  of this field on a border contour \n is non-zero. The field  27rH is a topologically distinctive, si ngular magnetic field. The gradient r  is \nconsidered as the velocity of the irrotational fl ow on a lateral surface of a ferrite disk. When we \nrepresent a single-valued membrane function for G modes as    ,rR r  , where \nfunction, ( ) Rr is described by the Bessel functions and () ~ie, 1, 2, 3...   , the \nvelocity of the irrotational flow on a lateral surfa ce of a ferrite disk is defined as [21 – 24, 94]  \n \n                                           iq\nrqfVi e e\n  \n     ,                          (82) \n \nwhere e is the unit azimuth vector. Evidently, 0 V\n, but the circulation of the velocity V \naround a closed contour  is a constant quantity. From Eqs. (80), (81), one has for the surface \nmagnetic charge density: \n \n                   () 1() ()44iv q m\nsa a rrvq f iCz Cz R e\n       \n    .            (83) \n \nBy multiplying the right-hand side of Eq. (83) with ite and taking a real part, one obtains the \nreal-time azimuth wave of a surface magnetic charge density.  \n    In our problem under consideration, the phase -derivative effect of di pole-dipole interactions \nremoves the rotational symmetry of the magnetic  collective oscillations  on a border ring of a \nferrite disk. It is well known th at in different struct ures of low-dimensi onal solids, which are \ndescribed by the complex order parameter, the phase derivative effects may play an essential \nrole. In linear-chain conductors, the time and sp atial derivatives of the phase of the complex \norder parameter can be related to the electron charge-density waves and the electric current \nwaves [95]. In quasi-one-dimensional metals, ther e are also so-called spin-density waves [96]. \nAll these fluctuations are due to broken-symmet ry ground states in metals. There are the ground \nstates of the coherent superposi tion of pairs (pairs of  electrons or pairs of  electrons and holes) \n[95, 96]. It was shown that the macroscopic e ffect of the electric charge density waves \n(conductivity oscillations) is also possible in  the Aharonov-Bohm-configuration structures due \nto the nontrivial real-space topol ogy [97, 98]. Magnons are also the collective excita tions of the \nground state. In absence of spin-o rbit and dipole-dipole interactions, the spin degrees of freedom \nare characterized by full rotational symmetry. Th is leads to excitations of a 1D Heisenberg \nantiferromagnet. Such excitati ons, considered as persistent  magnetization cu rrents around \nmesoscopic Heisenberg rings, were analyzed in  Refs. [72]. In our case, one has the broken-\nsymmetry states on a ferrite disk surface with  the low-dimensional dipole-dipole magnetic-\ncondensate waves. The magnetic charge densit y wave appears when the correlation length \nexceeds the circumference of the border ring of a ferrite disk. \n    For time varying G-mode fields, the azimuth waves of the surface magnetic charge density \n()m\ns, excited due to discontinuity of a normal component of the magnetic flux density, presume \nthe presence of waves of the su rface magnetic current density ()m\nsj. These are the circulating \nmagnetic current density waves. Both quantities, ()m\nsand ()m\nsj, have time- and space-dependent \nphases. With use of separation of variables on a cylindrical surface of a ferrite disk, one has the \ncontinuity equation for the m onochromatic wave process (~ite): \n                                       \n                                                 () () ()mm m\ns zs szjj i     ,                                          (84)                         28 \nwhere ()\n() 1m\nsm\nsj\nj\n \n \n and ()\n()m\nsm z\nzszj\njz\n \n. An azimuth component of the \nsurface magnetic current density  is an azimuth wave, whic h, at a given coordinate z, can be \nrepresented as  \n                                                    \n () ()() ()iv q mm i t\nssjzJ z ee\n,                                       (85) \n \nwhere ()()m\nsJz\n is an amplitude. Since function ( ) z  is a smooth function with a very small \nvariation on the thickness distance of a ferrit e disk [57] we assume here, as necessary \napproximation, that ()0m\nzszj . With this assumption, we obt ain from the above equations: \n \n                                                     () 1() ()4m\nsa rJz C z R f\n  .                                  (86) \n \n    Let us formally associate Eq. (83) for the surface magnetic charge density and Eqs. (85), (86) \nfor the surface magnetic current density. We can write    \n                                                         \n () () ()mm m\nssj\n\nV ,                                                     (87) \n \nwhere  \n                                                            \n()m\nq\n\nV                                                            (88) \n \nis a certain velocity. Eq. (87) shows that the ma gnetic charge density wave slides along a border \ncontour at a constant  \"drift\" velocity ()mV . In fact, this is the phase velocity for the magnetic- \ncharge-density wave along a border contour 2 . The velocities ()mV  are different for the \npositive (with the q wavenumber) and negative (with the q wavenumber) singular edge wave \nfunctions. It is worth noting that for a circul ating magnetic current density wave, a magnetic \nmoment (the magnetization vector) on contour  feels no force and undergoes the Aharonov-\nBohm-type interference effect.  \n    Similar to the vector potential ()m\n€\n on a singular contour 2 , the surface magnetic \ncurrent density ()m\nsj is related to double-valued functions and so can be obser vable only via the \ncirculation integr al over contour , not pointwise. Based on the in tegral relations, one finds \nsolutions for a vector potential ()\n€m\n and an electric field €\n in points exterior to a singular \ncontour . A region of a source is an infinitesima lly thin cylinder: delta-function magnetic-\ncurrent loops of radius  being summed up in a ferrite-disk height d. Since ()m\nsj is a circulating \nmagnetic current density wa ve with a time-dependent phase, in these solutions we have to take \ninto account the retardation effects. For th is reason (with certain similarity to the L-mode \nsolutions), one has the causal Green function wh ich means that the source-point time is always \nearlier than the observation-point time.  29     Non-zero circulation of the velocity V around a closed contour  results in angular \nmomentum of the G mode. There is an electri c (anapole) moment of the G mode originated from \nthis angular momentum. The anapole mome nt determines an interaction of the G mode with an \nexternal electric field [21, 53 – 55, 90]. To fi nd the anapole moment we  introduce the following \nintegral quantity: \n                                                  () ( )\n04( )  d\nem\ns aj z d d z\n  \n,                                             (89) \n \nThe integrand for this quantity is defined as ()4( )m\nrsej z\n, where re is a unit vector along a \ndisk radius. With use of Eqs. (85) and (86), Eq. (89) is written as   \n \n   2\n() 2 2\n00 0()  2 ()d d\niv q ei t it\nar arfaC R f e z d z e d i C R e z d zq\n       \n   \n     .      (90) \n \nAt the time reversal we change a sign of coefficient  f (we have f f ). Also the time reversal \nchanges a sign of an imagin ary unit. If we assume that the azimuth numbers are 1  and \n1\n2q , we can see that in this case vector ()ea does not change its sign at the time reversal. It \nmeans that ()eais a polar vector. \n    Let us rewrite Eq. (90) as follows  \n \n                                                             () 2\n()m\nseI\nac\n\n\n ,                                                       (91) \n \nwhere \n \n                                              ()\n02 ()d\nmi t\nsa rfI ic C Re z d zq \n\n.                          (92)  \n \nOne can see that the formally introduced quantity ()ea has physical meaning of the electric \nmoment originated from a loop of the az imuthally averaged magnetic-charge current ()m\nsI\n. \nThere is a parity-odd toroidal (o r anapole) moment [99, 100]. On e can interpret Eq. (91) as an \nexpression which describes the elec tric (anapole) moment of a fe rrite disk far away from the \nmagnetic current loop ( )   [21]. By analogy with the magnetic  field originated from a loop \nelectric current [34], one can  define the electric field € in spherical coordinates ( ,,) far \naway from a ferrite disk ( )  :  \n \n                                                         () 2\n3cos2m\nsI\n€c \n\n \n,                                       (93) \n  30                                                         () 2\n3sinm\nsI\n€c \n\n \n,                                          (94) \n \nwhere  is an inclination angle. One can s ee that no azimuth variation of the €\n field originated \nfrom magnetic-charge current ()m\nsI\n is assumed in Eqs. (93), (94). This can be correct when \none neglects the retardation effects in solutions for a vector potential ()\n€m\n and an electric field \n€\n. For a given loop current ()m\nsI\n, and in neglect of the retardation effects, one can formally \nmake use of the orthogonal and complete-set ve ctor spherical harmonics  which are utilized for \nthe magnetostatic problem solutions [34]. It  is necessary to note that frequency  corresponds to \na resonance frequency of a certain  MDM in a ferrite disk. So, th ere are spectra of azimuthally \naveraged magnetic-charge currents ()m\nsI\n and spectra of electric moments ()ea. It is worth \nnoting also that the electric fiel d for a \"pure\" (with tw o real point electric charges separated at a \ncertain distance) point-like electric dipole moment ()ep looks very similar to Eqs. (93), (94). In \nspherical coordinates ( ,,) far away from a dipole, one has well known equations for the \nelectric field [34]: \n \n                                                           ()\n3cos2eEp\n ,  \n \n                                                           ()\n3sineEp\n ,                                                               (95) \n \n                                                           0 E \n \nThere is, however, a fundamental di fference between the electric moment ()ea and the electric \ndipole moment ()ep. Consequently, there is a fundamental difference between the electric field \n€\n defined by Eqs. (93), (94) and the electric field E\n defined by Eq. (95). It is evident that there \nis no scalar electric potential used for representing the €\n-type electric field. \n    One of the main remarks of the G-mode analysis concerns the fl ux of a pseudo-electric field. \nIt is evident that the Berry c onnection shown in Eq. (77) should be extended for an entire MS-\npotential function of a MDM, the function . From the above equations, this function is \nrepresented as: \n \n                                        () () () ()Cz Cz Cz R r       ,                          (96) \n                 \nwhere functions  are defined on a singular contour 2  and function  is defined on a \nregion 2S . In connection with the function , a total flux of the pseudo-electric field \noriginated from a MDM ferrite disk should be written as: \n      31                                                      () 2 () *\n0()d\nee\nSCd S z d z  \n ,                                 (97) \nwhere *\nSdS  is a dimensionless norm of a certain G mode. The flux ()e\n  through the ring \n2  should be evaluated modulo the elem entary flux of the electric field ()\n0e.  \n    It is a property of a surface magnetic current ()m\nsj that the electric flux passing through an area \nbounded by such a circulating current  is a quantized quantity. This quantization occurs because \nthe MS-potential wave function must be single valued: its phase difference around a closed loop \nmust be an integer multiple of 2 π. We may predict that there exis ts a quantum of an electric flux \nwhich should be a physical constant. A total el ectric flux passing through a bounded area must \nbe a multiple of a quantum of an electric flux. However, in itself definition and, moreover, the value of a quantum of an elect ric flux are under a question. As  one of the versions, we can \nrepresent the elementary flux of the electric field as the quantity \n()\n04ee , where e is the \nelectron charge. It thus appears that we have \n \n                                                                    () 124eqe  .                                                (98) \n \nBased on Eqs. (97), (98), one defi nes the normalization coefficient K in Eqs. (77), (79) as \nfollows:   \n                                                               2\n*\n0()4d\nSCKd S z d ze  .                                          (99) \n \nIt means that spectral properties of the MDM ferrite disk are quantized with respect to \nelementary electric charge. Th is resembles the Dirac quantiza tion conditions. Dirac's proposition \nof a magnetic monopole appears from an idea of quant ization of a magnetic flux. In our case, we \nhave quantization of an electric flux. \n    Use of the quantity ()\n04ee  as an elementary flux of the electric field represents, however, \na much disputed problem. In our case we have quantization of a circular magnetic current, not \nthe magnetic charge. So an electric flux should be represented by a 2-form and not, like the \nelectric charge, by a 3-form. Accordingly, th ere should be essentia l differences between \nconservation laws for the 2-form and 3-form quan tities. This expresses the peculiarities of the \nmagnetoelectric-field phenomena. At the external mi crowave electric field, some of the electric \nfield lines may penetrate the ferrite in the form of thin threads of material that have turned \nnormal. These threads, which we can call \"electric fluxons\" because they carry an electric flux, are in fact the central regions (\"cores\") of vortices in the magne tic currents. Each electric fluxon \ncarries an integer number of elect ric flux quanta. The external elec tric field directly changes the \nphase of an MS-potential wave f unction, and it is these changes in phase that lead to measurable \nquantities.         For electron wave functions, the Ahar onov-Bohm principle tells us that the Hamiltonian \ndescribing the system is gauge invariant under a magnetic flux changes by integral multiple of \n()\n0mhc e , the elementary quantum of magnetic flux.  Therefore, an adiabatic increase of ()m \nby a single flux quantum is a cycle of the pump in a looped ribbon. For the G-mode  32magnetostatic wave functions, the system is gauge invariant under an electric flux changes by \nintegral multiple of the elementary quantum of  an electric flux. An adiabatic increase of ()e by \na single flux quantum is a cycle of the pump in  a looped ribbon – a latera l surface of a ferrite \ndisk.  \nVI. Discussions \n \nA. Connection between the L- and G-mode spectra . A torsion degree of freedom for ME fields \n \nTwo types of the MDM spectrum (the L and G modes) arise from the fact that the permeability \ntensor depends both on a frequenc y and on a bias magnetic field: 0 (, )H  . For both these \ntypes of the spectrum solutions, the resonances ta ke place at certain quantized states of the \npermeability tensor: \n0() |res H const for L modes and 0 |const resH  for G modes. Since \ncomponents of the permeability tensor depend non- linearly, both on a fr equency and on a bias \nmagnetic field [48], no linear correspondence exists betwee n the resonances of the L- and G- \nmode spectral solutions. The L- and G- mode spectra appear with different physical properties. \nThe G modes are characterized by hermitian differential operator ˆG. These modes are described \nby the complete-set orthogonal MS-wave functions  with the energy eige nstates [21, 57]. For L \nmodes one has pseudo-hermitian differential operator ˆL. The modes are described by the quasi-\northogonal MS-wave functions a nd are characte rized by the  -invariance properties [18, 24]. \nAs it was shown in Ref. [18], th ere exists a certain operator ˆ which provides us with \nconnection between the L- and G-mode spectra.  \n    The connection between the L- and G-mode spectra may manifest a certain contribution to a \ntorsion degree of freedom for MD M oscillations, and so for ME  fields. This concerns an \nadditional spin precession which can be considered in two aspects. Firstly, it is related to the \npresence of the \"spin-orbit\" interaction term: m€\n. This means coupling between the gauge \nfield €\n and magnetization m. Due to the term m€\n one has an interaction between a linear \nmagnetic current ()mj (L modes) and a circular surface magnetic current ()m\nsj\n (G modes). \nSuch an interaction results in the torsion degree of freedom of the near fields. There should be \ntwo vector quantities: the vector m€\n directed along the z axis and the vector m€\n \ndirected along the z axis. These two vectors give two diffe rent types of the near-field torsion \nstructures above and below the ferrite disk. Secondly, one can observe the gravitomagnetic \neffect of the rotating fields [101]. For the G-mode spectrum, one has an anapole-moment \"spin\" \n()es which is an intrinsic \"spin\" of the MDM disk  particle [21, 90]. At the same time, for the L-\nmode spectrum, we have a rotating magnetic assemb ly. With respect to the laboratory frame, the \nL-mode fields rotate at the RF frequency . As measured by the laboratory-frame observer the \n\"spin\" ()es must \"precess\" in a sense opposite  to the sense of rotation of the L-mode fields. The \nHamiltonian associated with such  motion would be of the form ()es\u0000 . The existence of \nsuch a Hamiltonian would show that the intr insic \"spin\" has rotational inertia. Such a \ngravitomagnetic effect of the rotating MDM fields  can be observable only via the circulation \nintegral over contour 2 , not pointwise. So there should be non-zero overlapping integral \nof double-valued and single-va lued functions along contour . This overlapping integral is \nexpressed as    2\n**\n01\n2d\n       , where asterisk means complex conjugation in  33frequency domain. It is evident that integrals    2\n**\n01\n2d\n           and \n   2\n**\n01\n2d\n        are different quantities.   \n \nC. Interaction of ME fields with dielectrics and biological-type samples \n \nMicrowave ME fields originated from ferrite particles with MDM oscillations are very sensitive \nto dielectric parameters of mate rials. Because of special symmetry  properties, these fields should \nbe also sensitive to a topological structur e of some chemical and biological objects. \n    A spectral theory of magnetic-dipolar (magne tostatic) resonances in small ferrite particles \ndoes not presume the presence of the el ectric displacement current. The vectors \n, \n, and \nD, \nreflecting different aspects of the electric field in the L-mode spectrum, as well as the pseudo-\nelectric field €\n for G modes, are polar vectors. At the same  time, in frames of the magnetostatic-\noscillation description (character izing by negligibly small variation of electric energy in small \nmagnetic objects with strong temporal dispersion of  permeability at microw aves), these electric \nfields cannot by related to the electric-polarizati on effects both inside a fe rrite and in abutting \ndielectrics outside a ferrite. So no transforma tions of the MDM spectra  due to variation of \ndielectric parameters of a sample should be observed in experiments and numerical-simulation \nresults. Nevetheless, recent microwave experiment s [55] clearly show an influence of dielectrics \non the MDM oscillations. In these experiment s, the MDM spectrum transformation due to \ndielectric samples abutting to the surface of a fe rrite disk has been de monstrated. As one can \nobserve, such dielectric loadings result in transformations of the MDM-resonance peak positions with very small variations of the peak amplitudes.  It was shown that for a higher permittivity of a \ndielectric loading one has bigge r spectrum expansion. To explai n the experiments with the MDM \nspectrum transformation, the G-mode analytical model has been used [55]. Based on this model, \nit was shown that the MDM spectrum is sensitive to the permittivity parameters of materials \nabutting to a ferrite disk due to the pr esence of the eigenelectric fluxes of the \n€\n fields. While an \nanalysis in Ref. [55] explains the MDM spectrum expansion with a dielectr ic loading, it cannot \ngive an answer why the entire MDM spectrum beco mes shifted (with respect to frequency or a \nbias magnetic field) at such a loading. This shift of an entire MDM spectrum, well observed in our numerical and experimental studies [32], ca n be explained based on an analysis of the \npresent paper.      From the L-mode solutions, one sees that (in neglec t of the effective electric polarization \noriginated from chiral-order magnetiza tion in a ferrite) the electric fields \n\n [see Eqs. (72), (73)] \nand the magnetic fields in classical magnetostatic s problems [34] are comp letely dual with each \nother. Because of such a duality, we can assume the existence of the mechanical torque when the \nelectric field \nexerts on a test point electric dipole ()ep. This mechanical torque is defined as a \ncross product of the electric field \n and the electric moment of the dipole: \n \n                                                                    () ()ee N p .                                                   (100) \n \nIt makes the electric field \n a physically observable quantity in  a local point, both inside and \noutside a ferrite.   34    Let us suppose that  a point electric dipole ()ep is initially oriented al ong a disk axis. From the \nabove analytical solutions for L modes, as well as from the nu merical-simulation results [17, 18, \n22, 23], it follows that in a MDM ferrite disk there is a rotating electric field \n. In a central \nregion of a ferrite disk one has a homogeneous rotation of the \n field, which lies in a disk plane. \nSince there is no electric polarizatio n effects due to the electric field \n, an action of this field \nwill result in precession of the electric dipole ()ep. The mechanical torque is equal to the time \nrate of change of angular momentum. As an ini tial assumption, we can suppose that the time rate \nof change of angular momentum is proportional to the time rate of change  of orientation of the \nelectric dipole ()ep. Based on Eq. (100), we can write: \n \n                                                       ()\n() ()\n()1e\nee\ned\ndt   pNp =  .                                        (101) \n \nwhere a quantity ()1\ne  is a certain coefficient of proportionality. A phenomenological \nparameter ()e we formally introduced in an analogy with the gyr omagnetic ratio  which \nrelates the electron sp in angular momentum and the el ectron magnetic moment [47, 48]. \n    A ferrite is a dielectric material with a sufficiently big value of a dielectric permittivity \n(1 5r). Suppose that we put a quasi-2D ferrite di sk in an external homogeneous DC electric \nfield 0E\n oriented along a disk axis. In this case,  we will have the homogeneous electric \npolarization ()ep inside a ferrite disk. A uniform array of  identical dipoles or iented along a disk \naxis is equivalent to surface electric charges on di sk planes which produce a depolarization field. \nEvidently, the DC electric field 0E\n (resulting in the constant electric polarization) does not cause \nany mechanical torque in the motion equation for polarization ()ep.  \n    Suppose now that we excite  the MDM oscillations in a ferrite  disk. When a ferrite disk is \nplaced in an external hom ogeneous DC electric field 0E\n, the electric field \n of MDM \noscillations will cause precession of electric polari zation about the direction of a disk axis. For \nthe electric polarization ()ep, one has the following precession equation: \n \n                                                                  ()\n() ()e\nee dppdt.                                              (102) \n \nHere we assume that at this precessi on there is a small deviation of vector ()ep from the direction \nof vector 0E\n. So, one can neglect variation of quant ity of the electric polarization.  \n    The torque exerting on the electric polarization ()ep due to the electric field \n should be \nequal to reaction torque ex erting on the magnetization m in a ferrite disk. In this reaction, \nhowever, one should take into account an \"orbital\" moment of the \n field. As we discussed \nabove, in Section III, the existence of the powe r-flow-density vortices of  the MDM oscillations \npresumes an angular momentum of the fields. So, the electric field \n of MDMs has both the \n\"spin\" and \"orbital\" angular momentums. It means that the electric polarizat ion in a disk (in the \npresence of an external electric  field) will be characterized with both the \"spin\" and \"orbital\" \nangular momentums. Due to the angular veloci ty of the \"orb ital\" rotation of  the electric  35polarization, the motion equation for magnetiza tion will be modified. This modification, \ndescribed by Eqs. (B5), (B6) Appendix B, will result in transformation of the MDM spectra.     When we put a dielectric lo ading above or (and) below a ferrite  disk and apply to a structure a \nDC electric field oriented along a disk axis, we have two (or three) capacitances connected in \nseries. The capacitance of a thin-film ferrite di sk is much bigger than the capacitances of \ndielectric samples. So, surface electric charges on ferrite-disk planes will be mainly defined by \nthe permittivity and geometry of dielectric sa mples. As a result, one will have MDM spectrum \ntransformation dependable on parameters of the di electric samples. In microwave experiments \n[32, 55], as well as in numeri cal analyses [17, 18, 22, 23, 32] , we do not have external DC \nelectric fields. In these studies, however, the electr ic polarization of a ferri te disk and dielectric \nsamples takes place due to RF electric fields  of electromagnetic waves propagating in a \nmicrowave waveguide. In such a case one has a rather more complicated process of an \ninteraction of MDMs with polariz ed dielectrics. Nevertheless, the main physical aspects of this \ninteraction discussed above for the DC electric pola rization, will be applicable also in a case of \nthe RF electric polarization. Furthe r discussions on precession of el ectric polarization in a case of \nthe RF electric fields and a role of this pr ecession on transformation of the MDM spectra are \ngiven in Ref. [32].       With use of the microwave ME near-field structures one acqui res an effective instrument for \nlocal characterization of topologi cal properties of matter. Non-zer o helicity density of the MDM \nnear fields allows precise spectroscopic analysis of natural and artificial chiral structures at \nmicrowaves. This paves a way to creating pure microwave devices for se paration of biological \nand drug enantiomers. This also may give an answer to a cont roversial issue whether or not \nmicrowave irradiation can exert a non thermal effect on biomolecule s [102].  In a view of these \ndiscussions, it is worth noting that in biologic al structures, microwav e radiation can excite \ncertain rotational transitions and some extraord inary effects, which cannot be explained as \nclassical heating effects, give the clearest ex amples of a possible specific action created by a \nmicrowave radiation field [102]. \n \nC. On magnetoelectric interact ions in artificial electrom agnetic materials: Do really \nmagnetoelectric interactions  exist in bianisotropic metamaterials.   \n \nA general idea of magnetoelectric (ME) metamateri als is to obtain coupli ng between the electric \nand magnetic fields separately from such a coupling in Maxwell's equations. So called \nbianisotropic metamaterials were conceived as such a kind of ME meta materials, where one \nsupposes existence of local cross-polarization te rms together with the electric- and magnetic-\npolarization terms. However, the shown peculiar properties of ME fields give us the possibilities \nfor some critical analysis of near-field \"ME interactions\" in such artificial electromagnetic materials as bianisotropic metamaterials.      In a case of a metallic bianisotropic partic le [such, for example, as a split-ring resonator \n(SRR) or an omega-particle], an incident EM wa ve experiences phase shift between the electric \nand magnetic fields. Every bian isoropic particle (BAP) behave s like a small electromagnetic \nantenna. In dilute structures, an interaction between BAPs is due to electromagnetic radiation . \nIn dense metamaterials, quasistatic interactions  between BAPs are the following: there are only \nquasielectrostatic (between the capacitance parts) and quasimagneto static (between the inductive \nparts) interactions. There are no quasistatic ME  interactions, since no internal ME energy are \npresumed in such elements. In a dense BAP la ttice, there are, in fact, only the electric and \nmagnetic interaction constants (ECand MC), but there are no ME-interaction constants (no \nMEC). The only way for ME coupling is via the radiation (retardation) effects.  So bianisotropy in \nthese metamaterials is possible only due to e ffects of nonlocality. Th ere are non-standing-wave  36currents inside a split-ring resonator or an  omega-particle. We cannot speak about the \nmicroscopic  electrodynamics of bianisotropic metamate rials, since there ar e no internal motion \nprocesses associated with microscopic ME fiel ds. One can adduce also other argumentation. Let \nus consider a small object. Ther e is an object with sizes much less than a free-space wavelength. \nWe know that inside this obj ect one has transformation of el ectric energy to magnetic energy \nand vice versa. The object is an electromagnetic LC oscillator. Since there is an open structure, \nwe can use point electric and ma gnetic probes (placed in definite positions of the object) to see \nthe electric-to-magnetic and ma gnetic-to-electric energy transf ormations. Suppose now that an \nelectromagnetic wave incident on our object. Can th e wave scattered from this particle bear the \nimprint of the LC-oscillation process of the electric-to-m agnetic and magnetic-to-electric energy \ntransformations occurring inside  the object? Certainly, can. But on sizes comparable, to some \nextent, with the free-space wavelength, sinc e only on such a scale there are energy \ntransformations in electromagne tic waves propagating in vacuum. There is an evident reason for \nthis. In classical electrodynami cs, one does not have locally coupled electric-plus-magnetic \nsources [34]. An array of such LC particles is just a classical diffractional structure with specific \nfield polarization effects. Let us consider now  a small open structure with the electric- and \nmagnetic-field oscillations, but not the LC oscillator. It means that we  cannot separate definitely \nthe regions of the electric and magnetic fields. There is a particle with specific intrinsic \ndynamical process in the material. The near-fie ld region of such a particle should be \ncharacterized by a specific field structure since the electric- and magnetic-field components of \nthe near fields are originated from intrinsic dynam ical process in material of the object. They are \nnot coupled via Maxwell equations. We call such near fields as the magnetoelectric fields. EM \nfields scattered from the ME-fie ld particles should have a topol ogical structure wi th the presence \nof geometrical phases.         In a classical consideration, ME fields or iginated from a lossless MDM ferrite particle are \npotential near fields. In the near-field regi on one has the Laplace equation for magnetostatic \npotential \n (20 ) as well as the Laplace equation for electrostatic potential ()e \n(2( )0e  ). At the MDM resonances, the potentials  and ()e are coupled due to \nmagnetization processes inside a ferrite disk  particle. The MDM magnetization processes \n(described phenomenologically by the Landau-Lifshitz equation) ar e related to electron mass via \ngyromagnetic ratio [48]. Since mass is a measure of inertia, one has causality in ME dynamical \nresponses of a ferrite particle. A composite ba sed on ferrite MDM partic les will behave as a \ncausal ME metamaterial. In a case of lossless metallic \"bianisotropic particles\" such, for \nexample, as SRRs, a character of a dynamical response is completely different. A \"magnetic \npart\" of a SRR is characterized by inertia du e to inductance (inductance is like mechanical \ninertia). If you have a wire in which you try to change the current, that change in current \nproduces an electric field back upon the wire whic h tries to oppose that in crease in current. The \nfield opposing the change does not tr avel at infinite velocity so it always lags a bit behind the \npotential driving the current. For a small metallic ring this lag is not due to non-locality. At the \nsame time, the \"electric part\" (in neglect of any plasmon resonances and when a SRR is at rest) \nhas no inertia in local dynamic response. The \"M E\" properties of a SRR particle are due to \nretardation effects. The fields surrounding su ch particles are described by the Helmholtz \nequation. One does not have the Laplace equations for electric and magnetic potentials and no \ncoupling between such potentials.      In a case of ferrite MDM disks we have a lo calized ME field. There is a \"microscopic\" field \nentity. We can create a ME lattice based on or G-mode interacting ME particles, or L-mode \ninteracting ME particles, or combined L- and G-mode interacting ME particles. In these \nstructures, electric fluxes of one  ME particle interacts with magnetization dynamics of another \nME particle. Also magnetic fluxes of one ME part icle interacts with magnetization dynamics of  37another ME particle. It means that in the inte raction process (between  ME particle) both the \nelectric and magnetic components of  the ME field take part. One cannot separate the regions of \n\"pure\" electric and \"pure\" magnetic fields. Here we stress on the following. Subwavelength \ninteraction between particles with ME propert ies – ME particles – can be realized only by \n\"microscopic\" ME fields. No such an interaction is possible via electromagnetic near fields.    \n \nVII. Conclusion  \n Interaction between electromagnetic waves and matter on a subwavelength scale opens a new \nfield of studies: near-field electrodynamics. In  the near-field electr odynamics, space and time can \nbe coupled in a manner different from the far-fie ld electrodynamics. This may create a new type \nof the field substance. In this paper, we showed that in a clos e proximity of a MDM ferrite disk \nthere exists a quantized near field which is char acterized by peculiar symmetry properties. This \nis a topological, curved space-time field.  Such an entity, differing from the known \nelectromagnetic near-field structures, we call a magnetoelectric field. A near field of the MDM \nparticle – the ME field – is a qua simagnetostatic field. This fiel d, however, cannot be considered \nas a field dual to known quasiel ectrostatic fields. We showed that there is a fundamental \ndifference between the observed ME fields and the near fields originated from plasmon-\noscillation particles. One of important distinctive features of the ME fields  is the presence of the \nhelicty structure in a vacuum near-field region.      The main properties of the ME fields become  clear when one analyses spectral solutions for \nthe MS-potential wave function \nin a ferrite-disk particle. To make the MDM spectral problem \nanalytically integrable, two a pproaches were suggested. Thes e approaches, distinguishing by \ndifferential operators and boundary conditions used for solving the spectral problem, give two \ntypes of the MDM oscillation spectra in a quasi-2D ferrite disk: The G- and L-mode spectra. The \nMS-potential wave function  manifests itself in different manners for every of these types of \nspectra. In this paper, we studied the field structures for the G- and L-mode spectra. We also \nanalyzed possible interactions be tween these two-type spectral solutions. Based on the spectral \nanalysis, we showed that for ME fields originat ed from MDM ferrite particles one can observe a \ntorsion degree of freedom.      We discussed the mechanisms of interact ion between microwave ME fields and dielectric \nsamples. We propose also that with use of th e microwave ME near-field structures one may \nacquire an effective instrument for local char acterization of special topological properties of \nmatter. This, in particular, will allow realizat ion of microwave devices for precise spectroscopic \nanalysis of materials with ch iral structures such, for example, as biological and drug \nenantiomers. In a view of physical  properties of the ME fields, we discussed the mechanisms of \nnear-field coupling between ME  particles. We stress on the fact that the subwavelength \ninteraction between particles with ME propertie s can be realized only by \"microscopic\" ME \nfields and that no such an interaction is possi ble in so called bianisotropic metamaterials.  \n \nAppendix A: Magnetic-current vector potentials in MS oscillations \n \nIn classical electromagnetism, the vector potential A\n is introduced for c onvenience in solving \nmagnetostatic problems with use of the repr esentation for the magnetic flux density as \nB A\n. Frequently, the term vector potential is re ferred as the magnetic vector potential. For \nthe magnetostatic problems, the re lation of the v ector potential A\n to the electric-current source \n()ej is [34]: \n  38                                                                2( ) 4eAjc .                                                       (A1) \n \nWhen the fields are time-varying, one can intro duce auxiliary potential f unctions and specify the \nE\n and B\n fields as  \n \n                                           1AEct \n;            B A\n,                                             (A2) \n \nwhere  is the scalar potential function and A\n is the vector potential function. In a case of the \ntime-varying fields, the relation of  the vector potential function A\n to the electric-current source \n()ej is [34]: \n \n                                                           2\n2( )\n2214e AAjct c   .                                              (A3) \n \nBecause of the electric-current sources for the vector potential,  both for the magnetostatic and \ntime-varying fields, one can al so call the vector potential A\n, as the electric-current vector \npotential. To stress on this defi nition, we rewrite Eq. (A3) as \n \n                                                    2( )\n2( ) ( )\n2214e\nee AAjct c    ,                                              (A4) \n \nwhere ()eA\n means the electric-curr ent vector potential. \n    In spite of the fact that no magnetic char ges and no motion equations for magnetic charges are \nknown in nature, because of the electromagneti c duality one can formally introduce magnetic \ncurrents in Maxwell equations. This formal pro cedure allows solving numerous electrodynamics \nproblems [78, 79].  For the electrodynamic v ector potential caused by a magnetic current ()mj \nwe have the wave equation [78, 80]:  \n \n                                                    2( )\n2( ) ( )\n2214m\nmm AAjct c    ,                                            (A5) \n \nwhere ()mA\n we call the magnetic-current vector potential. The Eq. (A5) was obtained when one \nuses the following representation fo r the electric displacement field \n \n                                                                ()mDA\n.                                                             (A6) \n \nWith such a representation, one has from th e Maxwell equations for the magnetic field: \n \n                                                           ()1mAHct \n,                                                      (A7) \n \nwhere  is the magnetic scalar potential.  It is worth noting that, referring to representation \n(A6), in some publications, the ma gnetic-current vector potential ()mA\n is called as the electric  39vector potential. There is, for examples, the study of the fields of toroidal  solenoids [81]. At the \nsame time, such terms as the electric-curre nt vector potential (regarding the vector ()eA\n caused \nby an electric current) and the magnetic-curre nt vector potential (regarding the vector ()mA\n \ncaused by a magnetic current), used also in  our study, one can find in Ref. [82]. \n     As we discussed above, in a case of MS oscillations there are no effects of the \nelectromagnetic retardation. The term ()mA\nt\n\n does not define the magnetic field H (in a case of \nthe MS oscillations the magnetic field is H\n) and thus Eq. (A5) is written as \n \n                                                              2( ) ( ) 4mmA jc  .                                                     (A8) \n \nThere is a dual situation with respect to the pr oblem described by Eq. (A1). Such an equation one \nobtains immediately from Eqs. (37) and (38) and taking into account a proper gauge (the \nCoulomb gauge [34]) transformation.      Based on Eqs. (37) and (38), we have  \n                                                         \n2( ) ( ) ( ) 40mm mAA jc      \n .                                     (A9)                       \n \nThis equation shows that formally two types of ga uges are possible. In the first type of a gauge \nwe have:  \n                                                                       \n()0mA\n                                                        (A10)  \n                                              \nand, therefore,  \n                                                                   \n2( ) ( ) 4 mmAjc  \n .                                               (A11)   \n                                                    \nThe second type of a gauge is written as  \n                                                                \n() () 40mmAjc   \n                                            (A12) \n                                               \nand, therefore,  \n                                                                     \n2( )0mA\n .                                                        (A13) \n \nThe last equation shows that any sources of the el ectric field are not define d and thus the electric \nfield is not defined at all. So only the first type of a gauge, described by Eq. (A10) and resulting in Eq. (A11), should be taken into account.   \nAppendix B:  Transformation of a magnetic suscepti bility tensor to a rotating reference \nframe \n  40Let ) ,,( zyxS  be a frame of reference rotating with respect to the laboratory frame ) ,,( zyxS  \nwith an angular velocity represented by a vector frame. The space-time transformation from the \nlaboratory frame S to the rotating frame S is given by (see e. g. [87])  \n \n                                                      )  ( sin ) ( cos t yt xxframe frame    ,           \n \n                                                      )  ( cos ) ( sin t yt x yframe frame    ,                                 (B1) \n \n                                                       zz,              tt. \n \nIt is evident that the vector differential operator \n is invariant under th e transformation (B1): \n \n                                                                        \n.                                                              (B2)          \n \n    According to the general law of relative mo tion, the time derivative  of any time-dependent \nvector ) (tC\n, computed in the laboratory frame S, and the time derivative computed in the \nrotating frame S, are related thr ough (see e.g. [88]) \n \n                                                           CdtCd\ndtCd\nframe\nrot lab \n\n\n\n\n\n\n .                                     (B3)    \n \nThe motion of the magnetic moment in the laboratory frame is described by the equation \n \n                                                                        dHdt  .                                               (B4) \n \nBased on Eq. (B1), we have the motion equation of the magnetic moment in the rotating frame \n[89] \n                                \n                                                            frame\nrotdHdt        .                                   (B5) \n \nFor the motion of the magnetic moment in the rotating frame we have an effective field \n \n                                                                      frame\neffH H\n ,                                               (B6) \n \nwhich is the sum of the laboratory-frame field H\n and a fictitious field frame\n .  \n    Now consider a ferromagnet. Because of the presence of the exchange  interaction between the \nspins of the individual atoms, the magnetic mome nt of the ferromagnet may be regarded as rigid \nprovided only the temperature of the ferromagnet is sufficiently small. For the magnetic moment \ndensity of the ferromagnet, M\n, we have the motion equation in the rotating frame similar to Eq. \n(B5):  \n  41                                                      frame\nrotdMMHdt      \n.                                         (B7)    \n \n    We represent the total field H\n and the magnetization M\n in the rotating frame as sums of the \nDC and RF components   \n                                                \n~ 0H HH\n ,                0 rot M Mm                                    (B8) \n \nand suppose that   \n                                                   \n0 ~ H H\n ,                   0 rotmM.                                       (B9) \n \nIn this case, Eq. (B7) acquires the form:  \n                                            \n~ 0 0   H H mdtmd frame\nrot\nrot\n\n\n\n\n\n .                         (B10)  \n  \n    Let ~H\n and rotm be the time harmonic functions  characterized by frequency  (~tie ). For \ncomplex amplitudes in the rotating frame, Eq. (B10) is rewritten as \n \n                                             ~ 0 0    H H m miframe\nrot rot\n\n\n  .                          (B11) \n \nWe consider the case when the vectors 0H\n and 0M\n are directed along the z axis and when the \nvector frame is oriented along the z axis as well. From Eq. (B11), one has the following \nequations in Cartesian coordinates: \n \n                                            00 ~    xf r a m e yrot y rotim H m M H        \n                                                                                                                                       \n                                          00 ~    frame x y rot x rotHm i m M H                          (B12) \n \n                                                                         0rotzm                       \n \nThis gives the following equation:  \n                                                                   \n~H mrot rot ,                                                     (B13) \n \nwhere  \n                                                    \n\n\n\n\n\n\n0 0 000rot rotarota rot\nrot ii\n\n                                           (B14)  42 \nand  \n                   \n0\n22 H frame\nrot\nHf r a m eM \n\n\n\n,       \n0\n22 \narot\nHf r a m eM\n\n,     0HH .       (B15)     \n \nWe have to note here  that the frequencyframe  is a positive quantity. In a particular case when \nH frame , we have 0 rot  and 0\narotM . \n    In the rotating frame, one has for magnetic flux density \n  \n                                                                 ~H Brot rot\n ,                                                       (B16) \n \nwhere the magnetic permeability tensor is defined as  \n                                                                \nrot rotI 4 .                                                      (B17) \n \nHere I\n is unit matrix. \n    We consider now the frequencies correspondi ng to the MDM resonances in a ferrite disk. At \nthese resonances, one has the rotating magnetic fields. For res frame  , we obtain \n \n                   \n0\n2 \n2H res\nrot res\nH Hr e sM \n\n,         \n0\n2 \n2res\narot res\nH Hr e sM\n .                     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Qualitative distributions of the helicity density F for the fields above 2dz and below \n2dz a ferrite disk for different orientations of a bias magnetic fi eld. The unit vector ze is \noriented along z-axis. \n            46\n \n \n \nFig. 1. Homogeneous rotation of the electric field \n. A bias magnetic field 0H\n is directed \ntoward the observer normally to a disk plane. Wh en (for a given radius and a certain time phase \n t) an azimuth angle  varies from 0 to 2, the electric-field v ector accomplishes the \ngeometric-phase rotation.  \n                                                                                     \n                                        \n  \n \nFig. 2. Qualitative distributions of the helicity density F for the fields above 2dz and below \n2dz a ferrite disk for different orientations of a bias magnetic fi eld. The unit vector ze is \noriented along z-axis. \n "    },    {        "title": "2002.00439v3.A_millimeter_wave_Bell_Test_using_a_ferrite_parametric_amplifier_and_a_homodyne_interferometer.pdf",        "content": "1 \n A m illimeter -wave  Bell Test using a ferrite  parametric amplifier and a homodyne \ninterferometer     \nNeil A. Salmon *,1, Stephen  R. Hoon2 \nManchester Metropolitan University, All Saints, Oxford Road, Manchester, M15  6BH , UK . \n________________________________________________________________________________ _ \nARTICLE INFO  \n_________________ __________________  \nKeywords:  \nSpin-wave  \nYttrium i ron garne t \nHomodyne interferometer  \nMillimeter wave  \nEntangled photons  \n \n \n \n \n ABSTRACT   \n___________________________________ ___________________________________________________  \nA combin ed ferrite  parametric amplifier and millimeter -wave homodyne interferometer are proposed as an \nambient temperature Bell Test. It is shown that t he non -linear magnetic susceptibility of the yttrium iron garnet \n(YIG) ferrite , on account of its narrow line -width Larmor precessional resonance , make it an ideal material for \nthe creation of entangled photons . These can be measured using  a homodyne interferometer , as the much larger \nnumber  of thermally generated photons  associated with  ambient temperature emission  can be screened out . The \nproposed architecture may enable YIG quantum technology -based  sensors to be developed , mimicking in the \nmillimeter -wave band the large number of quantum optical experiments in the near -infrared and visible regions \nwhich had been made possible by use of  the nonlinear beta barium borate ferroelectric , an analogue of YIG . It is \nillustrated here how the YIG parametric amplifier can reproduc e quantum optical Type I and Type II wave \ninteractions, which can be used to create entangled photons in the millimeter -wave band.  It is estimated that \nwhen h alf a cubic centimeter  of YIG crystal is placed in a magnetic field of a few Tesla and pumped with 5 Watt s \nof millimeter -wave radiatio n, approximately 0.5x1012 entangled millimeter -wave photon  pairs per second  are \ngenerated by the spin -wave interaction . This means an integration time of only a few tens of seconds is needed \nfor a successful Bell Test.  A successful demonstration of th is will lead to novel architectures of entanglement -\nbased  quantum technology room temperature sensors, re -envisioning YIG as a modern quantum material.  \n__________________________________________________________________________________  \n1. Introduction   \nThe Einstein -Poldolsky -Rosen (EPR) paper [1] from  1935 \nprovoked a  dialogue questioning  the nature of reality  at the quantum \nlevel , which  stimulat ed hugely the development of quantum theory . \nThe ensuing discussions around this are likely to extend well into the \n21st century , as they are sitting right at the heart of emerging quantum \ncomputers  and sensors .   \nJohn Bell in 1964 conceived a  simple test on the nature of local \nreality  [2] that centered on evaluating an  inequality. If the inequality \nis obeyed , future events in a system are completely determined by past \nhistory, this information being passed on  in postulated hidden \nvariables. The term hidden is used as these variables are not part of \nquantum theory. If the inequality is violated , any local hidden \nvariables are  not dictating future events , and the state of a system is \ndetermined only by its measurement.  \nJohn Clauser  experimentally embodied [3] the Bell Test by using \npolariz ed optical photon s; similar tests followed and these types of test \nare now  referred to as Bell Tests. The maturing technologies of \nnonlinear dielectric crystals, lasers and photo -diodes then enabled the \nfirst demonstrations of a Bell Test  that violated  the Bell inequality  by \n40 standard deviations  [4]. With the exceptions of a few loopholes, \nthis demonstrate d that at the quantum level , reality is defined by a \nmeasurement.  Since then  optical Bell Tests have closed many of the \nloopholes  [5] and reduc ed signal integration times from 200 hours in \nthe 19 70’s [6], to a few seconds with today’s  technology . This \ntechnology  in the optical band is now available commercially for \nundergraduate teaching laborator ies [7].  \nThe initial Bell Tests in the optical band were optimal, as detection \nof the weak entangled photon  flux went unhindered by spurious \nsignals from thermally generated photons. For operation in the \nmillimeter wave band the problem of the thermal photons has been \nsuccessfully comb ated by  cryogenically cool ing the sources and \nreceivers [8], [9].  \nFor operation at ambient temperature in the millimeter wave band, \na novel Bell Test is herein proposed. Such a system  would be  cheaper  \nand lead to easier  experimentation and in-field system deployments . \nThe thermal photon problem is circumvented by using a yttrium iron \ngarnet (YIG) ferrite parametric amplifier source of entangled photons \nwith a phase link to a homodyne interferometer performing  coherent \nintegration  [10]. A key point about this test is that the pump which \ngenerates the entangled photons is used to coherently integrate the \nentangled photon signature in quadrature space.  It is important to \nrecognize that the special phase r elationship between the pump, signal \nand idler is a fundamental enabler of th e coherent integration in the \nproposed Bell Test.  \n \n* Corresponding Author.  \n     E-mail address: n.salmon@mmu.ac.uk (N.A. Salmon).  \n1 ORCid: 0000 -0003 -4786 -7130.  \n2 ORCid: 0000 -0002 -1250 -9432. The proposed homodyne system  generat es non-degenerate \nentangled photons  over a quasi -continuous spectrum , and then  \nmeasure s their phase s and amplitude s, as proposed  in [11] for the \noptical band. This type of measurement is referred to as one of \ncontinuous -variable  (CV) . The other type of entanglement \nmeasurement is that of the discrete -variable  (DV) , where only two \nenergy states are involved. Successful Bell Test s (invalidating the Bell \ninequality) were first demonstrated using discrete -variable \nmeasurements.  Successful continuous -variable Bell Test s came later \n[12].  \nThe method section in this  paper analyses the numbers of \nmillimeter wave thermal photons  produced  at ambient temperature \nand explains how entangled photons could  be generated using \nnonlinear magnetic materials such as a YIG ferrite. The results section \ndetermines the flux of entang led photon pairs in relation to practicable \nlevels of parametric pump power. The discussion section describes the \ntypes of millimeter wave circuits that could be used to induce  Type I \nand Type II interactions and how these may be used to generate \nentangled  photons and the Bell States. Opp ortunities to close possible \nBell Test loopholes are also discussed.    \n2. Method  \nSemiclassical c onsiderations are now made as to  the numbers of \nmillimeter wave thermal photons that are generated at ambient \ntemperature and how entangled millimeter wave photons may be \ngenerated using nonlinear susceptible materials, in particular YIG \nferrites.   \n2.1. Thermally generated photons in the Rayleigh -Jeans and quantum \nsensing regimes  \nThe ratio of p hoton to thermal  energy ( hf/kT) makes a clear \ndistinction between the Rayleigh -Jeans regime ( hf/kT < 1 ) and the \nquantum regime ( hf/kT >1), where f is the photon frequency, T is \nambient temperature, h is Planck’s constant and k is Boltzmann’s \nconstant. At the ambient temperature (290 K) the photon energy \nbecomes  equal to the thermal energy ( hf=kT ) at 6 THz; below this \nfrequency is the Rayleigh -Jean’s regime and above  this is the quantum \nregime.  \nBeing either in the R ayleigh -Jeans  regime or the quantum  regime \nhas a major effect on the mean number of thermally generated photons \n𝑛̅ which occupy a mode in the sensor.  The mean  number of photons is \ngiven  [13] by the Bose -Einstein distribution function  \n 𝑛̅=1\n𝑒ℎ𝑓𝑘𝑇⁄−1 (1) \nwhich at ambient temperature in the optical band (~5x1014 Hz) \nbecom es exp ( -hf/kT) , of the order of  2x10-22 and in the microwave \nband (~10 GHz) becomes  kT/hf , of the order of 60 4. This indicates the 2 \n heart of the problem : In the millimeter wave band, the number of \nthermally generated photons is high and likely to swamp the relatively \nlow fluxes of entangled photons.  \nBecause of the large numbers of thermally generated photons at \nambi ent temperature the photon counting techniques used in the \noptical band are completely unsuitable in this frequency band. The \nabove figure of 604 photons indicates there are these numbers of \nphotons per unit bandwidth. This indicates if the classical radia tion \nfield were sampled at the Nyquist rate there would be this number of \nphotons in a single sample. If there were one o r more entangled \nphotons  also there, they would be completely swamped by the thermal \nphoton flux. For this reason,  the technique propos ed here uses radio \nreceivers to measuring in quadrature space , with a phase link to the \nparametric amplifier pump  to enabl e coherent integration .  \n2.2. Generation of entangled photons in a nonlinear medium  \nPair production is a  process whereby an incident photon vanishes \nand two photons appear in its place, whose energies sum to that of the \noriginal photon . It is a quantum process happening across the \nelectromagnetic spectrum , from microwave s to gamma ray s. The \nincident photon must interact with matter  for the process to proceed, \nas this provides energy levels with which the electromagnetic wave \ncan interact and generate entangled photons . A Feynman diagram for \nthe process is illustrated in Fig. 1.    \n \nFig. 1. Feynman diagram for SPDC pair production, where a pump (P) photon \nvanishes and a signal (S) and idler (I) photon appear in its place.   \nIn quantum optics entangled photons are  generated  when the \nelectric field E from a laser  beam  is sufficiently intense to force the \ndielectric susceptibility  into a nonlinear region. The dielectric \nsusceptibility E is defined by Eq.  (2) where  P is the material \npolarization  [14].  \n𝑷=𝜀0(𝝌(1)E𝑬+𝝌(2)E:𝑬2+𝝌(3)E∷𝑬3+⋯+𝝌(n)E:…:𝑬n) (2) \nAt low intensities only the linear susceptibility 𝜒(1)𝐸 is active, but \nat higher fields t he nth order nonlinear susceptibilities  𝜒(𝑛≥2)𝐸 become \nactive , this happening more readily in non-centrosymmetric crystal s. \nThe unit cells of these crystals can have permanent dipole electric \nmoments  and these may spontaneously align to form a  domain.  These \ntypes of materials are referred to as pyro - or ferro -electric  and are used \nto generate visible entangled photons . The most commonly used \nmaterial for this a t present is beta barium borate (BBO).  \nFerromagnetic material s are the magnetic equivalent of \nferroelectric ones, as magnetic moments in the unit cells \nspontaneously align  to form a domain . Upon the application of a \nmagnetizing  intensity  H, a magnetization  M results, the relationship \nbeing highly nonlinear and hysteretic, and described by  \n𝐌=𝛘(1)𝑀𝐇+𝛘(2)𝑀:𝐇2+𝛘(3)𝑀∷𝐇3+⋯+𝛘(𝑛)𝑀:…:𝐇𝑛 (3) \nwhere 𝜒(1)𝑀 is the linear magnetic  susceptibility and  𝜒(𝑛≥2)𝑀 are the \nhigher order nonlinear susceptibilities  [15]. These materials are not \ncurrently recognized  as being potential sources of entangled photons.   \nEntangled photon generation takes place by a process known as \nspontaneous parametric fluorescence or spontaneous parametric \ndown -conversion (SPDC). It is the second order (n=2) nonlinear \nsusceptibility which is responsible fo r this three -wave interaction that \ncreates a pair of entangled photons from a pump photon [16], the first \nexperimental proof of this being presented in [17].  \nThe semi -classical explanation for this process is that the nonlinear \nmedium generates a photon that has a difference frequency , this being \ndefined as the difference between the pump photon and a vacuum \nphoton from the quantum zero -point field energy. For the difference \nfrequenc y photons to be generated it is necessary for the second order \nnonlinear susceptibility 𝜒(2)𝐸 to be relatively large, typically in the region of a few pm/Volt for many ferro electric s when excited by laser \nradiation.   \n2.3. Energy and momentum conservation in entangled photon \ngeneration  \nConservation of energy dictates that th e vacuum photon must have \nan energy less than that of the pump photon. The energy from the \npump photon then passes to that of the signal and idler photons. Given \nthe Einstei n energy relation  𝐸=ℏ𝜔, conservation of energy relates \nthe frequencies of the pump P, signal S and idler I as  \n𝜔𝑃=𝜔𝑆+𝜔𝐼. (4) \nFrom photon ballistics , the process must also satisfy the \nconservation of momentum. Given the de  Broglie momentum \nrelation  𝐩=ℏ𝐤, where k is the direction propagation vector, \nconservation of momentum relates the propagation vectors of the \npump  kP, signal kS and idler KI as,  \n𝐤P=𝐤S+𝐤I, (5) \nand in the  vector triangle of Fig. 2, the magnitude of the propagation \nwave vector is related to the wavelength  of the photon as | k|=2/.  \n \nFig. 2. The momentum ( k) conservation vector triangle is shown for entangled \nphoton pair creation, where  is the angle between the pump and signal \npropagation vector.  \nSince the magnitude of the wave vector is related to its phase  𝜙, \nthrough the wave description  𝑒𝑥𝑝[𝑗(𝐤.𝒓− 𝜔𝑡 + 𝜙)], the \nconservation of momentum indicates the phases of the pump, signal \nand idler at the point of pair creation are related as   \nϕ𝑃=ϕ𝑆+ϕ𝐼. (6) \nThis phase relationship enables coherent integration to recover the \nsignature of the signal and idler photons in the radio Bell Test from \nthe high level of thermal noise characteristic of the Rayleigh -Jeans \nregion.  Without this basic and very fundamental phase relationship, \nno coherent integration would be poss ible. \n2.4. Phase matching to maximize the flux of entangled photon pairs  \nMost materials display properties of normal dispersion , where \nrefractive index rises with frequency. The refractive index usually \nvaries with the direction of propagation and it is often not possible to \nsatisfy completely the conservation of momentum of Eq. (5) depicted \nin Fig. 2. In these  cases,  there is a mismatch in the wave -vector k and \nthis is quantified as  \n∆𝐤=𝐤𝑃−𝐤𝑆−𝐤𝐼 . (7) \nSince the magnitude of the k-vector can be written 𝜔𝐧𝑐⁄, where n \nis the refractive index in the direction of propagation , the above k -\nvector mismatch can be written as   \nΔ𝐤=(𝜔𝑃𝐧𝑃−𝜔𝑆𝐧𝑆−𝜔𝐼𝐧𝐼)1𝑐⁄ . (8) \nSince matching the wave vectors is associated with the \nwavelengths of the pump, signal and idler radiation, achieving \nmomentum conservation can be achieve d by tailoring the refractive \nindex and angles of propagation. This is referred to as phase matching. \nIn cases where phase matching is achieved at an angle =0, it is \nreferred to as collinear phase matching . The oth er cases , where   0, \nare referred to as non -collinear phase matching.  \nIn the 1980’s attention to e ffective phase matching increased th e \nefficiency of entangled photon generation by several orders of \nmagnitude . As a result, f or a single crystal the probability of detecting \nan entangled pair per pump photon is now in the region of 3x10-11 [18]. \nPeriodic poling in lithium niobate has increased th is probability to \n4x10-6 per pump photon [19]. Other methods of generating entangled \nphotons in the visible band have been SPDC in an AlGaAs \nsemiconductor waveguide [20]. In the microwave band operating at \ncryogenic temperatures an entangled photon power of -110 dBm has \nbeen  generated by pumping a Josephson junction (acting as a \nkS, S \nkP, P \nkI, I \nTime  \nkP \nkI \nkS \n 3 \n parametric amplifier) with -75.6 dBm, indicating efficiency of the \norder of -34.4 dB (3.6x10-4) [9].   \nTwo relatively efficient interactions for generating entangled \nphotons  are referred to as  Type I and Type II [14]. In a Type I \ninteraction each photon of the entangled pair has the same (linear) \npolarization , which  is orthogonal to the pump polarization. This is \nreferred to as an ‘e -oo’ or an ‘o -ee’ type interaction, where ‘o’ refers \nto the ordinary or o -mode polarization (E -vector perpendicular to the \ncrystal incidence plane) and ‘e’ ref ers to the extraordinary or e -mode \npolarization (E -vector parallel to the crystal incidence plane) [14], the \ncrystal incidence plane being defined as that which contains the wave \npropagation vector k and the crystal optic axis.  In the designation ‘e -\noo’, the first letter ‘e’ refers to the polarization of the pump and the \nother two letters refer to the polarizations of the signal and idler \nphotons.  \nIn a Type II interaction,  the polarizations of each photon of a pair \nare orthogonal to one another, with one of these polarizations being \nidentical to that of the pump polarization , these interactions designated \nas ‘e-oe’, ‘e -eo’, ‘o -oe’ or ‘o -eo’.  \n2.5. Ferrite s as a nonlinear medium for entangled photon generation  \nThe potential of ferrites as a nonlinear medium for the generation \nof entangled photons can be  appreciated from  the nonlinear \ndependen ce of the magnetization M on the static magnetizing intensity \nH shown in Fig. 3. Ferrites , a well understood class of materials [21], \n[22], can have their  constitutive magnetic properties (eg. coercivity \nHC, remanence M R, saturation magnetization M S, susceptibility ) \nfinely tuned by appropriate doping and prepared in single crystal or \nthin film form. This is illustrated in Fig. 3 by the hysteresis loops of \nHo doped YIG, described by Y 3-xHo xFe5O12, where x=0 and 1.5  [23], \nthe curves being  modelled heuristically using a modified form of the \nLangevin function L(x), where 𝑀=𝑀𝑠(coth 𝜇0𝑎(𝐻−𝐻𝑐)−1/\n𝜇0𝑎(𝐻−𝐻𝑐)) and a is a fitting parameter , from data presented in [23].    \n \nFig. 3. The nonlinear magnetic susceptibility (M/H) evident from the hysteresis \nloop shown here is typical of a pure and Ho doped (x=1.5) YIG ferrite  \nillustrating the high field (1T) saturation magnetization (M S x=1.5  0.804 T (i.e. \n640 kA/m)) (remanent magnetization (M R x=1.5 =561 A/m) and coercivity (H C \nx=1.5 = 0.013 T).  \nAt the atomic level in ferrites , the Heisenberg uncertainty principle \nprevents th e net atomic or domain magnetic moment (related to \nangular momentum) aligning with the magnetizing intensity vector H, \nas angular momentum and its direction don’t commute. However, the \ntorque to align induces a precessional motion of the magnetic moment \naround the field direction at the (classical) Larmor precession \nfrequency [24] of  \nwhere  \nis the electron spin gyro magnetic ratio and e and me are the electron \ncharge and mass respectively, and 𝑔𝑒 is the electron Landé g -factor, \nhaving a value ~2.002319 . The Larmor precession frequency is \nequivalent to the frequency of the photons absorbed in the quantum \nmechanical representation as magnetic moments flip in an applied field during resonant absorpti on between spin up and spin down states  \n[25].  \nAs the conduction bands in ferrites contain no free electrons the ir \nresistivity is high , typically 107 m, so the  material is transparent to \nmicrowaves and millimeter waves. This means the ferrite interaction \nof radiation is governed b y the complex refractive index  given  [26] by  \nwhere r (=r‘- jr”) and r (=r‘ - jr”)) are the complex relative \npermittivity and permeability of the material, these being related to the \nfirst order (linear) susceptibility as r  = 1 + 𝜒(1)𝐸. For ferrites the \ncomplex relative permittivity ( r‘, jr”) is [27] isotropic and takes a \nscalar value, typically (12 -22, 0.05).  The permeability of the ferrite is \ndictated by the Polder Tensor [24] and the variation of this with \nfrequency over the millimeter wave band determines largely how \nthese materials behave and have been exploited as devices in a large \ncommercial market.  \nThe ferrite response to electromagnetic  radiation  falls into two \northogo nal propagation modes, one couple d strongly to the magnetic \ndipole s and the other weakly coupled , the strength being dependent on \nthe angle of propagation in the medium . For longitudinal propagation \n(along the static magnetic field lines)  the strongly coupled mode is \nright -hand circularly (RHC) polarized and the weakly coupled mode \nis left -hand circularly (LHC) polarized. For transverse propagation (at \nright angles to the static field lines) the strongly couple d mode  is the \ne-mode (the H -field of the wave being perpendicular to the static \nmagnetic field direction ), with the weakly coupled mode being the o -\nmode  (having the wave H-field parallel to the static field direction ). \nFor the strong mode the ferromagnetic resonance shifts from 0=L \nfor longitudinal propagation to (0(0+M)) where M=-SM, for \ntransverse propagation. At intermediate propagation angles the strong \nand weak modes are elliptically polarized.   \nGarnets are a class of ferrites of particular interest as they have \nvery low damping and hence little crystal lattice absor ption of the \nwave energy . This means the ferromagnetic resonance line is very \nnarrow and in yttrium iron garnet (YIG) it has a width [28] of ~1 MH z, \nwhich gives i t a highly non-linear magnetic susceptibility . \nA plot of the complex refractive index is shown in Fig. 4 for \ntransverse propagation  in an infinite solid of YIG for a Larmor \nprecession frequency (associated with the internal field H) o f 15 GHz \nand a  Larmor precession frequency (associated with the internal \nmagnetization M/2 ) of 6.9 GHz . The dielectric constant (’) is \nassumed to be 14.7, the dielectric loss tan ( ”/’) to be 0.0002 [29] and \nthe damping factor  to be (~1/Q) of 0.00007 (typical for YIG) [30]. \nFor the strongly couple d mode there is a (negative permeability) non-\npropagating region extending from the resonant frequency to  the cut -\noff frequency, 0+M, above which th e permeability becomes \npositive.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n -800-4000400800\n-0.4 -0.2 0 0.2 0.4M (kA/m)\nB(T)= 0HMS\nMR\n0HY1.5Ho1.5Fe5O12\nY3Fe5O12\n𝜔𝐿=−𝛾𝑆𝐻 ,  (9) \n𝛾𝑆=−𝑔𝑒𝜇0𝑒\n2𝑚𝑒 .     (10) 𝑛=√𝜀𝑟𝜇𝑟 , (11) 4 \n      \nFig. 4. The real (a) and imaginary (b) parts of the refractive index for the strong and weak transverse propagation modes in YIG.     \nIn general the refractive index is a function of the propagation \nangle in the YIG and its quantitative behavior is well known from the \nsolution to Eq. (11), as illustrated  to good effect  in [24]. From this \nequation two orthogonal elliptical polarization modes arise (which \nbecome linearly (circularly) polarized for propagation across (along) \nthe magnetic field lines. This behavior is similar to the  angular \nvariatio n in the  refractive index  for the (extra)ordinary mode in \nelectro -optic (ferroelectric) crystals , as can be appreciated in the \nrefractive index polar plots in [14]. There fore there is an opportunity \nto use the angular variation of refractive index in YIG for phase \nmatching  (by minimizing the momentum mismatch k from Eq. (8)) \nto maximiz e the  intensity of the entangled photons  in accordance with \nEq. (17). It may also be possible to increase the nonlinear interaction \nlength in YIG (whilst retaining good phase matching), to increase the \nefficiency of entangled photon generation , by periodically poling YIG \n(PPYIG), as is done in periodically polled lithium niob ate (PPLN)  \n[31].  \n3. Results  \nGiven the above mechanism for creating entangled photons, \nestimations are now made of the likely fluxes of entangled photons \nfrom YIG in the millimeter -wave band  using moderate pump powers \nthat are readily available. This will determine the practicability of the \nproposed Bell Test in this spectral band and therefore the likelihood \nthat the system could be used as the basis for a sensor.   \n3.1. The second order nonlinear magnetic susceptibility of YIG  \nOptimization of the second order nonlinear susceptibility of \nferrites has led to these devices being used as frequency doublers and \nparametric amplifiers  [15]. The second order nonlinear susceptibility \nof a frequency doubling ferrite is given  [32] as \nwhere M0 is the YIG static magnetization , L is the Larmor angular \nprecession frequency associated with the externally applied static field \nH0 (given by -SH0) and H is the ferromagnetic resonance linewidth \nin A/m.  It can be seen  from this equation  that a higher second order \nnonlinear susceptibility results from a narrow resonance width. Small \nresonance widths of the order of 0.36 Oersted (~28 A/m) can be \nachieved using a YIG crystal [32]. For a garnet with a static \nmagne tization of 238 kA (~0.3 Tesla) pumped at  a frequency of 20 \nGHz  (to resonate with the externally applied field)  the second order \nnonlinear susceptibility from  Eq. (12) is ~0.015 m/A.  \nFor frequency doubling a nonlinear response is achieved  [32] by \npumping in the transverse mode , with the B -field of the pump \nradiation being perpendicular to the static field. The frequency \ndoubled component is also taken off in the transverse propagation \nconfiguration, with the B -field parallel to the static field.  \nFor parametric amplification in YIG a number of modes are [15], \n[33] possible . These are analogous to the different types of optical \nentangled photon creation: just how these are excited is explained in \nthe following section on the circuits for millimeter wave entangled \nphoton creation.  The mode with the highest gain is the magneto -static \nmode  (or the spin-wave or three -magnon mode ) [34]. Here a linearly \npolarized pump wave at twice the res onance frequency propagates \ntransversely across the static magnetic field lines with its wave H -field \nparallel to the static field direction, almost an exact reverse process of \nthe above frequency doubling mechanism . This system creates \nparametric gain in ferrites at pump power levels of 0.5 W, generating \nsignal and idler photons that are linearly polarized with their H -fields perpendicular to the static field directions. Considering the direction \nof magnetization in a ferrite to be analogous to the optic a xis in a \nferroelectric , the mode is the magnetic equivalent of the Type I \ninteraction in ferroelectrics .  \nA further parametric amplification mode is the electromagnetic \nmode. Here a linearly polarized pump wave at the ferromagnetic \nresonance frequency propagates transversely across the static \nmagnetic field lines with its H -field perpendicular to the static fi eld \ndirection [35]. Signal and idler photons are created having orthogonal \nlinear polarizations, one having the H -field perpendicular and the \nother with the H -field parallel to the static  field direction. This mode \nis the magnet ic equivalent of the Type II entangled photon creation in \nferroelectrics .  \nMiller’s rule [36] is an empirical relationship from the field of \nnonlinear optics for non -centrosymmetric dielectrics stat ing that the \nratio of the second order nonlinear susceptibility to the product of the \nfirst order susceptibilities at the three frequencies (𝜔1+𝜔2,𝜔1,𝜔2) \nof a three -wave interaction, namely  \nis constant .  \nBoyd shows  [31] that the mathematical form of Miller’s rule is a \nconsequence of the anharmonic  motion of the electron  in the E -field \nof a passing electromagnetic wave, this anharmonicity being a direct \nresult of the local crystal E -field in the non -centrosymmetric material. \nHowever, there is only finite second order nonlinear electrical \nsusceptibility  (2)E, if the crystal is non -centrosymmetric. This suggests \nthat if YIG has a finite magnetic susceptibility (2)M, it should also \nfollow Mill er’s rule.   \nAssuming Miller’s rule is followed for the magnetic \nsusceptibilities, this suggests  that the nonlinear susceptibility  of Eq. \n(12) is very close to that for difference  frequency generation , provided \nthe first order susceptibilities are similar. The value from Eq. (12) of \n0.0015 m/A may therefore be used in the estimation of the nonlinear \nsusceptibility for the difference frequency generation in spontaneous \nparametric down -conversion.   \n3.2. The power of entangled photons  \nIn the process of entangled photon generation,  the pump photon \ninteracts with a vacuum photon to annihilate itself and create a pair of \nentangled photons. A high pump power parametrically amplifies  [37] \nthe vacuum photons. If the field-gain of the nonlinear medium is \ndesignated   and its length l, the spectral radiance of the entangled \nphoton flux (in W/m2/sr/(radians /s) is given [38] by  \n𝐼𝜔Ω(𝐤)=𝐼𝜔Ω𝑉𝐴𝑉𝑠𝑖𝑛ℎ2[(𝛾2−|Δ𝐤|2/4)1/2𝑙]\n(1−|Δ𝐤|2/4𝛾2),  (14) \nwhere k is the wavevector  (phase) mismatch  of Eq.  (8), and 𝐼𝜔Ω𝑉𝐴𝑉 is \nthe spectral radiance of the vacuum photon fluctuations, given by  \n𝐼𝜔Ω𝑉𝐴𝑉=ℏ𝜔3𝑛𝑆2\n8𝜋3𝑐2 , (15) \nwhere nS is the medium refractive index.  The vacuum photon \nfluctuations correspond to a single  photon in each mode.  The field -\ngain from  [38] is given as  \n𝜒(2)𝑀=𝛾𝑆𝑀0\n𝜔𝐿Δ𝐻 , (12) 𝜒(2)𝐸(𝜔1+𝜔2,𝜔1,𝜔2)\n𝜒(1)𝐸(𝜔1+𝜔2)𝜒(1)𝐸(𝜔1)𝜒(1)𝐸(𝜔2) (13) \n𝛾𝐸(𝑚−1)=(8𝜔𝑆𝜔𝐼𝐼𝑃𝜇0\n𝑐𝑛𝑃)1\n2\n𝜋𝜒(2)𝐸  , (16) (a) (b) 5 \n where 𝑛𝑃 is the medium refractive index for the pump radiation, which \nhas an intensity (or irradiance) in W/m2 given as 𝐼𝑃. Taking a pump \npower of 1 04 W/m2 (1 W/cm2), a nonlinear electrical susceptibility of \n5x10-12 m/V and a refractive index of 2.2, the field gain of the medium \nis ~ 1.2x10-5 per meter  for a signal and idler frequency of 10 GHz . For \na low gain medium in which  << k, Eq. (14) becomes  \nand if perfect phase matching is satisfied the entangled photon spectral \nradiance in the  nonlinear dielectric medium becomes   \nThis indicates that for the above pump power the spectral \nirradiance of the entangled photon flux would be 6.88 x 10-32 \nWatts/sr/m2/(radians/s)  for an interaction length of 10 cm. If it is \nassumed the bandwidth is 5 GHz and emission is measured over  \nsteradians , the entangled power flux is 6.79 x 10-25 Watts over an area \nof one square centimeter . \nMaking a similar analysis  to [38] for entangled photon generation \nfrom a magnetic material the field -gain is given by  \nwhere  𝜒(2)𝑀 is the 2nd order nonlinear magnetic susceptibility  and 𝑍𝑃 \nis the medium impedance to the pump radiation  𝑍𝑃=√𝜇𝑟𝜇0𝜀𝑟𝜀0 ⁄ . \nPumping YIG possessing  a nonlinear magnetic susceptibility of 0.015 \nm/A and a refractive index of 3.8 with a pump power of 104 W/m2 (1 \nW/cm2) at 20 GHz  results in a field gain from Eq. (19) of 630 per \nmeter . This is a considerably higher gain than that of a nonlinear \ndielectric. Assuming that the conditions of phase matching are \nsatisfied (ie k =0), then the entangled photon spectral radiance in the \nnonlinear magnetic medium from Eq. (14) becomes  \nGiven the estimated magnetic nonlinear susceptibility of 0.015 \nm/A from above  and a refractive index of 3.8  for YIG, the entangled \nphoton spectral irradiance becomes  1.80x10-19 Watts/sr/m2/(radians/s) \nfor an interaction length of 3 mm  at signal and idler frequencies of 10 \nGHz . Assuming a bandwidth of 10 GHz and that emission is collected \nover  steradians , the entangled photon flux f or a pump power of  \n5 W/cm2 is 3.56  10-12 Watts/cm2, which is approximately 0.5  \n1012 photons per second from an area of 1  cm2. This is also \nconsiderably higher than the above counterpart for a nonlinear \ndielectric medium.    YIG ferrites have many potential advantages when it comes to the \ngeneration of entangled states. YIG, as a bulk crystal, regular prism or \nthin film, can be immersed in an applied static bias field, H 0, close  to \nthe resonance radio frequency field, mounted in either a waveguide \n[39] or a very high quality factor (Q~5000) resonant cavity [40]. The \nhigh Q-factor raises  the magnetizing intensity of the applied radiation, \nincreasing  the probability of generating entangled photons.  \n4. Discussion  \nGiven the results on the  fluxes of millimeter  wave entangled \nphotons, a discussion is opened here on architectures for their \ngeneration and a homodyne receiver system for their detection.   \n4.1. Circuits for Type I & Type II ferrite source s of entangled photons  \nGiven that the  cross -sectional area of the YIG crystal  is envisaged \nto be in the region of 1 cm2, the pump beam would need to be either \nfocused from a waveguide or transmitted direct to the sample using a \ntransmission line. Because the sample cross -sectional  dimension is of \nthe order of the wavelength of the signal and idler photons, these will \nemerge from the YIG crystal into a wide angled cone  due to \ndiffraction . This cone would constitute  a single mode of the receiver \nsystem  so signal and idler may be captured by the receiver regardless \nof energy.  This represents an essential difference between what is \nproposed here and the quantum o ptics experiments in entanglement.    \nA circuit using the magneto -static mode of a parametric amplifier  \nto create entangled photons  is illustrated in Fig. 5. This  exploits the \nType I interaction  and is similar to  that in [41], but differs in that it \nuses a r esonant circuit . The system  comprises a pump generating \nlinearly polarized  radiation orientated at an angle of 45, which splits \ninto two orthogonal horizontally and vertically polarized  beams using \na vertically wired grid polarizer  [42]. The horizontally polarized  (E-\nfield) beam enters the YIG-1 sample with the radio -frequency H -field \nparallel to the static H -field in the ferrite. Signal and idler photons \ngenerated with  H-fields perpendicular to the static H -field exit via a \nreflection from the second vertically  wired grid. The vertically \npolarized  (E-field) beam enters the YIG-2 sample with the H -field \nperpendicular to the static H -field stimulating signal and idler photons \nwhich exit by transmission through the second wire grid.  \nThe unused horizontal and vertically polarized  pump radiation exit \nvia the second wire grid polarizer  with a relative phase (epoch) angle \ndefined by the difference in phase delay s between  the two arms of the \nsystem. An adjustable wave -plate then provides sufficient delay \nbetween these polarizations  to generate linear polarization  orientated \nat -45. This maximizes  the amount of radiation that reenter s the \ncavity by reflection from  the 45 -angled polarizer . The polarization  \nrotator is then adjusted to bring the orientation around to +45 , in order \nthat equal fluxes  of horizontal and vertical polarizations  can be \npresented to the two YIG ferrite crystals. The entangled photons exit \nthrough  the vertical wire grid polarizing beam splitter  and are then \nseparated in to two channels by the following beam splitter so they may \nenter the homodyne interferometer of Fig. 8.  \n \nFig. 5. Type I interaction  in a resonant circuit using the magneto -static parametric amplifier mode in a YIG ferrite pumping at twice the ferromagnetic resonance \nfrequency generat es signal and idler photons about the ferromagnetic resonance frequency  0 and the entangled state (|V S>1|Vi>1+ej|HS>2|Hi>2)/2. The orientation s \nof linear polarizations  are indicated as +45, -45, H and V .  \nThe beam splitter of Fig. 5 from  which the signal and idler emerge \ncan be realized in practice in several different forms. 1) It may be a \nnon-polarizing 50/50 beam splitter , similar to [43]. 2) It may be a \nhigh/low pass filter, passing photons of frequency > 𝜔𝑃/2 into one \nchannel and those <  𝜔𝑃/2 into the other channel. 3) It may be a \npolarizing grid orientated at 45 . The latter two forms  would however \nmake a preselection of photons on the basis of their momentum  or \npolarization and thereby may remove one of the degrees of \nentanglement. However, since signal and idler are initially entangled in both momentum and polarization, removing one of these may still \nenable a Bell Test to be constructed.   \nA circuit using the electromagnetic mode of a parametric \namplifier, which exploits the Type II interaction  for the creation of \nentangled photons , is shown in Fig. 6. Pumping the YIG at the \nferromagnetic resonance transversely with the H -field perpendicular \nto the static magnetic field direction , the signal and idler can be taken \noff transversely, one with the wave H -field perpendicular to the static \nfield and the other parallel to the static field. The signal and idler are 𝐼𝜔Ω(k)=𝐼𝜔Ω𝑉𝐴𝑉𝛾2𝑙2𝑠𝑖𝑛𝑐2[(|Δ𝐤|)𝑙/2] (17) \n𝐼𝜔Ω(𝐤)=ℏ𝜇0𝜔𝑆4𝜔𝐼𝐼𝑃𝑛𝑆2𝜒(2)𝐸2𝑙2\n𝜋𝑐3𝑛𝑃 . (18) \n𝛾𝑀(𝑚−1)=(8𝜔𝑆𝜔𝐼𝐼𝑃𝜇0\n𝑐𝑛𝑃)1\n2𝜋𝜒(2)𝑀\n𝑍𝑃 , (19) \n𝐼𝜔Ω(𝐤)=ℏ𝜔3𝑛𝑆2\n8𝜋3𝑐2 sinh2(𝛾𝑙) . (20) \nHP \nUnused pump having polarization epoch angle  \nVertical wire \ngrid polarizing \nbeam splitter  \n|H s >2 ,|H i>2 \n VP \n|Vs>1 ,|V i>1 \n+45 \n Linear  \n +45 \nYIG-1 \nYIG-2 \n45 angled wire \ngrid polarizer  \n \nVertical wire \ngrid polariser  \nWave -plate  \n(variable)  \nPump P =20 \n-45 \nPolarization \nrotator  \nBeam \nsplitter  \nOutputs  \n|Vs>1,|H s>2 \n|Vi>1,|H i>2 6 \n taken off using a dichroic mirror fabricated  from a mesh grid  [42]. The \nunused radiation gets reflected by the mesh filter and as the pump is \nhorizontally polarized  it passes through a half -wave plate to become \nvertically polarized , enabling  it to re-enter the resonance cavity by \nreflection off the gird. The polarization  rotator rotate s the plane of polarization  so that horizontally polarized  radiation is incident on the \nYIG.  \nOn exit via the dichroic mirror the entangled photons pass through \na beam splitter , similar to that in the Type I system above .      \n \nFig. 6. Type II interaction in a resonant circuit using the electromagnet ic mode parametric amplifier in YIG pumping at the ferromagnetic resonance frequency \ngenerating signal and idler photons about half the ferromagnetic resonance frequency , creating the entangled state (|Hs>1|Vi>1+ej|Vs>2|Hi>2)/2. The orientations of \nthe linear polarizations  throughout the system are indicated as H and V.  \nA non-resonant circuit to stimulate  Type II interactions is adopted \nfrom the Sagnac interferometer of [44] and depicted in Fig. 7. Here  \nlinearly polarized pump radiation orientated at 45  encounters a \npolarizing beam -splitter constructed from a vertical grid. A horizontal \npolarization component of the pump passes through the grid and is \nconverted to vertical polarization by the half wave -plate, before \nstimulating the YIG to generate entangled photons in mode 1 . A \nvertical polarization component is reflected from the grid and \nstimulates the YIG crystal directly from the other direction  to create \nentangled photons in mode 2 . The Type II interactions create \northogonal signal  and idler photons in these two modes which exit via the polarizing beam -splitter and the dichroic mirror. The dichroic \nmirror, manufactured from mesh grids  [42], is designed to pass the \npump radiation , but reflect the lower frequency signal and idler \nradiations. The outputs from the circuit pass directly into the channels \nof the homodyne interferometer of Fig. 8. The advantage of this \narchitecture is that the photons remain entangled in both momentum \nand polarization. The unused pump radiation from the Sagnac system \nexits in the direction of the pump, but  this could re -enter the system \nby a reconfiguration using addit ional  resonant circuit components , not \nshown here .   \n \nFig. 7. A circuit based on a Sagnac interferometer [44] using a Type II interaction of the electromagnetic mode in YIG to create the entangled state ( |Vi>1|Hs>1+ \nej|Hs>2|Vs>2)2. \n4.2. The homodyne interferometer  Bell Test   \nA single -channel Bell Test architecture, the type initially proposed \nin [3], but exploiting the attributes of millimeter wave receivers for an \nambient temperature Bell Test is shown in Fig. 8. It creates  the entangled photons by pumping a nonlinear YIG crystal , in the circuits \nof Fig. 5 (for entangled photons created by a Type I interaction ) and \nFig. 6 or Fig. 7 (for entangled photons created by a  Type II \ninteraction ), and then separates the pair  into two channels .  \n|Hs>1, |Vi>1 \n|Vs>2, |H i>2 \nUnused pump horizontal polarization H P \nHP \n HP \n Linear  \nHP \nDichroic Mirror \n(mesh grid filter)  \nYIG \nVertical grid  \nwire polari zer \n \nHalf-wave \nplate  \nHP \nPump P =0 \nVP \n<P \n~P \nPolarisation \nrotator  \nBeam \nsplitter  \nOutputs  \n|H s>1, |V s>2  \n|Vi>1|Hi>2 \nVP \n|H s>2, |V i>2 \n Polarizing \nbeam -splitter \n(vertical grid)  \n|H s>1, |V i>1 \nOutputs  \nVP \nHP \nYIG \nDichroic mirror \n(mesh grid filter)  \n45 linear  \nPump P =0 \nVP \nHalf wave -\nplate \nVP \n|H s>1, |V s>2 \n|Vs>2, |H i>2 \n|Vi>1, |HS>2 7 \n  \nFig. 8. The homodyne interferometer single -channel Bell Test is proposed to overcome the noise generated by thermal photons when operating in the Rayleigh -Jeans \nregime ( hf<kT ) [10], where 2 E is the angular frequency difference between the signal and idler photons and 2 SI is the assumed angular bandwidth of the signal \nand idler photons.  The box labelled ‘entangled photon state creation’ represents  any of the circuits from Fig. 5, Fig. 6 or Fig. 7. \nWave -plates are included in the circuit of Fig. 8 so that all Bell \nStates can be created , as indicated in [45]. Since the system is an \ninterferometer it is sensitive to phase, so adjustments to the phase can \nbe made by the variable phase shifter in one of the arms of the \ninterferometer. The effects of birefringence in components of the \nsystem could potentially lea d to a reduced level of entanglement, but \nthese may be compensated for by extra components in the circuity of \nFig. 8. Since  signal and idler photons need to arrive at more or less the \nsame time at the receivers , the transit time of photons could be \nbalanced by physical  movement or by insertion of low-loss refractive \nindex material into one of the arms of the interferometer , to increase \nits effective optical path length . The above adjustments would be \nmade to maximize the measured entanglement signature, similar to \nthat in the optical band described in  [46].   \nThe radiation in each channel is amplified to drive mixer -1 into its  \nnonlinear regime. The mixer output contains the multiplication of \nsignal and idler fields together with relatively large amount s of \nthermal noise on a 20 GHz carrier frequency . The phases  of the signal \nand idler are summed by this mixing process. The output from mixer -\n1 is then mixed with the original pump radiation at 20 GHz in mixer -\n2, thereby shifting the combined signal and idler signature down to \nbaseband.   \nThe mixer -2 output is lowpass filtered and sampled in a sine and \ncosine receiver to record a complex amplitude , which is then \ncoherently integrated  to recover the stationary complex value for the \nsignature of the signal and idler photon pairs  from the noise . The sine \nand cosine receiver is also referred to by some as an I,Q or a quadrature \nreceiver. It  integrate s to a stationary complex value because the p hase \nnoise n average s to zero, leaving t he phase relation of Eq. (6) to \nprevail.  The square of this stationary value is equivalent to the number \nof coincidence count s, usually designated as N, in the optical Bell Test , \nas indicated by Eq. (21).   \nThe circuit in Fig. 8 is the radio receiver equivalent of a single \nchannel Bell Test. The Bell Test test -statistic, usually designated as S, \nis formed by the usual combination of measurements of N at the Bell \nTest angles [3]. Measurements made at the Bell Test angles maximize \nthe difference between the classical and quantum results, this giving \nthe highest chance of demonstrating violation of the Bell inequality.  \nA twin -channel Bell Test architecture, also proposed in [3], but \nsuitable for ambient temperature operation in the millimeter wave \nband , is constructed b y replacing the polarization  rotators with \nrotating polarizing  beam splitter s, which are then  followed by two \nfurther receiver channels . The twin -channel Bell Test  architecture  has \nsmall er systematic uncertainties than the single -channel  test. The \nsingle -channel Bell Test was initially proposed as a more practicable instrument, but with more sophi sticat ed technology the twin -channel \nmeasurement with its superior performance is now the preferred  \noption , forming the basis for almost all Bell Tests today . As with the \nsingle -channel test, measurements are made at the Bell Test angle s. In \nthis case four  effective coincidence counts are formed  from Eq. (21), \nwhich are then used to determine the so -called quantum correlation, \nusually designated by E for each of the Bell Test angles, from which \nthe Bell Test test -statistic S is then formed  as detailed in  [12] and [47].     \n4.3. System s ignal -to-noise ratio s and integration time s     \nOne of the main sources of noise  comes from the amplifiers and \nthe thermal ( black body ) radiation entering the receiver from the \nambient temperature surroundings. This noise  power  in each channel , \nreferred to the amplifier input is just FkT 0B, where F is the amplifier \nnoise factor, T0 is ambient temperature ~290 K and B is the bandwidth \nof the system. For an amplifier noise factor of 2.5 and a 10 GHz \nbandwidth the noise is ~ 71 pW.  \nThe entangled photon power at the input to the  radio frequency  \namplifier is  𝑃𝑆/𝐿, where L is the loss in the entangled photon flux \narising from stimulated emission and absorption. Since the entangled \nphoton flux from the YIG has been estimate d at 3.56  10-12 Watts  and \nall front -end components in front of the waveguide horn can be \ndesigned to be low loss , it is reasonable to a ssume  L would be at the \nmost  a factor of two , which gives the entangled power flux at the \namplifier input as 1.78  10-12 Watt s, for 5 Watt s of pump power  over \none square centimeter . It is therefore also assumed that the loss of \nsignal due to absorption and s timulated emission would still permit \nsufficient information from the signal and idler to be coherently \nintegrate d for the Bell Test .  \nAs the YIG parametric amplification will also amplify thermally \ngenerated photons which have random phases, these will constitute \nextra noise in the system, having a power level   𝑛𝑃𝑆/𝐿 where 𝑛 is the \nmean photon number of Eq.(1). Since this is estimated at  604 at 10 \nGHz, the power flux from parametrically amplified thermal photons \nis 604  1.78  10-12 = 1.08  10-9 Watt s.  \nSummarizing  the above , the s ignal -to-noise on each of the signal \nand idler arms of the interferometer on the input to mixer -1 is given \nby  \nwhich has a numerical value of 1.55  10-3. Given that mixer one \nmultiplies the signal and idler with their independent noises together, \nthe signal -to-noise ratio output from mixer -1 is  \n 𝑁=𝐴2⇐|𝐴𝑒𝑗(𝜙𝑆+𝜙𝐼)|2⇐|<𝐴𝑛𝑒𝑗(𝜙𝑆+𝜙𝐼+𝜙𝑛)>|2 (21) \n𝑆𝑁𝑅1=𝑃𝑠/𝐿\n𝑛𝑃𝑆/𝐿+𝐹𝑘𝑇0𝐵  , (22) \nSum frequency \nsection P SI \n \nSignal  \n\nS = \nP/2±\nE \n \n  P \n \nP \nMixer -1 \n(taking sum \nfrequencies)  \nBand -pass filter:  \n \nP/2 (\nE +\nSI) \nAmplifier  \nAmplifier  \nPhase shifter \n(variable)  \nPump \noscillator  \nAttenuator \n(variable)  \n PSI \nBand -pass \nfilter  \nBaseband section  \nDC SI \n \nS \nSI  \n \nP \n \nBand -pass filter  \n \nP/2 (\nE +\nSI) \n \nI \nSI  \nIdler    \n \nI = \nP /2 ∓\nE \n \nWave -plate  \nRotatable \npolariser  \nrotat or \nRotatable \npolariser  \nLow-pass filter \n (DC to SI) \nADC sampling clock, sample \nfrequency  SI   \nA \nD \nRadio frequency (RF) section: DC to P \nIntegration \nfollowed by \ncomplex square  \nWave -plate  \nPolarisation \nselector  \nPolarisation \nselector  \nEntangled photon \nstate creation  \nN=||2 \n  \nMixer -2  8 \n which has  a numerical value of 2.41  10-6. \nThe final stage of the receiver mixes the combined and signal and \nidler signatures  (together with noise)  down to a base -band frequency \nby mixing with the pump radiation  in mixer -2. Coherent integration of \nthe output from mixer -2 brings an  improvement in the signal -to-noise \nratio proportional to the root of the number of samples. Sampling at \nthe Nyquist rate of 2B , the signal -to-noise ratio output after an \nintegration time of t INT is \nIt can be seen from this that the highest signal -to-noise ratios will \nbe achieved by using  the largest possible radiation bandwidth. \nInserting numerical values into this equation gives a signal -to-noise \nratio of unity  for an integration time of 8.59 s. Measurement of this \nsignature at each of the Bell Test angles will therefore take a few tens \nof seconds.  \nAs the parametric gain is increased , more entangled photons will \nbe created and with a sufficiently high gain the parametrically \namplified thermal photon fluctuations may dominate the noise \ngenerated by the RF amplifier. In this case 𝑛𝑃𝑆/𝐿≫𝐹𝑘𝑇0𝐵 and Eq. \n(24) simplifies to  \nTransposing the last equation indicates the integration time to achieve \na given signal -to-noise ratio is  \nThis indicates there would be benefits in moving up in frequency, \nso bandwidth B could be increased and the mean numbers of thermal \nphotons would be reduced. Doubling the frequency of the pump from \n20 GHz to 40 GHz  (also requiring the doubling of H 0) and increasing \nthe bandwidth from 10 GHz to 20 GHz, would shorten the integration \ntime by a factor o f 32.  \n4.4. Closing Bell Test loopholes  \nThe two main loopholes in Bell Tests are the ‘detector efficiency \nloophole’ and the ‘locality loophole’, the former being associated with \nthe fact that a photon counting detector does not ensure sufficient \nquantum efficiency and the latter associated with a local \ncommunication link between the two re ceivers which could  corrupt \nthe result.  \nFor the case of the detection efficiency loophole this should be less \nof an issue with radio receivers, as information is more easily captured \nwith homodyne quadrature receivers operating in the continuous -\nvariable mode  [12]. Noise is present due to the ambient thermal \nphotons, which is removed by coherent integration, but the essential \npoint is that the information about  the arrival of each photon of an \nentangled pair is captured.    \nFor the case of the locality loophole s, strategies to remove these \nwould be the following. With the envisaged integration times being a \nmatter of seconds and receivers placed several meters apart,  it might \nbe argued that there would be sufficient time for millimeter wave \ncommunication between receivers , potentially leading to corruption of \ndata. However, this loophole would be closed by randomly changing \nthe plane of polari zation  on a timescale sufficiently rapidly that \nmeasurement at one polarization is shorter than the time taken for \nsignals to  pass between the receivers and the source , as is done in the \noptical Bell Tests [48]. Rapidly switching the polarization can be done \nusing phase shifters and waveplates constructed from pin or \nmicrowave varactor diodes, as these devices switch on nanosecond \ntimescales. This switching would need to be done in synchronization \nwith the data acquisition. This is entirely possible as the sample times \nrequired to satis fy the Nyquist criterion associated with GHz \nbandwidths is sub -nanosecond, a sampling capability readily available \nnow.  \nIn the case  of the  locality loophole associated with the link between \nthe two receiver channels at the site of mixer -1 (before the sampling ) \nthis may be removed by sampling separately in each of the channels \nbefore mixer -1. The action of mixer -1 would then be performed digitally on uncorrupted samples, before a digitally downshifting \naction of mixer -2. In such a configuration it might be argue d there \nmay be a communication link via the sampling clock suppl ied to the \ntwo samplers. This link would be removed by having free -running \nsampling clocks at the sites of the samplers, which are only \nperiodically up -dated to keep them in synchronization.   \nIt is not being claimed here that this system offers a completely \nloophole free Bell Test, only that it offers a new potential direction for \nresearch into the phenomenon of entanglement , with attention paid to \nhow loopholes may be closed .  Furthermore, th e approach here is only \nsemiclassical and that  a full quantum operator evaluation is necessary \nto identify potential quantum effects that may thwart the function of \nthe proposed system. This would constitute a n essential  next phase of \nthe work.   \n5. Conclusion  \nThe proposed combination of a YIG parametric amplifier with a \nhomodyne interferometer enables the measurement of a flux of \nentangled photons in the presence of a much larger flux of thermally \ngenerated photons. Calcula tions indicate that approximately 5 Watt s \nof millimeter -wave radiation pumping 0.3 cm3 of YIG will generate \n~0.5  1012 entangled photon pairs per second. The proposed \narchitecture enables an ambient temperature Bell Test to be made \nwhich would require only tens of seconds of integration time. It is \nencouraging with this architecture of Bell Test that there are good \nopport unities to close the most severe loopholes of detector efficiency \nand locality. A successful demonstration of this test will lead to novel \narchitectures of entanglement -based  quantum technology system s, \npotentially for applications in computers  and millime ter wave \ncommunication systems and sensors .   \nData Availability  \nThe datasets generated during and/or analyzed during the current \nstudy are available from the corresponding author on reasonable \nrequest.  \nAcknowledgements  \nThe authors are grateful for the partial financial  funding  of this \nwork from the T ERANET  Seedcord funding scheme provided by the \nUK EPSRC , through the School of Electronic and Electrical \nEngineering, at the University of Leeds , managed by Professor John \nCunningham and Professor Martyn Chamberlain. The insightful \ncomments and critique provided by the referees has been a n enormous \nhelp in the refinement of the paper.   \nAuthor contributions  \nN.A.S. made the calculations of the entangled photon fluxes and \nthe signal -to-noise ratios in the receiver systems. S.R.H. provided the \ndetails on the YIG ferrites , their hysteresis  and use in resonance \ncircuits.  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The metamaterial consisted of an array of wires cladded in dielectric embedded in a magne tic ferrite.\nThe ferrite replaced the cut-ring structure usually used to create the negative permeability. The dielectric cladding decoupled\nthe ferrite from the wires, thereby allowing the wire array p ermittivity to be simultaneously negative with the permeab ility. The\nsimplicity of the design allows for miniaturization of pote ntial microwave devices based on this metamaterial.\nIndex Terms —Negative index of refraction, microwave magnetics, ferri te, reversible directional coupler.\nI. INTRODUCITON\nSince the ﬁrst demonstration of a two-dimensional metama-\nterial with a negative index of refraction [Smith 2000], the re\nhave been many advances in tailoring the electromagnetic\nresponse of man-made structures, including extending demo n-\nstration of the phenomenon from the microwave frequencies\nto the optical range [Shalaev 2007]. Besides verifying the f act\nthat a light ray passing through a prism having a negative\nindex of refraction bends in the opposite direction to that\nfound with an ordinary prism [Shelby 2001], sub-wavelength\nimaging with a spherical aberration free lens [Pendry 2000] is\nwell understood. In addition, modiﬁcation of the permittiv ity\nǫand permeability µin the region of space surrounding an\nobject has resulted in phenomena ranging from “cloaking”\n[Pendry 2006, Schurig 2006, Cai 2007, Wood 2009] of that\nobject to having the object become a super scatterer [Yang\n2008, Wee 2009]. Indeed, the concepts of cloaking and super\nscattering have been extended beyond Maxwell’s equations\nto the diffusion equation and the possibility of creating\nmetamaterials for mass separation [Guenneau 2013, Restrep o-\nFl´ orez 2016]. Other predictions [Veselago 1968] regardin g\nreverse Cherenkov radiation [Chen 2011a, Duan 2017] and the\nreverse Doppler effect [Chen 2011b] have been experimental ly\nveriﬁed. However, the reverse Casimir effect [Veselago 196 8,\nLeonhardt 2007] remains speculative.\nThe key ingredient to guarantee a negative index of re-\nfraction whereby the light wave’s phase fronts move in the\ndirection opposite to which energy propagates is that both\nthe permeability and permittivity are negative. Implicit i n\nthis is that the losses in the medium, usually described by\nthe imaginary parts of the permeability and permittivity, a re\nsmall; this ensures that the medium is reasonably transpare nt.\nThe negative response functions are typically obtained wit h\nresonant structures driven at frequencies somewhat above t heir\nnatural ones. Then the responses are more than 90oout of\nphase with the drives and, provided that responses are large\nenough, the permittivity and permeability are negative.\nOnly negative permeability is required to produce a materia l\nwhich has a negative phase velocity. See, for example, (21) a nd\nCorresponding author: G. Dewar (email: graeme.dewar@ndus .edu).(25) in Cochran et al. [1977]. If the permittivity is positiv e with\nµ <0then the index of refraction calculated from Maxwell’s\nequations is an imaginary number, hence the medium is highly\nreﬂective and does not permit light to freely propagate thro ugh\nit. Atypically, an array of ferrite rods [Gu 2013] alone can h ave\na negative effective permittivity simultaneously with µ <0\nbut this is not the usual case. Also, a metallic ferromagneti c\nconductor can have µ <0and hence n <0[Pimenov 2007]\nbut the inevitable attenuation associated with ohmic losse s\ndestroys the medium’s transparency. In the simplest case a\nnegative permittivity coupled with a negative permeabilit y,\nboth accompanied by low losses, results in a negative real\nvalued index of refraction and the negative index material\n(NIM) is transparent.\nThe ﬁrst demonstrated NIM used an array of conducting\nwires for which permittivity was negative for frequencies l ess\nthan the array’s plasma frequency. The negative permeabil-\nity was supplied by an array of cut-rings with a resonant\nfrequency slightly below the frequencies of interest. In th e\nexperiments described here the NIM was fabricated with a\nnonconducting magnetic host which supplied the negative\npermeability and a wire array, suitably decoupled from the\nmagnetic host, supplied the negative permittivity. This is a\nmuch simpler structure to fabricate than the wire/cut-ring\narrays. This metamaterial has the added features that the ra nge\nof microwave frequencies over which it exhibits a negative\nindex of refraction can be tuned by changing the biasing\nmagnetic ﬁeld acting on it and the negative index of refracti on\nis relatively isotropic for microwaves propagating in the p lane\nperpendicular to the ferrite’s magnetization direction.\nII. METAMATERIAL DESIGN AND FABRICATION\nSimply embedding an array of wires in a non-conducting\nmagnetic material does not lead to a NIM [Pokrovsky 2002,\nDewar 2002] because material having µ <0in close contact\nwith the wires causes the wire array to acquire a positive\npermittivity. The cure to restoring ǫ <0is to surround the\nwires with a non-magnetic dielectric [Dewar 2002, Dewar2\nFig. 1. (a) Photograph of the ferrite block with a square arra y of holes drilled\nthorough it. For the transmission experiments the holes wer e were threaded\nwith copper wires cladded in TeﬂonTMtubing and the block was inserted\ninto the section of waveguide shown. (b) Cut-away sketch of t he nickel zinc\nferrite block. The cladded wires formed a 9×5square array with lattice\nconstanta= 3.0mm.\n2005a]. For these structures the permittivity is determine d\nfrom the (angular) plasma frequency ωpby\nǫ=ǫ0/parenleftBigg\n1−ω2\np\nω2/parenrightBigg\n(1)\nand is negative for frequencies ω < ωp. The usual expression\nfor the plasma frequency of a metal,\nω2\np=ne2\nmeﬀǫ0, (2)\nis modiﬁed in two ways for the wire array [Pendry 1996]. First ,\nthe electron number density nfor the array of conducting wires\nis diluted by the volume between the wires and is much smaller\nthan for a conductor ﬁlling all space. Second, the electron\neffective mass meﬀis much enhanced by the wires. The\neffective mass relates the work done by the electric ﬁeld on\nelectrons to change their kinetic energy meﬀv2/2; for electrons\nin the ionosphere the effective mass is just the usual electr on\nmass while for electrons in a metal the effective mass is rela ted\nto the metal’s band structure. For our wire array, an electri c\nﬁeld that increases the speed vof the electrons (and increases\nthe electric current) creates an augmented local magnetic ﬁ eld\naround each wire. The energy associated with this near-zone\nmagnetic ﬁeld is much greater than the electronic kinetic\nenergy, thus meﬀ∝L, whereLis the inductance of the wire.\nBut ifµ <0thenL <0, leading to a negative meﬀin (2)\nand a positive ǫin (1), thus destroying the n <0property.\nSurrounding each wire with a non-magnetic dielectric creat es a\nvolume in which the inductive energy is positive and, provid ed\nthe magnitude of the (negative) permeability in the magneti c\nhost is not too large, the permittivity is negative, thus res toring\nthen <0property. Calculations show [Dewar 2002] that\nchoosing the outer radius of the cladding to be the geometric\nmean of the wire radius and the array lattice constant provid es\nan suitable frequency range over which both µandǫare\nnegative.\nThe structure described here offers advantages over other\nsuccessfully demonstrated n <0metamaterials employing a\nferrite for the µ <0property. The multi-layer wire and ferrite\nrod or plate conﬁgurations [He 2009, Zhao 2007, Zhao 2009,\nBi 2013, Bi 2014] is somewhat more complicated to construct\nand can result in a highly anisotropic index of refraction\n[Rachford 2007].\nFig. 2. Transmitted power in percent versus frequency of inc ident microwaves\nfor ferrite blocks in waveguide. The red square symbols repr esent the\ntransmission data acquired for the ferrite block/wire arra y in an applied ﬁeld\nof2.4×104A/m (300 Oe). The blue ﬁlled line represents the transmissio n\nthrough a similarly sized ferrite block which had no wires or holes in it and\nwhich was in an applied ﬁeld of 8.0×104A/m (1.0 kOe). The scale for the\nﬁlled line has been compressed by a factor of 3.0 for ease of co mparison.\nDue to the negative demagnetization factors associated wit h the holes, the\ninternal magnetic ﬁelds in both blocks was approximately th e same. For both\nsamples, the permeability was negative at frequencies less than 16 GHz. The\nsigniﬁcant transmission for the ferrite block/wire array i n the 12 - 14 GHz\nrange demonstrates transparency for n <0.\nFig. 1 is a photograph of the ferrite block/cladded wire\nused in the microwave transmission measurements described\nhere together with a schematic of the array of wires threadin g\nthe block. Ferrite blocks were fabricated by cold pressing\n(4.0×104N) powdered starting material (19% by weight NiO,\n66% Fe 2O3, 14% ZnO, and 1% MnO 2) in cylindrical molds\n39 mm in diameter. Rectangular blocks were cut from from\nthe pressed material and sintered at 1000oC for ﬁve hours with\nheating and cooling rates of 100 Co/hour. This low sintering\ntemperature resulted in blocks which were structurally str ong\nbut could be drilled with ordinary drill bits and were easily\nshaped by sawing and sanding into rectangular parallelepip eds\nof ﬁnal dimensions of 30×15.7×7.9mm3. The block shown\nin Fig. 1 had a 5×9square array of 1.7 mm diameter holes,\nlattice constant 3.0 mm, drilled into it. Each hole was threa ded\nwith a 0.29 mm diameter copper wire cladded with TeﬂonTM\ntubing having an inner diameter of 1.0 mm and outer diameter\n1.6 mm.\nIII. TRANSMISSION MEASUREMENTS\nFig. 2 displays the measured microwave transmission\nthrough the ferrite block with a wire array threaded through it\ntogether with the transmission through a similarly sized fe rrite\nblock which did not have an array of holes or wires in it. The\nferrite blocks each ﬁlled the lateral dimensions of a sectio n of3\nWR-62 wave guide and were mounted in the gap of a 9-inch\nVarian electromagnet. Additional blocks of ferrite which h ad\nundergone the same preparation history as the samples were\nmounted in the magnet’s gap. These blocks constituted part\nof a magnetic circuit that, in the absence of any holes in the\nsample, limited the static demagnetization factor acting o n the\nsamples to less than 10−2. The essential difference between\nthe two measurements is that the negative permittivity of\nthe wire array allowed signiﬁcant transmission at frequenc ies\nbelow14GHz for which the permeability was negative (and\nn <0) and reduced the transmission at frequencies >16GHz\nfor which µ >0. The sample without wires behaved in the\nopposite manner, having signiﬁcant transmission at higher fre-\nquencies with both response functions positive ( n >0). This\nis a clear demonstration that our sample exhibited a negativ e\nindex of refraction in a manner similar to the demonstration\n[Smith 2000].\nThe maximum transmission of our n <0metamaterial is\napproximately 1/3 that of the ferrite. Since both samples ha d\nmeasured reﬂectivity’s on the order of 50%, we attribute the\nlower transmission to resistive losses in the wire array. Th e\ninternal magnetic ﬁelds for the two measurements shown in\nFig. 2 are comparable because the holes in the ferrite gave ri se\nto a signiﬁcant negative demagnetization factor. We calcul ated\nthe demagnetization factor to range from -0.16 near the cent er\nof our sample to -0.05 near a corner. This inhomogeneous\ndemagnetization factor may have also led to a decrease in\nthe observed transmission relative to the ferrite block wit h no\nwires or holes in it.\nIV. PROPAGATION CONSTANT CALCULATION\nFig. 3 is a plot of the calculated [Dewar 2005b] propa-\ngation constant, k, versus frequency for waves of the form\nexp(ik·r−iωt)in an inﬁnite medium. Note that kplotted\nin Fig. 3 has been scaled by the lattice constant of the wire\narray. The values of the magnetization ( M= (3.80±0.21)×\n105A/m) and g-factor (g= 2.006±0.049) required in the\ncalculation were obtained from ferromagnetic resonance me a-\nsurements on a small cylindrically shaped sample of the ferr ite.\nThe ferrite’s intrinsic permittivity ( ǫ= (4.45±0.5)ǫ0) was de-\ntermined from the spacing in frequency at H= 8.8×105A/m\n(µ >0,ǫ >0) of transmission modes [Barzilai 1958] for the\nferrite block having no wires in it.\nThe relevant feature of Fig. 3 is that the real part of kis\nnegative for 11 GHz ≤f≤14GHz and the imaginary part\nofk(representing dissipation) is small. Over this frequency\ninterval the index of refraction is calculated to fall in the range\n−4.5≤n≤0. In the calculation µferrite<0forf <16GHz.\nThe relatively large transmission through our NIM sample fa lls\nin this frequency range. The “band gap” (large imaginary par t\nofkwith the real part near 0) between 14 and 18 GHz is due\nto two causes. First, the magnitude of the ferrite’s negativ e\npermeability is so small that the permeability averaged ove r\na cell of the photonic lattice is positive and second, the k’s\nmagnitude was reduced by a component equivalent to half a\nwavelength across the width of the waveguide representing t he\ntransmission cut-off for the waveguide. The calculation al so\nFig. 3. Plot of the propagation constant kversus frequency according to the\ncalculation outlined in Dewar [2005b]). The ferrite/wire a rray is assumed\ninﬁnite in extent and khas been scaled by the lattice constant aof the\nwire array. Propagation is along the (1,0)direction and khas been adjusted\nto account for the cut-off of long wavelength radiation in wa veguide; it is\nthe component of kparallel the waveguide transmission direction for the\nTE1,0mode. A negative index of refraction corresponds to the real part ofk\nnegative while the imaginary part, representing dissipati on, is small. Between\n11 and 14 GHz the index of refraction ranges from −4.5to approximately 0.\nImmediately below 11 GHz there is a Bragg reﬂection. The band gap between\n14 and 18 GHz is due to two causes: 1) kis below the cut-off for propagation\nin the waveguide and 2) the permittivity of the wire array is n egative while\nthe permeability averaged over a cell of the array, includin g the hole and wire,\nis small and positive. Between 18 and 27 GHz the inﬁnite ferri te/wire array is\ncalculated to exhibit a positive index or refraction but has a Bragg reﬂection\nabove 27 GHz.\nshows that, for f <11GHz, there is a Bragg reﬂection of\nthe microwaves and transmission is curtailed. In addition, the\nnegative permeability below 11 GHz is so large that it destro ys\nthe negative permittivity of the wire array, further enhanc ing\nattenuation of the microwaves. Any transmission through ou r\nferrite/wire array sample between 11 and 12 GHz could not\npropagate in the waveguide we used. The transmission we\nobserved through our ferrite with the wire array, and shown\nin Fig. 2, is large for k <0, equivalent to n <0, and small\notherwise.\nV. CONCLUSIONS\nWith proper impedance matching a microwave device based\non our NIM would have a 5 dB insertion loss. Further improve-\nment is possible by optimizing the wire array lattice consta nt\nand radii of the wires and holes. A potential application cou ld\nbe a directional coupler capable of reversing the direction\nfrom which microwaves are coupled. Our sample exhibited a\npositivenand transmission comparable to that shown in Fig. 2\nin an applied ﬁeld of 6.0×105A/m (7.5 kOe). All that need\nbe done to reverse the input coupling direction with a simila r\nmaterial in a directional coupler is change the biasing magn etic\nﬁeld on the ferrite. More generally, our transparent NIM has a\nsimpler structure than earlier NIM’s based in rods (or wires )\nand cut-ring structures. The waveguide used to house our NIM4\ncould, for example, be easily reduced to transverse dimensi ons\nof 5.0 mm by 1.0 mm. Further, the operating frequency of our\nNIM can be adjusted by changing the biasing magnetic ﬁeld\nacting on it, thus offering the prospect of more readily real ized\nnovel uses.\nThe success of using a magnetic material as one of the two\ncrucial components of a NIM suggests that it is possible to\nsimply achieve a NIM at much higher frequencies. Ferrimag-\nnets and antiferromagnets have a resonance in the terahertz or\nfar infrared regime whereby two magnetic sublattices exert\ntorques on each other [Moorish 1965]. 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Lett., vol. 91,\n131107, doi: 10.1063/1.2790500.\nZhao H, Zhou J, Kang L, Zhao Q (2009), “Tunable two-\ndimensional left-handed material consisting of ferrite ro ds\nand metallic wires,” Opt. Express, vol. 17, pp. 13373–13380,\ndoi: 10.1364/OE.17.013373."    },    {        "title": "2001.00741v1.Synthesis_and_characterization_of_Zn_doped_Mn_ferrites_nanostructures.pdf",        "content": "Synthesis and characterization of Zn doped Mn ferrites \nnanostructures  \nSaima Rani1 and Syed Shahbaz Ali1,* \nDepartment of Physics, The University of Lahore, Lahore Pakistan  \nAbstract  \nZn doped Mn ferrites nanopart icles were fabricated by using C o-precipitation. Variation in \nstructure, magnetic and optical properties of MnZn ferrites has been discussed . First of all, \nsamples were synthesized, annealed at (400  C, 500 C, 600 C & 700 C) and then characterized.  \nThe as -synthesized and annealed samples were  investigated by  X-ray diffraction (XRD), \nScanning electron microscopy (SEM), Energy Disper sive spectroscopy (EDX), Ultra V iolet \nvisible spectrometry (Uv -Vis spectrometry) and Vibrating s ample magnetometer (VSM). The \naverage crystallite size of MnZn ferrites nanoparticles determine d from XRD were in the range \nof 42 to 60 nm. These nanoparticle s possess normal spinel  structure. The SEM images showed \nthe physical shape  of the samples , which showed  that the as prepared samples are more \nagglomerated and having flake like shape rather than annealed at 700ºC while the samples have \nlongitudinal or rod like shape on annealing at 700ºC. The coercivity (Hc), saturation \nmagnetization (Ms), and remanen ce (Mr)  of Np’s were  also calculated. T he (Ms) value is \nincreasing from  26 to 65 emu/g, the co ercivity (Hc) is varying from  13 to 1 93 Oe and remanence \n(Mr) has also showing increasing trend although very less , from  0.031 to 0.798 emu/g which are \na little p art of their bulk counter parts. The band gap  energy of the samples was  showing \ndecreasing trend as with the increase of particle size which is of the order of 3.5 to 2.9 eV.  \n1. Introduction  \nA lot of substitutes have been added in ferrites (Fe 2O4) as doping to enhance the magnetic and \nelectric properties of ferrites. These ferrites can be prepared in different forms like in thin films \nor in powder form. A large number of methods have been developed to synthesize ferrites; some \nof them are the hydr othermal process, the co -precipitation method, the micro -emulsion method, \nand sol gel synthesis. After the study of literature about soft ferrites it has been cleared that chemical co -precipitation method is much easier, low cost, and convenient to control  particle \nsize and purity. Soft ferrites are a type of ferrites having superior magnetic properties used in \nelectrical and microwave industry. MnZn ferrites got a deep interest for research from \nresearchers’ just because of application point of view in pre vious years. Its preparation method, \ncomposition, doping, calcination or annealing changed its properties and application. That why \nit’s a deep ocean of properties variation. Soft ferries are used in stem up, step down transformers, \nin recording heads, in remote devices etc.  \nC. Venkataraju et al investigated the cation distribution effects on the structural and magnetic \nproperties of Nano particles of Mn (0.5–x) NixZn0.5Fe2O4 (where x varied from 0.0, 0.1, 0.2, 0.3) \nprepared by Co -precipitation method [1]. C .F. Zhang et al investigated the structural and \nmagnetic properties of MZn ferrite NP’s with different doping values of cobalt by Co -\nprecipitation method [2]. Darko Makovec et al studied the spinel structure of MnZn ferrite \nnanoparticles by Co -precipitatio n in reverse micro -emulsion method [3].  \nXie Chao et al studied the effect of pH on Mn –Zn ferrites properties synthesized from low grade \nmanganese ore (LMO). He prepared Mn -Zn ferrites powder from LMO by co -precipitation \ncombining with ceramic preparation m ethod [4].  N. D. Kandpal et al studied the synthesis and \ncharacterization of Iron oxide with the help of cost effective co -precipitation method. Different \ntechniques like XRD, TEM, VSM and FT -IR were followed to identify the lattice parameters, \ncrystal str ucture a nd the magnetic properties [5].  \n2. Experimental  \nSynthesis and annealing process of Mn 0.5Zn0.5Fe2O4 \nFor the preparation of Zn doped Mn (MnZnFe 2O4) ferrites nanoparticl es by C o-Precipitation . We \nused Fe2O 4, ZnCl2 and MnCl2 as the starting precursors. The 0.5 M aqueous solution of each \nsalt was prepared by adding these salts in 50 ml of distilled water. NaOH worked as precipitating \nagent, its 0.64 M aqueous solution was also prepared. All these three initial ly prepared precursor \nsolutions were mixed by using magnetic stirrer continuously. The 0.5 M aqueous solution of \neach salt was fabricated to get required atomic ratio of final product. During the mixing of three \ninitial precursor solutions, NaOH was added drop wise to the precipitation to control the pH of \nthe solution. During the mixing process a constant temperature of 80°C was provided \ncontinuously for 2 hours for better results of reaction process. After sufficient precipitation has been observed, the r eaction was stopped and the precipitates were allowed to settle down at the \nbottom of beaker. Later on the liquid was taken out by using a simple sucker and the precipitates \nwas washed 4 to 5 times with distilled water and then at least 2 times with ethano l. The \nprecipitates were dried out for about 4 -6 hours in a drying oven at a temperature of about 80°C \nand then the final product was grinded by using mortar and pestle to get uniform powder which \ncontained fine nanopa rticle of our desired materials.  \nFollowing reaction took place.  \nMnCl 2.3H 2O+ZnCl 2.H2O+2FeCl 3→MnZnFe 2O4+4H 2+5Cl 2 \nH2 and Cl 2 were washed out during the washing and sucking process to remove impurities.  \nThe prepared samples were annealed at 400˚C, 500˚C, 600˚C &  700˚C temperatures to study the \npossible crystalline structure modifications and variation in grain size, magnetic & optical \nproperties. These as prepared samples & annealed samples at different temperature were \ncharacterized and then compared their resul ts. \nCharacterization  \nThe 5 different samples including as prepared, annealed at 400˚C, 500˚C, 600˚C & 700˚C are \nused to characterized through different characterizing mechanisms like SEM, EDX, XRD, Uv -\nVis spectrometry & VSM. Scanning electron microscopy an d Energy Dispersive X -ray \nspectroscopy were used to analyze the morphology and to conform the prepared particle’s \npresence and their physical appearance. X -ray Diffraction spectroscopy was used to calculate the \nparticle size, lattice parameter and d spacin g. Optical properties like band gap  energy  was \ncalculated by carried out results of Uv -Vis spectrometry. Vibrating Sample Magnetometer results \nwere used to calculate the magnetic properties like Saturation Magnetization (Ms), Coercivity \n(Hc) & Ramanence (M r).  \nResults & Discussion  \n X-ray Diffraction  \nTo study the lattice parameter, phase structure and the particle size XRD analysis was carried out \nwith Cu -Kα (λ=1.5406 Å) radiations at room temperature.  The samples analysis conformed  the \ncubic spinel structure  of Particles. The experimentally generated peaks were matched with th e \ntheoretical peaks and indices . The particle sizes were calculated by The De-bye sheerer  formula  \nD = Kλ/βcosθ  Fig#1 shows a  pattern of as pr epared and annealed samples with  grain growth  from 42 to 60 nm  \nas with the increase of peak , the peak of as prepared and  with the increase of annealing from \n400˚C up to 700˚ C every peak  representing the grain growth and also indicating that the particles \nbecame more crystalline. Annealing make particles more refine and crystalline. The ca lculated \nparticle size is shown  in table#1.  \n \n \nFig#1 Indexed XRD patterns at different temperatures  \n \nTable#1 Brief description of Particle size (nm) of every peak at different annealing temperatures  \nLattices  \nLattices Particle size (nm)  \nmiller \nindices  As prepared  annealed at \n400˚C  annealed at \n500˚C  annealed at \n600˚C  annealed at \n700˚C  \n(220)  34 26 17 20 41 \n(311)  42 53 53 56 60 \n(400)  17 18 21 21 36 \n(511)  14 19 28 28 38 \n(440)  16 24 14 19 38 \n \n \n \nThe calculate particle sizes, lattice parameters and the d -spacing for max. peak is written in table \n#2. \nTable#2 Calculated results of XRD for index (311)  \nAnnealing \ntemperature  Particle size  \nD (nm)  Lattice \nparameters  \na=b=c (nm)  d spacing  \n(nm)  \nAs prepared  42 7.7 2.3 \nannealed at 400˚C  53 7.7 2.3 \nannealed at 500˚C  53 7.8 2.3 \nannealed at 600˚C  56 7.9 2.3 \nannealed at 700˚C  60 7.9 2.3 \n  \nThe graph between the particle size and th e annealing show the direct relation between the \nannealing and the particle size  as shown below .   \n SEM  \nThe SEM technique use to observe the morphology of the Mn -Zn ferrites NPs. The SEM images \nof the sampl es as prepared and annealed at 700ºC are shown below in Fig.#2(a) and 2(b).  \n(a) \n \n(b) \n \nThe SEM images show that the as deposited  particles are more agglomerated but when the \nsamples have been annealed at 700ºC the agglomeration reduces because of heat  treatment. As \nprepared samples looked having  flake like shapes while the samples annealed at 700ºC showed  a \nrod like shape morpholog y. So, this implies that because of heat treatment there happened a \nlongitudinal growth of the particles resulting in rod like structures at 700ºC.  \n EDX  \nEnergy dispersive graph shows the presence of Mn, Zn, Fe and O 2 atoms in the prepared \nsamples. All the g raphs were almost same so here we attach only one to avoid repetition.  \n Uv.Vis. spectroscopy  \nThe optical properties of Zn doped Mn ferrites are studied by using the UV -Vis \nspectrometry technique. The results were used  to study the absorption of photon \nand t he band gap energy. The graph between absorbance and wavelength indicat es \nthat annealing effected  the abso rbance. The graph showed  the maximum \nabsorbance from 364 nm to 435 nm (from near ultra to red shift) and after that \nabsorbance decreased  down graduall y. \n \n \nThe band gap energy Eg calculated by using Tauc relation which is mentioned below.    \nhʋɑ = A˟ (h ʋ - Eg)1/n \nIn this relation “ ɑ” is absorption coefficient, A is proportionality constant which is equals to 1 \nand Eg  is band gap corresponding to a particular transition of electrons while n=1/2, 2, 3/2 and 3 \ncharacterizes the nature of transition which may be direct allowed, indirect allowed, direct \nforbidden and indirect forbidden transitions respectively n=1/2 showed  that there is direct \nallowed transition occurred in Mn -Zn ferrite case.  \nTable# 3 UV-Vis spectroscopy results  \nAnnealing \ntemperature  Wavelength at max. \nabsorbance (nm)  Band gap energy  \n(Eg) eV  \nAs prepared  364 3.5 \n400˚C  364 3.25 \n500˚C  411 3.14 \n600˚C  435 3.01 \n700˚C  435 2.91 \n \nThe calculated band gap energy of every sample is shown in Table #4. The absorbance and band \ngap energy decreased d own as the annealing  increased. The XRD results and the Uv -Vis results \nindicates the  reverse quantization phenomena. Quantization states that “Band gap energy \nincreases with decrease in particle size due to electron confinement at nano -scale”.  While  in \ncalculated results the Eg decreased down with the increase of particle size.  \n \nFig#5: Graph betw een annealing temperature and band gap  \n VSM  \nVSM were used to investigated the magnetic properties of the samples. The M –H loops of the \nannealed Mn -Zn ferrites NPs with a maximum magnetic field of ±8 kOe at room temperature. It \nis cleared from results that a ll samples have different saturation magnetization Ms values. The \nparticles behave ferromagnetic behavior with Ms values increasing gradually  from 26.5 to 65.7 \nemu/g. The spins at A and B lattice sites are antiparallel to each other in spinel ferrites stru cture.  \n                 (a)                                                                   (b) \n      \n                 \nTable# VSM calculated resu lts of  Hc & Mr  \nAnnealing \ntemperature (˚C) Coercivity (Hc), \n(oe) Ramanence (Mr), \nµ(emu)  \nAs prepared  25.640  3.4489  \n400 44.059  0.015097  \n500 53.998  0.001475  \n600 110.95  0.001475  \n700 140.04  367.5  \n \n \nFig.#5 Graph between annealing temperature (˚C) and Retentivity (Mr ) µ emu  \n \nFig.#6: Graph between annealing temperature and coercivity Hc  \nWith high saturation magnetization Ms i.e. ±8 kOe, low coericivity and low remanence values \nthey work as soft ferrites (easily magnetize and demagnetize as well). These materials very \nuseful to prepare many of field applications like Ferro fluids, step up step down transformers, \nrecording heads etc.  \nConclusions  \nThe particles size increased with the annealing which is also confirmed by SEM results while \nthe band gap energy decreases with the annealing which is def ined as the reverse quantization \nprocess. VSM results indicates that the samples are ferromagnetic in behavior with a maximum \nmagnetic field of ±8 kOe at room temperature with high Ms value, low coericivity and low \nremanence values while they work as soft ferrites.  \n \nReferences  \n[1] C. Venkataraju, G. Sathishkumarb, K. Sivakumarc, Journal of Magnetism and Magnetic \n Materials 322 (2010), 230 –233. \n[2] C.F. Zhang, X.C. Zhong, H.Y. Yu, Z.W. Liu, D.C. Zeng , Physica B 404  (2009) , 2327 –\n 2331.  \n[3] DarkoMakovec, AlojzKodre, Iztok Arcˇon, Æ Miha Drofenik, Journal of Magnetism and \n Magnetic Materials 298 (2006) 83 –94. \n[4] Xie Chao, Xu Longjun, Ye Yongjun, Li Xiangyang, Wang Shuyun , Chin. J. Geochem. \n (2015) 34(2):219 –223. \n[5] N D kandpal , N Sah, R Loshali, J joshi, J Parsad, journal of scientific research Vol.73,  \n (2014),  pp.87 -90. "    },    {        "title": "1401.7643v1.Bimodal_island_size_distribution_in_heteroepitaxial_growth.pdf",        "content": "Bimodal island size distribution in heteroepitaxial growth\nP. V. Chinta and R. L. Headrick\nDepartment of Physics, University of Vermont, Burlington, VT 05405\n(Dated: June 29, 2021)\nA bimodal size distribution of two dimensional islands is inferred during interface formation in\nheteroepitaxial growth of Bismuth Ferrite on (001) oriented SrTiO 3by sputter deposition. Features\nobserved by in-situ x-ray scattering are explained by a model where coalescence of islands determines\nthe growth kinetics with negligible surface di\u000busion on SrTiO 3. Small clusters maintain a compact\nshape as they coalesce, while clusters beyond a critical size impinge to form large irregular connected\nislands and a population of smaller clusters forms in the spaces between the larger ones.\nPACS numbers: 61.05.cf, 68.47.Gh, 68.55.A-, 81.10.Aj, 81.15.Cd, 81.15.Kk\nControl of atomic-level processes in heteroepitaxial\nthin \flm growth is critically important for the forma-\ntion of interfaces in arti\fcially layered nanoscale struc-\ntures. In turn, growth modes determine or in\ruence im-\nportant interface properties such as roughness, chemi-\ncal intermixing, defects, and strain. Phenomena typi-\ncally observed in homoepitaxy arise from well-known pro-\ncesses of random atomic deposition, surface di\u000busion, and\nthe aggregation and coalescence of two-dimensional (2D)\nclusters.[1] At moderate growth temperatures, these pro-\ncesses lead to layer-by-layer (LBL) crystal growth. [2, 3]\nInheteroepitaxy , de\fned as layered crystal growth of\ntwo or more materials with compatible crystal structures\nand lattice constants, there are other modes than can\nbe observed.[4, 5] The best known of these are three-\ndimensional (Volmer-Weber) and 2D followed by a tran-\nsition to three-dimensional (Stranski-Krastanov). How-\never, there may be additional possible modes involving\nonly 2D structures during LBL growth.\nHere, we discuss a case of heteroepitaxy where inter-\nface formation is dominated by coalescence of 2D clusters.\nIn the case we will consider, surface di\u000busion on the sub-\nstrate surface is very low so that the mobility of single\nmonomers can be neglected. However, surface di\u000busion\nof deposited monomers that land on the overlayer and at\nthe boundaries of overlayer islands is fast in comparison.\nThis leads to e\u000ecient coalescence of compact 2D islands\nover a range of length scales, and the system exhibits\nkinetics that are more akin to droplet growth processes\n(Family and Meakin, Blackman and Brochard, Refs. 6{\n8) than to standard surface di\u000busion driven aggregation\nand coalescence.\nIn the case described above, clusters should theoret-\nically grow exponentially with deposition time.[6] The\nprocess would lead to a single cluster covering the en-\ntire sample surface, except for kinetic limitations that set\nin at a material dependent yet well-de\fned length scale.\nGrowth models incorporating the e\u000bects of kinetically\nlimited coalescence have previously been developed. For\nexample, the Interrupted Coalescence Model (ICM) and\nthe Kinetic Freezing Model (KFM) successfully repro-\nduce irregular or fractal patterns observed in vapor de-\nFIG. 1. X-ray re\rectivity data near the (0 0 0.5) re\rec-\ntion during RF sputter deposition of 7.5 UC of BFO. The\ngrowth temperature and pressure were 650\u000eC and 20 mTorr,\nrespectively (with Ar:O 2of 2:1). (a) Two dimensional image\nat a nominal \flm thickness of 0.5 UC. (b) Specular (blue cir-\ncles) and di\u000buse (red circles) integrated intensities. (c) Time\nresolved di\u000buse scattering map of circularly averaged pro\fles\nversus time. (d) Circularly averaged intensity at \u0012= 0:5. The\ndata (open circles) is \ft using a two component empirical ex-\npression (lines). Data points at 205 s in (b) have been \flled\nin green, and a corresponding green line in (c) mark the time\nslice corresponding to panels (a) and (d).\nposited metal thin \flms on inert substrates.[9{11] Similar\nobservations have been reported for ultra-thin epitaxial\nmetal \flms that initially grow in a Volmer-Weber mode\non single-crystal oxide substrates, followed by coalescence\ninto islands with a distinct bimodal distribution.[12]\nIn this letter we show that the ICM can be adapted to\ndescribe experimental observations of the layer-by-layer\ngrowth process in a case where the predictions of stan-\ndard LBL growth models fail. One important prediction\nof ICM is a bimodal distribution of 2D cluster sizes in\ngood agreement with the experimental data. This modelarXiv:1401.7643v1  [cond-mat.mtrl-sci]  29 Jan 20142\nFIG. 2. (a) Di\u000buse scattering line shapes (circles) during co-\nalescence of the \frst layer 0 :5\u0014\u0012\u00141:0 for BFO growth.\nThe solid lines are two component \fts to the data. (b) Esti-\nmated island separation obtained from the peak positions of\nthe di\u000buse lobes. Low Qrcomponent: red circles and solid\nline; highQrcomponent: green circles and dashed line.\nmay \fnd wide applicability in cases where there is a dis-\nparity in surface di\u000busion coe\u000ecients between the sub-\nstrate and the \flm. An intriguing example is for SrRuO 3\ngrowth on SrTiO 3where a change of the surface ter-\nmination during the \frst growth layer leads to a large\nenhancement of surface di\u000busion of monomers on the\noverlayer.[13]\nBFO has attracted much interest due to its high fer-\nroelectric polarization, coupled with antiferromagnetism\nand weak ferromagnetism.[14, 15]. For this study, epitax-\nial \flms were grown on TiO 2-terminated (001) SrTiO 3\n(STO) substrates using on-axis radio-frequency mag-\nnetron sputter deposition in a custom growth chamber\nsituated at beamline X21 at the National Synchrotron\nLight Source. Film growth was monitored by in-situ x-\nray scattering using radiation with \u0015= 0:124 nm. A fast\nsingle-photon counting x-ray area detector was used to si-\nmultaneously record the evolution of specular and di\u000buse\nintensities near the anti-Bragg scattering condition. The\nmorphology of the \fnal surfaces were also corroborated\nwith ex situ atomic force microscopy (AFM) measure-\nments and by additional x-ray di\u000braction measurements.\nFig. 1(a) shows a time slice from a series of im-\nages recorded during BFO deposition and correspondsto 0.5 unit cell (UC) nominal \flm thickness. The image\nshows an almost perfectly circular di\u000buse ring that forms\naround the specular re\rection, indicating the presence of\ncorrelated 2D islands on the surface. The specular spot\nnear the center of the image is elongated due to the ter-\nrace structure of the STO substrate, where a step spacing\nof\u0019700 nm was observed by AFM. Fig. 1(b) shows the\nintegrated specular and di\u000buse intensities at (0 0 0.5) for\ndeposition to 7.5 UC of BFO. In this letter, we focus\non the the submonolayer deposition regime. An impor-\ntant feature to note is that while the specular intensity\nreaches a minimum at coverage \u0012= 0.5, the di\u000buse in-\ntensity continues to increase monotonically up to \u0012=\n1.5. This unusual behavior is due to the late peaking of\nthe di\u000buse scattering during growth of the \frst unit cell,\nwhich we will explain in detail below, and also due to the\nnucleation of the second layer before the completion of\nthe \frst layer.\nMore detail is obtained in Fig. 1(c), where circularly\naveragedQrradial pro\fles vs. time are presented. The\nabsence of any strong features below \u0012= 0:5 indicates\nthat the nuclei formed on the surface are very small, <5\nnm. Broad features extending up to Qr\u00190:5 nm\u00001\nare found to appear for \u0012\u00150:5, indicating formation or\ncoarsening of clusters at very short length scales \u001420 nm.\nThe strong \\ring\" feature also becomes visible at about\nthe same coverage. The evolution of the di\u000buse intensity\nfor\u0012 > 1 appears to be compatible with the standard\nLBL growth mode via surface di\u000busion and aggregation.\nHowever, our observations for \u0012<1 are inconsistent with\nstandard LBL since it predicts strong di\u000buse scattering\nin the aggregation regime, peaking in intensity at\u00190:5.\nFig. 1(d) shows the radial pro\fle for a single frame\nat\u0012= 0:5. The data is \ft using the empirical form\nsuggested by Brock et al.[16]\nI(Qr) =I0\n[1 +\u00182(Qr\u0000Qr;0)2]3=2(1)\nwhereQr;0determines the peak position, and \u0018the peak\nwidth. Two di\u000berent di\u000buse components are observed\natQr;01= 0:09 nm\u00001andQr;02= 0:31 nm\u00001indi-\ncating surface features of di\u000berent length scales. The\nseparation between the two components is considerably\nlarger than what we expect for a single population of\ndisk-like features on the surface, based on calculated\nstructure factors. Speci\fcally, the disk structure factor\nS(Qr)/[J0(RQr)=RQ r]2produces a series of fringes,\nbut these are too closely spaced to produce both com-\nponents observed in the data of Fig. 1. In this expres-\nsion,Ris the disk radius and J0is the Bessel function of\nthe \frst kind. Fig. 2 shows how the peak positions and\nthe estimated length scales evolve with coverage between\n0.5 and 1.0 UC. The results show that the length scales\nare relatively constant for \u0012\u00140:65, and, only the broad\ncomponent coarsens for \u0012 > 0:65. The lack of signi\f-3\nFIG. 3. Simulated cluster maps for ICM on a 1000 \u00021000 grid with D= 2 and\u0019R2\nc= 200 monomers. (a) shows that at 0.50,\nthe clusters are still mainly isolated. (b) and (c) show large connected regions just before the percolation coverage is reached.\nThe color scale in each image is by the size of connected regions.\ncant coarsening of the sharp component indicates that\nsome mechanism plays a role to prevent the coarsening\nof large islands. Below, we discuss these results in terms\nof a coalescence-dominated model.\nStandard models of LBL growth generally involves\nthree regimes: nucleation ( \u0012\u00140:1), aggregation (0 :1<\n\u0012 < 0:4), and coalescence ( \u0012 > 0:5).[17] Initially, de-\nposited monomers di\u000buse on the substrate, and a sta-\nble nucleus is formed when a critical number of them\nmeet. Once a high enough density of nuclei is reached,\nthe monomer density drops dramatically and the nucle-\nation rate drops correspondingly. Aggregation thus refers\nto the growth of existing clusters at a nearly \fxed num-\nber density. Finally in the coalescence regime, the islands\nbegin to join together and eventually form a continuous\nlayer. Impingement is a special case of coalescence where\nthe redistribution of matter among islands does not take\nplace after their collision.[18]\nOur experimental observations lead us to a di\u000berent\nmodel: (a) very little surface di\u000busion, producing small\nlength scales in the early stages of monolayer forma-\ntion; (b) formation of compact clusters on the surface,\nso that the asymptotic form of the structure factor is\nS(Qr)/Q\u00003\nrasQr!1 , as for disks; (c) irreversible\nattachment of monomers to the islands, since relaxation\ne\u000bects after deposition is stopped are minimal. In order\nto model this process, we have performed Monte-Carlo\nsimulations on a 1000 \u00021000 array. Clusters are assumed\nto be perfectly compact disks with irreversible monomer\nattachment, and monomers landing atop existing islands\nmigrate instantaneously to the island edge. We assume\nthe critical cluster size i= 0 case with no di\u000busion. Con-\nsequently, there is no aggregation regime and coalescence\ne\u000bects dominate for all coverages.\nThe FM model has been studied for surfaces of dimen-\nsiondand droplets of dimension Dfor many combina-\ntions withD\u0015d.[6{8] Our experiments relate to the case\nD = d = 2, i.e. two dimensional clusters on a 2D sur-\nface. This leads to a situation where the mean cluster sizegrows exponentially. This behavior is inconsistent with\nour BFO radial pro\fles, where the low-Q peak shifts very\nlittle with coverage. In addition, FM with D= 2 does\nnot lead to a bimodal distribution of cluster sizes and we\n\fnd that the structure factors produced by this model\ndo not have a pronounced sharp component as we have\nobserved in our experiment [Fig. 1(d)].\nAfter considering several possible mechanisms to limit\nthe growth of the largest islands, we decided to add the\nICM mechanism to our model.[9] Monomers (or particles\nwith a size R0) are added to the surface in random loca-\ntions, as in FM. However, once clusters reach a certain\ncritical size Rcthey no longer coalesce by merging with\neach other, rather the clusters impinge without combin-\ning. Deposited monomers and small clusters below the\ncuto\u000b are still allowed to combine with larger clusters.\nThis model results in the formation of irregular connected\nislands composed of impinging 2D clusters.\nFig. 3(a) shows ICM results for \u0012= 0:5. At this stage,\nthe great majority of clusters have not reached the cuto\u000b\nsize. Small clusters are continually replenished because\nwhen clusters merge their centers move together, expos-\ning a region of the surface for new clusters to nucleate.\nThis regeneration e\u000bect is central to the FM mechanism.\nFig. 3(b,c) show two views of a cluster map for \u0012= 0:75.\nAt this stage the largest clusters have formed connected\nislands, while a second population of smaller clusters con-\ntinues to develop within the interstices of the larger ones.\nThus, a feature of ICM with D= 2 is the formation of\na bimodal cluster distribution. It is caused by a deple-\ntion of clusters just below the cuto\u000b size, which are most\nlikely to collide and coalesce with larger clusters. We also\nobserve that \u0012= 0:75 is close to the percolation thresh-\nold\u0012p, since the largest connected region nearly spans\nthe map. This is in agreement with the results of Yu et\nal., who \fnd \u0012p\u00190:78 for ICM with Rc=R0= 4.[9]\nFig. 4 shows results for structure factors generated\nfrom ICM cluster maps. Fig. 4(a) shows the total dif-\nfuse scattering as a function of coverage up to \u0012= 1,4\nFIG. 4. Calculated structure factors for ICM. (a) illustrates that rather than peaking at \u0012= 0:5 as the total di\u000buse scattering\nintensity does (line), the integrated intensity with Qr<0:8 reciprocal lattice units (r.l.u.) exhibits a delayed peak at \u0012= 0:65\n(circles). For comparison, the triangles show the rescaled total di\u000buse intensity from Fig. 1(b). Plot (b) shows the two\ncomponent \ft at \u0012= 0:70. (c) shows the evolution of the lineshape at three coverages, illustrating the sudden appearance of\nthe narrow component for coverages above \u0012= 0:5, and sharpening of the line shape near \u0012= 0:8.\nas well as the integrated intensity within a region of Qr\nmeant to illustrate the di\u000buse intensity striking a detec-\ntor of limited size. At the early stages when clusters are\nvery small, a small fraction of the total intensity reaches\nthe detector. This behavior reproduces our experiment,\nwhere very little di\u000buse signal is detected for \u0012 < 0:5,\nand the peak occurs late so that it merges into the di\u000buse\nsignal after the second layer has nucleated. We have in-\ncluded the experimental di\u000buse data from Fig 1(b) in Fig.\n4(a) for comparison, which is consistent with continued\ncoarsening for \u0012>0:65 as shown in Fig. 2, implying that\nthe approximation of instantaneous island coalescence is\ntoo drastic. Fig. 4(b) shows a two component \ft of the\nradial pro\fle of the structure factor for \u0012= 0:7, where\na pronounced second component is observed. Fig. 4(c)\nshows the evolution of the lineshape for disconnected is-\nlands (\u0012= 0:5), connected islands with a strong com-\nponent from the smaller islands ( \u0012= 0:65), and at per-\ncolation where a signi\fcant fraction of the small islands\nhave merged with the connected regions ( \u0012= 0:8). The\nresults reproduce the sudden appearance of the sharp\npeak, which has been one of the most puzzling aspects\nof our experimental data. We \fnd that tuning Rchas\nlittle e\u000bect on the shape of S(Qr) at a given coverage,\nbut simply changes the overall length scale.\nTo conclude, we \fnd that a coalescence-dominated\nmodel explains the structural evolution during interface\nformation in BFO layer-by-layer growth on STO(001).\nThe growth mode is distinguished from standard layer-\nby-layer growth by a bimodal cluster size distribution,\nwhich we have observed experimentally and con\frmed\nthrough simulations.\nThis work was supported by the U.S. DOE O\u000ece of\nScience, O\u000ece of Basic Energy Sciences under DE-FG02-\n07ER46380. Use of the NSLS was supported by the U.S.\nDOE, O\u000ece of Science, O\u000ece of Basic Energy Sciences,\nunder Contract No. DE-AC02-98CH10886.[1] J. G. Amar and F. Family, Phys. Rev. Lett., 74, 2066\n(1995).\n[2] D. H. A. Blank, G. Koster, G. Rijnders, E. van Setten,\nP. Slycke, and H. Rogalla, Applied Physics A - Materials\nScience & Processing, 69, S17 (1999).\n[3] J. D. Ferguson, G. Arikan, D. S. Dale, A. R. Woll,\nand J. D. Brock, Physical Review Letters, 103, 256103\n(2009).\n[4] M. Opel, Journal of Physics D-Applied Physics, 45,\n033001 (2012).\n[5] S. A. Chambers, Surface Science Reports, 39, 105 (2000).\n[6] F. Family and P. Meakin, Physical Review Letters, 61,\n428 (1988).\n[7] F. Family and P. Meakin, Physical Review A, 40, 3836\n(1989).\n[8] J. A. Blackman and S. Brochard, Physical Review Let-\nters, 84, 4409 (2000).\n[9] X. Yu, P. M. Duxbury, G. Je\u000bers, and M. A. Dubson,\nPhysical Review B, 44, 13163 (1991).\n[10] G. Je\u000bers, M. A. Dubson, and P. M. Duxbury, Journal\nof Applied Physics, 75, 5016 (1994).\n[11] R. F. Voss, R. B. Laibowitz, and E. I. Allessandrini,\nPhysical Review Letters, 49, 1441 (1982).\n[12] L. Zhang, F. Cosandey, R. Persaud, and T. E. Madey,\nSurface Science, 439, 73 (1999).\n[13] G. Rijnders, D. H. A. Blank, J. Choi, and C. B. Eom,\nApplied Physics Letters, 84, 505 (2004).\n[14] J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B.\nOgale, B. Liu, D. Viehland, V. Vaithyanathan, D. G.\nSchlom, U. V. Waghmare, N. A. Spaldin, K. M. Rabe,\nM. Wuttig, and R. Ramesh, Science, 299, 1719 (2003).\n[15] G. Catalan and J. F. Scott, Advanced Materials, 21, 2463\n(2009).\n[16] J. D. Brock, J. D. Ferguson, and A. R. Woll, Metallur-\ngical and Materials Transactions A-Physical Metallurgy\nand Materials Science, 41A, 1162 (2010).\n[17] J. G. Amar, F. Family, and P. M. Lam, Phys. Rev. B,\n50, 8781 (1994).\n[18] M. Tomellini and M. Fanfoni, Journal of Physics-\nCondensed Matter, 18, 4219 (2006)."    },    {        "title": "1106.1269v1.Effect_of_Zn_substitution_on_morphology_and_magnetic_properties_of_copper_ferrite_nanofibers.pdf",        "content": "Effect of Zn substitution on mo rphology and magnetic properties of \ncopper ferrite nanofibers  \nWeiwei Pan, Fengmei Gu, Kuo Qi, Qingfang Liu, Jianbo Wang a) \nInstitute of Applied Magnetics, Key Labor atory of Magnetism and Magnetic Materials \nof Ministry of Education, Lanzhou Uni versity, Lanzhou 730000, People’ s Republic of \nChina \nAbstract: \nSpinel ferrite Cu 1-xZnxFe2O4 nanofibers over a compositi onal range 0 < x < 1 were \nprepared by electrospinning combined with sol-gel method. The influence of Zn2+ \nions substitution on morphology, structure, and magnetic propertie s of copper ferrite \nhas been investigated. The resu lts show that surface of CuFe 2O4 nanofibers co nsists of \nsmall open porosity, while surface of doped na nofibers reveals smooth and densified \nnature. With increasing Zn substitution, sa turation magnetization initially increases \nand then decreases with a maximum valu e of 58.4 emu/g at x = 0.4, coercivity and \nsquare ratio all decrease. The influence of substitution on magnetic properties is \nrelated with the cation distraction and excha nge interactions between spinel lattices. \n \nKeywords:  Electrospinning; Spinel ferrites;  Nanofibers; Magnetic properties; \n \n \na)  Electronic mail: wangjb@lzu.edu.cn\nTel: +86-0931-8914171 and Fax: +86-0931-8914160  \n1 \n 1. Introduction \n  Interesting in spinel ferrites ( MFe2O4, M = Co, Ni, Mn, Mg, Z n, etc) has greatly \nincreased in the past few years due to thei r remarkable magnetic,  electrical, optical \nproperties and compelling applications in many areas [1-3]. In particular, \nnanostructure spinel ferrites have gained considerable attention owing to their \nphysical properties are quite different from  those of bulk and particle [4-5]. Among \nvarious nanostructures, nanofibers are a promising candidate as building blocks in nanoscale sensors, nanophotonics, nanocompu ters, and so forth. Spinel ferrites \nnanofibers therefore have become an im portant subject of many research groups. \nElectrospinning provides a versatile, low cost, and simple process for fabricating nanofibers with diameters in the range from  several micrometers down to tens of \nnanometers. It has been studi ed and patented in 1940s by Formhals [6], and it has \nbeen used to synthesize a va riety of materials, including  organic composites, \ninorganic composites, and ceramic magnetic materials from now. NiFe\n2O4, CoFe 2O4, \nMgFe 2O4, and MnFe 2O4 ferrites nanofibers have been  synthesized by electrospinning \nand their structural and magnetic propert ies are investigated in detail [7-10]. \n  Copper ferrite (CuFe 2O4) is an interesting material and has been widely used for \nvarious applications, such as catalysts fo r environment [11], gas sensor [12], and \nhydrogen production [13]. Magnetic and electrical  properties of spinel ferrites vary \ngreatly with the change ch emical component and cation distribution. For instance, \nmost of bulk CuFe 2O4 has an inverse spinel structure, with 85% Cu2+ occuping B sites \n[14], whereas ZnFe 2O4 is usually assumed to be a completely normal spinel with all \n2 \n Fe3+ ions on B sites and all Zn2+ ions on A sites [15]. Zn2+ ions occupy preferentially \nA sites  while Fe3+ ions would be displaced from A sites for B sites. Zn-substitution \nresults to a change of cations in chemical  composition and a different distribution of \ncations between A and B sites. Consequently  the magnetic and elect rical properties of \nspinel ferrites will change with changi ng cation distribution. The influence of \nZn-substitution on magnetic and structure of Co 1-xZnxFe2O4 and Ni 1-xZnxFe2O4 \nnanofibers have been reported [16-18]. Howeve r, there are no repor ts on the synthesis \nand characterize of Cu-Zn ferrites nanofibers in the literatures. \n  In this paper, we successfully prepared Cu 1-xZnxFe2O4 ferrites nanofibers via \nelectrospinning technique combined with sol-gel, and the influence of Zn2+ \nsubstitution on structure and magnetic pr operties of ferrite  nanofibers was \ninvestigated. Magnetic state of  spinel ferrite nanofibers changes from ferrimagnetic to \nparamagnetic was observed. \n \n2. Experimental \n  In a typical procedure, a appropr iate amount of copper nitrate (Cu(NO 3)2·3H 2O), \nzinc nitrate (Zn(NO 3)2·6H 2O), and ferric nitrate (Fe(NO 3)3·9H 2O) with 1-x:x:2 molar \nratios of Cu:Zn:Fe were dissolved in N ,N-Dimetylformamide (DMF), followed by \nmagnetic stirring for 3 h to ensure the completely dissolution of metal salt. \nSimultaneously, poly (vi nyl pyrrolidone) (PVP, M W ≈ 1, 300, 000,) was dissolved in \nethanol (1 mL) and magnetic stirred for 3 h. Then 1 mL metal ni trates/DMF solution \nwas added slowly to the PVP/ethanol solu tion under continuous stirred for 2 h with \n3 \n PVP concentration of 8 wt% for electr ospinning. The obtained electrospinning \nsolution was loaded in a plastic syringe with a stainless steel needle, which was \nconnected to high-voltage equipment. A piece  of aluminum foil used as the collector \nwas placed in front of the needle tip as the negative electrode. The applied voltage \nwas 12.5 kV and the distance between syringe  needle tip and co llector was 14 cm. \nThe collected as-spun Cu 1-xZnxFe2O4 nanofibers were dried at 50 oC for 5h. The dried \nnanofibers subsequently were calcined at 650 oC for 3 h under an ambient atmosphere \nwith heating rate of 1 oC/min.  \nX-ray diffraction (XRD) patterns were recorded by a PANalytical diffractometer \nusing Cu Kα radiation with λ = 0.15418 nm. Morphology, diameter, and chemical \ncomposition of calcined nanofibers were ch aracterized by field emission scanning \nelectron microscopy (FE-SEM, Hitachi S-4800 ) and Energy disper sive spectroscopy \n(EDS). Magnetization properties measurement of the calcined nanof ibers were carried \nout on a vibrating sample magnetomete r (VSM, Lakeshore 7403, USA) with a \nmaximum applied field of 12 kOe at room temperature. \n \n3. Results and discussion \n3.1 XRD analysis  \nFigure 1 shows XRD patterns of Cu 1-xZnxFe2O4 (x = 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0) \nferrites nanofibers , which were prepared by electrospinning at room temperature. \nLattice parameter ( a), average grain size ( D) and X-ray density ( ρx) are listed in Table \n1. XRD patterns show that all peaks indexed to pure cu bic phase, where (220), (311), \n4 \n (400), (422), (511), and (440) represent the main crystal phase in CoFe 2O4 spinel \nferrite. As shown in inset of Figure 1, the position of peaks shifts slightly to lower \nscattering angles in the series Cu 1-xZnxFe2O4 nanofibers in accordance with the slight \nincrease of the lattice parameter  (Figure 2). This can be  predominantly attributed to \nthe replacement of smaller Cu2+ ions (0.72 Å) by larger Zn2+ ions (0.74 Å). The X-ray \ndensity (ρx) was calculated according to following relation; ρx = ZM/Na3, where, Z is \nthe number of molecules per unit cell (Z = 8), M is the molecular weight, N represent \nthe Avogadro’s number and a is the lattice parameter of the ferrite [19]. Figure 2 \nshows that as zinc content increases, X -ray density increases linearly. It directly \ncorrelates with the incr ease of molecular weight. The average grain size D were \ncalculated from x-ray line broadening of the (311), (511), (440) diffraction peaks using Scherrer equation. The value of D varies from 26.1 nm (CuFe\n2O4) to 32.1 nm \n(Cu 0.2Zn0.8Fe2O4), which denotes that zinc substitution leads to an increase of average \ngrain size of Cu 1-xZnxFe2O4 nanofibers. \n \n3.2 Morphological and componential analysis \n  Figure 3 shows representative SEM images of Cu 1-xZnxFe2O4 nanofibers with \ndifferent Zn content calcined at 650 oC for 3 h in air. The surface of CuFe 2O4 \nnanofibers is relatively rough, with a small quantity of op en porosity (as shown in \nFigure 3 (a)). When Zn content is in creases, smooth surface and densified \nmicrostructure of nanofibers are predominant. The diameter of Cu 1-xZnxFe2O4 \nnanofibers dependent on zinc content is pres ented in Figure 4. It can be seen that \n5 \n diameters of CuFe 2O4, Cu 0.8 Zn0.2Fe2O4 and ZnFe 2O4 nanofibers are about 110, 80, \nand 60 nm, respectively, clearly indicating Zn2+ substitution leads to an obvious \ndecrease of diameter. EDS is used to confirm chemical composition of Cu 1-xZnxFe2O4 \nnanofibers as shown in Figure 5, detail data  is given in Table 2. The number of Fe \nelement do not change much with increasing zinc content, but the numbers of Cu and \nZn element change obviously with zinc content. For instance, Cu 0.6Zn0.4Fe2O4 \nnanofibers involve 9.6 at% of Cu ions  and 6.8 at% of Zn ions, while ZnFe 2O4 \nnanofibers contain only O, Fe, and Zn elem ents. Consequently with increasing zinc \ncontent, Cu content is decreases and Zn content is increases gradually. \n \n3.3 Magnetic properties \n  Tytical hysteresis loops for zinc substituting CuFe 2O4 nanofibers at room \ntemperature are shown in Figure 6. The values of saturation magnetization ( Ms), \ncoercivity ( Hc), remanence (M r) and square ratio Mr/Ms are given in Table 1. The \nmagnetic properties of nanofibers vary with changing zinc content. The variation of \nmagnetic properties of Cu 1-xZnxFe2O4 nanofibers can be unders tood in term of cation \ndistribution and exchange interac tions between spinel lattices.  \nThe Cu 1-xZnxFe2O4 nanofibers with x ≤ 0.6 exhibit ferromagne tic behavior, whereas \nother nanofibers display paramagnetic charact er with zero coercivity, zero remanence \nand non-saturated magnetization. The saturati on magnetization initially increases with \nincreasing zinc content to reach a maximum (58.4 emu/g) and then decreases. The \nincrease in saturation magnetization may be attributed to the fact that, small amount \n6 \n of Zn2+ ions substituted for Cu2+ occupy A sites displacing Fe3+ ions from A sites to B \nsites, which increasing the content of Fe3+ ions in B sites. This leads to an increase of \nmagnetic moment in B-site and a decrease of magnetic moment in A-site. So the net \nmagnetization increases, which is consistent with the increase of saturation \nmagnetization. With further increase nonmagnetic Zn2+ ions content, an increasing \ndilution in A sites is present, which results  to the collinear ferromagnetic phase breaks \ndown at x = 0.4. For Cu 1-xZnxFe2O4 nanofibers (x = 0.6, 0.8, and 1.0), the triangular \nspin arrangement on B-sites is suitable and th is causes a reduction in A-B interaction \nand an increase of B-B in teraction. Therefore, the decrease of saturation \nmagnetization can be explained on the basis of three sublattice Yafet-Kittle model [20].  \n  As shown in Table 1, coercivity ( Hc) and square ratio ( Mr/Ms) continuously reduced \nwith increasing Zn2+ ions content. These magnetic behaviors of ferrite depend \nintensely on the spinel structure. For instance, normal spinel ferrite shows an antiferromagnetically ordering, while inve rse spinel ferrite shows a ferromagnetic \nordering [21]. With increasing Zn\n2+ ions concentration, a transformation from inverse \nspinel structure of CuFe 2O4 ferrite to normal spinel structure of ZnFe 2O4 ferrite will \narises gradually. Consequently, the decrease tendency of Hc and M r/Ms is directly \nrelated to the paramagnetic relaxation effect. \n \n4. Conclusion  \n  In conclusion, a series of single-phase Cu 1-xZnxFe2O4 (x = 0.0, 0.2, 0.4, 0.6, 0.8 and \n7 \n 1.0) nanofibers were synthesized by elect rospinning combined with sol-gel method. \nThe lattice parameter, average grain size, and X-ray density all are found to increase \nwith increasing zinc content. The substitu tion of zinc ions imp roves the morphology \nof Cu-Zn ferrites nanofibers and decreases the diameter of nanofibers. The magnetic \nproperties of Cu 1-xZnxFe2O4 nanofibers depend on zinc co tent, which is consistent \nwith the cation distraction and exchange interactions between spinel lattices. \n \nAcknowledgements \n  This paper is supported by National Science Fund of China (11074101), and the \nFundamental Research Funds for the Central Universities (860080). \n \n \n \n \n \n \n \n \n \n \n \n \n8 \n References \n[1] J. Fang, N. Shama, L.D. Tung, E.Y . Shin, C.J. O’Connor, K.L. Stokes, et al., J. \nAppl. Phys. 93 (2003) 7483-7485. \n[2] Y .C. Wang, J. Ding, J.H. Yin, B.H. Li u, J.B. Yi, S. Yu, J. Appl. Phys. 98 \n(2005)124306-7.  \n[3] M.A. Ahmed, N. Okasha, S.I. Ei-Dek, Nanotechnology 19 (2008) 065603-7. \n[4] S. Ayyappan, S. Mahadevan, P. Chandramohan, M.P. Srinivasan, J. Philip, B. Raj, J. Phys. Chem. C  114 (2010) 6334-6341.  \n[5] V . Sepelak, I. Bergmann, A. Feldhoff, P. Heitjans, F. Krumeich, D. Menzel, et al, J. Phys. Chem. C 111 (2005) 5026-5033. \n[6] A. Formhals, U.S. Patent No. 1, 975 504 (1934). \n[7] D. Li, T Herricks, Y .N. Xia,  Appl. Phys. Lett. 83 (2003) 4586-4588. \n[8] M. Sangmanee, S. Maensiri, Appl. Phys. A 97 (2009) 167-177. \n[9] S. Maensiri, M. Sangmanee, A. Wiengmoon, Nanoscale Res. Lett. 4 (2009) 221-228. \n[10] J. Xiang, X.Q. Shen, X.F. Meng, Mater. Chem. Phys. 114 (2009) 362-366. \n[11] T. Tsoncheva, E. Manova, N. Velinov, D. Paneva, M. Popova, B. Kunev, et al., \nCata.  Commun. 12 (2010) 105-109. \n[12] S.W. Tao, F. Gao, X.Q. Liu, O.T. Sørensen, Mater. Sci. Eng., B 77 (2000) \n172-176. \n[13] K. Faungnawakij, Y . Tanaka, N. Shimoda, T. Fukunaga, R. Kikuchi, K. Eguchi, Appl. Catal. B 74 (2007) 144-151. \n9 \n [14] X. Zuo, A. Yang, C. Vittoria, V .G. Harris,  J. Appl. Phys. 99 (2006) 08M909-3.  \n[15] C.W. Yao, Q.S. Zeng, G.F. Goya, T. Torres, J.F. Liu, J.Z. Jiang, J. Phy. Chem. C \n111 (2007) 12274-12278. \n[16] X.Q. Shen, J. Xiang, F.Z. Song , M.Q. Liu, Appl. Phys. A 99 (2010) 189-195. \n[17] J.H. Nam, Y .H. Joo, J.H. Lee, J.H.  Chang, J.H. Cho, M.P. Chun, B.I. Kim, J. \nMagn. Magn. Mater.  321 (2009) 1389-1392. \n[18] J. Xiang, X.Q. Shen, F.Z. Song , M.Q. Liu, Chin. Phys. B 18 (2009) 4960-4965. \n[19] M.A. Gabal, Y .M. Al Angari, Mater. Chem. Phys.  118 (2009) 153-160. \n[20] M. Ajmal, A. Maqsood, J. Alloy. Comp.  460 (2008) 54-59. \n[21] M.H. Khedr, A. A. Farghali, J. Mater. Sci. Technol, 21 (2005) 675-680. \n \n \n \n \n \n \n \n \n \n \n \n \n10 \n  \nFigure 1. X-ray diffraction patterns of Cu 1-xZnxFe2O4 (0.0 ≤  x ≤ 1.0) nanofibers. The \ninset is XRD patterns from 2 θ = 33o to 40o \n \nFigure 2.  Lattice parameter a and X-ray density ρ x as a function of the zinc content \nx. \n \nFigure 3. SEM images of Cu 1-xZnxFe2O4 nanofibers with differen t Zn contents (x): (a) \n0.0; (b) 0.2; (c) 0.4; (d) 0.6; (e) 0.8 and (f) 1.0. \n11 \n  \nFigure 4. Diameter of Cu 1-xZnxFe2O4 (0.0 ≤  x ≤ 1.0) nanofibers depends on the zinc \ncontent.  \n \nFigure 5. EDS spectra of Cu 1-xZnxFe2O4 nanofibers: (a) 0.0; (b) 0.4; and (c) 1.0. \n \nFigure 6. Magnetic hysteresis loops for Cu 1-xZnxFe2O4 (0.0 ≤ x ≤ 1.0) nanofibers at \nroom temperature. \n \n12 \n Table 1: Table 1. Structural and magnetic parameters for Cu 1-xZnxFe2O4 nanofibers \nsystem: lattice parameter ( a), average grain size (D), X-ray density ( ρx), saturation \nmagnetization (M s), coercivity ( Hc) and square ratio ( Mr/Ms). \nSample \nCu1-xZnxFe2O4Lattice \nparameter \na (Å) Average \ngrain \nsize D \n(nm) X-ray \ndensity\nρx \n(g/cm3)Ms\n  \n(emu/g)Hc  \n \n(Oe) Mr/Ms\nx-0.0 8.3766 26.1 5.42 31.8 723.5 0.47 \nx-0.2 8.3810 26.9 5.46 55.3 127.8 0.28 \nx-0.4 8.3893 30.8 5.48 58.4 84.3 0.24 \nx-0.6 8.4088 31.3 5.50 32 35.2 0.13 \nx-0.8 8.4110 32.1 5.54 - - - \nx-1.0 8.4137 30.8 5.58 - - - \n \n \n \n \n \n \n \n \n \n13 \n Table 2: The elemental analysis results of Cu 1-xZnxFe2O4 (0.0 ≤  x ≤ 1.0) nanofibers by \nEDS.  \n \nElemental compositions (at %) Samples \nformula \nCu1-xZnxFe2O4Fe (1-x) (Cu) (x)(Zn) \nCuFe 2O4 29.3 14.1 0.0 \nCu0.8Zn0.2Fe2O4 30.0 13.2 3.8 \nCu0.6Zn0.4Fe2O4 29.6 9.6 6.8 \nCu0.4Zn0.6Fe2O4 22.7 4.7 7.7 \nCu0.2Zn0.8Fe2O4 21.7 2.8 9.1 \nZnFe 2O4 25.1 0.0 11.2 \n \n14 \n "    },    {        "title": "1712.05928v1.Contrasting_magnetoelectric_behavior_in_multiferroic_hexaferrites_as_understood_by_crystal_symmetry_analyses.pdf",        "content": "Contrasting magnetoelectric behavior in multiferroic \nhexaferrites as understood by crystal symmetry analyses  \nY . S. Chai,1,2,3* S. H. Chun,1 J. Z. Cong2, and Kee Hoon Kim1* \n1 Center for Novel States of Compl ex Materials and Rese arch (CeNSCMR) and \nInstitute of Applied Physics, Departmen t of physics and astronomy, Seoul National \nUniversity, Seoul 151-747, Republic of Korea \n2 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, \nChinese Academy of Sciences, Beijing 100190, China  \n3 Department of Applied Physics, Chongqing University, Chongqing, 401331, China \n \n*Correspondence and requests for materials should be addressed to K. H. Kim ( optopia@snu.ac.kr ) and Y . S. Chai \n(hawkchai@gmail.com) \n \nMagnetoelectric (ME) properties under rotating magnetic field H are \ncomparatively investigated in two representative hexaferrites Y-type \nBa0.5Sr1.5Zn2(Fe 0.92Al0.08)12O22 and Z-type Ba 0.52Sr2.48Co2Fe24O41, both of which \nhave exhibited a similar transverse conica l spin structure and giant ME coupling \nnear room temperature. When the external H is rotated clockwise by 2 π, \nin-plane P vector is rotated clockwise by 2 π in the Y-type hexaferrite and \ncounterclockwise by 4 π in the Z-type hexaferrite . A symmetry-based analysis \nreveals that the faster and opposite rotation of P vector in the Z-type hexaferrite \nis associated with the existence of a mirror plane perpendicular to c-axis. \nMoreover, such a peculiar crystal symm etry also results in contrasting \nmicroscopic origins for the spin-driven ferroelectricity; only the inverse DM \ninteraction is responsible for the Y-type hexaferrite while additional p-d \nhybridization becomes more important in  the Z-type hexaferrite. This work \ndemonstrates the importance of the cryst al symmetry in the determination of \nME properties in the hexaferrites and provides a fundamental framework for \nunderstanding and applying the giant ME coupling in various ferrites with hexagonal crystal structure.  Introductions \nMultiferroics are materials that combine the magnetic and ferroelectric orders1-3. \nTheir possible magnetoelectric (ME) effect s—the response of electric polarization P \nto a magnetic field ( H) or magnetization to an electric field ( E) —have attracted great \nattention because of the potential a pplication for novel electronic devices4-5. In \nparticular, the spin-driven multiferroics, of which ferroelectricity has a magnetic origin, have been a focus due to their strong and versatile ME effects\n6. Therefore, in \nterms of both fundamental and technological points of view, it will be necessary to \nunderstand the microscopic origin of ferroelectric polarization P as well as the \naccompanied ME behaviors.  \nIt is known that the microscopic origin of polarization and related ME responses \nare correlated with the magnetic orderi ng pattern or the a ssociated magnetic \nsymmetry6. On one hand, several microscopic ME  mechanisms based on the magnetic \nordering pattern have been proposed such as the spin-current (or inverse \nDzyaloshinskii-Moriya (DM) interaction) P∝PDMe12×(S1×S2)7,8, exchange-striction \nP∝ PESe12(S1·S2) where  e12 is connecting vector of adjacent spin pair S1 and S2 9,10, and \np-d hybridization P∝ Ppde1(e1•S1)2 where vector e1 connects the transition metal and \nits neighbor ligand atom11-13. On the other hand, direct relationship between the ME \neffect and a magnetic point group is es tablished since the initial work on the \nmagnetoelectric Cr 2O314. Nevertheless, the ME effect s in pure magnetoelectrics is \nvery weak while it is very strong in so me of the spin-driven multiferroics because \ntheir spin order sensitively responds to applied H. One of the highest ME coefficients (=d P/dH) in spin-driven multiferroics were \nfound in hexaferrite systems both at low temperatures and at room temperature. The crystal structure of hexafe rrites can be described as the stacking sequence of \nspinel-like S, tetragonal-like T,  and rhombohedral  like R blocks, i.e., S-block  \n(Me\n2+Fe4O8; Me2+= divalent metal ion), T [(Ba,Sr) 2Fe8O14] and R [(Ba,Sr)Fe 6O11]2- \nstructure blocks along [001] direction. According to the different sequences, \nhexaferrites are classified into six main types depending on their chemical formulas \nand stacking sequences: M-type [(Ba,Sr)Fe 12O19], W-type [(Ba,Sr)Me 2Fe16O27], \nX-type [(Ba,Sr) 2Me 2Fe28O46], Y-type [(Ba,Sr) 2Me 2Fe12O22], Z-type \n[(Ba,Sr) 3Me 2Fe24O41], and U-type [(Ba,Sr) 4Me 2Fe36O60]. In particular, Y-type and \nZ-type have STSTST  and STSRSTSR sequences, respectively (Fig. 1a)15. For every \nS and T block, there are space inversion centers in the middle while there is mirror \nplane ( ⊥[001]) for R block instead. Therefore, Y-type has a -3m point group while \nZ-type has a 6/ mmm  point group with the extra mirro r plane. Both of them are \ncentrosymmetric without ferroelectricity in high temperature. However, a so-called transverse cone spin configuration at low T and low H could break the space inversion \ncenter in those hexaferrites  and induce the in-p lane polarization. Due to the very \nsensitive response of transverse  cone to the small external H, strong ME effects are \nfound in M-type Ba(Fe,Sc)\n12O19, Y-type (Ba,Sr) 2Me 2(Fe,Al) 12O22, Z-type \n(Ba 1-xSrx)Co 2Fe24O41, and U-type Sr 4Co2Fe36O60 with transverse cone phase up to \nroom T. The accompanied converse ME effects, i. e., electric field controlled large \nmagnetization reversal or changes ar e also demonstrated in Y-type Ba0.5Sr1.5Zn2(Fe 0.92Al0.08)12O22 and Z-type Ba 0.52Sr2.48Co2Fe24O41, respectively. \nHowever, their ME behaviors ar e different—direct in-plane H reversal can fully \nreverse the P value in Y-type, but not in Z-type  when TC phases persist around zero \nfield, leading to a giant ME effect at zero- H for Y-type and a large ME effect at finite \nH for Z-type. \nTo thoroughly understand the physical orig in of the difference in their ME \nbehaviors, the lattice symmetry has to be considered. The ferroelectric polarization \narises from the reduced lattice symmetry and thereby its emergence is constrained by the original symmetry. However, this aspect has not been accounted for the description of ME characteristics based on the magnetic structure. We find that the \noriginal lattice symmetry in hexaferrite systems has a prominent role to determine dominance of ME behaviors originating fr om the possible microscopic mechanisms. \nIn this work, distinct ME behavi ors in the ME hexaferrites Y-type \nBa\n0.5Sr1.5Zn2(Fe 0.92Al0.08)12O22 (BSZFAO) and Z-type Ba 0.52Sr2.48Co2Fe24O41 (BSCFO) \nare observed and attributed to  the additional mirror plane in the crystal symmetry of \nZ-type, based on a local-symmetry-based theoretical analysis. The microscopic \norigins of dominant P are found to be different be tween two system s due to the \nexistence of additional mirror plane in Z. In  particular, the dominant polarization of \nZ-type is induced by non-spin current mechanisms.  \n \nResults Magnetic and ME behavior s of Y-type and Z-type.  The Y-type and Z-type \nhexaferrites have very similar in-plane  magnetic field induced commensurate \ntransverse conical (TC) magnetic phase (Fig. 1)16-18, but distinct cr ystal symmetries. \nTo understand their magnetic structures, we will follow an approximation used in the \nprevious studies15-19. All the spins are conveniently divided into alternating stacked L \nblocks with large spin moments ( μL) and S blocks having small spin moments ( μS) \n(Fig. 1), which are different from the R, S, and T structure  blocks. In each magnetic \nblock, all the spins are assumed to be parallel for simplicity. According to the previous neutron diffraction measurements around H = 0, Y-type hexaferri te at low temperature \n(< 100 K) after a high H history and Z-type below 400 K have similar commensurate \nTC phase with propagation vector k\n0 = (0,0,3/2) and k0 = (0,0,1) respectively (Fig. 1). \nAs can be seen in Fig. 1, both TC phase s have exactly the same magnetic block \nperiodicity and spin texture that L blocks and S blocks have antiferromagnetically \nordered in-plane components and out-of-pl ane components, respectively, whereas \nthey are antiparallel along the cone directions. In terms of the lattice symmetry of magnetic blocks, it is the same for their  S blocks but different for L blocks in that \nthere is a space inversion center in the L blocks of Y-type while a mirror plane \n(⊥[001]) in that of Z-type. \nRegardless of the differen ces in their crystal symmetr ies, previous studies found \nthat the TC phases in both systems are ferroelectric with in-plane polarization\n15,20. It \nwas widely believed that the microscopic origin of P in all the TC phases was the \nspin-current mechanism:  P∝Σk0×(μL×μS)/| k0|. This mechanism will lead to a P vector perpendicular to  k0 (//[001]) and cone axis (// H for large H, as shown in Fig. 1a), very \nsimilar to the case of CoCr 2O421. To verify such orthogonal relationship between P, H \nand k0 direction, there are two methods, 1) the reversing of in-plane H can lead to a \nreversal of the P vector (Fig. 2a) if the spin helicity Σk0×(μL×μS) could be preserved \non passing the H = 0 point22. 2) horizontally  rotating  H can lead to the projection of P \nvector along a fixed direction to show a sinusoidal behavior as the function of rotating \nangle φ (Fig. 2b)22. In some Y-type hexaferrites, the TC induced P has passed the \nabove two tests15,22. On the other hand, some recent in-situ X-ray diffraction studies \non the Z-type hexaferrite impl y a violation to these tests23. In this work, we performed \nsimilar experiments on both Y and Z-type hexaferrites. \nWe characterize the ME properties of the Y-type BSZFAO at 30 K where the TC \nphase is stabilized between ± 15 kOe. Figure 2c shows that the Py can be fully \nreversed by the revers al of perpendicular Hx, indicating a c onservation of P⊥H⊥k0 \nrelationship during such process (Fig. 2a). Figure 2d demonstrate the angle φ \ndependent Py(φ) under rotating H of 0.2 and 2 kOe for BSZFAO. Here, φ is defined as \nthe relative angle between H and x-axis in the clockwise direc tion. It is very clear that \nPy shows nearly cos φ dependent behaviors with small hysteresis for both field values, \nalso indicating the orthogonal relationship between P, H and k0 directions in Fig. 2b. \nThis behavior is essentially the same as that reported for other Y-type hexaferrites18,22. \nThe above ME behaviors strongly suggests th at the spin-current model holds for the \nTC induced P in our Y-type hexaferrite.  \nOn the contrary, the inverse of  P vector by H reversal has never been reported in any Z-type hexaferrites with  TC phase stabilized around H = 017,18,24. We also verify \nthis in our Z-type BSCFO at 305 K where the TC phase should be  stabilized within \nthe range of ± 20 kOe18,24. As shown in Fig. 2e, ± Py just quickly approach zero and \nrecovers to a slightly smaller magnitude with the same sign after Hx reversal, similar \nto the previous investigations. A possibl e explanation could be the effects from \nunknown magnetic phase or exotic  domain structure around Hx = 025. To eliminate \nthese possibilities, we performed the angle φ dependent polarization measurement \nalong y direction under hor izontally rotating H of 5 and 12 kOe for BSCFO at 305 K. \nHere, φ is defined as the re lative angle between H and x-axis in the clockwise \ndirection. These two magnetic fields are enough to keep the sample a single TC \ndomain state following the H direction during the rotation processes. Surprisingly, the \nPy show roughly the cos(2 φ) dependent behaviors for both H values. This novel \nbehavior is in sharp contra st to the case of Y-type hexaferrites. Due to the large \nbackground signal in ME current at 305 K, the absolute Py value cannot be reliably \nestimated in these measurements. However, we can still conclude that the dominant \ncos(2 φ) dependent Py in the TC phase of Z-type  BSCFO strongly indicates a \nmicroscopic origin different from the expected spin-current model. \nTo map out the complete trajectory of P vector under reversing and rotating H, \nwe polished the sample in roughly c ubic shape with two pairs of orthogonal \nelectrodes along x and y directions, respectively, as shown in Fig. 3c. Both Px and Py \ncan be monitored simultaneously  in this configuration. The P measurements were \nperformed at 10 K to reduce the backgr ound signal and obtain the absolute P values. As shown in Figs. 4a and 4b, neither Px nor Py reverses its signs after Hx reversal. The \nestimated directions of P vector are almost the same for ±4 kOe, consistent with the \nresults at 305 K. In addition, the magnitude of Py is much larger than that of Px, which \nroughly keeps the P⊥H (⊥k0) relationship. Then, to further test the orthogonal \nrelationship, we measured Px and Py simultaneously in horizontally rotating H of 4 \nand 10 kOe (Figs. 3c-g). More surprisingly, the Px and Py show dominating sin(2 φ) \nand cos(2 φ) behaviors respectively, with a re lative phase difference of about 45 \ndegree. As a result, the calculated P vector rotates in oppos ite direction and roughly \ntwice faster in speed than that of the H vector, as schematically illustrated in Fig. 3h. \nThere may be some small sin φ or cos φ components which are close to our technical \nlimitations. This ME behavior resembles the case of triangular-lattice helimagnet \nMnI 2, which exhibits the P vector smoothly rotates clockwise twice while the H \nvector rotates counterclockwise onc e at certain critical field region26. This feature has \nbeen interpreted in terms of H switching of the multiple in-plane propagation vectors \ndomains. However, BSCFO only hosts an out-of-plane k0 = (0,0,1) single domain state \nfor the above in-plane H values. Nevertheless, the ME behaviors and ferroelectricity \nin Z-type BSCFO are beyond the prediction of  spin-current mechanism and must have \nother physical origins.  \n Symmetry analysis of the electric dipole produced by two spins.  So far, no \npractical first-principle calcu lation approach is possible to resolve the microscopic \norigin of spin-induced ferroelectricity in any ME hexaferrite systems due to their extremely large unit cells and complex s ite-by-site spin structures. Instead, we will \nanalyze their microscopic origin of P and ME behaviors by some recent developed \nlocal symmetry theories27-29.  \nIn general, the local electric-dipole p caused by a spin pair μ1 and μ2 or by one of \nthe spins with its surroundings can be univers ally expressed as th e quadratic functions \nin Einstein convention:  \nβ α αβγ γμμj i ij ijP p=  andβ α αβγ γμμi i ii iiP p=                                     ( 1 )  \nwhere α, β, γ run over all the Cartesian coordinates, x, y, z; i, j run over the site or spin \nlabels, 1,2. αβγ\niiP andαβγ\nijP can be regarded as a kind of si ngle-spin tensor and two-spin \ntensor, respectively, accor ding to their definitions28. Then total polarization Ptot of a \nmultiferroics with spin-induced ferroelectric ity can be concisely expressed by those \nlocal ME tensors as:  \n \n≠ ≠+ = + ∝\njiii i ii j i ij\njiiii ij P P p p P\ntotβ α αβγ β α αβγ γ γ γμμ μμ                         ( 2 )  \nWhere α, β, γ run over all the Cartesian coordinates, x, y, z; i, j run over the site or \nspin pair labels in a magnetic unit cell of th e multiferroics. Note that the forms of the \nlocal ME tensors are site or site-pair-specific and dictated only by the lattice symmetry of the site or site-pair since they are third rank polar  tensors [see the \nSupplementary Note 1]. The matrix form of each tensor can be mathematically \ntransformed by symmetry operators and simplified according to the local crystal symmetries, see Supplementary Note 1 for deta iled calculations. Therefore, the Eq. (2) \nof a multiferroics can also be  simplified according to its crystal symmetry. Hexaferrite systems have very high lattice symmetry which will put severe symmetric constraints \nover the forms of their local ME tensors. Th erefore, we may adopt this method in both \nhexaferrite systems to calculate the Ptot with a much-simplified analytical form. \nMoreover, all three known mechanisms ar e the special cases of the local ME \nsingle spin tensor αβγ\niiP and two-spin tensorαβγ\nijP27,28. In particular, the spin-current \nmechanism P∝PDMe12×(μ1×μ2) will allow a two-spin tensorαβγ\nijP with antisymmetric \nmatrix form: \n\n\n\n\n− −− −− −\n0 , ,0 ,0,,,0 0 0,,,0, 0, , 0\n12 12 12 1212 12 12 1212 12 12 12\ny z x zy z x yx z x y\nDM\nee e ee e eee e e e\nP                                      ( 3 )  \nwhile exchange-striction P∝PESe12(μ1·μ2) allows a two-spin tensorαβγ\nijP with non-zero \ndiagonal form only:  \n\n\n\n\nz y xz y xz y x\nES\neeeeeeeee\nP\n12 12 1212 12 1212 12 12\n,, 0 00 ,, 00 0 ,,\n                                   ( 4 )  \nwhere  e12 =(e12x, e12y, e12z) is the connecting vector of μ1 and μ2. \nIn contrast, p-d hybridization mechanism P∝ Ppdei(ei•μi)2 gives a single spin \ntensorαβγ\niiP with the symmetric matrix form: \n\n\n\n\niziziz iyiziz ixiziz iziyiz iyiyiz ixiyiz izixiz iyixiz ixixiziziziy iyiziy ixiziy iziyiy iyiyiy ixiyiy izixiy iyixiy ixixiyizizix iyizix ixizix iziyix iyiyix ixiyix izixix iyixix ixixix\npd\neeeeeeeee eeeeeeeee eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee\nP\n, , , , , ,, , , , , ,, , , , , ,\n          (5) \nwhere vector ei = (eix, eiy, eiz) connects the μi and its neighbor ligand atom ( i = 1,2). \nTherefore, from the non-zero components of the simplified Ptot, we could also deduce \nits microscopic origin by comparing it with Eqs. (3)-(5).  \n Calculation of spin-driven polarization and re lated ME properties of both \nHexaferrites.  To conveniently compare with the experimental observations, we will \ncalculate the angle dependent Ptot of both hexaferrites under in-plane rotating  H since \nthe spin configuration is fixed to a single domain TC state. Instead of using the full \natom-by-atom crystal and spin models, we first adopt two simplified generic models \nfor the crystal structures of the Y-type and Z-type respectivel y, which preserve the \ncrystal point groups in the paramagneti c phases with the essential symmetric \noperations (Fig. 4a). The lattices are divide d according to the magne tic blocks instead \nof structure blocks, where each magnetic bl ock is represented by two atomic layers \nconnected by either a spatial inversion cente r or a mirror in the middle (Fig. 4). Each \natomic layer consists of three identical tr ansition metal ions forming an equilateral \ntriangle (we don’t consider the difference between Fe and Co ions to simplify the models). Moreover, a three- fold rotation along [001] di rection and three mirrors \nincluding [001] axis are also allowed fo r each layer, and subsequently for these \nmodels, as shown in Figs. 1a, 1b and 4b.  \nThen, we assume that for the initial H\nx, the initial TC configuration would have a \nspin configuration in one magnetic unit cell: \nμS1= ) ,0, (z\nSx\nS μ μ − − , \nμS2= ),0, (z\nSx\nSμ μ−  \nμL1=) 0, ,(y\nLx\nLμ μ−  \nμL2=) 0,,(y\nLx\nLμμ                                                               ( 6 )  Where μS1, μS2, μL1 and μL2 are the total spin vectors in S1, S2, L1 and L2 blocks within \none magnetic unit cell respect ively, the coefficientsx\nSμ− ,z\nSμ,x\nLμandy\nLμ are the initial \nspin components in μS2 and μL2 along three Cartesian coordinates. As to describe their \nrotating angle dependent spin patterns, we a ssume that TC is rigid and the cone axis \nfollows precisely the in-plane H direction:  \nμS1(φ)= ) , sin, cos (z\nSx\nSx\nS μφ μφ μ − − , \nμS2 (φ)= ),sin, cos (z\nSx\nSx\nS μφ μφ μ−  \nμL1 (φ)= ) 0, cos sin ,sin cos( φ μφ μφ μφ μy\nLx\nLy\nLx\nL − − −  \nμL2(φ)= ) 0, cos sin ,sin cos( φ μφ μφ μφ μy\nLx\nLy\nLx\nL + − +                                ( 7 )  \nwhere φ is the angle between H and x-axis defined in Fig. 3c. Finally, we assume that \nall the six atoms in each block have exactly the same spin for simplicity, or 1/6 of \ntotal moment within a block. With the above  lattice and spin configurations for both \nY and Z-type hexaferrites, we could calculate their φ dependent Ptot according to Eq. \n(2). \nWe find that the summation of si ngle-spin tensor in one block \niiiPαβγhas a \nsimplified matrix form according to the la ttice symmetries, as shown the Table 1 (See \nSupplementary Note 2 for detailed discussion). The matrix forms of \niiiPαβγin S2 and \nL2 can be calculated  in that they ar e connected with S1 and L1 respectively by space \ninversion operator. In  this case, only the \niiiPαβγof L blocks in Z-type allow nonzero \nmatrix components with one independent co efficient due to the existence of mirror \nm⊥[001] in the L blocks of Z-type. Every summation of \niiiPαβγin the  blocks with \nthe existence of a space inversi on center are exactly zero.  Table 1. The simplified \niiiPαβγin S1 and L1 magnetic blocks for both hexaferrite \nsystems. \n S1 L 1 \nY-type \n\n\n\n\n0,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,0 \n\n\n\n\n0,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,0 \nZ-type \n\n\n\n\n0,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,0 \n\n\n\n\n− −−\n0,0,0 0,0,0 0,0,00,0,00,0, 0,,00,0,00,,00,0,\n60 00 0\na aa a \n \nFor the summation of two-spin tensor \n''\niiiiPαβγ, there are inter-block and \nintra-block cases, see Fig. 4c. Whatsoever, \n''\niiiiPαβγhave simplified matrix forms \naccording to the lattice symmetries, as show n the Table 2 (see also the Supplementary \nNote 2). Other inter and intra-block tensor  summations can be inferred accordingly \nvia space inversion or mirror operation. \nTable 2. The simplified \n''\niiiiPαβγin S1 and L1 and between S1 and L1 magnetic \nblocks for both hexaferrite systems. \n S1-S1 L 1-L1 S 1-L1 \nY-type \n\n\n\n\n− − 0,0,00,,00,0,0,,00,0,0 0,0,00,0, 0,0,0 0,0,0\n3\n0 000\nc ccc \n\n\n\n\n− − 0,0,00,,00,0,0,,00,0,0 0,0,00,0, 0,0,0 0,0,0\n3\n1 111\nc ccc \n\n\n\n\n− −−\n2 3 32 2 2 22 2 2 2\n,0,0 0,,0 0,0,0,,0 ,0, 0,,00,0, 0,,0 ,0,\n3\nd c cc b a ac a b a \nZ-type \n\n\n\n\n− − 0,0,00,,00,0,0,,00,0,0 0,0,00,0, 0,0,0 0,0,0\n3\n0 000\nc ccc \n\n\n\n\n− −− −−\n0,0,00,,00,0,0,,00,0, 0,,00,0, 0,,00,0,\n3\n1 11 1 11 1 1\nc cc a ac a a \n\n\n\n\n− −−\n2 3 32 2 2 22 2 2 2\n,0,0 0,,0 0,0,0,,0 ,0, 0,,00,0, 0,,0 ,0,\n3\nd c cc b a ac a b a \n \nNext, by substituting Eq. (7), Tables 1 and 2 into Eq. (2) and summing over one \nmagnetic unit cell of TC phases, we calculated the angle φ dependent total \npolarization Ptot(φ). Contribution of each term to the polarization can be obtained \nseparately, as shown in Tables 3 and 4. \nTable 3. The calculated net polarization in each kind of magnetic block according \nto the simplified \niiiPαβγin Table 1 for both hexaferrite systems.  S  L  \nY-type 0 0 \nZ-type 0 )0,2cos,2sin(32\n0 φ φ μμ −y\nLx\nLa  \n \nTable 4. The calculated net pol arization from the inter and intra-blocks according \nto the simplified \n''\niiiiPαβγin Table 2 for both hexaferrite systems. \n S-S  L-L  S-L  \nY-type 0 0 )0, cos, (sin31\n3 φ φ μμy\nLz\nSc  \nZ-type 0 )0,2cos,2sin(31\n1 φ φ μμ −y\nLx\nLa  \n)0, cos, (sin31)0,2cos,2sin(61\n32\nφ φ μμφ φ μμ\ny\nLZ\nSy\nLx\nS\nca\n+− −\n \n \nFor the Y-type hexaferrite with TC phase, the angle dependent polarization PY(φ) \nonly comes from inter-block two-spin term: \n)0, cos, (sin31)( )(3 φ φ μμ φ φy\nLz\nS SL Y c P P = =                                        (8)  \nwhere c3 has the form2' '\n3zyy\niizxx\nii P Pc+= , i and i’ is a spin-pair between adjacent L and S \nblocks (see Supplementary Note 2). Equation (8) predicts that the P vector rotates \ncoordinately under in-plane H-rotation with P⊥H⊥[001] relationship, fully consistent \nwith the experimental observations in Figs. 2c&d for BSZFAO as well as many other \nY-type ME hexaferrite systems. Note that, non-zero c3 only allows in the spin-current \nmechanism (see Eq. (3)), not in the exchange -striction mechanism (see Eq. (4)), ruling \nout the possible cooperative contribution of the magnetostriction to  the polarization of \nY-type18.  \n For TC phase induced polarization PZ in the Z-type hexaferrite, things are quite different: 1) The single-spin term in L blocks contributes nonzero polarization PL with \nsinusoidal 2 φ dependent:   \n)0,2cos,2sin(32)(0 φ φ μμ φ − =y\nLx\nL L a P                                  ( 9 )  \nwhere a0 has the form\n40yyx\niiyxy\niixyy\niixxx\nii P P P Pa− − −= , i is a site in L block. 2) The \ntwo-spin terms also generate  non-zero polarization between L-L (PLL) with sinusoidal \n2φ dependence and L-S blocks ( PLS) with both  sinusoidal φ and 2 φ dependences:  \n)0,2cos,2sin(31)(1 φ φ μμ φ − =y\nLx\nL LL a P , \n)0, cos, (sin31)0,2cos,2sin(61)(3 2 φ φ μμ φ φ μμ φy\nLZ\nSy\nLx\nS LS c a P + − −=           ( 1 0 )  \nwhere a1, a2 and c3 have the form \n41yyx\nijyxy\nijxyy\nijxxx\nij P P P Pa− − −= , \n4' ' ' '\n2yyx\niiyxy\niixyy\niixxx\nii P P P Pa− − −=  and 2' '\n3zyy\niizxx\nii P Pc+= ,respectively, ij is a spin-pair \nbetween two layer of L block and ii’ is a spin-pair between two S and L blocks. Then, \nthe φ dependent total polarization PZ is: \n)0, cos, (sin31)0,2cos,2sin() 2 4(61)(3 2 1 0 φ φ μμ φ φ μμ μ μ φy\nLz\nSy\nLx\nSx\nLx\nL Z c a a a P + − − + =    ( 1 1 )  \nThis formula predicts that if H rotates clockwise in th e plane with an angular \nspeed of ω, both Px and Py in TC phase will  have sinusoidal φ and 2 φ dependent \nbehaviors together, leading to a clockwise rotating of P component with a speed of ω \nand a counter clockwise rotating of P component with the double speed of 2 ω. \nHowever, from our experiment results in Figs. 2f and 3d-g, both Px and Py shows \ndominant sinusoidal 2 φ behavior and the dominant P vector rotates counterclockwise \nwith nearly double rotating sp eed (Fig. 3h), indicating a relatively small sinusoidal φ \ncomponent. This fact is also re vealed by the weak asymmetric P(H) profile in Figs. 2e \nand 3a-b where the effect of Hx reversal can be regarded as φ=0→φ=π in Eq. 11. \nIndeed, the weak asymmetric P(H) is a universal feature in our Z-type BSCFO and \nmany other Z-type hexaferrite systems.  Then, we will try to understand the \nmicroscopic origin of each P contribution. \nIt should be mentioned here that th is kind of comparison between model calculation and H rotating experiment may be applicable to other multiferroic \nhexaferrites like M-type and U-type hexaferrites to check any possibility of \nnon-spin-current mechanism since they all have structure R block in their lattice. \nEspecially, under certain circumstances, they  have shown irreve rsible polarization \nunder H reversal25,30. \n \nDiscussion  \nTo have the single-spin  mechanism contributed PL, 40yyx\niiyxy\niixyy\niixxx\nii P P P Pa− − −=  \nat some low symmetric Fe/Co sites in L blocks should be non- zero. There are 10 \ndifferent atom positions for Fe/Co ions, whic h are labeled as Me1 to Me10 (see Table \n5 and Fig. 1b). We only have to consid er sites Me3-10 which belong to the L block. \nTo deduce the non-zero components ofαβγ\niiP for each site, one has to consider their \nglobal site symmetry of each Wyckoff position instead of their local environment. As \nshown in Table 5, there are th ree kinds of local environmen t, octahedral, tetrahedron \nand bipyramid which allows very high loca l symmetries. If MeO polyhedrons are far \naway from each other, then those local symmetries would be a good enough \napproximation to calculate theαβγ\niiP and net PL from each site. This is exemplified in \nthe case of Ba 2CoGe 2O7 where the CoO 4 tetrahedrons are separated by Ba2+ ions so \nthat the net P within a CoO 4 due to p-d hybridization mechanism can be reliably \ncalculated without considering global symmetry13. However, it is not the case in the \nZ-type hexaferrites owing to the compact edge or corner sharing between those \npolyhedrons. From the site symmetry shown in Table 5, all the Me3 to Me10 sites \nseem to be able to have non-zero0a(see Supplementary Note 3) unless there are other \nhidden or accidental symmetry constraints. Th erefore, at least each Fe/Co site in the L \nblock can generate a non-zero net P via single spin mechanism, i.e., p-d hybridization \nmechanism. However, due to the quenched orbital moments in the tetrahedral and centered octahedral31, the p-d hybridization in Me5-8 and Me10 would be very weak, \nleading to the negligible polarization by this  mechanism. But, the off-center Fe/Co in \noctahedral can induce large orbital moments31 which may enhance the extent of p-d \nhybridization. Therefore, Me3, Me4, a nd Me9 sites may provide larger a0 and \nsubsequently larger P via the p-d hybridization mechanism. Th is is consistent with the \nobservations from in-situ  X-ray diffraction23. However, we do not exclude the \ncontributions to PL from other exotic single-spin mechanisms. \nTable 5. The ten independent Wyckoff positions of the Fe/Co ions and their \nglobal and local symmetries. \nAtom \nposition Wyckoff \nletter Site \nsymmetry Local environment \nMe1 2a -3m octahedral \nMe2 4f 3m tetrahedral \nMe3 4e 3m octahedral \nMe4 12k m octahedral \nMe5 4e 3m tetrahedral \nMe6 4f 3m octahedral \nMe7 4f 3m tetrahedral \nMe8 12k m octahedral \nMe9 4f 3m octahedral \nMe10 2c -6m2 bipyramid \n \nTo have the two-spin mechanism contributed PLL and PLS, \n41yyx\nijyxy\nijxyy\nijxxx\nij P P P Pa− − −= ,\n4' ' ' '\n2yyx\niiyxy\niixyy\niixxx\nii P P P Pa− − −= and2' '\n3zyy\niizxx\nii P Pc+= should be \nnon-zero, where ij is a spin-pair between two layer of L block and ii’ is a spin-pair \nbetween two S and L blocks. As discussed in the case of Y-type, only the sinusoidal φ \ncomponent and3ccorresponds to the spin-c urrent generated polariza tion. In the case of \nZ-type, the sinusoidal φ component and3cis very weak which may be due to the much \nlower volume density of the L-S block boundaries in the Z-t ype than that of Y-type. \nAs for the a1 and a2, they are exactly zero for both the exchange-striction and \nspin-current mechanisms (see Eqs. (3) and (4 )). That means, other exotic mechanisms must be the origins of sin2 φ/cos2 φ components in PLL and PLS if they are non-zero in \nreality. Actually, a recently proposed anis otropic symmetric exchange mechanism due \nto spin orbital coupling allows non-zero a1 and a232. To examine this mechanism, a \nfirst principle calculation on BSCFO is highly desired. \nIn conclusion, we compared the respons es of polarization to the external \nmagnetic field between Y-type and Z-type he xaferrites, and reveal ed a direct link \nbetween the symmetry in magnetic L blocks and the microscopic mechanisms of \ntransverse cone induced polar ization and related ME behaviors. To prove this, we \nonly rely on a rigorous lat tice symmetry-based local ME tensors approach without \nrequiring any first principal calculation. This  means the lattice symmetry, especially \nlocal symmetry, is also crucial in determine the microscopic mechanism of polarization and related ME behavior in a spin-driven multiferroics. More important, we have suggested the unexpected contribution of polarization from p-d hybridization \nand other exotic mechanisms in Z-type hexaferrites which is beyond any previous \nexpectations for multiferroic with conical spin-order.    ACKNOWLEDGMENTS We thank H. J. Xiang and J. H. Han fo r enlightening discussions. This work was \nsupported by the National Creative Res earch Initiative (2 010-0018300) through the \nNational Research Foundation (NRF) funded by the Korea government. The work at \nChina was supported by the National Natu ral Science Foundation of China under \nGrant Nos. 11374347, 11674384.   \nMethods \nSample preparation and x-ray diffraction . Single crystals of Y-type hexaferrite \nwith a nominal composition of Ba 0.5Sr1.5Zn2(Fe 0.92Al0.08)12O22 and Z-type hexaferrite \nwith a nominal composition of Ba 0.52Sr2.48Co2Fe24O41 were grown from the \nNa2O-Fe 2O3 flux in the air.  The crystals were collected by checking the c-axis lattice parameter from the X-ray diffraction study. The orientation of the single crystals was \ndetermined using back-reflection X-ray La ue photographs. Before any magnetic and \nelectrical measurements, all the samples we re heat-treated to remove oxygen vacancy \nat 900  ˚C under flowing O 2 for 8 days. \nMagnetic and electric measurements . The sample was cut into a rectangular \nparallelepiped shape with the large su rfaces normal to the [100], [120] or both \ndirections. Each face was mechanically polished to obtain a flat smooth surface. \nElectrodes were formed on two faces normal to  [100] or four faces perpendicular to \n[100] and [120] directions. To measure th e polarization, each specimen was subjected \nto an ME annealing procedure starting at 120 K for Y-type and 305 K for Z-type \nhexaferrites. Here, we introduce a Cartesia n coordinate as shown in Fig. 2a; the x, y, \nand z axes are parallel to the [100], [120] a nd [001] directions, respectively. 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Magnetic origin of  giant magnetoelectricity in doped Y-type \nhexaferrite Ba 0.5Sr1.5Zn2(Fe 1−xAlx)12O22. Phys. Rev. Lett.  114, 117603 (2015). \n[32] Feng, J. S. and Xiang, H. J. An isotropic symmetric exchange as a new \nmechanism for multiferroicity. Phys. Rev. B  93, 174416 (2016). Figure Captions \nFig. 1  The crystal and magnetic structure of a Y-type hexaferrite and b Z-type \nhexaferrite. The Schematic models of the tr ansverse cone are shown for both systems. \nIn the left panel of b, 10 different Wyckoff positions for Fe/Co ions are labeled as \nMe1 to Me10. Fig. 2  Schematics showing relationshi p between the direction of P, H, k\n0, the \ncrystallographic axes, and th e Cartesian coordinate under  a H reversal and  b H \nrotation, respectively. Py curves of Y-type BSZAFO measured under c Hx sweeping \nand d H rotation ( H = 0.2 kOe and 2 kOe) at 30 K. ΔPy curves of Z-type BSCFO \nmeasured under e Hx sweeping and f H rotation ( H = 5 kOe and 12 kOe) at 305 K. \nFig. 3  H dependent P vector measurements at 10 K. a Px and b Py curves of Z-type \nBSCFO measured under Hx sweeping. c Schematics of H rotating measurement \nconfiguration. Angle dependent Px and Py curves measured under rotating H of (d, e) \n4 kOe and ( f, g) 10 kOe. h Schematic angle dependent relationship between H and P \nvectors. Fig. 4  Crystal structure models of the Y-type and Z-type. a Simplified generic models \nfor the crystal structures of the Y-type a nd Z-type on the basis of the magnetic block \napproximation. The trajectory of the tran sverse cone spin configuration in yz plane for \nH//x condition is shown in the right panel. Th e schematic illustration of the symmetry \noperations for  b one of the layers in the S block as a representative, c two layers in the \ndifferent blocks or in the same block. \n  \n   \nFig. 1 Y . S. Chai et al.  \nFig. 2 Y . S. Chai et al.  \n \nFig. 3 Y . S. Chai et al. \n  \n \nFig. 4 Y . S. Chai et al. \n \n  \nSupplementary Note 1: PROPERTI ES OF THE ME LOCAL TENSORS \nAccording to the definition ofαβγ\niiP andαβγ\nijP in Eq. (1), they are the even function of \ntime because electric polarization is an ev en function of time. Therefore, both ME \ntensor are third rank polar  tensors with the transformation laws: \nlmn\npp kn jmilijk\npp Paaa P='  and lmn\npq kn jmilijk\npq Paaa P='                               ( 1 )  \nwhere ail, ajm and akn are the direction cosines relati ng the two coordinate systems. \nFrom Supplementary Eq. (1), we could calcu late the transformed matrix form of the \nαβγ\niiP andαβγ\nijP under a symmetry operation. In Suppl ementary Table 1, we list the \ntransformed matrix forms of αβγ\nijP under the selected symmetry operations required \nfor all the 32 crystal classes. The transformation ofαβγ\niiP under any symmetry \noperation can be deduced sinceβαγ αβγ\nii ii P P = . These are the transformation operations \nneeded to develop the local ME matrices for various multiferroic systems if people \nwant to perform similar calculations in systems other than hexaferrites. From \nSupplementary Table 1, the matrices form ofαβγ\niiP andαβγ\nijP can be simplified by \napplying these symmetry operations a ccording to Neumann’s Principle: \nijk\nppijk\npp P P='   and ijk\npqijk\npq P P='                                            ( 2 )  \nif the single spin or the spin-pair possesses the symmetry operation used in the \ntransformation. \nSupplementary Table 1. The transfor med matrices of a two-spin tensorαβγ\n12P under \nvarious symmetry operations.  Transformedαβγ\n12P  in the matrix form  \n1 \n\n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n, , , , , ,, , , , , ,, , , , , ,\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n-1 \n\n\n\n\n− − − − − − − − −− − − − − − − − −− − − − − − − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n2//x \n(2x) \n\n\n\n− − − − −− − − − −− − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n2//y (2y) \n\n\n\n\n− − − − −− − − −− − − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n2//z (2z) \n\n\n\n\n− − − −− − − − −− − − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \nm⊥x \n(mx) \n\n\n\n− − − −− − − −− − − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \nm⊥y \n(my) \n\n\n\n− − − −− − − − −− − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \nm⊥z \n(mz) \n\n\n\n− − − − −− − − −− − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n3//[111] \n\n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n, , , , , ,, , , , , ,, , , , , ,\nyyy xyy zyy yyx xyx zyx yyz xyz zyzyxy xxy zxy yxx xxx zxx yxz xxz zxzyzy xzy zzy yzx xzx zzx yzz xzz zzz\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n4//z \n\n\n\n\n− − − −− − − −− − − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n, , , , , ,, , , , , ,, , , , , ,\nzzz xzz yzz zzx xzx yzx zzy xzy yzyzxz xxz yxz zxx xxx yxx zxy xxy yxyzyz xyz yyz zyx xyx yyx zyy xyy yyy\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n -4//z \n\n\n\n\n− − − − −− − − − −− − − −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n, , , , , ,, , , , , ,, , , , , ,\nzzz xzz yzz zzx xzx yzx zzy xzy yzyzxz xxz yxz zxx xxx yxx zxy xxy yxyzyz xyz yyz zyx xyx yyx zyy xyy yyy\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n3//z (3z) \n833 3 3 3 3 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nxxx P P P P P P P PP+ − − + − + + −=\n83 3 33 3 3 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nxxy P P P P P P P PP− + − + + − + −=\n83 3 3 33 3 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nxyx P P P P P P P PP− + + − − + + −=\n83 3 3 3 3 33 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nxyy P P P P P P P PP+ − + − + − + −=\n83 33 3 3 3 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nyxx P P P P P P P PP− − + + + + − −=\n83 3 3 33 3 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nyxy P P P P P P P PP+ + + + − − − −=\n83 3 3 3 33 3 3')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nyyx P P P P P P P PP+ + − − + + − −=\n83 3 3 3 3 3 33')(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12yyy xyy yyx xyx yxy xxy yxx xxx\nyyy P P P P P P P PP− − − − − − − −=\n43 3 3'43 3 3'\n)(\n12)(\n12)(\n12)(\n12 )(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12\nyzy xzy yzx xzx\nxzxyyz xyz yxz xxz\nxxz\nP P P PPP P P PP\n+ − −=+ − −=\n \n43 3 3'43 3 3'\n)(\n12)(\n12)(\n12)(\n12 )(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12\nyzy xzy yzx xzx\nyzxyyz xyz yxz xxz\nyxz\nP P P PPP P P PP\n− − +=− − +=\n \n43 3 3'43 3 3'\n)(\n12)(\n12)(\n12)(\n12 )(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12\nyzy xzy yzx xzx\nxzyyyz xyz yxz xxz\nxyz\nP P P PPP P P PP\n− + −=− + −=\n \n43 3 3'43 3 3'\n)(\n12)(\n12)(\n12)(\n12 )(\n12)(\n12)(\n12)(\n12)(\n12 )(\n12\nyzy xzy yzx xzx\nyzyyyz xyz yxz xxz\nyyz\nP P P PPP P P PP\n+ + +=+ + +=\n \n43 3 3')(\n12)(\n12)(\n12)(\n12 )(\n12zyy zyx zxy zxx\nzxx P P P PP+ − −=  43 3 3')(\n12)(\n12)(\n12)(\n12 )(\n12zyy zyx zxy zxx\nzxy P P P PP− − +=  \n43 3 3')(\n12)(\n12)(\n12)(\n12 )(\n12zyy zyx zxy zxx\nzyx P P P PP− + −=  \n43 3 3')(\n12)(\n12)(\n12)(\n12 )(\n12zyy zyx zxy zxx\nzyy P P P PP+ + +=  \n23' ,23')(\n12)(\n12 )(\n12)(\n12)(\n12 )(\n12zzy zzx\nzzxzyz zxz\nzxz P PPP PP+ −=+ −=\n23' ,23')(\n12)(\n12 )(\n12)(\n12)(\n12 )(\n12zzy zzx\nzzyzyz zxz\nzyz P PPP PP− −=− −=\n23')(\n12)(\n12 )(\n12yzz xzz\nxzz P PP+ −= ,23')(\n12)(\n12 )(\n12yzz xzz\nyzz P PP− −= ,)(\n12)(\n12'zzz zzzP P =  \n \n \nSupplementary Note 2: THE SIMPLIFICATION OF LOCAL ME TENSORS IN \nHEXAFERRITES \nAs shown in Fig. 4(b) and according to Eq. (1), the total local el ectric dipole from \none S layer due to three identical spins μS/6 at site 1, 2 and 3 via single spin tensor \nterm can be expressed in Einstein convention as:  \n) ( )6/1( )6/1(2 2   = =\niii S S S S\niii\niii P P pαβγ β α β α αβγ γμμ μμ                        ( 3 )  \nwhere i = 1, 2 and 3. We further noticed that  this layer has a 3 z and my a three-fold \nrotation along z-axis and three mirrors including z-axis (one of them is my) are also \nallowed for. From Neumann’s Principle (S upplementary Eq. 2), the transformed and \nuntransformed sum of the single-spin tensor \niiiPαβγin one layer must be equal:  \n =\niii\niii z P P ) () (3αβγ αβγand  =\niii\niii y P P m ) () (αβγ αβγ                       ( 4 )  Therefore, from Supplementary Table 1, we could deduce a much-simplified \nmatrix of\niiiPαβγ:   \nzzzyzy xzx yyz xxz yyx yxy xyy xxxiii\nPdP PcP PbP P P Pad c cc ba ac a ba\nP\n1111 11 11 11 11 11 11 11,2,2,4,\n,0,00,,0 0,0,0,,0,0, 0,,00,0, 0,,0 ,0,\n3\n=+=+=− − −=\n\n\n\n− −−\n=αβγ\n              ( 5 )  \nwhereαβγ\n11Pare the matrix components of single-spin tensor at site 1.  \nOn the other hand, the total local di poles by three inter- block spin-pairs 11', 22' \nand 33' via two-spin tensor t erm shown in the upper panel of Fig. 4(c) can be \nexpressed as:  \n) ( )6/1( )6/1(\n''2\n''2\n''    = =\niiii L S L S\niiii\niiii P P pαβγ β α β α αβγ γμμ μμ                        ( 6 )  \nwhere ii' = 11', 22' and 33'. Similarly, after considering the 3 z and my symmetries and \napplying Neumann’s Principle, th e sum of the two-spin tensors \n''\niiiiPαβγbetween the \ntwo layers in the upper panel of Fig. 4(c) can be simplified as \nzzzzyy zxx yzy xzx yyz xxz yyx yxy xyy xxxiiii\nPdP PcP PcP PbP P P Pad c cc ba ac a ba\nP\n'11'11 '11 '11 '11 '11 '11 '11 '11 '11 '11''\n',2'',2',2',4'',0,00,'',0 0,0,''0,',0',0,' 0,',00,0,' 0,',0',0,'\n3\n=+=+=+=− − −=\n\n\n\n− −−\n=αβγ\n  ( 7 )  \nwhereαβγ\n'11Pare the matrix component s of two-spin tensor of site-pair 11 '.  \nMoreover, there are extra symmetry ope rations between two layers within a \nmagnetic block: space inversion for S & L blocks in Y-type hexaferrite and S block in \nZ-type hexaferrite, mirror symmetry mz for L block in Z-type hexaferrite, as shown in \nFig. 4. The sum of the single-spin tensor \niiiPαβγat two layers in one block can be \nsimplified further. If there is a space inversion at the block center, the sum of\niiiPαβγin the two layers of the block will be exactly oppos ite because they can be \nmutually transformed by -1 symmetry. Then, the total summation matrix \nof\niiiPαβγwithin one block will be exactly zer o for every component. Whereas, if \nthere is a mirror in the middle of the block, the matrix of \niiiPαβγbecome: \n\n\n\n\n− −−\n=\n0,0,00,0,0 0,0,00,0,00,0, 0,,00,0,00,,00,0,\n6 a aa a\nP\niiiαβγ                                            ( 8 )  \nThe sum of the two-spin tensor \n''\niiiiPαβγbetween two layers in one block can be \nalso be simplified further. If there is a space inversion at  the block center, as shown in \nthe middle panel of Fig. 4(c), the matrix of \n''\niiiiPαβγwithin one block will be simplified \nas: \n\n\n\n\n− −=\n0,0,00,',00,0,'0,',00,0,0 0,0,00,0,' 0,0,0 0,0,0\n3\n''\nc ccc\nP\niiiiαβγ                                           ( 9 )  \nNote that the number of spin-pair can be six in this case. However, this will not affect \nthe number of independent coef ficient, the form of matrix and the net polarization of \ntwo hexaferrite systems in Eqs. (8) and (11). While if there is an mz at the block center, \nas shown in the lower panel of Fig. 4(c), the matrix of \n''\niiiiPαβγwithin one block will \nbe simplified as:  \n\n\n\n\n− −− −−\n=\n0,0,00,',00,0,'0,',00,0,' 0,',00,0,' 0,',00,0,'\n3\n''\nc cc a ac a a\nP\niiiiαβγ                                    ( 1 0 )  \nwhere the summation goes over every site i or site pair ii' in one block.  \nFrom the above procedures, we have grea tly simplified the matrix forms of the \nsummation of local ME tensors \niiiPαβγand \n''\niiiiPαβγfor both inter-block and \nintra-block cases. The forms of local ME te nsor matrices for intra-block summations \nin S1 and L1 blocks, and inter-block summation between S1 and L1 block are \ndetermined in Tables 1 and 2. Then, those of  other inter and intra- block summations in two hexaferrite systems can be derived by applying space inversion or mz symmetry \noperator according to Supplementary Table 1. \n \nSupplementary Note 3: THE SIMPLIFI CATION OF SINGLE-SPIN TENSORS \nWITH WYCKOFF SITE SYMMETRY OPERATIONS \nWe could calculate the two-spin ME tensor matrices ofαβγ\n12P for the 32 symmetry \npoint groups From Neumann’s Principle (Supplementary Eq. 2) and Supplementary \nTable 1, as summarized in Supplementary Table 2 . The single-spin ME tensor \nmatrices ofαβγ\n11P  for the various symmetry point groups can be deduced similarly.  \nSupplementary Table 2. The matrices of two-spin tensorαβγ\n12P in 32 point groups. \n Simplifiedαβγ\n12P in matrix form  \n1 \n \n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n, , , , , ,, , , , , ,, , , , , ,\nzzz yzz xzz zzy yzy xzy zzx yzx xzxzyz yyz xyz zyy yyy xyy zyx yyx xyxzxz yxz xxz zxy yxy xxy zxx yxx xxx\nP P P P P P P P PP P P P P P P P PP P P P P P P P P\n \n2y \n\n\n\n\n0, ,0 ,0, 0, ,0,0, 0, ,0 ,0,0, ,0 ,0, 0, ,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nyzz zzy xzy yzxzyz xyz yyy zyx xyxyxz zxy xxy yxx\nP P P PP P P P PP P P P\n \nmy \n\n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0, 0, ,0 ,0,0, ,0 ,0, 0, ,0,0, 0, ,0 ,0,\nzzz xzz yzy zzx xzxyyz zyy xyy yyxzxz xxz yxy zxx xxx\nP P P P PP P P PP P P P P\n \nmm2 \n\n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0,00, ,00,0,0, ,0 ,0,0 0,0,00,0, 0,0,0 ,0,0\nzzz yzy xzxyyz zyyxxz zxx\nP P PP PP P\n \n222 \n\n\n\n\n0,0,0 0,0, 0, ,00,0, 0,0,0 ,0,00, ,0 ,0,0 0,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nxzy yzxxyz zyxyxz zxy\nP PP PP P\n \n3z \n\n\n\n\n−− − − − −−\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0,0 0, , 0, ,0, , , , , ,0, , , , , ,\nzzz xzx yzx yzx xzxxxz yxz zxx yxx xxx zxy xxx yxxyxz xxz zxy xxx yxx zxx yxx xxx\nP P P P PP P P P P P P PP P P P P P P P\n 3z2x \n\n\n\n\n−− − − −−\n0,0,0 0,0, 0, ,00,0, 0,0, , ,00, ,0 , ,0 0,0,\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nyzx yzxyxz xxx zxy xxxyxz zxy xxx xxx\nP PP P P PP P P P\n \n3zmy \n\n\n\n − −−\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0,0 0, ,0 0,0,0, ,0 ,0, 0, ,00,0, 0, ,0 ,0,\nzzz xzx xzxxxz zxx xxx xxxxxz xxx zxx xxx\nP P PP P P PP P P P\n \n4z, 6z \n\n\n\n\n−− −\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0,0 0, , 0, ,0, , ,0,0 ,0,00, , ,0,0 ,0,0\nzzz xzx yzx yzx xzxxxz yxz zxx zxyyxz xxz zxy zxx\nP P P P PP P P PP P P P\n \n-4 \n\n\n\n\n−− −\n0,0,0 0, , 0, ,0, , ,0,0 ,0,00, , ,0,0 ,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nxzx yzx yzx xzxxxz yxz zxx zxyyxz xxz zxy zxx\nP P P PP P P PP P P P\n \n4mm, 6mm \n\n\n\n\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\n,0,00, ,00,0,0, ,0 ,0,0 0,0,00,0, 0,0,0 ,0,0\nzzz xzx xzxxxz zxxxxz zxx\nP P PP PP P\n \n422, 622 \n\n\n\n\n−− −\n0,0,0 0,0, 0, ,00,0, 0,0,0 ,0,00, ,0 ,0,0 0,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nyzx yzxyxz zxyyxz zxy\nP PP PP P\n \n-42m \n\n\n\n\n0,0,0 0,0, 0, ,00,0, 0,0,0 ,0,00, ,0 ,0,0 0,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nyzx yzxyxz zxyyxz zxy\nP PP PP P\n \n-6 \n\n\n\n − − −−\n0,0,0 0,0,0 0,0,00,0,00, , 0, ,0,0,00, , 0, ,\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nyxx xxx xxx yxxxxx yxx yxx xxx\nP P P PP P P P\n \n-6my2 \n\n\n\n − −−\n0,0,0 0,0,0 0,0,00,0,00,0, 0, ,00,0,00, ,00,0,\n)(\n12)(\n12)(\n12)(\n12\nxxx xxxxxx xxx\nP PP P\n \n-43m, 23 \n\n\n\n\n0,0,0 0,0, 0, ,00,0, 0,0,0 ,0,00, ,0 ,0,0 0,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzxy zxyzxy zxyzxy zxy\nP PP PP P\n \n432 \n\n\n\n\n−−−\n0,0,0 0,0, 0, ,00,0, 0,0,0 ,0,00, ,0 ,0,0 0,0,0\n)(\n12)(\n12)(\n12)(\n12)(\n12)(\n12\nzxy zxyzxy zxyzxy zxy\nP PP PP P\n others \n\n\n\n\n0,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,00,0,0  \n \nFrom the Supplementary Table 2, we c ould deduce the simplified single-spin ME \ntensor matrices ofαβγ\n11P for 3 m, m and -6 m2 respectively in Supplementary Table 3: \nSupplementary Table 3. The matrices of single-spin tensorαβγ\n11P in selected point \ngroups \n Simplified matrices ofαβγ\n11P   \nm \n\n\n\n\n)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11\n,0, 0, ,0 ,0,0, ,0 ,0, 0, ,0,0, 0, ,0 ,0,\nzzz xzz yyz zxz xxzyyz zyy xyy yxyzxz xxz yxy zxx xxx\nP P P P PP P P PP P P P P\n \n3m \n\n\n\n − −−\n)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11)(\n11\n,0,0 0, ,0 0,0,0, ,0 ,0, 0, ,00,0, 0, ,0 ,0,\nzzz xxz xxzxxz zxx xxx xxxxxz xxx zxx xxx\nP P PP P P PP P P P\n \n-6m2 \n\n\n\n − −−\n0,0,0 0,0,0 0,0,00,0,00,0, 0, ,00,0,00, ,00,0,\n)(\n11)(\n11)(\n11)(\n11\nxxx xxxxxx xxx\nP PP P\n \n \nFrom the site symmetries shown in Table 5,  all the site symmetries at Me3 to Me10 \nfall in the above three symmet ries. According to Supplemen tary Table 3, those sites \nshould have non-zero a0 in Eq. 9 if there is no extra or hidden symmetry constraints. \n "    },    {        "title": "1112.3929v1.On_the_Angular_Width_of_Diffractive_Beam_in_Anisotropic_Media.pdf",        "content": "1 \n \nOn the Angular Width of Diffractive Beam in Anisotropic Media \nEdwin H. Lock \n(Kotel’nikov Institute of Radi o Engineering and Electronics of  Russian Academy of Sciences, \nFryazino branch)  \n2-D diffraction patterns arising in the far-field region were investigated theoretically for \nthe case, when the plane wave  with non collinear group and phase velo cities is incident \non the wide slit in opaque scr een with arbitrary orientation. This investigation was \ncarried out by consideration as an example of  magnetostatic surface  wave diffraction in \ntangentially magnetized ferrite slab. It was deduced the universal analytical formula, \nwhich one can use to calcula te the angular width of di ffractive beam in any 2-D \nanisotropic geometries for the waves of va rious nature. It was shown, that in 2-D \nanisotropic geometries this  width may be not only more or less then the value λ0/D (λ0 – \nwavelength of incident wave, D – length of slit), but it also may be equal to zero in \ncertain conditions.  \n \n \nIntroduction \n \nAs it is known, different kind of waves, propa gating in various media,  are characterized by \ncommon physical laws. An examples of such we ll-known common laws for is otropic media are the \nlaws of geometrical optics and the formula, descri bing angular width of diffr active beam (arising as \na result of incidence of plane wa ve on the wide slit in opaque sc reen) as a ratio of incident \nwavelength λ0 and slit length D.  \nFor anisotropic media the laws of wave propa gation, reflection, and refraction (in geometrical \noptics approximation) are determined by such ma thematical properties of the wave isofrequency \ndependence (ID)1 as the existence of asymptotes, centers of symmetry, extremum and inflection \npoints on this dependence, the number of symmetry axes and the single- or multi-valuedness of this \ndependence. If ID of the wave possesses by some of these properties, then  propagation, reflection, \nand refraction of waves can be accompanied by such phenomena as the impossibility of wave \npropagation in some directions or within some  angular sector, nonrecip rocality of propagation \n(when counter-propagating rays ha ve different parameters), unidi rectional propagation (when for a \ngiven ray there exists no counter -propagating ray and, in some  cases, there is only a single \ndirection, in the vicinity  of which the energy can be transfer red), negative reflection and refraction \n(when the incident, reflected, and refracted rays ar e on the same side of th e normal to the interface), \nthe emergence of two (or more) reflected or refrac ted rays, the absence of reflection altogether, and \nthe irreversibility of ray paths in  reflection or refraction (see [1] and references, reported there).  \nIt is evidently to suppose, that mathematical properties of the wave ID also determine the \nangular width of diffractive beam in anisotropic medi a and structures. So, the question is appears: is \nit possible to deduce some  universal formula, describing a ngular width of diffractive beam in \nanisotropic media 2-D geometries (similar to the fo rmula for isotropic media)? How it is seen from \nanalysis of scientific literature, diffractive phe nomena in anisotropic media were investigated \nmainly for electromagnetic wave in plasma, for light  in optical crystals, fo r acoustic waves and for \ndipole spin waves, named usually magnetostatic waves (MSWs) [2]. However, it seems that such \nformula is absent for all these waves at th e moment (probably, because of mathematical \ndifficulties). So an attempt is taken to obtain such universal formula through the study of MSW \ndiffraction in ferrite slab.  \n                                                           \n1 ID is also known as “section of wa vevector surface”, “section of the isoene rgy surface” and “equifrequency line” [1].  2 \n \nStatement of the Problem and Main Results.  \n \nLet’s consider an infinite ferrite slab of thickness s magnetized to saturation by a tangent \nhomogeneous magnetic field H0. The ferrite slab is charac terized by well-known magnetic \npermeability tensor, and ambient half-spaces have magnetic permeability equaling to 1. To describe \nMSW in ferrite slab let’s use magnetostatics equations rot h = 0 and div b = 0 and introduce \nmagnetostatic potential Ψ in agree with formula h = gradΨ [2]. Then consider diffraction pattern, \narising in the most general case – when plane surface MSW with frequency f0 and non collinear \nwave vector k0 and group velocity vector V0 is incident on the slit of length D in opaque thin screen \nwith arbitrary orientation (Fig.1). For convenient study of this problem there are used Cartesian \nΣC = {x; y; z} and Polar ΣP = {x; r; φ} coordinate systems, coupled with ferrite slab anisotropic \ndirections (with direction of external magnetic field H0), and another Cartesian and Polar systems, \ncoupled with orientation of  the slit and screen ( Σ'D = {x; y′; z′} and Σ'P = {x; r; φ′} respectively).  \nDue to potential Ψ is scalar function, the study of MSW diffraction becomes significantly \nsimpler and we can follow in general by the widely  known analytical ways, used for isotropic media \n[3 - 5]. However, it is necessary to take into  account anisotropic character of MSW propagation in \nferrite slab. In particular, the next important mentions must be made.  \n1. Let’s suppose, that the result  of plane MSW incidence onto slit  is equal to appearance of \ndistribution of secondary elementary MSW sources  with various phases along slit line, and to \nestimate the resulting action of these sources in th e far-field region we must  integrate contributions \nfrom all these elementary sources (i.e. calcu late superposition of all these sources).  \n2. Action of every secondary source is defined by magnetic potential excited by this source, and, \nconsequently, to calculate resulting action of all so urces in the far-field region it is possible to \nsummarize potentials excited by all sources (wher eas every potential is scalar function).  \n3. As a distinct from the solving of similar problems in isotropic media, we always will to deal \nwith two directions φ and ψ (or with φ′ и ψ′ directions in Σ'D and Σ'P coordinate systems) \ncorresponding to orientation of the wave vector k and group velocity vector V respectively. While \nconditions, determining constructive  interference of the secondary sources, are formulated for their \nwave vectors (i.e. for φ or φ′ directions), the MSW energy is tr ansferred (when this constructive \ninterference is realized) along the corresponding group ve locity vector direction (i.e. along ψ or ψ′ \ndirections). So for mathematical description of studied diffractive pr oblem it is necessary to use two \npoints at infinity - Рk and РV: the direction from the slit centre to the Рk point is coincided with \nsome orientation φ of secondary sources wave vectors, and the direction from the slit centre to the \nРV point – with corresp onding group velocity vector orientation ψ (Fig.1). Thus , ID of MSW k(φ) in \nPolar coordinate system (see inset on Fig.1) and corresponding dependence ψ(φ) are important for \nour further consideration. The sake of si mplicity let’s assume, that dependencies k(φ) and ψ(φ) are \nsingle-valued functions (for surf ace MSW in free ferrite slab this assumption is always correct), \nand, therefore there is only one point РV, corresponding to every point Рk. Let’s assume also, that \ninverse dependence φ(ψ) is single-valued function too.  \nWith these assumptions for the case D/λ0 >> 1 after some analytical ca lculations it is  possible to \nfind that angular distribution of total (summa ry) magnetic potential from all secondary MSW \nsources is determined by expression of ~ sin Ф/Ф (similar to analogous expression for isotropic \nmedia), but the phase function Ф is more complex, than in isotropic media  \n \n⎥⎦⎤\n⎢⎣⎡− −− = ) )( sin())(() sin( ))(( θψϕψϕθϕπψϕ\n00\n0 kk\nλDФ ,     ( 1 )  \nwhere k(φ) and φ(ψ) – dependences, characterized certain wave  in anisotropic medium or structure \n(for considered example this wave is surface MSW in ferrite slab), φ0 and k0 = 2π/λ0 – parameters of \nincident plane MSW, D and θ – length and orientation of the slit (see Fig.1). Since sin Ф/Ф is \nrapidly oscillating function we ma y neglect by the slowly changi ng factor (similar to Kirchhoff \nfactor) in beam width ca lculations. Finding difference between directions ψ, corresponding to the  3 \n \n \n \nk0V0k(ϕ')lsinϕ'\nk0lsinϕ'0yy'\nz'\ndl l=0H0\nθ\nχ0ϕ'0ϕ'0ϕ'ϕ'\nϕ\nψ'0ψ'ψ\nr2 r1 rl rPk\nPV\nl=D\n400\n400-400\n0800\nk0\nϕ0χ0ψ0\nV0\nSH0ky, cm -1\nzkz, cm -1y' y\nz'θ\nϕ'0\n \n \nFig.1. Geometry of the problem in the plane of ferrite slab. Phas e fronts of plane incident MSW \nare shown by dashed lines. Inset: half of ID for surface MSW with f0 = 2900 MHz in free ferrite \nslab (point S corresponds to incide nt MSW with non collinear vectors k0 and V0).  \n 4 \n \ncases Ф = 0 (sinФ/Ф = 1) and Ф = π (sinФ/Ф =0), it is possible to deduce formula, describing \nabsolute  angular width of ma in diffractive beam Δψ (at level 0.5). However in anisotropic media it \nis more useful relative  angular width of ma in diffractive beam σ, which is related with Δψ by the \nsimple formula σ = Δψ/(λ0/D). Relative  width σ show, how much absolute angular beam width Δψ \ndiffers from the similar width λ0/D in isotropic media, if slits lengt hs and lengths of incident waves \nare the same. Relative  angular width σ of main diffractive beam is described by the formula \n \n0 0\n00 0\n0' 'kddkdd\nkddkdd\nϕϕϕϕϕϕψ\nθϕθϕϕϕϕϕψ\nσ\ncos sin)()(\n) cos() sin()()(\n00\n00\n+=\n−+−= ,  (2) \n \nwhere dψ/dφ and dk/dφ – derivatives valu es of functions ψ(φ) and k(φ) respectively at φ = φ0. There \nis used sign of modulus, because it is convenien t to characterize angular width by positive numbers, \nlike a distance. If the wave vector k0 of incident MSW is perpendicular to the slit line ( φ′0 = 0) \ndenominator in (2) is equal to un ity and formula (2) looks most simp le. In this case for isotropic \nmedia (whose ID is circumference, dependence  ψ(φ) has the form ψ = φ and dψ/dφ ≡ 1) we get σ =1 \nand arrive to well-known expression Δψ = λ0/D.  \nIn spite of deduction method there is a hope, that formula (2), deduced for MSW, will be valid \nfor any anisotropic media and structures for 2-D geometries, including metamaterial structures \n(whose IDs is differ from circumference too). As  it is seen from the formula (2) an unusual \nphenomenon may be appear in anisotropic 2-D geom etries: if incident wave is characterized by \nsuch value φ0 that dψ/dφ = 0 at φ = φ0, then σ = Δψ = 0! It means, that diffractive beam conserve its \nwide during propagation! Mention mu st be made, that not any wave has ID with point(s), where \ndψ/dφ = 0, but such points present on ID for surface MSW in free ferrite film. Calculations, based \non formula (2) together with num erical computations (when the difference between directions ψ, \ncorresponding to Ф = 0 and Ф = π are calculated on agree with (1))  are shown in the Fig. 2 for the \ncase ψ′0 = 0, that corresponds to all geometries, where incident MSW vector V0 is normal to the slit \nline and orientation of incident MSW wave vector k0 is changed (i.e. angle χ0 is varied from -90o to \n+90o). MSW’s ID for f0 = 2330 MHz has the points, where dψ/dφ = 0 (at χ0 = ± 73о) and, as it is \nseen from Fig. 2, if incident MSW is characterized by angle  χ0 = ± 73о, then relative angular beam \nwidth σ = 0. Mention must be made also, that σ → 0 at χ0 → ± 90о due to dk/dφ → ∞ in (2) near ID \nasymptote.  \n  \nSummary \n \n2-D problem of diffraction on th e slit for MSW with non collinea r group and phase velocities is \nsolved. It is shown, that angular width of diffr active beam is determined by both parameters of \nincident wave and mathematic properties of wave ’s ID. It is deduced th e universal analytical \nformula, which one can use to calculate the angular  width of diffractive beam in any 2-D \nanisotropic geometries for the wa ves of various nature (including waves in metamaterials). It is \nshown, that in anisotropic media (including metama terials) the angular wi dth of diffractive beam \nmay be equal to zero at the certa in conditions (when the incident wave is characterized by such \npoint on the wave’s ID, where dψ/dφ = 0).  \n 5 \n \n- 80 - 60 - 40 - 20 0 20 40 60 800123\nχ0, degreeσ\n1, 2\n3\n4, 5\n6\n \n \nFig.2. Relative  angular width σ of transmitted diffractive beam versus angle χ0 between vectors \nV0 and k0 of incident MSW for ψ′0 = 0 and H0 = 300 Oe, saturation magnetization 4πM0 = 1750 Gs, \ns = 10 μm: 1 – 3 – for f0 = 2900 MHz, 4 – 6 – for f0 = 2330 MHz. 1 and 4 – calculations, using \nformula (2); 2, 5 and 3, 6 – numerical computations for λ0/D = 0.01 and λ0/D = 0.1 respectively.  \n \n \nThis work is partially supporte d by the Program “Development of  the Scientific Potential of \nHigh School” (project No. 2.1.1/1081).   \nReferences \n[1] Lock E.H. Physics - Uspekhi 51 (4) 375 - 393 (2008).  \n[2] R.W. Damon, J.R. Eshbach, Magnetostatic modes of a ferromagnet slab, J. Phys. Chem. \nSolids 19 (1961) 308-320.  \n[3] Born M, Wolf E. Principles of Optics (Oxford: Pergamon Press, 1969).  \n[4] F.S. Crawford, Berkeley Physics Course Vol. 3 Waves, New York: McGraw-Hill, 1968  \n[5] G.S. Landsberg, Optics, Moscow: Nauka, 1976.  "    },    {        "title": "1209.4546v1.Nucleating_Acicular_Ferrite_with_Galaxite__Al2O3MnO__and_Manganese_Oxide__MnO2__in_FeMnAlC_steels.pdf",        "content": "Nucleating Acicular Ferrite with Galaxite (Al 2O3MnO) and Manganese Oxide (MnO 2) in FeMnAlC \nsteels \n \nMC McGrath, DC Van Aken, K Song \n \nABSTRACT \n \n Effectiveness of galaxite (Al 2O3MnO) and manganese oxide (MnO 2) in nucleating acicular \nferrite was demonstrated in a steel with co mposition of Fe-13.92Mn-4.53Al-1.28Si-0.11C.  The hot \nrolled and quenched steel had a duplex microstructure of δ-ferrite and partially transformed austenite.  \nThermal treatments were used to manipulate the con centration and type of oxides and the ferrite plate \ndensity was found to correlate with in clusions of low misfit.  Both ba initic and acicular ferrite were \nobserved and the prior austenite had a grain diamet er of 16.5 µm. Upon tensile  testing the retained \naustenite transformed to α–martensite.  Ultimate tensile streng th and elongation we re 970 MPa and 40%. \nThe yield strength was found to be dependent on the fe rrite plate size, which varied linearly with the \nstrength of the austenite.  Ductility decr eased as the strength  disparity between į-ferrite and \ndeformation-induced Į-martensite increased. \n \nI.  INTRODUCTION \n \n I n t e r e s t  t o  d e v e l o p  n e w  a d v a n ced high strength steel for automo tive applications is a result of \nincreasing CAFE (corporate average fuel economy) standards where by 2025 an average fuel economy \nof 54.5 mpg is expected.  Automotive steels will need to have ultimat e tensile strengths greater than 900 \nMPa to remain competitive with magnesium and aluminum when these automotive materials are \ncompared on a strength to dens ity (specific strength) basis.1  Two approaches are being used to increase \nthe specific strength of  steel: increasing strength to contain larger proportions of martensite and \ndecreasing density through the additi on of aluminum.  Dual phase steels  with strengths greater than 900 \nMPa have been achieved by increasing the volume frac tion of martensite and au stenite (MA).  Addition \nof aluminum to lower the density of steel is also well documented.2-5  The aluminum atom is 12.6% \nlarger than that of iron and 48% the atomic ma ss of iron.  Thus, steel can be lightweighted by a combination of lattice dilatation a nd mass reduction.  Frommeyer et al.2 have shown that weight \nreductions as great as 16-17% can  be obtained using aluminum cont ents of 12 wt.%, but addition of \naluminum may also lead to the formation of ordere d intermetallic compounds such as FeAl (B2) or \nFe3Al (DO 3),6 which can reduce ductility.  Large additions of manganese , greater than 15 wt.%, and \ncarbon up to 1wt.% are often employ ed to stabilize the austenitic  phase.  These austenitic high \nmanganese and aluminum steels are referred to as second generation a dvanced high strength steels and a \ntwin-induced plasticity (TWIP) grade is now commercially produced by POSCO for automotive \nstructures.  \n Development of third generation advanced high stre ngth steels (AHSS) has been initiated to find \nlower cost steels than the austen itic second generation AHSS, but with  properties exceeding those of \nfirst generation AHSS that are dual phase or martensitic.7   Target properties considered break-through \nfor these new automotive steels would be combinati ons of ultimate tensile st rength and elongation to \nfailure of 1000 MPa and 30% or 1500 MPa and 20%.8  Matlock et al.7 using a mechanics approach have \nshown that a combination of martensite and austenite  would be capable of achieving the targeted goals \nand the quench and partition process appears to be a viable approach.9  Acicular ferrite and austenite \nmicrostructures are also a possibility  for meeting the targeted properties;10-12 however, the nucleation of \nintragranular ferrite or ac icular ferrite is thought to be restrict ed by grain size and a critical austenite \ngrain size of greater than 20-35 µm has been reported.13,14  Intergranular nucleati on of bainitic ferrite \noccurs in preference to acicular ferr ite at these finer grain structures.13-18  It should be noted that the \nstructure of these transformation products are identical  and the nomenclature relates to where the ferrite \nis nucleated. Experimental studies have shown that  acicular ferrite is ba inite and heterogeneously \nnucleated within the austenite grain i.e. intragranular.19  The term acicular ferrite is reserved for those \nplates of ferrite nucleated within the austenite grain.  \n A c i c u l a r  f e r r i t e  i s  i m p o r t a n t  f o r  a c h i e v i ng good combinations of strength and ductility.20,21  \nNonmetallic inclusions that are chemically active or have low misfit with ferrite are considered effective nucleation sites for acicular ferrite.22-26  Some nonmetallic inclusions like manganese sulfides (MnS) are \nchemically active and deplete manganese from the surrounding austenite, which promotes the nucleation \nof acicular ferrite.22,27-32   Park et al.33 observed galaxite (Al 2O3MnO), aluminum oxide (Al 2O3), \naluminum nitride (AlN), manganese sulfides (M nS), and complex aluminum and manganese-rich \ninclusions formed in an austenitic steel with  3 wt% aluminum and 10 wt% manganese. The misfit \nbetween galaxite and ferrite is 1.8 % and galaxite was reported as a pot ent nucleation site for acicular \nferrite by Mills et al.26  \n Cerium, calcium, and misch metal additions have al so been explored to inoculate steels to form \nacicular microstructures.  Bin et al.34 showed in situ observations of acicular ferrite nucleation on \ncerium-rich inclusions with low disregistry betwee n ferrite in a 0.9 wt% manganese steel with a prior \naustenite grain size of 120 µm.  Deoxidation and desulfurization of lightwe ight FeMnAlC steels was \ninvestigated to determine the eff ectiveness of calcium and misch meta l additions to inoculate acicular \nferrite.35  The microstructures and inocul ation practices of the FeMnAlC st eels are presented in Figure 1.  \nFigure 1(a) highlights the correlatio n of ferrite plate density with th e measured density of inclusions \nhaving good lattice registry; total inclusion density did not show a direct correlation.  The highest \nacicular ferrite density was obtained in the calcium tr eated alloy without any ar gon stirring (process A).  \nArgon stirring was shown to remove the sulfides with low misfit (52%  reduction) and process D samples \nexhibited 34% lower plate density.  A ll of the steels exhibited an equiva lent austenite grain size with a \nmean linear intercept dimension of approximately 15 µm, which is well below a size where acicular \nferrite would be observed.  Thus, the correlation betw een ferrite plate density and inclusion density was \nunexpected.  The alloy produced by pr ocess E was 90% transformed and showed larger ferrite plate size \ncompared to the other steels even th ough the density of ferrite plates re mained similar. The argon stirred \nsamples also had a high density of galaxite inclusions.   \n Producing an inoculated automotive steel can  present manufacturing difficulties for the \nsteelmaker.  Nonmetallic inclusions usually asso ciated with acicular ferrite are produced during solidification and as a resu lt the distribution within th e austenite is le ss than ideal to produce acicular \nferrite.  This paper investigated the nature of the nonmetallic incl usions that formed in a duplex \nFeMnAlC steel of composition Fe -13.92Mn-4.53Al-1.28Si-0.11C and their role in nucleating acicular \nferrite.  The inclusion types and fract ion were manipulated with heat trea tments to elucidate their role as \nnucleating agents. Mechanical testing was performed to demonstrate that a duplex ( δ+γ) steel containing \na combination of bainitic and acicular ferrite is capable of achieving the targeted goals for 3rd generation \nAHSS and at a reduced density.10,35 \n \nII.  EXPERIMENTAL PROCEDURE \n \n A  s t e e l  w i t h  c o m p o s i t i o n  o f  F e - 1 3 . 9 2 M n - 4.53Al-1.28Si-0.11C was produced to study the \neffectiveness of galaxite and manga nese oxide in nucleating acicular ferrite.  High purity induction iron, \nelectrolytic manganese, aluminum , ferrosilicon and carbon were melte d in a 45 kg (100 lb) induction \nfurnace under an argon protective atmosphere.  Desla gging was performed with a low-density granular \ncoagulant.  Plates were cast into  phenolic no-bake olivin e sand molds designed to produce 12.5 cm x 6 \ncm x 1.7 cm blocks.  Chemical analysis was perfor med by ion coupled plasma spectrometry after sample \ndissolution in perchloric acid.  Nitrogen, phosphorus, and sulfur con centrations were 450 ppm, 170 ppm, \nand 110 ppm, respectively.  \n Cast materials were homogenized  at 1373 K (1100 ºC) for 2 hours before being air cooled to \nambient temperature.  The castings were subseq uently milled to produce 13.6 mm x 126 mm x 50 mm \nplates for hot rolling.  Hot rolling was performed by reheating the machined castings to 1173 K (900 ºC) \nand incrementally reducing the ingo t by rolling, i.e. star ting at 1173 K (900 ºC) w ith reheating between \nreductions when the temperature fell below 973 K (700 ºC ).  The plates were re duced 80% to a thickness \nof 2.8 mm.  After the final rolling pa ss the plates were either water que nched (method A) or reheated to \n1173 K (900 ºC) for 10 minutes prior to water quen ching (method B).  Specimens from each method \nwere isothermally held at 1123 K (850 ºC) for 2 hours and 47 hours and then water quenched.  An additional heat treatment, 15 minutes at 1373 K (1100°C) and water quenching, was performed on a \nmethod B specimen that was isothermally held 47 hours at 1123K (850°C).  High temperature heat \ntreatments were performed on samples se aled under vacuum in quartz tubes. \n Light optical microscopy was used to characteri ze the microstructures.  Standard metallographic \npractices were employed to polish specimens and each  was etched with 2% nital and subsequently \netched with 10% sodium metabisulfat e to contrast the differences between ferrite and retained austenite \nin the microstructure.  Ferrite plate density was determined based on ASTM E 112-96.36  Five \nrepresentative microphotographs at 500x were used from each sample with ten different measurements \nfrom each photograph to determine plate density.  The Į-ferrite plate density included both bainite and \nacicular ferrite since the fine austenite grain structur e made it difficult to distinguish intergranular ferrite \n(bainite) from intragranular ferrite (acicular).  The ferrite plate size was determined as the mean linear \nintercept measured on random sections.  X-ray diffrac tion was utilized to identify the presence of ț-\ncarbide.  Characterization was performed w ith a Scintag 2000 diffractometer using CuK Į radiation.  \nScans were run from 30-80 degrees with a scan step size of 0.03 degrees.  An FEI Tecnai F20 \nscanning/transmission electron microscope (S/TEM ) operated at 200 kV was used to compare the \ndislocation structures de veloped from the two hot rolling methods .  The S/TEM samples were prepared \nwith an FEI Helios NanoLab 600 FIB/FESEM and specifi c regions in the samples were cross-sectioned. \n Nonmetallic inclusions were characterized by size and composition with automated feature \nanalysis (AFA) using a scanning el ectron microscope controlled with  ASPEX PICA 1020 software. An \naccelerating voltage of 20 keV and a working distance of 20 mm were used.  Samples were \nmetallographically prepared, not etched, and character ized in the short-longitudinal plane of the hot \nrolled plate.  Contrast threshol ds, magnification, spot size, electron beam size, scanni ng speed and image \nsize were kept constant for all samples.  For each  sample, the AFA application was applied to two \ndifferent regions of approximately 4 mm2 each.  The AFA is limited to inclusions between 0.5 ȝm and \n30 ȝm in diameter.  T e n s i l e  t e s t  s p e c i m e n s  w e r e  m a c h i n e d  f r o m  t h e hot rolled products in accordance to ASTM E8-\n0837 with a gage length of 50 mm a nd width of 12.5 mm.  Tensile tests were performed with the load \naxis parallel to the rolling dire ction.  Tests were conducted at room  temperature using a displacement \nrate of 0.01 mm/s.  Microhardness measurements were  performed with an app lied 98.1 mN load for 5 \nseconds.  All of the uncertainties reported in th is paper are given at a 95% confidence level. \n \nIII.  RESULTS \n \n Hot rolled microstructures of the steel processed by the two methods are shown in Figure 2. Both \nmethods produced microstructures consisting of primary į-ferrite stringers aligned parallel to the rolling \ndirection, retained austenite, and a co mbination of bainitic and acicular ferri te plates within the austenite.  \nThe density and size of the ferrite plates varied  between methods A and B with method A having a \ngreater density of ferrite plates.  Figure 3 compares the x-ray diffr action (XRD) patterns obtained from \nthe plates processed by methods A and B.   Method B processed plates showed {111} ț-carbide \ndiffraction intensity while no evidence of ț-carbide was observed in the diffraction pattern from the \nplate processed by method A.  Diff raction peak broadening, as exhib ited in Figure 3(a) for method A, \nsuggests that recrystallization was suppressed by the low finishing temperature and that the \nmicrostructure retained some level of cold work.  S/TEM images of re presentative dislocation structures \nin į-ferrite of the hot rolled plates processed by me thods A and B are shown in Figure 4.  Polishing \nartifacts from ion milling during specimen prepar ation were observed as dark features in the į-ferrite \nand S/TEM images were used to show greater contrast  of the dislocation densit y. The dislocation density \nis qualitatively observed to be greater in the plate processed by method A and confirms that some level \nof cold work was retained.  The dislocation structure in  the plate processed by method B consisted of \nsmall angle boundaries with long st raight dislocations distribute d in a regular network in the į-ferrite \n(Figure 4 (b)).    T h e  h o t  r o l l e d  p r o d u c t s  w e r e  s u b s e q u e n t l y  h e at treated at 1123 K (850 ºC) for 2 hours and 47 \nhours and these microstructure s are shown in Figure 5. A qua ntitative comparison of the Į-ferrite plate \ndensity, Į-ferrite plate size, and volum e fractions of austenite and δ-ferrite in the processed materials is \nshown in Table 1.  The Į-ferrite density was 53,300 ± 10,900 mm-2 in the plate processed by method A.  \nIn contrast, the Į-ferrite density decreased almost 60% to 21,600 ± 6,000 mm-2 when the hot rolled plate \nwas processed by method B and the Į-ferrite plate size increased fr om 0.75 ± 0.19 µm to 1.01 ± 0.32 µm \n(see Table 1).  Heat treating the plates at 1123 K (850 ºC) also produced a 60% decrease in the Į-ferrite \nplate density while the Į-ferrite plate size increased with incr eased holding time at 1123 K (850 °C). \nThere was no statistical difference in  the volume fraction of austenite or į-ferrite between the two plates; \nhowever, the austenite measured in the plate pro cessed by method B was inhomogeneous with slightly \nhigher volume fractions of  retained austenite in some measured  regions. Figure 6 shows that the {111} \nț-carbide peak was present in the XRD pa tterns for all heat treated products.   \n Nonmetallic inclusions in the various hot rolled  and heat treated products were analyzed and the \naverage inclusion diameters are summarized in Tabl e 2. The inclusion size in the method A plate was \nsignificantly less compared to the method B plate; the inclusion size in the heat  treated plates for both \nmethods were all indistinguishable statistically.  Specifically of interest to this study, the average \ndiameters of galaxite and manga nese oxides increased from 0.6 ȝm and 0.7 ȝm from method A to 1.3 \nȝm and 1.1 ȝm for method B, respectively.  Process met hod did not appear too affect the average \ndiameter of aluminum nitrides, which measured 4.5 ȝm.    \n Figure 7 compares inclusion density and incl usion volume fraction for the hot rolled products \nand the heat treated plates.  The total inclusion density and volume fraction of the hot rolled plate \nprocessed by method A were 400 mm-2 and 1 x 10-3, respectively.  The inclus ion density decreased to \n200 mm-2 and the volume fraction increased to 1.3 x 10-3 when the plate was processed by method B. \nThe nonmetallic inclusion density decreased after isothermal heat treatment at 1123 K (850 ºC) as \nshown in Figure 7.  The increase in  inclusion volume frac tion may suggest that the measurable inclusion content increased as a re sult of coarsening rather than by additi onal precipitation. The majority of the \ninclusions analyzed in the me thod A plate were galaxite (Al 2O3MnO, 140 mm-2) and a combination of \naluminum silicates and silica surrounding alumina (Al-Si-O, 85 mm-2).  On the other hand, the method B \nplate contained a low dens ity of galaxite (20 mm-2) and a high density of aluminum nitrides (70 mm-2 vs. \n27 mm-2).  The volume fraction of galaxite decreased to 2 x 10-5 from 6 x 10-4 and the volume fraction of \nnitrides increased 60% to 8 x 10-4 from 5 x 10-4 in the method A plate that was isothermally held at 1123 \nK (850 ºC) for 47 hours.  In contrast, the volume fraction of nitrides was not affected by heat treating the \nplate processed by method B at 1123 K (850 ºC).      \n F i g u r e  8  ( a )  d i s p l a y s  a  l i g h t  o p t ical micrograph of a plate pro cessed by method B, subsequently \nisothermally treated at 1123 K (850 ºC) for 47 hours, water quenched, and then heat treated at 1373 K \n(1100 ºC) for 15 minutes prior to be ing water quenched.  In the 1373 K (1100 ºC) heat treated plate, the \nĮ-ferrite plate density increased to 31,300 ± 5,300 mm-2 from 7,600 ± 2,200 mm-2 measured when the \nsample was held at 1123 K (850 ºC) for 47 hours.  The Į-ferrite plate dimension also decreased by 67% \nto 1.2 µm from 3.65 µm (see Table 3).  In addition, the volume fraction of the į-ferrite decreased to 0.20 \n± 0.02 in the 1373 K (1100 ºC) heat treated plate fr om 0.25 ± 0.04 as observed in the 1123 K (850 ºC) \nisothermally held plate.  Figure 8 shows ț-carbide was present after th e 15 minute hold at 1373 K (1100 \nºC) and water quenching the sample.  Diffraction intensity for {220} ț-carbide in addition to the {111} \nwas observed (see Figure 8 (b)). \n N o n m e t a l l i c  i n c l u s i o n  a n a l y s e s  ( s e e  F i g u r e  9 )  revealed a 46% reduction in the total inclusion \nvolume fraction when the plate was heated to 1373 K (1100 ºC) relative to the method B plate held at \n1123 K (850 ºC) for 47 hours.  The to tal volume fraction of nitrides in the 1373 K (1100 ºC) heat treated \nplate was reduced by 40%.  The total volume fraction of  the oxides remained relatively constant even \nthough the volume fraction and number density of galax ite increased after heat treatment.  The volume \nfraction of oxides containing silicon decreased to 2 x 10-5 from 1 x 10-4.    Microhardness measurements of į-ferrite and regions of Į-ferrite and austenite are compared in \nTable IV for steels processed by method A, method B, and method A heat treated for 2 hours at 1123 K \n(850 ºC).  Hardness of the austen ite region was 9.1% higher than į-ferrite when the plate was processed \nby method A.  In contrast, the austen ite region was 8.3% softer than the į-ferrite in the plate processed \nby method B.  The microhardne sses of the austenite and į-ferrite decreased from 431 and 395 to 265 and \n326 after a plate processed by method A was heat  treated for 2 hours at 1123 K (850 ºC).   \n Tensile test results for the materials listed in Table IV are shown in Figure 10 and Table V \nprovides a summary of the mechan ical properties including strain  hardening exponents measured \nbetween strain levels of 0.1 and 0.15.  The sample  processed by method A e xhibited discontinuous \nyielding (649 MPa upper yield stress and 637 lower yi eld stress) and yield point  elongation for up to \n12% strain prior to rapid work hardening.  Serrate d stress flow was also observed during strain \nhardening of the method A sample. When the produc t was processed by method B the work hardening \nbehavior appeared linear after yi elding at approximately 400 MPa and the strain hardening exponent was \n27% lower than the method A (see Table V).  A pl ate processed by method A and heat treated for 2 \nhours at 1123 K (850 ºC) showed simila r tensile behavior to the method B prepared specimen with linear \nwork hardening behavior after yielding (305 MPa) a nd an absence of serrated flow stress during work \nhardening.  Light optical  microscopy revealed Į-martensite in the tensile test specimens as shown in \nFigure 11.  Table IV shows the hard ness values measured of the mart ensite was approximately 530 HV. \n \nIV.  DISCUSSION \n \n A combination of bainite and acicular ferrite was observed in the micr ostructures of the Fe-\n13.92Mn-4.53Al-1.28Si-0.11C steel studied.  Literature shows that acicular ferrite rather than bainite is \nobtained when inclusion density in creases and grain size increases (i .e. the fraction of grain boundary \nnucleation sites decrease).13-18  In the work presented here the austenite grain size is limited to the \nspacing of the į-ferrite stringers, which measured 16 to 17 µm  and remained statistically constant within 1.5 to 2.5 ȝm after the various isothermal heat treatments  as shown in Table VI.  Thus, the austenite \ngrain size is eliminated as an experimental vari able in the thermal processes investigated.   \n Thermodynamic predictions for nonmetallic inclusi on formation in the steel composition studied \nare shown in Figure 12.  Equilibrium solidificat ion predictions (Figure 12 (a)) show galaxite \n(Al 2O3MnO) is not expected to form.38  However, when accounting for solute partitioning during \nsolidification using a method of Sche il segregation modeling, galaxite a nd silica become favorable in the \nlast 15% of the liquid to solid ify (Figure 12 (b)).  FactSage38 was used to predict that the final liquid \ntransformed to austenite and containe d lower aluminum concentration.  For this last portion of austenite, \ngalaxite is predicted to be stable  and should react with nitrogen and si licon in solid solution to form SiO 2 \nand AlN at temperatures approximately below 1133 K (860 ºC).  Figure 13 (a ) shows verification of \nSiO 2 nucleating on an aluminum oxide particle in a method A plate heat treate d at 1123 K (850 ºC).  \nThese thermodynamic calculations ag ree with the observed reduction in  galaxite particles and an \nincrease in silicon-aluminum oxi des during heat treatment at 1123 K (850 ºC).  Galaxite was nearly \nabsent ( f = 9.2 x 10-6) in the method B processed plate after 47  hours at 1123 K (850 ºC).  In contrast \nwhen the 47 hour heat treated plat e was reheated to 1373 K (1100 ºC) the volume fraction of galaxite \nincreased to 2.5 x 10-5.  Galaxite was cited as being an effect ive nucleation site for acicular ferrite26 and \nmany of the austenite grains showed  evidence of ferrite nucleation towa rds the center of the grain as \nshown in Figure 13 (b). However, it is difficult to claim intragranular nucleation when the austenite \ngrain size is less than 20µm.  \n D i f f e r e n t  m e c h a n i s m s  h a v e  b e e n  p r o p o s e d  t o  explain the role of nonmetallic inclusions in \nnucleating acicular ferrite.22-32, 39,40 Recent literature34, 41 appears to reach a consensus that inclusions are \nchemically active and cause mangane se solute depletion, which reduces  the stability of  the austenite; \nand these inclusions generally have  a small lattice disregistry with the ferrite which would reduce the \ninterfacial energy.  Literature on aci cular ferrite nucleation suggest that  a nonmetallic inclusion with a \nmisfit less than 6% is an effective nucleation site fo r ferrite; an inclusion with a misfit between 6-12% is moderately effective; and an inclusion with greater than 12% misfit would not be  expected to contribute \nto nucleation.25  Table 7 lists the misfit of inclusions th at are probable nucleation sites for acicular \nferrite.23, 26  Barbaro et al.14 noted that curvatur e of larger inclusions increased the probability to nucleate \nacicular ferrite due to the decrease in the interface curvature between the ferrite  and inclusion interface \nto provide better epitaxy.  The potency of the inclus ion as a nucleation site is  not linearly dependent on \nthe inclusion size.  An inclusion with a diameter gr eater than 0.6 µm had a highe r probability to nucleate \nacicular ferrite even though an in clusion with a diameter of 0.4 ȝm was noted as a possible nucleation \nsite for acicular ferrite.  Inclusions less than 0.6 µm in diameter were omitted from the analyses \ndiscussed below.    \n F i g u r e  1 4  t e s t s  t h e  r e l a t i o n s hip between the density of Į-ferrite and the density of nonmetallic \ninclusions listed in Table VII.  Densities of ma nganese oxide (including manganese oxides nucleated on \nmanganese sulfides, Figure 14 (a)) an d galaxite (Figure 14 (b)) suggest  a direct correlation with the \ndensity of ferrite plates.  These inclusi ons have less than 2% lattice misfit with Į-ferrite (see Table VII).  \nGalaxite and manganese oxide  were cited in past studi es as having a low misfit with ferrite, which led to \nacicular ferrite nucleation.23, 26    Manganese oxide was also cited as  depleting solute from the austenite \nand would thus promote the nucleation of acicular ferrite.42,43  In this study ga laxite and manganese \noxide (including these same oxides nucleated on manganese sulfides) were shown to have a strong \ncorrelation with the Į-ferrite density and therefore are viable nucleation sites for ferrite. \n I n  c o n t r a s t  t h e r e  w a s  no relationship between the Į-ferrite density and the densities of aluminum \noxide and manganese sulfide as shown in Figures 14 (c) and 14 (d).  In literature manganese sulfide was \nreported as a possible nucleation site  for acicular ferrite.  A manganese  depleted region surrounding an \ninclusion was hypothesized to ca talyze the ferrite nucleation.27-32  Mabuchi et al.27 showed manganese \ndepleted zones surrounded manganese  sulfides but these regions were  homogenized after the completion \nof precipitation in a steel cont aining 1.4 wt% manganese. The mangane se sulfides were ineffective \nnucleation sites for acicular ferrite after the elimination of the manganese depletion zone.27  In the study presented here, the volume fraction and density of manganese sulfides remained constant after the heat \ntreatments at 1123 K (850 ºC)  for 2 and 47 hours and we suggest that any ma nganese depleted zone \nwould be eliminated by the 47 hour heat treatment time. However, no difference in correlation between \nheating time and ferrite plate dens ity was observed and all measurements fell on the same trend line.  \nThe manganese profile across the interface of manganese  sulfide and austenite in a sample heat treated \nat 1373 K (1100 ºC) was acquired using energy dispersi ve spectroscopy in a S/TEM.  An TEM image in \nFigure 15 (a) shows morphology of the manganese sulfid e and the austenite grain was contrasted dark \nsince the sample was tilted to satisf y a <100> zone axis.  Figure 15 (b ) shows austenite was not depleted \nof manganese near the interface, which would suppor t the observation that manganese sulfides were \nineffective nucleation sites for acicula r ferrite nucleation.  The same shou ld be true for the galaxite and \nmanganese oxides, but the correla tion between ferrite plate density and oxide number density remains \nstrong even after 47 hours of isotherm al heat treatments suggesting that lattice registry is playing a more \nimportant role. \n \n F i g u r e  1 6  s h o w s  a  s t r o n g  correlation between the ferrite pl ate density and the density of \ninclusions determined viable based on Fi gure 14.  The relationship determined was ȡĮ = 440 ȡinclusion + \n2,800, where ȡĮ i s  t h e  Į-ferrite density and ȡinclusion i s  t h e  v i a b l e  i n c l u s i o n  d e n s i t y .   T h e  r e l a t i o n s h i p  \ndeveloped from Figure 16 suggests that th e bainitic ferrite contribution was 2,800 mm-2.  The austenite \ngrain size was approximately 16.5 µm so th ere would be approximately 2,810 grains/mm2 a n d  t h i s  \ncorresponds to an average of 1 bainite plate per grain.44  Additionally this relationship suggests that a \nviable inclusion greater than 0.6 ȝm will cause a nucleation event for 440 acicular ferrite plates/mm2 per \ninclusion.  However, the measured inclusion density is much lower than the austenite grain density and \ncorrelates to less than 1 inclusion (greater than 0.6 ȝm) per grain. Inclusion di ameters less than 0.5 ȝm \nare difficult to detect with the instrument used fo r automated feature analysis and sectioning effects may \nhave led to a smaller than actual measured in clusion density.  Incl usions less than 0.6 ȝm may also have contributed to the nucleation of acic ular ferrite, but to a lesser eff ect based upon the work of Barbaro et \nal.14 \n The density of acicular ferrite pl ates was a magnitude larger than the density of viable inclusions, \nwhich is similar to observations made by Barbaro et al.14  and Ricks et al.45 suggested nucleation \noccurred sympathetically on the broad ferrite plat e after the inclusion surface became saturated.  A \nrecent study by Wan et al.41 investigated the mechanism for the in terlocked microstruc ture of acicular \nferrite using three-dimensional reconstruction techni ques.  Multiple ferrite plates nucleating on a single \ninclusion, sympathetic nucleation along the broad face of a previous formed ferrite plate, and hard \nimpingement with intersection between ferrite pl ates were noted as being responsible for the \ninterlocking ferrite microstructure .  Sympathetic nucleation would re sult in the ferrite density being \nlarger than the inclusion densit y, which was also observed in the results reported he re.  The strong \ncorrelation between ferrite plate density and inclus ions of galaxite and manganese oxide support a \nconclusion that acicular ferrite can be formed in small grained austeniti c structures.  Furthermore, this \nstudy shows that a steel chemistry ca n be formulated where the inocula ting inclusions can be produced \nby heat treatment is the solid state rather than  relying on their formati on during solidification. \n M e t h o d  A  p r o d u c e d  a n  u l t i m a t e  t e n s i l e  s t r e n g t h  o f  970 MPa and an elongation to failure of 40% \nwhere the tensile strength is slightly below that  considered a break-throu gh for a third generation \nadvanced high strength steel.  The mechanical propert ies were significantly lower in the steel processed \nby method B and in the steel produced by method A af ter heating for 2 hours at  1123 K (850 ºC).  For \nexample, the yield strength decreased to 400 MPa when the plate was processed by method B (method A \n– 642 MPa) and to 305 MPa when the method A plat e was held at 1123 K (850 ºC) for 2 hours.   \n F o r  s t e e l s  w i t h  a c i c u l a r  m i c r o s t r u c t u r e s ,  t h e  y i e l d  strength is related to the inverse of the ferrite \nplate size.46  Dallum et al.47 observed coarser acicular ferrite in an  HSLA steel when the prior austenite \ngrain size was reduced which resulted in  fewer nucleation sites.  It was reasoned that the acicular ferrite \nwas coarser due to a reduced nuc leation rate.  Singh et al.48 modeled the thickness of ferrite plates in steels with an average composition of Fe-2 Si-2Mn-0.25C by accounting for the transformation \ntemperature, composition, and amount of supercooling re lative to the transformati on temperature.  It was \nshown that the strength of austen ite at the transformation temperat ure and the chemical free energy \nchange were the major contributors to  the thickness of ferrite plates.  The average size of the ferrite in \nthis study generally increased as the ferrite density decreased (Figure 16 (a)), wh ich was related to fewer \nviable nonmetallic inclusions.  Figur e 16 (a) suggests the ferrite size was controlled by a factor other \nthan just the ferrite nucleation rate since two separate  trend lines can be drawn for the data.  The ferrite \nplate size data can be reduced to a single trend li ne by plotting the data as a function of austenite \nhardness as shown in Figure 16 (b).  Here room temperature hardness is  used as a proxy for the strength \nof the austenite at the transformation temperature.  Figure 16 (c) shows an i nverse linear relationship \nbetween the hardness of austenite an d the fraction of nitrides.  It should be noted that the method A \nprocessed material was excluded from  these correlations since the material was in a work hardened state \nprior to transformation. Increased ni trogen in solution, as evidenced by a decrease in volume fraction of \nnitrides in method A processed steel,  would result in solid solution st rengthening of th e austenite.  The \nincrease in the ferrite plate size (method A: 0.75 ȝm vs. method B: 1.01 ȝm vs. method A heat treated 2 \nhours: 3.12 ȝm) observed in this study may have led to some  of the decrease in yield strength.  Tensile \ntests were repeated for a number of  the heat treated samples and the yi eld strength as a function of the \nferrite plate width is  shown in Figure 17. \n T h e  t e n s i l e  d u c t i l i t y  a s  m e a s u r e d  b y  e l o n g a t i o n  t o  failure of the plates processed by method B \nand method A plates that were heat  treated for 2 hours at 1123 K ( 850 ºC) decreased to  34% and 32%, \nrespectively.  The loss in ductility may be explained by considering the change in strength of the various \nmicrostructures, δ-ferrite in particular, with processing relative to the hardness of the martensite formed \nupon deformation. Sun et al.49 modeled the ductile failure mechanism in dual phase steels and showed \nthat the strength disparity between mechanically stable ferrite phase and deformation-induced martensite \nphase adversely affects the ductilit y.  A large discrepancy between the ferrite and martensite would lead to early void nucleation and reduced tensile ductility.49  Hasegawa et al.50 has also observed minimizing \nthe hardness differences between ferrite and marten site would lead to higher formability in a DP980 \nsteel. The retention of the cold wo rked dislocation structure in the į-ferrite when the plate was processed \nby method A (Figure 4 (a)) and the po ssibility of strain aging as di scussed below led to the higher \nhardness of the į-ferrite, decreased the har dness disparity with the mart ensite, and thus increased \nelongation. \n S e r r a t i o n s  i n  t h e  s t r e s s - s t r a i n  curve and high strain hardening exponents are characteristic of \ndynamic strain aging51,52 and were observed only in the steel processed by method A.  Serrated flow \ndisappeared and strain hardeni ng exponents decreased from 0.48 to 0.37 when a method A plate was \nheat treated at 1123 K (850 ºC) for 2 hours.  The steel processed by meth od B also had a reduced strain \nhardening exponent (0.36) and did no t show serrated flow.  Mn-C dipol es have been shown to aid in \ndynamic strain aging and rapid work hardening in high manganese steels like Hadfield steels.51,53  ț-\ncarbide, (Fe,Mn) 3AlC, has been reported to form in duplex fe rritic and austenitic steels with aluminum \nand carbon concentrations ranging from 5 to  7 wt% and 0.1 to 0.4 wt%, respectively.54  ț-carbide \nformation was suppressed in the plate processed by method A where the cold worked dislocation \nstructure was remnant.  ț-carbide was observed in the heat treated plates. A consequence of ț-carbide \nformation could be the removal of carbon from the austenite, which would reduce the Mn-C dipole \nconcentration and lead to lower strain harden ing exponents.  In a study by Zudiema et al.55 the addition \nof aluminum to a Hadfield steel caused the dynamic strain aging behavior to be suppressed at room \ntemperature.  It has been  shown by Medvedeva et al.56 that Fe-Al-C defect structures with the same \ncoordination as κ-carbide are energetically pr eferred.  This may suggest that deformation processing \nbelow the recrystallization temperature may destroy th ese Fe-Al-C clusters and inhibit the nucleation of \nκ-carbide as evidence in Figure 3a.   \n D i s c o n t i n u o u s  y i e l d i n g  w a s  o b s e r ved only in the plate processed by method A, which indicates a \nstrain aging behavior.  According to  literature discontinuous yielding is attributed to in terstitial nitrogen in solution.57-59  A plate processed by method A s howed a low volume fraction (6.2 x 10-4) of nitrides \nwhich suggest that nitrogen remained in solution.  The į-ferrite had a high ha rdness and dislocation \ndensity, which is prerequisite for nitrogen pinning and strain aging.  The volume fraction of nitrides \nincreased to 8.8 x 10-4 after a plate processed by method A wa s heat treated at 1123 K (850 ºC) for 2 \nhours.  The removal of nitrogen from solution and rec overy of the dislocation st ructure resulted in the \ndisappearance of the di scontinuous yielding. \n \nV.  CONCLUSION \n \n This study shows that an acicula r ferritic steel can be formulated  where nonmetallic inclusions \ncan be precipitated by thermal processing rather than formation during solidification.  Thermal \ntreatments were used to manipulate the concentration and type of oxides and the ferrite plate density was \nfound to correlate with inclusions  of low misfit. Specifically, manga nese oxide and galaxite were \neffective nucleation sites for aci cular ferrite in a duplex FeMnAl C steel of composition Fe-13.92Mn-\n4.53Al-1.28Si-0.11C.  The ferrite plate density did not de pend on the density of manganese sulfide.  The \nheat treatments employed eliminated any manganese depl eted zones next to these sulfides and as a result \nthe MnS was ineffective in nucleatin g acicular ferrite.  However, the sa me rationale should be true for \ngalaxite and manganese oxides, but the correlatio n between ferrite plate density and oxide number \ndensity remained strong even after 47 hours at  1123 K (850 °C) suggesting that the low lattice \ndisregistry between ferrite and eith er galaxite or manganese oxide wa s sufficient to encourage acicular \nferrite formation.   \n Both bainitic and acicular ferrite were observed in austenite with an av erage grain diameter of \n16.5 µm, which was below the previously reported cr itical austenite diameter for acicular ferrite \nformation.  The strong correlation between ferrite plate density and inclusions of galaxite and \nmanganese oxide support a conclusion that acicular fe rrite can be formed in small grained austenitic \nstructures that might be produced for automotive applications.     T h e  d u p l e x  s t e e l  a c h i e v e d  a n  u l t i m a t e  t e n s i l e  strength and elongation of  970 MPa and 40% when \nthe steel was finished at a low rolling temperat ure and possessed some degree deformation.  Upon \ntensile testing the retained  austenite transformed to Į-martensite.  Reheating the steel after hot rolling \nreduced the tensile ductility as a result of a greate r strength disparity between the deformation-induced \nĮ-martensite and the heat treated microstructure.  The yield strength was f ound to be dependent on the \nferrite plate size, which va ried linearly with the strength of the austenite.   \n T h e  s t r a i n  h a r d e n i n g  e x p o n e nt increased to 0.48 when ț-carbide was suppressed.  Rolling \ndeformation below the recrystallization temperature might have destroyed the energetically favorable \nFe-Al-C clusters that would have led to nucleation of ț-carbide upon quenching.  Discontinuous \nyielding was also observed when nitride volume frac tion was low.  Nitrogen in solution along with a \nhigh dislocation density would prom ote the discontinuous yielding of th e ferritic structures.  Conversely \nthe discontinuous yielding disappeared when th e volume fraction of n itrides increased. \n \nACKNOWLEDGEMENTS \n \n This work was supported in part by the Nationa l Science Foundation (NSF ) and the Department \nof Energy under contract CMMI 0726888. The FEI Helio s NanoLab dual beam FIB was obtained with a \nMajor Research Instrumentation grant from NSF under contract DMR-0723128 and the FEI Tecnai F20 \nscanning/transmission electron microscope was obtain ed with a Major Research  Instrumentation grant \nfrom NSF under contract DMR-0922851. The author s gratefully acknowledge the support of the \nGraduate Center for Materials Research and in partic ular Eric Bohannen for help with x-ray diffraction.  \nMeghan McGrath was supported by a Department of Education GAANN fellowship under contract \nP200A0900048.   \n \n \n REFERENCES \n1.   J. Fekete and J. Hall: NIST Internal Repor t 6668, National Institute of Standards and Technology, \nWashington, DC, May 2012. \n2.   G. Frommeyer and U. Brux:  Steel Research Int., 2006, vol. 77, pp. 627-633. \n3.  S. Allain, J.-P. Chateau, O. Bouaziz, S. Mi got and N. Guelton:  Mater. Sci. Eng., 2004, vols \n387-389A, pp.158-162.  \n4. A. Dumay, J.-P. Chateau, S. Alla in, S. Miget, and O. 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Leslie:  Acta. Metal., 1972, vol. 20, no. 10, pp. 1157-1167. \n54.  K-G. Chin, H-J. Lee, J-H, Kwak, J-Y Kang, a nd B-J. Lee:  J. Alloys and Compounds, 2010, vol. \n505, pp. 217-223. 55.  B.K. Zuidema, D.K. Subramanyam, and W. C. Leslie:  Metall. Trans. A, 1987, vol. 18, pp. 1629-\n1639. \n56.  N. I.  Medvedeva,  M. S. Park,  D. C. Van Aken,   and J. E. Medvedeva: arXiv:1208.0310v1 \n[cond-mat.mtrl-sci], 2012. \n57.  J.R. Low, and M. Gensamer:  Trans. Am er. Inst. Min (metal.) Engrs., 1944, vol. 158, p 207. \n58.  D.S. Wood and D.S. Clark:  Tran s. Amer. Soc. Metals, 1952, vol. 44, p. 726. \n59.  M.L.G. Byrnes, M. Grujicic, and W.S.  Owen: Acta Mater., 1987, vol. 3, no. 7, pp 1853-1862. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n LIST OF FIGURE CAPTIONS \n \n \nFig. 1 – (a) Ferrite plate density decreas es with decreasing amount of vi able nonmetallic inclusions for \nnucleation in six different FeMnAlC steels (lab eled A through E) with varying deoxidation and \ndesulfurization practices.15 Deoxidation/desulfurization practices: A- Ca additions; B- no Ce additions; \nC- Ce additions D- Ca additions and 30s Ar-stirri ng; and E- calcium additi ons and 60s Ar-stirring.  \nMicrostructures from the steels processe d by (b) A, (c) B, and (d) E are shown.35  \n \nFig. 2 – Light optical micrographs of Fe-13.92Mn- 4.53Al-1.28Si-0.11C plates hot rolled at 1173 K (900 \nºC) and were either (a) water quenched immediately (method A) or (b) reheated at 1173 K (900 ºC) for \n10 minutes before being water quenched (method B). \n \nFig. 3 – XRD patterns of hot rolled products (a) water quenched immediately (method A) and (b) \nreheated for 10 minutes at 1173 K (900 ºC ) prior to water quenching (method B). \n \nFig. 4 – S/TEM images of the dislocation structure of į-ferrite in the hot rolled plates processed by (a) \nmethod A and (b) method B.  Polishing artifacts from ion milling during preparation of S/TEM \nspecimens were observed in the į-ferrite.   \n \nFig. 5 – Light optical micrographs  from hot rolled plates that we re water quenched immediately and \nthen subsequently heat treated at 1123 K (850 ºC ) for (a) 2 hours and (b) 47 hours; and micrographs \nfrom plates that were reheated pr ior to water quenched and then heat  treated at 1123 K (850 ºC) for (c) 2 \nhours and (d) 47 hours.  Specimens were water quenched after isothermal holds. \n \n Fig. 6 – XRD patterns of hot rolled products processe d by (a) method A then heat treated at 1123 K (850 \nºC) for 2 hours; (b) method A then heat treated at 1123 K (850 ºC) for 47 hours; (c) method B then heat \ntreated at 1123 K (850 ºC) for 2 ho urs; and (d) method B then heat treated at 1123 K (850 ºC) for 47 \nhours.  Specimens were water quenched after annealing.  ț-carbide peaks were observed in the patterns \nof the heat treated plates. \n \nFig. 7 – (a) Comparison of inclusion density between hot rolled products that were rolled at 1173 K (900 \nºC) and processed by method A.  Plates processed by method A were then heated at 1123 K (850 ºC) for \n2 hours and 47 hours. (b) Comparison of volume frac tion of inclusions for the various hot rolled \nproducts processed by method A.  (c) Comparison of in clusion density between hot rolled products that \nwere processed by method B and then subsequently  heat treated at 1123 K (850 ºC) for 2 hours and 47 \nhours.  (d) Comparison of the volume fraction of inclusions in plates processed by method B and \nsubsequently heat treated.  Inclusions identified in le gend reading left to right a nd are plotted in the same \norder from top to bottom.  \n \nFig. 8 – (a) Light optical micrograp h of method B plate that was h eat treated at 1373 K (1100 ºC) (15 \nminutes) after heat treating at 1123 K (850 ºC) fo r 47 hours.  (b) XRD pattern showing diffraction \nintensity for κ-carbide.  Sample was water quenc hed after each heat treatment. \n \nFig. 9 – (a) Comparison of inclusion density between pl ates that were hot roll ed at 1173 K (900 ºC) and \nthen processed by method B; heat treated at 1123 K (850 ºC) for 47 hours; and heat treated at 1373 K \n(1100 ºC) for 15 minutes. (b) Comparison of volume frac tion of inclusions for the various hot rolled \nproducts.  Inclusions identified in le gend reading left to right  and are plotted in th e same order from top \nto bottom. \n Fig. 10 – Representative tensil e tests of Fe-13.92Mn-4.53Al-1.28Si- 0.11C after being processed by \nmethod A (black curve); method B (red curve); a nd method A heat treated for 2 hours at 1123 K (850 \nºC) (blue curve).   \n \nFig. 11 – Light optical micrograph from gage secti on of a plate that was processed by method A and \npulled in tension to failure at 970 MPa with 44% elongation. \n \nFig. 12 – FactSage38 predictions for nonmetallic inclusions in steels with com position (a) Fe-13.92Mn-\n4.53Al-1.28Si-0.045N-0.11C-0.05O-0.05S (bulk); (b) Fe -20.3Mn-0.7Al-3.2Si- 1.2N-0.25C-0.05O-0.05S \n(last 15% to solidify). \n \nFig. 13 – Secondary electron images of (a) silica nuclea ting on an aluminum oxide particle in a sample \nprocessed by method A and heat treated at 1123 K (850 ºC) for 47 hours and (b) galaxite observed \nwithin a cluster of acicular ferrite  in a sample processed by method B and held at 1373 K (1100 ºC) after \nbeing heat treated at 1123 K (850 ºC) for 47 hours. \n \nFig. 14 – Relationships between acicular ferrite density and density of (a) MnO 2 and MnO 2 nucleated on \nMnS; (b) galaxite; (c) Al 2O3; and (d) MnS for inclusi ons larger than 0.6 µm. \n \nFig. 15 – (a) TEM image showing boundary between manganese sulfide (MnS) and austenite (FCC).  (b) \nManganese profile across the MnS and FCC bou ndary.  There was no depletion of manganese. \n \nFig. 16 – A positive relationship between acicular ferrite  plate density and inclusions with low misfit \nthat were larger than 0.6 ȝm was observed. \n Fig. 17 – Į-ferrite plate size was a function of (a) ferrite plate density a nd (b) austenite hardness.  (c) \nThe hardness of the austenite was a function of the total nitrides.  The datum of processing by method A \nwas removed due to this steel being in a wo rk hardened state prior to transformation.  \n \nFig. 18 – Yield strength was inversely related to the Į-ferrite plate size.  The plate processed by method \nA had retention of the dislocation structure from cold working which increased the strength of the \nmaterial.  This data point was excluded from dete rmining the linear relationshi p between yield strength \nand the inverse of the size of the acicular ferrite. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n LIST OF FIGURES \n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\n\u0003\nLIST OF TABLE CAPTIONS \nTable I.  Summary of Density and Size of Į-Ferrite Plates and Volume Fractions of Phases  \n \nTable II. Summary of Aver age Inclusion Diameter  \n \nTable III. Summary of Į-Ferrite Plate Density and Size and Volume Fraction of Phases  \n \nTable IV.  Summary of Microhardness  \n \nTable V.  Summary of Mechanical Properties \n \nTable VI.  Comparison of Spacing between į-ferrite Stringers  \n \nTable VII. Lattice Misfit between Different Inclusions and Ferrite23, 26 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \nLIST OF TABLES \n \n \nTable I.  Summary of Density and Size of Į-Ferrite Plates and Volume Fractions of Phases  \n \n Į-Ferrite Plate \nDensity, mm-2 Į-Ferrite Plate \nsize, ȝm Vf of \nAustenite Vf of į-\nFerrite \nMethod A  53,300 ± 10,900 0.75 ± 0.19 0.35 ± 0.06 0.16 ± 0.05\nHeat Treated at 1123 K (850 ºC) for:   \n2 hrs 20,100 ± 4,800 3.12 ± 0.40 0.35 ± 0.08 0.16 ± 0.02\n47 hrs 10,900 ± 3,800 5.6 ± 0.85 0.38 ± 0.07 0.22 ± 0.04\nMethod B 21,600 ± 6,000 1.01 ± 0.32 0.40 ± 0.07 0.17 ± 0.05\nHeat Treated at 1123 K (850 ºC) for:   \n2 hrs 13,800 ± 3,300 2.51 ± 0.20 0.47 ± 0.03 0.19 ± 0.02\n47 hrs 7,600 ± 2,300 3.65 ± 0.99 0.49 ± 0.06 0.25 ± 0.04\n \n \nTable II. Summary of Aver age Inclusion Diameter  \n \n Inclusion Diameter, µm \nMethod A  0.9  ± 0.2 \nHeat Treated at 1123 K (850 ºC) for:   \n2 hrs 2.2 ± 0.2  \n47 hrs 2.9 ± 0.2  \nMethod B 2.2 ± 0.2  \nHeat Treated at 1123 K (850 ºC) for:   \n2 hrs 2.2 ± 0.2  \n47 hrs 2.1  ± 0.2  \n \n \n \n \n \n \n Table III. Summary of Į-Ferrite Plate Density and Size and Volume Fraction of Phases  \n \n Į-Ferrite Plate \nDensity, mm-2 Į- Plate \nsize, ȝm Vf of \nAustenite Vf of į-\nFerrite \nMethod B 21,600 ± 6,0000 1.01 ± 0.32 0.40 ± 0.07 0.17 ± 0.05 \nHeated at 1123 K \n(850 ºC) for 47 hrs 7,600 ± 2,300 3.65 ± 0.99 0.49 ± 0.06 0.25 ± 0.04 \nHeat Treated at \n1373 K (1100 ºC) 28,300 ± 3,800 1.20 ± 0.21 0.39 ± 0.05 0.20 ± 0.02 \n \n \nTable IV.  Summary of Microhardness  \n \nProcessing Sequence į, \nVickers \nHardnessĮ + Ȗ, \nVickers \nHardness Ȗ, \nVickers \nHardnessĮ-martensite (post \ntensile testing), \nVickers Hardness \nMethod A 395 ± 37 453 ± 32 431 ± 23 530 ± 19 \nMethod A Heated at 1123 \nK (850 ºC) for 2 hours  326 ± 12 291 ± 31 265 ± 20 526 ± 68 \nMethod B 322 ± 18 332 ± 15 295 ± 28 532 ± 32 \n  \n \n \n \nTable V.  Summary of Mechanical Properties \n \nProcessing Sequence Yield \nStrength, MPaUltimate Tensile \nStrength, MPa % \nElongation n \nMethod A 637 970 44 0.48 \nMethod A Heated at 1123 \nK (850 ºC) for 2 hours  305 682 34 0.37 \nMethod B 400 748 32 0.35 \n \n \n \n \n \n \n \n Table VI.  Comparison of Spacing between į-ferrite Stringers  \n \n į-ferrite Stringer \nSpacing, ȝm \nMethod A  17.1 ± 1.0 \nHeat Treated at 1123 K (850 ºC) for: \n2 hrs 16.0 ± 1.6 \n47 hrs 17.3 ± 1.4 \nMethod B 16.4 ± 2.2 \nHeat Treated at 1123 K (850 ºC) for: \n2 hrs 16.0 ± 1.5 \n47 hrs 16.3 ± 2.6 \nHeat Treated at 1373 K (1100 ºC) for: \n15 minutes 16.3 ± 1.7 \n \n \n \n \n \n \nTable VII. Lattice Misfit between Different Inclusions and Ferrite23, 26 \n \nInclusion Misfit % \nĮ-MnS 8.8 \nȖ-Al 2O3 3.2 \nAl2O3MnO (Galaxite) 1.8 \nMnO 2 1.5 \n \n\u0003\n\u0003\n \n \n \n \n \n \n \n "    },    {        "title": "0911.4227v1.Electronic_structure_and_magnetic_properties_of_Ni0_2Cd0_3Fe2_5_xAlxO4__x___0_0___0_4__ferrite_nanoparticles.pdf",        "content": "Electronic structure and magnetic properties of Ni 0.2Cd0.3Fe2.5-xAlxO4 \n(0≤x≤0.4 ) ferrite nanoparticles  \nShalendra Kumar1*, Khalid Mujasam Batoo2, S. Gautam3, B. H. Koo1, Alimuddin2, K. H \nChae3, Chan Gyu Lee1 \n1School of Nano and Advanced Materials Engineerin g, Changwon National University, 9 \nSarim dong,  Changwon -641-773, Republic of Korea  \n2Department of Applied Physics, Aligarh Muslim University, Aligarh, UP 202002, India  \n3Nano Materials Analysis Center , Korea Institute of Science and Technology , Seoul \n136-791, Republic of Korea  \nStructural, magnetic and electronic structural properties of Ni0.2Cd0.3Fe2.5-xAlxO4 ferrites \nnanoparticles have been studied x-ray diffraction  (XRD) , transmission electron microscopy  \n(TEM) , dc magnetization, and near edge x -ray absorpt ion fine structure spectroscopy  (NEXAFS)  \nmeasurements.  Nanoparticles of Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) ferrite  were synthesized using \nsol-gel method. The XRD and TEM measurements show that all samples have single phase \nnature with cubic structure and hav e nanocrysta lline behavior. From the XRD and TEM analysis , \nit is observed that particle size increases with Al doping . DC magnetization measurements infer \nthat magnetic moment decreases whereas blocking temperature increases with increase in Al \ndoping.  It is observed  that the magnetic moment decreases with Al doping which may be due to  \nthe dilution of the sublattice by the doping  of Al ions. The NEXAFS measurements performed at \nroom  temperature indicates that Fe exist  in mix ed valence state.   \nKeywords: Ferrite Nanoparticles , XRD, TEM, DC Magnetization , NEXAFS   \nCorresponding author  \nE-mail: shailuphy@gmail.com  (S. Kumar) ; chglee@changwon.ac.kr  (C. G. Lee)  \nPh: +82 -55-213-3703; Fax: +82 -55-261-7017  1. INTRODUCTION  \nIn the recent years , magnetic nanoparticles have been a subject of great interest from both \ntechnological and fundamental points of view.1-7 Among the magnetic materials, ferrites \nnanoparticles  found to  play crucial roles in the fast -packed miniaturization of modern electronic \ndevices and biomedical applications. The origin for the spacious applications is related to the \ndiversity of transition metal cations which can be integrated into the lattice of the parent \nmagnetic st ructure.  Particularly , mixed spinel ferrite nanoparticles have generated a large \nresearch effort because their magnetic properties differ markedly from single ferrite. Spinel \nferrites crystallize into a cubic close packed structure of oxygen ions. The cati ons occupy two \ntypes of interstitial sites. One of them is called a tetrahedral (A) site with the cation surrounded \nby the four oxygen ions in tetrahedral coordination. The other interstitial position is known as \noctahedral (B) site with cation coordinated  by six oxygen ions in octahedral symmetry .8 In \ngeneral, the cation distribution in spinel lattice has the form: (D1-xTx) [DxT2-x] O4, where D and T \nare divalent and trivalent ions, respectively and x is so called the degree of inversion. The round \nand squ are brackets denote the cations located at the center of the tetrahedral lattice of oxygen \n(A) and those at the octahedral ( B) lattice, respectively. The main source  of the magnetic \nproperties of spinel ferrites  is the spin magnetic moment of the unpaired 3d electrons of the \ntransition metal cations  coupled by the superexchange interaction via the oxygen ions separating \nthem. The magnetic properties of spinel ferrites, such as transition temperature and saturation \nmagnetic moment are strongly dependent on t he distribution of cations , type of doping atom  and \ncrystallite size . Therefore, the magnetic  properties of these materials can  be tailor ed by using the \ndoping of the different transition metal cations as well as by varying  the particle size.  The X -ray abs orption spectroscopy (XAS) is recognized  to be a powerful and most insightful  \ntechnique  for probing the chemical valency of the cations in the transition metal compounds .6 \nThe sensitivity of this technique to a crystal electric field with specific point gr oup symmetry can \nbe used to extract a number of  significant  parameters, such as crystal field strength ( 10 Dq ) and \nhybridization etc. from a single experiment. The NEXAFS has been widely  used to identify the \nvalence states of ions in a material . In this procedure , the photons of specific characteristic \nenergy are  absorbed to produce the transition of a core electron to an empty state above the \nFermi level which is governed by a set of dipole selection rules.  In the present work, we have \nstudy the effect of Al doping on structural, magnetic and electronic  structural  properties of \nNi0.2Cd0.3Fe2.5O4 ferrite nanoparticles.  \n2. EXPERIMENTAL DETAILS  \nSol-gel technique has been used to fabricate the nanoparticles of Al doped Ni 0.2Cd0.3Fe2.5O4 \nferrite. The analytical gra de chemical reagents used to prepare these materials were Ni(NO 3)2 . \n6H2O, Cd(NO 3)2 . 2H2O, Al(NO 3)3 . 9H2O, and Fe(NO 3)2 . 9H2O. Details  of the preparation of \nthese mat erials are reported elsewhere .9 The stoichiometric amounts of metal nitrates were \ndisso lved in deionized water and then few drops of ethyl alcohol were added to this solution. In \norder to get the fine crystalline particles, few drops of N,N -dimethylformamide C 3H7NO (M.W. \n73.10) were also added in the solution. The solution was put on the mag netic stirrer at 75 oC with \nconstant stirring until the gel was obtained. The gel formed was annealed at 90 oC for 19 h \nfollowed by grinding for half an hour. Finally the powder was calcinated at 400 oC for 36 h to \nremove any organic material present in the system and then grinded for half an hour.  \nThe prepared nano particles were characterized using X -ray diffraction (XRD) , \ntransmission electron microscopy  (TEM ), dc magnetization  and near edge x -ray absorption fine structure spectroscopy (NEXAFS) measuremen ts. Philips x -pert x -ray diffractometer with Cu Kα \n(λ = 1.54 Å) was used to study single phase nature of the samples at room temperature. TEM \nmeasurements were performed using FE -TEM (JEM 2100F). DC magnetization measurements \nwere performed using Quantum Design physical properties measurement setup. The NEXAFS \nmeasurements of these samples, along with reference compounds of FeO  (Fe2+), Fe 2O3 (Fe3+) \nand Fe3O4 (Fe2+/Fe3+) , at Fe L3,2-edge s were performed at the soft X -ray beam line 7B1 XAS \nKIST (Korea Institute of Science and Technology) of the Pohang Acce lerator Laboratory (PAL), \noperating at 2.5 GeV with a maximum storage current of 200 mA. The spectra were \nsimultaneously collected in the total electron yield (TEY) mode and the fluorescence yield (FY) \nmode at room temperature in a vacuum of better than 1. 5×10-8 Torr. The spectra in the two \nmodes turned out to be nearly identical, indicating that the systems were so stable that the \nsurface contamination effects were negligible even in the TEY mode. The spectra were \nnormalized to the incident photon flux, an d the energy resolution was better than 0.2 eV. The \ndata were normalized and analyzed using Athena 0.8.054.  \n3. RESULTS AND DISCUSSIONS  \nFig. 1 represents the θ-2θ XRD patterns of the Ni 0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) nanoparticles.  \nThe reflection peaks observed from the samples ca n be well indexed with the standard pattern of \nthe cubic spinel ferrites. The broadening of the reflection peaks suggests that samples  have \nnanocrystalline behavior.  The particle size of Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) nanoparticle was \ncalculated from the most intense peak (311) of XRD data using Debye  Scherrer formalism10 \n                                            \n                                                                      1 where  = (M2-i2)1/2. Here  is x-ray wavelength (1.54 Å for Cu K ), M and i is the measured \nand instrumental broadening in radians respectively and  is the Bragg`s angle in degrees.  The \ncalculated value of particle size was found to increase  from 4 to 7 nm with Al doping, which \nimplies that Al doping favor  the particle growth.  \nThe size distributions and presence of any impurity phase were further studied by TEM images \nand electron diffr action patterns. Ima ge J 1.3.2 J software was used to determine the average \nparticle size and the size distribution, by analyzing approximately 200 particles. Figure 2 (a ) and \n(b) displays TEM micrographs of Ni0.2Cd0.3Fe2.5-xAlxO4 nano particles  for x = 0.0  and 0.4.  It is \nclearly evident from the micrographs that the prepared samples are composed of nanoparticles. \nFrom the particle size distribution histograms (see upper right inset in Fig. 2  (a) and (b) ), the \naverage diameters are in the range from 3.5 – 7.5 nm for different Al concentrations. The particle \nsize measured from the TEM micrographs is in  excellent agreement with the calculated values by \nScherrer’s formula. Upper left inset in Fig. 2 (a) and (b)  shows the  selected area electron \ndiffraction (SAED)  patterns observed from the Ni 0.2Cd0.3Fe2.5-xAl xO4 (0≤x≤0.4 ) nanoparticles. \nRings in these patterns indicate clearly the randomly oriented single crystals and hence ruled out \nthe presence of any impurity phase. SAED pattern (see upper left inset in Fig. 2 (a) and (b))  \ndemonstrate that each nanoparticle is i ndeed in single phase.  \nFig. 3 presents the magnetic hysteresis loop measurements of Al doped Ni0.2Cd0.3Fe2.5O4 \n(0≤x≤0.4 ) ferrite nanoparticles  at room temperature. An absence of hysteresis, almost \nimmeasurable coercivity and remanence represents the charac teristic of super -magnetic behavior \nof the samples . In order to get  more insight of the magnetic behavior of Al doped \nNi0.2Cd0.3Fe2.5O4 nanoparticles, we had performed zero field cooled (ZFC) and field cooled (FC) \nmagnetization. Inset in Fig. 3 shows the Z FC and FC magnetization measurements curve for x = 0.0 and 0.4. In the ZFC cycle , the sample was cooled from 320 to 20 K in the absence of \nmagnetic field and after stabilization of the temperature a magnetic field of 500 Oe was applied. \nThe data were then recorded while heating the sample. In FC cycle the sample was cooled from \n320 to 20K in the presence of a magnetic field of 500 O e and then the measurement s were \ncarried out while heating the sample in the same field.  The presence of the cup in the Z FC and  \nthe bifurcation between ZFC and FC curve at certain temperature shows the characteristic feature \nof the superaparamagnetic system. The presence of bifurcation in the ZFC and FC curve at \ncertain temperature shows the characteristic feature of  a superparama gnetic behavior .11,12 The \ntemperature at which magnetization start decreasing is know n as blocking temperature (T B).The  \nbroad maxima observed in ZFC curve (denoted as  TB) indicates a certain particle size \ndistribution  in the system.  Such type of magnetic b ehavior can be explained in the light of \nStoner -Wohlfarth theory. According to Stoner -Wohlfarth theory, the magnetocrystalline \nanisotropy EA of a single domain particle is expressed as follows:  \n                                                  EA = K DS Sin2θ,                                                                     (2) \nwhere K is magneto crystalline anisotropy constant, D S is size of nanoparticles, and  θ is the angle \nbetween magnetization direction and ea sy direction of nanoparticles .13 When ther mal activation \nenergy ( kBT, where k B is Boltzman n constant)  is comparable with EA, the direction of the \nmagnetization of the nanoparticles starts fluctuate and goes through superparamagnetic  \nrelaxation.  As the temperature increases over T B, the magnetocrys talline anisotropy  is overcome \nby the thermal energy, the direction of the magnetization of the nanoparticles follows the \ndirectio n of the applied magnetic field which results  in the superparamagnetic behavio r of the \nnanoparticles. However , below  TB therma l energy is no longer competent to overcome the \nmagnetocrystalline anisotropy of the nanoparticles as a result the magnetization direction of the nanoparticles rotates from field direction to its own easy axis. In the present studied samples, the  \nbroad max ima is observed at a slightly l ower temperature (T B) than T IRR (TIRR is the temperature \nat which irreversibility start between ZFC and FC magnetization). This behavior results from a \ncertain parti cle size distribution in system and reflects that a fraction  of large particles freezes at \nTIRR whereas the majority of nanocrystallites are being blocked at T B. The T B of Al doped \nNi0.2Cd0.3Fe2.5O4 nanoparticles were found to increase from at 88  K to 110K with increase in Al \ndoping.  The increase in the T B reflect s that size of the nanoparticles increases with Al doping \nwhich is in well agreement with XRD and TEM results. Moreover, it can be clearly seen from \nthe magnetization versus field measurements that t he magnetic moment was found to decrease \nwith Al doping. The decrease in magnetic moment may be due to the dilution of the sub -lattice \nby the substitution of magnetic i ons by the non -magnetic ions.5 This decrease in magnetic \nmoment infers that the Al ions occupies the octahedral site of the spinel lattice thereb y \ndecreasing the number of magnetic ions at B site and thus decreases the magnetic moment of the \nsystem that is evident from the magnetization measurements.  \nThe near edge x -ray absorption fine structure (NEXAFS) study is most trustworthy method for \nthe ele ment specific characterization. The NEXAF S measurements were performed to \ninvestigate the ele ctronic structure and chemical environment of Fe ions in  Ni0.2Cd0.3Fe2.5_xAlxO4 \n(0≤x≤0.4 ) ferrite nanoparticles . Fig. 4 shows the Fe L3,2 –edge NEXAFS spectra of \nNi0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) along with the reference compounds Fe 2O3(Fe3+), FeO(Fe2+) and \nFe3O4(Fe2+/Fe3+) . The NEXAFS spectra at Fe L3,2 –edge find out the 3d occupancy of the Fe \nions and provide information about valence state of Fe ions in the studi ed samples. The Fe L3,2 –\nedge spectra are primarily due to the Fe 2 p-3d hybridization and are affected  by the core -hole \npotentials. The intensity of these lines represents  the total unoccupied Fe 3 d states. Th e two broad multiple structures  L3 and L2 obser ved in Fe L3,2 spectra are well known for reference \ncompounds Fe 2O3, FeO, and Fe 3O4. The main difference in  the reference  spectra can  be clearly \nseen at the L3 feature. The difference in L3 features can be  attributed to the variety of 3 d electron \nconfigura tion of Fe ions (Fe2+ or Fe3+) and the indication of local symmetry (tetrahedral or \noctahedral). The L3 feature of Fe 2O3 is characterized by a well developed doublet, a small \nintensity peak marked as A and a main peak marked as B, while in FeO the first pe ak becomes a \nshoulder of the main peak. These two spectral features in the L 3 region were assigned to Fe t 2g \nand e g sub-bands, respectively.  It can be clearly seen from the NEXAFS spectra that the spectral \nfeature of Ni 0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4) nanoparticles resembles with Fe 3O4 and FeO spectra \nindicates that Fe is in mix valence (Fe3+ & Fe2+) state. A difference spectra is also drawn (see \ninset in Fig. 4) by subtracting  the x=0.0 spectra with respective spectra. It is observed that pre -\nedge peak is decreased with Al -ion doping and main peak is also shifted slightly, which shows \nan increase of Fe2+ions in the system. In order to obtain a better insight of the valence state of Fe \nions, a linear combination fitting (LCF) at Fe K -edge NEXAF S spectra of the samples is carried \nout ( see Fig. 5  (a) and (b) ) from -20 to +40 eV taking as Fe(Fe0), FeO(Fe2+), Fe 2O3(Fe3+) and \nFe3O4(Fe2+/Fe3+) as standard spectra. The calculated parameters clearly show the existence of \nmixed valence state in Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4) nanoparticles .  \n4. CONCLUSIONS  \nWe have successfully synthesis  single phase polycrystalline  nanoparticle s of           \nNi0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) ferrites. XRD and TEM measurements indicate that particle size \nincreases with Al dop ing. Magnetic measurements reflects that Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) \nferrites nanoparticles have the superparamagnetic behavior at room temperature and blocking \ntemperature increases whereas magnetic moment decreases with Al doping. The NEXAFS measure ments performed at Fe L3,2 – edge and linear combination fitting (LCF) done at Fe K -\nedge spectra show that Fe is in mix valence state  (Fe3+/Fe2+).       \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n References:  \n1. S. C.  Bhargava and N. Zeman Phys. Rev. B  21, 1717 (1980).  \n2. K. Murali dharan, J. K. Srivastava, V. R Moratha and  R. J. Vijayaraghavan  Phys. C  18, \n5897 (1985).  \n3. R. A. Brand, J.  Lauer and D. M. Herlach J. Phys.  F14 (1984).  \n4. J. L. Dormann and M. Nogues  J. Phys. Condense Matter  2, 1223 (1990).  \n5. Shalendra Kumar, Alimuddin, Ravi  Kumar, A. Dogra, V. R. Reddy, A. Banerjee, J. Appl. \nPhys; 99 (2006) 08M910.  \n6. Shalendra Kumar, Alimuddin, Ravi Kumar, P. Thakur, K. H. Chae, Basav arav Angadi \nand W. K. Choi   J. Phys. Condense Matter  19, 476210 (2007).  \n7. S. K. Sharma, Ravi Kumar, Shalendra Kum ar, M. Knobel, C. T. Meneses, V. V. Siva \nKumar, V. R. Reddy, M. Singh and C. G. Lee, J. Phys. Condense Matter  20, 235214 \n(2008).  \n8. Shalendra Kumar, Ravi Kumar, S. K. Sharma, V. R. Reddy, and Alimuddin, Solid State \nCommun. 142, 706 (2007).  \n9. Khalid Mujasam Bato o, Shalendra Kumar, Chan Gyu Lee, Alimuddin, Current Appl. \nPhys. 9, 826 (2009).  \n10. S. K. Sharma, Ravi Kumar, Shalendra Kumar, V. V. Siva Kumar, M. Knobel, V. R. \nReddy, A. Banerjee, M. Singh, Solid State Commun. 141, 203 (2007).  \n11. V. Kumar, A. Rana, M. S. Yadav,  R. P. Pant, J. Magn. Magn. Mater. 320, 1729 (2008).  \n12. M. Sorescu, Diamandesca, R. Swaminathan , M. E. Mchenry, M. Fedar, J. Appl. Phys. 97, \n10G105 (2005).  \n13. H. S. Choi, M. H. Kim, H. J. Kim, J. Mater. Res., 9, 2425 (1994).   \nFigure Captions  \nFig. 1  (Colour Onlin e) X -ray diffraction patters of Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤ x ≤0.4) ferrite  \nnanoparticles.  \nFig. 2  (Colour Online) TEM micrograph of Ni0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4 ) ferrite  nanoparticles \n(a) x = 0.0, (b) x = 0.4. Upper left insets show  the corresponding SAE D pattern  for  x = 0.0 and \n0.4. Upper right insets show the particle size distribution histograms for x = 0.0 and 0.4  \nFig. 3 (Colour Online) Magnetization versus magnetic field measurements of \nNi0.2Cd0.3Fe2.5_xAlxO4 (0≤ x ≤0.4) ferrite  nanoparticles. Inset  shows the ZFC and ZC curve for x \n= 0.0 and 0.4.  \nFig. 4 (Colour Online)  Normalized Fe L3,2 -edge spectra Ni 0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4) ferrite \nnanopart icles  plotted with reference s pectra FeO(Fe2+), Fe 2O3 (Fe3+) and Fe 3O4(Fe2+/Fe3+) for \ncomparison.  Inset shows difference spectra obtained after subtracting x=0.0 spectra from \nrespective spectra, to observe the difference with Al –ion doping.  \nFig. 5 (Colour Online) (a) Normalized Fe K -edge spectra Ni 0.2Cd0.3Fe2.5-xAlxO4 (0≤x≤0.4) ferrite \nnanoparticles  p lotted with reference spectra FeO(Fe2+), Fe2O3 (Fe3+) and Fe3O4(Fe2+/Fe3+) for \ncomparison. (b) Linear combination fitting (LCF) for x=0.0, 0.1, 0.2, 0.3 and 0.4 samples plotted \nwith experimental spectra. Inset as table shows the calculated parameters.  \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1  \n \n  \n \n \n \nFig. 2  \n \n \n \n \n \n \n \n \n \nFig. 3 \n \n 700 710 720 730700 710 720 730 740\nBIntensity (arb. units)\nPhoton Energy (eV)Fe L3,2 -edge\nNi0.2Cd0.3Fe2.5-xAlxO4\nx=0.4\nx=0.3\nx=0.2\nx=0.1\nFeOFe2O3Fe3O4x=0.0A\nDifferance spectra  (arb. units) x=0.1-x=0.0\n x=0.3-x=0.0\n x=0.4-x=0.0 \n \n \n             \n    \n \nFig. 4  \n 7080 7110 7140 7170 7200 7230 7100 7125 7150 7175(b) \n Ni0.2Cd0.3Fe2.5-xAlxOIntensity (arb. Units)Fe K-edge\nFeO\nFe Foilx=0.4\nx=0.3\nx=0.2\nx=0.1\nFe2O3Fe3O4x=0.0(a) \nPhoton Energy (eV)fitting rangex=0.4\nx=0.3\nx=0.2\nx=0.1\nx=0.0\nSample Fe2+(%) Fe3+(%) R-factor\nx=0.0 10 90 5.9E-5\nx=0.1 22 78 2.2E-5\nx=0.2 23 77 3.5E-5\nx=0.3 24 76 1.6E-5\nx=0.4 25 75 1.5E-5 \n \nFig. 5  "    },    {        "title": "1201.0091v1.Large_magnetic_circular_dichroism_in_resonant_inelastic_x_ray_scattering_at_the_Mn_L_edge_of_Mn_Zn_ferrite.pdf",        "content": "PHYSICAL REVIEW B 74, 172409 (2006) \n1 Large magnetic circular dichroism in resonant inelastic x-ray scattering at the Mn L-edge of Mn-Zn ferrite M. Magnuson1, L.-C. Duda1, S. M. Butorin1, P. Kuiper2 and J. Nordgren1 1Department of Physics, Uppsala University, Ångstrom Laboratory, Box 530, S-75121 Uppsala, Sweden 2Department of Physics, Växjö University, Vejdes plats 6, S-351 95 Växjö, Sweden  Abstract We report resonant inelastic x-ray scattering (RIXS) excited by circularly polarized x-rays on Mn-Zn ferrite at the Mn L2,3-resonances. We demonstrate that crystal field excitations, as expected for localized systems, dominate the RIXS spectra and thus their dichroic asymmetry cannot be interpreted in terms of spin-resolved partial density of states, which has been the standard approach for RIXS dichroism. We observe large dichroic RIXS at the L2-resonance which we attribute to the absence of metallic core hole screening in the insulating Mn-ferrite. On the other hand, reduced L3-RIXS dichroism is interpreted as an effect of longer scattering time that enables spin-lattice core hole relaxation via magnons and phonons occurring on a femtosecond time scale.     The prediction of magnetic circular dichroism (MCD) in x-ray emission at the L-edge of ferromagnetic 3d-metals such as iron[1] has triggered much experimental effort to study magnetic effects in x-ray fluorescence spectroscopy[2, 3, 4, 5, 6]. Initial interpretations[1] of MCD in x-ray fluorescence centered around the notion that the MCD spectra of itinerant ferromagnetic metals reflect the occupied partial (e.g. Fe 3d) density of states (pDOS) and thus treated as a bulk-sensitive complement to spin-resolved photoelectron spectroscopy. On the other hand, one quickly realized that the observed dichroic asymmetries i.e., the relative difference in magnetization specific spectra, are an order of magnitude smaller than theoretically expected. This is puzzling since the initial spin-polarization of the core hole [7, 8, 9] produced by the interaction of the circularly polarized x-rays removing an electron from a 2p core level should match the spin polarization of the outgoing photoelectron. Moreover, other issues, such as dichroic saturation or dichroic self-absorption that arise at the magnetic transition metal L-edges have hampered development. Instead, much work has been devoted to studying competing atomic like Ll(2p−3s)-decay or using other experimental geometries [10, 11]. Only a few attempts have been made to explain the intriguing blatant discrepancy [12, 13] in the magnitude of theoretically expected and experimentally observed dichroic asymmetries in x-ray emission. It has been shown that it is important to take into account the spin-orbit interaction and the fact that spin is no longer a good quantum number for the core-excited 2p PHYSICAL REVIEW B 74, 172409 (2006) \n2 level. However, this effect is too small to account for the entire reduction of the dichroic asymmetry [14, 15]. Recent x-ray absorption magnetic circular dichroism (XAS-MCD) experiments of several magnetic systems using the integrated transition metal core-to-core 2p-3s-scattering show that spin-selective core hole screening is substantial [11] and is also important for resonant inelastic x-ray scattering (RIXS). The core hole screening is due to spin-flip processes [4] that we denote by L3-L'3M4,5 (L2-L'2M4,5) for excitation at the L3(L2)-resonances. Core hole spin-flips between exchange split 2p3/2(2p1/2)-sublevels produce low energy electron/hole pairs that only occur in metals due to their lack of a bandgap. In metals, in contrast to insulators, the energy gain of such spin flips (on the order of some 0.1 eV) can be transferred to low energy electron/hole pairs close to the Fermi level thus increasing the number of core holes with spin of lowest energy. Although this could explain the reduction in the dichroic assymmetry one may ask whether lattice relaxations of the core hole via phonons and magnons are of significance too. Consider first MnO which is an antiferromagnetic insulator with a ground state close to the ionic low spin 3d5-configuration [16, 17]. In principle, the simplicity of the 3d5-configuration (one half of the 3d-band is filled with majority spin electrons, the other half is empty) and the large magnetic moment would make this system an ideal insulating magnetic compound in which metallic core hole screening is quenched. However, due to the superexchange mechanism, pure MnO is an antiferromagnet for which dichroic effects cancel. On the other hand, in Mn-ferrites the Mn-spins are ferromagnetically aligned and offer a close approximation to a MnO-sublattice. Thus the Mn-ferrite, as a magnetic insulator, offers an ideal system to study MCD in x-ray emission in the absence of core hole spin-flip processes, which are present in itinerant systems. Moreover, many interesting materials of scientific and technological importance today have magnetic properties of localized nature and thus it is timely to extend the scope of MCD in RIXS to this kind of materials. In this Letter, we investigate magnetic circular dichroism at the Mn L-edge of Mn0.6Zn0.4Fe2O4 using resonant x-ray emission spectroscopy excited with circularly polarized x-rays with energies at the Mn 2p resonances. We observe that the x-ray spectra are dominated by dd-excitations and have no resemblance with spin-resolved density of states. Moreover, the dd-excitations show large dichroism at certain excitation energies, well exceeding that found in metallic systems. However, comparison to atomic crystal-field multiplet calculations shows that some other process reduces the dichroic asymmetry in this insulating magnetic system. We discuss a depolarization mechanism due to magnon and phonon coupling to the core hole excited state. The experiments were performed at the helical undulator beamline ID12B at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France [21, 22]. This beamline consists of a Dragon-like spherical grating monochromator producing 83% circularly polarized x-rays. The XAS spectra were measured in the total electron yield mode. The Rowland-type x-ray emission spectrometer [23, 24] had a 40 µm entrance slit and a spherical grating with 1800 lines/mm in the first order of diffraction, resulting in energy resolutions of 0.6 eV and 0.9 eV for XAS and RIXS, respectively. The incidence angle of the photon beam was 17o and the optical axis of the spectrometer was adjusted to the surface normal thus eliminating dichroic self-absorption effects. The Mn-Zn ferrite sample was a grown single crystal thin film of 5 × 5 mm2 surface area [25] and the measurements were made at room temperature. It was magnetized by using two Nd-Fe-B permanent magnets situated directly behind the sample with a magnetic field strength of 0.2 T. PHYSICAL REVIEW B 74, 172409 (2006) \n3 Figure 1 (top) shows the measured dichroic Mn 2p3/2,1/2 XAS spectra for magnetization parallel (I+, solid curve) and antiparallel (I−, dashed curve) to the photon spin. The filled curve is the MCD difference XAS spectrum. Calculated magnetic Mn2+ XAS spectra for the parallel and antiparallel magnetizations are shown in the lower panel. The calculated spectra have been shifted by -2.05 eV to coincide with the experiment. The 2p3/2,1/2 peaks in the MCD spectra at 640.5 eV and 651 eV are split by approximately 11 eV by the spin-orbit interaction. The fine structures of the 2p3/2 and 2p1/2 groups consists of the crystal-field split 2p53d6 configuration. For clarity, the main final states in spherical symmetry are indicated. The 2p3/2 and 2p1/2 thresholds are dominated by sextuplets while quadruplets dominate above. The calculated results are in good overall agreement with the experimental results although charge-transfer effects are not included. We find that the 2p absorption spectra are typical for divalent Mn in tetrahedral (Td) symmetry and the calculations indicate that the spectra are strongly influenced by the relatively weak crystal-field interaction (optimized to -0.8 eV) between the 3d5 ions. The XAS spectra are dominated by strong multiplet effects due to Coulomb and exchange interactions between the 2p core holes and the 3d electrons. Note that the experimental MCD difference signal is large and that the 2p3/2 and 2p1/2 MCD peaks are opposite to each other as also predicted by our ionic model calculations. This is also the case for other Mn doped ferrites [31, 30], where the ionic model predicts a dominant single spin-down MCD peak accompanied by a weak low-energy pre-peak at 639 eV. The success of the crystal-field multiplet theory shows that the majority of the Mn2+ ions indeed occupy the Td sites. The relative amplitudes of the calculated MCD peaks are sensitive to both the symmetry and the superexchange field. X-ray emission (leaving a valence excitation in the final state) excited resonantly e.g. at the 3d-transition metal L-edge of materials with localized states has been shown to be very sensitive to excitation energy. This has been explained by describing resonant x-ray emission as a scattering process involving two dipole transitions where energy transferred from the photon to the atom (i.e. inelastic scattering, thus called RIXS) is reflected as spectral weight at a corresponding energy from the elastic peak. Selection rules and large transition probabilities  Intensity (arb. units)\n660655650645640635Photon Energy (eV)Mn 2p XMCD I+ I- I+ - I-2p3/22p1/2Experiment\nCalculationXASXMCD\nXASXMCDA\nTd: 10Dq=-0.8 BCDEFG\n6P4D4F6F4F4D\n Figure 1: (Color online) Top, MCD in the Mn 2p x-ray absorption of Mn0.6Zn0.4Fe2O4 (top). I+ represents the XAS for the magnetization parallel to the photon-spin and I- represents the antiparallel. The excitation energies used for the dichroic RIXS measurements in Fig. 2 are shown by the arrows and denoted by the letters A-G. Bottom, calculated Mn2+ XAS and difference spectra in the tetrahedral (Td) symmetry.  PHYSICAL REVIEW B 74, 172409 (2006) \n4 lead to the domination of crystal field excited final states in RIXS spectra. Divalent model calculations for RIXS of MnO in octahedral (Oh) symmetry have shown to be very successful in reproducing the observed excitation energies and transition intensities of dd- and (metal-to-ligand) charge-transfer excitations [18]. In the Mn-Zn ferrite Mn0.6Zn0.4Fe2O4, the magnetic Mn2+ and Fe3+ ions are both in the 3d5 state where the Mn2+ Td and mixed valent Oh Fe2,3+ magnetic moments are antiparallel to each other [19]. However, quantitative information about the dichroism of the Mn2+ ions can be obtained since Mn- and Fe-RIXS spectra are energetically well separated by their core electron binding energies. Figure 2 shows the MCD in a set of experimental RIXS spectra plotted on a photon energy loss scale, with excitation energies denoted by A-G from 640.75 eV up to 652.9 eV. We observe strong energy dependent dichroism in the Mn L2,3 RIXS-spectra of Mn0.6Zn0.4Fe2O4 and compare the spectra by performing crystal-field multiplet calculations using the same set of parameters as in XAS. The RIXS spectra can be interpreted by assigning the structures to three different categories; the recombination peak, the resonating loss structures due to dd and charge-transfer excitations, and the normal Lα,β x-ray emission which is very weak at resonant energies. The elastically scattered recombination peak disperses with the excitation energy, and has a width of 0.9 eV. The recombination peak is strongest at the 2p3/2 resonance and decreases with increasing excitation energy due to the decreasing absorption cross-section. A notable success of the calculation is the relative intensity of Intensity (arb. units)\n-15-10-50Energy Loss (eV)GABCDFEx10 x2 x2 x2 \nx2 Mn 2p dichroic RIXS\nM hiinhiout17o4P4G4D4F6S4D4F4D\n4F4F4P4G4D4G4F\n4D4F Figure 2: (Color online) Measured Mn L2,3 dichroic RIXS spectra denoted A-G compared to crystal-field multiplet calculations in Td symmetry, plotted on an energy-loss scale. The measured spectra were excited at 640.75 eV, 642.1 eV, 642.4 eV, 643.0 eV, 644.4 eV, 651.1 eV and 652.9 eV photon energies, indicated by the arrows in Fig. 1. For each photon energy, the corresponding calculated spectrum is shown below.  PHYSICAL REVIEW B 74, 172409 (2006) \n5 the dd-excitations relative to the elastic peak. However, charge-transfer processes (3d6L where L denotes a hole in the O 2p band) causing the broad structures at 10-15 eV loss energy in spectra F and G are not taken into account in the calculations. The Mn 3d5→2p53d6→3d5 transitions in the dichroic RIXS process were calculated as a coherent second-order optical process employing crystal-field multiplet theory in Td symmetry using the Kramers-Heisenberg formula [26]. The values of the core-level lifetime Γis used in the calculations were 0.4 eV and 0.6 eV for the 2p3/2 and 2p1/2 thresholds, respectively [27]. The Slater integrals, describing 3d−3d and 3d−2p Coulomb and superexchange interactions, and spin-orbit constants were obtained by the Hartree-Fock method [28]. The effect of the configurationally dependent hybridization was taken into account by scaling the Slater integrals Fk(3d3d), Fk(2p3d) and Gk(2p3d) to  80 %. The ground state of the Mn2+ ion was derived from the atomic 6S5/2 high-spin state. The crystal-field splitting 10Dq, was optimized to -0.8 eV in the Td symmetry and the superexchange field to 10 meV. Calculations were made both in the Oh and Td symmetries with 3d5 valency, with a clear preference for the Td symmetry. A direct comparison of the calculated spectra with the measured data was finally achieved by taking into account the instrumental and final state broadenings [29]. Our crystal-field multiplet calculations generally reproduce the spectral shapes of the RIXS spectra very well and the trend of the dichroic asymmetries follow the experimental ones. The final states of the dd-excitations are dominated by quadruplets of 4P, 4D, 4F and 4G symmetry. Note that the elastic recombination peak (6S5/2) is dichroic as a result of the dichroism in the first step of the scattering process, i.e. the absorption step. The RIXS cross-section is therefore a combination of absorption dichroism and emission dichroism, where both have been taken into account in the calculations. Comparison between experiment and calculation reveals the important observation that, in spite of the quenching of the metallic core hole screening channel, the dichroic asymmetry at the L3-resonance is reduced. Strikingly, the dichroic asymmetry is largest at the Mn L2-edge (spectra F and G); this seems to be a universal effect that is observed in several metallic [3] and half-metallic [5] magnetic materials. In order to understand the difference in reduction of the dichroic asymmetry at the L3- vs. L2-resonance, we recall that the scattering time (also called “core hole clock” [11]), i.e. the time that the electronic system is allowed for its rearrangement, is proportional to the core hole lifetime. At the L2-resonance the core hole lifetime is about 50% shorter than at the L3-resonance. Hence, x-ray scattering at different resonances offers a means of studying relaxation dynamics. Previously disregarded effects, such as spin-orbit splitting, 2p3/2 life-time broadening and the insufficient treatment of the spin quantum number, are explicitly taken care of in our calculations whereas lattice relaxation processes are not taken into account. In analogy to the metallic case (L3-L'3M4,5), lattice relaxations of the core-excited intermediate state can be denoted by L3-L'3PM and L2-L'2PM, where PM stands for either a phonon or a magnon. This includes nonlocal spin-flips occuring in the core-excited intermediate state (similarly as known for electronic screening processes [33]) as opposed to the localized spin-flip excitations in the final state discussed recently by van Veenendaal [34]. Note that the intermediate Mn 3d6-state is Jahn-Teller active implying that the neighboring atoms apply a torque on the 3d-shell at this PHYSICAL REVIEW B 74, 172409 (2006) \n6 site, as a function of its total magnetic moment. This interaction could produce magnons or optical phonons entailing a reduction of the dichroic asymmetry as a function of the available scattering time [35]. The corresponding RIXS loss energies are likely to be smaller than discernable with present instrumentation. On the other hand, using our observation we already can estimate that lattice relaxation time of a spin polarized core hole has an upper limit below 1 fs which is similar to the metallic core hole screening process. In conclusion, we report valence RIXS dichroism at the Mn 2p-resonances of Mn-Zn ferrite. At resonant excitation, dd-excitations dominate the RIXS spectra due to the localized nature of the intermediate state and no resemblance to spin-resolved pDOS is found. We note that also 3d-orbitals of ferromagnetic metals have a certain degree of localization that could be of significance for their dichroic asymmetry in RIXS. The observed magnitude of the dichroic asymmetry is found to be larger in the Mn-Zn ferrite than in metallic magnetic systems, an effect of quenching of metallic core hole screening via the L3-L'3M4,5 and L2-L'2M4,5 decay channel. The spectral shapes and intensity trends are well reproduced by our model calculation assuming atomic-like Mn2+-ions residing in a tetrahedral spin component of the ligand field. However, the calculated dichroic asymmetry is still larger than experiment, pointing to residual core hole relaxation mechanisms. We interpret this as existence of substantial spin-lattice interactions at the excited Mn-atom on a femtosecond time scale. Our MCD in RIXS investigation of a localized magnetic system provides an important starting point for further investigations of related ferromagnetic systems containing localized magnetic ions such as dilute magnetic semiconductors that are currently receiving strong attention for use in nanostructured hybrid materials and spintronic applications. We acknowledge the Swedish Research Council (VR) and the Göran Gustafsson Foundation for financial support and we thank R. B. van Dover, Cornell University, for providing the sample and L. Qian and the staff at ESRF for experimental support.   References [1] P. Strange, P. J. Durham and B. L. Gyorffy; Phys. Rev. Lett. 67, 3590 (1991). [2] C. F. Hague, J.-M. Mariot, P. Strange, P. J. Durham and B. L. Gyorffy Phys. Rev. B 48, 3560 (1993); C. F. Hague, J.-M. Mariot, G. Y. Guo, K. Hricovini and G. Krill; Phys. Rev. B; 51, 1370 (1995). [3] L.-C. Duda; J. Elec. Spec. Relat. Phen. 110-111, 287 (2000). [4] S. Eisebitt, J. Luning, J.-E. Rubensson, D. Schmitz, S. Blugel and W. Eberhardt; Solid State Commun. 104, 173 (1997). [5] M. V. Yablonskikh, Yu. M. Yarmoshenko, V. I. Grebennnikov, E. Z. Kurmaev, S. M. Butorin, L.-C. Duda, J. Nordgren, S. Plogmann and M. Neumann; Phys. Rev. B 63, 235117 (2001). [6] G. Ghiringhelli, M. Matsubara, C. Dallera, F. Fracassi, A. Tagliaferri, N. B. Brookes, A. Kotani and L. Braicovich; Phys. Rev. B 73, 035111(2006). [7] A. K. See and L. E. Klebanoff; Phys. Rev. Lett. 74, 1454 (1995). [8] P. D. Johnson; Rep. Prog. Phys. 60, 1217 (1997). [9] C. Bethke, E. Kisker, N. B. Weber and F. U. Hillebrecht; Phys. Rev. B 71, 024413 (2005). [10] L. Braicovich, M. Taguchi, F. Borgatti, G. Ghiringhelli, A. Tagliaferri, N. B. Brookes, T. Uozumi and A. Kotani; Phys. Rev. B 63, 245115 (2001). [11] L. Braicovich, G. Ghiringhelli, A. Tagliaferri, G. van der Laan, E. Annese and N. B. Brookes; Phys. Rev. Lett. 95, 267402 (2005). [12] T. Jo and J.-C. Parlebas; J. Phys. Soc. Jpn. 68, 1392 (1999). [13] T. Jo; J. Elec. Spec. Relat. Phen. 86, 73 (1997). PHYSICAL REVIEW B 74, 172409 (2006) \n7 [14] P. Kuiper; J. Phys. Soc. Jpn. 69, 874 (2000). [15] P. Kuiper, B. G. Saerle, L.-C. Duda, R. M. Wolf and P. J. Van Der Zaag; J. Elec. Spec 86, 107 (1997). [16] Ferromagnetic materials: a handbook on the properties of magnetically ordered substances; E. P. Wohlfarth and K. H. J. Buschow, Amsterdam: North-Holland, (1995). [17] P. A. Cox, Transition Metal Oxides, Clarendon, Oxford, 1995, Chapter 5.2.2. [18] S. M. Butorin, J.-H. Guo, M. Magnuson, P. Kuiper and J. Nordgren; Phys. Rev. B 54, 4405 (1996). [19] A. H. Morrish and P. E. Clark; Phys. Rev. B 11, 278 (1975). [20] E. Rezlescu and N. Rezlescu; J. Magn. Magn. Materials 193, 501 (1999). [21] P. Elleaume; J. Synchrotron Radiat. 1, 19 (1994). [22] J. Goulon, N. B. Brookes, C. Gauthier, J. Goedkoop, C. Goulon-Ginet, M. Hagelstrin and A. Rogalev; Physica B 208-209, 199 (1995). [23] J. Nordgren and R. Nyholm; Nucl. Instr. Methods A 246, 242 (1986). [24] J. Nordgren, G. Bray, S. Cramm, R. Nyholm, J.-E. Rubensson and N. Wassdahl; Rev. Sci. Instr. 60, 1690 (1989). [25] Y. Suzuki, R. B. van Dover, E. M. Gyorgy, J. M. Phillips, V. Korenivski, D. J. Werder, C. H. Chen, R. J. Felder, R. J. Cava, J. J. Krajewski and W. F. Peck Jr.; J. Appl. Phys. Lett. 79 5923 (1996). [26] H. H. Kramers and W. Heisenberg; Zeits. F. Phys. 31, 681 (1925). [27] O. Keski-Rahkonen and M. O. Krause; At. Data Nucl. Data Tables 14, 139 (1974). [28] R. D. Cowan, The Theory of Atomic Structure and Spectra; University of California Press, Berkeley, Calif., 1981). [29] A. Santoni and F. J. Himpsel; Phys. Rev B 43, 1305 (1991). [30] S. Imada, A. Kimura, T. Muro, S. Suga and T. Miyahara; J. Electr. Spec. 88-91, 195 (1998). [31] S. Suga and S. Imada; J. Electr. Spectr. 92, 1 (1998). [32] L.-C. Duda, T. Schmitt, M. Magnuson, J. Forsberg, A. Olsson, J. Nordgren, K. Okada and A. Kotani; Phys. Rev. Lett. 96, 067402 (2006). [33] M. van Veenendaal and G.A. Sawatzky; Phys. Rev. Lett. 70, 2459 (1993). [34] M. van Veenendaal; Phys. Rev. Lett. 96, 117404 (2006). [35] T. Privalov, F. Gel’mukhanov, and H. Ågren; Phys. Rev. B 59, 9243 (1999).    "    },    {        "title": "1907.01052v1.Structural_phase_diagram_and_magnetic_properties_of_Sc_substituted_rare_earth_ferrites_R1_xScxFeO3__R_Lu__Yb__Er__and_Ho_.pdf",        "content": "1 \n Structural phase diagram and magnetic properties of Sc -substituted  rare earth ferrites  R1-\nxScxFeO 3 (R=Lu, Yb, Er, and Ho)  \nJason White1, Kishan Sinha1, Xiaoshan Xu1,2 \n1Department of Physics and Astronomy, University of Nebraska, Lincoln, NE 68588, USA  \n2Nebraska Center for Materials and Nanoscience, University of Nebraska, Lincoln, NE 68588, \nUSA \n \n \nAbstract  \n \nWe have  studied the structural stability of Sc-substituted  rare earth  (R) ferrites R1-xScxFeO 3, and \nconstructed a structural phase diagram for different R and x. While RFeO 3 and ScFeO 3 adopt the \northorhombic and the bixbyite structure  respectively , the substitute d compound R1-xScxFeO 3 may \nbe stable in a  different structure . Specifically, for R0.5Sc0.5FeO 3, the hexagonal structure can be  \nstable for small R, such as Lu and Yb, while the garnet structure is stable for larger R, such as Er \nand Ho.  The formation of garnet structure of the R0.5Sc0.5FeO 3 compounds which requires that Sc \noccupies  both the rare earth and the Fe sites, is corroborated by their magnetic properties.  \n  2 \n Introduction  \nRare earth ferrites include  insulating materials of magnetic orders significantly above room \ntemperature, making them  suitable for various applications. For example, orthoferrites, or  \northorhombic  RFeO 3 (R: rare earth), are weakly ferromagnetic based on an antiferromagnetic order \nbelow 600-700 K, which  combined with  their transparency , are great  for magneto optical \napplications.1–3 Rare earth iron garnet, or R3Fe5O12, are unique insulating materials of \nferrimagneti c order below about 550 K , which  are critical for spintronics applications4,5. \nHexagonal rare earth ferrites (h-RFeO 3, R: Er-Lu) exhibit coexisting ferroelectric and magnetic \norders6,7, making them multife rroics with application potential in novel sensors, actuators, and \nenergy -efficient information storage and processing.  The stable structure of RFeO 3 is \northorhombic [Fig. 1(a)]2,3, which is  the origin of the name orthoferrite  (o-RFeO 3). The hexagonal \nRFeO 3 (h-RFeO 3) [Fig. 1(b)] are unstable due to the expanded lattice compared with that of  o-\nRFeO 3. Typically , h-RFeO 3 can be stabilized in epitaxial films  with the aid of interfacial energy  \nwith the substrates6,8–15. Recently , it was demonstrated that the stability of the hexagonal structure \nof h-LuFeO 3 can be enhanced  by doping Sc on the Lu site16–18. ScFeO 3 is unstable in  the \northo rhombic structure due to the small size difference between Sc and Fe compared with that \nbetween rare earth and Fe . Instead, it adopts a bixbyite structure [Fig . 1(c)] containing many open \nspaces .19–21 Therefore, doping Sc on the R site of h-RFeO 3 reduces the stability of the orthorhombic \nstructure , which  affects the competition between the orthorhombic and the hexagonal phases, \nleading to the enhanced stability of  the hexagonal phase. Indeed, it was shown that by doping Sc, \nh-Lu0.5Sc0.5FeO 3 can be synthesized in  bulk powder and  even single -crystalline form16–18. \nThe enh anced stability in h -Lu0.5Sc0.5FeO 3 has made the investigation of its properties more \nefficient.22 On the other hand, the effect of Sc doping on the stability of other members of h -RFeO 3, \nas well as whether the stability can introduce the new members to the h -RFeO 3 family, have not \nbeen reported. In this work, we studied the effect of Sc doping in rare -earth ferrite compounds R1-\nxScxFeO 3, where R=Ho, Er, and Yb.  The results show that h-Yb0.5Sc0.5FeO 3 can be stabilized in \npolycrystalline powders , but with impurity phases; for larger R, h-R0.5Sc0.5FeO 3 crystalize in a pure \ngarnet phase.  \nExperimental  \nPowder  samples  were  synthesized using solid -state reactions starting from R2O3, Fe 2O3, and Sc 2O3 \npowders  purchased from Alfa Aesar. The samples were mixed stoichiometrically using a mortar \nand a pestle before being calcinated at 900 C for 12 hours  in a tube furnace with a c onstant oxygen \nflow. The samples were then crushed and ground again before being pressed into pellets  and \nsintered at 1200 C for 18 hours.  The orthoferrite RFeO 3 pellets were made first by reacting R2O3 \nand Fe2O3 using the above method. The ScFeO 3 pellets were made similarly by reacting Sc2O3 and \nFe2O3 powders . The R1-xScxFeO 3 compounds were then made by reacting RFeO 3 and ScFeO 3 \npowders  according to the corresponding stoichiometry.  Ho3Fe5O12 garnet sample s were  also made \nfrom Ho2O3 and Fe 2O3. X-ray diffraction measurements were carried out using a Rigaku D/Max -\nB diffractometer with Co -K radiation (  = 1.789 Å) . The temperature and magnetic -field \ndependence of the magnetization were measured using a vibrating sample magnetometer  (VSM ). 3 \n Results and discussion  \nStructural properties  \nTo investigate the effect of Sc doping on the stability of h -RFeO 3, we synthesized polycrystalline \nR0.5Sc0.5FeO 3 compound for R = Yb, Er, and Ho. As shown in Fig. 2, For R=Yb, hexagonal phase \nwas observed in the x -ray diffraction spectra; i n addition, the orthorhombic phase and a garnet \nphase were  also observed . When R = Er and Ho, no hexagonal, orthorhombic, or bixbyite structures \nwere  observed. The solid -state reaction  appears to  result in a pure garnet structure phase, which \ncan be seen by the comparison between the x -ray diffraction spectra with that of Ho 3Fe5O12, as \nshown in Fig. 2 (a). \nThe observation of pure garnet phase in both Ho0.5Sc0.5FeO 3 and Er0.5Sc0.5FeO 3 suggests  the \nstability of the garnet structure. To further investigate the effect of Sc doping on the stability of \nthis structure, we synthesized Ho1-xScxFeO 3 of different x between 0.1 and 0.9. As shown in Fig. \n3, when x is small, orthorhombic structure (stable for RFeO 3)2,3 dominates, while when x is large, \nthe bixbyite structure (stable for ScFeO 3)19,20 domi nates, as expected. On the other hand, the garnet \nphase always exists for 0.1 x0.9. When x=0.5, both bixbyite and orthorhombic phases disappear, \nleaving a pure garnet phase.  \nWe constructed the structural phase diagram (Fig. 4) of R1-xScxFeO 3 using the data from this and \nthe previous work19. Two variables in the phase diagram are x and R, representing the Sc/rare earth \nstoichiometry and the species of the rare earth respectively. According to Fig. 4, the hexagonal \nphase can be stabilized in  R1-xScxFeO 3 of smaller rare earth (Yb and Lu) . In particular, pure \nhexagonal Lu1-xScxFeO 3 (0.4x0.6) phase can be obtained. R1-xScxFeO 3 of larger rare earth (Er \nand Ho) favor the garnet phase instead.  \nThe appearance of the garnet phase is surprising since all the competing structural phases \nconsidered previously, including the bixbyite, the orthorhombic , and the hexagonal structural \nphase s have the RFeO 3 chemical formula, while the garnet has a R3Fe5O12 chemical formula. This \npuzzle may be resolved if we rewrite R0.5Sc0.5FeO 3 as (R2Sc)(Fe 4Sc)O 12, assuming that Sc can \noccupy both the rare earth site and the Fe site.  \nTo verify this proposal that Sc can occupy both the rare earth site and the Fe site, we carried out  \ncrystal structure refinement  of R0.5Sc0.5FeO 3 and Ho3Fe5O12 by fitting the diffraction spectra with \nthe FullProf program23,24 using the garnet structure which has a cubic symmetry ; the comparison \nbetween the observation and the fit are displayed in Fig. 2(b -d); the refinement parameters are \nshown in Table I. The refinement on Ho 3Fe5O12 results in a lattice constant 12.365 Å, in good \nagreement with the previous work25,26. The closeness of expect ed and weight -profile R-factor ( Rexp \nand Rwp) suggests fair agreement within the experimental uncertainty. For Er 0.5Sc0.5FeO 3 and \nHo0.5Sc0.5FeO 3, we assumed that the Sc may substitute R on the “ c” site and Fe on the “ a” site \nrespectively because Sc is too large to fit in the “ d” site26,27. As shown in Table I, in Ho0.5Sc0.5FeO 3, \nthe best fit co rresponds to an occupancy of 0.74 and 0.26  0.1 for Ho and Fe on the “ c” sites, and \n0.44 and 0.56   0.05 for Fe and Sc on the “ a” sites respectively;  in Er 0.5Sc0.5FeO 3, the occupancy \nis 0.76 and 0.24  0.1 for Er and Sc on the “ c” sites respectively, and 0.5 and 0.5   0.05 for Fe and \nSc on the “ a” sites respectively. In both Ho0.5Sc0.5FeO 3 and Er 0.5Sc0.5FeO 3, within the experimental 4 \n uncertainty, the site occupancy agrees with the proposed chemical formula (R2Sc)(Fe 4Sc)O 12, \nwhere Sc substitutes 1/3 of the R atoms on the “ c” sites and ½ of the Fe atoms on the “ a” sites on \naverage. Therefore, the result in Table I provides direct evidenc e for the garnet structure  and \nchemical composition of Ho 0.5Sc0.5FeO 3 and Er 0.5Sc0.5FeO 3. \nTo understand why Sc can occupy both the rare earth site s and the Fe site s, we surveyed the \neffective ionic radii of trivalent transition metal and rare earth  elements .28 As shown in Fig. 5(a), \nthe R3+ ions are larger than the transition metal ion s, which is why R3+ tend to have a coordination \nnumber (CN) 8, i.e., surrounded by 8 oxygen ions. In contrast, the coordination numbers of \ntrivalent transition metal ions are smaller; typically, CN=6. On the other hand, the coordination \nnumber of Sc3+ can be  both 6 and 8. When CN=8, the radius of Sc3+ is close to that of R3+; when \nCN=6, the radius of Sc3+ is close to that of trivalent transition metal  ions. Therefore, it is possible \nthat Sc3+ ion c an occupy both the rare earth site and the Fe site.  \nAs shown in Fig. 5(b), the lattice constant of the R0.5Sc0.5FeO 3 garnet structure ( ag) decreases when \nthe R changes from Ho to Yb, due to a decreasing ionic radius from Ho3+ to Yb3+. In comparison \nto the lattice constants of R3Fe5O12 (from the previous work26), the  ag values of R0.5Sc0.5FeO 3 \ngarnet s are systematically smaller. Since replacing R atoms  with Sc is expected to make the lattice \nconstant smaller because Sc3+ is smaller than Ho3+ [Fig. 5(a)], this observation corroborates  the \nsubstitution of Fe with Sc , which is expected to expand the lattice constant since Sc3+ is larger than \nFe3+. \nWe also analyzed the lattice constant s of the garnet  (ag) and the bixbyite (ab) structural phases for \nthe Ho 1-xScxFeO 3 samples ; both structures have a cubic symmetry.  As shown in Fig. 5(c), when x \nis small, the lattice constant of the garnet phase  ag is larger than that of Ho3Fe5O12, indicating Sc \nis going into the Fe sites first. When x increases, corresponding to more Sc, ag decreases \nmonotonically, suggesting that Sc atoms start to occupy more on the Ho sites. At x = 0.8, a rapid  \ndrop of ag is observed, which is accompanied by the appearance of the bixbyite structural phase  \n[see Fig. 5(c)]. The lattice constant of the garnet phase  ag drops to values close to  that of Ho3Fe5O12. \nAt the same time, the lattice constant of the bixbyite structural phase  ab is also smaller than that of \nScFeO 3. These  observation s indicate that w hen x is too large, the garnet structure  for Ho1-xScxFeO 3 \nis unstable against the decomposition into garnet Ho3Fe5O12 and Fe -rich ScFeO 3. \nThe lattice constants for the hexagonal Sc -substitute  YbFeO 3 phase was also analyzed , resulting \nah = 5.83 Å and ch = 11.69  Å. These numbers are slightly smaller than the lattice constants of h -\nLu0.5Sc0.5FeO 3 (ah = 5.86 Å and ch = 11.71  Å), but close r to those of Lu 0.4Sc0.6FeO 3 (ah = 5.83 Å \nand ch = 11.705  Å) reported previously17. This observation may indicate that more Sc doping is \nneed ed to stabilize hexagonal  YbFeO 3 compared with that of hexagonal LuFeO 3. \nMagnetic properties  \nNext , we turn to the magnetic properties of Ho 0.5Sc0.5FeO 3 and Er0.5Sc0.5FeO 3 garnet s, as well as \nthat of Ho3Fe5O12 as a reference. Figure 6(a) shows  the temperature dependence of magnetization  \nM(T) measured in a 700 Oe magnetic field , where T is temperature.  All three compounds share the \ntypical features of rare earth iron garnets: an above -room -temperature magnetic ordering  \ntemperature  (Curie temperature TC), and a maximum and a downturn of magnetization  at low 5 \n temperature  which comes from the compensation  between the Fe and the rare earth magnetic \nmoment s that causes a minimum of magnetization at the compensation temperature (Tcmp).4 From \nFig. 6(a), we found  TC  600 K and Tcmp  130 K  for Ho 3Fe5O12, TC  500 K  and Tcmp  63 K  for \nHo0.5Sc0.5FeO 3 garnet,  and TC  500 K  and Tcmp < 50 K  for Er0.5Sc0.5FeO 3 garnet.  By comparing \nthe results of Ho 0.5Sc0.5FeO 3 garnet and Ho3Fe5O12, Er0.5Sc0.5FeO 3 garnet and Er3Fe5O12 (data from  \nthe previous work TC  560 K  and Tcmp  80 K)4, one observes  that both TC and Tcmp are reduced \nafter substituting Sc in  Ho and Er iron garnet . \nFigure 7(a) and (b) shows the  dependence of the magnetization on magnetic field M(H) for various \ntemperatures , where H is the  external  magnetic field.  For both Ho 0.5Sc0.5FeO 3 and Er0.5Sc0.5FeO 3 \ngarnets,  the coercive field appears to be smaller than 500 Oe. At high field, the magnetization rises \nlinearly with  H, as observed in other iron garnets.4 By fitting the linear part of data in Fig. 7(a) and \n(b) (|H| > 5 kOe ) using the formula M(H) = Mint + (dM/dH) H, the slope d M/dH and the intercept \nMint can be found, as displayed in Fig. 7(c) and (d).  While dM/dH decreases monotonically with \ntemperature, Mint shows a maximum  and a downturn at low temperature , similar to the trend of \nM(T) displayed in Fig. 6(a).  \nTo understand the temperature and magnetic -field dependence of magnetization of Ho 0.5Sc0.5FeO 3 \nand Er 0.5Sc0.5FeO 3 garnets, and the effect of Sc-doping , we summarize  the basics of crystal  and \nmagnetic structure s of R3Fe5O12. Displayed in Fig. 6(b)  is 1/8 of the unit cell of R3Fe5O12, \ncontaining two “a” sites, three “d” sites, and three “c” sites which are occupied by the metal atoms. \nThe two “ a” sites (octahedra local environment  with six -fold oxygen coordination ), and the three \n“d” sites ( tetrahedral  local environment  with four -fold oxygen coordination ) are occupied by Fe \natoms, while the three “ c” sites ( dodecahedral local environment  with eight -fold oxygen \ncoordination ) are occupied by the rare earth atoms.29 When only the nearest  neighbor s are  \nconsidered, t he Fe magnetic moment s on the “ a” sites couple to those  on the “d” sites \nantiferromagnetically  to form a ferrimagnetic order below TC.4 Because there  are more “ d” sites  \nthan “a” site s, the net magnetic moment  is parallel to those on the “ d” sites.  The coupling between \nthe rare earth and the Fe magnetic moments  is much weaker than that between the Fe sites . For \nnot-so-low temperature ( above a few K), t he rare earth sites can be approximately described in the \nmean field theory as individual paramagnetic moments magnetized by the exchange field from the \nFe sites ( HR-Fe) which is opposite to the Fe moments.5 \nTo analyze the result in Fig . 5 and Fig. 6, we write the  total magnetization as \nMtotal = MFe + MR,  (1) \nwhere MFe and MR are magnetization of Fe and rare earth ions respectively .  \nTreating the contribution of rare earth ions as individual par amagnetic ions,  one has  \nMR = NR R Br(y),     (2) \nwhere y  0 R(H+ HR-Fe)/(kBT), NR and R are the density and the magnetic moment of the rare \nearth sites, 0 and k B are the vacuum permeability  and the Boltzmann constant , H and T are the \nexternal magnetic field and temperature  respectively , and Br(y) is the Brillouin function.  6 \n Since Br(y)  y/3 for y << 1, if 0 R(H+ HR-Fe)/(kBT) << 1, Eq. (2) can be simplified as  \nMR = NR 0 R2 (H+ HR-Fe)/(3kBT)  (3) \nAssuming HR-Fe = -MFe, where  > 0,  (4) \nMtotal = MFe + NR 0 R2 (H-MFe)/(3k BT) \n= MFe [1-NR 0 R2/(3k BT)] + NR 0 R2 H/(3k BT).   (5) \nWhen 1 -NR 0 R2/(3k BT) = 0, the total magnetization Mtotal has a minimum, corresponding to the \ncompensation temperature.  In addition, if 1 -NR 0 R2/(3k BT) < 0, to maximize magnetization or \nto minimize the total energy,  MFe becomes antiparallel to the external field.  \nAt high enough H, MFe is expected to saturate. According to the second term in Eq. (5), t he total \nmagnetization Mtotal is then linearly proportional to H, as also observed in Fig. 7(a) and (b).  In \naddition, the slope d M/dH is inversely proportional to T at high  field according to Eq. (5),  as also \nobserved in Fig. 7(c) and (d).  The intercept for the linear part of the  high-field magnetization  \ncorre spond s to the first term of Eq. (5), which is expected to reflect the magnetization \ncompensation, i.e., a  maximum and  down turn at low temperature  of the magnetization , as \nobserved in Fig. 7(c) and (d).  \nThe reduction of Tcmp by replacing rare earth with  Sc is expected  from Eq. (5) , which leads to  Tcmp \n= NR 0 R2/(3k B)  NR. Since Sc3+ has no magnetic moment, the density of magnetic R3+ (NR) \nis reduced  when they are replaced by Sc3+. Higher magne tization  of rare earth ions is then needed \nto compensate the Fe magnetization , which requires lower temperature.  Therefore,  replacing \nmagnetic R3+ with Sc3+ is expected to lower Tcmp,  as observ ed in Fig. (5)  as well as previously in \nY-substituted  (on Gd sites)  Gd3Fe5O12.30 A quantitative estimation of the reduction of Tcmp can be \ncarried out using the site occupancy found in Table I . Clearly, NR is proportional to the site \noccupancy of the rare earth  atoms;  is proportional to  coordination number of rare earth atoms in \nthe Fe environment, which is proportional to the Fe site occupancy. Therefore, using the data in \nTable I, we estimate that Tcmp(Ho0.5Sc0.5FeO 3)/Tcmp(Ho 3Fe5O12) = 0.57  0.08 and \nTcmp(Er 0.5Sc0.5FeO 3)/Tcmp(Er 3Fe5O12) = 0. 59  0.08, which leads to the prediction of \nTcmp(Ho 0.5Sc0.5FeO 3) = 75  10 K and Tcmp(Er 0.5Sc0.5FeO 3) = 49  6 K, consistent with the \nobservation in Fig  6(a) where Tcmp(Ho 0.5Sc0.5FeO 3)  63 K and Tcmp(Er 0.5Sc0.5FeO 3) < 50 K.  \nThe reduction of TC by doping Sc can be understood considering that Sc atoms also occ upy the Fe \nsites.  The replacement of Fe with Sc disrupts  the Fe-Fe exchange coupling for the ferrimagnetic \norder . Because Sc3+ has no magnetic moment, replacing Fe with Sc leads to a reduction  of the \neffective Fe-Fe coordination number , which is expected to lower the magnetic ordering \ntemperature  TC31, as also observed in Ga-substituted  (on Fe sites)  Y3Fe5O12 and Al -substituted (on \nFe sites) Gd 3Fe5O12.4,32 \nConclusio n \nWe have studied the stabilization of the hexagonal structural phase of RFeO 3 by partially replacing \nthe rare earth elements with Sc  to destabilize the competing orthorhombic structural phase . The 7 \n results indicate that for smaller rare earth , such as Lu and Yb, the stability of  the hexagonal phase  \nis enhanced , leading to the existence of hexagonal Lu 0.5Sc0.5FeO 3 and Yb 0.5Sc0.5FeO 3. In contrast, \nfor larger rare earth, such as Ho and Er, the destabilization of orthorhombic phase accompanies \nthe stabilization of the garnet phase instead of the hexagonal phase . The effect of doping Sc leads \nto the reduction of both Curie and compensation temperature s of the garnet phase, which can be \nunderstood by the occupancy of Sc on both the rar e earth and the Fe sites.  These results suggest \nstrong tunability of rare earth ferrites both in crystal structure and magnetic properties, which could \nbe useful for future material design and engineering.  \n  8 \n Acknowledgement  \nThis work was primarily supported by the National  Science Foundation (NSF), Division of \nMaterials Research  (DMR) under Grant No. DMR -1454618. The research was performed in  part \nin the Nebraska Nanoscale Facility: National  Nanotechnology Coordinated Infrastructure and the  \nNebraska  Center for Materials and Nanoscience, which  are supported by the National Science \nFoundation under  Grant No. ECCS -1542182, and the Nebraska Research  Initiative.  \nCharacterization analysis were also performed  in part  at the NanoEngineering Research Core \nFacility, University of Nebraska -Lincoln, which is partially funded from Nebraska Research \nInitiative funds.         \n  9 \n Reference  \n1 N.P. Cheremisinoff, Handbook of Ceramics and Composites  (M. Dekker, New York, 1990).  \n2 R.L. White, J. Appl. Phys. 40, 1061 (1969).  \n3 D. Treves, J. Appl. Phys. 36, 1033 (1965).  \n4 R. Pauthenet, J. Appl. Phys. 29, 253 (1958).  \n5 Ü. Özgür, Y. Alivov, and H. Morkoç, J. Mater. Sci. Mater. Electron. 20, 789 (2009).  \n6 W. Wang, J. Zhao, W. Wang, Z. Gai, N. Balke, M. Chi, H.N. Lee, W. Tian, L. Zhu, X. Cheng, \nD.J. Keavney, J. 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Uemura, S. Ebisu, S. Chikazawa, and S. Nagata, \nPhilos. Mag. 85, 1819 (2005).  \n31 N.W. Ashcroft and N.D. Mermin, Solid State Physics  (Holt, Rinehart and Winston, New York, \n1976).  \n32 P. Hansen, P. Roschmann, and W. Tolksdorf, J. Appl. Phys. 45, 2728 (1974).  \n \n  11 \n  \nFigure 1 . Crystal structures of (a) the orthorhombic  and (b) the hexagonal  RFeO 3 showing the unit \ncells. (c) Crystal structure of the bixbyite ScFeO 3, with t wo types of metal sites ( “b” and “d”) \nwhich are occupied by both Sc and Fe . (d) Crystal structure of the garnet R3Fe5O12 showing 1/8 of \nthe unit cell  and three types of metal sites ( “a”, “d”, and “c”). The “a” and “d” sites are occpupied \nby Fe ; the “c” sites are occupied by rare earth.  \n \n \n  \n12 \n  \n \n \nFigure 2.  (a) Powder x -ray diffraction spectra for R0.5Sc0.5FeO 3 with different R, as well as \nHo3Fe5O12 as a reference . H and O stand for the hexagonal and orthorhombic structures \nrespectively.  (b)-(c) Structural refinement of Ho 3Fe5O12, Ho 0.5Sc0.5FeO 3, and Er 0.5Sc0.5FeO 3 \nrespectively using the garnet structure  (see text).  \n  \n13 \n  \n \nFigure 3 . Powder x -ray diffraction spectra for Ho 1-xScxFeO 3 with different x, where B, G, O \nrepresents bixbyite, garnet, and orthorhombic structures.  \n \n  \n14 \n  \nFigure 4 . Structural phase diagram of Sc -substituted  rare earth ferrites. The two dimensions are \nSc/rare earth ratio x and the rare eart h species R. \n  \n15 \n  \n \nFigure 5 . (a) The ionic radius as a function atomic number. The hexagons correspond to  \ncoordination number (CN)  = 6, while the circles correspond to  CN=8. (b) The  lattice constants of \nthe R0.5Sc0.5FeO 3 garnet (measured  in this work) and R3Fe5O12 garnet (from previous work26) for \ndifferent R. (c) The lattice constants of the garnet and bixbyite structural phases in Ho 1-xScxFeO 3 \nas a function of x. The lattice constants of Ho 3Fe5O12 and ScFeO 3 are indicated by  the dashed lines.  \n  \n16 \n  \n  \n \nFigure 6 . (a) Temperature dependence of  magnetization of Ho 0.5Sc0.5FeO 3, Er0.5Sc0.5FeO 3, and \nHo3Fe5O12 garnets measured in a 700  Oe magnetic field , respectively . The arrows indicate \ncompensation temperature s. (b) Crystal structure and magnetic structure of iron garnet s shown \nusing 1/8 of the unit cell. The arrows indicate the magnetic moments on the metal sites . While the \nmagnetic moments on the “a” sites couple to those on the “d” sites antiferromagnetic ally, the \nmoments on the “c” sites are aligned by the exchange field from the “a” and “d” sites \nparamagnetically.  \n \n  \n17 \n  \nFigure 7 . The magnetic -field dependence of the magnetization M(H) of Ho0.5Sc0.5FeO 3 (a) and \nEr0.5Sc0.5FeO 3 (b) at various temperatures. (c) and (d) are the slope (d M/dH) and the intercept ( Mint) \nof the high -field linear part of M(H) in (a) and (b) respectivel y (see text) . \n  \n18 \n Table I . Parameters found from structural refinement using the FullProf program .  \n Ho 3Fe5O12 Ho 0.5Sc0.5FeO 3 Er0.5Sc0.5FeO 3 \nLattice constant (Å)  12.365  12.478  12.458  \nFe1/Sc1  \n(16a)  Position  (0, 0, 0)  (0, 0, 0)  (0, 0, 0)  \nOccupancy  1 0.44/0.56  0.05 0.5/0.5  0.05 \nFe2 \n(24d)  Position  (3/8, 0, 1/4)  (3/8, 0, 1/4)  (3/8, 0, 1/4)  \nOccupancy  1 1 1 \nR/Sc \n(24c)  Position  (1/8, 0, 1/4)  (1/8, 0, 1/4)  (1/8, 0, 1/4)  \nOccupancy  1 0.74/0.26  0.1 0.76/0.24  0.1 \nO \n(96h)  Position  (0.1509, -0.0236, \n0.0626)  (0.1471, -0.0292, \n0.0692)  (0.1500, -0.0262, 0.0707)  \nOccupancy  1 1 1 \n2 2.16 1.66 3.50 \nRwp (%) 22.0 29.5 39.1 \nRexp (%) 20.9 22.93  20.9 \n \n "    },    {        "title": "1707.04287v1.Towards_the_ab_initio_based_theory_of_the_phase_transformations_in_iron_and_steel.pdf",        "content": " \n  \n1 \n STRUCTURE, PHASE TRA NSFORMATIONS \nAND DIFFUSION  \n \n \nTOWARDS THE AB INITIO BASED THEORY OF PHASE \nTRANSFORMATIONS IN IRON AND STEEL  \nI. K. Razumov1,2, Yu. N. Gornostyrev1,2, M. I. Katsnelson3 \n1Micheev I nstitute of Metal Physics, UB of RAS, 18 S. Kovalevska ya st., Ekaterinburg , 620990, Russia  \n2Institute of Quantum Materials Science,  Ural Hi -Tech Park, 5 Konstruktorov st,  Ekaterinburg , 620072, Russia  \n3Radboud University, Institute for Molecules and Materials, Heyendaalseweg 135, Nijmegen, 6525 AJ, Netherlands  \ne-mail: rik@imp.uran.ru  \nDespite of the  appearance of numerous new materials, the iron based alloys and steels continu e to play an \nessential role in modern technology. The properties of a steel are determined by its structural state  (ferrite, \ncementite, pe arlite, bainite, martensite , and their combination ) that is formed  under thermal treatment as a \nresult of  the shear lattice reconstruction γ (fcc) → α (bcc) and carbon diffusion redistribution. We present a \nreview on a recent progress in the development of a quantitative  theory of the phase transformation s and \nmicrostructure formation in steel that is  based on an ab initio parameterization of  the Ginzburg -Landau free \nenergy functional.  The results of computer modeling describ e the regular  change of transformation scenario \nunder cooling from ferritic (nucleation and diffusion -controlled growth of the α phase) to martensitic ( the \nshear lattice i nstability γ → α ). It has been  shown  that the increase in short -range magnetic order with \ndecreasing the temperature plays a key role in the change of transformation scenarios. Phase -field modeling \nin the framework of a discussed approach  demonstrates the typical transformation patterns.   \nKeywords: ab initio, short -range magnetic order, steel, ferrite, pearlite, bainite, martensite, eutectoid.  \nCONTENTS  \n1. Introduction.  \n2. Phase transformation s and microstructure formation \nin steel.  \n3. Current understandin g of phase transformations in \niron and steel . \n4. Theoretical approaches to thermodynamics and \nkinetics of phase transformations in steel.   \n5. The ab initio based model of shear -diffusion phase \ntransformations in iron and steel .  \n5.1. Ab initio parameteriza tion of Bain \ntrans formation path.  \n5.2. Generalized Ginzburg -Landau functional for \nthe  –  transformation in steel .  \n5.3. Description of  transformation kinetics .  \n6. Construction of transformation diagram of steel . \n7. Modeling of the phase transformations  kinetics . \n7.1. Athermal and isothermal martensite  \ntransformation .  \n7.2. Kinetics of pearlite transformation. Globular \nand lamellar structures.  \n7.3. Scenarios of ferrite and  bainite  (intermediate ) \ntransformation s.  \n8. Effect of external magnetic field on  the start of  \nphase transformations . \n9. Conclusions and outlook.  \n10. References.  1. INTRODUCTION  \nDespite a broad distribution of numerous new \nmaterials , steel known from ancient times remains the \nmain structural  material of our civilization [ 1], due to \nhigh availability of its main components (Fe and C) \nand diversity of properties reached by a realization of \nvarious (meso)structural states [ 2–6]. One can control \nthe structural state of steel due to a rich phase diagram \nof iron with several structural transfor mations at \ncooling from moderately high temperatures (  δ → γ → \nα); the presence of carbon adds carbide phases, \ncementite Fe 3C being the most important one.  \nDevelopment of the phase transformations in steel \nincludes two main types of physical processes, namely \nthe crystal lattice reconstruction and redistribut ion of \ncarbon between the phases. Depending on the rates of \nthese processes and the morphology of decomposition \nproducts , metallurgists separate several main types of \nthe transformations, namely, ferrite, pearlite, bainite , \nand martensite formation, which follow one another  \nwith decreas ing temperature . All these \ntransformations (except the martensitic one) involve \nboth shear and diffusion mechanisms, their relative \nimportance is changed with the temperature increase.  \nThe interplay of  differen t types of tran sformations \ndetermines the diversity of properties of steel and  \n  \n2 therefore it is of a crucial importance for our \nunderstanding of metallurgical processes.  \nRegardless of the great practical significance and \ncomprehensive experimental study, the mechanisms \nof phase transformations in steel are not fully \nunderstood. Firstly, t here is still no commonly \naccepted quantitative theory that could describe the \nchange of the transformation mechanism with \ntemperature increase from martensitic ( the lattice \ninstability γ → α  over the entire volume) to ferritic \n(diffusion -controlled nucleation and growth of \nprecipitates of α-Fe). Secondly , the properties  of steel \nare due to the precipitates  morphology , understanding \nof which requires the development of quite a \ncomplicated  kinetics theory of phase transformations, \nwhich takes into account simultaneously the lattice \ndegree of freedom and carbon diffusion.   \nBased on state -of-the-art first -principle \ncalculations [ 7, 8] and combining them  with the \nprevious models [ 9–11], we hav e recently proposed a \nconsistent model of phase transformations in steel [ 12, \n13] that includes a generalized Ginzburg -Landau \nfunctional with ab initio parameterization, and \nnonlinear elasticity equations for the shear \ntransformation and diffusion equation  for the carbon \nconcentration. In the framework of this model it was \nshown that the main factor determining scenarios of \nthe phase transformations in steel is the magnetic state \nof Fe and its temperature dependence. The constructed \ncurves of the start of f errite, bainite , and martensite \ntransformations ( A3, T 0, M S) coincided with the experimentally known ones with good accuracy, and \nthe phase field simulations reproduced the typical \ntransformation patterns. In Ref. [ 14] this model was \ngeneralized, with taking into account the cementite \nformation, and it was shown that the pearlite colony \ncan emerge by an autocatalytic mechanism at \novercooling below the critical temperature.  \nHere we review the earlier obtained and new \nresults in the framework of this model. In comparison \nwith previous publications, we consider in more detail \nthe results of phase field simulation of transformation \nkinetics. Also , we discuss the effect of external \nmagnetic field on the curves of transformations  \ndiagram.   \n2. PHASE TRANSFORMATIO NS AND  \nMICROSTRUCTURE FORMATION  IN STEEL   \nFig.1а presents  the experimental transformation  \ndiagram  of the Fe-C system,  and Fig.1b shows the \ntypical microstructures  arising during these \ntransformations . The boundaries of two -phase regions \n\"austenite -ferrite \" (A1, A3) and \"austenite –cementite \" \n(Acm),\nas well their  metastable extensions , - are \nconstructed according to data  [15–17]. The lines of \nthe start of the bainitic (BS) and martensitic ( MS) \ntransformations  is drawn  following results in Ref.  [18, \n19]. Also , the eutectoid  temperature  Teutec (~1000 K) is \nindicated . \nAt high temperatures (T>A 3, T>A cm) the fcc \ncrystal structure o f iron ( γ-Fe, austenite) with \n \nFig.1. (a) Schematic transformations diagram and (b) main scenarios of phase transformations in steel.   The lines A1, \nA3 and Acm are the boundaries of two -phase regions α + γ and γ + θ , as well their metastable extensions below the \neutectoid temperature Teutec [16, 17]; BS and MS are lines of start of bainit ic and martensit ic transformations , \nrespectively [ 4, 18].  \n  \n3 homogenous carbon distribution is stable . Small \novercooling  below A3, results in diffusion -controlled \nprecipitation of ferrite ( α-Fe, almost  pure bcc iron)  \nand the precipitation of cementite (orthorhombic θ \nphase with 25% at. of carbon) takes place below Acm. \nThere  are several  morphological types  of ferrite  [20]; \nallotriomorphic  ferrite  is usually located at  the grain \nboundaries , whereas the needle crystals of \nWidmanst ätten and acicular  ferrite  form  in the bulk .   \nIf both condi tions  T<A 3 and T<A cm are fulfilled \nsimultaneously,  austenite usually decomposes into \nalternating la mellae of ferrite and cementite , and the  \ninterlamellar spacing decrease s with overcooling \ntemperature , \n) /(1~eutecTT . The resulting  regular  \ndisper sed structure is known as  pearlite. The kinetics \nof pearlite transformation  includes the autocatalytic \ngeneration of new lamellae, – usually on the grain \nboundary, – and the growth of the lamellar colony into \nthe bulk  [6].   \nOvercooling below the temperatu re BS results in a \nbainitic  transformation , which develop s by \nautocatalytic nucleation and growth of successive sub -\nunits [ 4]. In the case of upper bainite (which form s in \nthe temperature interval 800K –670K ) the ferrite  \nplatelets with the same  crystallogra phic orientation are \nseparated by cementite precipitates . In the case of  \nlower bainite , which is form ed at a lower temperature, \nthe individual ferrite subunits contain the small -\ncarbide precipitates (usually transforming into \ncementite in the later stage s) in addition to the inter -\nplatelet cementite laths. Presumably, a crucial role in \nthe start of bainite transformation is played by a \ntemperature of paraequilibrium T0 where the free \nenergies of the α and γ phases with the same carbon \nconcentration become equal. Temperature T0 was \nintroduced in Ref. [ 21] as a pre -condition for the start \nof bainit e transformation  by displacive mechanism  \n(cooperative displacements of iron atoms) . It is \nassumed in [ 4, 21] that the diffusion is slower than the \nshear transformation and therefore there is no \nredistribution of carbon between the α  and γ phases \nduring the growth of α phase plates.   \nAt deeper overcooling (below the start \ntemperature of martensitic  transformatio n, MS) the \ndiffusionless γ→α  lattice reconstruction occurs  by \ndisplacive  mechanism , and cementite does not appear . \nHerewith the twinned structure of martensitic plates \ncompensates for the elastic stresses accompanying the \nlattice reconstruction. Experiment al evidence of the \nexistence of two type s of martensite, namely, \nisothermal (scenario of nucleation and growth of the \ncolony of α-Fe plates with different orientation) and \nathermal (scenario of spontaneous lattice instability development over the entire vo lume simultaneously) – \nwere given in [ 22–25], wherein the first scenario is \nrealized at a higher temperature.  \n3. CURRENT UNDERSTANDING OF PHASE \nTRANSFORMATIONS IN IRON AND STEEL  \nAs it accepte d now  [4], the displacive mechanism  \nof transformation  plays essential role in realization not \nonly the  martensitic and the bainitic transformation , \nbut in the formation of high temperature structural \nstate,  such as  Widmanstätten and acicular ferrite , as \nwell. Thus, u nderstanding of physical mechanisms of \nthe lattice instability of fcc ( γ) iron is essential  part of \nour general view on the phase transformations in steel. \nTwo possible mechanisms of γ→α  lattice  \nreconstruction  were proposed , which correspond to  \nthe Bain  (tetragonal  distortion ) [26] and Kurdjumov -\nSachs  (two shear)  [27] deforma tions  schemes . Despite \nthe fact that the Kurdjumov -Sachs  scheme is better in \ncorrespond ing to experiment, the Bain deformation  \n(see Fig.2)  usually  is considered  in the most  of \ntheoretical approaches . Reason  for this is due to the \nfact that  Bain scheme give s a simple st transformation \nway that capture s, however, important transformation \nfeatures . Based on the Bain transformation scheme, i n \nRefs. [28, 29] there has been proposed a \nphenomenological model , being further developed  in \nRefs. [ 9, 11], which describe s the main features of the \nmartensitic  transformation , including  the formation of \ncoherent systems of twin -like domains.   \nAn overwhelming majority of mater ials \ndemonstrating the martensitic transformation can be \ntreated as the Hume -Rothery alloys where a p articular \ncrystal lattice corresponds to some interval of electron \nconcentration per atom [ 30]. Electronic mechanisms \nof the crystal lattice instabilities for these cases are \nreasonably well understood. They are related to \nenhanced van Hove singularities i n the electron \nenergy spectrum and to the energy gain arising when \nFe\nCFe\nC \nFig.2. Lattice γ → α  reconstruction  due to Bain \ndeformation.   \n  \n4 the Fermi energy lies in a pseudogap  [31]. In the new \ncrystal structure,  the geometry of the Bri llouin zone \nallows to accommodate all electrons with the essential \ndecrease of the total ener gy. Since the position of the \nFermi energy is determined by the number of \nelectrons per atom , the Hume -Rothery alloys are \ncalled also electronic phases. Typically, in these \nalloys the transformations are close to the second -\norder phase transitions with a v ery small hysteresis, \nthe low -temperature phase being more close packed \nthan the high -temperature one. For these alloys a soft -\nmode picture of phonon spectra is typical [ 29, 32, 33]. \nHowever, the iron -based alloys belong to a group \nof rare materials where the high -temperature phase \n(fcc, γ) is close packed and the low -temperature phase \n(bcc, α) is not. Neither experimental data [ 34] or \nrecent first -principle calculations [ 35, 36] show soft -\nmode phonons in fcc Fe  above the start temperature of \nmartensitic transformation MS (see Fig.3) . The \nquestion whether a soft-mode in phonon spectra \nappears under cooling , or the mechanism of transition \nis more co mplicated than for the Hume -Rothery alloys  \nand cannot be described in terms of individual phonon \nsoft modes, is unclear. The situation looks \nparadoxical: the γ→α  transfo rmation in iron was \nhistorically a prototype of martensitic transitions at all, but this case r emains still rather poorly \nunderstood, in comparison with many cases \ndiscovered later.   \nStarting  from  the seminal  works  by Zener  [37], it \nis commonly  accepted  that magnetism  plays a crucial \nrole in the phase equilibrium of iron and its alloys, \nincluding the basic fact that the temperature of the \nγ→α  transformation in elemental Fe is close to the \nCurie temperature of α-Fe, and  bcc iron is stable at \nlow temperatures (see, e.g, [ 38, 39]). Moreover, the \nrecent first -principles calculations [ 7, 8, 40] show ed \nthat magnetic and lattice degrees of freedom are \nstrongly co upled in γ-Fe. Ther efore, it can be \nexpected , the classical martensitic scenario (through \nlattice instability over the entire volume) of the γ→α  \ntransformation is realized  at overcooling below some \ntemperature where a strong enough short -range \nferromagnetic order arises in γ-Fe (see Section 6 for \nfarther discussion).  \nThe mechanism of bainite trans formation  that \nappears for temperature just above MS remains a \nsubject of debate  so far  [4, 19]. Tw o competing \ntheories (diffusion -controlled growth [ 41–45] and \ndisplacive diffusionless nucleation [ 4, 21, 46]) have \nbeen proposed to explain this transformation. This \nproblem has not been solved until now; perhaps the \nupper bainite formation is a diffusion -controlled \nprocess and the lower bainite form s via lattice \nshearing, as it was assumed in Ref. [ 47]. Interestingly, \nin hyper -eutectoid steels the upper bainite is observed \neven at T>T 0 [45, 48], where T0 is the paraequilibrium \ntemperature at which  the fre e energies of the α  and γ \nphase s with initial carbon concentration are equal  [21, \n49]; that is in clear disagreement with the displacive \nmodel and allows us to consider the upper bainite as a \ndiffusion -controlled nonlamellar eutectoid \ndecomposition product . At the same time, the lower \nbainite is always form ed below T0 [19]. However, in \nhypo -eutectoid steels the curve of the start of bainitic \ntransformation , BS, is lower than T0, therefore the \nthermodynamic possibility of shear  transformation \ndoes not always  lead to the formation of upper or \nlower bainite. Thus, the problem which mechanism \ncontrol s the bainite transformation is very obscure.   \nThe other sophisticated  problem  is nucleation and \ngrowth of pearlite colonies , which is a particular case \nof a more g eneral issue  of eutectoid decomposition. \nTransformations of this type are also observed in Zn -\nAl [50], Cu -Al [51], Au -In [52], Cu -Zn, Al -Mn, Cu -\nSn, Cu -Be, etc., and  the precipitates morphology \n(lamellar or globular structure) depends on the type of \nalloy a nd the position of alloy parameters on the phase \ndiagram.  Although the pearlite transformation ( PT) in \n \nFig.3. Phonon dispersion curves and corre sponding \nphonon density of states of paramagnetic fcc Fe as \ncalculated within the nonmagnetic GGA (top) and \nDMFT (bottom) [ 35]. The DMFT result is further \ninterpolated using a Born -von Kármán model with \ninteractions expanded up to the fifth nearest -\nneighbo r shell. The results are compared with \nneutron inelastic scattering measurements at 1428K .    \n  \n5 steel is studied experimentally in detail [ 53–55], the \nprocess of lamellae colony formation remains unclear.  \nThe well -known spinodal decomposition kinet ics \n[56] is not applicable to PT because the mixing energy \nof carbon in γ-Fe is positive [ 57, 58], so that  the γ \nphase is stable with respect to small fluctuations of the \ncomposition. Thus, more advanced approaches are \nrequired for understanding of the PT kinetics. \nTheoretical stud ies have been focused on determining \nthe interlamellar spacing and its temperature \ndependence in a steady  state growth of the colony , as \nwell as the problem of stability of the transformation \nfront [21, 59–65]. In Refs. [21, 59] it has been shown  \nthat the interlamellar spacing in this case  obeys the \nlaw \nT/1~ , where  \neutecTTT . As it was found  \n[61], the interlamellar spacing must ensure a \nmaximum growth rate; thin lamellae dissolute  and \nwide ones split during the growth of colony, thus \noptimum interlamellar spacing is achieved . Herew ith, \nthere  was supposed an acceleration of diffusion on the \ntransformation  front . The recent results of  phase -field \nsimulations  [63–65] confirm  the necessity of the \nabove a ssumption. This essential result describes the \ncondition of  steady  state growth , but it does not \nconcern the problem of nucleation of the pearlite \ncolony , which remains in shadow .  \nBy now , a few important questions are still open . \nOne of them is what  phase  (α or θ) is appears first or \nthey both form  together [ 66–68]. The question wh at \nfactors ensure the stability of the front of colony is \nremain ed open to discussion [ 61–65]. The two \ncompeting  mechanisms of lamellae multiplication by \nlateral replication [ 3, 53, 69] and splitting of existing \nlamellae [ 70] have been proposed. In addition, the \nreasons for the transition from lamellar to globular \npearlite structure with increasing temperature is a \nmatter of considerable interest [71–75]. There is no \ntheory to exp lain the appearance of pearlite  type \ncolonies under realistic parameters.   \nEven well-known  kinetics of ferrite  / cementite \nprecipitation from a supersaturate d austenite includes \nsome unresolved problems.  In particular, the \nmechanism of  the lattice rearrang ement γ→θ  is under \ndiscussion . As is proposed  in Ref. [ 76], the γ→θ  \ntransformation is realized through an intermediate \nmetastable ε-cementite with hexagonal close -packed \n(hcp) crystal structure  which  is closer to γ-Fe than the  \northorhombic θ phase . The re cent a b initio \ncalculations [ 77] indicate, that lattice γ→θ  \nreconstruction can be  implemented through a specific \nMetastable Intermediate Structure (MIS)  that develop s \nnear the boundary of ferrite plate  when the carbon \nconcentration is about 15%at., i.e. far from  the \nstoichiometric composition of cementite.  The change of mechanical properties of pearlitic steel after \nannealing indicates the existence of metastab le \ncementite states in the \"fresh\" pearlite  [3]. \nIn the case of ferrite transformation,  the attent ion \nof the researchers is drawn  to the di fference  of several  \nmorphological forms , polygonal , Widmanstätten , and \nacicular ferrite  (WF, AF) [20, 78, 79]. The polygonal \nand the Widmanstätten ferrite  are realized at a little \nundercooling (i.e. above the T0 temperature)  and, \ntherefore , they both are diffusion -controlled \ntransformations . However,  in the first case the lattice \ncoherence on the γ/α interface is lost so that  elastic \nstresses are absent, whereas in the second case the \nelastic stresses relax as a resu lt of twinning of α phase \nplates . Unlike the two cases  mentioned , the acicular \nferrite appears below T0 and it grows by the displacive \nmechanism [ 4, 20]. Thus, WF and AF can’t be \ndescribed in the framework of simple  models with one \norder parameter.  Phase -field simulations of WF \nformation [80] led to a controversial conclusion that \nthe growth of the WF plates requires high anisotropy \nof interfacial energy , but the possible role of elastic \nstresses has not been considered  in this work .  \nThus, both shear and d iffusive scenarios of phase \ntransformations in steels require detailed theoretical \nstudy. First, it is necessary to explain the mechanisms \nresponsible for the change of transformation scenarios \n(ferrite  \n pearlite  \n bainite  \n martensite) with \ndecreasing temperature. Secondly, the precipitates \nmorphology in the decomposition (including the \nnucleation and growth of the pearlite and bainite \ncolonies, conditions of lamellar or globular pearlite, \nupper and lower bainite formation , etc.) is a subject of \nconsiderable interest. Besides, in some cases (such as \nbainite or WF transformation) shear and diffusion \nkinetics should be described together.  Discussion of \nthese problems is a matter of the rest pa rt of  the \npresent review.   \n4. THEORETICAL APPROACHES TO \nTHERMODYNAMICS AND KINETICS OF PHASE \nTRANSFORMATIONS  IN STEEL  \nIn the framework of a phase -field approach [ 81] \nthe evolution of microstructure during the martensitic \ntransformation  (MT)  can be describe d by the Allen -\nCahn equation [ 9, 11, 82, 83] for a non-conserved \norder parameter in the capacity of which is the \ntetragonal deformation \nte  is chosen :   \ntt\neF\nte\nδδ\n                                                      (1)  \n  \n6 \n   \n rd ek f eFt t t2\n21\nel,                          (2) \nwhere \nteF  is the Ginzburg -Landau free energy \nfunctional, \ntk  is the parameter determining the \ninterface energy, and \nelf  is the nonlinear elastic  free \nenergy contribution  [84, 85] that is  presented as \npolynomial over \nte : \n6\n64\n42\n2 t t tteA eA eA f f el el\n                      (3) \nThe transformation mechanism  in this model  switch es \nfrom the normal type (nucleation and growth) to a \nmartensi tic scenario (lattice inst ability) when  the \nparam eter A2 decrease s, so one can accept  \nM MT TTA A /20 2\n, where TM is the start \ntemperature of MT [10]. However, as was shown in \n[86], all components of the deformation tensor should \nbe taken into accou nt for a proper descri ption of \nelastic energy at the polymorphic transformation , \nbecause they should satisfy a set of Saint -Venant’s \ncompatibility conditions \n0e  [87], which  \ncan be written in 2D case as:  \n 0 82t yy xx xy v e e es\n,                  (4) \nwhere \n2/yy xx te is a tetragonal \ndeformation, \n2/yy xx ve   is a dilatation, \nxy se\n is a shear (trigonal) deformation, \nijε  are the \ncomponents of deformation s tensor , \n2/) (, , , , jkik ij ji ij uu u uε\n and \nj i ji x u u  /, , and \nui are displacements. Therefore, Eq.( 3) should include  \nadditional terms :    \n),(s veef f ft\nel el el\n,                                    (5) \n2/) (),(2 2\nss vv s v eA eA eef el\n.                                   \nThe coefficients \n,vA\nsA  are e xpressed in terms of \nelastic moduli [ 86], \n12 11C C Av , \n444C As , and \n) (12 11 2 C C A\n. As was shown in Refs. [ 9, 11], by \nvirtue  of Saint -Venant’s compatibility conditions ( 4), \nthe Eq.(1) can be converted to a integro -differential \nform taking into account the effective long -range \ninteractions for the field of order parameter. Due to \nthese long -range effects, the transformation occurs \nconsistently in different microvolumes and i s \naccompanied by the pattern formation that is really \ncharacteristic of the MT. Also it was reported on the \nspecific tweed structure that appears as a result of \ncompositional fluctuations and long -range effects at a \nmoderately high temperature.  \nIt should be noted, that an energy -controlled \neffect of accomodation of elastic stresses  that leads to the formation of modulated structures was earl y \nconsidered in Ref. [88]. At the present time the role of \nlong-range interactions in the pattern formation is well \nknown [ 89, 90, 91] and was discussed many times for \nvery different systems, from stripes in high -\ntemperature superconductors [ 92–94] to stripe \ndomains in ferromagnetic films [ 95–97].  \nThe equations  of motion for the atomic \ndisplacements  u(r,t) in the form [ 98] \n\n\n\nj jij i\nrt\ntt u ),( ),(\n22r r σ\nρ\n,                                (6) \n      \n),(),(tFt\nijijrrδδσ , \nare more convenient  than Eq. (1) for numerical \nsimulation  of the transformations kinetics . Here ρ is \nthe density,  \n),(tijrσ  are the components of elastic \nstresses, i,j={x,y}. The solution of Eq.( 6) satisfies the \nSaint -Venant's compatibility conditions automatically  \nand can contain also  the lattice vibrations (lattice \ntemperature). This approach has been used in the \nsimulations of MT  in Refs. [10, 99].  \nFirst theoretical  description of  the martensitic type \nphase transformation  in 3D case  was proposed  in \nRefs.  [28, 29]. This approach is based  on the \nexpansion  of the Ginzburg -Landau  functi onal over \ndeformations  relevant  for the lattice  transformation  \nand taking into account only the deviatoric \ncomponents (\ntt tee, ) of the deform ations tensor:  \n22 2 2 22 2\n) ( )3 () ( ),(\ntt t t tt tttt t ttt\ne eC e e Bee eA eef\n\n                        (7) \nгде \n2/yy xx te , \n 6/ 2zz yy xx tte  . \nWithin this model, the transformation mechanism is \nchanged from “nucleation  and grows”  to lattice \ninstability development ( martensitic scenario ) when  \nparam eter B decrease s.  \nIn Ref.  [10] the contributio ns related to carbon \nconcentration and its interplay with deformations have \nbeen added to the free energy , and the system of \nequation s for atomic displacements ( 6) and the Cahn -\nHilliard equation for carbon diffusion [ 56] was \nresolved by phase -field simulat ions for the martensitic \nand pearlitic scenarios of phase transformations. The \nequation for the carbon diffusion has a form:  \nItc\n,     \n\n\ncFc ckTDIδδ)1( ,             (8) \nwhere c is a local carbon concentration, D is a carbon \ndiffus ion coefficient.  Herewith the free energy  \nfunctional has a form  of:   \n  \n7 \n\n ,]21\n21                    [ ,,,\n2 2rdc k ekf f f ceeeF\nc t ts vt\n cpl ch el        (9)          \nwhere \nelf  is the density of elastic free energy , which \ninclude s the term given by  Eq. (5)  and contribution of \nconcentration  expansion , \n4\n42\n2 cv cv f ch  is the \nchemical  contribution to the  free energy , and the  \nterm\n22)(\ntc\ntecA fcpl  takes  into account the  coupling \nbetween elastic distortions and composition  changes . \nTypical  transformation patterns  obtained in the \nframework of this approach  are shown in Fig.4.  The \nmodel  [10] is one of the first attempt s to take into \naccount the interplay between the diffusive and \ndisplacive mechanisms of phase transformation. \nHowever,  this approach  is pure phenomenological and \ncontai ns assumption s that are  incorrect  for steels . For \nexample,  the coupling contribution \ncplf  did not \ninclude  the linear concentration term , although the \nsolution energies of carbon are different in  the γ and α \nphases [100–102]. Besides , the proposed model  \nassumes  that mixing energy of carbon in the γ phase is \nnegative in disagreement with experiment [ 57] and ab \ninitio calculations [ 58]. Finally,  precipitates \nmorphologies of pearlite  obtained in the simul ations \nwere far from those observed in experiments [ 3, 53, \n69].   \nIt should be noted  that the mechanism of colony \nformation in the system with a positive mixing energy \n(such as the γ phase) is unknown.  The proposed \napproaches  considered  mostly  the evolutio n of \nexisting  colony  of alternating plates  of ferrite and \ncementite placed on the flat grain boundary [ 62–65]. \nThe model of eutectoid transformation in a system \nwith a symmetric phase diagram was considered in \nRef.[ 62] where the growth of two phase lamella e was \nobserved in the case of equal diffusion coefficients in \nthe different phases (see Fig.5) . In a more realistic \ncase the widths of cementite and ferrite lamellae are \ndifferent, the carbon diffusion coefficient in ferrite is \nmuch more larger than this o ne in cementite and \naustenite,  therefore an assumption of the diffusion \nacceleration on the colony front is required to provide \nsteady state grow th [63, 64].  \nThe problems of early stages of the colony  \nformation  and the multiplication  of lamellae remain \noutside of the scope of proposed models . The similar \nissues  exist  in the eutectic colonies growth , where the \nmetastable liquid phase decomposes into two new \nphases at the solidification under a temperature \ngradient [ 103–107] or without it [ 108–110].  As was discussed above, the regular martensite \npattern formation is driven by the  accommodation  of \nelastic stresses to minimize the energy.  In last decade  \nthe attention of researchers was attrac ted to the \nproblem of the plastic accommodation of  \ntransformation  strain s that provide s another relaxation \nchannel  of elastic energy minimization [111–114]. It \nwas shown  that accounting for the plastic relaxation \nprocesses results in the possibility of the  easier \nmartensitic  transformation and a more complex and \ncoarser microstructure (see Fig.6).  It should be noted  \nthat the essential role of plastic deformation in a phase \ntransformation was early predicted in Ref. [115, 116] \nwhere a single ellipsoidal nucl eus has been  \nconsider ed. The general phase -field approach \n \nFig.4. The appearance and evolution of martensitic \nstructure to pearlite -like one in the model taking into \naccount an interplay between diffusive and \ndisplacive phase transformations; (a) t=4000, (b) \n6000, (c) 12000  [ 10]. \n \nFig.5. The stru cture of stationary growing colonies \nin eutectoid system with a simmetric phase diagram \nat different temperatures; T/T eutec= (a) 0.59, (b) 0.70, \n(c) 0.82 [ 62]. \n \nFig.6. Evolution of martensite in 2D case with only \n(a)–(c) elastic and (d)–(f) with elasto -plastic \ndeformations; t= (a,d) 0, (b,e) 25, (c,f) 100 [ 111].  \n  \n8 including a system of the coupled equations for the \norder parameters  of phase transformation and the \nmechanics equation for dislocation -assisted  plasticity \nwas pr opos ed in Ref. [114].  \nThe main features of the pattern formation in the \ncourse of  the martensitic -type structural phase \ntransitions  proved possible to describe  within the \nframework of the models proposed in Refs. [9–11, \n112, etc.]. The scenarios of athermal [ 10, 86] (lattice \ninstability ove r the entire volume in the case of rapid \nquenching), isothermal [ 9, 11, 117] (autocatalytic \ngeneration of martensitic plates in the case of holding \nthe steel at a moderate temperature), stress -assisted ,  \nand strain -induced [ 112, 113, 118] MT were \ndiscussed . However, it remained unclear how to apply \nmore correctly these model approaches to  the real  iron \nand steel.  \nThe general disadvantage  of the theoretical \napproaches considered above is the phenomenological \nform of free energy density . In particular, t he authors \ndo not distinguish the enthalpy and entropy \ncontributions to the free energy  density , therefore the \nmicroscopic meani ng of parameters appears  lost and \ntheir correct choice is impracticable .     \n5. THE AB INITIO BASED MODEL OF \nSHEAR -DIFFUSION PHASE \nTRANSFORMATIONS IN  IRON AND STEEL  \nThe consistent model of phase transformations in \nsteel should take into account (1) the lat tice \nreconstruction  γ→α  during cooling to the critical \ntemperature  (Bain [26] or Kurdjumov -Sachs  [27] \ndeformation ); (2) Saint -Venant ’s compatibility  \nconditions  [87] for the components of deformations \ntensor  leading to the appearance  of effective  long-\nrange  intera ctions  for the field of the order  parameter  \n[9–11]; (3) redistribution of carbon between the \nphases , including the formation of cementite . \nHerewith,  the Ginzburg -Landau free energy functional  \nshould include  the magnetic energy contribution .  \n5.1. Ab initio parameteriza tion of Bain \ntrans formation path.  \nThe total energy per atom along the Bain \ndeformation path was calculated for both \nferro magnetic ordered and paramagnetic (disordered \nlocal moment, DLM) states of iron [7, 8]. The \ndifference between energies in these  states  is related \nto magnetic exchange energy. The ab initio \ncomputational results show that the appearance of \nferromagnetic order leads to the change of the \npreferable crystal structure  of Fe  from fcc to bcc (see \nFig.7). In [8] was also shown that there is stro ng \ncoupling between the magnetic and lattice subsystem s in fcc Fe so that exchange energy drastically changes \ndue to the volume increase or tetragonal distortions \n(see Fig. 8). In addition, the ferromagnetic ordered fcc \nstructure is unstable in respect to f cc → bcc \ntransformation. These result s suggest  that the \nmartensitic transformation  of Fe can appear as a result \nof lattice instability due to the increase in short -range \nmagnetic order under the cooling.    \nThe first -principles computational results allow  us \nto find an explicit expression for the density of free \nenergy for pure Fe, which takes into account both \n \nFig.7. Variation in the total energy per atom along \nthe Bain deformation path for different magnetic \nstates. FM (empty triangles) and AFM (sol id \ntriangles) label collinear ferromagnetic and \nantiferromagnetic structures, SS (empty circles) \ncorresponds to the  spin-spiral state, DLM (crosses) \nbelongs to the  disordered local moments  \napproximation of paramagnetic state , DLM 0.5 (empty \ndiamonds) stands  for the  DLM state with a total \nmagnetic moment equal to half of that for the FM \nstate [7]. \n \nFig.8. Exchange parameters  Jn for n=1,2,3,4,5 and \nthe total exchange parameter J0 in dependence on \natomic  volume for (a) fcc and (b) bcc Fe  [8].  \n  \n9 deformations and ma gnetic degrees of freedom. For \nthis purpose , we represent the magnetic -dependent \npart of the total internal  energy in He isenberg -like \nform  \n)()ˆ( )ˆ(, TQ J EEij\njijiε εPM\n\n,                        (10) \nwhere \nj i ijTQ mm)(  is the correlation function \nof magnetic moments on sites i and j, EPM is the \nenergy of paramagnetic state, and the brackets <…> \nmean the average over an ensemble of magnetic \nconfigur ations at a given temperature. Assuming that  \nthe nearest -neighbor contribution is dominate in \nexchange interactions , the energy density can be \npresented as   \n)()ˆ(~)ˆ( ),ˆ( TQJ g Tg εε εPM\n,                          (11) \nwhere \n /~ 2JmJ  , \n is the volume per atom and m \nis the magnetic moment , \n2\n1 0 / )( m TQ  m m  is \nthe spin correlation function dependent on \ntemperature ; \n0Q stands for the totally di sordered \nparamagnetic (PM) state and \n1Q , for the \nferromagnetic (FM) ground state. The exchange \nenergy \n)ˆ(εJ  can be e xtracted from the computational \ndata [ 7,8],  \n)ˆ( )ˆ( )ˆ(~ε εεPM FMg g J \n.                                 (12) \nIt is assume d here that \n)ˆ(εJ  depends on the Bain \ntetragonal  deformation et only, and the value of \ndilatation is chosen from the minimum of e nergy at a \ngiven et.  \nIn order to determine the spin correlation \nfunction , the model  proposed by Oguchi  for the total \nspin equal to 1/ 2 [119] was employed  as a benchmark  \nin Refs. [12,13 ]. In this model  \njj\ne he hTQ22\n)1)(2(3)1)(2()(\n\nchch\n,                         (13) \nj z hh )1(0\n,  \nkTHh0\n0gβ ,  \nTJjzk~\n ,    (14) \nwhere  \n2g  is the Lande  factor , \nβ is the Bohr  \nmagneton , H0 is the external  magnetic  field (if it is \npresent ), \n is the reduced magnetization determined \nfrom the transcendental equation : \n1)(2)(2\n2h eh\njchsh\n,                                     (15) \nThe essential  advantage  of the Oguchi  model  \n(compared  with the well-known  Langevin  formula  for \nthe magnetization)  is the account ing for the short -range magnetic order at  T>TC, where  TC is the Curie  \ntemperature .  \nBased on these formulas, in Ref. [ 13] there was \naccept ed that \nT TQ /1~)(  (without  an external \nmagnetic field ) for T>T C, and the empirical \ndependence  of magnetization [ 120] was used for \nT<T C. It was assumed there that  Q(TC)~0.4, according \nto Ref. [ 119]. The Curie temperature TC is related to \nthe exchange parameter as kTC(et)\n )(~\nteJλ , with the \nnumerical factor \nαλ =0.472  for α-Fe; this choice of \nαλ\n provides an agreement of the Curie temperature \nwith the experiment, TC=1043K. The co rrelator for γ-\nFe is chosen in a similar way, with the Curie \ntemperature \nγ\nCT ≈300K, according to the calculations \n[8] for the fixed atomic volume \n12 Å3; \n606.0γλ\n according to Ref. [ 121] (see [13] for \ndetails). The nonphysical ferromagnetic long -range \nordering in γ-Fe is not essential for our model, but the \nhigh-temperature short -range ordering in both α and γ \nphases  is important enough for the transformation \nkinetics.  \nAssuming that tetragonal  deformation  is counted \nfrom an fcc phase  (et=0 in the γ  phase and \n2/11te\n in the α  phase ) we consider the order \nparameter \n1 1  which is related to the Bain \ndeformation as \nte12/2  . Positive and \nnegative values of \n  correspond to the two possible \n(mutually orthogonal) directions of the Bain \ndeformation in two -dimensional case.  \nThe energies found from the first -principle \ncalculations [7, 8, 13] for pure iron were \napproximated by the following polynom ials:  \n\n\n\n \n\n\n\n\n\n \n2 3 262)(~\n4 6 4\n2  \nPM(FM)PM(FM)\nPM(FM)\nγPM(FM)\nαPM(FM)\nγPM(FM)\nccg gg g\n                (16) \nIts form guarantees an extremum  at the points \n0  \nor \n1 , and parameters \nPM(FM)\nγ[α]g , \nPM(FM)c  were \nfound by fitting to the ab initio comp utational results  \n(see [ 13] for the details) .  \n  \n10 The Bain path energies  are modified  in the \npresence of carbon:  \n\n   2/)( 12/)(~),(\n22\ncv vcf cvc gc g\nγαPM(FM)\nγPM(FM)\nαγPM(FM)\nγPM(FM) PM(FM)\n    \n \ns\n          (17) \nwhere function  \nsf  have been chosed in the form \n221sf\n. Thus, in our approximation the \ndependence of B ain path energy on carbon \nconcentration is reduced to the accounting for the  \nsolution energies of carbon , \nPM(FM)\nγ[α] , and the energies \nof carbon -carbon interactions (mixing energies)  \nγ[α]v  \nin the γ  and α phases, but not  in the intermediate \nstates. The solution energies were obtained from  ab \ninitio calculations  (see [ 13] for the details) and mixing \nenergies from  Refs.  [57, 58]. It should be noted that \nthe known estimates  of \nv  vary widely from 1 to 3 \neV/at, i.e. the \n  phase is  to be  stable with respect to \ncarbon decomposition . If the changes of carbon \nconcentration are small or moderate (as in the cases of \nferrite, martensite , and in the early stages of bainite \ntransformations)  the contribution of carbon -carbon \ninteractions  can be neglected [13]. However, in the \ncases of the cementite formation (i.e. in case of \npearlite and at later stages of bainite transformations) \nthe carbon concentration increases essentially (up to с=0.25); the carbon -carbon interactions must  be taken \ninto account  in this case .  \nDependences of the energy density g(c/a) on \ntetragonal distortion calculated  according to the \nformulas ( 16),(17) are shown in Fig .9a. One can see \nthat γ-Fe is  stable in paramagnetic state but it loo ses \nits stability with respect to tetragonal (B ain) \ndeformation when becoming ferromagnetic. \nTher efore, the classical martensitic scenario (through \nlattice instability over the entire volume) of the γ→α  \ntransformatio n can emerge  at the overcooling below \nsome temperature where a strong enough short -range \nferromagnetic order arises in γ-Fe. The doping by \ncarbon does not change this important feature . \nMoreover, carbon decreases the energy of \nferromagnetic γ-Fe, with the solution  energy  of the \norder of -0.2 eV per carbon atom. It is not surprising , \nsince carbon creates a strong local ferromagnetic \norder in PM or AFM  γ-Fe [ 40]. It is a common \nwisdom that interstitial impurities (including carbon) \nalways prefer fcc surroundi ng compared to bcc, just \nfor geometric reasons [ 122] (the voids are larger in fcc \nlattice than in bcc with the same density). This is for \nsure correct, also for carbon in iron and results in a \nmore pronounced effect of carbon addition on energy \nof α-Fe. Wh at is much less trivial is that carbon \nsolubility in γ-Fe is sensitive to the magnetic state \nbeing maximal in ferromagnetic surrounding.  \n \n \nFig.9. (a) Energy resulting from the first -principle calculation for the Bain path at T=0K (curves 1,1'), 800K (curves \n2,2'), 1400K (curves 3,3') and in paramagnetic states (curves 4,4') and (b) the free ene rgy density as a function of \ntetragonal deformation for the temper atures T=600K (curves 1,1’), 800K (2,2’), 1000K (3,3’), 1400K (4,4’) found \nfrom Eqs. (17). The carbon concentration is C = 0 (1 –4) and C = 3at% (1' –4'); circles  correspond to ab initio \ncalcu lated results.   \n  \n11 5.2. Generalized  Ginzburg -Landau functional for  \nthe  –  transformation in steel . \nBain path energy is important part  to construct a \nquantitative theory of phase transformation s in steel. \nThe Ginzburg -Landau (G. -L.) functional of free \nenergy  of iron and steel  should include also \ncontrib utions related to the magnetic, phonon, \nelectron , and carbon configuration entropy, energy \ngain during formation of cementite, energy of elastic \nstresses , and interphases energies . The G. -L. \nfunctional can be written in the form similar to Eq.(9) : \n    ),( ,,,s t eef T ecf Fv el\n                   (18) \n    \n rd ekek\ntt\n\n2 2\n2 2 , \nwhere \nT ecft,,,  is local density of free energy  \ndepending on the carbon concentration c, tetragonal \ndeformation parameter \nte , temperature T, and order \nparameter \n  characterizing the transformation of \naustenite to cementite at the point r; \n),(seefv el  is the \nelastic energy determined by Eq. (5) ; \ntk and \nk are \nthe parameter s determini ng the width of ferrite or \ncementite interphase boundary , respectively  [86]. \nLet us first consider  the local  density  of free \nenergy  in the absence of cementite . This situation is \ntypical of the ferritic ( in low -carbon steel at T>T eutec) \nand martensitic (i. e. below the temperature MS) \ntransformations.  Using the Hellmann -Feynman \ntheorem and Eq.( 11) one can represent the free -energy \ndensity for the elemental  Fe as \n \n J\nts t JdTJQ efTs g Tef~\n00~),~( ),(PM\n,  (19) \nwhere s0 is the high-temperature  limit  of the entropy  \ndifference  between  the fcc and bcc phases , including  \nphonon  contribution ; \ntsef  is a function provided a \ngradual switching of the entropy contribution from fcc \nto bcc (\n1tsef  in fcc and \n0tsef  in bcc ) phase . \nAccordi ng to existing  concepts  (see, for example , \n[123]) the value s0 depends slightly on the temperature \nat T>T D, where  TD is the Debye temperature (473K in \nbcc and 324K in fcc phase). It has been chosen such \nthat the start of the transformation determined by th e \ncondition \n0)( )( )(  Tf Tf Tfα γ  agrees with the \nexperimental value for elemental Fe, T0 = 1184K. This \ngives us the value s0= -0.19k, quite close to the \nexperiment [ 124].  \nThe temperature dependence of the energy \n)( )( )( Tg Tg Tgα γ\n and free energy difference \n)( )( )( Tf Tf Tfα γ for the pure Fe agrees well \nwith the results of CALPHAD [ 124] within the \ntemperature range 600÷1200K  (see Fig. 10). Herewith, \nthe magnetic contribution dominates at T ≤ TC and is \ncompensated essentially by the phonon contribution at \nT > TC. \nThe configurational entropy of carbon is found \nfrom the model of ideal solutions, assuming that for T \n>300K carbon is equally distributed among all three \ninterstitial sublattices in  α-Fe, whereas in γ-Fe carbon \natoms can occupy only quarter of the interstitial \npositions [ 88,102 ]. As a  result, the local density of \nfree energy  can be presented as:   \n\n  , 1~),~( ),,(~\n00\ntsJ\nts t\nef S S STJdTJQ efTs g Tecf\n  \nγαγPM\n    (20) \nwhere \nα(γ)S  is the configurational ent ropy of carbon \nin the α(γ)  phase,  \n3/lnc kc Sα , and \n 4/)41ln()41()4ln(4 c c c ck S  γ\n. \nDependences of the local density of free energy  \non tetragonal distortion , calculated  according to the \nformulas ( 20), are shown in Fig. 9b. It can be seen \nfrom a comparis on of Fig.9a and Fig. 9b that the \n \nFig.10. The energy difference \n)( )( )( Tg Tg Tgα γ\n (curve 1) and free energy \ndifference \n)( )( )( Tf Tf Tfα γ  (curve 2) at the  \n→  transition in elemental iron in comparison with \nknown data ( dotted lines 1’,2’) [ 124]; contribution  \nof magnetic entropy to the free energy (curve 3) a nd \nthe contribution from phonon entropy (curve 4) .  \n  \n12 curves \n)(teg  and \n)(tef  are qualitatively similar , but \nthey differ in the depth of the minima corresponding \nto the phases α and γ. In particular, the minimum \ncorresponding to  the γ phase e xists on the cur ve of \n)(tef\n up to  the sufficiently low temperature  about \n400K.  It means that  lattice reconstruction requires an \novercome of some energy barrier  at an experimental \ntemperature  MS, and one follow s to expect  that the \nmartensi tic transformation occur s by the nucleation \nand growth of an embryo in this case .  \nThe lattice  reconstruction  γ→θ  leading  to \ncementite  formation is another structural \ntransformation , which is controlled by carbon \ndiffusion . Herewith , the order  parameter  \n in Eq.(18) \ndescribes a preferred trajectory of the transition  \n\n including the Metastable  Intermediate  \nStructure  (MIS) [77]. According to these ideas , the \nMIS appears in the thin ferromagnetic layer  existing \nnear the ferrite plate.  The subsequent  lattice \nreconstruction MIS→θ occurs by the cooperative \ndisplacements  mechani sm when the local carbon \nconcentration increases to a threshold value с~0.18.  \nThen the θ phase is saturated with carbon to the \nstoichiometric composition of cementite (\ncemC =0.25). \nAs a result , the lattice coherence is maintained , \nwhereas  the elastic stresses are well compensated  at \nthe interface  α/θ.  \nSince  the lattice  reconstruction  is a rather fast  \nprocess  (unlike  diffusion ), γ→θ  can be considered as \nimmediately occu rring as soon as the free energies of \naustenite and cementite become eq ual. Then, the local \ncarbon concentration is a single order parameter \ncharacterizing the cementite , and the density of its \nfree energy [14] can be written  as \n  ),( ) ( )()( )( ),(\n)1( )1(T f cfcfTf T f Tcfe\nbound\nθ cemθ θαθ Fα θ\n  \n             (21) \nwhere  \n)(T fFeα  is the free energy of the p ure α-Fe, \n)(Tfαθ\n is free energy of formation of cementite \nfrom the pure compound s (α-Fe and graphite) known \nfrom  CALPHAD  and ab initio calculations  [125, 126], \nccem is the stoichiometric composition of cementite  \n(ccem=0.25) , \n)()1(cfθ  is the concentration dependence \nof free energy of cementite  [127]. The value of \nbound\nθf\n~ -0.02eV/at  is a shift of free energy of \ncementite due to magnetization induced by the \nadjacent ferrite plate ; \nbound\nθf =0 if an isolated \ncementite nucleus is considered.   \n \n 5.3. Description of transformation kinetics .   \nBy using the approach  proposed in Ref. [ 10] the \ntransformations kinetics can be described  by the \nsystem of coupled equations for the atomic \ndisplacement s (6) and carb on redistribution  (8). The  \ncarbon  diffusion  coefficient  we define  as: \n  \n), ( ) () ( )(\n1 10\nT TT\nCchDc Chc ChD D D cD\n \nθγα γ\n           (22) \nwhere  h(x) is a smoothed  Heaviside  function , \n0TС\n,\n1TС  are the carbon  concentrations  corresponding  \nto the conditions  of paraequilibrium , namely , \n),( ),( Tcf Tcfα γ\n and \n),( ),( Tcf Tcfθ γ ,  \nrespectively . Eq. (22) provide s that  the carbon  \ndiffusion  coefficient  is equal to \nαD ,\nγD,\nθD in the \nbulk of the respective phases  and it takes the \nintermediate values  \n)(cD  at the interfaces . The ratio s \nof the coefficients  \nγαD D/ , \nθγDD/  are 102 or 103 \n[128, 129], thus the simulation  with realistic  diffusion  \ncoefficients  is unfeasible , but the qualitative trends \ncan be derived  by choosing a reasonably large value \nof ratios of the diffusion coefficients .  \n6. CONSTRUCTION OF TRANSFORMATIONS \nDIAGRAM OF STEEL  \nThe model  proposed in Refs [12 –14], which \ninclude s the lattice and magnetic degrees of freedom , \nallows  to construct  the transformation  diagram of the \nsystem Fe -C. This diagram (Fig.11) includ es the \nboundaries of two -phase regions γ/(α+γ) , γ/(θ+γ)  \n(lines A3 and Acm respectively , see Fig.1 ), as well their  \nexten sions  into metastable region below the eutectoid \ntemperature Teutec, and also the lines of instability in \nrespect of the γ→α  and γ→θ  transitions  (T0 and T1, \nrespectively ).  \nThe lines  A3 and Acm are defined by the equality  of \nthe chemical potentials of carb on, and the lines T0 and \nT1 are determined b y the equality of  the free energies \nof corresponding  phases at a fixed carbon \nconcentration : \n1 31 3\n3 1)( ) () ( )(\nA AA A\nA AC CCf CfCdcdfCdcdf\n α γ γ α\n,   (23) \ncem cmcemα cmγ\ncemθ\ncmγ\nC CCf Cf\nCdcdfCdcdf\nAA\nA\n ) ( ) (\n) ( ) (\n \n) ( ) (0 0 T T Cf Cfγ α\n,     \n)( )(1 1 T T Cf Cfθ γ ,      (24) \nwhere  \nθ)α(γ,f  are the free energy density of  the α(γ,θ)  \nphases , wherein  \nα(γ)f  are determined by the Eq. (20)  \n  \n13 at \nte=\nγ\nte=0 in the γ phase and \nte =\nα\nte\n2/11  in \nthe α phase , respectively , and \nθf is determined by the \nEq.(21) at \n0bound\nθf . The line  TF was construct  by \nusing  the condition  \nT cef Tceft t ,0, ,,0 α γ , \nwhere с0 is the initial (average over the sample) \ncarbon concentration ; and the line MS (the start of \nmartensitic transformation ) is defined by the \ndisappearance of the barrier on the Bain deformation \npath, \n 0 /,,2 2 t t e Tcef . The line  A1 is close to a \nzero carbon concentration  and is not presented here . \nThe transformation diagram at a low carbon \nconcen tration  (Fig.11a) , without the formation of \ncementite , was constructed in Ref.  [13] and at a high \ncarbon concentration (Fig.11b) discussed in Ref. [ 14].   \nLet us first consider an expect ed qualitative \npicture of the ferritic and martensitic transformation \nscenarios in low - and medium -carbon steel  [13]. At a \nlow enough overcooling below the temperature А3 the \nferrite transformation proceeds slowly , since its \ndriving force  is small. The nucleation  of ferrite as a \nresult of thermal fluctuations is scarcely probable , \nbecause the critical size of nucleus (determined by the \nratio of surface energy to the bulk energy ga in) is very \nlarge . This is more likely at the grain boundaries \nwhere the chemical potential of carbon is changed. \nHerewith, t he growth rate of ferrite is limited by carbon diffusion in  the γ phase  in this case , because \nthe energy of the α phase without carbon is higher \nthan the energy of the γ phase with an initial \ncomposition . In the case  T ≤ TF the ferrite nucleus can \ngrow even if the composition of the γ phase remains \nunchanged  and the growth rate of ferrite accelerates \nessentially . \nA further decrease of temperature results in a \nslowdown of carbon diffusion and enhancement of the \ntransformation driving force. At intermediate \ntemperatures, a crucial role in determining of the start \nof trans formation is played by the  temperature of \nparaequilibrium T0 (see formula (24)), below which \nthe free energy density of the α phase is less than the \nfree energy density of the γ phase with the same \ncarbon concentration . Temperature T0 was introduced \nin Ref. [ 21] as a pre -condition for the start of bainite \ntransformation. Since diffusion is slower than the \nshear transformation [ 4, 21], there is no redistribution \nof carbon between the α  and γ phases during the \ngrowth of α phase plates. In low -carbon steels  the \nrelation Teutec<T<T 0  is possible, when the shear lattice \nreconstruction  is realized in the ferrite region of the \ntransfor mations  diagram. The corresponding \ntransformation scenario can be interpreted as acicular \nferrite, which is realized by displacive mechanism \n[20]. As can be seen in Fig. 11a, the calculated va lue \n \n \nFig.11. (a) Calculated lines (solid) corresponding to the start of ferrite  formation , shear nucle ation (line of \nparaequilibrium) , and martensitic transformation  [13]. MS and MS’ are the start temperatures of the absolute lattice \ninstability and martensitic colonies nucleation. Dashed lines show experimental boundary of two -phase region ( A3) \n[16], expe rimental paraequilibrium temperature ( T0Z) [49], and experimental temperature of the start of martensitic \ntransformation (\nexp\nSM ) [18]. The circles correspond to the start of martensitic transformation in phase -field \nsimulations with therm al fluctuations.  (b) Transformation diagram taking into account the formation of cementite  \n[14]. The boundaries of two -phase regions γ/(α+γ) , γ/(θ+γ)  (lines A3 and Acm, respectively) with their metastable \nextensions and the curves of paraequilibrium γ/α,  γ/θ (T0 and T1 respectively) are calculated.   \n  \n14 of A3 and T0 agrees well with the known experimental \nquantity  \nexp\n3A and T0Z [16,49 ]. \nThe condition \n 0 /,,2 2 t t e Tcef , \ncorresponding to the disappearance of the barrier  on \nthe Bain deformation path,  is attained by quenching of  \nthe γ phase to the temperature MS where \nferromagnetic short range order in  the γ phase \nbecomes important. Below this temperature the \nabsolute lattice instability of the γ phase  should take \nplace, and the obtained martensite can be called as the \nathermal one. I t can be seen  that the temperature MS \nfound in this way is actually lower than the \nexperimental value. However, according to  the \nconcept of isothermal martensitic transformation [ 22–\n25] the condition of the martensite start may be taken \nas \nkTC f0~αγ\nbarrier , where parameter \n0~C =0.04 is \nchosen by fitting to the experiment for pure Fe [ 18]. \nThe temperature MS’ determined in this way agrees \nwell with the experiment in a broad interval of carbon \nconcentration s.   \nThe possibility  of overcooling  of austenite  to the \nliquid nitrogen temperature, with the formation of \nmartensite only during the subsequent heating, was \nfirst discovered in Ref. [22]. This observation is \nclearly points to thermally  activated character  of the \ntransformation with  a rather small activation energy \nvalue of about 0.04eV/at  [22]. It has been  later shown  \nthat the isothermal kinetics changes to an athermal \none in some Fe-based alloys at overcooling below \nsome  critical temperature [ 2]. The avoiding of \ndiscussion of this problem can lead to some kind of \nmisunderstanding.  For example , the athermal \nmartensite was in focus of the model  used in  Ref. \n[10], whereas the kinetics of the nucleation and \ngrowth of isothermal martensite has been  investigated  \nin Ref. [ 11].  \nThe transfo rmations  diagram shown in Fig. 11b \ntakes in addition  into account the formation of \ncementite  (for details of parameterization  see Ref. \n[14]). The line Acm is the boundary of the two-phase \nregion γ+θ. The intersection of A3 and Acm lines is the \neutectoid point ( ceutec,Teutec); below  the themperature  \nTeutec the pearlite transformation  (PT) occurs . In \naccordance with a traditional point of view , the PT is \nrealized between the A3 and Acm lines  extrapo lated into \na T < Teutec region (\"Hultgren extrapolation\"  [130]), \nwhere the simultaneous nucleation of α and θ from the \ninitial γ phase is possible  (the possibility of PT outside \nof Hultgren extrapolation is also discussed [131]). The \nline Acm intersects also the paraequilibrium line T0, so \nthe different transformation kinetics in the bainitic \nregion above and below Acm can be expect ed. This \nmay be relevant to the formation  of several microstructures below the line T0, such as acicular \nferrite and various mo rphologies of bainite. At last, \nthe line T1 describing the start condition of the γ→θ  \ntransformation lies in the region of high carbon \nconcentration ( c~0.20), which  causes a problem in \ndescribing the nucleation of cementite. The possible \nmechanism of facilitation of cementite formation due \nto local  magnetization and appearance of the \nintermediate state ( MIS) near the boundary of ferrite \nplate is discussed  in Section 7.2.    \nTo conclude this section, it should be stressed  that \nthe curves A3, Acm, T0, T1, TF do not depend on the \nenergy relief along the Bain path and are determined \nonly by the terminal values \nPM(FM)\nγ[α]g . On the c ontrary, \nthe martensitic curves MS and MS’ depend on the \ntransformation energetics at intermediate  deformation  \nte\n. For the carbon concentration under consideration \nthe magnetic order effects in γ-Fe are negligible, for \nthe temperatures above T~400K. Therefore,  the \ntransformations diagram is determined, first of all, by \nthe evolution of magnetic state in α-Fe. In particular, \nthe γ→α  transition turns out to be possible above \nCurie temper ature (\nα\nCT ≈1043K) in pure iron due to \nthe short -range ferromagnetic order in α-Fe. The short \nrange magnetic order in γ-Fe becomes important at T \n≈ 400K, which determines the start temperature of the \nathermal martensitic transformation MS, developing \nvia the lattice instability.  Thus, the temperature \ndependence of magnetic short -range order is the key \nfactor determining the diversity of phase \ntransformation s in iron and steel. The closeness of the \nCurie temperature in α-Fe to the temperat ure of \nstructural transformation is not accidental , but is \nrelated with the essence of phase transformations  in \niron and steel .  \n7. MODELING OF THE PHASE \nTRANSFORMATION KINETICS  \nThe phase diagram discussed above  determines  \nthe conditions of the start of phase transformations \nand the fraction of a new phase at large times . \nHowever,  it does not allow to understand the \nintermediate stages  of transformation and features of  \nmicrostructure formation . Namely , the microstructure \nformed in the intermediate stages of  the \ntransformation is a matter of the greatest interest to \nachieve  desirable properties.  \n7.1. Athermal and isothermal martensite  \ntransformation .  \nPhase-field simulation of transformation kinetics \nin the framework of proposed model, i.e. numerical  \n  \n15 solutio n of Eq s. (6), (8) with G. -L. functional (1 8), is \nrequired for the analysis of a martensitic \ntransformation  (MT) . Herewith, the shear \ntransformation occur s with a velocity ~ 103 m/sec, i.e. \nin a time much shorter than the characteristic diffusion \ntimes . Thus, the carbon distribution can be considered \nas a \"frozen\" , and the Eq.( 8) may be neglected. The \nmodeling of MT kinetics was carried out on a square \ngrid by the classical Runge -Kutta method with \nperiodic boundary conditions  [12].  \n As was mentioned above, the formation of \nmartensite does not require thermal activation at \nT<M S, while the appearance of isothermal martensite \nis expected in the temperature range MS<T<M S’ after \naging as a result of thermal fluctu ations. Therefore,  \nthe simulation of MT must take into account the \nthermal lattice vibrations. The lattice temperature was \nintroduced in the framework of microcanonical \nensemble. First , the system is heated to a high \ntemperature ( Т=1200K) by the small random forces \n\n(r,t) (the corresponding term is appended to the \nright side of Eq.(6) leading to the Gibbs distribution of atomic displacements). Then the random forces are \nshut down and the equilibrium state is a ttained after \naging at this temperature. Finally,  the lattice \ntemperature is reduced to the desired value from the \ninterval 400K...1000K by rescaling of velocity field. \nHerewith, the estimation of lattice temperature was \ncarried out by calculation of the a verage kinetic \nenergy per degree of freedom, \n2/2 v kTρ , \nwhere < v2> is the average square of the velocity over \nthe calculation area. The spin temperature was chosen \nin the region of stability of the γ phase during th ese \npreparation procedures and then it switches to the \nvalue of lattice temperature.   \nThe typical patterns of the order parameter  \ndistribution \n  depending on time are shown in  \nFig.12–14. Black and white colors  corresp ond to the \ntwo possible values of order parameters  for the α \nphase  in two -dimensional case, \n1 , i.e. to the two \nmutually orthogonal directions of the Bain \ndeformation. Time is given in dimensionless units , \nρα2/~LJtt\n. At significant overcooling  (T<M S) \n \nFig.12. Kinetics of a thermal martensitic transformation; T=400K, c=0. [12] \n \nFig.13. Kinetics of isothermal martensitic transformation; T=800K, c=0. [12] \n \nFig.14. Heterogeneous nucleation of isothermal martensite , with taking into account the relaxation of elastic stresse s; \nT=700K, c=0.01 . [13]  \n  \n16 the homogeneous transition is realized by \ndevelopment of all fluctuations inherited upon cooling \nfrom a high temperature state (see Fig. 12). In the \ntemperature range  MS<T<M S2 the system remains \nstable with respect to small flu ctuations , and the phase \ntransformation starts  with the appearance of critical \nfluctuation after the incubation period  (a few \nnanoseconds ) and it is realized by replication of \ntwinned plates  (see Fig. 13). The similar mechanism \nwas early observed in Ref.[ 11] at phase field \nsimulation of MT in the system with an α  phase \nnucleus initially introduced in it. The difference of \norientation of the neighboring domains reduces the \nelastic energy of the system, but raises its surface \nenergy. As a result, the character istic size of domain is \ndetermined by minimization of the sum of elastic and \nsurface contribution to the total energy. The \ntemperature  MS2 is defined in this case as a result of \nphase -field simulations in 2D model and it does not \nnecessarily coincide with  MS', which was early \nobtained (see Fig. 11a) from the fitting to experimental \ndata. Nevertheless , these temperatures are close ; the \nvalues of  MS2 obtained from simulations are \ndesignated by the solid circles in Fig.11a. \nThe elastic stresses play a crucial  role in \nmartensite transformation.  As it is well known, the \ncomponents of the deformations tensor are coupled to \neach other  by Saint -Venant's compatibility equations \n(4). It leads to an additional contribution to the G. -L. \nfunctional and results in  effectiv e long -range \ninteractions in distribution of order parameters , which \nplay a crucial  role in pattern formation upon  the phase  \ntransition [ 9–11].  \nAs was shown in  [111–114], the relaxation of \nelastic stresses during the transformation is an \nimportant factor determining the martensite \nmorphology . The main relaxation channel is the \nplastic deformation arising under  local  stresses \nexceed ing the yield stress. A consequent description \nof the plastic deformation requires an essential \ncomplication of the model by including additional  \norder parameters. Instead, in Ref. [ 13] the plastic \ndeformation has been taken into account in a \nphenomenological way. Since the contribution of the \nelastic stresses to the Ginzburg -Landau free energy \nfunctional is determined by the coef ficients Av, As, the \nreal values of these parameters were replace d by some \neffective, temperature -dependent values, 0<\neff\nvA <A v, \n0<\neff\nsA <A s. The thermal lattice fluctuations can not \nbe taken into account in this scheme, s ince the \nrenormalization of Av, As leads to the incorrect change \nof fluctuations amplitude , affecting the start condition \nof homogeneous  transition and the morphology of \nmartensite . This approach  can be considered as reasonable  for the stage of growth of  isothermal \nmartensite . It provides the  fast stress relaxation and \nthe lattice remains coherent during the whole \ntransformation process.  \nFig.14 show s the MT kinetics  when choosing \nparameters \neff\nvA ,\neff\nsA  in such a way that the average \nelastic energy over the sample is equal to the \nexperimental value of the stored energy in martensite, \n0.007eV/at [ 132] (i.e. ~10% from the nominal values ). \nHerewith , the heterogeneous nucleation is provided by \nadditional contribution to the fre e energy near grain \nboundary  (see details in Ref.[ 13]). In this case the \nmartensite is formed as a lenticular colony of twinned \nplates.   \nThus, the proposed model [ 12,13 ] reveal s two \ntypes of MT kinetics, athermal and isothermal  at \ndifferent temperatures , in accordance with existing \nconcepts  [133, 2, 22–25]. The e xperimentally known \nline of the start of MT [ 18] corresponds to isothermal \nMT, whereas the athermal scenario is more \nhypothetical and it can develop  due to short -range \nmagnetic order in the γ phase.    \n \n7.2. Kinetics of pearlite transformation. Globular \nand lamellar structures.  \nThe pearlite morphology is similar to that formed \nduring discontinuous decomposition  [134–136], when  \nthe supersatured mother phase \n0α  decompose s into a \ntwo-phase structure \nβαα0 , where the phases \n0α\n and \nα  have the same crystal structure , but with \ndiffer ent composition . The model of the spinodal \ndecomposition (SD) provoked by grain boundarie s \nwas proposed for explanation of this phenomenon \n[137]. However, the SD kinetics is not applicable to \nPT, because the mixing energy of carbon in γ-Fe is \npositive ( v>0) [ 57,58 ] that prevent of the carbon \ninhomogeneity  formation in the γ  phase . PT is also \nsimilar  to the eutectic colony growth in the absence of \na temperature gradient [ 108,109 ], a process that \npresupposes a  realization of the condition  of the solid \nsolution decomposition . As it was shown  recently \n[14], the lamellar structures can arise by an \nautocatalytic mechanism below the critical \ntemperature  even if the mother phase  (austenite)  is \nstable with respect to decomposition  (v>0) and the \ntransition from lamellar structure to globular one takes \nplace with temperature increase.  \nFirst of all , let us discuss the possible \ntransformation scenarios of decomposition by using \nschematic Fig. 15 where free energies of involved  \nphases at different temp eratures are presented. At a \nhigh temperature the stable equilibria of α/γ  or γ/θ  \n  \n17 exist in  the system  and ferrite or cementite will \nprecipitat e from  a γ matrix in different concentration \nintervals  (curve a).  At a lower temperature, T<T eutec, \nthe γ phase i s metastable with respect to the \ndecomposition α+γ or γ+θ and, in addition, the stable \nequilibrium of α/θ  appears (curve b). The uncorrelated \nnucleation and growth of the α  and θ phases inside a \nγ- phase matrix are expected  in this case.  \nA further decrease  in temperature  leads  to the loss \nof one or both metastable equilibria at a preservation \nof thermodynamic equilibrium between the α  and θ \nphases (curves c and d). As a result, the change in the \ndecomposition kinetics of austenite is expected, \nbecause the cooperative formation of the α  and θ \nphases become s preferable. As was shown in Ref.  \n[14], in this case the pearlite colony can emerge by \nsome kind of autocatalyt ic mechanism when the \nappearance of one of the phases ( or ) stimulate s \nthe nucleation of next one, and the lamellar or \nglobular structure  can form  in dependence o f the \ntemperature.  The similar autocatalytic decomposition \nscenario was earlier considered for the system  with \nthe metastable phase and a symmetric phase diagram, \nand the possibility of such a mechanism of PT was \ndiscussed in  Refs [138,139 ].  \nA key component  of the model  PT is the start \ncondition  of the cementite  formation . Indeed , \naccording  to the transformation diagram in Fig.11b , \nwhen a carbon concentration increases  the metastable \nequilibrium of α/γ is achieved before then the \ncementite formation is realized, since the line A3 goes \nmuch  left of T1. To solve  this problem, in Ref. [ 14] it \nwas accepted that cementite nucleation is facilitated in \nthe thin ferromagnetic region near the ferrite plate \nwhere so called the Metastable Intermediate Structure \n(MIS) [ 77] exists.  As a result, the line T1 is shifted to the left by the value \nboundc ~0.05 and crosses the line \nA3 at approximately 15% carbon . (see Fig.16). Thus , \nthe occurrence of MIS [77], which is a precursor of \ncementite formation, is very important for the kinetics \nof PT.   \nOn the transformation diagram in Fig.16 there can \nbe identified three regions I –III where different PT \nscenarios can be realized in dependence o f \ntemperature.  These regions  are determined by \nintersection points of the lines A3 and Acm with \nbound\n1T  \nand T0 and correspond to free energies curves b,c,d  in \nthe Fig.15, respectively .  \n \nFig.15. Varian ts of phase equilibrium in the system \nwith the triple -well potential f(c).  \n \nFig.16. Transformation diagram of carbon steel \ntaken into account the facilitation of cementite \nformation near the ferrite boundary.  The temperature \nregions I –III are determined by intersection points of  \nthe two phase region boundaries A3 and Acm with the \nparaequilibrium lines  \nbound\n1 0,TT . \n \nFig.1 7. Temperature dependences of pearlite \ncolonies (1) nucleation  rate, (2) growth rate, and (3) \nthe effe ctive transformation rate [ 3].  \n  \n18 The formation of the regular lamellar pearlite by \nautocatalytic mechanism is to be  expected in region \nIII, wherein  the instability of austenite in respect to \nthe γ→α+θ  decomposition appear s stepwise with \ndecreasing temperature. This agree s with the \nexperiment [ 3]; the  temperature dependence of \npearlite colony nucleation rate, growth rate , and the \neffective transformation rate ( see Fig.1 7) have a \nmaximum near the te mperature 820K, which indicates \nthe thermodynamic instability of austenite.  Moreover, \nthe pearlite nucleation rate (in contrast to the growth \nrate) is close to zero at \neutec p TT Texp  and changes \nabruptly at \nexp\npT 820K  (similar re sults were found \nearlier in Refs . [53, 66]). So, the nucleation rate \nappears very slow above 820K, while existing pearlite \ncolony can grow. Therefore, the temperature T~820K \nmay be considered as an experimental estimation of \nthe value \n)2(\npT  in Fig.16 .  \nSince PT is realized above the paraequilibrium \ntemperature ( T>T 0), it is controlled by carbon \ndiffusion. In this case  Eq. (8) can be solved under the \nassumption  that the lattice reconstruction is a rather  \nfast process in comparison  with the characteristic \ndiffusion times . In this case , the fast variables \nte ,\n \ncan be avoided by minimization  of the  local free \nenergy density over these ones, so \n),( ),,,( Tcf Tc eft eff\n. In result, the G.-L. \nfunctional  have a form : \n \n\n  drckf Tcf Fc 2\n2),(el eff\n              (25) \n ),(),,(),,( min),( TcfTcfTcf Tcfθ γ α eff\n,                                                        \nwhere \nTc f ,γ(α,θ)  is the local density of the free \nenergy of austenite (ferrite, cementite).  Since the α  \nand θ phases in pearlite colonies are usually \nconjugated with small mismatch and the coherency is \nlost mostly on the transformation front [ 140] the \nelastic energy contribution \nelf  was neglected in [ 14].  \nFig.1 8 shows the typical  evolution of  \ntransformation  patterns  arising at overcooling  of \nausteni te into the region  III of the transformation \ndiagram . Carbon is pushed out from an embryo of \nferrite because its solubility in the α phase is  much  \nlower than in the γ phase . Since c(A3)>c(\nbound\n1T ) (see \nFig.16 , the region III ), the local metastable phase \nequilibrium of α/γ  can no t be reached, and the \nformation of cementite takes place.  The growth of the \narising  cementit e nucleus leads to depletion of carbon \nin surrounding austenite. Since c(Acm)<c(T0) (see \nFig.16, the region III), the local metastable phase \nequilibrium of θ/γ  also can no t be reached , and the new ferrite layer is formed near the θ phase.  The \nprocess descr ibed above is repeated , so the \ncorresponding mechanism can be considered  as \nautocatalytic. Phase -field simulation shows that a fine \nlamellar structure is formed in this case  and the \nmovement of the front of pearlite colony is \naccompanied by  increasing its transverse size. As a \nresult, the pearlite colony becomes  of a fan -type shape \nin accordance with experiment [ 3, 53, 74]. Note that a \nsimilar fan -type pearlite structure appears , if we start \nfrom one cementite embryo  instead of the case of \nferrite .  \nFig.19 shows  the decomposition  kinetics  in the \ncase, where  the metastable  equilibrium  of the γ phase  \nwith cementite  exists in the region II, but its \nequilibrium with ferrite is impossible, i.e. \n)2( )1(\np p T T . \nIn this case the PT starts  only with ferrite  embryos, \nsince they alone can not be in equilibrium with \naustenite. The condition of autocatalytic \nmultiplication of lamellae is violated and the phase -\nfield simulation demonstrate s the relevance of  \nglobular structure. As in the previous case, carbon is \npushed out from the embryo of ferrite and the \nnucleation of cementite takes place.  However, in this \ncase the line Acm is achieved before the critical \nconcentration c(T0) does, so that the metastable phase \nequilibrium of γ/θ is realized, and the new ferritic \nlayer does not appear. As a result, the other scenario \nof transformation take s place, which results in \nnumerous small cementite precipitates in the single \nferritic matrix.   \nIn the region I in Fig. 16 austenite is decom posed \nby conventional nucleation -and-growth mechanism , \nas was discussed in Ref.[ 13], and we  do not show \ncorresponding  pictures here. Carbon is pushed out \nfrom the ferrite embryo and its concentration near \nferrite interface reaches the value determined by A3 \ncurve. Since c(A 3)<c(\nbound\n1T ), the metastable phase \nequilibrium of α/γ is reached, and the formation of \ncementite does not occur in this case.  And vice versa, \nif we start from one ce mentite embryo, the metastable \nphase equilibrium of γ/θ  is realized and ferrite does \nnot occur because c(Acm)>c(T0).  \nThus, the two possible scenarios of pearlite \ntransformation, lamellar and globular, are possible \nwithin the model  presented in [ 14], and second one is \nrealized at a higher temperature.  The autocatalytic \ndecomposition described above differs from the well - \nknown spinodal decompos ition (SD) by the fact that \nthe γ phase loses its stability in respect to large \ncomposition deviations ( near the  existing \nprecipitates ), so that decomposition is realized by the \nscenario  of colonies growth, while during SD the  \n  \n19 homogeneous instability of solid solution in respect to \nsmall  compositional fluctuations develops in the bulk.  \nThe nucleation  of globular  pearlite , also known  as \nDivorced  Eutectoid  Transformation  (DET ), attracts an \nessential interest [71–75]. This state is usually \nproduced by the heating of the existing lamellar  \npearlite  above the temperature  Teutec until the \ncementite is almost completely dissolved, and then the \ncooling below the temperature Teutec is carried out. As \na result, the observed PT morphologies is similar to \nthe Fig.1 9, wherein the numerous precipitations of \ncementite are immersed in the single  α matrix with a \npronounced transformation front.  According t he \nconventional point of view,  the cementite nucleuses \nare storing in  the γ matrix after the heating  and grow \nupon  a subsequent small overcooling below Teutec , \nwhile the nucleation of lamellar st ructure do not occur before completion of DET . This scenario is consistent \nwith the transformation in the region I (see Fig. 16). \nMoreover , in Ref.[72] it was pointed  out that the \nglobular pearlite is realized  in hypoeutectoid  steels  \neven at overcooling fro m an almost homogeneous \nstate, thus the number of cementite globules after the \nDET is much more than the number of potential  \nnucle i. In the context  of presented  phase -field \nsimulations (Fig.19) , this fact may indicate  that the \nkinetics of globular pearlite  includes the autocatalytic \nnucleation of the new cementite globules, as it occur s \nin the region II  of transformation diagram .     \nVariation of the parameters leads to some  changes \nof the precipitates morphology. We only discuss the \ngeneral trends observed  in the calculations. The \ninterlamellar spacing decreases with the decreasing of \ntemperature T in accordance with  known  classical \n \nFig.18. Kinetics of lamellar structure growth from a nucleus placed on the grain boundaries junctions (ferrite nucleus \non the bottom left and cementite nucleus on the upper right are indicated by arrows) ; T=675K,  c0=0.06 , \nγv=1.5 eV/at  \n[14].  The carbon concentration is indicated by the gray scale; the black color corresponds to ferrite, and the white to \ncementite. The time is given in dimensionless units, \nαDL/2 . The  embryos of ferrite and cementite are introduced \ninto the initial state, lower left and upper right corner of the calculation square , respectively.  \n \nFig.19. Kinetics of globular colony growth from a ferrite nucleus; T=800K,  c0=0.06 , \nγv=1.5 eV/at   [14]. \n \nFig.20. Kinetics of lamellar structure growth from a nucleus of ferrite; T=675K,  c0=0.06, \nγv =2 eV/at.   \n  \n20 concepts  [3]. The ratio of the temperatures \n)1(\npT ,\n)2(\npT  \ncan be change d by varying of  the p arameters \nFM(PM)\nγ  \nand \nγv . The tendency of lamellar  structure  formation \nincreases with increasing \nγv  (see Fig. 20), however , \nthe morphology of lamellae differs from the \nconventional pearlit ic structur e (the concentric layers \ninstead of radial strips  are observed) . In our opinion , \nthe elastic stresses  can play an essential role in  the \norientation of lamellae , which are not taken into \naccount in the simplified G.-L. functional,  Eq.(25).  \nThe qualitative conclusions  presented  here are \nquite  general  and they can be attributed to other \neutectoid systems , for example to the alloy  Zn-Al \n[50], where the  lamellar  structures  are also formed . In \nthe same  time, the proposed  model  does not explain \nthe appearance of a small number of colonies  of \ncoarse  lamellar  pearlite , which is observed in the \ntemperature range  \neutec p TT Texp  [3], i.e. together \nwith DET . So, additional factors should be taken into \naccount (such as incompatibility elastic stresses) to \nprovid e more reliable results of calculations.  \n7.3. Scenarios of ferrite and intermediate \ntransformation s.  \nThe ferrite transformation (FT) starts just after the \ncooling below the line A3 and results in the appearance \nof almost pure bcc -iron ( α phase).  Because driving \nforce in this case is rather small, the transformation \nusually starts on grain boundaries where nucleation is \nfacilitated. S ince carbon solubility in  the α phase is \nvery small, the carbon  is pushed out in to the γ matrix , \nwhich  results in the appear ance of the regions \ndepleted or enriched in carbon. FT is a diffusion -\ncontrolled phase transformation, so that nucleus of the \nα phase can not grow without carbon redistribution  in \nthis case . The temperature region of diffusion \ncontrolled growth of the α phase is T0<T<A 3 (see \nFig.11b)  and the condition  where ferrite can grow \nwithout cementite formation is T>T eutec.  \nIt is necessary to pay attention to the two \nimportant features of FT . At first, the gain of free \nenergy  in the formation  of ferrite  is small  (see Fig.9b), \ntherefore the realization of FT  requires  almost \ncomplete relaxation of elastic stresses . Secondly , FT \nis even observed  experimentally  above  the Curie  \ntemperature , T>TC, thus it is due to short -range \nmagnetic order in the absence of long -range or der.   \nFig.21 shows  the kinetics of FT  when the solution \nof the complete s et of equations for  shear -diffusion  \ntransformations is carried out . The upper and lower rows of images  correspond to  the shear  order  \nparameter  and carbon  concentration , respectively . \nTime  is given  in dimensionless  units , \nρα2/~LJtt\n. It was supposed , the elastic  stresses  \nare absent ,  \nffe\nvA =\neff\nsA =0, and the additional  \ncontribution  to the free energy  exist near the grain \ntriple junctions a nd boundaries  (see [13] for details ). \nThe growing polygonal ferrite precipitates, \nsurrounded by a carbon shell, are observed in \naccordance with experiments [6].  \nIn the temperature  range  MS’<T<T0 the model \ndemonstrates several possible scenarios . As was no ted \nin the discussion of a transformation diagram , at \ntemperature  T<T0 the lattice  reconstruction  γ→α  can \noccur  even  if homogeneous  carbon  distribution is \nretained . However , the diagram  (Fig.11) was \nconstructed without taking into account the \ncontribution of elastic stresses to the free energy . As \nwas discussed  in [13], the nominal  contribution of \nelastic stresses is very large , but it decreases due to \nplastic deformation . The effective values  \nffe\nvA ,\nffe\nsA  \ncan be roughly estimated  from experimental data  on \nthe residual stresses  [4]. As a result, the start \ntemperature o f shear transformation  decreases , \nTstart<T0. Therefore,  transformation  scenario  in the \ntemperature range  MS’<T<T0 depends on degree of \nundercooling . Here  we restrict  ourselves  to discussion  \nof the scenarios of ferrite nucleation and growth in the \npresence of elastic stresses , without the cementite \nformation , and we call them scenarios  of intermediate  \ntransformation . \nAt elevated temperatures the transformation is \ncontrolled by carbon diffusion, as in the case of ferrite \nformation, but the precipitate of α phase acquires a \nplate form, like bainitic or Widmanstatten ferrite \n(Fig. 22). At lower temperatures the ferrite nucleus \nappears and grows by shear mechanism, up to some \ncritical size, determined by increasing elastic stresses \n(Fig. 23). In this case the deple tion of carbon of an α \nplate and its diffusive growth occurs in the second \nstage. Besides, at the same temperatures, but at a large \namplitude of the initial perturbation on GB, the sheaf \nof ferrite plates can be formed, in which the elastic \nstresses produced by tetragonal defo rmations are \ncompensated (Fig. 24). For large time s, the carbon \naccumulates at the boundaries of ferrite plates, where \nits concentration reaches the big values, and cementite \nnucle i may appear.   \n  \n21 The scenarios  of intermediate  transformation  \nshown in  Fig.22–24 may be asso ciated with upper and \nlower bainite . Indeed , the opinion  that upper bainite \nemerges by diffusion and lower bainite by shear \nmechanism is widespread  [19]. However,  the stopping  \nof growth  of plates  in the presented  calculation  is due \nto the boundary conditio ns, whereas the stopping of \nbainitic  plate growth  is explained by lattice coherency \ndisturbance on the boundary  γ/α after the plastic \ndeformation  [4]. Herewith , all subunits  in the bainite  \ncolony  have  the same  orientation ; this is problematic \nwithout  coher ency disturbance on the colony front . \nThus , the proposed  model  describes  the main  \npeculiarities  of ferrite  and some  features  of initial  \nstages  of bainite transformations.  The consistent  \nmodel  of bainite  transformation  should take into \naccount  the plastic d eformation more correctly , \nincluding  the loss of  lattice coherency at the  γ/α  \nboundary when  the critical  size of ferrite  subunits is \nreached . 8. EFFECT OF EXTERNAL MAGNETIC FIELD \nON THE START OF PHASE \nTRANSFORMATIONS  \nThe effect  of powerful  pulsed  magnetic  field on \nthe martensitic transformation (MT) in steel was first \ndiscovere d in Ref.[141]. In Ref. [142, 143] it was \nshown  that the magnetic  field linearly shifts  the start \ntemperature of  MT (\nexp\nSM  increase s by about 0.5 \ndegree in the field  H=1kOe). In Ref. [144, 145] it was \nconcluded  that the pulsed  magnetic  fields do not affect  \nthe isothermal  MT, but can provoke  the athermal  MT, \nleading to the specific distinctive morphology of \nmartensitic crystals . This was explained as follows: \nthe rate of athermal MT is close to the impulse \nduration ( ~10-3 sec), while the ra te of isothermal MT \nis much less (ten minutes), so that isothermal MT can \nbe realized only in a powerful static field; such fields \nwere not available in Ref. [144, 145]. In Ref. [146] it \n \n  \nFig.21. Kinetics of formation of polygonal ferrite at \ntriple grain junctions, T=1000K, c0=0.01  [13]. Fig.22. Diffusion -controlled nucleation and growth of \nbainitic ferrite plate with taking into account the elastic \nstresses, T=850K, c0=0.01.  \n \n  \nFig.23. Shear -controlled nucleation and growth of \nbainitic ferrite  plate with taking into account the elastic \nstresses, T=800K, c0=0.01.  Fig.24. Shear -controlled nucleation and growth of the \nsheaf of bainitic ferrite plates, T=800K, c0=0.01 [13].  \n  \n22 was shown  that the static field shifts the start \ntemperature of isoth ermal MT. Futher investigation \nhas pointed out  that the static  magnetic  field 50kOe \nalso accelerates  the pearlite  and bainite  \ntransformations , herewith  the start temperature  shifts  \nby 10 degrees  [145]. The interest to the effect of \nexternal magnetic field on the kinetics of diffusion  \ncontrolled  transformations  increase s in recent years \n[147–150]. In particular,  it was found  that the \nmagnetic field enhance  the mass fraction  of \nproeutectoid  ferrite  and influences  the morphology  of \ncementite  precipitates .   \nTo explain  the effect of magnetic field on MT , the \nKrivoglaz -Sadovsky  equation  was proposed  [145, \n151]. According  to this formula , the magnetic  field \nshifts  the thermodynamic  equilibrium  to the formation  \nof a magnetic α phase   \nqHMVTT /0 αα\n,                                        (26) \nwhere  T0 is the start temperature of  the γ→α  \ntransformation , \nαV ,\nαM  are the  volume and the \nmagnetization of the α phase,  H is the magneti c field , \nq is the heat of transformation . Also, the formula for \nthe shifting of solubility limits was obtained from the \ncondition of equality of chemical potentials of the \nphases:  \n\n\n\ncVM\nRTH\ncc\nHcHc ) (\n)0()0(\n)()(11\n21\n21Exp\n                   (27) \nwhere  \n)(Hci , \n)0(ic  are solubility  limits in the \nmagnetic field and without it.  \nIt should  be noted  that the Eqs (26),  (27) \ncorrespond to the lines  T0, A3 (see Fig.11) determined \nby equilibrium conditions , while the lines  MS, MS' are \nrelated to  the barrier on the Bain  path. Thus, formulas \n(26),  (27) can not be used for the analysis of a \nmartensitic transformation, contrary to the popular \nbelief. According to the model  [13], the athermal MT \nis due to the appearance of short -range magnetic order \nin the γ phase , and isothermal MT depends on the \nenergy of some intermediate state  near the γ phase on \nthe Bain path.   \nThe transformation  diagram  in the case of the \npresence of external magnetic field is presented in \nFig.25. Here we  used the general formulas  (10)–(11) \nwith an additional contribution \n0Hgβ  and formulas \n(13) –(15) for the spin correlation function . The \nexternal magnetic field increases the degree of order \nnear the Curie temperature , which  result s in increase \nof the spin correlato r magnitude  and the shift of lines \nof the transformation  diagram . Herewith , the change  \nof the magnitude  of the  correlator  depends  on a \ntetragonal  deformation ,  \n)( 1) ( ),(  s γ α γ f Q Q Q TQH H H H , (28) \nwhere  \nHQγ(α)  is the change  of correlator  in γ(α) \nphases , and \n)(sf  is the function characterizing the \nmagnetic susceptibility in intermediate lattice states.  \nFig.26 shows the transformation  diagram  \"under \" the \nfield H=50kOe in the case  of \n22) 1()(sf  \n(curves  A3(1), T0(1), MS(1), MS'(1)) and \n1)(sf  \n(curves  A3(2), T0(2), MS(2), MS'(2)). One can see  that the \nlines A3, T0 do not depend on the choice of \n)(sf , \nwhereas the lines  MS, MS' are very sensitive to this .  \nThus , the proposed  model  allow us to find the \nshifts  of the lines  A3, T0 in agreement  with Refs.[145], \nand it also shows  that the formulas (26),  (27) can not \nbe used for the lines MS, MS'. The construction of \nthese lines under external magnetic field  is to be a \nseparate problem and  requires a justification of the \nform of \n)(sf . \n9. CONCLUSION S AND  OUTLOOK  \nThe problem of the phase transformations and \nmicrostructure formation in iron and steel is  to be in \nscope of interest for a long time and is actively \ndiscussed by now [1–6, 19, 22]. Nevertheless, despite \ngreat efforts , the number of  important questions  are \nstill under  debates . One of the reasons  for this  is the \ncomplexity of the phase transformation s in iron based \nalloys that involve both lattice and magnetic degrees  \n \nFig.25. The transformation diagram \"under \" external \nmagnetic field, H=50kOe. The lines  A3(1), T0(1), MS(1), \nMS'(1) and A3(2), T0(2), MS(2), MS'(2) correspond  to the \nchoice  of \n22) 1()(sf  or \n1)(sf , \nrespectively . The dotted lin es A3, T0, MS, MS'  \ncorrespond to the absence of external field .  \n  \n23 of freedom , as well  the carbon redistribution , which \nalso plays an important role.  Besides, the processes  of \ntransformation  involv e several spatial scale  levels, \nfrom microscopic (atomistic) to macroscopic (at the  \nlevel of the grain size).   \nStarting with a conceptual work by Zener [ 37], it \nis believed that magnetism  plays a crucial role in the \nphase transformations in  iron and steels . However, all \nearly  proposed models are too phenomenological, so \ntheir correct choice is impossible . In this review we \nhave present ed recent progress in understanding of \nmicroscopic mechanisms of phase transformations in \niron and steel . This progress was possible , on the one \nhand , due to the widely  using of ab initio methods for \ncalculation of the electronic structure and total  energy \nin different structural and magnetic states of iron [7, 8, \n35, 36, 58, 77, 120, 126, 152], and , on the other hand, \ndue to applications of the atomistic simulations within \nthe phase -field approach  [81] to the transformations  \nkinetics [9–14, 48, 74, 75, 111, 112].  \nThe rapid development of computing technology \noffers the prospect for the research of realistic \ntransformations kinetics in 3D -models depending on \ncooling rate and concentrations  of allo ying elements . \nIn that connection the task of construc ting of  the \nconsistent ab initio based model of phase \ntransformations in steel, describing the shear -diffusion \ntransformation  kinetics and taking into account the \nmagnetic degree of freedom is very relevant.  \nThe recently  proposed  model  [12–14] agrees  well \nwith the known  experimental  data and predicts the \nstart temperatures of different transformations (ferrite , \npearlite , bainite , martensite ). It was shown  that the \nmagnetism  provides  the main  contribution  to the \nchange  of free energy  at the γ → α  transformat ion. \nTherefore,  the increase of short -range magnetic order \nplays a key role in the change of transformation \nscenarios (from ferrite to martensite ) under cooling . \nPhase-field simulation  carried out in the framework of \nthe proposed model reproduces the typic al precipitates \nmorphology, including ferrite, twinned martensite , \nand pearlite colonies.  \nThe ferrite transformation starts  at a temperature \nbelow A3 due to the short -range magnetic order (with \na possible absence of long -range order) and requires \nthe esse ntial relaxation of the elastic stresses. The \npearlite transformation  results in the formation of a \nregular structure  due to  autocatalytic mechanism, \nwhich is realized in the absence of thermodynamic \nequilibrium between initial austenite and \ntransformation  products (ferrite and cementite). Two \ntypes of autocatalysis were revealed leading to a \nlamellar or globular pearlite structure depending on \nthe temperature. Also two types of intermediate  (bainite)  transformations were observed below the \nparaequilibrium temperature T0 in phase -field \nmodeling . These are  diffusion  and shear -controlled  \ntransformations , which  can be associated  with upper  \nand lower  bainite , respectively . The experimental \ncurve of the start of martensitic transformation (MT) \ncorresponds to the conception of isothermal \nmartensite, whereas the classical (athermal) scenario \nof MT is due to the short -range magnetic order in γ-\nFe, which arises at lower temperature. The model \nallows us to consider  an effect of  external magnetic \nfield on the curves of the start of ferritic and bainitic \ntransformations in agreement with the Krivoglaz -\nSadovsky  concept  [145], and reveals inapplicab ility of \nthis concept to a martensitic transformation.  \nDespite the significant progress in recent years,  a \nnumber of problems remain unresolved, including  \npeculiarities of bainitic microstructure with taking into \naccount the cementite formation, the role o f elastic \nstresses , and their plastic relaxation in growth kinetics \nof the pearlite and bainite colonies, the effect of \nalloying elements on the thermodynamics and kinetics \nof phase transformations . Also the  dimension  of \nmodel  (2D or 3D) is essential for t he kinetics [ 28, 29, \n87, 112], so that more realistic simulations should be \nbased on 3D models . \nAccording to modern views  [4] the role of plastic \ndeformation increases with temperature; this is a \nprincipal channel of elastic energy relaxation in the \ncase o f ferrite transformation , while the relaxation of \nelastic energy in the case of martensit ic \ntransformation is provide d by twinning of plates. \nHerewith, plasti c deformation causes the bainite \nmorphology , since the characteristic  size of the \nbainitic  subunit s is determined by the start condition \nof plastic deformation , disturbing the lattice \ncoherency at the interface γ/α.  \nThus , the essential contours of an ab initio based \ntheory of phase transformations in iron and steel  are \nformed . 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Shimotomai M., Maruta K.  Aligned two -phase \nstructures in Fe -C alloys. // Scripta Mater. 2000. V.42. \n№5. P.499 –503 \n148. Shimotomai M., Maruta K., Mine K., Matsui M. \nFormation of aligned two -phase microstructures by \napplying a magnetic field during the austenite to ferrite \ntransfor mation in steels //  Acta Mater. 2003. V.51. \n№10. P.2921 –2932  \n149. Zhang Y.D., Gey N., He C.S., Zhao X., Zuo L., Esling \nC. High temperature tempering behaviors in a \nstructural steel under high magnetic field // Acta Mater. \n2004. V.52. №12. P.3467 –3474  \n150. Zhang Y.D., Esling C., Zhao X., Zuo L.  Solid State \nPhase Transformations under High Magnetic Fields in \na Medium Carbon Steel // Mat.Sci.Forum. 2005. \nV.495 –497. P.1131 –1140  \n151. Krivoglaz M.A., Sadovskii V.D.  On strong magnetic \nfield effect on phase transfo rmations //  Fiz. Met. \nMetalloved. 1964. V.18. №4. P.502 –505 \n152. Abrikosov I.A., Ponomareva A.V, Steneteg P., \nBarannikova S.A., Alling B.  Recent progress in \nsimulations of the paramagnetic state of magnetic \nmaterials. // Current Opinion in Solid State and  \nMaterials Science. 2016. V.20. P.85 –106. \n          \n \n \n "    },    {        "title": "1404.2658v1.Design_and_performance_of_LLRF_system_for_CSNS_RCS.pdf",        "content": "Submitted to ‘Chinese Physics C ' \n1 Design and performance of LLRF system for CSNS/RCS* \nLI Xiao1)   SUN Hong    LONG Wei     \nZHAO Fa -Cheng    ZHANG Chun -Lin \nInstitute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China  \nAbstract :  The rapid cycling synchrotron (RCS) is part  of China S pallation Neutron Source (CSNS) . The RCS provides  \n1.6GeV protons with a repetition rate of 25Hz . The RF system in RCS is mainly composed of a ferrite loaded RF cavity, a high \npower tetrode amplifier, a bias supply of 3 300A an d a digital low level RF (LLRF)  system based on FPGA. The major cha l-\nlenge of the LLRF system is to solve problems caused by rapid frequency sweeping and heavy beam loading effect . This paper \nwill present the design and structure of the LLRF system, and show results of performance tests.  \nKey Words :  LLRF, CSNS , RCS, RF  \nPACS :  29.20.dk, 07.57.Kp\n                                                             \n* Supported by National Natural Science Foundation of China ( 11175194 ) \n1) Email: lixiao@ihep.ac.cn  \n 1   Introduction  \nChina Spallation Neutron Source (CSNS)[1,2] is \ncomposed of an H - linac  and a proton rapid cycling \nsynchrotron (RCS). It is designed to accelerate proton \nbeam pluses to 1.6 GeV, striking a metal target to \nproduce spallation neutron for scientific research. The \ninjection and extraction beam energy in CSNS/RCS \nare 80MeV and 1.6GeV respectively. Table 1 summ a-\nrizes the primary parameters related to RF system.  \nTable 1.  RF system parameters for CSNS/RCS.  \nParameters  Value  \nBeam power (kW)  100 \nCircumference (m)  229.7  \nEnergy (GeV)  0.081~1.6  \nIntensity (ppp)  1.87E13  \nCirculating dc current (A)  1.5/ 3.6  \nRepetition rate (Hz)  25 \nHarmonic number  2 \nRF frequency (MHz)  1.02~2.44  \nPeak RF voltage (kV)  165 (h=2)  \nA total of  8 fundamental RF systems are used in \nCSNS/RCS to provide a peak  RF voltage of 165kV . The \nRF systems operate on the harmonic number h=2 , and \nthe frequency sweeps  from 1.02 2MHz at injection to \n2.444MHz at ex tractio n with a repetition rate of 25Hz . \nFig.1 shows the frequency , amplitude and synchr o-\nnous  phase  patterns during an accelerating time of \n20ms.  According to the working pattern, the frequency, \nthe amplitude and the phase of the RF signal should \nbe carefully controlled. The CSNS/RCS LLRF control \nsystem is designed to achieve required acceleration \nvolta ge amplitude and phase regulation of ± 1% and \n± 1deg ree respectively. The resonant state of the cavity and the high beam loading effect should also be taken \ncare of by LLRF system, to  make sure of the stability \nof the RF system.  \n \nFig. 1.  RF frequency  (dot-dash line) , accelerating  voltage  \n(solid line)  and synchronous phase  (dot line) in one cycle . \n2   CSNS/RCS RF system  \nIn CSNS/RCS, each  RF system is mainly composed \nof a cavity, a bias sup ply, a high power RF ampli fier \nand a digital LLRF  system[3]. The cavity used is a fe r-\nrite-loaded coaxial resonator with 2 accelerating gaps \nand single ended . The loaded material  is Ferroxcube \n4M2.  The inductance  of cavity can be shifted as a bias \nfield alter ing. Corresponding to the bias current va r-\nying range of 2 00~3000A, the resonant frequency of \nthe cavity can sweep from 1.02 to 2.44MHz to satisfy \nthe CSNS /RCS operation . For one gap, a nominal peak \nRF gap voltage of more than 11.8kV is required.  In \norder to satisfy the requirement of cavity dynamic \ntuning  caused by the nonlinear characteristics of the \nferrite  material during rapid sweeping of RF fr e-\nquency and voltage , a 3300A switch type power supply \nSubmitted to ‘Chinese Physics C ' \n2 and a 150A linear type power supply are used  in bias \nsupply. Two types power supply are connected in par-\nallel, and a big range more than 3000A and a high \nbandwidth up to 20  kHz can be achieved respectively.  \nThe RF amplifier consists of a three stage amplifiers \nchain. The final stage amplifier using a tetrode TH558 \noperated in class AB1, with a configuration of cath-\node-grounded. The maximum plate dissipation of the \ntube is 500kW. The tetrode is driven by a feedback (FB) \namplifier  of 800W located  in the rack of final stage \namplifier adjacent to the cavity. The RF drive signal \nfrom a wide -band solid -state preamplif ier (SSA) of \n500W will be combined with a RF signal extracted \nfrom the cavity which having a proper amplitude and \ndelay to drive FB amplifier.  Fig.3 shows the simplified  \nschemati c of CSNS/RCS RF system.  \n+\n－+Bias Supply\nBias SupplyTH 558800W 500WFrom LLRF\nFrom LLRF\nFrom LLRFBeamBus Bar\nBus Bar HV\n17kVFBSSA3300A\n10A\n \nFig. 2.  CSNS/RCS RF system simplified  schemati c. \n3   LLRF system overview[4] \nConsidering system compatibility and bandwidth, \nCPCI bus is adopted. The hardware platform includes \nthe CPCI6 200 CPU, the CSNS standard timing board \nto receive system clock and event trigger, the custom \nCPCI carrier to realize RF signal digitization , data \nprocessing and control arithmetic for control function . \nThe CPCI hardware is arranged into two CPCI creates. \nOne is for acceleration voltage control and cavity d y-\nnamic tuning. Another is for beam orbit correction and \nbeam loading compensation.  \nThe custom CPCI carrier is the heart of LLRF co n-\ntrol system . It features one Stratix IV EP4SGX530  \nFPGA which provides 531.2 K equivalent logic el e-\nments. It also includes high performance \nTMS320C6 655 DSP, PCI9656 CPCI bridge chip, high \nspeed ADC and DAC  with sampling rate  up to \n125MHz , optical interface , ethernet  interface , and so \non. Operation command and control data exchange \nprotocol between the CPCI6 200 and the CPCI carriers \nwere carefully defined. The custom driver was deve l-\noped to access specified  registers on CPCI carriers \nfrom CPCI6 200 running the VxWorks operating sy s-\ntem.  Adhering to the CSNS control standard, the \nLLRF control system is Experimental Physics and Industrial Control System (EPICS) -based. EPICS \ndrivers have been developed to provide  user interface \nthrough  standard Channel Access protocol. As shown \nin Fig. 3, each custom CPCI carrier has an allocated \ntime slot  of 2.25ms for transmission data in 25Hz \nsystem operation and the actual bandwidth is greater \nthan 200MB/s. A self -defined bus based on LVDS is \nadopted to transmit control data, which is shared by \nthe CPCI carriers. Event trigger and following clock \nare used to ensure the reliability  and stability of \ntransmission, and the maximum bandwidth is greater \nthan 3.84  Gb/s. Fig. 4 shows the configuration of the \nLLRF control system.   \n12345678 CPCI Bus\nRF system data uploadWaiting time (18ms) Operating time (22ms)\n1 cycle (40ms)2.25ms\nCPCI Carrier 1-8\nRF system data storage\n \nFig. 3.  CPCI bus operation timing  \nSFF\nFPGA\nEP4SGX530AD9268AD9268AD9747\n16 1616 16\n1616\nDDR3\nSDRAM32FLASH16\nPCI9656\n64Bit/33MHz CPCIEEPROMTMS\n320C6655SDRAM\nFLASHLevel\nShiftDigital I/OFiber \nInterface\n64\n8Ethernet \nInterface\nCPCI 6200\nVxWorks\nEPICS driverEPICS RecordEthernet\nCPCI\nInterface\nCPCI InterfaceCPCI J4, J510MHz CLK Input\nCLK Output to Devices4\nRapid IO\n10\nSelf-defined BusPHYRJ45\nSGMII\nSGMII\n \nFig. 4.  LLRF system configuration.  \n4   Control architecture of LLRF system  \nEach RF system has inde pendent LLRF  system \nbased on FPGA which is composed of 8 control loops \nincluding cavity voltage loop, cavity phase  loop, syn-\nchronous  phase loop, cavity tune loop, tetrode grid \ntune loop, beam loading compensation loop, orbit \nfeedba ck loop, and direct RF feedback loop[5,6].Fig.5 \nshows the block diagram of the LLRF control system.  \n4.1   RF signal processing  \nFor most control loops in CSNS/RCS LLRF control \nsystem, the procedure of signal processing works in \nmuch the same way. The RF and reference  signal  areSubmitted to ‘Chinese Physics C ' \n3 \n                              Cavity\nBPM FCT WCMTetrode\nBias SupplyDDS\nDirect Digital \nSynthesizer\nBeam\nPhaseCavity Voltage \nControl Loop\nIQ \nDemodulation\nCavity Tune \nControl \nLoopRF Phase\nGrid \nVoltage \nPhaseRF Voltage\nCavity Voltage Detector Frequency\nSet\nPhase Modulation\nOrbit\nFeedback\nBeam\nFrequencyFrequency\nCorrectAmplitude\nModulation\nBeam Loading \nCompensation\nBeam \nCurrent \nAmplitudeDirect RF\nFeedback\nAnode \nVoltage \nPhaseSolid-State \nAmplifier\nRF Feedback \nAmpilifier－       \n+\n \n      \n+\nFront Tune \nControl \nLoop\nSynchronous \nPhase LoopCavity Phase\nControl Loop\nPhase Set\n \n      \n+ \nFig. 5.  Block diagram of the LLRF control system.\nboth generated by DDS (Direct Digital  Synthesizer) \nmodule. To satisfy the sweeping freq uency work status \nof RF system, a direct demodulation method is \nadopted in LLRF system. All RF signals are sampled \nwith 40MHz, and then digital signals are separately \nmultiplied by two reference signals for orthogonal \ndemodulation. A 70 taps FIR low -pass filter is adopted \nwith 1.75 μs delay. The C oordinate Rotation Digital \nComputer (CODIC) algorithm is also used to comp ute \ntrigonometric function and get the amplitude and \nphase of the signals. Fig. 6 shows the procedure of \nsignal processing.  \nFIR\nFIRI\nQADRefrence \nSignalsFrequency Set\nCordic\nModulePhase\nAmplitude RF Signal\nASin (ωt+θ)Sampling Rate\n40MHz\nCos (ωt)Sin (ωt)\n \nFig. 6.  Procedure of signal processing  \n4.2   Cavity Voltage Loop  \nThe cavity voltage loop exists individually  for each \ncavity at (h=2), and makes the cavity voltage follow a \npattern shown in Fig.1. Each cavity has two acceler a-\ntion gaps in parallel. For one gap, the minimum vol t-\nage is 1.3kV and the maximum voltage is 10.3kV.  \nThere is a feedforward compensation option to d e-\ncrease the following error induced by the rapid change \nof cavity voltage. The feedforward compensation \nmainly includes two parts, one is average of the a m-\nplitude modulation values of the past few cycles, and \nthe other is the amplitude error mul tiplied by a ce r-\ntain factor.  \n4.3   Cavity phase  Loop  \n   The cavity phase loop locks the phase between ac-\ncelerating voltage  and RF reference signal. The initial phase of RF reference signal is triggered by the timing \nsystem, when the beam is injected into R CS. The ca v-\nity phase loop should compensate the phase shift \nmainly caused by 2 stages tune loop s and frequency \nresponse in amplification chains.  The bandwidth of \ncavity phase loop should be much higher than the \ntune loops to avoid  loops coupling. When detu ning \nhapp ens, the cavity phase loop must immediately  ad-\njust the phase of RF driving signal to maintain  the \ncorrect phase of accelerating voltage .  \n4.4   Synchronous  Phase Loop  \nThe s ynchronous  phase loop damp s synchrotron \noscillation. The beam phase is detected by Fast Cu r-\nrent Transformer (FCT) , and compared with the phase \nof accelerating voltage . According to the result of \ncomparison, the synchronous  phase loop  provides a \ndamping component with 90 degree phase shift into \nthe RF diver signal.  \n4.5   Cavity Tune Loop  \nThe cavity tune loop is fed by the phase between \ngrid voltage of tetrode and the cavity voltage. The \nchange of resonant frequency of the cavity from \n1.022MHz to 2.44 4MHz corresponds to the bias cu r-\nrent changing from about 200A to 3000A.  \nDue to the bandwidth limit ation of bias supply, a \nfeedforward compensation is used for the 25Hz system \noperation. The algorithm is similar  to cavity voltage \nloop. The actual bias current of the past few cycles and \nthe tuning error are used to generate a feedforward \ncorrection table.  \n4.6   Tetrode Grid Tune Loop  \nThe tetrode grid tune loop  is used to compensate \nthe parasitic capacitance of the tetrode grid, and e n-\nsure system gain and stability in operating frequency  \nrange. a linear bias supply of 10A is used to change Submitted to ‘Chinese Physics C ' \n4 the inductance of a  ferrite -loaded  low-Q resona nt cir-\ncuit, as shown in Fi g.2. There is only a feedback loop \nfor the tetrode grid tuning. Except for this, both of the \ntetrode grid tune loop and the cavity tune loop have \nthe same operating principle .  \n4.7   Beam Loading Compensation Loop  \nThe circulating current in CSNS/RCS is fai rly high, \nand the beam loading effect must be carefully  consi d-\nered. A classic feedforward algorithm is adopted for \nthe beam loading compensation. The beam signal is \npicked up by Wall Current Monitor (WCM), and added \ninto the RF drive signal with an opposit e phase. The \nfundamental component of the beam signal is taken \nout, and given the proper gain and phase[7]. \nBecause of two stage of tuning loops, some phase \nerror will be introduced. An adaptive algorithm is now \nbeing developed for adjusting the phase sett ing of the \nfeedforward algorithm.  \n4.8   Orbit Feedback Loop  \nThe orbit feedback loop is reserved to adjust the \ninitial frequency setting. The beam signal is picked up \nby Beam Position Monitor (BPM), and used to co m-\npute the modification value.  \n4.9   Direct R F Feedback Loop  \nThe RF feedback is an analog control loop. On a c-\ncount of the characteristic s of low latency and high \nbandwidth, i t is adopted to reduce transient beam \nloading and enhance the dynamic perfor mance of the \nLLRF  system. A small fraction of cavity voltage is \npicked up and fed back to the feedback amplifier.  the \nphase shift in the feedback path is guaranteed  by the \ntwo stage of tuning loops.  Since there are no filters \nand signal demodulation, the RF feedback loop c an \nresponse disturbance of arbitrary frequency. In prac-\ntice, the feedback gain is about 20db.  \n5.   System testing results  \n \nFig. 7.  View of the CSNS /RCS RF system  prototypes . \nThe R&D prototype of RCS RF system have been \ndeveloped, including a full size ferrite -loaded RF ca v-\nity, a set of RF amplifier, a 3300A bias supply and a \ndigital LLRF system, as shown in Fig.7 .  The high power integration test was completed on \nthe proto type of RF system , with the cavity voltage \nloop, cavity phase  loop, cavity tuning loop, tetrode grid \ntuning loop, and RF feedback loop . Except for the \nbeam related control loops , the entire LLRF system is \nverified by the integration test . A schematic diagram \nof the test is shown in Fig.6. The maximum cavity \nvoltage up to  12kV is achieved , with the RF frequency \nsweeping from 1.022 to  2.444MHz  and a repetition \nrate of 25Hz . The waveform of cavity voltage and grid \nvoltage of the tetrode is shown in Fig. 8. \n \n \nFig. 8.  Waveform of the cavity voltage (above) and grid \nvoltage of the tetrode (below) on oscilloscope . \nAs shown in Fig.9, due to the dynamic range of \ncavity voltage, the amplitude error is more than 3% at \nbeginning  of one cycle  with only feedback control, and \ndecrease to  less than ±0.4% with the feedforward \ncompensation . \n \n(a) \n \n(b) \nFig. 9.  Cavity voltage error in one cycle without (a) and \nwith (b) feedforward compensation.  \nFig.10 shows the phase error between the cavity \nvoltage and the RF reference signal. With the cavity \nphase loop, t he phase shift  introduced by amplific a-\ntion chains including 2 stages tune loops, decrease  to \n±0.6 degree.\nSubmitted to ‘Chinese Physics C ' \n5  \nFig. 10.  Cavity phase error in one cycle . \n \n(a) \n \n(b) \nFig. 11.  Cavity tuning error in one cycle without (a) and \nwith (b) feedforward compensation.  The algorithm  of feedforward compensation  used \nin cavity tune loop is similar  to the cavity voltage loop. \nThe maximum cavity tune error decreases from about \n30 degree to 2 degree , as shown in  Fig.11 . \nBecause the Q value of the resona nt circuit is much \nsmaller than the cavity, the bandwidth of grid tune is \nhigher than the cavity. Therefore, there is only a \nfeedback control adopted. As shown in Fig.12, the \nmaximum grid tune error  is about 1.3 degree.  \n \nFig. 12.  Grid tuning error in one cycle.  \n5.   Summary  \nThe CSNS/RCS LLRF system provides a total so-\nlution for the RF system in RCS  with rapid frequency \nsweeping and heavy beam loading effect . The cavity \ntuning and beam loading compensation are the key \nissues, and need to be properly handled. Feedforward \ncompensation is widely used to  improve dynamic p er-\nformance and stability of the system.  The bandwidth \nof different control loops should be carefully set to \navoid coupling oscilla tion.\nReferences : \n1  WANG  Sheng et al. Chinese Physics C, 2009, 33(Suppl. II): 1 -3. \n2  WEI J et al. Nucl. Instrum. Methods A, 2009, 600: 10-13. \n3  SUN Hong et al. IPAC2013, 2765 -2767.  \n4  LI Xiao  et al. IPAC2013, 2977 -2979.  \n5  Li Xiao.  Study on the digital low level RF  control system of CSNS/RCS  (Ph.D. Thesis) . Beijing: Graduate University of Chinese \nAcademy of Sciences, 2010  (in Chinese).  \n6  F. Tamura et al. PAC2005: 3624 -3626.  \n7  LI Xiao et al. High Power Laser and Particle Beams , 2013, \n25(10): 2671 -2674 (in Chinese).  \n \n"    },    {        "title": "1908.08257v3.Influence_of_structure_and_cation_distribution_on_magnetic_anisotropy_and_damping_in_Zn_Al_doped_nickel_ferrites.pdf",        "content": "arXiv:1908.08257v3  [cond-mat.mtrl-sci]  7 Apr 2020Inﬂuence of structure and cation distribution on magnetic a nisotropy and damping in\nZn/Al doped nickel ferrites\nJulia Lumetzberger,1,∗Martin Buchner,1Santa Pile,1Verena Ney,1Wolfgang Gaderbauer,2\nNi´ eli Daﬀ´ e,3Marcos V. Moro,4Daniel Primetzhofer,4Kilian Lenz,5and Andreas Ney1\n1Johannes Kepler University Linz, Institute for Semiconduc tor and\nSolid State Physics, Altenberger Strasse 69, 4040 Linz, Aus tria\n2Christian Doppler Laboratory for Nanoscale Phase Transfor mations, Johannes Kepler University Linz,\nCenter for Surface and Nanoanalytics, Altenberger Strasse 69, 4040 Linz, Austria\n3Swiss Light Source (SLS), Paul Scherrer Institut, 5232 Vill igen PSI, Switzerland\n4Department of Physics and Astronomy, ˚Angstr¨ om Laboratory,\nUppsala University, Box 516, SE-751 20 Uppsala, Sweden\n5Helmholtz-Zentrum Dresden – Rossendorf, Institute of Ion Be am Physics and Materials Research,\nBautzner Landstr. 400, 01328 Dresden, Germany\n(Dated: April 8, 2020)\nAn in-depth analysis of Zn/Al doped nickel ferrites grown by reactive magnetron sputtering\nis relevant due to their promising characteristics for appl ications in spintronics. The material is\ninsulating and ferromagnetic at room temperature with an ad ditional low magnetic damping. By\nstudying the complex interplay between strain and cation di stribution their impact on the magnetic\nproperties, i.e. anisotropy, damping and g-factor is unrav elled. In particular, a strong inﬂuence\nof the lattice site occupation of Ni2+\nTdand cation coordination of Fe2+\nOhon the intrinsic damping\nis found. Furthermore, the critical role of the incorporati on of Zn2+and Al3+is evidenced by\ncomparison with a sample of altered composition. Especiall y, the dopant Zn2+is evidenced as a\ntuning factor for Ni2+\nTdand therefore unquenched orbital moments directly control ling the g-factor.\nA strain-independent reduction of the magnetic anisotropy and damping by adapting the cation\ndistribution is demonstrated.\nI. Introduction\nIn spintronics one aims to obtain pure spin currents\nas an additional degree of freedom in logic circuits aside\nfrom electric charges [1]. By spin pumping [2] it is fea-\nsible to induce a spin current from ferromagnetic mate-\nrials into adjacent non-magnetic layers. To ensure pure\nspin currents and exclude charge currents ferromagnetic\ninsulators are the materials of choice. However, ferro-\nmagnetic insulators with low intrinsic magnetic damp-\ning are sparse [3]. The most commonly used material is\nyttrium iron garnet (YIG) [4, 5]. YIG has two major\ndrawbacks, i.e. the complex garnet structure, which is\nalmost exclusively grown on gadolinium gallium garnet\n(GGG) substrates and its weak magnetoelastic response.\nIt could be replaced by cubic Zn/Al doped nickel fer-\nrite (NiZAF) thin ﬁlms grown on MgAl 2O4, which were\nreported to have comparably favorable magnetic proper-\nties [6] as YIG: ferromagnetic at room temperature and\na low intrinsic damping. Furthermore, the static mag-\nnetic analysis shows a soft magnetic behavior with a low\ncoercive ﬁeld, which makes the material relevant for high\nfrequency applications. The combination of these prop-\nertiesmakesit especiallysuitableforapplicationsinmag-\nnetoelectric and acoustic spintronics.\nNiZAF is based on nickel-ferrite NiFe 2O4, which ideally\ngrows in the cubic inverse spinel crystal structure, i.e. Ni\n∗Electronicaddress: julia.lumetzberger@jku.at; Phone: + 43-732-2468-9651; FAX: -9696occupies octahedral (Ni Oh) lattice sites and Fe is located\nin a 1:1 ratio either at tetrahedral (Fe Td) or octahedral\n(FeOh) lattice sites. This is explained in more detail in\nsection VI [see Fig. 5(a)]. Thin ﬁlm growth of NiZAF on\na MgAl 2O4(001) substrate with a normal spinel crystal\nstructure gives rise to disorderbetween the tetrahedral A\n(Ni2+) and octahedral B (Fe3+) sites. Therefore Ni and\nFe with a valency of 2+ (Ni2+\nTdand Fe2+\nOh) are introduced\ninto the lattice. Previous work on the single ion model of\nferrite magnetism [7] shows a negative impact of tetrahe-\ndrally coordinated Ni2+\nTdon the magnetic damping due to\nunquenched orbital moments. This eﬀect has not been\nfound for octahedrally coordinated Ni2+\nOh. Additionally,\nFe2+\nOhincreases the magnetic damping due to hopping.\nThin ﬁlm growth of co-sputtered NiFe 2O4on the same\nsubstrate conﬁrms the presence of Fe2+\nOhin this system\n[8].\nThe structural and magnetic propertiescan be controlled\nby doping with Zn and Al, which predominantly sub-\nstitute Ni and Fe, respectively. Since Zn2+prefers the\ntetrahedralcoordinationit preventsNi2+fromoccupying\nthese sites and therefore, predominantly Ni2+\nOhis formed\n[9]. Growth of highly strained NiFe 2O4thin ﬁlms [8]\nshowed an increase in damping caused by defects. Thus,\nto reduce the strain, smaller Al3+cations are added to\nsubstitute for Fe3+. A doping percentage of 0.8 was\nreported to be the best compromise between magne-\ntostriction and saturation magnetization for bulk poly-\ncrystalline NiZAF [10].\nSubstituting ferromagnetic with non-ferromagnetic ele-\nments in NiZAF results in a reduced saturation magne-2\ntizationMS∼110kA/m and Curie temperature ∼450K\n[6] compared to undoped NiFe 2O4withMSup to\n300kA/mandasigniﬁcantlyhigherCurietemperatureof\nTC= 860K [11]. In turn, NiZAF exhibits a smaller ferro-\nmagnetic resonance (FMR) linewidth down to ∼0.82mT\nresulting in a Gilbert damping parameterof α∼3×10−3\n[6] in comparison to a linewidth µ0∆Hpp= 35mT of\nNiFe2O4[11]. Furthermore, in contrast to a coercivity of\nHc= 100mT [8] for undoped NiFe 2O4a reduced coer-\nciveﬁeldof ∼0.2mT[6]hasbeenreported. Zn/Aldoping\nof NiFe 2O4is, therefore, an advantageous trade-oﬀ since\nlow damping and coercivity are desired for applications\nin spintronics.\nNiZAF was studied for the correlation between composi-\ntion, strain, cation distribution, magnetic anisotropyand\nintrinsic damping. This manuscript is structured as fol-\nlows: in section II the experimental details of the sample\nfabrication are provided together with the experimental\ndetails of the various techniques used throughout this\nwork. At ﬁrst the structural characterizationis discussed\ninsectionIIIfollowedbythestaticanddynamicmagnetic\nanalysis in section IV. In section V the cation distribu-\ntion in the material is studied in detail by comparing\nthe experimental spectra with multiplet ligand ﬁeld sim-\nulations. A comparison to a sample with intentionally\naltered stoichiometry is given in section VI to highlight\nthegoverningmechanisminthe materialsystem. Finally,\na short conclusion is given in section VII.\nII. Experimental Details\nIn this work NiZAF was fabricated using reactive\nmagnetron sputtering (RMS) with a nominal target\ncomposition of Ni 0.65Zn0.35Al0.8Fe1.2O4as suggested in\nRef.[6], in which pulsed laser deposition (PLD) was\nused. The epitaxial thin ﬁlms were grown in an ul-\ntra high vacuum (UHV) chamber with a base pres-\nsure of 2 ×10−9mbar. A doubleside polished single\ncrystalline spinel [MgAl 2O4(001)] was chosen as a sub-\nstrate. The data of the growth optimization includ-\ning temperature, Ar:O 2ratio and thickness variation\nand annealing can be found in the data repository [16].\nThe best results were achieved with the highest possi-\nble heater temperature corresponding to a sample tem-\nperature of ∼525◦C, an Ar:O 2ratio of (10:0)sccm to\n(10:0.1)sccm, a magnetronpowerof30W and aworking\npressure of 4 ×10−3mbar. Additionally, a sample with\nan adjusted composition, i.e., increased Zn while main-\ntaining the Al concentration was fabricated. During that\ngrowth a sample temperature up to ∼600◦C could be\nachieved.\nThe structural characterization of the samples was done\nby X-raydiﬀraction (XRD) measurementswith a Panan-\nalytical X’Pert MRD and aSeifert XRD3003 recording\nω−2θscansandsymmetricaswellasasymmetricrecipro-\ncal space maps (RSM). The surface roughness was char-\nacterized with an atomic force microscope (AFM). Theinterface on an atomic scale was checked by transmis-\nsion electron microscopy (TEM) using a Jeol JEM-2200\nFS. Furthermore, the chemical composition wasobtained\nwith energy dispersive X-ray spectroscopy (EDX) in the\nTEM system. Additionally, the composition was deter-\nmined moreaccuratelybyionbeamanalysis, i.e. Ruther-\nford backscattering spectrometry (RBS) using 2MeV\nHe+and 10MeV12C3+primary beams at the Tandem\nLaboratory at Uppsala University. To disentangle the el-\nement speciﬁc contributions, the spectra were analyzed\nusing the SIMNRA software [12]. Details of the exper-\nimental setup are described elsewhere [13]. The static\nmagnetic properties were measured with integral super-\nconducting quantum interference device (SQUID) mag-\nnetometrybya Quantum Design MPMS-XL5 systemap-\nplying the magnetic ﬁeld in the ﬁlm plane (IP) as well\nas out-of-plane (OOP). The magnetic behavior in a ﬁeld\nrange of ±5T from room temperature (RT) down to 2K\nandthe temperaturedependence ofthe magnetizationup\nto 400K were measured. The data were background cor-\nrected for the contribution of the diamagnetic substrate\nand known artefacts were carefully avoided [14, 15].\nThe dynamic magnetic properties were analyzed with a\nbroadband vector network analyzer ferromagnetic reso-\nnance (VNA-FMR) setup at room temperature at the\nHZDR, researching the polar and azimuthal angular de-\npendence of resonance position and linewidth. Further-\nmore, the frequency dependence was measured to verify\nthe g-factor and disentangle the diﬀerent contributions\nto the magnetic damping. The angular dependence was\ncross-checked with a conventional X-band FMR setup\nand the data can be found in [17].\nFinally, X-ray absorption (XAS) and X-ray magnetic cir-\ncular dichroism (XMCD) spectra at the Ni and Fe L 3,2\nedges were measured at the X-Treme beamline at the\nSwiss Light Source (SLS) [18]. The XMCD spectra were\nobtained in total electron yield (TEY) with 20◦grazing\nincidence at RT and a ﬁeld of 6 .8T. The XMCD(H)\ncurves were measured in a ﬁeld range of ±6.8T in total\nﬂuorescence yield (TFY) at normal incidence, RT and at\nthe maximum of the XMCD peaks sensitive to the site\noccupancyof the cation in the crystallographicstructure.\nThe raw data of the measurements can be found in the\ndata repository [17].\nIII. Structural Properties\nThe crystalline structure of the material is routinely\nanalyzed by symmetric ω−2θscans using XRD. Re-\nsults for twosamples with diﬀerent thicknessesareshown\nin Fig.1(a). Both samples have the (004) reﬂection at\n42.67◦with respect to the substrate and show Laue os-\ncillations. The ﬁlm reﬂection with a small full width\nat half maximum (FWHM) around 0 .13◦for the 70nm\nsample corresponds to a perpendicular lattice parameter\nofa⊥= (8.47±0.01)˚A. The symmetric RSM along the\n(004) plane and the asymmetric RSM along the ( ¯1¯15)3\n(115)(c)\n(004)(b)(a)\nFIG. 1: Structural properties of NiZAF measured with XRD\non a 70nm sample. (a) shows a symmetric ω−2θscan in\ncomparison to a 16nm sample, (b) symmetric RSM along\n(004) and (c) asymmetric RSM along ( ¯1¯15).\nplane are shown in Figs. 1(b) and1(c), respectively. The\nreﬂection from the MgAl 2O4substrate corresponds to\na lattice parameter of asub= 8.08˚A, which has a lat-\ntice mismatch of ∼2% to bulk NiZAF ( a0= 8.24˚A)\n[10]. The asymmetric scan reveals an in-plane lattice pa-\nrameter of a/bardbl= (8.09±0.01)˚A, i.e. fully strained with\nrespect to the substrate. A calculation of the tetrago-\nnal distortion yields c/a= 1.047±0.003 conﬁrming a\nclearly strained material. Additionally, the unit cell vol-\nume of the strained material is a2\n/bardbl·a⊥= (554±2)˚A3,\nwhich is equal to a reduction of the unit cell volume by\nonly∼1% with respect to the bulk a3\n0=559.5˚A3. Even\nthough the samples are signiﬁcantly strained, they main-\ntain an excellent crystal quality evidenced by the well\npronounced Laue oscillations visible in the line scan as\nwell as the RSM indicating a fully strained homogeneouscrystal growth for a thickness of up to 70nm.\nIn Fig.2(a) the high resolution cross-section TEM im-\nage along the /angbracketleft100/angbracketrightdirection shows a sharp interface\ntransition between ﬁlm and substrate on an atomic scale.\nNo indication for dislocations or defects can be found\nin the measurements on various length scales, further\nproving the excellent structural quality of the coherently\nstrained material. A clear separation between ﬁlm and\nsubstrate is visible at the sample interface on a larger\nscale, depicted in Fig. 2(b). However, the ﬁlm exhibits a\nclearly enhanced strain contrast, which corroborates the\nﬁndings of the RSM [see Fig. 1(b)]. Additional annealing\nhad no inﬂuence on the structural quality nor the mag-\nnetic properties further supporting a low-defect material,\nin contrast to previous works on epitaxially grown spinel\nferrite thin ﬁlms [11, 19].\nThe analysis of the chemical composition with EDX in\nthe TEM system (not shown, see repository [17]) re-\nvealed a ratio of (Fe : Ni : Zn)(1 : 0 .55 : 0.22). In\ncomparison to the nominal values of (Fe : Ni : Zn :\nAl)(1 : 0 .54 : 0.29 : 0.66) (all values are normalized\nwith respect to the Fe amount in the material) a clear\nreduction of Zn content is observed. The deﬁciency\nin Zn can be explained by its high volatility, which is\neven more enhanced by the high growth temperatures\n[20]. The aluminium content cannot be reliably deter-\nmined by EDX due to the diﬀerence in atomic number\nin comparison to the other metals. Therefore, the ele-\nmental composition was reﬁned by an in-depth ion beam\nanalysis using RBS with 2MeV He+primary particles,\nwhich makes an accurate determination of the concen-\ntration depth proﬁle of lighter elements like Al feasi-\nble. From these measurements a chemical composition\nof (Fe : Ni : Zn : Al)(1 : 0 .62 : 0.26 : 0.71) is determined\n[see Fig.2(c)]. The Zn deﬁciency is less than suggested\nfrom EDX, however the amount of Ni is increased com-\npared to the nominal values from the composite target.\nThe Al content is slightly increased, indicating only a\nsmall deviation from the nominal values. These ﬁndings\nsuggest that the stoichiometric transfer for this composi-\ntion by RMS is comparable to PLD [6] since both results\nare in reasonable agreement with the target composition.\nNote that the Al content cannot be correlated due to the\nlack of data from the PLD samples in Ref.[6]. Further-\nmore, by analyzing the sample with a 10MeV12C3+ion\nbeam, a more reliable result for the heavier elements of\n(Fe : Ni : Zn : Al)(1 : 0 .6 : 0.23 : 0.83) could be obtained\ndue to the better mass separation. Thus conﬁrming pre-\nvious results, i.e. a good incorporation ofNi with a slight\ndeﬁciency in Zn. The increased Al content is conﬁrmed\nfrom this measurement. From the spectrum in Fig. 2(c)\nno drastic gradient is visible in the element concentra-\ntion. This behavior is supported for Ni, Zn and Fe by\nthe C ion beam [17].\nIn Fig.2(d) an AFM image of the surface of a 550 ×\n550nm2area is shown. A ﬂat surface with neither cracks\nnor holes and a low average maximum roughness peak\nheightRpm= 40pm as well asa root mean squarerough-4\n(a)\n (c)\n(b)\n(d)\nFIG. 2: Surface and interface properties of a 70nm NiZAF samp le. (a) a high-resolution cross section TEM image showing th e\ninterface of ﬁlm and substrate on an atomic scale, (b) the ﬁlm on a larger scale. The RBS spectrum from a 2MeV He+beam\nto determine the chemical composition analysed by SIMNRA is depicted in (c). (d) shows an AFM scan of the surface.\nness ofRq= 24pm is found. This is further supported\nby the maximum height deviation of 2nm within the im-\nage, which corresponds to only 2 monolayers of NiZAF.\nThe smooth surface is ideal for growing heterostructures\nmaking NiZAF a perfect choice for spin pumping devices\nwith an insulating ferromagnetic material.\nIV. Magnetic Properties\nThe static and dynamic magnetic behavior of the ma-\nterial is measured by SQUID magnetometry and FMR,\nrespectively. In Fig. 3(a) the magnetic hystereses at se-\nlected temperatures are shown. The inset depicts the\ndependence of the coercive ﬁeld for all investigated tem-\nperatures between 300K and 2K. A small coercive ﬁeld\nofHc= 1.2mT at 300K is determined, which is fur-\nther evidence for a material devoid of pinning centers.\nA steady increase from Hc= 1.2mT at 300K up to\nHc= 40mT at 2K was determined, as can be seen in\nthe inset of Fig. 3(a). For low temperatures the typical\nincrease of remanence and coercivity is observed.\nThe volume of the ﬁlm is taken into account for the mag-\nnetization values by thickness determination from the\nLaue oscillations [see Fig. 1(a)] and the area of the used\nsample piece. Accordingly, a saturation magnetization ofMS= (118±12)kA/m is obtained from the SQUID data\nat RT. This value matches the bulk value of ∼120kA/m\n[10] and previous values for thin ﬁlms ∼120kA/m [6].\nThe Curie temperature of TC= (375±2)K was esti-\nmated from the M(T) curve at 10mT shown in the inset\nof Fig.3(b). Compared to PLD samples with TC=450K\n[6] the Curie temperature is slightly reduced. Neverthe-\nless, it is well above 300K making ferromagnetic appli-\ncations at RT feasible.\nIn addition, the directionaldependence ofthe staticmag-\nnetic propertieswasstudied. In Fig. 3(c) the diﬀerence in\nsaturation ﬁeld between IP /angbracketleft100/angbracketrightand OOP /angbracketleft001/angbracketrightM(H)\ncurves measured by SQUID is shown in comparison with\nXMCD (H) curves of Fe Oh, FeTd, Ni and an average of\nall three. For now, only the comparison of the diﬀerent\ncontributions to the average with respect to the SQUID\ndata is of importance and the XMCD spectra will be\ndiscussed in more detail further below. A large uniaxial\nOOP anisotropy ﬁeld of µ0Hk>3T is obtained, which\ncannot be solely explained by shape anisotropy, which is\nµ0MS=0.15T. Furthermore, the high OOP anisotropy\nﬁeld of more than 3T is evident in the XMCD(H) curves.\nThe average of the element selective hystereses shows\nthe same behavior as the OOP M(H) curve of the inte-\ngral SQUID measurements. Fully strained NiFe 2O4with\nlower crystal quality, on the other hand shows no con-5\n(a)\n(b)\n(c)\nFIG. 3: Static magnetic properties obtained from integral\nSQUIDmagnetometry. (a) shows the hystereses at exemplary\ntemperatures between 300K and 2K with an inset giving a\ndetailed look at the coercive ﬁeld in this temperature range .\n(b) depicts the temperature dependence of the magnetizatio n\nfrom 2K to 400K at 10mT. An estimation of the Curie\ntemperature is given in the inset. In (c) the XMCD (H) of\nFeOh, FeTd, Ni and an average of all three in comparison with\nthe integral IP /angbracketleft100/angbracketrightand OOP /angbracketleft001/angbracketrightSQUID measurements\nare plotted.\ntribution to the OOP anisotropy ﬁeld except for shape\n[8]. The large µ0Hkpresumably originates from the\nmagnetocrystalline anisotropy of the highly strained in-\nverse spinel crystal structure. This is supported by pre-\nvious work on less-strained Zn/Al doped nickel ferrite\n(c/a= 1.035) with a smaller out-of-plane anisotropy\n(µ0Hk>1T) in Ref.[6]. At ﬁrst sight the major change\nin anisotropy from µ0Hk>1T [6] to µ0Hk>3T seemsto be caused by a change in the clattice parameter from\na⊥= 8.36˚A [6] toa⊥= 8.47˚A for the present ﬁlm. The\nfact that, in the present sample the Al content is slightly\nhigher than nominal together with the lack of knowledge\nabout the actual Al content in Ref.[6] indicates that an-\nother factor could play a decisive role for the increased\ndamping.\n(a)\n(b)\n(c)[100]\n[110][001]\nBθ\nFIG. 4: Dynamic magnetic properties including (a) the polar\nangular dependence from −20◦to 90◦and (b) the azimuthal\nangular dependence from 0◦to 360◦with an anisotropy ﬁt\n(redline)for theresonanceposition inbothdependences. T he\ninset in (a) depicts the smallest FMR line measured with a\nconventional setup at f= 9.5GHz in the easy axis. (c) shows\nthe frequency dependence of the resonance position as well a s\nlinewidth (inset) in a range from 5GHz to 40GHz.\nAngle- and frequency-dependent FMR measurements\nhave been performed at RT to obtain accurate values6\nfor magnetic anisotropy and damping. The FMR spec-\ntra were ﬁtted by Lorentzians to obtain the resonance\nﬁeld (µ0H) and peak-to-peak linewidth ( µ0∆Hpp). Fig-\nure4(a)showsthepolarangulardependencemeasuredat\nf= 9.5GHz by VNA-FMR, where θH= 0◦denotes the\nout-of-plane direction, i.e. the hard axis [see Fig. 4(a)].\nThe resonance ﬁeld in this direction was larger than the\navailable ﬁeld range of the electromagnet. Hence, no res-\nonance could be found around the hard axis. The inset\nshows a typical FMR spectrum taken for ﬁeld in-plane,\ni.e.,θH= 90◦, using a conventional X-band resonator\nwith ﬁeld-modulation technique. The ﬁt reveals a res-\nonance ﬁeld of µ0H= (36.2±0.2)mT and a peak-to-\npeak linewidth of µ0∆Hpp= (12.0±0.2)mT. Compared\nto that the linewidth of the bulk material is four times\nhigher, i.e., µ0∆Hpp= 43mT[10]. Figure 4(b) showsthe\ncorresponding azimuthal angular dependence clearly in-\ndicating a cubic (fourfold) magnetocrystallineanisotropy\nwith easy axes along the /angbracketleft110/angbracketrightdirections. A comparison\nwith bulk showsthatthe IPeasyaxisswitchesfrom /angbracketleft111/angbracketright\nto/angbracketleft110/angbracketrightdirectionsfor thin ﬁlm growthin agreementwith\n[6].\nTo ﬁt the angular dependences in order to determine\nthe cubic IP ( K4||/Ms) and OOP ( K4⊥/Ms) anisotropy\nﬁelds as well as the intrinsic uniaxial OOP ( K2⊥/Ms),\nIP (K2||/Ms) and shape contributions, the following free\nenergy density function was used [21]:\nF=−MsH0[sinθsinθHcos(φ−φH)+cosθcosθH]\n−(2πM2\ns−K2⊥)sin2(θ)\n−K2/bardblsin2(θ)cos2(φ)\n−K4⊥\n2cos4(θ)\n−K4/bardbl\n8[3+cos(4 φ)]sin4θ . (1)\nHereθ,θHandφ,φHdenote the OOP and IP angles of\nthe magnetization and external magnetic ﬁeld, respec-\ntively. To account for a possible deviation from the cu-\nbic system due to the high strain in the material the free\nenergy density of the lower-symmetric tetragonal crys-\ntal structure was used. The ﬁts of the experimental\npolar and azimuthal angular dependences to the reso-\nnance condition [red curves in Figs. 4(a) and4(b)] result\nin the following anisotropy ﬁelds: The eﬀective magne-\ntization is µ0Meﬀ=µ0Ms−2K2⊥\nMs= 2.5T. Using Ms\ndetermined by SQUID the uniaxial OOP anisotropy ﬁeld\n2K2⊥/Mscan be calculated as 2 .35T, which correlates\nwell with the anisotropyﬁeld estimated from SQUID [see\nFig.4(b)]. Thisis in agreementwith the expectation that\na strong hard axis in a material with a low saturation\nmagnetization has to stem from a strain induced perpen-\ndicular uniaxial anisotropy component. The uniaxial IP\nanisotropy ﬁeld K2/bardbl/Msis disregarded, since the visible\nscatter of the data points makes potential contribution\nnot signiﬁcant. Furthermore, the ﬁts revealthat K4/bardbland\nK4⊥are the same. Hence, the uniform cubic anisotropy\nﬁeld is 2 K4/Ms=−4mT. Previous work showed aneven stronger preference for the /angbracketleft110/angbracketrighteasy axis with an\nin-plane anisotropy of ∼10mT for a less-strained ma-\nterial (c/a= 1.035) grown by PLD [6]. The g-factor\nofg= 2.18±0.19 was obtained from the frequency-\ndependence of the resonance ﬁeld depicted in Fig. 4(c)\nby using the Kittel equation [22]. This infers to a lower\ncontribution of the orbital momentum in comparison to\nRef.[6], where g= 2.29±0.09 was reported. The in-\nset of Fig. 4(c) depicts the frequency dependence of the\nlinewidth. A monotonic non-linear dependence is visible,\nwhich is a clear sign for a two-magnon scattering (TMS)\ncontribution to the damping in addition to the intrinsic\nGilbert damping [23, 24], which is linear in frequency.\nThe ﬁt (red line) includes both components resulting in\na Gilbert damping parameter of α= 1×10−2and a TMS\nfactor of Γ 100= 0.04×10−8s−1, respectively. Even ne-\nglectingthe TMScontribution, the pureGilbertdamping\nis still higher by an order of magnitude than previously\nreported [6]. Additionally, an inhomogeneous contribu-\ntion ofBinhom=7.8mT is apparent from the linewidth\ndependence. Since the material and substrate are both\ninsulating and without additional cap layers, broadening\nby eddy current damping and/or spin pumping can be\nexcluded.\nIn a ﬁrst evaluation of the magnetic properties in com-\nparison to the structural analysis the following observa-\ntion is made: The higher strained RMS material ( c/a=\n1.047±0.001) has a lower cubic anisotropy 2 K4/Msof\n4mT, yet shows a signiﬁcant increase in magnetocrys-\ntalline anisotropy evidenced by the high 2 K2⊥/Msof\n2.35T. In turn, this leads to a broadening of the\nlinewidth and increased damping with a non negligible\ncontribution of TMS at higher frequencies. Even though,\nthe structural analysis shows an excellent crystal struc-\nturewith alowamountofdefectsat leaston alocalscale,\na signiﬁcant inhomogeneous component is detectable by\nsensitive magnetic measurements.\nV. Cation distribution\nTo verify if strain is causing the increased damping,\nthe cation occupation is studied in more detail, since it\nreportedly [6, 7] has a strong inﬂuence on the intrinsic\ndamping. The possible coordination in the case of an\nideal inverse spinel crystal structure, as it is given for\nundoped bulk NiFe 2O4, is shown in Fig. 5(a). Half the\nFe3+occupiesoctahedral, theotherhalftetrahedralsites,\nwhereas Ni2+only sits at octahedral sites. Ideally, the\ndopant Zn2+would substitute for Ni2+and the smaller\nAl3+goes to octahedral Fe3+sites according to the nom-\ninal composition of Ni 0.65Zn0.35Al0.8Fe1.2O4maintaining\nthe inverse spinel. However, due to thin ﬁlm growth on a\nnormal spinel substrate MgAl 2O4a mixed spinel crystal\nstructure leading to a small degree of A/B disorder, is\nmost likely.\nFor experimental evidence, total electron yield XAS and\nXMCD spectra in 20◦grazing incidence at the Ni and Fe7\n(b)\n(c)(a)\n(d)\nFIG. 5: (a) Sketch of the inverse spinel crystal structure of NiFe2O4as the underlying structure for NiZAF. Total electron\nyield XAS and XMCD spectra in grazing incidence of 20◦at RT and simulations done with CTM4XAS [25] are depicted at\n(b) the Ni L 3,2and (c) the Fe L 3,2edges. (d) shows the cation distributions of the simulated w eighted linear combination of\nthe XMCD at the Fe L 3,2edges.\nL3,2edges shown in Fig. 5are recorded. Respective sim-\nulations are done with CTM4XAS [25] using multiplet\nligand ﬁeld theory. The chosen parameters are adapted\nfrom nickel ferrite [26], which shows similar XAS spectra\nand resulting XMCD curves. Scaling the percentages for\ntheFdd,Fpd, andGpd, Slater integral reductions of 70%\nand 80%, respectively, are used to consider interatomic\nscreening and mixing. For the octahedral coordination a\ncrystal ﬁeld splitting of 10 Dq=1.2meV and a positive\nexchange ﬁeld of J=48meV is used to ﬁt the obtained\nCurie temperature of TC= (375±2)K [see Fig. 3(c)]\n[27]. The tetrahedral coordination is simulated with a\nsplitting of 10 Dq=−0.6meV and a negative exchange\nﬁeldJ=−48meV. Charge transfer is not taken into\naccount, because no inﬂuence on the absorption spectra\nhas been found [25]. Furthermore, the instrumental and\nintrinsic broadening is included by a Gaussian function\nofσ=0.25eV and a Lorentzian function with a range of\nΓ[0.3eV to 0 .5eV], respectively. Note that the simula-\ntion is shifted in photon energy to ﬁt the experimental\nspectra.\nA comparison between the experimental and simulated\nXAS and XMCD spectra at the Ni L 3,2edges [see\nFig.5(b)] suggests percentages of 94% Ni2+\nOh, and 4%\nNi2+\nTd, showing a slight deviation from the inverse spinelcrystalstructure. TheassumptionofA/Bdisorderissup-\nported by evaluation of the simulation matching the ex-\nperimental XAS and XMCD spectra of Fe [see Fig. 5(c)].\nThe Fe contributions are more complex, since not only\nFe3+occupying Oh and Td sites but also typical spec-\ntroscopic signatures of Fe2+\nOhwere identiﬁed in the ex-\nperimental data. The best match between experiment\nand simulation was determined by relying on the main\npeaks of the XMCD at the L 3edge as shown in Fig. 5(d).\nPercentages of 33% Fe2+\nOh, 45% Fe3+\nOhand 22% Fe3+\nTdare\nobtained as best ﬁt. From this, a mixed state between\nthe inverse and normal spinel crystal structure is appar-\nent, as has been evidenced for thin ﬁlm nickel ferrites in\nearlier works [6, 8, 28]. The element selective magnetic\ncontributions from Ni and Fe indirectly infer the occu-\npation of Zn and Al. According to the single ion model\nof ferrites [7], Ni2+\nOhdoes not enhance the damping, but\nNi2+\nTdcontributes by an unquenched orbital momentum.\nA non-negligible amount of 4% Ni2+\nTdcan be correlated\nwith the reduced amount of Zn2+[7] as evidenced by\nEDX and RBS. Additionally, the strong imbalance be-\ntween octahedral and tetrahedral coordinated Fe as well\nas the high amount of Fe3+\nOhsuggest a deﬁciency of Al.\nThis is further supported by the ﬁndings of bulk NiZAF\nreported by Ref.[10] since a higher amount of Al doping8\nlowers the saturation magnetisation and the magnetic\ndamping. However, the concentration of Al relative to\nFe determined from RBS ﬁts rather well with the nomi-\nnal values. Therefore the diﬃculties of Fe incorporation\ncannot solely be explained or solved with the Al concen-\ntration. Additionally, the large amount of ∼33% Fe2+\nOh\nhas a negative impact on the damping due to hopping\n[Fe2+→Fe3++ e-] [6, 7].\nTheresultsareingoodagreementwiththepreviousanal-\nysis of the material system and give an explanation on\nthe atomic level for the increased intrinsic damping. The\nmain cause being a non-negligible amount of Ni2+\nTddue to\na low incorporation of Zn2+and large amounts of Fe2+\nOh.\nThe role of Ni2+\nTdis even more important, since it directly\ninﬂuences the g-factor by orbital momentum. In addi-\ntion, thestrongA/Bdisorderpromotesscatteringcenters\nresulting in the TMS contribution to magnetic damping.\nFurthermore, the imbalance between tetrahedral and oc-\ntahedral coordinated Fe3+supports the assumption of a\nmixed spinel structure as suggested by growth and chem-\nical composition.\nVI. Governing Mechanism\nTo evidence the crucial role of the Ni2+\nTdin conjunction\nwith the Zn deﬁciency a sample with a diﬀerent com-\nposition of (Fe : Ni : Zn : Al)(1 : 0 .43 : 0.39 : 0.71)\nwas fabricated. The composition was determined using\nRBS with a 2MeV He+beam in analogy to Fig. 2(c).\nFor this purpose the Al content was kept constant to\nmaintain strain and directly see the inﬂuence of the Zn\ndopant. By comparing the composition with the previ-\nous sample a strong change in the (Ni : Zn) ratio from\n(0.62 : 0.26) to (0 .43 : 0.39) relative to Fe is apparent.\nFurthermore,thehigherenergetic10MeV12C+ionbeam\nanalysis of (Fe : Ni : Zn : Al) = (1 : 0 .48 : 0.42 : 0.76)\nrevealed an even more drastic change of the Ni-to-Zn ra-\ntio to (0.48 : 0.42). However, structural analysis shows\na comparably excellent crystal quality with well pro-\nnouncedLaueoscillationsanda(withinerrorbarssimilar\nor even slightly increased) perpendicular lattice parame-\nter ofa⊥= (8.49±0.01)˚Ainstead of a⊥= (8.47±0.01)˚A\n[see inset of Fig. 6(a)]. Thus a tetragonal distortion of\nc/a= 1.049±0.003 is determined.\nIn Fig.6(a) a comparison between data and simulation of\nthe XAS and XMCD spectra at the Ni L 3,2edges is de-\npicted resulting in percentages of 2 .5% Ni2+\nTdand 97.5%\nNi2+\nOh. A reduction of almost 50% in tetrahedrally coor-\ndinated Ni either suggests an improved incorporation of\nZn2+or an overabundance of Zn in this sample leading\nto less possibilities for Ni2+to occupy Td sites. Further-\nmore, the correlation between data and simulation of the\nXAS and XMCD spectra at the Fe L 3,2edges shown in\nFig.6(b) conﬁrms the results obtained for Ni. The simu-\nlation is ﬁtted with respect to the primary peaks of the\nXMCD at the L 3edge, since the magnetic information\nof the material is mainly contained in the XMCD spec-tra. Percentages of 28% Fe2+\nOh, 28% Fe3+\nOhand 44% Fe3+\nTd\narecalculatedindicatingamoreevendistributionofFein\ntheinversespinelcrystalstructurein contrasttoprevious\nvalues [see Fig. 5(c)]. From this a better incorporation of\nthe Al3+in the crystal structure can be inferred, even\nthough the analysis of the chemical composition revealed\nthe same amount of Al relatively to Fe. This further\nhighlights the sensitivity of Fe to disorder in the crystal\nlattice. Additionally, the reduction of Fe2+\nOhsuggests a\npositive impact on the intrinsic magnetic damping due\nto reduced hopping. This interpretation is conﬁrmed by\nthe improved magnetic properties of the material indi-\ncated by an increase in Curie temperature by 10K up\ntoTC= (385±2)K at 100mT as shown in the inset of\nFig.6(c). Furthermore, the hystereses at exemplary tem-\nperatures in a range of 300K to 2K shown in Fig. 6(d)\nindicate a transition to an even softer magnetic material\nthan previously determined [see Fig. 3(a)]. From a closer\nlookatthe coerciveﬁelds dependenceonthe temperature\ndepicted in the inset, a coercive ﬁeld of Hc∼0.2mT at\n300K is determined. At temperatures as low as 100K\nsimilar values for the coercivity as in previous samples of\nHc= 1.2mT are reached, increasing up to Hc= 4mT at\n2K. Additionally, an increase of the saturation magneti-\nzation to MS= (195±20)kA/m at RT (not shown, see\nrepository [17]) in comparison to MS= (118±12)kA/m\nwas measured.\nIn Fig.7the VNA-FMR analysis is shown. In the in-\nset of Fig. 7(a) the FMR line at f=9.5GHz is shown\nindicating an increased resonance position of µ0H=\n(57.8±0.2)mT and a reduced linewidth of µ0∆Hpp=\n(4.4±0.2)mT conﬁrming the improved magnetic proper-\nties. Since the resonanceposition depends on the magne-\ntocrystalline anisotropy this coincides with the decreased\nanisotropy of 2 K2⊥/Ms= 1.53T determined by the po-\nlar angular dependence. Additionally, a lower eﬀective\nmagnetization of µ0Meﬀ=1.78T is calculated from the\nﬁt to the angular dependences. The azimuthal angu-\nlar dependence conﬁrms the cubic (fourfold) symmetry\nof the crystal. The easy axis in the /angbracketleft110/angbracketrightdirection co-\nincides with NiZAF [see Fig. 4(b)], however the cubic\nin plane anisotropy has two components 2 K4⊥/Ms=\n−(420±100)mT and 2 K4/bardbl/Ms=−1.2mT, indicat-\ning a slight shift towards a tetragonal crystal symme-\ntry. Additionally, a slight uniaxial in plane anisotropy\ncomponent of 2 K/bardbl/Ms=−1.4mT is needed to ﬁt the\ndata. The frequency dependences of the resonance po-\nsition and the linewidth are shown in Fig. 7(c) and its\ninset, respectively. Due to the larger FMR signal, the g-\nfactor could be determined with high precision yielding\ng= 2.109±0.002. A comparison with previous work [6],\nwherevaluesof g= 2.29±0.09areobtained,showsadras-\ntic change in orbital momentum. These ﬁndings match\nthe decreased Ni2+\nTd, since these contribute to damping by\nunquenched orbital momentum. Furthermore, the fre-\nquency dependence of the linewidth shows that in the\nZn-rich sample no two magnon scattering is apparent,\nresulting in a Gilbert damping of α= 6.8×10−3. Addi-9\n(c) (d)\n(b) (a)\nFIG. 6: Analysis of a Zn-rich NiZAF showing the TEY XAS and XMC D spectra in grazing incidence of 20◦at RT and\nsimulations done with CTM4XAS [25] at (a) the Ni L 3,2and (b) the Fe L 3,2edges. The inset in (a) depicts a symmetric\nω−2θscan. (c) the M(T) curve at 100mT from 2K to 390K with an inset magnifying the hi gh temperature range. (d) shows\nhystereses at exemplary temperatures between 300K and 2K wi th an inset giving a detailed look at the coercive ﬁeld in this\ntemperature range.\ntionally, the lack of an inhomogeneous contribution sug-\ngests an excellent crystal growth without any defects.\nAlthough the strain has slightly increased, by improving\nthe cation distribution and deviating from the reported\nbest stoichiometry [6, 10] the magnetic anisotropy and\nthe linewidth broadening eﬀects could be reduced re-\nmarkably. These results demonstrate the relevance of\nthe cation distribution, in particular the amount of Ni2+\nTd\nand Fe2+\nOhas another major criteria for the optimisation\nof the magnetic properties. In particular, the reduction\nof Ni2+\nTdand A/B disorder, lowered the g-factor and mag-\nnetic damping and eliminated the TMS contribution.\nSo far, only the inﬂuence of Al on bulk NiZAF [10] has\nbeen studied systematically. The present investigation\nshows the importance of tuning both the Al and the Zn\ncontent. Therefore, for a full understanding of this com-\nplex material system the determination of the whole pa-\nrameter space of Al and Zn concentration together with\nthe site occupancy in NiZAF thin ﬁlms is required.VII. Conclusion\nIn this work the growth of Zn/Al doped nickel ferrite\nthin ﬁlms with excellent crystal quality was achieved by\nreactive magnetron sputtering from a target with nomi-\nnal composition of Ni 0.65Zn0.35Al0.8Fe1.2O4. The chem-\nical composition of (Fe : Ni : Zn : Al)(1 : 0 .62 : 0.26 :\n0.71) obtained by RBS evidenced a rather good stoichio-\nmetric transfer with a slight Zn deﬁciency. The material\nis highly, but coherently strained and no indication for\ndefects or dislocations can be observed from a structural\nanalysis. A comparison with bulk [10] shows a shift of\nthe in-plane magnetic easy axis from /angbracketleft111/angbracketrightto the/angbracketleft110/angbracketright\nin agreement with the thin ﬁlms in [6]. Furthermore,\nthe frequency dependence revealed a reduced g-factor,\nan increased damping and additional contributions from\ntwo magnon scattering and inhomogeneity compared to\nRef.[6] indicating a lower amount of orbital momentum.\nThese changes are caused by the cation distribution,\nwhich has a signiﬁcant inﬂuence on the magnetocrys-\ntalline anisotropy and the intrinsic magnetic damping.\nA comparative sample with increased Zn content shows\nareductionofNi2+\nTd, whichinfersabetterincorporationof10\n(b)\n(c)[100]\n[110][001]\nBθ\nFIG. 7: Dynamic magnetic properties including (a) the polar\nangular dependence from −20◦to 90◦and (b) the azimuthal\nangular dependence from 0◦to 360◦with an anisotropy ﬁt\n(redline)for theresonance position inbothdependences. T he\ninset in (a) depicts the smallest FMR line at f= 9.5GHz\nin the easy axis. (c) shows the frequency dependence of the\nresonance position as well as line-width (inset) inarange f rom\n5GHz to 40GHz.Zn2+leading to a decrease in coercivity, anisotropy and\ndamping. Due to the reduction in orbital momentum\nthe g-factor is lowered signiﬁcantly. Furthermore, less\nFe2+\nOhresulted in a smaller linewidth and an even lower\nmagnetic damping. By improving the A/B disorder, the\ncontributions of two magnon scattering and inhomogene-\nity could be eliminated.\nTo conclude, in contrast to initial assumptions, strain is\nnot the sole mechanism to control the magnetic prop-\nerties in this complex material system. In addition,\nthe cation distribution, i. e. the amount of Fe2+\nOhand\nNi2+\nTdwas found to have a major impact on the magnetic\nanisotropy and damping independent of strain. The in-\nﬂuence of the doping can be divided into two aspects: To\nﬁrst order the Ni:Zn ratio controls the magnetic damp-\ning and g-factor and the Fe:Al ratio the strain. Sec-\nondly, the resulting A/B disorder promotes the forma-\ntion of scattering centers leading to TMS contributions\nand inhomogeneity. While previous work on bulk NiZAF\nhave already evidenced the importance of the Al concen-\ntration, the present work demonstrates that for future\noptimization the crucial role of the Zn concentration has\nto be taken into account.\nAcknowledgement\nThe authors gratefully acknowledge funding by FWF\nproject ORD-49. The X-ray absorption measurements\nwere performed on the EPFL/PSI X-Treme beamline\nat the Swiss Light Source, Paul Scherrer Institut, Villi-\ngen, Switzerland. In addition, support by VR-RFI (Con-\ntract No. 2017-00646 9) and the Swedish Foundation\nfor Strategic Research (SSF, Contract No. RIF14-0053)\nsupporting accelerator operation at Uppsala University\nis gratefully acknowledged. The ﬁnancial support by the\nAustrian Federal Ministry for Digital and Economic Af-\nfairs and the National Foundation for Research, Tech-\nnology and Development in the frame of the CDL for\nNanoscale Phase Transformations is gratefully acknowl-\nedged. Further the authors thank Werner Ginzinger for\nTEM sample preparation.\n[1]S. Wolf, D. Awschalom, R. Buhrman, J. Daughton, S.\nVon Molnar, M. Roukes, A. Y. Chtchelkanova, and D.\nTreger, Science 2945546, 1488 (2001).\n[2]Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. B6622, 224403 (2002).\n[3]B. M. Howe, S. Emori, H.-M. Jeon, T. M. Oxholm, J. G.\nJones, K. Mahalingam,Y. Zhuang, N. X. Sun, and G. J.\nBrown, IEEE Magn. Lett. 6, 1 (2015).[4]M. C. Onbasli, A. Kehlberger, D. H. Kim, G. 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Baberschke, K.\nNagy and A. Jnossy, Phys. Rev. B 73, 144424 (2006).\n[25]E. Stavitski, and F.M.F. de Groot, Micron 41, 687\n(2010).\n[26]R. A. D. Pattrick, G. van der Laan, M. B. Henderson, P.\nKuiper, E. Dudzik, and D. J. Vaughan, Eur. J. Mineral\n14, 1095 (2002).\n[27]C. Piamonteze, P. Miedema, and F.M.F. de Groot, Phys.\nRev. B80, 184410 (2009).\n[28]M. Hoppe, M. Gorgoi, C. M. Schneider, and M. M¨ uller,\nIEEE Trans. Magn. 50, 2506204 (2014)."    },    {        "title": "1408.3275v2.Role_of_magnetic_degrees_of_freedom_in_a_scenario_of_phase_transformations_in_steel.pdf",        "content": " 1\nDecisive role of magnetic degree s of freedom in a scenario of \nphase transformations in steel \n \nI. K. Razumov1,2*, D. V. Boukhvalov3, M. V. Petrik2, V. N. Urtsev4, A. V. Shmakov4, M. I. \nKatsnelson5,6, and Yu. N. Gornostyrev1,2 \n \n1Institute of Quantum Materials Scie nce, Ekaterinburg, 620075, Russia \n2Institute of Metal Physics, Russian Academy of  Sciences-Ural Division, Ekaterinburg, 620041, \nRussia. \n3School of Computational Sciences, Korea Institu te for Advanced Study (KIAS) Hoegiro 87, \nDongdaemun-Gu, Seoul, 130-722, Republic of Korea  \n4Research and Technological Center Ausferr, Magnitogorsk, 455000, Russia   \n5Radboud University Nijmegen, Institute for Mo lecules and Materials, Heyendaalseweg 135, \nNijmegen, 6525AJ, Netherlands \n6Dept. of Theoretical Physics and Applied Mathema tics, Ural Federal University, Mira str. 19, \nEkaterinburg, 620002, Russia  \n \nAbstract \nDiversity of mesostructures formed in steel at cooling from high temperature austenite ( γ) \nphase is determined by interplay of shear reco nstructions of crystal lattice and diffusion of \ncarbon. Combining first-principle calculations w ith large-scale phase-field simulations we \ndemonstrate a decisive role of magnetic degrees of freedom in the formation of energy relief \nalong the Bain path of γ-α transformation and, thus, in this interpla y. We show that there is the \nmain factor, namely, magnetic stat e of iron and its evolution with temperature which controls the \nchange in character of the transformation. Ba sed on the computational results we propose a \nsimple model which reproduces, in a good agreement with experiment, the most important \ncurves of the phase transformation in Fe-C, namely, the lines relevant to a st art of ferrite, bainite, \nand martensite transformations. Phase field simulations within the model describe qualitatively typical patterns at these transformations.  \n \nPACS: 63.70.+h, 64.60.qe, 75.50.Bb, 71.15.Nc \n \n*e-mail:  rik@imp.uran.ru   \n \n1. Introduction \nDespite a broad distribution of numerous new materials steel known from ancient times \nremains the main construction material of our civili zation [1], due to high availability of its main \ncomponents (Fe and C) and diversity of prope rties reached by a realization of various \n(meso)structural states [2,3]. One can control the structural state of steel due to a rich phase \ndiagram of iron with several structural tran sformations at cooling from moderately high \ntemperatures ( α γ δ → → ); the presence of carbon adds  carbide phases, cementite Fe 3C being \nthe most important one. Development of the phase  transformations in steel includes two main \ntypes of processes, the crysta l lattice reconstruction and redi stribution of carbon between the \nphases. Depending on the rates of th ese processes metallurgists separate three main types of the  2\ntransformations, namely, ferrite, bainite, and mart ensite [2,3,4]. In practice, all transformations \n(except the martensitic one) involve both sh ear and diffusion mechanisms, their relative \nimportance being changed with the temperature in crease [4]. The difference between these types \nof transformations determines the diversity of properties of steel and th erefore is of crucial \nimportance for our understandin g of metallurgical processes. However, there is still no \ncommonly accepted quantitative theory which c ould describe the change of transformation \nmechanism with temperature from martensitic (l attice instability) to fe rrite (nucleation and \ngrowth).  \nHere we demonstrate that the main fact or determining scenario of the phase \ntransformations in steel is the magnetic state of  Fe and its temperature dependence. Empirically, \nthe temperature of γ-α transformation in elemental Fe is close to the Curie temperature of α-\nFe; therefore the idea on the decisive role of  magnetism in phase transf ormations for pure iron \nlooks natural and was discussed many times; for review, see Ref. [5].  \nBased on state-of-the-art first-principle calcu lations and combining it with the phase field \nsimulations [6] we build a consistent model which allows us to estimate (with a surprisingly high \naccuracy, keeping in mind its simp licity) temperature ranges corr esponding to the three types of \nthe transformations. This model includes a genera lized Ginzburg-Landau functional for the Bain \ntransformation path with ab initio parameteri zation and nonlinear elasticity equations for the \ntetragonal deformation, as well as diffusion equation for the carbon concentration. Therefore it \ntakes into account both carbon di ffusion and lattice and magnetic degrees of freedom of iron.  \n \n2. Methods \n 2.1. Generalized \nGinzburg-Landau functional for the Bain transformation path \nThe minimal set of variables which is necessary to describe the γ-α transformation in \nsteel includes Bain tetragonal deformation and carbon concentration. Other relevant degrees of \nfreedom are volume per atom and magnitude of magnetic moment but we assume (following \nRef. [5]) that they are fast and can therefore be taken into account just by optimization of the \ntotal energy along the Bain transformation path. Th e parameter of short-range magnetic order is \nintroduced as for the case of pure iron [5].  \nA generalized Ginzburg-Landau f unctional for the total energy can be represented in the \nform [7]:  \ndr ekg G tte∫ ⎟\n⎠⎞⎜\n⎝⎛∇ + =2) (2,                                                 (1) \nwhere eg is the energy density of  lattice deformations, tk is a parameter determining the width \nof interphase boundary [7].  We restrict ourselves by a two-dimensional model when eg can be \nchosen as [8,9]: 2/) ( yy xx ve ε ε+ =  \n2 2\n2 2),,( ssvvtt e eAeATceg g + + = ,                                           (2)  \nwhere 2/) ( yy xx ve ε ε+ =  is dilatation, 2/) ( yy xx te ε ε− =  tetragonal deformation, xy se ε= \nshear deformation, and ),,( Tcegtt  is the energy density depending on the tetragonal deformation \nparameter, local carbon concentration, and temp erature. Using two-dimensional model is, of  3\ncourse, a simplification which does not provid e the complete picture of morphology after \ntransfromation since we have two orientation options for α-phase. Nevertheless, this model gives \ncorrectly thermodynamic condition of transformatio n and describes main qualitative features of \nmicrostructure formation [5,8,9]. Similar to Ref. [5] we assume that in γ-phase (initial phase for \nthe transformation) 0 =te  and in α-phase 2/11−=te . The coefficients s vAA,  are expressed via \nelastic moduli [7], 12 11C C Av + = , 444C As= . Following Ref. [5] we determine the energy density of \ntetragonal deformation as  \n                  )(),(~),( ),,( TQceJce g Tcegt t PM t t − =         (3) \nwhere ),( ),( / ),(~\n112ce gce g Jzm ceJt FM t PM t − =Ω =   is exchange energy, Ω is the volume per \natom, 1z is the nearest-neighbor number, J1 is the exchange integral, m is the magnetic moment, \nand  \n2\n1 0 / )( m TQ >⋅ ≡< mm                                                       (4) \nis the spin correlation function dependent on temperature. We ha ve improved our model for the \ntemperature dependence of th e nearest-spin correlator )(TQ  in comparison with our previous \nwork [5]. Namely, we use Oguc hi model [10] and determine )(TQ  as T TQ /1~)(  for T>T C; for  \nT<T C we use the empirical formula for magnetizati on [11], choosing parameters in such a way \nthat ) (CTQ ~0.4, according to Ref. [10]. Thus, at T=T C the dependence )(TQ  has a cusp. Curie \ntemperature TC is related to the exchange parameter as Ω = )(~)(t t C eJ e kT λ , with the numerical \nfactor for α-Fe αλ=0.472; this choice of λα provides an agreement of the Curie temperature \nwith the experiment, TC=1043K.  The correlator for γ-Fe is chosen in a similar way, with the \nCurie temperature fcc\nCT≈ 300K, according to the calculations [12] for the atomic volume \n12≈Ω Å3; 606.0=γλ  according to Ref. [13]. The temperat ure dependences of the correlators \nare shown in Figure 1.  \n                \nFigure 1.  Temperature dependences of the spin correlator )(TQ  for α-Fe (1) and γ-Fe (2).  \n 4\nTo calculate free energy, we have to add en tropy contributions to Eq.(1). The magnetic \nentropy is calculated, as in Ref. [5], from Hellmann-Feynman theorem. The configurational \nentropy of carbon is found from the mode l of ideal solutions, assuming that for T > 300K carbon \nis equally distributed among all th ree interstitial sublattices in α-Fe whereas in γ-Fe carbon \natoms can occupy only quarter of the interstitial  positions [14]. As a result, we obtain the \nfollowing formula for the local density of free energy:  \n() ( ) () ( )()⎥⎦⎤\n⎢⎣⎡−⎟\n⎠⎞⎜\n⎝⎛− + +′ ′ − − = ∫ t sJ\nt s PM t ef c ccc c ckTJdTJQ efTs g Tecf 1 4ln3ln 4ln~),~( ),,(~\n00      (5)\n \nHere ()t sef  is a function provided a gradual switchi ng of the entropy contribution from fcc to \nbcc ( () 1=t sef  in fcc and () 0=tsef  in bcc phase); s0 is the high-temperature limit of the entropy \ndifference between the phases including phonon contribution. It is commonly accepted (see, e.g., \nRef. [15]) that the value s0 is almost temperature independent at T>T D, where TD is Debye \ntemperature (equal to 473K and 324 K for bcc and f cc phases, respectively). We will assume that \nit is a constant. The latter has been chosen such that the start of the transformation determined by \nthe condition 0 )( )( )( ≡ − = Δ Tf Tf Tfbcc fcc agrees with the experimental value for elemental \nFe, T0 = 1184K. This gives us the value s0= -0.19 k, quite close to the experimental estimate [16].  \nThe resulting Ginzburg-Landau functiona l for the free energy reads:   \n() () dr ekeAeATecf F ttssvvt∫ ⎟\n⎠⎞⎜\n⎝⎛∇ + + + =2 2 2\n2 2 2,,                                     (6) \nThe quantities ) ,( ),,( ce gce gt FM t PM  are found from the energy curves along the Bain path \nfor para- and ferromagnetic states, respectively.  Carbon shifts the thermodynamic potentials of γ \nand α phases of Fe in accordance wi th its solution enthalpy. As was shown in Ref. [17], carbon \nturns out to effect dramati cally on magnetic state of γ-Fe; it can create a locally \nferromagnetically polarized region with tetrag onal distortions. Thus, thermodynamics of γ-Fe-C \nsystem, in particular, solution enthalpy of carbon, should be strongly dependent on local \nmagnetic order. Here we include th e dependence of the energies of γ and α phases on carbon \nconcentration into the model base d on first-principles electronic structure calculations of the \nsolution enthalpy. \n \n2.2. First-principle calculations \nThe calculations of energetics of Fe-C system  were performed by dens ity functional theory \nin the pseudopotential code SIES TA [18], similar to our previous work [ 17]. All calculations \nwere carried out using the generalized grad ient approximation (GGA-PBE) with spin-\npolarization [19]. Full optimization of the atomic positions was performed. During the \noptimization, the ion cores were described by norm-conserving pseudo-potentials [20] and the \nwave functions were expanded with a double- ζ plus polarization basis of  localized orbitals for \niron and carbon. Optimization of the forces and to tal energy was performed with an accuracy of \n0.04 eV/Å and 1 meV, respectively.  All calculations were carried  out with an energy mesh cut-\noff of 300 Ry and a k-point mesh of 4×4×4 in the Mokhorst-Park scheme [21]. For the modeling \nof all configurations 3×3×3 supe rcell of 108 iron atoms in fcc configuration was used. Varying \nof the concentration of carbon was realized by th e change of the number of interstitial carbon \natoms in the voids from one (~1 at%) to th ree (~3 at%). For the modeling of paramagnetic \nconfiguration five possible spec ial quasi-random structures (S QS [22]) of magnetic moments  5\nwere generated by reinitialize each time the ps eudo-random number generator. The structure \nwith the lowest total energy have been define d as a ground state and energy difference per iron \natoms have been used to estimate the error of the modeling of paramagnetic iron. The modeling \nof the Bain pathways was performed by the met hod previously employed for the pure iron [12]. \nIn contrast to Ref. [12], to take into account thermal expans ion effects the elementary cell \nvolume was chosen close to experimental values for γ- and α-Fe at the temperature of γ-α \ntransition and linearly interpolated for 1 / 2/1 < <ac  (actually, the change of the lattice \nconstant along the path is within 1%). The di fference of the energies between ferromagnetic and \nparamagnetic state agrees well with the “exchange  energy” calculated in Ref. [12], thus, the \ndifferent choice of the lattice c onstant is not essential.   \nThe energies found from the first-principle cal culations for pure iron were approximated by \nthe following polynoms:  \n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛− +⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛+ − + =2 3 2 62 )(~4 6\n) (4\n2 ) (\n) ( ) ( ) ( ) (φ φ φφ φFM PMFM PM fcc\nFM PMbcc\nFM PMfcc\nFM PM FM PM ccg g g g     (7) \nHere the order parameter 1 1 <<− φ  related to the Bain tetragonal deformation as \n( )te12/2 − =φ . Positive and negative values of φ correspond to two possible (mutually \northogonal) directions of the Ba in deformation in two-dimensi onal case. Its form guarantees \nextrema at the points 0=φ  or 1±=φ , parameters fcc\nPMg, fcc\nFMg, bcc\nPMg, bcc\nFMg, PMc, FMc were found \nby fitting to the ab initio  computational results. \nWe do not take into account ca rbon-carbon interactions, due to a smallness of carbon \nconcentration. Thus, its contri bution was taken as linear:  \n() () )( 1 )(~),() ( ) ( ) ( ) ( ) ( φ ε ε ε φ φsfcc\nFM PMbcc\nFM PMfcc\nFM PM FM PM FM PM f c c g c g − − + + =                (8) \nFunction ()φsf  have been chosed in form ()()221φ φ −=sf . Within the approximation (8) the \neffect of carbon on Bain-path en ergetics is determined only by carbon solution energies in γ- \nand α- phases. We deal with the temperatures T > 400K where carbon fills equally all three \nsublattices of octahedral intersti tials and therefore we do not take s into account te tragonality of \nmartensite which arises at T ≈ 300K [14]. \nParameterization of these formulas from ab  initio calculations leads to the following \nvalues: bcc\nPMg=0.19, fcc\nPMg=0.14, fcc\nFMg=0.095, bcc\nFMg=0 (in eV/at) and PMc=0.05, FMc= - 0.08 (all in \neV/at). These data were slightly different from those calculated by us ea rlier [12] by VASP (the \nenergy bcc\nPMg coincides with Ref. [12], the energy fcc\nPMg differs by -0.02eV/at). The solution \nenergies of carbon in different phases,bcc\nFMε=0.8, bcc\nPMε=0.7, fcc\nFMε= - 0.2, fcc\nPMε=0.22 (in eV/at) were \nchosen on the base of similar ca lculations for iron w ith carbon concentration ~1% at. The value \nbcc\nFMε agrees with the result of the previous first-principle calculations [23]; fcc\nPMε agrees with the \nresult [17], but lower than the e xperimental value 0.4eV/at [24]. \n \n2.3. Kinetic equations \nIt was shown in Refs. [7,25] that at the desc ription of atomic disp lacements in solids one \ncannot take into account only the order-parameter  (in our case, tetragonal deformation) since \nother components of the deformation tensor are coupled to the order parameter by Saint Venant \ncompatibility equations. The latter result in effective long-range interactions which are crucial  6\nfor the pattern formation at the transition [6,8, 25]. Therefore, following Ref. [6] we write the \ndynamical equations for atomic displacements in a form similar to Newton equations rather than \nAllen-Cahn time dependent Ginz burg-Landau relaxation equation  for the order parameter \n[26,27]. It allows taking into ac count automatically the Saint Ve nant compatibility equations.  \nWe exploit the equations of motion used by us earlier for elemental iron [5] plus the \nequation of carbon diffusion: \n∑∂∂=∂∂\nj jij i\nrt\ntt u ),( ),(\n22r r σρ\n              Ι−∇=∂∂\ntc                                               (9) \nHere ρ is the mass density of iron; elastic stresses ijσ and a flow of carbon atoms I are \ncalculated via variational derivatives of the Ginzburg-La ndau functional:  \n),(),(tFt\nijijrrδεδσ =  ,         \n   ⎟⎠⎞⎜⎝⎛∇− −=cFc ckTD\nδδ)1( I                                           (10) \nwhere D is carbon diffusion coefficient (s ee Appendix); the deformations ijε introduced above \nare connected with the variable of our model as   \n( )te12/2 − =φ ,        () 2/yy xx ve ε ε+ = ,      () 2/yy xx te ε ε− =                             (11) \nxuxxx∂∂=ε ,   yuy\nyy∂∂=ε ,   ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\n∂∂+∂∂=xu\nyu y xxy 5.0 ε                                             (12) \nAs a result,  \n()φφφσ2~ ~),,(\n121∇− +\n−=t vv xx keAdTcdf,                                                          (13) \n()φφφσ2~ ~),,(\n121∇+ +\n−−=t vv yy keAdTcdf,                                                          (14)   \n ss xy eA=σ                                                                                                             (15) \nwhere 2/~\nv vA A= ,  2/)12(~− =t tk k , tk=10-3 (in the units of αJL~2Ω  where L is the sample size, \nαJ~Ω =0.19eV/at). We pass further to dimensionless units Lr ri i / → , Lu ui i / → , \nρα\n2~\nLJtt→ , \n1→ρ , αρ\nJLD D ~2→ , ασ σ Jij ij~/ → . \nTo maximize the size of the system under si mulation for given computer resources we \nrestrict ourselves to the two-dime nsional case. It is enough to dist inguish clearly pa tterns typical \nfor different transformations in Fe-C. The details of the simulations are presented in the Appendix.  \n \n3. Results and discussion \n \n3.1. Bain path and free energy in Fe-C \nThe Bain path is the tetragonal deformation accomplished γ-α lattice reconstruction, \nwhich change from 1 /=ac  for fcc ( γ) to 2/1 /=ac  for bcc ( α) structures. Dependences of  7\nthe energy ( g) and free energy ( f) per Fe atom on tetragonal distortion calculated according to the \nformulas (3–5), (7–8) are shown in Figure 2. One can see th at the ratio of the energies for α and \nγ phases changes strongly with the temperature decrease and α-phase becomes preferable at T < \nTC. Figure 3  displays the temperature dependence of the energy ) ( )( )( T g T g Tgbcc fcc− = Δ  and \nfree energy difference ) ( )( )( Tf Tf Tfbcc fcc− = Δ  in comparison with the data [16] for elemental \niron. One can see on this figure that the model constructed with correlator )(TQ  (see Figure 1) \ndescribes correctly thermodynami cs of both phases of pure Fe within the temperature range \n600÷1200K and agrees well with th e results of CALPHAD [16]. It turns out that the magnetic \ncontribution dominates at T ≤ TC and is compensated essentially by the phonon contribution at T \n> TC. \n \n \n \n   \n \n    \n \n     \n \n \nFigure 2.  (Color online) Energy (a) resulting from the first-principle calculation for the Bain path \nin ferro- (curves 1,2) and paramagnetic  (3,4) states for carbon concentration C = 0 (1,3) and C = 3 at% \n(2,4). Free energy (b) as functions of tetragonal deformation for temperatures T=600K (curves 1,1’), \n800K (2,2’), 1000K (3,3’), 1400K (4,4’) found from Eqs.  (5) and the first-principle computational results \nfor carbon concentration C = 0 and C = 3 at%, respectively. Symbols correspond to the computational \nresults, solid lines are approximations used in the model.  \n \nIn elemental Fe, for ferromagnetic state γ-phase corresponds to th e maximum of the total \nenergy, instead of local minimum and therefore the transition to α-phase happens without barrier \n[28]. It turns out that doping by carbon does not ch ange this important peculiarity. Moreover, \ncarbon decreases the en ergy of ferromagnetic γ-Fe, with the enthalpy solu tion of the order of -0.2 \neV per carbon atom (Figure 2). It is not surprising since ca rbon creates a strong local \nferromagnetic order in parama gnetic or antiferromagnetic γ-Fe [17]. For the other cases ( α-phase \nand paramagnetic γ-Fe) the solution enthalpy of carbon is positive. It is a common wisdom that \ninterstitial impurities (including ca rbon) always prefer fcc surrounding compared to bcc, just for \ngeometric reasons (the voids are larger in fcc lat tice than in bcc with the same density) [29]. This \nis for sure correct, also for carbon in iron and results in a mo re pronounced effect of carbon \n 8\naddition on energy bcc-Fe. What is much less trivia l is that carbon solubility  in fcc iron is very \nsensitive to the magnetic state being maximal in ferromagnetic surrounding. \n \n \nFigure 3.  The energy difference )( )( )( T g T g Tgbcc fcc− = Δ  (curve 1) and free energy difference \n)( )( )( Tf Tf Tfbcc fcc− = Δ  (curve 2) at γ → α transition in elemental iron in comparison with known \ndata  (dotted lines 1’,2’) [16]; contribution  of magnetic entropy to the free energy (curve 3) and the \ncontribution from phonon entropy (curve 4) . \n \n3.2. Construction of the phase diagram a nd scenarios of transf ormations in steel \nNow we are ready to discuss the difference be tween scenarios of phase  transformations in \nour model. This difference originates from th e strong temperature dependence of driving force \nfor the transformation, the rate of carbon diffusion and plastic relaxation of transformation strain. \nAs discussed above, the strong temperature dependence of the fo rmer (followed from the strong \ntemperature dependence of the potential transformation relief) is magnetic in origin: the temperature enters our model mainly via the parameter of a short-range magnetic order. \nFerrite transformation kinetics is controlled by carbon diffusion. Wit hout the redistribution \nof carbon, α-phase is not thermodynamically favorab le and therefore ferrite formed by the \nmechanism of heterogeneous nucleation, usually at grain boundaries. At a low enough \novercooling below the temperature А\n3 [2,3], determined by the condition of equality of chemical \npotentials for α-phase depleted by carbon and γ-phase enriched by carbon,   and restricting the \ntwo-phase γ+α region, the ferrite transformation proceed s slowly since its driving force is \nsmall and a realization of the transformation requir es a redistribution of ca rbon at large distances. \nThermodynamic potentials of α-phase without carbon and γ-phase with nominal carbon \nconcentration are equali zed at a temperature TF < A3, when () ( ) T cef Tceft t ,0, ,,0 = =α γ, с0 is \ninitial (average over the sample) carbon concentration. One can expect that at T ≤ TF the γ-α \ntransformation accelerates essentially since in  this case the short-range carbon diffusion is  9\nsufficient. Therefore we identify the temperature TF with the start of rapid ferrite transformation. \nIt should be noted that TF appears to be close to Curie temperature TC in a broad range of carbon \nconcentration.  \nFurther decrease of temperature result s in a slowdown of carbon diffusion and \nenhancement of the transformation driving force. At intermediate temperatures, a crucial role in determining of the temperature of start of tr ansformation [4] is played by a temperature of \nparaequilibrium T\n0 where the free energies of α- and γ-phases with the same carbon \nconcentration become equal, () () Tcef Tceft t ,, ,,0 0α γ= . Temperature T0 was introduced in Ref. \n[30] as a pre-condition for the start of bainite tran sformation. In this case, as it assumed in [4,30], \nthe diffusion is slower than the shear transforma tion and therefore there is  no redistribution of \ncarbon between α- and γ-phases during the growth of α-phase plates. The value of A3 and T0 \ncalculated by us agrees well with the experimental quantity exp\n3Aand T0Z (Figure 4).  \nAt last, the martensite transformation is ch aracterized by mechanical instability of γ-phase \nwith carbon, that is, the free energy as a function of tetragonal deformation should have a \nmaximum instead of minimum at the fcc point, ( ) 0 /,,2 2=∂ ∂t t e Tcef . This condition is attained \nby quenching of γ-phase to the temperature MS where ferromagnetic short range order in γ-\nphase becomes important. One can see that the temperature MS found in this way is actually \nlower than the experimental value (see Figure 4). One has to keep in mind, however, that the \nmartensitic transformation observed in steel do not fo llow the scenario of latt ice instability and is \ndeveloped, rather, by heterogeneous nuclea tion and “replication” mechanism discussed \npreviously [5]. Indeed, it was shown in Ref. [5] that above MS a broad temperature range exists \nwhere the transformation is martensite-like but includes nucleation and growth processes. We \nfollow the concept of isothermal martensitic transformation [31–34] and accept the condition of \nmartensite start as kTC fbarrier 0=→αγ where parameter C0=0.04 is chosen by fitting to the experiment \nfor pure Fe [35]. The temperature MS’ determined in this way agrees well with the experiment in \na broad interval of carbon concentration.  \nWith these definitions, the curves A3,T0,TF do not depend on the energy relief along the \nBain path and are determined only by terminal values fcc\nPMg, fcc\nFMg, bcc\nPMg, bcc\nFMg. Contrary, the \nmartensitic curves MS’,MS depend on the energetics at intermediate te. For the concentration \nrange under consideration the magnetic order effects in γ-Fe are negligible, for the temperatures \nabove T~ 400K. Therefore the general shape of the phase diagram (the lines A3 , Tf , T0, M S’) are \ndetermined, first of all, by the evolution of magnetic state in α-Fe. In particular, the α γ→  \ntransition turns out to be possi ble above Curie temperature (bcc\nCT≈1043K) due to the short-range \nferromagnetic order in α-Fe (see also Ref. [5]). The s hort range magnetic order in γ-Fe \nbecomes important at T ≈ 400K, which determines the temperat ure of start of the martensitic \ntransformation MS, developing via the lattice instability. \nThe results presented in  Figure 4 are purely ther modynamic for the lines A3,T0,TF and do \nnot take into account the internal strain produced  by transformation which plays a crucial role in \nphase morphology and transformation kinetics. Due to requirements following from Saint \nVenant compatibility equations, the resulting Ginzburg-Landau functional for the free energy \nshould include different components of the deforma tion tensor as well as their gradients (6). \nBesides, the plastic relaxation of elastic stresse s accompanying the formation of the new phase is  10\nanother important factor which was taken into account in a m odel way (see Appendix). We use \nthe phase-field model formulated earlier for the elemental iron [5], genera lizing it with taking \ninto account diffusive redistri bution of carbon. Therefore, we  describe the transformation \nkinetics by equations for atomic displacements , plus diffusion equation for carbon, using the \nGinzburg-Landau functional (6), se e Methods section for details. \n  \n \n  \nFigure 4.  The left panel shows calculated lines (solid) corresponding to the start of ferrite \ntransformation, paraequilibrium, and the start of martensitic transformation. Ms and Ms’ are temperatures \nstart of lattice instability and martensitic-like transf ormation. Dashed lines show  experimental boundary \nof two-phase region ( A3) [36], experimental paraequilibrium temperature ( T0Z) [37], and experimental \ntemperature of start of martensitic transformation (exp\nsM ) [35]. The right panel shows microstructures \nforming as a result of transformation at various temperatures: T0<T<A 3 (1,2), Ms’<T<T 0 (3,4; 5,6), T<M s’ \n(7,8). The left and right columns at this panel co rrespond to tetragonal strain (black and white are two \northogonal directions of tetra gonal deformation in bcc phase, gr ey shows fcc regions) and carbon \ndistribution (the darker the smaller), respectively.   \n \nIn the right side of Figure 4 we show typical patterns of tetragonal deformation (first \ncolumn) and carbon distribution (second column) obt ained in our phase field simulations for \ndifferent temperatures. The ferrite transforma tion starts as a heterogeneous nucleation of α-\nphase. One can see, indeed, that at the temperature T<A 3 carbon leaves α-phase and this process \ncontrols a formation of the new phase. For this  situation we see arising polygonal particles of \ncarbon-free α-phase surrounded by carbon-r each shell that are real ly typical for ferrite \ntransformation [3].  \n 11\nFor the temperature range MS’<T<T 0 the model demonstrates several possible scenarios. At \nsmall cooling below T0 the transformation develops by the di ffusion mechanism. In this case, due \nto incomplete relaxation of the internal stresses  α-phase has a shape of plate. For deeper cooling \na fast growth of the plate via the shift m echanism is possible and redistribution of carbon \nbetween α- and γ-phases happens only after the plates ar e formed.  Most of carbon atoms sit at \nthe interphases of plates with different orientations. All these features are indeed characteristic of \nearly stage of bainite transformation [2,4]. For a longer exposure time the formation of the \ncementite particles takes place that is beyond a scope of our consideration. Finally, at T<M S’ \nlocal fluctuations initiate martensite transformation which results in a formation of lenticular \ncolony of tweens with carbon homogeneous ly distributed over the sample.  \n \n4. Discussion \nTo conclude, we propose a microscopic model describing, in agreement with experiment, \nthe curves at Fe-C phase diagram separating regi ons of ferrite, paraequilibrium (bainite), and \nmartensite transformations. We were able not only describe these phenomena but, to some \nextent, understand them separating the main f actor, namely, the temperature dependence of \nmagnetic short-range order. The curves of star t of ferrite, paraequili brium (bainite), and \nmartensite transformations are shown in Figure 4,  together with known experimental data. This \nis the main result of our work. Keeping in mi nd that our model is ab-initio based (does not \ncontain fitting parameters except the thresh old value of energy ba rrier for martensitic \ntransformation) one can consider  the agreement as amazingly good. One should stress that this \nagreement is reached for the model where main temperature dependence en ters via the degree of \nshort-range magnetic order. Thus , the closeness of the Curie temperature in bcc iron to the \ntemperature of structural transformation is not acc idental but is related with the essence of phase \ntransformations both in elem ental iron [5] and steel.  \n \nAppendix. Simulation of transformation kinetics  \nRelaxation processes of elastic fields play an essential role in transformation kinetics and \nmorphology of the new phase. The main channel of such a relaxation is a plastic deformation \narising when the stresses exceed the yield stre ss. A consequent description of the plastic \ndeformation requires an essential complication of the model, by adding parameters describing \nthe plastic deformation to the corresponding dyna mic equations [38,39]. Instead, we take into \naccount the plastic deformation in a phenomenological  way. Since the contri bution of the elastic \nstresses to the Ginzburg-Landau functiona l is determined by the coefficients Av,As, we replace \nthe real values of these parameters by some e ffective, temperature dependent values. The scheme \nproposed provides the stress rela xation assuming that the relaxati on processes are faster than \ntypical times of development of the transformati on and that the lattice remains coherent during \nthe whole process. We assume that for ferritic temperatures ( T>T 0) where the transformation \nvelocity is limited by the carbon diffusion the stresses have enough of time to relax completely, \nchoosing therefore 0 ≈ =eff\nseff\nv A A . Contrary, the martensitic tr ansformation occurs with the \nvelocities comparable with the speed of s ound and therefore for T < M S there is no relaxation \nwithin the relevant time interval, therefore veff\nv A A= , seff\ns A A= . The values of the parameters Av, \nAs where chosen as in Ref. [5]. For the temperature range MS < T < T 0 intermediate values of the  12\ncoefficients eff\nvA, eff\nsAwere used (see Table 1, lines I and II). Parameters eff\nvA, eff\nsA for \nmartensite (Figure 4, fragments numbered 7 and 8) were chosen in  such a way that the average \nelastic energy over the sample was equal to the experimental value of the stored energy in martensite, 0.007eV/at [40]. For the other temper ature ranges these parameters were chosen \naccording to the experimentally known values of the stored energies for Widmanstaetten ferrite, \nbainite and martensite [41].  \nThe diffusion coefficient of carbon D is different in \nα- and γ-phases and temperature \ndependent. We use a simple expression () ) 2(2 2φ φγ α γ − − + = D D DD ,   where αD, γD are \nhandbook data [42], for which we use approximations (m2/s): ) / 18530 exp( 105.45T D − ⋅ =−\nγ , \n231061.1 52.09.4 lg X X D−⋅ + − −=α , T X /104= . In particular, at T=1000K the \nratioαD/γD≈300, that is, at th e precipitation of α-phase carbon is expelled into the boundary \nlayer but only weakly diffuse into the bulk of γ-phase.  \nWe do not take into account te mperature-induced lattice fluctu ations. The latter are mostly \nimportant for homogeneous nucleation whereas we  deal with inhomogeneous  nucleation at grain \nboundaries. Indeed, it is known experimentally th at ferrite nucleates preferably at grain \nboundaries and their triple joints. To describe this process we c onsider a region with two triple \njoints of grains and introduce an additional contribution to th e free energy near the grain \nboundary,  \n                     () )( 2 )(2 2 0xP f xfGB GB φ φ − Δ= Δ ,       ()44\n1334)(\nxxxP\nλλ\n+=                         (16) \nwhere x is the distance from the grain boundary (in dimensionless units as described above) in \nthe direction perpendicular to the boundary, 0\nGBfΔ  is the maximal amplitude of the perturbation, \nλis the parameter characterizing the width of the grain boundary.  This means that a near-\nboundary region is favorable for the transformation but its pene tration through the boundary is \nsuppressed by the change of crys tal lattice orientation. Apart from this, we use the local \nperturbation initiating the start of the transformation as ()( )6 01/ )( r f rfloc loc λ φ+ Δ= Δ , where r is \nthe distance from the center of perturbation region. \nThe phase field simulations show that th e ferrite transformation observed in the \ntemperature range T0<T<A 3 is controlled by the diffusion of  carbon and requir es an essential \nstress relaxation; for homogene ous distribution of carbon and wit hout stress relaxation the ferrite \nembryos has no thermodynamic motivation to grow. In this case, we restrict ourselves by the \nconsideration of diffusive kine tics only and calculate the dist ribution of deformations from \nquasistationary equations 0),(=∂∂∑\nj jij\nrtrσ for a given (time-dependent) carbon distribution \n(Figure 4, fragments numbered 1 and 2). Since αD>>γD, a carbon shell is formed around \nprecipitates of α-phase during the tr ansformation.   \nFor the temperatures T<T 0 the transformation can proceed even for homogeneous \ndistribution of carbon. Howeve r, to find the temperature T0 from thermodynamic condition \n),1,( ),0 ,(0 0 T cf T cf = = = φ φ  is not enough since this conditi on does not take into account the \ncontribution of elastic stresses to the free energy. Our simulations show that the stresses shift the  13\nstart of the shear transformation towards lower temperatures T<T 0. The transformation scenario \nis dependent on the degree of overcooling. For highe r temperature when the relaxation is strong \nenough  (the case I in Table 1) the transformation is  developed similar to ferrite one; it is \ncontrolled by redistri bution of carbon but α-phase has a shape of a plate similar to \nWidmanstaetten ferrite [41,43] (Figure 4, fragme nts numbered 3 and 4). For lower temperatures \n(weaker stress relaxation, case II in Table 1), γ-α transformation starts as a shear one and is \ndeveloped with a formation of one or seve ral twinned plates depend on magnitude  0\nlocfΔ. By \nanalogy with bainite transformati on, one should expect  that the plate stops  its growth after \nreaching a critical size due to the loss of coherence at γ-α interface after a plastic deformation; \nfurther evolution is determined by diffusion of ca rbon, up to formation of a new plate. In this \ncase, we perform simulations in two stages. At the first stage ( t<5, dimensionless time was \ndetermined in section 2.3) we solve the fu ll set of equations with real parameters αD,γD; only \nweak redistribution of carbon takes place at  this stage. At the second stage ( t>5) the distribution \nof deformations is frozen and on ly diffusive part of the dynamical  problem is considered. At this \nstage, carbon moves from the bulk of α-plates to the host of γ-phase. (Figure 4, fragments \nnumbered 5 and 6).  \nIn reality, in steel within the temperature range MS’<T<T 0, apart from Widmanstaetten \nferrite, the bainite are observed, with coexiste nce of shear transformation and carbon diffusion, \nas well as a formation of cementite [41]. To simula te the growth of bainite colony one needs to \ninclude cementite in the model and to consider in a more consistent way plastic relaxation. This issue is therefore beyond a scope of our considerat ion. Nevertheless, our model is applicable at \nthe stage of nucleation and pred icts two possible scenarios of the transformation. Depending on \ntemperature, it can follow either shear or diffusive mechanisms. \n  \n T, \nK с L, \nnm 0\nGBfΔ , \neV/at 0\nlocfΔ , \neV/at λ \nveff\nv\nAA \nseff\ns\nAA αD, \nm2/s γD, \nm2/s \nT0<T<A 3 1000  \n \n0.01 \n  \n \n500  \n \n0.01  \n \n0.03  \n \n50 0 0 1.2E-10 4.0E-13 \nMS’<T<T 0 (I) 850 0.005 0.005 1.6E-11 1.5E-14 \nMS’<T<T 0 (II) 800 0.015 0.015 7.1E-12 3.9E-15 \nMS<T<M S’  700 0.050 0.050 1.1E-12 1.4E-16 \nT<M S 350 1 1 3.5E-19 4.6E-28 \nTable 1. The parameters used in the simulations.  \n \nReferences    \n[1] O. Kwon, Nature Materials  6, 713 (2007). \n[2] W. C. Leslie, and E. Hornbogen. Physical metallurgy of steels , in Physical Metallurgy. V. 2.  Ed. by \nCahn R.W., Haasen P., (Elsevier, 1996), pp.1555–1620. \n[3] R. W. K. Honeycombe and H. K. D. H. Bhadeshia,  Steels: Microstructure and Properties \n(Butterworth–Heinemann, Oxford, ed.2, 1995). \n[4] H. K. D. H. Bhadeshia, Bainite in steels  (IOM Communications Ltd, London, 2001), 460 pp. \n[5] I. K. Razumov, Yu. N. Gornostyrev, and M. I. Katsnelson, J. Phys.: Condens. Matter. 25, 135401 \n(2013).  14\n[6] M. Bouville, and R. Ahluwalia,  Phys. Rev. Lett.  97, 055701 (2006). \n[7] S. Kartha, J.A. Krumhansl, J. P. Se thna, and L. K. Wickham,  Phys. Rev. B 52, 803 (1995). \n[8] K. Ø. Rasmussen, T. Lookman, A. Saxena, A. R. Bi shop, R. C. Albers and S. R. Shenoy, Phys. Rev. \nLett. 87, 055704 (2001). \n[9] J. A. Krumhansl, and R. J. Gooding, Phys. Rev. B 39, 3047 (1989). \n[10] T. Oguchi, Progr. Theor. Phys. (Kyoto) 13, 148 (1955). \n[11] F. Kormann,  A. Dick, B. Grabowski et al., Phys. Rev. B 78, 033102 (2008). \n[12] S. V. Okatov, Yu. N. Gornostyrev, A. I. Li chtenstein, and M. I. Katsnelson,  Phys. Rev. B 84, \n214422 (2011)  \n[13] J.M.Ziman. Models of Disorder  (Cambridge University Press, 1979). \n[14] A. G. Khachaturyan, Theory of structural transformations in Solids (Dover Publication Inc., New \nYork, 2008). \n[15] M.Yu.Lavrentiev, D.Nguyen-Manh, S.L.Dudarev, Phys. Rev. B 81, 184202 (2010) \n[16] Q. Chen, and B. Sundman, Journal of Phase Equilibria 22, 631 (2001). \n[17] D. W. Boukhvalov, Yu. N. Gornostyrev, M. I. Katsnelson, and A. I. Lichtenstein, Phys. Rev. Lett. \n99, 247205 (2007). \n[18] J. M. Soler, E. Artacho, and J. D. Gale et al., J.Phys.: Condens. Matter.  14, 2745 ( 2002). \n[19] J. P. Perdew, K. Burke, and M. Ernzerhof,  Phys. Rev. Lett. 77, 3865 (1996). \n[20] O. N. Troullier, and J. L. Martins, Phys. Rev. B  43, 1993,  (1991). \n[21] H. J. Monkhorst, and J. D. Park, Phys. Rev. B  13, 5188,  (1976). \n[22] A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard, Phys. Rev. Lett. 65, 353 (1990).  \n[23] E. Jiang, and E. A. Carter,  Phys. Rev. B 67, 214103 (2003). \n[24] J. A. Lobo,  and G. H. Geiger, Metallurgical Transaction A 7 , 1359  (1976). \n[25] S. R. Shenoy, T. Lookman, A. Saxena, and A. R. Bishop,  Phys. Rev. B 60, R12537 (1999). \n[26] S. M. Allen, and J. W. Cahn,  Acta Metall. 24, 425 (1976). \n[27] Y. Wang, and A.G. Khachaturyan, Acta Mater. 45, 759 (1997). \n[28] S. V. Okatov, A. R. Kuznetsov,  Yu. N. Gornostyrev, V. N. Urtsev, and M. I. Katsnelson,  Phys. Rev. \nB 79, 094111 (2009). \n[29] W. F. Smith, and J.  Hashemi,  Foundations of Materials Science and Engineering  (McGraw-Hill,  \nAllas, USA, ed.4, 2005). \n[30] C. Zener, Trans. Amer. Inst. Min. Met. Eng. 167, 550 (1946). \n[31] G.V. Kurdumov, Doklady AN SSSR  60, N9, 1543 (1948). \n[32] M. Cohen, E. S. Machlin, and V. G. Paranjpe, in Thermodynamics in Physical Metallurgy  (Am. Soc. \nMetals, Cleveland, 1950), 242 pp.  \n[33] C. H. Shih, B. H. Averbach, and M. Cohen, Trans. AIME 203, 183 (1955). \n[34] R. F. Cech, and D. J. Turnbull, Metals 8, 124 (1956). \n[35] C. Liu, Z. Zhao, D. O. Northwood, and Y. Liu,  J. of Mat. Processing Technology 113, 556 (2001). \n[36] E. Scheil, T. Schmidt, and J. Wünning, Arch. Eisenhüttenw, 32, 251 (1961). \n[37] H. I. Aaronson, The Mechanism of Phase Transfo rmations in Crystalline Solids  (The Institute of \nMetals, London, 1969), 270 pp. \n[38] M. Rao, and S. Sengupta, Phys. Rev. Lett.  91, 045502 (2003). \n[39] A. Malik, H. K.Yeddu, G. Amberg, A. Borgenst am, and J. Agren, Materials Science & Engineering \nA 556, 221 (2012). \n[40] J. W. Christian, Proc. Int. Conf. on “Marte nsitic transformations”, ICOMA T 1979, Boston, USA, \n220. \n[41] H. K. D. H. Bhadeshia, Bainite in steels  (IOM Communications Ltd, London, 2001), 460 pp. \n[42] Diffusion In Solids Metals and Alloys. Landolt-Börnnstein, New Series  III/26  (1990). \n[43] H. I. Aaronson, and C. Wells. Trans. AIME  206, 1216 (1956).  "    },    {        "title": "1308.0171v1.Control_of_magnetization_reversal_in_oriented_Strontium_Ferrite_thin_films.pdf",        "content": "Control of magnetization reversal in or iented Strontium Ferrite thin films. \n \nDebangsu Roy* and P S Anil Kumar \nDepartment of Physics, Indian Institute of Science, Bangalore 560012, India \n \nOriented Strontium Ferrite films with the c axis orientation were deposited with varying \noxygen partial pressure on Al 2O3(0001) substrate using PLD technique. The angle \ndependent magnetic hysteresis, remanent coercivity and temperature dependent coercivity had been employed to understand the magnetization reversal of these films. It was found that the Strontium Ferrite thin film grown at lower (higher) oxygen partial pressure shows \nStoner-Wohlfarth type (Kondorsky like) reversal . The relative importance of pinning and \nnucleation processes during magnetization revers al is used to explain the type of the \nmagnetization reversal with different oxygen partial pressure during growth. \n \n \n \n \n \n  \n \n \n   \n \n \n \n*debangsu@physics.iisc.ernet.in \nIn today’s world permanent magnets become ubi quitous in terms of its application in the \nmodern technologies like motors, actuators, sensors, holding devices, ultrahigh density \nrecording etc1. The thin magnetic films with strong perpendicular anisotropy are central \nto the investigation of the different applications. This is due to the reduction of the \ndemagnetizing field present in the system, thus allowing sharp magnetic transition from one direction to the other. The response of a magnetic material in the external magnetic \nfield can be understood if one considers the underlying magnetization reversal process in \nthe system. In this regard the realization of the coercivity mechanism in the \nperpendicularly magnetized thin film system becomes important. The functionality of \ntheses system can be tailored by considering th e relative importance of the nucleation and \npinning, which governs the coercivity mechanism in these systems. In the literature there \nare numerous reports about the investigation of the coercivity mechanism of the several hard magnetic material thin films most of wh ich contains at least one of the rare earth \nelement like Sm, Nd, Pr etc. \n2-5 In this letter we have made an attempt to understand the \nmagnetization reversal and the coercivity mechanism of the perpendicularly magnetized \noriented hexagonal Strontium Ferrite thin film.  \n In the literature, there are numerous articles on the thin films of hexagonal ferrites like \nBarium Ferrite and Strontium Ferrites.\n6, 7-9 Several techniques such as Pulsed Laser \nDeposition (PLD)8, 9, sputtering 10, and plasma spraying 11 are used for the deposition of \nhexagonal ferrite thin films. Out of all these techniques, the PLD has emerged as an \neffective deposition technique for the oriented or epitaxial growth of the oxide hexagonal \nferrite thin film8, 9. In the PLD technique a large number of processing parameters \ndetermine the crystalline quality of the deposited thin film which in turn governs the \ncoercivity mechanism of the studied system. In  this letter we report about the growth of \nthe Strontium Ferrite thin film with a single orientation along c-axis on c-plane alumina \nby PLD and subsequently investigate the magnetization reversal mechanism with \ndifferent oxygen partial pressure during the growth. However there is a lack of \nunderstanding about magnetization reversal mech anism in the oriented Strontium Ferrite \nthin film. It has been found that depending on the oxygen partial pressure, the deposited \nthin films of Strontium Ferrite exhibit different magnetization reversal mechanism and \nthus exhibit different coercivity value. Thus  the coercivity mechanism in these Strontium Ferrite thin films are investigated by the angular dependent and temperature dependent \nmagnetic measurements and analyzed in accordance to the S-W model12, Kondorsky \nmodel13 and micromagnetic model14. The relative importance of the pinning and \nnucleation present in these systems are further discussed in the article along with the \nstructural information analyzed using thin film XRD.  \n The Strontium Ferrite thin film of composition SrFe\n12O19 was deposited on the c-plane \nalumina substrate (0001) using Pulsed Laser Deposition technique in a HV chamber (base \npressure < 2X10-5 mbar). During pulsed laser deposition, the substrate temperature was \nmaintained at 800°C. In this article, we discuss the results of the Strontium Ferrite thin \nfilm which was deposited at two different oxygen partial pressure of 0.1 mbar and 0.4 mbar, retaining the substrate temperature at 800\n°C. The thin film of Strontium Ferrite \ndeposited at 0.1 mbar and 0.4 mbar O 2 partial pressure has been named as SFO-LO and \nSFO-HO respectively. The thin film of Stron tium Ferrite was further annealed in-situ at \n500°C at 500 mbar oxygen pressure for one hour to avoid the oxygen deficiency in the \ndeposited film. It has to be noted that 4000 number of laser pulses with 300 mJ energy \nhaving fluence 2.5 J/cm2 was used to have a thickness of ~ 20 nm for the Strontium \nFerrite thin film.  The crystallographic structure, growth and the quality of the thin \nStrontium Ferrite films deposited at two different  oxygen partial pressure are analyzed by \nthin film x-ray diffraction and ω-scan about a particular Strontium Ferrite peak using \nBruker D8 discover with Cu-K α source. The magnetization measurement is performed at \nLakeshore 7404 VSM with angular variation set up and PPMS set up by Quantum Design \nwhich operates in field upto 14T. \nWe have used X-ray diffraction studies to understand the structural orientation of the \ndeposited Strontium Ferrite on the (0001) or iented alumina. The alumina is having \ntrigonal crystal structure with the lattice parameter of a= 4.754 Å and c= 12.99 Å. The lattice parameters for the Strontium Ferrite are a= 5.882 Å and c= 23.023 Å. Figure 1 \nshows the typical θ-2θ XRD scan of SFO-HO and SFO-LO thin film (SrFe\n12O19 (20 nm)/ \nAl2O3). It is evident from the figure 1 that the X-ray diffraction study of SFO-HO and \nSFO-LO thin film indicates the presence of the strong (000 l) reflection of Strontium \nFerrite indicating c-axis oriented growth of the Strontium Ferrite on c-plane alumina \nsubstrate. This is in accordance to the fact that the hcp oxygen plane of the Strontium \nFerrite aligns preferentially along the hcp oxygen plane of the alumina.  \nThe peak which was marked as (**) corresponds to the peak intrinsic to the instrument. \nThus the X-ray diffraction pattern for the Str ontium Ferrite thin film on c-plane alumina \nsubstrate confirms the oriented grow th of the Strontium Ferrite along (000 l) direction. In \norder to understand the relative importance of the mosaicity and the crystalline quality of \nSFO-HO and SFO-LO, we have undertaken XRD ω-scan about the Strontium Ferrite \n(00014) reflection as shown in the inset of the figure 1. Generally the mosaicity of a \ncrystal is the width of the distribution of the mis-orientation angle of the crystal plane in \nthe crystal and is directly related to the crystalline quality. The presence of the mosaic \ndefects in the crystal corresponds to the br oadening of the rocking curve. Thus the \nestimation of the width i.e. the FWHM of the rocking curve for both SFO-HO and SFO-LO is a measure of the crystalline quality of the thin film. The inset of the figure 1 shows \nthe Gaussian fitting of the rocking curve for both SFO-HO and SFO-LO and it was found \nthat the peak position remains same in both the cases. The FWHM for SFO-HO and SFO-\nLO can be estimated as 0.231\n0 and 0.140. The FWHM for the c-plane alumina obtained \nfrom the rocking curve about (0006) reflection of the substrate can be calculated as \n0.0143 0. Thus the lesser value of FWHM for the SFO-LO compared to SFO-HO \ncorresponds to the better crystallinity in SFO-LO compared to SFO-HO. \n The figure 2 shows the magnetic hysteresis loop for SFO-LO corresponding to the \nmagnetic field applied at different angle Ф, with respect to the film normal direction \n ((0001) direction of the c-plane Al\n2O3 substrate). All the magnetic measurements were \ndone in a Lakeshore 7404 VSM. The image inside the figure 2 shows the angle Ф \nbetween the applied field and the film normal. In addition, the angular dependence of the \nmagnetization of SFO-HO exhibits corres pondence to the magnetization behaviour of \nSFO-LO with respect to the (0001) direction of  the c-plane alumina substrate at different \nangles. (Supplement 1). \n It has been found that for both the film, the coercivity value decreases as the angle \nbetween the applied magnetic field and the film normal increases from 0\n0 to 900. From the \nmagnetization measurement for both the thin film, we can conclude that both the film \nexhibit perpendicular anisotropy corresponding to the Ф =00 configuration. The \nmagnetization behaviour at Ф= 900 for both the thin film shows a characteristics of the \nmagnetic hysteresis loop along the hard axis . Thus the deposited Strontium Ferrite on c-plane alumina, is having the easy axis of ma gnetization along the c axis of the Strontium \nFerrite and the corresponding hard axis lies in the film plane. The inset of the figure 2 \nillustrates the variation of the coercivity of SFO-HO and SFO-LO with different angle \nbetween applied field and (0001) direction of the substrate. It has also been found from \nthe inset of the figure 2 that the coercivity for SFO-HO and SFO-LO are ~2600 Oe and \n~3820 Oe respectively along the easy direction of the magnetization.  The coercivity of the SFO-LO is always higher compared to SFO-HO for all the angles Ф. Generally in the \nthin film of hard magnetic material, domain  forms with the application of the reverse \nmagnetic field\n1. Subsequently the movement of the domain walls determine the coercivity \nof the system. However the motion of this domain wall is affected by the anisotropy of \nthe system, density of pinning state, surface asperities etc. Thus for better understanding of the coercivity mechanism in SFO-HO and SFO-LO, all the above mentioned parameter \nshould be taken into account. \n \nIn order to understand the role of the defect  site in the coercivity mechanism for SFO-HO \nand SFO-LO, we have obtained DC demagnetization remanence for the film at various \nangles between the applied field and the ea sy direction of the magnetization using \nLakeshore 7404 VSM. From the corresponding DC demagnetization remanence curve we \nhave calculated the corresponding remanent coercivity H\ncr .The corresponding field value \nfor which the remanent magnetization becomes zero can be defined as the remanent \ncoercive field. This depends only on the irreversible part of the magnetization and \nindependent of the magneto-static interaction in the system as the net magnetization is \nzero at this point on the reversal curve. Us ing this procedure, we have calculated the \ncorresponding remanent coercive field value for both SFO-HO and SFO-LO at various \nangles Ф. Generally the simplest way to explain the angular dependence of the remanent \ncoercivity is attributed to Kondorsky model13 and the same can be expressed \nas() 1(0) coscr\ncrH\nHΦ=Φ. Here Ф is the angle between the film normal and the \napplied field as denoted by the diagram in the figure 2. Here H cr(0) corresponds to the \nremanent coercivity along the easy direction of the magnetization and the expression for \nthe Kondorsky model has been normalized using the value of H cr(0). According to this \nmodel, the magnetization reversal occurs through the domain wall formation and its \nsubsequent motion. The physical picture for th is model can be visualized to the situation \nwhen an external magnetic field is applied to the thin film at an arbitrary angle apart from the easy direction of the magnetization. In this scenario, if the applied magnetic field is \ninsufficient to rotate the magnetization of the film from the easy axis direction, then the \ncomponent of the applied magnetic field along the easy direction of the magnetization can \npush the domain wall from one meta-stable en ergy minimum to the other. According to \nthe Kondorsky model the obtained H cr should always be lesser than the corresponding \nanisotropy field of the system. The Kondorsky model is well applicable for the switching of the longitudinal recording media.\n1 \n \nAnother model which explains the variation of the remanent coercivity is the Stoner \nWohlfarth (S-W) Model12. This model considers the coherent rotation as the process for \nmagnetization reversal for the uni-axial, singl e domain non-interacting particles. The S-W \nswitching field can be expressed as\n()2/3 2/33/2() 1(0)cos sinsw\nswH\nHΦ=\nΦ+ Φ, here \n(0)swH is equal to the anisotropy field of the system. In the present context, we have \nconsidered that the remanent switching field is as same as remanent coercivity of the \nsystem. Thus in this model the remanent coercivity becomes maximum when the applied \nmagnetic field lies either parallel to the easy direction of the magnetization or to the hard \ndirection. The remanent coercivity according this model is found to be minimum at the \nangle Ф= 450. \n \nFigures 3(a) and 3(b) illustrate the variati on of the remanent coercivity for SFO-LO and \nSFO-HO. The variation of the simulated H cr/Hcr(0)  of the S-W model and the Kondorsky \nmodel with the different angle Ф has also been shown in the figure 3(a) and 3(b) \nrespectively. It has been found from the figure 3(a) that for SFO-LO, the angular dependence of the reversal resembles the functional form of the S-W model. It has also \nbeen found that the minimum of the normalized remanent coercivity occurs at 45\n0 for \nSFO-LO which is identical to the angle at which switching field becomes minimum for \nthe S-W model. This suggests that the magnetization reversal is coherent in nature for the \nhard SFO-LO. However, we have found that the magnitudes of the remanent coercivity for SFO-LO differs from the value predicte d from the S-W model for different angle Ф \nw.r.t. film normal. But the nature of the variation is same in both the cases. In addition, \nthe depth of the minimum of the normalized remanent coercive field is less compared to \nthe depth of the normalized remanent coercive field at an angle 45\n0 as predicted by the S-W model. This deviation from the ideal S-W model for the 20 nm thick Strontium Ferrite \nfilm can be explained by considering the ex istence of the large grain of the Strontium \nFerrite. The large grain will introduce the domain wall inside the grain and thus the \ncoherence of the magnetization in SFO-LO ge ts reduced. However, we also have to \nconsider the presence of the defects for the understanding of the magnetization reversal in \nSFO-LO. Thus the apparent contradiction of the magnetization reversal in the hard magnetic SFO-LO which shows S-W like magnetization reversal behaviour can be \nexplained according to a two step process of magnetization switching.\n15 The sanity of this \nmodel is further verified from the other studies that will be discussed in the later part of \nthis article. According to this process the reverse domain forms by nucleation and \nsubsequently these domains propagate. So the relative importance of the nucleation field and the domain wall propagation field controls the switching behaviour in the system. \nGenerally the existence of the defects in the system controls the domain wall propagation \nfield as these defects acts as a pinning centre. So when the nucleation field value is higher \ncompared to the domain wall propagation fiel d value, magnetization does not flip until \nthe applied field overcomes the nucleation field value. As soon as the domain gets \nnucleated, the domain wall forms and propaga tes resulting in the variation of the \nremanent coercive field with the angle Ф resembling S-W like switching. Thus in SFO-\nLO, the magnetization switching mechanism is nucleation dominated and the field at which the nucleation occurs is higher than the corresponding domain wall propagation \nfield.  \n \nIt has been found that the angular variation of the normalized remanent coercivity for SFO-HO exhibit Kondorsky type reversal. But after an angle Ф> 30\n0, the angular \ndependence of the normalized remanent coerci ve field value deviates from the remanent \ncoercive field value obtained from the simulated Kondorsky model. This deviation can \nalso be explained by considering the tw o step magnetization reversal process15 which has \nbeen used for the understanding of the magn etization switching in SFO-LO. In this \nsystem if the nucleation field is lesser than the domain wall propagation field, the \nmagnetization of the system undergoes nucleati on at the beginning. But the magnetization \ndoes not flip with the application of the reve rse field, until the energy corresponding to \nthe domain wall propagation field is overcome.  In this case the magnetization reversal is \ncontrolled by the Kondorsky model. The highe r concentration of the defects present in \nSFO-HO compared to the SFO-LO which is evident from the inset of the figure 1 corresponds to the higher domain wall propaga tion field compared to the nucleation field \nin SFO-HO. Thus in this system the magnetization reversal is primarily controlled by the \ndomain wall motion but the rotation mode owing to the nucleation also exists. This two \nstep model also corroborates well to the fact  that the coercivity is higher in SFO-LO \ncompared to SFO-HO as the nucleation field value is higher in SFO-LO compared to the \ndomain wall propagation field in SFO-HO. This result in a lesser coercivity in SFO-HO compared to SFO-LO. In the present context of the coercivity mechanism, we have \nconsidered negligible effect of the thermally activated magnetization reversal process \nwhich is relevant in some particular system. \n \nIn order to understand the quantitative import ance of the magneto-static interaction owing \nto the size of the defects and the respective pinning or nucleation strength which together \ndetermines the coercivity of the system, a micromagnetic model has been used\n14. \nAccording to this model, the coercivity of the system can be expressed \nas () ()cn s eff eff H HT NMT =α − .16 Here the coercivity cHis described as the reduction \nof the nucleation field nHowing to the magneto-static interactions effNM . Theoretically \nthe nucleation field can be expressed as the field for which the magnetization reversal \nwould take place for the SW particle and it is  an intrinsic property of the material. But in \nreality, the magnetization reversal is affected by the misalignment of the grains, surface \ndefects, stray field (magneto-s tatic interaction), structural inhomogenities etc. Generally \nthe defects are the source for the stray field as the magnetic induction creates a strong local demagnetizing field in the vicinity of these defects. These defect can acts as a \nnucleation or pinning centre depending on thei r respective size and thus become a major \nparameter for deciding the coercivity of the system. These extra effects have been \nincluded in the expression of coercivity as an extra parameter \neffαand effN which are \ntemperature independent. The parameter effα stands for the microstructure related \ndependence factor and can further be split as k eff Φ =α αα where the corresponding \nparameter kαis Kronmuller parameter 16-18 which is related to the structure of the sample. \nThe parameter kα is also independent of the temperature in the present context as the \nsize of the microstructure of the deposited hard Strontium Ferrite does not change with \nthe temperature. The other parameter Φαdepends on the easy axis misorientation. The demagnetizing parameter effNdepends on the distribution of the grain shape in the \nmaterial, structural defect owing to the growth of the thin film etc. \nTo estimate the parameter effNand the effα , we have done hysteresis measurement for \nSFO-HO and SFO-LO at different temperatures starting from 300K to 5K with an interval \nof 25K at Quantum Design PPMS. During this measurement, the magnetic field was \nalways applied along the c-axis of the Strontium Ferrite i.e. along the easy axis of \nmagnetization. It has been found that, the shap e of the hysteresis does not change as the \ntemperature has been varied from 300K to down 5K, apart from the individual change in \nsaturation magnetization and coercivity for SFO-HO and SFO-LO. The maximum applied field is 14T and the saturation magnetization has been measured at 14T after eliminating \nthe diamagnetic contribution of the substrate. It has to be noted that during this \nmeasurement configuration when the magnetic field is applied along the easy direction of \nthe magnetization of SFO-HO and SFO-LO, the magnitude of the parameter \nΦαcan be \nconsidered as unity.  \n According to literature \n19, the nucleation field in bulk Strontium Ferrite crystal can be \ndetermined by the first anisotropy constant K 1 and the nucleation field can be expressed \nas 12\nnsKHM= . So after incorporating the value of the nucleation field in the \nmicromagnetic equation, the resultant equation can be written as                       \n21 ()\n() ()2c\nKs\ns SeffHTMMT TKNM=α −               (1)  \nFigure 4(a) and 4(b) shows the variation of the ()\n()c\nsHT\nMT vs.21\n()2\nSTK\nM for SFO-LO \nand SFO-HO. It has to be noted that, we have  considered the first anisotropy constant \nvalue of 3.5×106 erg/cc for the calculation of the nucleation field. We have done the \nlinear fitting of the variation of the ()\n()c\nsHT\nMTvs. 21\n()2\nSTK\nM using the equation (1). \nThe obtained fitting parameter are αk= 0.103±0.009, N eff = 1.753±0.258 and αk= \n0.147±0.005, N eff = 2.904±0.219 for SFO-HO and SFO-LO respectively. Generally the \ndemagnetization parameter N eff can attain values between 0 and 1 for the homogeneously \nmagnetized sample. But for both the sample SFO-HO and SFO-LO, we have obtained the \nvalue of the N eff higher than 1. This corresponds to the presence of the enhanced stray field, which is higher than the saturation magnetization of the magnetic phase. The \ncorresponding higher value of the N eff i.e. the measure of the stray field for SFO-LO \ncompared to SFO-HO indicates that the defect size is higher in SFO-LO compared to \nSFO-HO. During the deposition of the Strontium Ferrite thin film on c-plane alumina \nsubstrate, the growth mode becomes important from structural as well as magnetic point \nof view. Generally during pulsed laser deposition of the oxide thin film, the layer by layer growth is desirable as only then one can obtain atomically smooth surface and superior \nphysical property. Generally the high temper ature of the substrate along with the low \noxygen pressure during deposition, enhance the mobility of the adatoms on the substrate \nthus increasing the possibility of the 2D layer by layer growth. As a result the probability \nof nucleation of the next layer on the top of 2D island becomes minimum and the growth of the next layer is only possible after the completion of the previous layer. However, the \nhigher oxygen pressure during deposition leads to a multilevel growth.  In the present \ncontext of SFO-HO and SFO-LO, one would expect multilevel growth mode for SFO-HO \nleading to higher structural defects and pinning sites. But for SFO-LO the concentration \nof the defect is less as evident from the rocking curve measurement in the inset of the \nfigure 1. It can also be concluded that the size of the defect is much larger than the \ndomain wall width of the system; otherwise SFO-LO would exhibit a pinning dominated \nmagnetization reversal process. Thus these defects in SFO-LO acts as a nucleation centre whereas the defects present in SFO-HO pinned the magnetization in the system. \nAccording to the literature the critical value of 0.35 \n18 of the microstructure parameter αk \ncorresponds to the nucleation dom inated magnetization reversal. The present value of the \nαk confirms that both the pinning and nucle ation coexist for both SFO-HO and SFO-LO. \nHowever, the lesser value of  αk for SFO-HO than SFO-LO indicates that the pinning is \nstronger in SFO-HO in comparison with SFO-LO. This is in well agreement to the \nprevious findings regarding the quantitative estimation of the magneto-static interaction. \n \nWe have presented a detailed structural analysis using thin film XRD, angle dependent \nmagnetic hysteresis and remanent coercivity measurement and coercivity mechanism by \nmicromagnetic analysis for SFO-HO and SFO-LO. We have initially deposited SFO-LO and SFO-HO on (0001) alumina, with different oxygen pressure of 0.1 mbar and 0.4 mbar \nkeeping the substrate temperature at 800\n°C. The rocking curve analysis of the oriented \nSFO-HO and SFO-LO reveals the better crystallinity in SFO-LO compared to SFO-HO. The angular dependence of the coercivity for both the SFO-HO and SFO-LO indicates \nthat the deposited films are having easy axis of magnetization along the growth direction \ni.e. along the film normal direction. Correspondi ngly it was found that the hard axis lies \nin the film plane. We have found that there is a significant variation of the coercivity \nbetween SFO-HO and SFO-LO. We have made an attempt to understand the factors \ngoverning the coercivity mechanism in these films by considering the pinning and nucleation centres. It has been found that for SFO-HO, the coercivity mechanism follows \na pinning dominated process where the field corresponding to the domain wall \npropagation is higher than that of the nucleation. But for SFO-LO, the coercivity \nmechanism is governed by nucleation of the domains and corresponding motion of the \ndomain walls. The micromagnetic analysis of the coercivity in SFO-HO and SFO-LO confirms that the size of the defects is  bigger in SFO-LO compared to SFO-HO. \nHowever, it has also been confirmed that the density of the defects are more in SFO-HO \ncompared to SFO-LO. The quantitative information regarding the strength of the \nmagneto-static interaction for SFO-HO and SFO-LO reveals that, the strength of the \nnucleation field in SFO-LO is more compared to the strength of the domain wall \npropagation field which governs the coercivity in the respective systems.  \n \nReferences: \n1 J. M. D. Coey, Magnetism and Magnetic Materials  (Cambridge University Press, \n2010). \n2 A. Singh, V. Neu, S. Fähler, K. Nenkov, L. Schultz, and B. Holzapfel, Physical Review B 79, 214401 (2009). \n3 M. Seifert, V. Neu, and L. Schultz, Applied Physics Letters 94, 022501 (2009). \n4 A. Singh, V. Neu, S. Fähler, K. Nenkov, L. Schultz, and B. Holzapfel, Physical \nReview B 77, 104443 (2008). \n5 S. L. Chen, W. Liu, and Z. D. Zhang, Physical Review B 72, 224419 (2005). \n6 H. J. Masterson, J. G. Lunney, J. M. D. Coey, R. Atkinson, I. W. Salter, and P. \nPapakonstantinou, Journal of Applied Physics 73, 3917 (1993). \n7 P. Papakonstantinou, M. O'Neill, R. Atkins on, I. W. Salter, and R. Gerber, Journal \nof Magnetism and Magnetic Materials 152, 401 (1996). \n8 M. E. Koleva, et al., Applied Surface Science 168, 108 (2000). \n9 M. E. Koleva, R. I. Tomov, P. A. Atanasov, C. G. Ghelev, O. I. Vankov, N. I. \nMihailov, J. Lancok, and M. Je linek, Applied Surface Science 186, 463 (2002). 10 H. L. Glass and J. H. W. Liaw, Journal of Applied Physics 49, 1578 (1978). \n11 T. Okuda, N. Koshizuka, K. Hayashi, T. Takahashi, H. Kotani, and H. Yamamoto, \nMagnetics, IEEE Transactions on 23, 3491 (1987). \n12 E. C. Stoner and E. P. Wohlfarth, Philosophical Transactions of the Royal Society \nof London. Series A, Mathematical and Physical Sciences 240, 599 (1948). \n13 E. Kondorsky, in . Phys. (Moscow) II , 1940), Vol. 2, p. 161. \n14 W. F. Brown, Jr., Reviews of Modern Physics 17, 15 (1945). \n15 G. Hu, T. Thomson, C. T. Rettner, and B. D. Terris, Magnetics, IEEE \nTransactions on 41, 3589 (2005). \n16 H. Kronmüller, K. D. Durst, and G. Martinek, Journal of Magnetism and \nMagnetic Materials 69, 149 (1987). \n17 H. Kronmüller, physica status solidi (b) 130, 197 (1985). \n18 H. Kronmüller, K. D. Durst, and M. Sagawa, Journal of Magnetism and Magnetic \nMaterials 74, 291 (1988). \n19 B. T. Shirk and W. R. Buessem, Journal of Applied Physics 40, 1294 (1969).  \nList of Figures: Figure 1:  θ-2θ XRD pattern for SrFe\n12O19 (20 nm) film grown on Al 2O3 (0001) with \ndifferent oxygen concentration. The red and black coloured graph denotes the SrFe 12O19 \nthin film deposited at 0.4 and 0.1 mbar O 2 pressure. The (**) peak corresponds to the \nintrinsic peak of the instrument. (inset) Rocking curve and the corresponding Gaussian \nfitting for sample SFO-HO and SFO-LO.  \nFigure 2:  Magnetization vs. Applied Field for SFO-LO at various applied field angle \nwith respect to the (0001) direction of the c- plane alumina substrate. (inset):Variation of \nthe coercivity of SFO-HO and SFO-LO with angle between applied magnetic field and film normal. \nFigure 3(a):  Variation of the normalized remanent coercivity for SFO-LO with the angle \nФ. (b): Variation of the normalized remanent coercivity for SFO-HO with the angle Ф. \nFigure 4 (a):  Variation of the ()\n()c\nsHT\nMTvs. 21\n()2\nSTK\nMfor SFO-LO. The dotted line \nshows the corresponding linear fit according to the eqn(1). (b): Variation of the \n()\n()c\nsHT\nMTvs. 21\n()2\nSTK\nMfor SFO-HO. The dotted line shows the corresponding \nlinear fit according to the eqn(1). 15 20 25 30 35 40 45 50 55 60 65 70110100100010000100000\n27.75 28.00 28.25 28.50 28.7501903805707609501140 SFO-HO\n SFO-LO\n ω (degree)Intensity (arbitrary unit)\nkβ of Al2O3 ** 000100004\n0006\n0008\n0001600014Al2O3 (0006)Intensity (arbitrary unit) SFO-HO\n SFO-LO\n2θ (degree)\n\nFigure 1 -18000 -12000 -6000 0 6000 12000 18000-1.8x10-4-1.2x10-4-6.1x10-50.06.1x10-51.2x10-41.8x10-4M (emu)\nApplied Field in OeSFO-LO\nΦ = 300-18000 -12000 -6000 0 6000 12000 18000-2.2x10-4-1.4x10-4-7.2x10-50.07.2x10-51.4x10-42.2x10-4M (emu)\nApplied Field in OeSFO-LO\nΦ = 00\n-18000 -12000 -6000 0 6000 12000 18000-1.5x10-4-1.0x10-4-5.0x10-50.05.0x10-51.0x10-41.5x10-4M (emu)\nApplied Field in OeSFO-LO\nΦ = 150\n-18000 -12000 -6000 0 6000 12000 18000-2.6x10-4-1.7x10-4-8.6x10-50.08.6x10-51.7x10-42.6x10-4M (emu)\nApplied Field in OeSFO-LO\nΦ = 450\n-18000 -12000 -6000 0 6000 12000 18000-1.3x10-4-8.8x10-5-4.4x10-50.04.4x10-58.8x10-51.3x10-4M (emu)\nApplied Field in OeSFO-LO\n600degree-18000 -12000 -6000 0 6000 12000 18000-2.0x10-4-1.3x10-4-6.7x10-50.06.7x10-51.3x10-42.0x10-4M(emu)\nApplied Field in OeSFO-LO\n75 0degree-18000 -12000 -6000 0 6000 12000 18000-1.6x10-4-1.1x10-4-5.3x10-50.05.3x10-51.1x10-41.6x10-4M(emu)\nApplied Field in OeSFO-LO\n90 0degree\n0 1 53 04 56 07 59 0056011201680224028003360\n|| SFO-HO\n SFO-LOCoercivity in Oe\nAngle (Φ) w.r.t Film normal ⊥\n \nFigure 2 0 1 32 63 95 26 57 80.9461.0321.1181.2041.2901.3761.462\n⊥SFO-HOHcr/Hcr(00)\nAngle (Φ) w.r.t Film normal0.521.041.562.082.603.123.64Simulated Kondorsky Model\nHcr/Hcr(00)\n(b)0 1 32 63 95 26 57 80.6400.7040.7680.8320.8960.9601.024\n⊥ SFO-LO\nSimulated   S-W Model Hcr/Hcr(00)\nAngle (Φ) w.r.t Film normal(a)\n \nFigure 3 \n 20.8 23.4 26.0 28.6 31.2 33.8 36.43.904.164.424.684.945.205.46\nαΚ= 0.103 ± 0.009\nΝeff = 1.753 ± 0.258HC(T)/MS(T)\n2K1/M2\ns(T)SFO-HO\n Linear Fitting of the eqn(1)\n(b)19.0 28.5 38.0 47.5 57.0 66.5 76.05.26.57.89.110.411.713.0 SFO-LO\n Linear Fitting of the eqn(1)HC(T)/MS(T)\n2K1/M2\ns(T)αΚ= 0.147 ± 0.005\nΝeff = 2.904 ± 0.219(a)\nFigure 4 "    },    {        "title": "1605.07508v2.Autocatalytic_mechanism_of_pearlite_transformation_in_steel.pdf",        "content": " 1\nAutocatalytic mechanism of pear lite transformation in steel \nI.K. Razumov1,2,*, Yu.N. Gornostyrev1,2,4, and M.I. Katsnelson3,4 \n \n1Institute of Metal Physics, UB of RAS, 18 S.  Kovalevskaya st., Ekaterinburg 620990, Russia \n2Institute of Quantum Materials Science, 5 Konstruktorov st., Ekaterinburg 620072, Russia \n3Radboud University, Institute for Molecules and Material s, Heyendaalseweg 135, Nijmegen, 6525 AJ, Netherlands \n4Ural Federal University, Department of  Theoretical Physics and Applied Mathem atics, 19 Mira st., Ekaterinburg \n620002, Russia \n \nA model of pearlite colony formation in carbon steels with ab-initio parameterization is proposed. The model describes the \nprocess of decomposition of austenite and cementite formation th rough a metastable intermediate structure by taking into accoun t \nthe increase of the magnetic order under the cooling. Autocatalytic mechanism of pearl ite colony formation and the conditions f or \nits implementation have been analyzed. We demonstrate that p earlite with lamellar structure is formed by autocatalytic \nmechanism when thermodynamic equilibrium between the ini tial phase (austenite) and the products of its decomposition \n(cementite and ferrite) does not take place. By using model expression for free energy with first-principles parameterization w e \nfind conditions of formation of both lamellar  and globular structures, in agreement w ith experiment. The transformation diagram  \nis suggested and different scenarios in the kinetics of decomposition are investigated by phase field simulations.  \n \nPACS: 64.60.-i, 64.60.My, 75.50.Bb \nI. INTRODUCTION \nPearlite is one of the main structural units of carbon \nsteels which have a significant effect on their properties \n[1,2]. It is formed by decomposition of austenite ( γ, fcc Fe-\nC solid solution) into ferrite ( α-phase, bcc Fe) and \ncementite (orthorhombic θ-phase, Fe 3C) during slow \ncooling or annealing at temperature 7200 – 5000C. The \nbrightest feature of pearlite is  a rather regular lamellar \nstructure in which α and θ phases are regularly alternated. \nPearlite transformation (PT) is an example of eutectoid \ndecomposition which was observed also in many non-ferrous \nalloys [3–5] below some criti cal (eutectoid) temperature. \nDespite numerous studies of PT motivated by its great \npractical importance for meta llurgy, the mechanism of \nformation of the regular lamell ar structure remains unclear.  \nThe proposed theoretical models of PT are focused \nmostly on the stage of steady-st ate growth of the pearlite \ncolony and on the problem of stability of the transformation front [6–13]. At the same time, the problems with early \nstages of the colony forma tion such as nucleation of \ncementite remain out of scope of the proposed models. Besides, the mechanism of lamellae multiplication by \nreplication [1,14] or splitting [15], which plays an important \nrole in PT is still under discu ssion. Also, moving factors of \nthe transition from lamellar to globular pearlite structure \nwith increasing temperature is currently not well understood \n[16–20].  \nThere is a certain similarity between PT and other \ndiffusion phase transformations which result in the formation \nof the lamellar structure. One of them is eutectic colony \ngrowth which appears behind the front of solidification and \nis driving by temperature gradient [21,22]. This transformation is determined by fast diffusion at \nsolidification front and/or decomposition of some \nintermediate states [23,24]. Another example is spinodal decomposition driven by the moving grain boundary (GB) in \nsystem with negative mixing energy, \nv<0 [25]. The stability \nof transformation front is guaranteed automatically in this \ncase and lamellar structure formation is controlled by the \nredistribution of alloying elements along GB. These \nobservations point out an impo rtance of the acceleration of \ndiffusion at the transformation front as was discussed in \nrelation to the PT problem (see Refs. [6,8,11]). Note, though \nredistribution of carbon plays a decisive role in the PT the \naustenite remains stable with respect to the carbon \ndecomposition, 0>γv  [26].  \nDespite a great practical importance of PT and longtime \ninterest of researchers the mechanisms of this transformation \nare still poorly understood. In this work we demonstrate that the formation of lamellar structure is a natural part of \nscenario of PT when the free energy of the system has a \nspecial form in which thermodynamic equilibrium between \nparent \nγ-phase and both transformation products ( α and \nθ) is impossible. In this case the pearlite colony can emerge \nby some kind of autocataly tic mechanism when appearance \none of the phases ( α or θ) stimulates the nucleation of the \nother one.  \nWe employ the previously proposed model of phase \ntransformation in iron and steel [27,28] and generalize it taking into account the cementite formation. Following Ref. \n[29] we also assume that Metastable Intermediate Structure \n(MIS) exists at \nαγ/  interface due to magnetization \ninduced by an adjacent ferr ite plate and nucleation of \ncementite occurs as result of MIS →θ lattice reconstruction \nwhen MIS is saturated by carbon. Thus, according to the \nscenario developing here the PT at undercooling is primarily  2\ninduced by arising magnetic order in α-phase. It is \nworthwhile to note that the formation of MIS is closely \nconnected to the known fact (see Refs. [30,31]) that the \nground state of ferromagnetically ordered γ-Fe has a strong \ntetragonal distortion.  \nII. METHODS \nA. Effective free energy functional \nHere we generalize the previously proposed model \n[27,28] of γ – α transformation by taking into account the \ncementite formation. In this approach all relevant degrees of freedom (lattice and magnetic) as well as the carbon \ndiffusion redistribution during \nγ – α and γ –θ phase \ntransformations should be included in consideration. We \nassume that the nucleation of cementite occurs at the γ/α \ninterface and γ–θ lattice reconstruction follows the \ntransformation path, which includes the formation of the \nmetastable intermediate structure (MIS) [29].  \nThe pearlite formation is controlled by the carbon \ndiffusion [1,2] which is slow process in contrast with γ–\nαand γ– MIS lattice reconstruction carried out by the fast \ncooperative displacements of Fe atoms. Therefore, we \nassume that the variables describing the lattice reconstruction \ntake quickly their equilibrium values and the local carbon \nconcentration )(rc remains a single variable which \ndetermines the slow evolution.  \nSince α and θ phases in pearlite colonies are usually \nconjugated with small mismatch, whereas the lattice \ncoherency is lost on the transformation front [32], we neglect \nthe elastic energy contribution within the simple model \nunder consideration. Thus, after excluding the fast variables the effective free energy functional can be written in form \n[33]: \n() ∫ ⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛∇ + = rdckTcf Fc\neff2\n2),(                         (1) \nwhere ),(Tcfeff  is effective free energy density for a \nhomogeneous state: \n{ }),( ),,( ),,( min),( TcfTcfTcf Tcfeff θ γ α =       (2) \nand ()Tc f ,),(θαγ  is local density of free energy of austenite \n(ferrite, cementite) at a carbon concentration с and \ntemperature T. This means, the phas e with lowest energy \nwith a fixed value of local carbon concentration is quickly \nrealized at a given point in space. A similar approach for \npearlite free energy was previously used in [11]. To determine the energies ()Tc f ,)(αγ  we use the earlier \nproposed model [28] which takes into account both lattice \nand magnetic degrees of freedom. According to this model: \n) (~),~( ),(0~\n0γ γ γγ γγ\nS sT JdTJQ g TcfJ\nPM + −′ ′ − =∫          (3) \nα α αα αα\nTS JdTJQ g TcfJ\nPM −′ ′ − =∫~\n0~),~( ),(                    (4) \nwhere s0 is the high-temperature limit of the entropy \ndifference between the phases including phonon \ncontribution, )(γαS  is configurational entropy of carbon in \n)(γα  phase; 2\n1 0 / )( m TQ > ⋅ ≡< m m  is the spin correlation \nfunction dependent on temperature according to Oguchi \nmodel [34], )( )( )(~ )( )(\n)( c gc g c JFM PMαγ αγ\nαγ − =  is exchange energy,  \n    \n2/~)(2/~)(\n2\n) ( ) ( ) (2\n) ( ) ( ) (\ncvc g c gcvc g c g\nFM PM FM PM FM PMFM PM FM PM FM PM\nα α α αγ γ γ γ\nεε\n+ + =+ + =\n,    (5) \n)(\n) (~αγ\nFM PMg  are energies of para (ferro)magnetic pure Fe found \nfrom the fitting to ab initio computational results [31,35]; \n)(\n) (αγεFM PM and )(αγv  are solution and mixing energies of \ncarbon in fcc (bcc) lattice. During PT, the carbon \nconcentration becomes rather high and reaches the value \nс=0.25. Therefore, the contribution proportional c2 which \ncharacterizing carbo n-carbon interaction was taken into \naccount in (5). Note that, w ithin this model, a strong \ntemperature dependence of the free energy of α-Fe \noriginates rather from the increase of degree of \nferromagnetic order during the cooling than from phonon \nentropy. \nWe use the traditional lattice-gas model to describe \nstatistical entropy of carbon randomly distributed over \ninterstitials of α-Fe. It is a rather good approximation due to \nvery low solubility limit of carbon in α phase, so correlation \neffects can be neglected.  On the other hand, solubility of  \ncarbon in γ phase is much higher. As in the previous work \n[28], we assume that carbon atoms may occupy only a part \nof interstitial positions in γ phase. Following Refs. [36,37], \nwe accounted for the repulsive interactions between nearest \nneighbor’s carbon atoms by excluding the part interstitial \npositions in fcc Fe.  \nThus, the configurational entropy of carbon in α and γ \nphase is  \n      ()\n[] 4/)41ln()41()4ln(43/ln\nc c c ck Sc kc S\n− −+ −=−≈\nγα           (6)  3\nFollowing Ref. [41] the concentration dependence of the \nfree energy of cementite can be presented as \n),( ),( )( )( ),(int)1(Tcf Tcf Tf Tf TcfFe + Δ+ Δ+ =θ αθ α θ     (7) \nwhere ),0( )( T f T fFe α α =  is the free energy density of pure \nα iron, )(TfαθΔ  is the free energy density of formation of \nstoichiometric cementite fro m pure components (bcc Fe and \ngraphite) which is known from  CALPHAD [38] or ab initio \ncalculations [39,40], cemc= 0.25 is the stoichiometric \ncomposition of cementite, ),()1(TcfθΔ  is the variation of free \nenergy density of cementite due to deviation from \nstoichiometry calculated in [41] within the model of the \nregular carbon-vacancy solution, ),(int Tcf  is an additional \ncontribution to the free energy caused by the interactions of \nover-stoichiometric carbon atoms. The details of parameterization of the formulas (3)–(7) see in Appendix. \nFollowing the results of Ref. [29] we assume that the \ncementite nucleation occurs by displacive mechanism in the ferromagnetic region which exists near the ferrite plate and \npropagates further into the bulk. Herewith, the MIS formed \nat \nγα/ interface provides the easer and faster realization of \nγ – θ phase transformation and maintains the lattice \ncoherence. Without describing in detail the process of \nnucleation, we accept that cementite emerges in the bulk \nwhen the local carbon concentration reaches the value of \nc≈0.20 (at T=0K) [29]. To take into  account MIS effect near \nthe ferrite boundary, we replace the concentration \ndependence )()1(cfθ [41] by an effective one, so that an \nintersection point of free energies )(cfγ  and )(cfθ  shifts to the left by the value boundcΔ ~0.05. We also assume that the \ncarbon concentration c=0.25 is reached primarily in the bulk \nphase with higher carbon solubility, so the energy ) (cemcfθ  \nremains unchanged.  \n \nB. Simulation of transformation kinetics  \nTo study evolution of the microstructure during PT we \nsolved numerically the nonlin ear diffusion type equation \ndescribing the distribution of carbon c(r,t) \nI−∇=∂∂\ntc,    ⎟\n⎠⎞⎜\n⎝⎛∇− −=cFc ckTcD\nδδ)1()(I           (8) \nwhere F(с) is determined by Eq. (1), )(cD  is the diffusion \ncoefficient which is suppo sed to be different in γ and α \nphases. The simulation was performed at the square grid \n800x800 with mirror- symmetric boundary co nditions [42] by \nusing Runge-Kutta procedure. Such choice of the boundary conditions allows to model the formation of a single isolated \npearlite colony in the considered area.  To simulate \nnucleation of colony, we have chosen the initial state as the \nγ-phase with a homogeneous  carbon concentration and \nintroduced there a sm all embryo of ferrite or cementite \nphase. \nIII. RESULTS  \nA. Transformation diagram \nThe local density of free energy of each phases \n)(),(c fθγα calculated for different temperatures by using \nFig.1. Variants of phase equilib rium in Fe-C system with triple-well thermodynamic potential f(c). (a) Calculated density of free \nenergy of θγα,,  phases in the model with ab initio parameterization at T=1050K (1,1',1''), 900K (2,2',2''), 750K (3,3',3''). (b) \nChange in the conditions of cementit e formation near the ferrite boundary at 750K; free energy density of  θγα,, -phase (1,2,3) \nand the effective free energy density of cementite (3'). (c) Resulting effective density of free energy as a function of carbon \nconcentration; dotted lines are tangents to the free energies of the phases.  4\nthe model described above (Eqs. (3)-(7)) are shown in \nFigs.1a,b together with effectiv e local density of free energy \n{ })(),(),( min)( cfcfcf cfeff θ γ α =  (Fig.1c). \nAfter exclusion of the fast variables, carbon \nconcentration is the only quantity  that determines the phase \nstate of the Fe-C. In this ca se, carbon concentration can be \nconsidered as an order parameter, at least in case of ferrite \nand pearlite transformation. As one can see from Fig. 1a, at \nT=750K ferrite and cemen tite are preferred for c < 0.027at% \nand c > 0.20 at%, respectively; w ithin the interval 0.027 at% \n< c < 0.20 at% austenite is energetically preferable.  \nThe realization of MIS on γ→θ transformation \npathway [29] facilitates the nucl eation of cementite near the \nγα/ interface due to magnetization induced by an adjacent \nferrite plate. As a result, the θ γ→  transformation starts \nnear the ferrite plate wh en reaching smaller carbon \nconcentrations (about 15 at %, see Fig.1b) in comparison with bulk. We assume that  stoichiometric cementite \n(c=0.25) exists in paramagnetic state and have the same \nenergy in the bulk and near the ferrite plate as well. Thus, \nwe consider an effective \nθ γ→  transformation pathway, \ntaking into account the nucleati on of cementite on the ferrite \n(curve 3’ in Fig.1b).  \nAbove the eutectoid temperature Tevtec=1000K two-\nphase equilibria γ+θ and γ + α take place (curve 1 in \nFig.1c). When temperature decreases, γ-phase becomes \nmetastable with respect  to decomposition on θ and α \nphases, wherein a stable equilibrium α/θ  arises (curve 2 in \nFig.1c). With further decrease of temperature, below some \ncritical value all metastable e quilibria disappear  (curve 3, \nFig.1c) and only stable two-phase α+θ state survives.  \nThese changes in equilibrium  conditions will result in \ndifferent scenarios of austenite decomposition. It is convenient to present them using the transformation diagram \n(Fig.2) proposed earlier in Ref. [28] which takes into account \nadditionally PT. In this diagram the lines A\n3 and Acm are \nboundaries of two phase regions αγ+ and θγ+ \nrespectively, are constructed fr om the condition of equality \nof chemical potentials of carb on in the corres ponding phases. \nThe lines T0 and bulkT1 correspond to αγ/ and \nγ/θ paraequilibrium condition when the free energies of γ, \nα or γ, θ phases with the same carbon concentration become \nequal; these lines are coinci ding with the temperature \ndependence of intersection points  of free energy densities of \nconsidered phases (see Fig.1a). The line boundT1  is obtained \nby shifting of about 5at% the line bulkT1 and describes the nucleation of cementite near the α/γ interface provided by \nMIS.  \nBelow eutectoid temperature Teutec the decomposition \nθα γ + →  is possible. As was suggested in Refs. [43,44] \nthe development of PT is expected below Teutec within a \nwindow between the metastable  extensions of the lines A3 \nand Acm (corresponding to metastable equilibria αγ/ and \nθγ/, respectively) where austenite is supersaturated with \nrespect to both α and θ phases. In this re gion the formation \none of the phases ( α or θ) will stimulate the appearance of \nanother one and therefore re sults in pearlite colony \nformation. Here we develop this view and show that this region can be divided into three subdomains I–III where the \nkinetics of PT is rather different. In the region I both \nmetastable equilibria \nαγ/ and θγ/ can be reached (see \nFig.1c, curve 2). In the regi on II only the metastable \nequilibrium θγ/ survives. Finally, in  the region III the \nmetastable equilibria between austenite ( γ) and both \ntransformation products ( α and θ) are impossible (Fig.1c, \ncurve 3). It is in the latter case, we can expect of the \naustenite decomposition when appearance one phase ( α or θ) \nwill stimulates the fast fo rmation of the another one. \n \nFig.2. The calculated transf ormation diagram. The lines A3 \nand Acm are the boundaries of two-phase regions γα+ \nand θγ+ as well their metastable extensions below the \neutectoid temperature Tevtec; the lines T0 and T1 are lines of \ninstability in respect α γ→   and θ γ→  \ntransformation, respectively. The temperature regions I–III \nare determined by intersection points of these lines.  5\nHowever, the simulation of the decomposition kinetics is \nrequired to study microstructure morphology.  \n \nB. Simulation of PT. Lamellar and globular pearlite.  \nTo specify morphology of the transformation product we \nhave carried out phase field si mulations of PT starting from \nhomogeneous initial state with a single small ferrite nucleus. \nWe found that below a temperature )1(\npT two scenarios of PT \nare possible, leading to the formation of either globular (region II) or lamellar structure (region III). The \ncorresponding results are presented in Figs. 3–6. The \ndifferent levels of carbon concentrations are shown by \ngrayscale, wherein cementite is white and ferrite is black. \nThe time is given in dimensionless unitsαDL/2, where the \nsquare side is L~1mkm (see Appendix). \nIn the region III, the fine and rather regular lamellar \nstructure is formed regardle ss of the location of initial \nembryo (Figs.3,4). In this case, as the first step, carbon is pushed out from embryo of ferrite and its concentration near \nFig.3.   Kinetics of lame llar structure growth from a single ferr ite nucleus placed on the grain boundary; T=675K,  c0=0.06; \nT=675K,  c0=0.06.  The numbers under each fragments corre spond to the dimensionless simulation time. \nFig.4. Kinetics of lamellar structure grow th from a nucleus placed on the grain bounda ries junctions (ferrite nucleus in the bo ttom \nleft and cementite nucleus in the upper ri ght corner are indicated by arrows); T=675K,  c0=0.06. \nFig.5. Kinetics of lamellar structure growth at shifting the line T1 to the left by cδ=0.03. The other parameters are the same as in \nFig.4. \nFig.6. Kinetics of globular structure growth;  T=800K, the other parameters are the same as in Fig.4.   6\nferrite interface reaches the threshold value c(boundT1 ). After \nthis the MIS →θ transformation occurs at interface and \ncarbon flow from γ-matrix provides saturation of θ-phase \nand depletion of carbon in surrounding austenite. Since \ncementite cannot be in equilibrium with γ-matrix in this \nregion of diagram, the proce ss continues until the critical \nconcentration c(T0) is reached.  After th at a new ferrite layer \nis formed near θ-phase and the process described above is \nrepeated, so the corresponding mechanism can be called \nautocatalytic. Phase field simu lations show that the front \nmovement of the pearlite colony is accompanied by \nincreasing its transver se size. As a result, the pearlite colony \ngets a fan-type shape in accordance with experiment [1,19]. Herewith, the lamellae do not ha ve a well-marked tendency \nof normal orientation to the tr ansformation front. At the late \nstages the space is filled by th e domains and the allocation of \nlamellae is well correlated w ithin each domain. One can \nassume that the elastic stresse s which were not taken into \naccount here, will provide even  more regular structures. \nNote that a similar pearlite structure can also arise in \nregion III, if we start from  one cementite embryo instead \nferrite (see Fig.4, upper right corner).  \nThe position of the start line \nboundT1  of θ γ→  \ntransformation (see Fig.2) is pa rtly controversial, since the \nresults [29] were obtained at T=0K. Therefore, we have \nperformed the calculation at various positions of boundT1 . \nFig.5 represents the simulation results at choosing the \nparameters analogous to Fig.4, but the line boundT1  is \nadditionally shifted to the left. It results in the formation of \nmore regular lamellar structure with a smaller interlamellar \nspacing. Otherwise, th e shift of the line boundT1  to the right \nleads to decreasing of critical temperature of autocatalysis \n} , min{)2( )1(\np p p T T T=  and to the coarsening of \nmicrostructure; the corresponding kinetics pictures are not \npresented here. \nIn the region II PT starts only with ferrite embryos, since \nthey alone can not be in equilib rium with austenite. In this \ncase the condition of autocatalytic multiplication of lamellae is violated and the phase field simulation demonstrates a \ncoarse globular structure (Fig.6 ). As in the previous case, \ncarbon is pushed out from embryo of ferrite and the chain of \ntransformations \nγ → MIS →θ is realized. However, in this \ncase the line Acm is achieved before th e critical concentration \nc(T0), so that the metastable phase equilibrium γ/θ is \nrealized, and the new ferritic layer does not appear. As a \nresult, the other scenario of transformation takes place which \nresults in numerous small cemen tite precipitates in the single \nferritic matrix.    In the region I in Fig.2 austenite is decomposed by the \nconventional nucleation-and-grow th mechanism as discussed \nin Ref. [28] (the corresponding pictures are not shown here). \nCarbon is pushed out fro m ferrite embryo and its \nconcentration near ferrite interface reaches the value \ndetermined by A3 curve. Since c(A3) < c(boundT1 ), the local \nmetastable phase equilibrium  α/γ is reached, and the \nformation of cementite does not occur in this case. And vice \nversa, if we start from on e cementite embryo, the local \nmetastable phase equilibrium γ/θ is realized and ferrite \ndoes not occur because c(Acm) > c(T0).  \n \n \nIV. DISCUSSION \nThe proposed model based on  ab-initio parametrization \ndescribes all the most essential features of PT. In particular, \nthe model predicts autocatalytic  scenario of quite regular \npearlite colonies formation and change in pearlite \nmorphology from lamellar to globular when temperature \nincrease above some  critical value )2(\npT . It is in agreement \nwith the experimental observations of globular and lamellar \npearlite transformati ons [16–19] and the former take place at \nsmaller undercooling temperat ures. Note that the free \nenergies of each phase depend on steel composition and \nalloying will affect the pos ition of regions I and II.  \nThe suggested mechanism of the autocatalytic PT has \nsome similarity to the spinodal decomposition (SD) of alloys but has also essentially new f eatures. Namely, the austenite \nremains stable with respect to  small fluctuations of carbon \nconcentration and only the fo rmation of ferrite plates \nstimulates the local satura tion of carbon and cementite \nnucleation. Note, that autoca talytic decomposition of some \nmetastable phases was earlier co nsidered in Refs. [45,11]. \nWithin the approach under co nsideration, the instability \nof austenite w ith respect to \nγ→α+θ decomposition \ndevelops stepwise with the temperature decrease. Existence \nof the threshold temperature of  autocatalysis is consistent \nwith the available experimental  data [46,47]. For example, \naccording to Ref. [47] the pearlite nucleation rate (in contrast \nto the growth rate) is close to zero at evtec p TT T <<exp and \nincreases abruptly at ≈exp\npT 820K. Similar behavior was \ndiscussed in Ref. [46] where a narrower temperature interval \nwith the close to zero nucl eation rate was observed.   \nAn important element of the proposed model is the \nassumption that nucleation of ce mentite is facilitated near the \nγα/ interface due to magnetization induced by an adjacent \nferrite plate. We assume that  the Metastable Intermediate \nStructure (MIS) emerging at γα/ interface due to  7\nincreasing of magnetic order in α-phase, plays a decisive \nrole in the formation of ceme ntite and provides the fast \nlattice reconstruction during the autocatalytic PT.  \nThe model is rather simple an d do not take into account \nreal geometry of conjugation α and θ phases as well as \nelastic strain due to lattice mismatch. Nevertheless, this \napproach allows us to constr uct a realistic transformation \ndiagram and investigate the ki netics of pearlite colonies \nformation. The results may also be important to eutectic or \neutectoid growth of colonies in other systems.  \nACKNOWLEDGMENTS \nThe research was carried out within the state assignment \nof FASO of Russia (theme “Magnet” N01201463328), and \nwas partially supported by Act 211 Government of the \nRussian Federation, contract № 02.A03.21.0006. MIK \nacknowledges a support from Nederlandse Organisatie voor \nWetenschappelijk Onderzoek (NWO) via Spinoza Prize. \n \nAPPENDIX: DETAILS OF PARAMETERIZATION \nWe use the dissolution and mixing energies of carbon \nclose to those known from expe rimental data and ab initio \ncalculations: αεFM=0.9 [37,47,48], αεPM=0.9 [48], γεPM=0.3 \n[50,51], γεFM= -0.4 [28,48], αv=6 [26], γv=1.5 [26,51] (in \neV/at). The estimations of γv are vary greatly from 1 to 3 \neV/at; the values αεPM,γεFM are known only from ab initio \ncalculations; and the difference between ab initio and \nexperimental estimations of αεFM,γεPM is about 0.2eV/at. \nNote, γ-phase is stable with resp ect to small concentration \nfluctuations because γv>0; and the value of αv is not \nessential since car bon solubility in α-phase is very small. We \nneglect the dependence of energies )(\n) (αγεPM FM on temperature, \nthat is partly compensated by the choice of its changed \nvalues (within a pointed out error, 0.2eV/at). \nThe temperature dependence of the difference between \ncementite and ferrite free energies )(TfαθΔ  was chosen \nclose to CALPHAD and ab initio  calculations [38–40], with \nthe additional conditio n, that the curve Acm passes through \nthe eutectoid point ( с=0.034, T=1000K); )(TfαθΔ =0.109-\n0.173cτ+0.0782\ncτ (in eV/at), where cτ=T/T c, Tc=1043K. \nWe also assume the value of ),(int Tcf  is so large for c \n>cemc that deviation from stoich iometry in this case can be \nneglected. The ratios of diffusion coefficients γ αD D/, θ γD D/ \nare 102÷103 [52,53], thus simulation with realistic diffusion \ncoefficients is impossible, but the qualitative tendencies may \nbe revealed when this ratio is choosing sufficiently large. 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B 52, 803 (1995).\n "    },    {        "title": "2102.03020v1.Inter_valence_charge_transfer_and_charge_transport_in_the_spinel_ferrite_ferromagnetic_semiconductor_Ru_doped_CoFe__2_O__4_.pdf",        "content": " 1 Inter-valence charge transfer and charge transport in the spinel ferrite ferromagnetic semiconductor Ru-doped CoFe2O4  Masaki Kobayashi1,2,*, Munetoshi Seki1,2, Masahiro Suzuki3, Miho Kitamura5, Koji Horiba5,  Hiroshi Kumigashira5,6, Atsushi Fujimori3,4, Masaaki Tanaka1,2, and Hitoshi Tabata1,2 1Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 2Center for Spintronics Research Network, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 3Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4Department of Applied Physics, Waseda University, Okubo, Shinjuku, Tokyo 169-8555, Japan 5Photon Factory, Institute of Materials Structure Science, High Energy Accelerator Research Organization (KEK), 1-1 Oho, Tsukuba 305-0801, Japan 6Insitute of Multidisciplinary Research for Advanced Materials (IMRAM), Tohoku University,  Sendai 980–8577, Japan *Email: masaki.kobayashi@ee.t.u-tokyo.ac.jp  Abstract Inter-valence charge transfer (IVCT) is electron transfer between two metal M sites differing in oxidation states through a bridging ligand: Mn+1 + M’m à Mn + M’m+1. It is considered that IVCT is related to the hopping probability of electron (or the electron mobility) in solids. Since controlling the conductivity of ferromagnetic semiconductors (FMSs) is a key subject for the development of spintronic device applications, the manipulation of the conductivity through IVCT may become a new approach of band engineering in FMSs. In Ru-doped cobalt ferrite CoFe2O4 (CFO) that shows ferrimagnetism and semiconducting transport properties, the reduction of the electric resistivity is attributed to both the carrier doping caused by the Ru substitution for Co and the increase of the carrier mobility due to hybridization between the wide Ru 4d and the Fe 3d orbitals. The latter is the so-called IVCT mechanism that is charge transfer between the mixed valence Fe2+/Fe3+ states facilitated by bridging Ru 4d orbital: Fe2+ + Ru4+ ↔ Fe3+ + Ru3+. To elucidate the emergence of the IVCT state, we have conducted x-ray absorption spectroscopy (XAS) and resonant photoemission spectroscopy (RPES) measurements on non-doped CFO and Co0.5Ru0.5Fe2O4 (CRFO) thin films. The  2 observations of the XAS and RPES spectra indicate that the presence of the mixed valence Fe2+/Fe3+ state and the hybridization between the Fe 3d and Ru 4d states in the valence band. These results provide experimental evidence for the IVCT state in CRFO, demonstrating a novel mechanism that controls the electron mobility through hybridization between the 3d transition-metal cations with intervening 4d states.    I. Introduction Inter-valence charge transfer (IVCT) occurs in mixed-valence coordination complexes, and is an electron transfer between two metal M sites differing in oxidation states through a bridging ligand: Mn+1 + M’m à Mn + M’m+1, and is usually related to photoexcitation phenomena in these compounds [1, 2]. Additionally, magneto-optical properties of ferrites or iron oxides that seem to originate from IVCT such as magneto-optical Kerr rotation and photo-induced magnetization have been reported so far [3, 4, 5, 6, 7, 8]. In semiconducting or insulating materials with mixed-valence states, especially for metal-organic frameworks, IVCT is considered to be related to the transport properties with hopping conduction [6, 9, 10, 11, 12, 13]. Then, it is expected that one can control magneto-optical and transport properties of mixed-valence semiconductors or insulators through IVCT.  Ferromagnetic semiconductors (FMSs) having both semiconducting and ferromagnetic properties are key materials for spintronics, which is a research field to use both the charge and spin degrees of freedom for functional electronic devices [14, 15, 16]. Controlling the conductivity of FMSs with keeping their ferromagnetic properties is useful for utilizing FMSs to spintronics devices such as magnetic random-access memories (MRAMs) and spin transistors. Spinel ferrites MFe2O4 [M = 3d transition metals (TMs)] are promising magnetic materials for spintronics because of their chemical stability and high Curie temperatures [17]. Indeed, spinel ferrite layers in magnetic tunnel junctions have been demonstrated to act as spin filters [18, 19]. CoFe2O4 (CFO) with the inverse spinel structure shows ferrimagnetism with the Curie temperature (TC) of 793 K and CFO epitaxial thin films have been studied for the application to spintronics devices [10, 20, 21, 22, 23]. In CFO, the stoichiometric valence states of the constituent Co and  3 Fe ions are Co2+ at the octahedral crystal-field (Oh) site and Fe3+ at both the Oh and the tetrahedral crystal-field (Td) sites, as shown in Fig. 1. In contrast to the semiconducting CFO, magnetite Fe3O4 shows metallic transport property because of double-exchange interaction between the Fe2+ and Fe3+ sites. Recently, Iwamoto et al. [10] have succeeded in increasing the conductivity of CoFe2O4 by Ru doping, where the Ru4+ ions are considered to preferentially occupy the Co2+ sites in CFO [7, 10]. Figure 2 shows the transport properties of Ru-doped CoFe2O4 thin films [10]. The results indicate that the Ru doping increases the carrier density and the electron mobility. The doped Ru4+ ions in CFO are expected to act as double donors and that the Ru doping increases not only the carrier concentration but also the hopping probability though hybridization between the Fe 3d and Ru 4d orbitals. It is considered that the carriers supplied by Ru doping generate the mixed valence state of Fe2+/Fe3+ in CFO, leading to the improvement of the conductivity through double-exchange interaction like Fe3O4. First-principles band calculation suggests that the increase of the hopping conduction occurs through hybridization between the Fe 3d and Ru 4d orbitals following the IVCT mechanism Fe2+ + Ru4+ ↔ Fe3+ + Ru3+.   To prove the existence of such an IVCT state in CRFO, two key factors should be confirmed experimentally: (1) the Fe2+/Fe3+ mixed-valence state and (2) the hybridization between the Fe 3d and Ru 4d orbitals. In this letter, we have conducted x-ray absorption spectroscopy (XAS) and resonance photoemission spectroscopy (RPES) studies of non-doped CFO and Ru-doped CoFe2O4 thin films in order to obtain experimental evidence for the IVCT state in CRFO. XAS and RPES enable us to probe element- and orbital-specific electronic structures. The XAS spectra reveal the valence states of Fe and Co in CRFO. The RPES spectra indicate that the partial density of states (PDOS) of Fe 3d change with Ru doping. These experimental findings suggest that the IVCT state is realized in Ru-doped CFO. The manipulation of the conductivity through IVCT may become a new approach of band engineering in FMS materials.  II. Experimental CoFe2O4 (CFO) and Co0.5Ru0.5Fe2O4 (CRFO) epitaxial thin films were grown on single crystal a-Al2O3(0001) substrates using the pulsed laser deposition (PLD) technique with an ArF-excimer laser with the wavelength was 193 nm, the frequency was 5 Hz, and the  4 fluence was E = 60 mJ. The details of the thin-film growth are described elsewhere [10]. The PES and XAS measurements were performed at BL-2A of Photon Factory (PF), High Energy Acceleration Research Organization (KEK) [24]. The total energy resolution was set to be 100-250 meV for PES measurements using photon energy of 400-1200 eV . The binding energies were calibrated by measuring the EF of a gold foil which was electrically contacted to the samples. The PES measurements were conducted with an SES2002 electron analyzer at room temperature under the base pressure below 2.0 ´ 10-8 Pa. The XAS spectra were measured in the total electron yield (TEY) mode.   III. Results and discussion Basically, XAS spectra reflect the local electronic structures of specific elements. We have performed XAS measurements on CRFO thin films to elucidate the valence state of each element. Figure 3 shows the Co L2,3 XAS spectra of parent CFO and Ru-doped one. The spectra of CFO and CRFO show multiplet structures that are different between them. Comparing the observed Co L2,3 spectra with those of reference compounds CoO (Co2+ Oh) and LaCoO3 (Co3+ Oh) [25], the spectrum of CRFO is similar to that of CoO and the spectrum of CFO seems to be a superposition of them. Here, the Co3+ Oh spectrum represents cation inversion or Co-antisite defects (the inter-site cation exchange between the A Td and B Oh sites, and the on-site cation replacement at the A sites) because the XAS spectra of the Co3+ Oh is similar to that of the Co3+ Td [26]. Here, the cation inversion defects is represented by the inversion parameter y defined by the chemical formula [Fe1-yCoy]Td[Fe1+yCo1-y]OhO4. To estimate quantitatively the ratio of these oxidation (valence) states, the Co L2,3 XAS spectra of CFO and CRFO are fitted by a linear combination of the reference spectra of CoO and LaCoO3. Figures 3(b) and 3(c) show such decomposition analyses for the Co L2,3 XAS spectra. Figure 3(b) shows that the Co L2,3 XAS spectrum of CRFO is fitted by the spectrum of CoO and that there is nearly no contribution of the Co3+ state to the experimental spectrum within the accuracy of the analysis. In contrast, the Co L2,3 XAS spectrum of CFO is decomposed into the ~75% Co2+ and the ~25% Co3+ components, as shown in Fig. 3(c), indicating that part of the Co2+ ions at the B site becomes Co3+ at the B site or move to the A site. The latter case means the existence of antisite defects, namely, Co ions at the A site and excess Fe ions at the B site. This is consistent with that non-negligible cation inversion defects usually exist in spinel ferrite thin films [22, 23]. In CRFO, on the other hand, electron-rich  5 environment induced by Ru doping suppresses the occurrence of Co3+ ions and possibly the cation inversion. It should note here that the Co2+ components possibly include both the Co2+ at the B site and the antisite defect of the Co2+ at the A site. The chemical compositions are discussed below considering the decomposition analysis of the Fe L2,3 XAS spectra.  Figure 4 shows the Fe L2,3 XAS spectra of the parent CFO and CRFO thin films. The Fe L2,3 XAS spectra show multiplet structures and the spectral line shapes are similar to that of Fe2O3, as shown in Fig. 4(a). In contrast to the Co L2,3 XAS spectra, the Fe XAS spectrum of CRFO film looks nearly identical to that of CFO one. To elucidate the Fe components precisely, the Fe L2,3 XAS spectra are decomposed into various valence and crystal-field states using linear combinations of reference spectra of Fe2O3 (Fe3+ Oh) [27], YBaCo3FeO7 (Fe3+ Td) [28], and FeO (Fe2+ Oh) [29]. As show in Figs. 4(b) and 4(c), the linear combinations of the reference spectra well reproduce the experimental spectra of CRFO and CFO samples. The ratios of the components are listed in Table I. As expected from the line shapes, the ratio among the Fe components of CRFO is similar to that of CFO. Considering the possible existence of the cation inversion defects in the CFO thin film observed in the Co L2,3 XAS spectrum, the Fe2+ component in the CFO film likely originates from the Fe-antisite defect, where the excess Fe ions substitute for Co2+ ions at the Oh sites, as in the case for the Co3+ component tentatively assigned to cation inversion defects [Fig. 3(c)]. Based on the analysis, the chemical composition of CFO is estimated as [Fe0.67Co0.33]Td[Fe1.33Co0.67]OhO4 (y = 0.33). On the other hand, the presence of the Fe2+ component in the CRFO film would be induced by the electron doping caused by the Ru substitution because there will be few cation inversion defects in the CRFO film. Although the ratio of Fe2+ is expected to be 25% in CRFO from the nominal stoichiometry assuming Ru4+, the deduced value is ~10.5%. Since there is nearly no Co3+ component in CRFO, this quantitative difference may come from Ru-antisite defects substituting for the Fe sites. These results provide spectroscopic evidence for the Fe2+/Fe3+ mixed-valence state in the CRFO thin film caused by the Ru doping, even though there is a quantitative difference.   To elucidate the possible hybridization between the Fe 3d and Ru 4d orbitals, which is another key factor for the IVCT, the Fe and Co 3d PDOS in the valence band (VB) that  6 arise from the different valence states have been obtained using RPES. Figure 5 shows the RPES spectra of the CFO and CRFO thin films. When incident photon energy hn corresponds to that for the core-hole excitation, the photoemission intensity of the orbitals into which the excited electron orbital is resonantly enhanced. Difference between the on-resonance spectrum and off-resonant spectrum reflects the 3d PDOS corresponding to those orbitals. Since the 2p-3d excitation energy for each component is different, one can obtain the PDOS dominated by each component by tuning hn (it is difficult to separate these PDOS perfectly because of the overlap of the XAS spectra between these components). For instance, as shown in Fig. 5(a), the Fe2+ and Fe3+ PDOS of the thin films are obtained from the on-resonance spectra measured at hn of 708.5 eV and 710 eV , respectively. Figure 5(b) shows the Fe and Co 3d PDOS and off-resonance spectra of CFO and CRFO films. Here, the on-resonance spectra are taken at the core-hole excitation energies for the Co2+, Co3+, Fe2+, and Fe3+ states, and the spectra are normalized to the spectral area. For the Co-3d PDOS, the differences of the spectra between the samples reflect the effect of Ru substitution for the Co sites. Since the Co 3d spectral intensity near the valence-band maximum (VBM) decreases with Ru doping, we conclude that the Co 3d PDOS hardly contributes to the increase of the conductivity. Similarly, the Fe3+ PDOS shows nearly the same spectral changes as Co 3d with Ru doping, suggesting few direct contributions of the Fe3+ component to the improvement of the conductivity.   It should note here that the differences of the Co off-resonance spectra between the CFO and CRFO films reflect the appearance of the Ru 4d PDOS and the decrease of the Co-3d PDOS with Ru doping. Considering the cross section of the atomic orbitals around this incident-photon energy region, the VB spectra without resonance mainly reflect the Ru 4d PDOS [30]. Since the spectral intensity near the VBM increases with Ru doping, the Ru 4d PDOS predominantly contributes to the DOS near the VBM, qualitatively consistent with the increase of the conductivity with Ru doping. To reveal the 3d component hybridized with the Ru 4d orbitals, the additional 3d PDOS near VBM induced by Ru doing are examined. Figure 5(c) shows comparison of the various PDOS in the vicinity of VBM that have been normalized to the intensity at EB~1.5 eV . The difference in the off-resonance spectra indicates that the Ru 4d PDOS is located near the VBM. Note also that additional Fe2+ PDOS appears with Ru doping and the position of the Fe2+ PDOS is nearly identical to that of the Ru 4d PDOS [see arrows in Fig. 5(c)]. In  7 contrast, the Co3+, Co2+, and Fe3+ PDOS have the same slopes irrespective to the Ru doping. These observations are consistent with the first-principle calculation that the B-site Fe 3d orbitals hybridizes with the Ru 4d orbital except for the opening of the conductivity gap in experiment [10]. The band gap may come from Coulomb interaction, i.e., a Mott gap. Thus, the present result is consistent with hybridization between the Fe 3d and Ru 4d orbitals in CRFO.   The present findings, that is, the mixed-valence state of Fe2+/Fe3+ and the hybridization between the Fe 3d and Ru 4d orbitals, provide spectroscopic evidence for the IVCT state in Ru-doped CFO. In CRFO, the O 2p orbitals that are located at the nearest neighbor atoms of the B sites make bonding states directly with the Fe 3d eg and Ru 4d eg orbitals. It is likely that the ligand O 2p orbitals bridge electron hopping between the Fe and Ru sites such as Fe2+ + Ru4+ ↔ Fe3+ + Ru3+. Here, while IVCT for photoexcitation is usually expressed with a single-headed arrow between M atoms, we use a double-headed arrow to IVCT for transport. In contrast to the M-M’ distances for IVCT in coordination complexes reportedly in the range of 10~25 Å [2], the effective distance for the IVCT possibly becomes longer in single crystalline TM compounds like spinel ferrite oxides because the ligand bands bridging the IVCT are expanded the entire crystal. Compared with the M-M’ distance in coordination complexes, in which the photo-excitation induces charge transfer in the environment of highly insulating nature, the longer effective distances for the IVCT in the TM compounds lead to the increase of the hopping probability (or electron mobility) through IVCT as a consequence of the reduction of energy barrier for the charge transfer. This may explain the increase of the conductivity by Rh doping in Fe2O3 (Fe1.8Rh0.2O3) without mixed-valence states [31], where the doped Rh3+ ions isovalently substitute for the Fe3+ sites. The same mechanism may also explain the increase of the hopping probability through IVCT of FeTiO3 ilmenite [9], Fe2+ + Ti4+ ↔ Fe3+ + Ti3+, without double-exchange interaction [since the Ti4+ ion (d0 configuration) is non-magnetic]. As in the cases for Fe3O4 and (La,Sr)MnO3, double-exchange interaction between magnetic ions with the mixed-valence states stabilizes the ferromagnetism and contributes to the increase the metallic conductivity [32]. In contrast, the conduction mechanism through IVCT in solids will be applicable for ferromagnetic semiconducting materials having low carrier concentrations with hopping conduction. Furthermore, as described in the introduction, the IVCT has the capability to modify the  8 optical properties of semiconducting materials. It follows from the above arguments that control of the IVCT in FMSs with mixed-valence states will be a new approach manipulating the electron mobility and magneto-optical properties through hybridization between the TM d orbitals with maintaining the ferromagnetism.   IV. Summary In this study, we have investigated the electronic structure of Ru-doped CoFe2O4 using XAS and RPES to elucidate the emergence of IVCT in semiconducting crystal. The Co L2,3 XAS spectra suggest that although the 25% of the total amount of Co atoms exist as Co3+ in the CFO thin film, there are only Co2+ ions in the Ru-doped CFO thin film. The result that there is almost no Co3+ ions in the Ru-doped CFO thin film provides an important aspect to control defect level for materials growth of spinel ferrites. The Fe L2,3 XAS spectra indicate that the Fe2+/Fe3+ mixed-valence state is realized in CRFO. The observation using RPES demonstrates that the Ru 4d PDOS appears near the VBM and hybridizes with the Fe2+ 3d state, and not with the Co ones. These findings provide spectroscopic evidence for the IVCT state between the Ru 4d and Fe 3d orbitals. Controlling the IVCT through the hybridization will open a new way to manipulate both the magneto-optical properties and the carrier mobility of FMSs.   Acknowledgment This work was supported by a Grant-in-Aid for Scientific Research (Nos. 15H02109,16H02115, 17H04922, 18H05345) and Core-to-Core Program from the Japan Society for the Promotion of Science (JSPS), CREST (JPMJCR1777), and the MEXT Elements Strategy Initiative to Form Core Research Center. This work was partially supported the Spintronics Research Network of Japan (Spin-RNJ) and Basic Research Grant (Hybrid AI) of Institute for AI and Beyond for the University of Tokyo. This work at KEK-PF was performed under the approval of the Program Advisory Committee (Proposals 2015S2-005 and 2018G114) at the Institute of Materials Structure Science at KEK. A.F. is an adjunct member of Center for Spintronics Research Network (CSRN), the University of Tokyo, under Spin-RNJ.     9 Reference [1] B. S. Brunschwig and N. Sutin, Coord. Chem. Rev. 187, 233 (1999). [2] J.-P. Launay, Chem. Soc. Rev. 30, 386 (2001). [3] W. F. J. Fontijn, P. J. van der Zaag, M. A. C. Devillers, V . A. M. Brabers, and R. Metselaar, Phys. Rev. B 56, 5432 (1997). [4] Y . Muraoka, H. Tabata, and T. Kawai, J. Appl. Phys. 88, 7223 (2000). [5] B. Zhou, Y .-W. Zhang, Y .-J. Yu, C.-S. Liao, and C.-H. Yan, L.-Y. Chen, and S.-Y. Wang, Phys. Rev. B 68, 024426 (2003). [6] M. Seki, A. K. M. A. Hossain, T. Kawai, and H. Tabata, J. Appl. Phys. 97, 083541 (2005).  [7] T. Kanki, Y . Hotta, N. Asakawa, M. Seki, H. Tabata, and T. Kawai, Appl. Phys. Lett. 92, 182505 (2008).  [8] M. Seki, “Iron Ores and Iron Oxide Materials”, (edited by V olodymyr Sharokha, InTechOpen), in press.  [9] B. Zhang, T. Katsura, A. Shatskiy, T. Matsuzaki, and X. Wu, Phys. Rev. B 73, 134104 (2006). [10] F. Iwamoto, M. Seki, and H. Tabata, J. Appl. Phys. 12, 103901 (2012). [11] L. E. Darago, M. L. Aubrey, C. J. Yu, M. I. Gonzalez, and J. R. Long, J. Am. Chem. Soc. 137, 15703 (2015). [12] C. Hua, P. W. Doheny, B. Ding, B. Chan, M. Yu, C. J. Kepert, and D. M. D’Alessandro, J. Am. Chem. Soc. 140, 6622 (2018). [13] L. S. Xie, L. Sun, R. Wan, S. S. Park, J. A. DeGayner, C. H. Hendon, and M. Dincă, J. Am. Chem. Soc. 140, 7411 (2018). [14] H. Ohno, Science 281, 951 (1998). [15] S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Molnár, M. L. Roukes, A. Y . Chtchelkanova, and D. M. Treger, Science 294, 1488 (2001). [16] I. Zutic, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 322 (2004). [17] M. Sugimoto, J. Am. Ceram. Soc. 82, 269 (1999). [18] U. Lüders, A. Barthélémy, M. 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Tabata, Appl. Phys. Express 5, 115801 (2012). [32] C. Zener, Phys. Rev. 82, 403 (1951).         11 Table Table I. Decomposition analysis for the Co L2,3 and Fe L2,3 XAS spectra. It should note here that the Co2+ components include both the Co2+ at the B site and the possible antisite defect of the Co2+ at the A site.    A(Td) site B(Oh) site Co3+ Fe3+ Co2+ Fe3+ Fe2+ CoFe2O4 25% 33.6% 75% 57.9% 8.5% Co0.5Ru0.5Fe2O4 0% 37.2% 100% 52.3% 10.5%     12 Figures  \n   FIG 1: Crystal structure of inverse spinel CoFe2O4. (a) Unit cell of CoFe2O4. The magnetic sublattice of the A(Td) site antiferromagnetically couples with that of the B(Oh) site. (b) Spin configurations of the A and B sites.      \n 13  \n FIG. 2: Transport properties of Co1-xRuxFe2O4 thin films [10]. (a) Temperature (T) dependence of resistivity r. (b), (c) Compositional dependence of the carrier density and the Hall mobility, respectively.     10-210-1100101102103Resistivity ρ (Ωcm)\n350300250200150100Temperature (K)\n681020246810212Carrier density (1/cm-3)\n765431000/T (K-1)6810-2246810-124Hall mobility (cm2/Vs)\n765431000/T (K-1) x = 0.2 x = 0.5 x = 0.8(a)Co1-xRuxFe2O4\n(b) x = 0.2 x = 0.5 x = 0.8(c) x = 0.2 x = 0.5 x = 0.8 14   \n FIG. 3: Co L2,3 XAS spectra of Co1-xRuxFe2O4 thin films. (a) The XAS spectra of parent CFO and CRFO thin films. The spectra of CoO (Co2+) and LaCoO3 (Co3+) are also shown as references [25]. The arrows denote excitation energies for RPES, i.e., 775 eV for the off-resonance, 778.7 eV for the Co2+ on-resonance, and 780 eV for the Co3+ on-resonance. (b), (c) Decomposition of the Co L2,3 spectra for the CRFO and CFO thin films, respectively.    Intensity (arb. units)800795790785780775Photon Energy (eV)Intensity (arb. units)784780776772Photon Energy (eV)784780776772Photon Energy (eV)Co2+ [CoO]Co3+ [LaCoO3] CRFO CFOCo L2,3 XAS(a)\n(b) Exp. Fitted Co2+ Co3+CRFOCFO(c) Exp. Fitted Co2+ Co3+ 15   FIG. 4: Fe L2,3 XAS spectra of Co1-xRuxFe2O4 thin films. (a) The XAS spectra of parent CFO and CRFO thin films. The reference spectra of Fe2O3 (Fe3+ Oh) [27], YBaCo3FeO7 (Fe3+ Td) [Hollmann_PRB_09], and FeO (Fe2+ Oh) [29] are also shown. The arrows denote excitation energies for RPES, i.e., 705 eV for the off-resonance, 708.5 eV for the Fe2+ on-resonance, and 710 eV for the Fe3+ on-resonance. (b), (c) Decomposition of the Fe L2,3 spectra for the CRFO and CFO thin films, respectively.    Intensity (arb. units)730725720715710705Photon Energy (eV)Intensity (arb. units)716712708704Photon Energy (eV)716712708704Photon Energy (eV)Fe L2,3 XAS CRFO CFOFe3+ Oh [Fe2O3]Fe3+ Td [YBaCo3FeO7]Fe2+ Oh [FeO] Exp. FittedFe3+ OhFe3+ TdFe2+OhCRFO(a)\n(b) Exp. FittedFe3+ OhFe3+ TdFe2+ Oh(c)CFO 16  \n FIG. 5: Resonant photoemission spectra taken at Fe L3 and Co L3 edges of Co1-xRuxFe2O4 thin films. (a) On- and off-resonance spectra at the Fe L3 edge of CFO. (b) Partial density of states of the Fe 3d and Co 3d states. The shaded areas are difference of PDOS between CFO and CRFO films. The spectra have been normalized to the intensity integrated from EF to EB ~ 11 eV . (c) Fe 3d and Co 3d PDOS near the valence band maximum. The spectra have been normalized to the intensity at EB = 1.5 eV in order to emphasizing the spectral change near the VBM.    1086420Binding Energy (eV)Intensity (arb. units)\nNormalized Intensity (arb. units)\n1086420Binding Energy (eV)Normalized Intensity (arb. units)1.51.00.50Binding Energy (eV)(a)Fe L3 RPES On-Reso. Fe3+ On-Reso. Fe2+ Off-Reso.(b)\nCRFOCFOPDOS\nCo2+Co3+\nOffFe2+Fe3+CRFOCFOCo3+Co2+Fe2+Fe3+Off(c)"    },    {        "title": "1507.01964v1.Fast_and_rewritable_colloidal_assembly_via_field_synchronized_particle_swapping.pdf",        "content": "Fast and rewritable colloidal assembly via \feld synchronized particle swapping\nPietro Tierno,1, 2,a)Tom H. Johansen,3, 4and Thomas M. Fischer5\n1)Estructura i Constituents de la Mat\u0012 eria, Universitat de Barcelona, Spain\n2)Institut de Nanoci\u0012 encia i Nanotecnologia IN2UB, Universitat de Barcelona\n3)Department of Physics, University of Oslo,P. O. Box 1048, Blindern, Norway\n4)Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong, NSW 2522,\nAustralia\n5)Institut f ur Experimentalphysik V, Universit at Bayreuth, 95440 Bayreuth, Germany\n(Dated: 22 November 2021)\nWe report a technique to realize recon\fgurable colloidal crystals by using the controlled motion of particle\ndefects above an externally modulated magnetic substrate. The transport particles is induced by applying a\nuniform rotating magnetic \feld to a ferrite garnet \flm characterized by a periodic lattice of magnetic bubbles.\nFor \flling factor larger than one colloid per bubble domain, the particle current arises from propagating defects\nwhere particles synchronously exchange their position when passing from one occupied domain to the next.\nThe amplitude of an applied alternating magnetic \feld can be used to displace the excess particles via a\nswapping mechanism, or to mobilize the entire colloidal system at a prede\fned speed.\nThe \row of electrical current in a conductor arises\nwhen electrons at the Fermi level are scattered from oc-\ncupied states into unoccupied ones due to the interaction\nwith an external electric \feld. Following this electronic\nanalogy, it is a compelling idea to \fnd ways to separate\nan ensemble of identical colloidal particles into a set of\nimmobile low energy particles, and colloids which can\nbecome mobile due to an external \feld. When applied\nto microtechnological devices like lab-on-a-chip, this con-\ncept has demonstrated precise single particle operations\nbased on the selective motion of colloidal inclusions.1{3\nParamagnetic microspheres, which can be remotely con-\ntrolled via non invasive magnetic \felds, are currently em-\nployed in biotechnological applications.4{6The surfaces\nof these particles can be chemically functionalized allow-\ning to bind selectively to de\fned targets.7For an ensem-\nble of monodisperse particles, the formation of a thresh-\nold energy where only a fraction of particles will move\nin response to an applied magnetic \feld is di\u000ecult to\nachieve. In most cases, the \feld-induced interactions be-\ntween the particles favor collective motion rather than\nselected displacements. Magnetic patterned substrates\nhave shown considerable potential to overcome the above\nlimitation. These patterns can generate strong localized\n\feld gradients, allowing controlled particle trapping and\ntransport along prede\fned magnetic tracks. Recent ex-\nperiments include the use of arrays of permalloy ellip-\ntical islands8, cobalt microcylinders,9domain wall con-\nduits,10magnetic wires,11exchange bias layer systems,12\nand magnetic micromoulds.13\nAn alternative method consists in using epitaxially grown\nferrite garnet \flms (FGFs), where magnetic domains\nwith the width of few microns, i.e. on the particle\nscale, self-assemble into patterns of stripes or bubbles.14\nOriginally developed for magnetic memory applications15\nand magneto-optical imaging,16the FGFs are ideal to\na)Electronic mail: ptierno@ub.edumanipulate paramagnetic colloids,17or superconducting\nvortices.18The highly localized driving force originates\nfrom the stray \feld gradient generated by the Bloch\nwalls (BWs) in the FGF. When properly synthesized,\nthe displacement of the BWs caused by an external \feld\nis smooth and reversible, with absence of wall pinning,\nthus creating a precise and controllable driving force for\nthe particle motion. Here, we use a FGF to manipulate\nand transport an ensemble of paramagnetic particles or a\nfraction of it, demonstrating a technique to dynamically\norganize a colloidal system into trapped immobile parti-\ncles and particles which become mobile above a threshold\n\feld.\nA bismuth-substituted ferrite garnet of composition\nY2:5Bi0:5Fe5\u0000qGaqO12(q= 0:5\u00001) was prepared by liq-\nuid phase epitaxial growth on a (111)-oriented gadolin-\nium gallium garnet (GGG) substrate. Oxide powders of\nthe constituent elements, as well as PbO and B 2O3, were\ninitially melted at 1050\u000eCin a platinum crucible while\nthe GGG wafer was located horizontally just above the\nmelt surface. After lowering the temperature to 700\u000eC,\ngrowth of the FGF was started by letting the substrate\ntouch the melt. Keeping it there for 8 minutes pro-\nduced a FGF of 5 micron thickness, more details can\nbe found in a previous work.19At ambient temperature\nthe FGF has a spontaneous magnetization perpendicular\nto the plane of the \flm. To minimize the magnetic en-\nergy, the FGF breaks up into domains characterized by\na labyrinth stripe pattern, easily observed by polarized\nlight microscopy due to the large Faraday e\u000bect in this\nmaterial. High frequency magnetic \felds were used to\ntransform this pattern into a regular triangular lattice of\n\"magnetic bubbles\". These are cylindrical ferromagnetic\ndomains magnetized oppositely to the remaining contin-\nuous area of the FGF, and separated by BWs. The mag-\nnetic bubbles have in zero \feld a diameter of 6 :4\u0016mand\na lattice constant of l= 8:6\u0016m, Fig.1(a).\nTo avoid particle adhesion to the \flm, the FGF was\ncoated with a 1 \u0016mthick polymer layer composed of\na positive photoresist (AZ-1512, Microchem).20Param-arXiv:1507.01964v1  [cond-mat.soft]  7 Jul 20152\nFIG. 1. (a) Sketch of the magnetic bubble lattice (lattice con-\nstantl= 8:6\u0016m) with paramagnetic colloids and subjected\nto a magnetic \feld rotating in the ( x;z) plane. Schematic on\nthe right shows four magnetic bubbles (B) with two intersti-\ntials regions ( I;\u0016I) illustrating the two possible particle path-\nways (dashed lines) and corresponding crystal directions. (b)\nSequence of images showing the transport via particle swap-\nping. (c) A particle defect transported via particle swapping,\nVideoS1. Superimposed are the trajectories of the particles,\nin green (blue) are trajectories along the BIB (B\u0016IB) path-\nway, VideoS1 (Multimedia View). (d) Filling of a lattice va-\ncancy in the colloidal crystal, VideoS2 (Multimedia View). In\nthese experiments, the \felds have amplitudes Hx= 0:7kA=m ,\nHz= 1:0kA=m and frequency != 18:8s\u00001.\nagnetic microspheres with 2 :8\u0016mdiameter (Dynabeads\nM-270) were diluted in highly deionized water (milli-Q,\nMillipore), and deposited above the FGF. After \u00185 min\nof sedimentation, the particles became two-dimensionally\ncon\fned above the FGF due to balance between repul-\nsive electrostatic interaction with the polymer layer and\nmagnetic attraction.\nThe magnetic \feld applied in the ( x;z) plane was gener-\nated by using two custom-made Helmholtz coils assem-\nbled above the stage of an upright microscope (Eclipse\nNi, Nikon) equipped with a 100 \u00021:3 NA objective and\na 0:45\u0002TV lens. The coils were arranged perpendicular\nto each other and connected to two independent power\nampli\fers (KEPCO BOP) driven by an arbitrary wave-\nform generator (TTi-TGA1244).\nIn the absence of an external \feld, the BWs of the mag-\nnetic bubbles attract the paramagnetic colloids and with-\nout the polymer coating, the particles sediment above\nthe BWs. However, due to the polymer \flm, the parti-\nFIG. 2. (a) Particle \rux Jversus normalized density ^ \u001afor dif-\nferent frequencies. (b) Jversus amplitude of the perpendicu-\nlar \feldHz(Hx= 0:7kA=m ). Dashed red lines separate the\nregimes where all particles are immobile ( Hz<0:7kA=m ),\nonly propagation of excess particles occurs (0 :7kA=m<H z<\n1:2kA=m ), and all particles are mobilized ( Hz>1:2kA=m ).\n(c)Jversus amplitude of the in-plane \feld Hxfor three di\u000ber-\nent values of Hz(d) Dependence of Jon the driving frequency\n!for ^\u001a= 1:27.\ncles have an higher elevation from the surface of the FGF\nand the magnetic potential become smoother, featuring\nenergy minima at the centers of the magnetic bubbles.21\nThus, once deposited above the FGF, the particles form\na perfect triangular lattice for a normalized areal density\n^\u001a\u0011\u001aa= 1, Fig.1. Here \u001a=N=A is the particle number\ndensity,Nis the number of particles located within the\nobservation area A= 140\u0002105\u0016m2anda= 64\u0016m2\nis the area of the Wigner Seitz (WS) unit cell around\none bubble. For ^ \u001a>1, the excess particles redistributed\nwithin the magnetic domains, and each bubble became\npopulated by colloidal doublets, triplets or larger clus-\nters.\nWe induce particle motion by applying an external\nmagnetic \feld rotating in the ( x;z) plane, H\u0011\n(Hxsin (!t);0;Hzcos (!t)), with angular frequency !=\n2\u0019\u0017and amplitudes ( Hx;Hz), Fig.1(a). In most of the\nexperiments, we keep \fxed the amplitude of the in-plane\ncomponent Hx= 0:7kA=m , and change the ellipticity of\nthe applied \feld by varying the amplitude of the perpen-\ndicular component, Hz.\nFor amplitudes 0 :8kA=m < H z<1:2kA=m , and load-\ning ^\u001a > 1, the particle transport takes place via a\nswapping mechanism in doubly occupied domains, where\nadjacent particles synchronously exchange their posi-\ntions. The bubble lattice thereby preserves the over-\nall occupancy of one particle per domain. Increasing\nthe particle density, the swapping motion occurs in the3\nform of creation/destruction of doublets, triplets or even\ntetramers. However, for a wide range of particle densi-\nties, we \fnd that colloidal defects propagate mainly via\ndoublets swapping motion, and the latter is illustrated in\nFig. 1(b).\nTo explain the mechanism leading to the defect propa-\ngation, let us consider the arrangement of four bubbles\nwith their interstitial regions, as shown in right part of\nFig. 1(a). Energy calculations21show that when the ro-\ntating \feld becomes anti-parallel with respect to the bub-\nble magnetization, it generates in the interstitial regions\ntwo energy wells with triangular shape and opposite ori-\nentations,Iand\u0016I, with corners pointing towards the 11\nand\u00001\u00001 directions, respectively. In this situation,\nthere are two equivalent pathways along which an excess\nparticle can propagate towards the \u000011 direction, either\nalong theB\u0016IBpathway (dashed blue line), or along the\nBIB pathway (dashed green line). Both pathways are\nenergetically equivalent, and the particle's choice is dic-\ntated by the initial orientation of the doublet in the bub-\nble domain. A doublet initially oriented along the \u000021\ndirection will send a particle along the BIB pathway\n(Fig.1(b), \frst three images). Afterwards, this particle\nwill form another doublet oriented along the \u000012 direc-\ntion, which will send a particle along the complementary\nB\u0016IBpathway (Fig.1(b), last three images). When the\nmoving particle encounters a vacancy in the colloidal lat-\ntice, it will \fll it and the defect propagation will end\nthere, as shown in Fig.1(c).\nWe characterize the system conduction along the driv-\ning direction by measuring the particle \rux as J=\u001ahvi,\nwherehviis the average speed as determined from par-\nticle tracking. Fig.2(a) shows Jversus the dimensionless\ndensity ^\u001a, for three di\u000berent frequencies. Below the load-\ning ^\u001a= 1, i.e., having less than one particle per unit cell,\nthere are no excess particles, and thus J= 0. For ^\u001a>1,\nwe measure a net colloidal current which grows linearly\nwith the loading up to ^ \u001a\u00181:6. In this regime, only\nthe excess particles contribute to the current, while one\nparticle per magnetic bubble does not reach the mobility\nthreshold. The speed of the particles inside the WS unit\ncell is phase-locked with the driving \feld, and given by\nv=l!=2\u0019wherelis the lattice constant. Thus, increas-\ning the driving frequency increases the average speed and,\nin turn, the slope of the curve in Fig.2(a). The direction\nof motion of the excess particles is dictated by the chi-\nrality of the rotating \feld, thus changing the polarity of\none of the \felds ( HxorHz) allows to invert the entire\n\row of particles across the \flm.\nIncreasing the loading further (^ \u001a > 2), the current\nreaches a maximum, and jamming between closely mov-\ning colloids forbids further particle transport. Shown\nin Fig.2(b) is the e\u000bect that the perpendicular compo-\nnent of the applied \feld Hzhas on the particle current.\nThis component acts directly on the size of the mag-\nnetic domains, since the diameter of the magnetic bub-\nbles increases (decreases) when Hzis parallel (antipar-\nallel) to the bubble magnetization. The graph, obtained\nFIG. 3. (a)-(f) Sequence of images showing the reversible\nassembly of a colloidal lattice by switching the applied \feld\nbetween the two threshold \felds, Hc\n2andHc\n2. First, the mag-\nnetic particles were driven toward the 1 \u00001 direction (green\narrow in (b) by an applied \feld rotating in the ( x;z) plane\nwith components Hx= 0:7A=m andHz= 1:3A=m > Hc\n2,\nand frequency != 18:8s\u00001. In (d) the direction of motion\nis inverted and the perpendicular component of the \feld de-\ncreased to Hz= 0:9kA=m < Hc\n2. In (e) the \feld rotates\nin the (y;z) plane, and the excess particles is transported to-\nwards the \u00001\u00001 direction, \flling the whole lattice of bubbles,\nVideoS3 (Multimedia View).\nfor a \fxed density of ^ \u001a= 1:27, shows that the parti-\ncle \rux grows in discrete steps as the amplitude of Hz\nincreases. Below Hz=Hc\n1= 0:7kA=m , no current is\nobserved. Increasing Hzthe excess particles start to be\nmobilized, and the \rux raises till reaching the constant\nvalueJ= 6:1\u0016m\u00001s\u00001forHz>0:7kA=m . Increasing\nHzfurther, reveals a second \feld value, Hc\n2= 1:2kA=m ,\nwhere the periodic displacements of the BWs are able\nto drive all particles synchronously across the \flm. The\n\feld values Hc\n1andHc\n2are therefore the mobility edges\nwhere di\u000berent sub-ensembles of particles can be set into\nmotion.\nThe e\u000bect of the in-plane component of the applied \feld\nHxon the \rux Jis shown in Fig. 2(c). While Hzcon-\ntrols the size of the magnetic domains, the e\u000bect of Hxis\nto break the spatial symmetry of the potential, inducing\na net particle current towards a de\fned direction. When\nHz<Hc\n1, no current is observed for any value of Hx. For\nHz= 1:0kA=m the \rux is composed of excess particles,\nwhile forHz= 1:3kA=m all particles are mobilized and,4\nin both cases, when Hx>0:7kA=m it becomes indepen-\ndent on further increasing of Hx.\nFig.2(d) shows the dependence of Jon the driving\nfrequency for ^ \u001a= 1:27. The current displays a\nfull participation of the excess particles up to !\u0018\n44s\u00001, where the \rux reaches its maximum value, J=\n13:8\u0016m\u00001s\u00001corresponding to an average defect speed\nofv= 60\u0016ms\u00001. Beyond this frequency, the participa-\ntion of excess particles becomes partial, and the current\ndecreases monotonously, reaching zero near != 110s\u00001.\nIn this second regime, the high frequency motion of the\nparticles is found to be intermittent, with the excess par-\nticles randomly switching between being immobile for\nsome period, for afterwards becoming fully mobile again,\nreducing the e\u000eciency of our magnetic device.\nControlling the motion of the excess particles makes it\npossible to easily create or destroy a colloidal lattice\nby switching the applied \feld between the two mobil-\nity thresholds. This concept is demonstrated in Fig.3\n(Video3). Starting from a triangular arrangement with\none particle per WS cell (a) ( t= 0s), we apply an ellip-\ntically polarized magnetic \feld with components Hx=\n0:7kA=m ,Hz= 1:3kA=m>Hc\n2, and angular frequency\n!= 18:8s\u00001, which mobilize the whole lattice at a speed\nof 25:8\u0016ms\u00001towards the 1\u00001 direction, Figs.3(b,c),\nleaving the bubble substrate almost un\flled. Since the\nmoving particles are phase-locked with the driving \feld\nfor the used frequency, the translating lattice is stable\nand preserves the initial triangular order during motion.\nAftert\u001810s, we change the polarity of the in-plane \feld\n(Hx!\u0000Hx), in order to invert the particle \rux towards\nthe\u000011 direction. We also decrease the \feld amplitude\ntoHz= 0:9kA=m < Hc\n2, inducing a current composed\nonly of excess particles which start \flling again the bub-\nble lattice in the bottom part of the \flm, Figs.3(d,e).\nSince, during these operations the top part of the \flm is\nleft un\flled, we change the orientation of the applied \feld\naftert= 26:7s. The \feld now rotates in the ( y;z) plane,\nand transports the excess particles toward the \u00001\u00001\ndirection, reforming the colloidal crystal after 31 s. The\ncolloidal assembly demonstrated here can be further opti-\nmized by either controlling the particle density, and thus\nthe amount of excess particles propagating through the\nlattice, or by increasing the driving frequency, up to a\nmaximum speed of v= 60\u0016ms\u00001, which will further\nspeed up the re-writing process.\nIn conclusion, we demonstrate a technique to remotely\ngenerate and control the motion of defects in two dimen-\nsional lattices, while keeping track of the position of the\nindividual particles, which could be used as model system\nto study the dynamics of impurities in crystalline materi-\nals. The energy scales involved in our system ( \u0018150kBT,\nwithT\u0018293K) are much beyond the e\u000bect of ther-mal \ructuations, which could interfere with the colloidal\ntransport process. This makes our magnetic device fully\ncontrollable, ensuring an extremely precise tuning of the\nparticle speed and dynamics in real time and space. Al-\nthough our experiments focus on using FGF \flms as func-\ntional platform, the technique reported here should be\napplicable, within the constraint of colloidal particle size\nand lattice wavelength, to other platforms where mag-\nnetic patterns are created by \"top-down\" fabrication pro-\ncesses.\nP. T. acknowledges support from the ERC starting grant\n\"DynaMO\" (No. 335040) and from the programs No.\nRYC-2011-07605, FIS2011-15948-E. T. H. J. thanks the\nResearch Council of Norway. T. M. F. acknowledges sup-\nport by the DFG via the center of excellence SFB 840.\n1A. Terray, J. Oakey, and D. W. M. Marr, Science 296, 1841\n(2002).\n2T. Sawetzki, S. Rahmouni, C. Bechinger, D. W. M. Marr, Proc.\nNatl. Acad. Sci. USA 105, 20141 (2008).\n3B. Kavcic, D. Babic, N. Osterman, B. Podobnik, I. Poberaj,\nAppl. Phys. Lett. 95, 023504 (2009).\n4U. H afeli, W. Schutt, J. Teller, M. Zborowsk, Scienti\fc and Clin-\nical Applications of Magnetic Carriers , Plenum Press: New York\nand London.\n5M. M. Miller, P. E. Sheehan, R. L. Edelstein, C. R. Tamanaha,\nL. Zhong, S. Bounnak, L. J. Whitman, R. J. Colton, J. Magn.\nMater., 225, 138144 (2001).\n6D. L. Graham, H. A. Ferreira, P. P. Freitas, Trends. Biotechnol.\n22, 455 (2004).\n7U. Jeong, X. Teng, Y. Wang, H. Yang, and Y. Xia, Adv. Mater.\n19, 33 (2007).\n8K. Gunnarsson, P. E. Roy, S. Felton, J. Pihl, P. Svedlindh, S.\nBerner, H. Lidbaum and S. Oscarsson, Adv. Mater. 17, 1730\n(2005).\n9B. Yellen, O. Hovorka and G. Friedman, Proc. Natl. Acad. Sci.\nUSA 102, 8860 (2005).\n10M. Donolato, P. Vavassori, M. Gobbi, M. Deryabina, M. F.\nHansen, V. Metlushko, B. Ilic, M. Cantoni, D. Petti, S. Brivio\nand R. Bertacco, Adv. Mater. 22, 2706 (2010).\n11T. Henighan, D. Giglio, A. Chen, G. Vieira, and R. Sooryakumar,\nAppl. Phys. Lett. 98, 103505 (2011).\n12A. Ehresmann, D. Lengemann, T. Weis, A. Albrecht, J. Langfahl-\nKlabes, F. G ollner, D. Engel, Adv. Mater. 23, 5568 (2011).\n13A. F. Demir ors, P. P. Pillai, B. Kowalczyk, B. A. Grzybowski,\nNature 503, 99 (2013).\n14R. Seshadri and R. M. Westervelt, Phys. Rev. Lett. 70, 234\n(1993).\n15A. H. Eschenfelder, Magnetic Bubble Technology (Springer Berlin\nHeidelberg, 1980).\n16T. H. Johansen and D. V. Shantsev, Magneto-Optical Imaging ,\nNATO Science Series II: Mathematics, Physics and Chemistry\nvol. 142 (Dordrecht, Kluwer Academic 2004).\n17P. Tierno, F. Sagu\u0013 es, T. H. Johansen, T. M. Fischer, Phys. Chem.\nChem. Phys. 11, 9615 (2009).\n18L. E. Helseth, P. E. Goa, H. Hauglin, M. Baziljevich, T. H. Jo-\nhansen, Phys. Rev. B 65, 132514 (2002).\n19P. E. Goa, H. Hauglin, A. A. F. Olsen, D. V. Shantsev, and T.\nH. Johansen, Appl. Phys. Lett. 82, 79 (2003).\n20P. Tierno, Soft Matter, 8, 11443 (2012).\n21See supplementary material at [URL] for details on the theoret-\nical model used in the text to calculate the energy landscape."    },    {        "title": "2310.15867v1.Complex_Poynting_vector_in_gyromagnetic_media_and_its_impact_on_power_flow_in_guided_modes.pdf",        "content": "Complex Poynting vector in gyromagnetic media and its impact on power flow in\nguided modes\nRajarshi Sen1and Sarang Pendharker1\n1Department of Electronics and Electrical Communication Engineering,\nIndian Institute of Technology Kharagpur, Kharagpur, West Bengal, India\n(Dated: Wednesday 25thOctober, 2023)\nIn this paper, we show the relation between the time-varying spinning nature of the instantaneous\nPoynting vector and the phasor form of the complex Poynting vector in the bulk of the gyromag-\nnetic medium. We show the presence of a transverse reactive power component in the bulk of the\ngyrotropic medium, even for plane wave propagation. We use a simple quantification technique of\nPoynting vector spin to analyze the rotation of the instantaneous Poynting vector using the com-\nplex phasor form of the time-averaged Poynting vector. For a transverse electric mode, we show\nthe similarity between the Poynting vector spin and the photonic spin of the magnetic field. The\nPoynting vector spin is then used to represent the transverse power transfer across a ferrite-air\ninterface supporting TE surface wave modes. Considering a gyromagnetic ferrite-filled rectangular\nwaveguide following the TE mode profile, we show the correspondence between the Poynting vector\nspin and the overall positive and backward power flow. We analytically propose a mechanism to\nengineer the region of the waveguide supporting backward and forward power propagation.\nI. INTRODUCTION\nGyrotropic materials form an important class of\nmaterials encompassing naturally occurring and engi-\nneered materials. Gyrotropic materials can be broadly\nclassified into gyroelectric and gyromagnetic materials.\nAmong these two classes, gyromagnetic materials have\nseen widespread applications due to their availability\nin the microwave frequency region. Microwave ferrites\nhave been extensively used to realize isolators[1, 2],\ncirculators[3–5], and phase shifters[6–8]. Apart from\nthese conventional applications, they have been recently\nused to realize nonreciprocity in numerous applications,\nsuch as topological insulators[9, 10], thin films[11, 12],\nand mode-converting waveguides[13, 14]. Despite the ex-\ntensive use of gyromagnetic media in various microwave\nand millimeter wave applications, an understanding of\nimportant physical phenomena corresponding to such\nmedia was still missing. Researchers consider isofre-\nquency surfaces helpful in investigating the wave propa-\ngation phenomena in complex media. These isofrequency\nsurfaces have been used to investigate wave propagation\ncharacteristics in both natural[15–17] as well as engi-\nneered materials[18–21].\nWe recently performed a thorough investigation of the\nnonreciprocal photonic spin profile in the bulk of the me-\ndia using isofrequency surfaces for gyrotropic media[22].\nThis investigation widened our understanding of the im-\npact of gyrotropy over manipulating the plane wave prop-\nagation in the bulk of the media and the existence of\nasymmetrical photonic spin profiles for guided modes.\nHowever, a comprehensive understanding of power prop-\nagation in gyrotropic media is still missing. We believe\na detailed investigation of the Poynting vector investiga-\ntion will help us reveal key insights into the wave propa-\ngation behavior in gyrotropic materials and provide gov-\nerning principles for engineering novel applications.\nPoynting vector, which corresponds to the power flux\ndensity, has been used extensively in different media andstructures [23–32] to observe the propagation of power.\nConventionally, the Poynting vector is represented us-\ning the instantaneous and time-averaged forms. The\ntime-averaged Poynting vector gives us an idea about\nthe overall power flow in the medium. The most general\nform of this time-averaged Poynting vector is its complex\nform, consisting of both real and imaginary components,\nwhere the imaginary component corresponds to the reac-\ntive power component. However, we mostly consider only\nthe real power component because, for most applications,\nthe investigation of real power itself is sufficient, such as\nreflection of power from metasurfaces[33, 34], coupling\nof power for guided modes[35–37], and trapping of light\nfor zero and negative Poynting vector[38, 39]. However,\nneglecting the imaginary (reactive) part of the complex\nPoynting vector leads to an incomplete picture of the\npower flow in the medium.\nIt has been shown that the complex Poynting vector,\nwhich includes both real and reactive power component\nplays an important role in the investigation of propa-\ngating beams[40], the near field of Hertzian dipoles[41],\nevanescent waves[42, 43] and optical forces over particles\n[40, 44, 45]. Recently, we have seen research efforts being\nmade to link this complex time-averaged Poynting vec-\ntor with the time-varying instantaneous Poynting vector.\nLitvin has demonstrated the relationship between com-\nplex time-averaged Poynting vector and the elliptical ro-\ntation of the instantaneous Poynting vector in the case of\nGaussian and Bessel beams[46]. Kim et al. made similar\nobservations in [47], where the longitudinal and trans-\nverse components of the time-averaged Poynting vector\nare used to explain the spinning nature of the instan-\ntaneous Poynting vector across an interface-supporting\nsurface wave. This transverse reactive power is crucial as\nit can explain the source of backward power propagation\nobserved in guided modes. Such investigation into the\nnature of the Poynting vector in gyrotropic materials is\nmissing.\nIn this paper, we show that gyrotropic material sup-arXiv:2310.15867v1  [physics.optics]  24 Oct 20232\nTABLE I. YIG ferrite material specifications\nParameter Specification\nMagnetic saturation (4 πM s) 1800 Gauss\nMagnetic bias ( H0) 3570 Oersted\nDielectric permeability ( ϵf) 14\nports a complex Poynting vector in the bulk for plane\nwave propagation. We further analyze the complex\nPoynting vector with the help of isofrequency surfaces\nand link it with the spinning nature of the instanta-\nneous Poynting vector for plane waves in the bulk of the\nmedium. We quantify this rotation of the instantaneous\nPoynting vector by introducing a Poynting vector spin.\nIn the special case of transverse electric mode, we show\nthat the Poynting vector spin is directly proportional to\nthe photonic spin of the magnetic field (Im( ⃗H∗×⃗H)).\nWe then use this Poynting vector spin to explain the in-\nstantaneous transverse power flow across an interface be-\ntween gyromagnetic ferrite and air. Further, considering\na ferrite-filled rectangular waveguide, we show the corre-\nspondence between the backward propagation of power\ndensity and the opposite sense of Poynting vector spin.\nWe provide guiding principles, using which the zero-\npower density point in the cross-section of the waveguide\ncan be engineered. The results presented in this paper\nplay a crucial role in understanding the nature of the\ninstantaneous Poynting vector in gyrotropic media and\nthe corresponding relation between the reactive and real\ncomponents of the time-averaged Poynting vector.\nThis paper is organized as follows. In section II, we\nanalyze the power flow phenomenon in the bulk of the\ngyromagnetic medium. Considering the transverse elec-\ntric mode, in section III, we quantify the temporal rota-\ntion of the instantaneous Poynting vector. In section IV,\nusing the understanding of the temporal behavior of the\nPoynting vector for TE mode in gyromagnetic media, we\nconsider two different cases of surface wave propagation.\nWe reveal the role of transverse reactive power in sup-\nporting the unidirectional surface wave propagation in\ngyromagnetic ferrite. Further, in section V, we report the\npresence of negative power flow in a ferrite-filled waveg-\nuide, which is validated using full-wave simulations. Our\nanalytical expressions help in engineering the Poynting\nvector distribution across the cross-section of the waveg-\nuide. Finally, we conclude our results in section VI.\nII. POYNTING VECTOR IN BULK OF\nGYROMAGNETIC MEDIA\nFor a comprehensive investigation of power flow over\ngyrotropic material, we select YIG ferrite as our material\nof choice. The material specifications are given in Ta-\nble I. Using the YIG ferrite material parameters, we can\nfind the permeability and gyrotropy terms, µ′andκ′, re-\nspectively, corresponding to the operating frequency. We\nFIG. 1. (a) Graphical illustration of the temporal variation of\nthe instantaneous Poynting vector (blue conical trace) in an\nelliptical manner around the time-averaged Poynting vector\n(red arrow). (b), (c) Isofrequency curves corresponding to the\nelliptical and hyperbolic regimes at frequencies 6 and 11 GHz,\nrespectively. The isofrequency curves are color-mapped with\nthe transverse reactive power. The arrows represent the time-\naveraged Poynting vector normal to the isofrequency surfaces.\ncan express the anisotropic permeability tensor for the\nˆz-biased ferrite using these anisotropy terms as\n↔µr=\nµ′−jκ′0\njκ′µ′0\n0 0 1\n. (1)\nConsidering the uniaxial nature of gyromagnetic ferrite,\nwe can simplify the investigation by analyzing the isofre-\nquency surfaces corresponding to a 2D propagation sce-\nnario. However, the selected 2D plane should be parallel\nto the bias-axis, which in this case is the ˆ z-axis. Here, we\nselect the X−Zplane as the plane of wave propagation,\nand correspondingly, we will have wave vector compo-\nnent ky= 0 and non-zero value of in-plane wave-vector\ncomponents kxandkz. For this work, we exclude all the\nlosses for simplicity. Thus, the material parameters µ′\nandκ′, and the wave-vector components kxandkzalong\nthe isofrequency curves will be purely real quantities.\nOnce we have computed the isofrequency contour, we\ncan find the field vector intensities for the bulk of the\nmedium using the method followed in [22]. Using the\nfield intensities, we can finally compute both the instan-3\ntaneous pointing vector ⃗Pins= Re( ⃗E)×Re(⃗H) and time-\naveraged Poynting vector ⃗Pavg= 0.5⃗E∗×⃗Hin its Phasorform (see Appendix A).\n⃗Pins=H2\n0k2\n0  \nkx(ϵfk2\n0κ′2(k2\nx−ϵfk2\n0)2+k2\nz(k2\nx+k2\nz−ϵfk2\n0µ′)2)\n2ωϵ0(k2x−ϵfk2\n0)2(k2x+k2z−ϵfk2\n0µ′)2\n+kx(−ϵfk2\n0κ′2(k2\nx−ϵfk2\n0)2+k2\nz(k2\nx+k2\nz−ϵfk2\n0µ′)2)\n2ωϵ0(k2x−ϵfk2\n0)2(k2x+k2z−ϵfk2\n0µ′)2cos(2 ωt)!\nˆx−κ′kx(ϵfk2\n0−k2\nx−k2\nz)\n2ωϵ0(ϵfk2\n0−k2x)(k2x+k2z−ϵfk2\n0µ′)sin(2ωt)ˆy\n+ \nkz((k2\nx+k2\nz−ϵfk2\n0µ′)2+ϵfk2\n0κ′2(ϵfk2\n0−k2\nx))\n2ωϵ0(ϵfk2\n0−k2x)(k2x+k2z−ϵfk2\n0µ′)2+kz((k2\nx+k2\nz−ϵfk2\n0µ′)2−ϵfk2\n0κ′2(ϵfk2\n0−k2\nx))\n2ωϵ0(ϵfk2\n0−k2x)(k2x+k2z−ϵfk2\n0µ′)2cos(2 ωt)!\nˆz!\n(2)\n⃗Pavg=H2\n0k2\n0 \nkx(ϵfk2\n0κ′2(k2\nx−ϵfk2\n0)2+k2\nz(k2\nx+k2\nz−ϵfk2\n0µ′)2)\n2ωϵ0(k2x−ϵfk2\n0)2(k2x+k2z−ϵfk2\n0µ′)2ˆx\n+jκ′kx(ϵfk2\n0−k2\nx−k2\nz)\n2ωϵ0(ϵfk2\n0−k2x)(k2x+k2z−ϵfk2\n0µ′)ˆy+kz(ϵfk2\n0κ′2(ϵfk2\n0−k2\nx) + (k2\nx+k2\nz−ϵfk2\n0µ′)2)\n2ωϵ0(ϵfk2\n0−k2x)(k2x+k2z−ϵfk2\n0µ′)2ˆz!\n(3)\nHere, k0denotes the free-space propagation constant. We\nobserve that the instantaneous Poynting vector for real\npower ⃗Pinstin the bulk YIG ferrite contains both time-\nvarying and time-invariant terms, which we can see in\neq. (2). Interestingly, the time-invariant terms exist only\nalong the plane of wave propagation, i.e., having x- and\nz-components only. On the contrary, the time-varying\ncomponent of ⃗Pinstexists along the X−Zplane as well as\ntransverse ˆ y-direction. The transverse component, how-\never, is 90◦out of phase with respect to the in-plane\ncomponent, as indicated by their sine and cosine terms,\nrespectively. This phase difference between the orthogo-\nnal components leads to an elliptical sweep of ⃗Pinstwith\nrespect to time. This spatio-temporal sweep of ⃗Pinstis\nillustrated in Fig. 1(a). For simplicity, we have consid-\nered only one direction of propagation having positive\nvalues of kxandkz. The rotational phenomenon is de-\npicted in this figure with the help of a dashed green curve\naround the elliptical trace. Also, we have taken four time-\nsamples t= 0, t=T/8,t=T/4, and t= 3T/8, which\ncorrespond to the phases 0, π/4,π/2, and 3 π/4, respec-\ntively. The ⃗Pinstat these four time-instances are shown\nalong the elliptical trace.\nAt this point, we make an important observation from\neq. (2) and Fig. 1(a). On time-averaging the instanta-\nneous Poynting vector, the net real power component\nalong the transverse direction vanishes, and only a real\ncomponent remains along the plane of wave propagation.\nThis non-zero real component of time-averaged power is\nrepresented using a red arrow in Fig. 1(a). However, on\nconsidering the complex time-averaged ⃗Pavgof eq. (3),\nwe find that there exists a purely imaginary transverse\nreactive component along ˆ y-axis. The magnitude of thistransverse reactive power component of ⃗Pavgin eq. (3) is\nsimilar to the magnitude of the transverse component of\n⃗Pinstin eq. (2), and is important in defining the axis of\nthe elliptical trace along ˆ y-axis. Thus, we can say that,\nby considering both the real and reactive power compo-\nnents of the complex ⃗Pavg, the complete time-varying be-\nhavior of ⃗Pinstcan be understood. Considering the cho-\nsen ferrite medium, we now compute the time-averaged\nPoynting vector ⃗Pavgfor two frequencies f= 6 GHz and\n11 GHz using eq. (3) while considering H0= 1 for sim-\nplicity. These two frequencies correspond to the ellip-\ntical and hyperbolic regimes with positive and negative\nvalues of µ′, respectively. The ( µ′,κ′) values for these\ntwo frequencies are (1 .79, 0.47), and ( −1.38,−2.62), re-\nspectively. In these two regimes, the values of µ′andκ′\nare suitable for supporting two independent isofrequency\nsurfaces[22]. Hence, with the help of these two cases, we\ncan completely observe the power flow mechanism in the\nbulk of gyromagnetic media.\nThe real part of ⃗Pavgdenoting the real power in the\nmedium is shown using arrows over the isofrequency\ncurves, and the imaginary component corresponding to\nthe reactive power is represented using colormap in\nFig. 1(b) and (c) for these two frequencies, respectively.\nWe can observe the direction of the real component of\n⃗Pavgis always normal to the isofrequency curve. This\ngeometrical relation of the time-average real-power prop-\nagation being normal to the isofrequency surface can\nbe verified throughout the isofrequency surface cover-\ning all propagation directions. Further, we observe that\nthe time-average Poynting vector for closed isofrequency\ncurves such as ellipses diverges outward. Whereas, for\nopen curves such as hyperbolas, the Real time-average4\nPoynting vector tends to converge along the transverse\naxis of the hyperbola.\nThis phenomenon of time-averaged Poynting vector\n(which only has a real component) being normal to the\nisofrequency surface has already been investigated for\nnon-gyrotropic anisotropic media [48–50]. However, for\ngyrotropic material, the scenario is more complex with\nthe presence of the imaginary component of ⃗Pavg, which\naccounts for the overall reactive power in the bulk of the\nmedium. The transverse reactive power (aligned along ˆ y-\naxis) for propagation along X−Zplane changes its sign\nfor forward and backward propagation corresponding to\n+ˆx−and−ˆx−direction, respectively. In uniaxial media\nsuch as gyromagnetic ferrite, considering the symmetry\nof the isofrequency surface around the bias axis, the 2D\nanalogous isofrequency curves portrayed the complete in-\nformation. Now, since the transverse reactive power fol-\nlows opposite sign along the opposite sides of the 2D\nisofrequency curve, the rotation of the isofrequency curve\nto form a complete isofrequency surface will give us an\nimpression of the reactive power flux to encircle the bias\naxis of the gyromagnetic medium. This intrinsic reactive\npower in plane waves in the bulk of the medium results\nfrom the gyrotropy of the material. The imaginary reac-\ntive power component of eq. (3) is linearly proportional\nto the gyrotropy term κ′. Thus, we can infer that in the\nabsence of gyrotropy, the reactive power in the bulk of the\nmedium vanishes. In the next section, we shall quantify\nthe rotation of the instantaneous Poynting vector ⃗Pinst.\nIII. QUANTIFICATION OF POYNTING\nVECTOR ROTATION\nThe temporal behavior of ⃗Pinsthas a strong relation\nwith the phasor form of complex time-averaged Poynt-\ning vector ⃗Pavg. Understanding the time-varying nature\nof⃗Pinstin complex media such as gyrotropic media is\na challenge. To solve this challenge, we quantify the\nrotation of ⃗Pinstusing the definition of Poynting vec-\ntor spin ⃗Sp= Im( ⃗P∗\navg×⃗Pavg). This method of defin-\ning the Poynting vector spin is similar to the way pho-\ntonic spin is defined for the electric and magnetic fields\nasSe= Im( ⃗E∗×⃗E) and Sh= Im( ⃗H∗×⃗H), respectively.\nNote that ⃗Spshall be non-zero only when ⃗Pavgis com-\nplex, which signifies the quantification of the rotation of\nthe instantaneous Poynting vector ⃗Pinstusing the real\nand reactive power components.\nLet us now consider a special condition of wave prop-\nagation in the bulk of YIG ferrite when the direction of\npropagation is along ±ˆx-axis. Under this condition, one\nof the propagation modes corresponds to the transverse\nelectric (TE) mode, supporting transverse electric Ez,\ntransverse magnetic Hy, and longitudinal magnetic field\ncomponents Hx. In this special case of TE propagation,\nthe⃗Spbecomes proportional to ⃗Sh(see Appendix B),\nas electric field spin in this mode is absent. This ab-\nsence of electric field spin simplifies our investigation asthe computation of photonic spin corresponding to the\nmagnetic field shall give us the Poynting vector spin ⃗Sp.\nWe will now use this approach of quantification of rota-\ntion of the Poynting vector to investigate the real and\nreactive power propagation in ferrite-associated surface\nwaves and waveguide modes.\nIV. POYNTING VECTOR SPIN ACROSS\nFERRITE-AIR INTERFACE\nSurface wave modes propagate along the interface be-\ntween two media while being evanescent in the direc-\ntion transverse to the interface. This surface wave con-\nfinement is typically realized in an interface between\na dielectric and a medium supporting either negative\npermittivity[51–53] or permeability value[16, 54]. In [47],\nKim et al. considered surface wave propagation along\nan interface between a dielectric and a negative permit-\ntivity medium. They showed the relation between the\nopposite sense of rotation of ⃗Pinstand the direction of\nreal power flow Re( ⃗Pavg) across the interface. However,\nsuch negative permittivity materials are reciprocal. Gy-\nromagnetic ferrite supports negative µeff= (µ′2−κ′2)/µ′\ncorresponding to the TE propagation mode for a par-\nticular frequency region, which depends on the material\nparameters and applied bias. We can facilitate highly\nnonreciprocal surface wave propagation along a ferrite-\nair interface in this frequency range.\nWe now consider two separate interfaces for investigat-\ning power flow for surface wave propagation: (a) nega-\ntive permeability medium (NPM)-air and (b) ferrite-air\ninterface. The medium parameters considered for the\nNPM-air interface are ϵnp= 1, µnp=−3. Whereas, for\nthe YIG ferrite, we consider material parameters given in\nTable I, except the 4 πMs, which is taken as 3750 Gauss.\nThe magnetic bias applied in the ferrite material is along\nthe−ˆz-axis. For this material configuration of the YIG\nferrite, the effective permeability in ferrite for TE mode\nof propagation µeffbecomes negative within a particular\nfrequency band. We shall only consider this special fre-\nquency band in this investigation of surface wave modes\nalong the ferrite-air interface.\nThe schematic representation for the surface wave\npropagation along the NPM-air and ferrite-air interface\nis shown in 2(a) and (f), respectively. The X−Zplane\nis the plane of the interface at y= 0. The waves will\nencounter evanescence conditions along |y|>0. Next,\nwe see the dispersion relation in these two cases. First\nwe find the field components Ez,Hx, and Hzcorre-\nsponding to TE mode, The field components are as-\nsigned eαydy+j(kxx−ωt)ande−αyuy+j(kxx−ωt)fory≤0\nandy≥0, respectively. Here, kxis the longitudinal\npropagation constant, αydis the attenuation constant in\nthe air, and αyuis the attenuation constant for the NPM\nand ferrite regions.\nNext, we find the dispersion relations by applying the\nboundary conditions. For the NPM-air interface, the dis-\npersion curve is simpler, and we get a closed-form expres-5\nFIG. 2. Surface wave propagation along (a-e) NPM-air and (f-j) ferrite-air interfaces. (a,f) Schematic representation of the\nsurface wave propagation, (b,g) dispersion curves, (c,h) Re( Ez) as colormap showing transverse electric field component for the\nsurface wave and traces of ⃗Pinst, (d,i) Real power component Re( ⃗Pavg), and (e,j) Poynting vector spin ⃗Sp\nsion for kxas\nkx=±k0s\nµaµnp(ϵnpµa−ϵaµnp)\nµ2a−µ2np(4)\nwhereas for the ferrite-air interface, we get the following\ncharacteristic equation\nH0k0(kx(αyu+αydµ′) +κ′(αydαyu−ϵfk2\n0)) = 0 (5)\nwhere αydand αyuare functions of kx,k0and the\nmedium parameters (Appendix D). k0is the free space\nwavelength. We now find the wave-vector value kxfor\nthe propagation mode along the NPM-air interface and\nferrite-air interface from eq. (4) and (5), respectively.\nThe dispersion curves for these two cases are shown in\nFig. 2(b) and (g), respectively. We observe that the dis-\npersion relation for the NPM-air interface is linear and\nreciprocal, whereas there is significant nonreciprocity for\nthe ferrite-air interface. Backward propagation ( kx<0)\nexists only for a small frequency region, and most of this\nspecial frequency region supports unidirectional surface\nwave propagation. We now select frequencies f1= 5 GHz\nandf2= 15.31 GHz to investigate further the Poynting\nvector corresponding to the NPM-air and ferrite-air in-\nterface, respectively. These frequencies are marked in the\ndispersion curves using dotted lines.\nThe electric field component Ezfor the NPM-air and\nferrite-air interfaces are shown in Fig. 2(c) and (h), re-\nspectively. We can observe strong confinement along the\ninterface for the surface mode in both cases. To inves-\ntigate the time-varying nature of ⃗Pinst, we select five\ntime-steps t= 0, 0 .08T, 0.16T, 0.34T, and 0 .42T.⃗Pinst\ncorresponding to these time-steps are overlayed over the\nelectric field plot. Interestingly, we observe a contrastingbehavior for the sense of rotation of ⃗Pinst. The sense of\nrotation of the instantaneous Poynting vector is opposite\nacross the NPM-air interface. Whereas, for the ferrite-air\ninterface, we observe that ⃗Pinstrotation follows the same\nsense.\nUsing the expression for the field intensities, we find\nthe analytical expressions for real and reactive compo-\nnents of the time-averaged Poynting vector. These real\nand imaginary power components for the NPN-air inter-\nface are (Appendix C)\nRe(⃗Pavg) =\n\nk2\n0µa(α2\nyd+ϵak2\n0µa)e2αydy\n2ωϵ0kxα2\nydˆx ify≤0\nk2\n0µnp(α2\nyu+ϵnpk2\n0µnp)e−2αyuy\n2ωϵ0kxα2yuˆxify≥0\n(6)\nIm(⃗Pavg) =\n\n−k2\n0µae2αydy\n2ωϵ0αydˆyify≤0\nk2\n0µnpe−2αyuy\n2ωϵ0αyuˆyify≥0(7)\nand for the ferrite-air interface (Appendix D)\nRe(⃗Pavg) =\n\nk2\n0(α2\nya+ϵak2\n0)e2αyay\n2ωϵ0kxα2yaˆx ify≤0\nk2\n0(α2\nyn+ϵfk2\n0µ′)(αynκ′+kxµ′)e−2αyny\n2ωϵ0(ϵfk2\n0κ′−αynkx)2 ˆx\nify≥0\n(8)\nIm(⃗Pavg) =\n\n−k2\n0e2αyay\n2ωϵ0αyaˆy ify≤0\n−k2\n0(αynκ′+kxµ′)e−2αyny\n2ωϵ0(ϵfk2\n0κ′−αynkx)2ˆyify≥0(9)\nFrom eq. (6) and (8), the real component of ⃗Pavgis com-\npletely along ˆ x-axis. This real component of ⃗Pavgis6\nshown in Fig. 2(d) and (i) corresponding to the NPM-air\nand ferrite-air interfaces, respectively. Additionally, for\nthe NPM-air interface, the analytical results are verified\nusing CST Microwave Studio simulations. We observe\nthat for the NPM-air interface, the real power propagates\nalong opposite directions on either side of the interface.\nIn contrast, the direction of real power propagation re-\nmains consistent for the ferrite-air interface. This behav-\nior of real power flow being opposite in the two opposite\nsides across the NPM-air interface and in the same direc-\ntion for the ferrite-air interface is in agreement with the\ncorresponding sense of rotation of ⃗Pinst. This matching\nbehavior of propagation direction of real power and rota-\ntional sense of ⃗Pinstrequires the computation of Poynt-\ning vector spin across the NPM-air interface ( ⃗Sp,np) and\nferrite-air interface ( ⃗Sp,f).\nThe analytical equations for ⃗Sp,npand⃗Sp,fafter nor-\nmalizing the magnetic field spatially are\n⃗Sp,np=\n\n−2αydkx(α2\nyd+ϵaµak2\n0)\nα4\nyd+ϵ2aµ2ak4\n0+α2\nyd(k2x+2ϵaµak2\n0)ˆz ify≤0\n2αyukx(α2\nyu+ϵnpµnpk2\n0)\nα4yu+ϵ2npµ2npk4\n0+α2yu(k2x+2ϵnpµnpk2\n0)ˆzify≥0\n(10)\n⃗Sp,f=\n\n−2αydkx(α2\nyd+ϵak2\n0)\nα4\nyd+ϵ2ak4\n0+α2\nyd(2ϵak2\n0+k2x)ˆz ify≤0\n−2(ϵfk2\n0κ′−αyukx)(α2\nyu+ϵfk2\n0µ′)\nα4yu−2αyuϵfk2\n0κ′kx+α2yu(k2x+2ϵfk2\n0µ′)+ϵ2\nfk4\n0(µ′2+κ′2)ˆz\nify≥0\n(11)\nThe subsequent plots of the Poynting vector spin ⃗Sp,np\nand ⃗Sp,facross the NPM-air and ferrite-air interfaces\nare shown in Fig. 2(e) and (j), respectively. As expected\nfrom the sense of rotation of ⃗Pinst, we observe the cor-\nrespondence between the sign of ⃗Spand the direction of\nRe(⃗Pavg).\nFor the ferrite-air interface in the specific frequency\nrange having negative µeff,⃗Spmaintains the same sign\nbut a different magnitude across either side of the in-\nterface. The inequality of the magnitude of ⃗Spcorre-\nsponds to the discontinuity of the real power propaga-\ntion at the interface. Further, the ⃗Spwill also indicate\nthe propagation of imaginary reactive power across the\ninterface. Previously, it was observed that the material-\ninduced photonic spin is locked to ferrite material for\nbidirectional wave propagation[22]. Similarly, the Poynt-\ning vector spin also remains locked in the ferrite medium\nirrespective of the direction of propagation. In the next\nsection, we consider a ferrite-filled rectangular waveg-\nuide and observe this phenomenon of opposite signs of\n⃗Spfor forward and backward real power propagation in\nthe waveguide.\nFIG. 3. (a) Illustration of a ˆ z-biased ferrite waveguide with\ncross-section of dimension a×b. The homogeneous bias H0\nis represented using red arrows. The direction of propagation\nis along ˆ x-axis. (b) and (c) shows the ⃗Spand Re( ⃗Pavg) along\nthe cross-section plane for TE 10mode. The plots include both\nanalytical and simulation results.\nV. POWER FLOW DYNAMICS IN\nFERRITE-FILLED WAVEGUIDE\nPreviously, we have performed a detailed study of the\nrole of gyrotropy in imposing a dominant photonic spin\nover the cross-section of a YIG ferrite-filled waveguide\n(section II.C of [22]). The dominant photonic spin re-\nsults from the interaction of the material-induced and\nstructure-induced photonic spins. Here, we understand\nthe resulting effect of the gyrotropy over the power flow\nin the waveguide. Let us consider a rectangular waveg-\nuide of cross-section a= 5, b= 3 mm, and length l= 30\nmm as shown in Fig. 3(a), which is filled with ˆ z-biased\nYIG ferrite (specifications similar to Table I). The oper-\nating frequency is selected as 7 GHz. The waveguide\nwalls are assigned PEC boundary conditions to avoid\nlosses. In this waveguide, TE 10is the dominant mode,\nand we shall select this mode for our power-flow anal-\nysis. Correspondingly, the field variations for the TE 10\nmode shall be along the broad dimension of the waveg-7\nFIG. 4. (a) 2D cross-section view of the waveguide depicting\nthe arrangement of the real power and the null-power point\nyc. The variation of the null-power point ycas a ratio with the\nhalf broad-side width a/2, corresponding to (b) the variation\nof broad-side width aand (c) frequency of operation f. Both\nanalytical and simulation results are shown.\nuide, i.e., along ˆ y-axis. The field equations for this YIG\nferrite-filled waveguide in TE 10mode of operation can be\nwritten as\nEz=Aωcoskyy\nkyejkxx(12)\nHx=jA(κ′kxcoskyy−µ′kysinkyy)\nkyµ0(κ′2−µ′2)ejkxx(13)\nHy=A(µ′kxcoskyy−κ′kysinkyy)\nkyµ0(κ′2−µ′2)ejkxx(14)\nSimilar to a dielectric-filled waveguide, we have ky=π/a\nin this ferrite-filled waveguide. The normal component of\nmagnetic flux density is zero on the surface of the PEC\nwall, i.e., ˆ y·⃗B= 0. This condition would also make the\nnormal component of the magnetic flux density equal to\nzero along the waveguide wall in dielectric-filled waveg-\nuides. However, in a gyromagnetic ferrite medium, due\nto the off-diagonal terms of the permeability tensor, we\nhave a non-zero normal component of magnetic field in-\ntensity along the waveguide wall at y=±a/2. This non-\nzero component can be observed in eq. (14) by puttingy=±a/2. At y=±a/2 we have sin kyy=±1 and\ncoskyy= 0. Hence, the non-zero HxandHyat the\nwaveguide wall correspond to non-zero Poynting vector\nspin ⃗Spalong the waveguide wall. The mathematical ex-\npression for the ⃗Spalong the ˆ y-axis of the waveguide is\n⃗Sp=2A2\nk2yµ2\n0(µ′2−κ′2)2(kyµ′sinkyy\n−kxκ′coskyy)(kxµ′coskyy−kyκ′sinkyy)ˆz(15)\nUsing this equation of ⃗Sp, we analytically compute the\nPoynting vector spin across the waveguide cross-section,\nwhich is further validated using CST Microwave Studio\nin Fig. 3(b). We observe a contrasting behavior of the\nPoynting vector spin in this ferrite-filled waveguide with\nrespect to the conventional dielectric waveguide. In the\nlatter case, an equal amount of positive and negative ⃗Sp\nexists across the center of the cross-section of the waveg-\nuide. Whereas, in this ferrite-filled waveguide, the ⃗Sp\ncrosses the zero-line in two places. The first cross-over\npoint is now offset from the center of the waveguide, and\nthe second cross-over point (non-existent in the case of a\ndielectric-filled waveguide) exists in the other half of the\nwaveguide. The analytical expressions for the first ( ya)\nand second cross-over ( yc) points along the cross-section\nare\nya=a\nπtan−1 \nκ′\nµ′r\nϵfµeffk2\n0a2\nπ2−1!\n(16)\nyc=a\nπtan−1 \nµ′\nκ′r\nϵfµeffk2\n0a2\nπ2−1!\n(17)\nFrom eq. (16) and (17), we can understand the role of\ngyrotropy over this shift of the cross-over points. For\na non-gyrotropic material, the gyrotropic parameter κ\nwould be 0, which will lead to ya= 0 and yc=a/2,\nsimilar to the case of a dielectric-filled waveguide.\nThese cross-over points having ⃗Sp= 0 exist due to\nthe absence of transverse reactive power and longitudi-\nnal real power at yaandyc, having Hx= 0 and Hy= 0,\nrespectively. We are more interested in the second cross-\nover point ycwhere the real power is absent, and be-\nyond which ⃗Spmakes a second transition into the neg-\native region. This point is marked using dotted lines in\nFig. 3(b). Correspondingly, we plot the real power prop-\nagation (Re ⃗Pavg) along the forward ˆ x-axis of the waveg-\nuide using analytical equation and CST Microwave Stu-\ndio in Fig. 3(c). The cross-over point ycis marked using\na dotted line. We observe that the region supporting\nadditional negative ⃗Spbeyond yccontains a backward\npropagation of the real power. This backward power is\nabsent for a dielectric-filled waveguide, as ycitself shall\nlie in the outermost wall at a/2.\nUsing the expression of ycin eq. (17), we have the\nmeans to engineer this cross-over point and alter the\nratio of negative to positive propagation of real power.8\nIn Fig. 4(a), the movement of the cross-over point yc\nalong ˆ y-axis is graphically illustrated. The variation of\nthe broad-width dimension aand the frequency of opera-\ntionfchanges yc. At this point, we have to note that the\nsecond cross-over point can be moved leftward maximum\nup to the center of the waveguide yc= 0. At this point,\nthere is an equal amount of forward and backward propa-\ngation of real power, and a cut-off condition is enforced in\nthe waveguide. This gyrotropy-enforced cut-off condition\nwas first reported in Section IV(C) of [22]. Figure 4(b)\nand (c) show the analytical and CST Microwave Studio\nplots of the ratio of ycwith half the broad width dimen-\nsiona/2 for variation of aandf, respectively. We can\nsee significant movement of ycusing variations of both\nparameters. Note that this zero cross-over point will be\nlocated on the other side of the waveguide for a back-\nward propagating wave. The tunability of the cross-over\npoint ycwill be beneficial for engineering nonreciprocal\ngyrotropic guiding structures.\nVI. CONCLUSION\nThis paper establishes the strong relationship between\nthe phasor form of the time-averaged Poynting vector\nand its instantaneous form, which has an intrinsic sense\nof rotation in a bulk gyrotropic media. We capture the\nsense of rotation of the instantaneous Poynting vector\nusing the Poynting vector spin, which can be computed\nusing the complex time-averaged Poynting vector. Our\nresults reveal the existence of reactive power components\nin the bulk of gyrotropic material for plane waves, which\ncan be represented aptly by the Poynting vector spin.\nFor the specific case of transverse electric mode propa-\ngation, the Poynting vector spin is shown to be identical\nto the photonic spin of the magnetic field component.\nThe photonic spin of the magnetic field is induced by\nthe gyrotropy. Further, we investigate the TE-guided\nmodes (i) at the ferrite-air interface and (ii) in the ferrite-\nfilled waveguide to reveal the impact of Poynting vector\nspin on the positive and negative power flow in these\nguiding structures. We show that the gyrotropy-induced\nasymmetry and nonreciprocity in Poynting vector spin\nand the resultant negative power flow in the ferrite-filled\nwaveguide can be controlled by the waveguide and gy-\nrotropic parameters. This work provides the analysis us-\ning which new avenues of designing gyrotropic material-\nbased guiding-wave structures can open up.\nAppendix A: Derivation of Poynting vector for bulk\nferrite\nUsing the mechanism used in [22], we find the magnetic\nfield in the bulk of ferrite\n⃗Hbf=H0(ˆx+ ((jϵfk2\n0κ′)/(k2\nx+k2\nz−ϵfk2\n0µ′))ˆy\n+ ((kxkz)/(k2\nx−ϵfk2\n0))ˆz)ej(kxx+kzz−ωt)(A1)The corresponding electric field vector can be found\nusing Maxwell’s equation as ⃗Ebf=−(1/(ωϵ0ϵf)↔\nk·⃗Hbf)\nNow we find the time-averaged Poynting vector ⃗Pavg\nas\n⃗Pavg=⃗E∗\nbf×⃗Hbf\n2(A2)\nUsing the instantaneous form of the magnetic field and\nelectric field component, we can find the instantaneous\nPoynting vector ⃗Pinstas\n⃗Pinst= Re( ⃗Ebfe−jωt)×Re(⃗Hbfe−jωt) (A3)\nThe real and imaginary components of equation A2\ngive us the time average real and reactive power in the\nmedium. In the instantaneous form, the time-varying\nnature of real power flow in the medium can be analyzed\nusing eq. A3.\nAppendix B: Poynting Vector and Photonic spin\nThe rotation of the field intensity vector ( ⃗Hor⃗E) can\nbe properly measured using the definition of Photonic\nspin and mathematically expressed as ⃗Sh=ℑ(⃗H∗×⃗H)\nfor magnetic field and ⃗Se=ℑ(⃗E∗×⃗E) for the electric\nfield. This definition of Photonic spin is analogous to\ncomputing the third Stoke’s Parameter. Considering a\ntransverse electric (TE) mode having field components\nEz,Hx, and Hypropagating along ˆ xaxis, the photonic\nspin for the electric field is absent, and only ⃗Shremains.\nFurther, the components of the time-averaged Poynting\nvector ⃗Pavgfor this TE mode will exist only in the ˆ xˆy\naxis, the latter being the reactive power direction. The\ntime-averaged Poynting vector in phasor form can be ex-\npressed as\n⃗Pavg=1\n2(⃗E∗×⃗H) =Pxˆx+Pyˆy (B1)\nNow, following an approach similar to computing the\nphotonic spin, we can quantify the rotation of the in-\nstantaneous Poynting vector and call it Poynting vector\nspin ( ⃗Sp). As we have real and imaginary component\nof⃗Pavgalong ˆ xand ˆy-directions, respectively, ⃗Spcan be\ndefined as\n⃗Sp=ℑ(⃗P∗\navg×⃗Pavg)\n=ℑ(P∗\nxPy−P∗\nyPx)\n= 0.25ℑ(−E∗\nzH∗\nyEzHx+E∗\nzH∗\nxEzHy)\n= 0.25|Ez|2ℑ(H∗\nxHy−H∗\nyHx)\n= 0.25|Ez|2ℑ(⃗H∗×⃗H)\n= 0.25|Ez|2⃗Sh(B2)9\nSince we only have non-zero HxandHycomponents, the\nphotonic spin ⃗Shis along ˆ z-axis, similarly, the Poynting\nvector spin ⃗Spis also along ˆ z-axis.\nWe have the Poynting vector spin ⃗Sprelated with the\nphotonic spin ⃗Shas⃗Sp= 0.25|Ez|2⃗Sh. As the photonic\nspin is normalized spatially, the term |Ez|does not hold\nany significant information, and we can safely relate ⃗Sp∼\n⃗Sh.\nAppendix C: Derivations for air-NPM interface\nTo support the wave confinement in the interface y=\n0, we define the TE mode field components as\nEz=\n\njH0k2\n0µaeαydy\nωϵ0αydify≤0\n−jH0k2\n0µnpe−αyuy\nωϵ0αyuify≥0(C1)\nHx=(\nH0eαydyify≤0\nH0e−αyuyify≥0(C2)\nHy=\n\n−jH0(α2\nyd+ϵak2\n0µa)eαydy\nαydkxify≤0\njH0(α2\nyu+ϵnpk2\n0µnp)e−αyuy\nαyukxify≥0(C3)\nFor simplicity, we omitted the basis function ekxx−ωt\nfrom the above field equations. On application of the\nboundary condition of Hxbeing continuous at the inter-\nfacey= 0, we obtain the dispersion relation as\nkx=±k0s\nµaµnp(ϵnpµa−ϵaµnp)\nµ2a−µ2np(C4)\nNext, we compute the real (0 .5ℜ(⃗E∗×⃗H)) and reactive\npower (0 .5ℑ(⃗E∗×⃗H)) using the field equations (eq. C1-\nC1) as\n0.5ℜ(⃗E∗×⃗H) =\n\nH2\n0k2\n0µa(α2\nyd+ϵak2\n0µa)e2αydy\n2ωϵ0kxα2\nydˆx ify≤0\nH2\n0k2\n0µnp(α2\nyu+ϵnpk2\n0µnp)e−2αyuy\n2ωϵ0kxα2\nyuˆxify≥0\n(C5)\n0.5ℑ(⃗E∗×⃗H) =\n\n−H2\n0k2\n0µae2αydy\n2ωϵ0αydˆyify≤0\nH2\n0k2\n0µnpe−2αyuy\n2ωϵ0αyuˆyify≥0(C6)We can substitute H0= 1 A/m to simplify these equa-\ntions of the time-averaged Poynting vector. Further, we\nnormalize the field component spatially for the computa-\ntion of photonic spin ⃗Sh. Correspondingly, we can com-\npute the Poynting vector spin as ⃗Sp,np=ℑ(⃗P∗\navg×⃗Pavg).\nAppendix D: Derivation for ferrite-air interface\nThe field intensities corresponding to the TE mode of\nwave propagation along the interface are defined as\nEz=\n\njH0k2\n0eαydy\nωαydϵ0ify≤0\njH0k2\n0(αyuκ′+kxµ′)e−αyuy\nωϵ0(ϵfk2\n0κ′−αyukx)ify≥0(D1)\nHx=(\nH0eαydyify≤0\nH0e−αyuyify≥0(D2)\nHy=\n\n−jH0(α2\nyd+ϵak2\n0)eαydy\nαydkxify≤0\n−jH0(α2\nyu+ϵfk2\n0µ′)e−αyuy\nϵfk2\n0κ′−αyukxify≥0(D3)\nUsing the boundary conditions of the tangential electric\nand magnetic field to be continuous at the interface, we\nobtain the characteristic equation for the solution of the\npropagation constant kxas\nH0k0(kx(αyu+αydµ′) +κ′(αydαyu−ϵfk2\n0)) = 0 (D4)\nThe terms αydandαyudenotes the decay of the field\ncomponents along −ˆyand ˆ ydirections, respectively.\nThese attenuation constants can be mathematically de-\nfined as αyd=p\nk2x−ϵak2\n0andαyu=p\nk2x−ϵfµeffk2\n0,\nwhere µeff= (µ′2−κ′2)/µ′. 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